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this is a review of what we covered in this tutorial on object-oriented design . when we create programs , we often find that we want to create many different objects that all share similar properties - like many cats , that have slightly different fur color and size , or many buttons , with different labels and positions . we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages . it then sets the initial properties of the object based on those arguments , using the this keyword . when we use this in an object , we are referring to the current instance of an object , referring to itself . we need to store the properties on this to make sure we can remember them later . to create an instance of a book object , we declare a new variable to store it , then use the new keyword , followed by the constructor function name , and pass in the arguments that the constructor expects : var book = new book ( `` robot dreams '' , `` isaac asimov '' , 320 ) ; we can then access any properties that we stored in the object using dot notation : println ( `` i loved reading `` + book.title ) ; // i loved reading robot dreams println ( book.author + `` is my fav author '' ) ; // `` isaac asimov '' is my fav author let 's contrast this for a minute , and show what would have happened if we did n't set up our constructor function properly : var book = function ( title , author , numpages ) { } ; var book = new book ( `` little brother '' , `` cory doctorow '' , 380 ) ; println ( `` i loved reading `` + book.title ) ; // i loved reading undefined println ( book.author + `` is my fav author '' ) ; // undefined is my favorite author if we pass the arguments into the constructor function but do not explicitly store them on this , then we will not be able to access them later ! the object will have long forgotten about them . when we define object types , we often want to associate both properties and behavior with them - like all of our cat objects should be able to meow ( ) and eat ( ) . so we need to be able to attach functions to our object type definitions , and we can do that by defining them on what 's called the object prototype : book.prototype.readitall = function ( ) { this.currentpage = this.numpages ; println ( `` you read `` + this.numpages + `` pages ! `` ) ; } ; it 's like how we would define a function normally , except that we hang it off the book 's prototype instead of just defining it globally . that 's how javascript knows that this is a function that can be called on any book object , and that this function should have access to the this of the book that it 's called on . we can then call the function ( which we call a method , since it 's attached to an object ) , like so : var book = new book ( `` animal farm '' , `` george orwell '' , 112 ) ; book.readitall ( ) ; // you read 112 pages ! remember , the whole point of object-oriented design is that it makes it easy for us to make multiple related objects ( object instances ) . let 's see that in code : `` ` var pirate = new book ( `` pirate cinema '' , `` cory doctorow '' , 384 ) ; var giver = new book ( `` the giver '' , `` lois lowry '' , 179 ) ; var tuck = new book ( `` tuck everlasting '' , `` natalie babbit '' , 144 ) ; pirate.readitall ( ) ; // you read 384 pages ! giver.readitall ( ) ; // you read 179 pages ! tuck.readitall ( ) ; // you read 144 pages ! `` ` that code gives us three books that are similar - they all have the same types of properties and behavior , but also different . sweet ! now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death . they 're both mammals , and that means they share a lot in common , even if they 're also different . in that case , we want to use the idea of object inheritance . an object type could inherit properties and behavior from a parent object type , but then also have its own unique things about it . all the cats and dogs could inherit from mammal , so that they would n't have to invent eat ( ) ing from scratch . how would we do that in javascript ? let 's go back to our book example , and say that book is the `` parent '' object type , and we want to make two object types that inherit from it - paperback and ebook . a paperback is like a book , but it has one main thing different , at least for our program : it has a cover image . so , our constructor needs to take four arguments , to take in that extra info : var paperback = function ( title , author , numpages , cover ) { // ... } now , we do n't want to have to do all the work that we already did in the book constructor to remember those first three arguments - we want to take advantage of the fact that the code for that would be the same . so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages ! calvin.burn ( ) ; // omg , you burnt all 256 pages ! `` ` ( well , we 're not really going to burn it , because that 's an amazing book , but perhaps if we were stuck in a glacial desert and desperate for warmth and about to die . ) and now you can see how we can use object-oriented design principles in javascript to create more complex data for your programs and model your program worlds better .
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this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages . it then sets the initial properties of the object based on those arguments , using the this keyword .
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var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; var book = new book ( `` robot dreams '' , `` isaac asimov '' , 320 ) ; 2 ) same question with the prototype and the methods.. why should i choose them over a regular function , which also be global and not only for a specific thing ?
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this is a review of what we covered in this tutorial on object-oriented design . when we create programs , we often find that we want to create many different objects that all share similar properties - like many cats , that have slightly different fur color and size , or many buttons , with different labels and positions . we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages . it then sets the initial properties of the object based on those arguments , using the this keyword . when we use this in an object , we are referring to the current instance of an object , referring to itself . we need to store the properties on this to make sure we can remember them later . to create an instance of a book object , we declare a new variable to store it , then use the new keyword , followed by the constructor function name , and pass in the arguments that the constructor expects : var book = new book ( `` robot dreams '' , `` isaac asimov '' , 320 ) ; we can then access any properties that we stored in the object using dot notation : println ( `` i loved reading `` + book.title ) ; // i loved reading robot dreams println ( book.author + `` is my fav author '' ) ; // `` isaac asimov '' is my fav author let 's contrast this for a minute , and show what would have happened if we did n't set up our constructor function properly : var book = function ( title , author , numpages ) { } ; var book = new book ( `` little brother '' , `` cory doctorow '' , 380 ) ; println ( `` i loved reading `` + book.title ) ; // i loved reading undefined println ( book.author + `` is my fav author '' ) ; // undefined is my favorite author if we pass the arguments into the constructor function but do not explicitly store them on this , then we will not be able to access them later ! the object will have long forgotten about them . when we define object types , we often want to associate both properties and behavior with them - like all of our cat objects should be able to meow ( ) and eat ( ) . so we need to be able to attach functions to our object type definitions , and we can do that by defining them on what 's called the object prototype : book.prototype.readitall = function ( ) { this.currentpage = this.numpages ; println ( `` you read `` + this.numpages + `` pages ! `` ) ; } ; it 's like how we would define a function normally , except that we hang it off the book 's prototype instead of just defining it globally . that 's how javascript knows that this is a function that can be called on any book object , and that this function should have access to the this of the book that it 's called on . we can then call the function ( which we call a method , since it 's attached to an object ) , like so : var book = new book ( `` animal farm '' , `` george orwell '' , 112 ) ; book.readitall ( ) ; // you read 112 pages ! remember , the whole point of object-oriented design is that it makes it easy for us to make multiple related objects ( object instances ) . let 's see that in code : `` ` var pirate = new book ( `` pirate cinema '' , `` cory doctorow '' , 384 ) ; var giver = new book ( `` the giver '' , `` lois lowry '' , 179 ) ; var tuck = new book ( `` tuck everlasting '' , `` natalie babbit '' , 144 ) ; pirate.readitall ( ) ; // you read 384 pages ! giver.readitall ( ) ; // you read 179 pages ! tuck.readitall ( ) ; // you read 144 pages ! `` ` that code gives us three books that are similar - they all have the same types of properties and behavior , but also different . sweet ! now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death . they 're both mammals , and that means they share a lot in common , even if they 're also different . in that case , we want to use the idea of object inheritance . an object type could inherit properties and behavior from a parent object type , but then also have its own unique things about it . all the cats and dogs could inherit from mammal , so that they would n't have to invent eat ( ) ing from scratch . how would we do that in javascript ? let 's go back to our book example , and say that book is the `` parent '' object type , and we want to make two object types that inherit from it - paperback and ebook . a paperback is like a book , but it has one main thing different , at least for our program : it has a cover image . so , our constructor needs to take four arguments , to take in that extra info : var paperback = function ( title , author , numpages , cover ) { // ... } now , we do n't want to have to do all the work that we already did in the book constructor to remember those first three arguments - we want to take advantage of the fact that the code for that would be the same . so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages ! calvin.burn ( ) ; // omg , you burnt all 256 pages ! `` ` ( well , we 're not really going to burn it , because that 's an amazing book , but perhaps if we were stuck in a glacial desert and desperate for warmth and about to die . ) and now you can see how we can use object-oriented design principles in javascript to create more complex data for your programs and model your program worlds better .
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in that case , we want to use the idea of object inheritance . an object type could inherit properties and behavior from a parent object type , but then also have its own unique things about it . all the cats and dogs could inherit from mammal , so that they would n't have to invent eat ( ) ing from scratch .
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how to bring out something from a object prototype and enter a new type of it ?
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this is a review of what we covered in this tutorial on object-oriented design . when we create programs , we often find that we want to create many different objects that all share similar properties - like many cats , that have slightly different fur color and size , or many buttons , with different labels and positions . we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages . it then sets the initial properties of the object based on those arguments , using the this keyword . when we use this in an object , we are referring to the current instance of an object , referring to itself . we need to store the properties on this to make sure we can remember them later . to create an instance of a book object , we declare a new variable to store it , then use the new keyword , followed by the constructor function name , and pass in the arguments that the constructor expects : var book = new book ( `` robot dreams '' , `` isaac asimov '' , 320 ) ; we can then access any properties that we stored in the object using dot notation : println ( `` i loved reading `` + book.title ) ; // i loved reading robot dreams println ( book.author + `` is my fav author '' ) ; // `` isaac asimov '' is my fav author let 's contrast this for a minute , and show what would have happened if we did n't set up our constructor function properly : var book = function ( title , author , numpages ) { } ; var book = new book ( `` little brother '' , `` cory doctorow '' , 380 ) ; println ( `` i loved reading `` + book.title ) ; // i loved reading undefined println ( book.author + `` is my fav author '' ) ; // undefined is my favorite author if we pass the arguments into the constructor function but do not explicitly store them on this , then we will not be able to access them later ! the object will have long forgotten about them . when we define object types , we often want to associate both properties and behavior with them - like all of our cat objects should be able to meow ( ) and eat ( ) . so we need to be able to attach functions to our object type definitions , and we can do that by defining them on what 's called the object prototype : book.prototype.readitall = function ( ) { this.currentpage = this.numpages ; println ( `` you read `` + this.numpages + `` pages ! `` ) ; } ; it 's like how we would define a function normally , except that we hang it off the book 's prototype instead of just defining it globally . that 's how javascript knows that this is a function that can be called on any book object , and that this function should have access to the this of the book that it 's called on . we can then call the function ( which we call a method , since it 's attached to an object ) , like so : var book = new book ( `` animal farm '' , `` george orwell '' , 112 ) ; book.readitall ( ) ; // you read 112 pages ! remember , the whole point of object-oriented design is that it makes it easy for us to make multiple related objects ( object instances ) . let 's see that in code : `` ` var pirate = new book ( `` pirate cinema '' , `` cory doctorow '' , 384 ) ; var giver = new book ( `` the giver '' , `` lois lowry '' , 179 ) ; var tuck = new book ( `` tuck everlasting '' , `` natalie babbit '' , 144 ) ; pirate.readitall ( ) ; // you read 384 pages ! giver.readitall ( ) ; // you read 179 pages ! tuck.readitall ( ) ; // you read 144 pages ! `` ` that code gives us three books that are similar - they all have the same types of properties and behavior , but also different . sweet ! now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death . they 're both mammals , and that means they share a lot in common , even if they 're also different . in that case , we want to use the idea of object inheritance . an object type could inherit properties and behavior from a parent object type , but then also have its own unique things about it . all the cats and dogs could inherit from mammal , so that they would n't have to invent eat ( ) ing from scratch . how would we do that in javascript ? let 's go back to our book example , and say that book is the `` parent '' object type , and we want to make two object types that inherit from it - paperback and ebook . a paperback is like a book , but it has one main thing different , at least for our program : it has a cover image . so , our constructor needs to take four arguments , to take in that extra info : var paperback = function ( title , author , numpages , cover ) { // ... } now , we do n't want to have to do all the work that we already did in the book constructor to remember those first three arguments - we want to take advantage of the fact that the code for that would be the same . so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages ! calvin.burn ( ) ; // omg , you burnt all 256 pages ! `` ` ( well , we 're not really going to burn it , because that 's an amazing book , but perhaps if we were stuck in a glacial desert and desperate for warmth and about to die . ) and now you can see how we can use object-oriented design principles in javascript to create more complex data for your programs and model your program worlds better .
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now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death .
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how would i code a cannon that shot in the direction of a player ?
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this is a review of what we covered in this tutorial on object-oriented design . when we create programs , we often find that we want to create many different objects that all share similar properties - like many cats , that have slightly different fur color and size , or many buttons , with different labels and positions . we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages . it then sets the initial properties of the object based on those arguments , using the this keyword . when we use this in an object , we are referring to the current instance of an object , referring to itself . we need to store the properties on this to make sure we can remember them later . to create an instance of a book object , we declare a new variable to store it , then use the new keyword , followed by the constructor function name , and pass in the arguments that the constructor expects : var book = new book ( `` robot dreams '' , `` isaac asimov '' , 320 ) ; we can then access any properties that we stored in the object using dot notation : println ( `` i loved reading `` + book.title ) ; // i loved reading robot dreams println ( book.author + `` is my fav author '' ) ; // `` isaac asimov '' is my fav author let 's contrast this for a minute , and show what would have happened if we did n't set up our constructor function properly : var book = function ( title , author , numpages ) { } ; var book = new book ( `` little brother '' , `` cory doctorow '' , 380 ) ; println ( `` i loved reading `` + book.title ) ; // i loved reading undefined println ( book.author + `` is my fav author '' ) ; // undefined is my favorite author if we pass the arguments into the constructor function but do not explicitly store them on this , then we will not be able to access them later ! the object will have long forgotten about them . when we define object types , we often want to associate both properties and behavior with them - like all of our cat objects should be able to meow ( ) and eat ( ) . so we need to be able to attach functions to our object type definitions , and we can do that by defining them on what 's called the object prototype : book.prototype.readitall = function ( ) { this.currentpage = this.numpages ; println ( `` you read `` + this.numpages + `` pages ! `` ) ; } ; it 's like how we would define a function normally , except that we hang it off the book 's prototype instead of just defining it globally . that 's how javascript knows that this is a function that can be called on any book object , and that this function should have access to the this of the book that it 's called on . we can then call the function ( which we call a method , since it 's attached to an object ) , like so : var book = new book ( `` animal farm '' , `` george orwell '' , 112 ) ; book.readitall ( ) ; // you read 112 pages ! remember , the whole point of object-oriented design is that it makes it easy for us to make multiple related objects ( object instances ) . let 's see that in code : `` ` var pirate = new book ( `` pirate cinema '' , `` cory doctorow '' , 384 ) ; var giver = new book ( `` the giver '' , `` lois lowry '' , 179 ) ; var tuck = new book ( `` tuck everlasting '' , `` natalie babbit '' , 144 ) ; pirate.readitall ( ) ; // you read 384 pages ! giver.readitall ( ) ; // you read 179 pages ! tuck.readitall ( ) ; // you read 144 pages ! `` ` that code gives us three books that are similar - they all have the same types of properties and behavior , but also different . sweet ! now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death . they 're both mammals , and that means they share a lot in common , even if they 're also different . in that case , we want to use the idea of object inheritance . an object type could inherit properties and behavior from a parent object type , but then also have its own unique things about it . all the cats and dogs could inherit from mammal , so that they would n't have to invent eat ( ) ing from scratch . how would we do that in javascript ? let 's go back to our book example , and say that book is the `` parent '' object type , and we want to make two object types that inherit from it - paperback and ebook . a paperback is like a book , but it has one main thing different , at least for our program : it has a cover image . so , our constructor needs to take four arguments , to take in that extra info : var paperback = function ( title , author , numpages , cover ) { // ... } now , we do n't want to have to do all the work that we already did in the book constructor to remember those first three arguments - we want to take advantage of the fact that the code for that would be the same . so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages ! calvin.burn ( ) ; // omg , you burnt all 256 pages ! `` ` ( well , we 're not really going to burn it , because that 's an amazing book , but perhaps if we were stuck in a glacial desert and desperate for warmth and about to die . ) and now you can see how we can use object-oriented design principles in javascript to create more complex data for your programs and model your program worlds better .
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we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages .
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any suggestions how to add new talk function to a child-object ?
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this is a review of what we covered in this tutorial on object-oriented design . when we create programs , we often find that we want to create many different objects that all share similar properties - like many cats , that have slightly different fur color and size , or many buttons , with different labels and positions . we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages . it then sets the initial properties of the object based on those arguments , using the this keyword . when we use this in an object , we are referring to the current instance of an object , referring to itself . we need to store the properties on this to make sure we can remember them later . to create an instance of a book object , we declare a new variable to store it , then use the new keyword , followed by the constructor function name , and pass in the arguments that the constructor expects : var book = new book ( `` robot dreams '' , `` isaac asimov '' , 320 ) ; we can then access any properties that we stored in the object using dot notation : println ( `` i loved reading `` + book.title ) ; // i loved reading robot dreams println ( book.author + `` is my fav author '' ) ; // `` isaac asimov '' is my fav author let 's contrast this for a minute , and show what would have happened if we did n't set up our constructor function properly : var book = function ( title , author , numpages ) { } ; var book = new book ( `` little brother '' , `` cory doctorow '' , 380 ) ; println ( `` i loved reading `` + book.title ) ; // i loved reading undefined println ( book.author + `` is my fav author '' ) ; // undefined is my favorite author if we pass the arguments into the constructor function but do not explicitly store them on this , then we will not be able to access them later ! the object will have long forgotten about them . when we define object types , we often want to associate both properties and behavior with them - like all of our cat objects should be able to meow ( ) and eat ( ) . so we need to be able to attach functions to our object type definitions , and we can do that by defining them on what 's called the object prototype : book.prototype.readitall = function ( ) { this.currentpage = this.numpages ; println ( `` you read `` + this.numpages + `` pages ! `` ) ; } ; it 's like how we would define a function normally , except that we hang it off the book 's prototype instead of just defining it globally . that 's how javascript knows that this is a function that can be called on any book object , and that this function should have access to the this of the book that it 's called on . we can then call the function ( which we call a method , since it 's attached to an object ) , like so : var book = new book ( `` animal farm '' , `` george orwell '' , 112 ) ; book.readitall ( ) ; // you read 112 pages ! remember , the whole point of object-oriented design is that it makes it easy for us to make multiple related objects ( object instances ) . let 's see that in code : `` ` var pirate = new book ( `` pirate cinema '' , `` cory doctorow '' , 384 ) ; var giver = new book ( `` the giver '' , `` lois lowry '' , 179 ) ; var tuck = new book ( `` tuck everlasting '' , `` natalie babbit '' , 144 ) ; pirate.readitall ( ) ; // you read 384 pages ! giver.readitall ( ) ; // you read 179 pages ! tuck.readitall ( ) ; // you read 144 pages ! `` ` that code gives us three books that are similar - they all have the same types of properties and behavior , but also different . sweet ! now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death . they 're both mammals , and that means they share a lot in common , even if they 're also different . in that case , we want to use the idea of object inheritance . an object type could inherit properties and behavior from a parent object type , but then also have its own unique things about it . all the cats and dogs could inherit from mammal , so that they would n't have to invent eat ( ) ing from scratch . how would we do that in javascript ? let 's go back to our book example , and say that book is the `` parent '' object type , and we want to make two object types that inherit from it - paperback and ebook . a paperback is like a book , but it has one main thing different , at least for our program : it has a cover image . so , our constructor needs to take four arguments , to take in that extra info : var paperback = function ( title , author , numpages , cover ) { // ... } now , we do n't want to have to do all the work that we already did in the book constructor to remember those first three arguments - we want to take advantage of the fact that the code for that would be the same . so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages ! calvin.burn ( ) ; // omg , you burnt all 256 pages ! `` ` ( well , we 're not really going to burn it , because that 's an amazing book , but perhaps if we were stuck in a glacial desert and desperate for warmth and about to die . ) and now you can see how we can use object-oriented design principles in javascript to create more complex data for your programs and model your program worlds better .
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so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages !
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are exampleobject.prototype and object.create ( someobjectprototype ) supported only by processingjs library , or they are also available in the javascript standard library ?
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this is a review of what we covered in this tutorial on object-oriented design . when we create programs , we often find that we want to create many different objects that all share similar properties - like many cats , that have slightly different fur color and size , or many buttons , with different labels and positions . we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages . it then sets the initial properties of the object based on those arguments , using the this keyword . when we use this in an object , we are referring to the current instance of an object , referring to itself . we need to store the properties on this to make sure we can remember them later . to create an instance of a book object , we declare a new variable to store it , then use the new keyword , followed by the constructor function name , and pass in the arguments that the constructor expects : var book = new book ( `` robot dreams '' , `` isaac asimov '' , 320 ) ; we can then access any properties that we stored in the object using dot notation : println ( `` i loved reading `` + book.title ) ; // i loved reading robot dreams println ( book.author + `` is my fav author '' ) ; // `` isaac asimov '' is my fav author let 's contrast this for a minute , and show what would have happened if we did n't set up our constructor function properly : var book = function ( title , author , numpages ) { } ; var book = new book ( `` little brother '' , `` cory doctorow '' , 380 ) ; println ( `` i loved reading `` + book.title ) ; // i loved reading undefined println ( book.author + `` is my fav author '' ) ; // undefined is my favorite author if we pass the arguments into the constructor function but do not explicitly store them on this , then we will not be able to access them later ! the object will have long forgotten about them . when we define object types , we often want to associate both properties and behavior with them - like all of our cat objects should be able to meow ( ) and eat ( ) . so we need to be able to attach functions to our object type definitions , and we can do that by defining them on what 's called the object prototype : book.prototype.readitall = function ( ) { this.currentpage = this.numpages ; println ( `` you read `` + this.numpages + `` pages ! `` ) ; } ; it 's like how we would define a function normally , except that we hang it off the book 's prototype instead of just defining it globally . that 's how javascript knows that this is a function that can be called on any book object , and that this function should have access to the this of the book that it 's called on . we can then call the function ( which we call a method , since it 's attached to an object ) , like so : var book = new book ( `` animal farm '' , `` george orwell '' , 112 ) ; book.readitall ( ) ; // you read 112 pages ! remember , the whole point of object-oriented design is that it makes it easy for us to make multiple related objects ( object instances ) . let 's see that in code : `` ` var pirate = new book ( `` pirate cinema '' , `` cory doctorow '' , 384 ) ; var giver = new book ( `` the giver '' , `` lois lowry '' , 179 ) ; var tuck = new book ( `` tuck everlasting '' , `` natalie babbit '' , 144 ) ; pirate.readitall ( ) ; // you read 384 pages ! giver.readitall ( ) ; // you read 179 pages ! tuck.readitall ( ) ; // you read 144 pages ! `` ` that code gives us three books that are similar - they all have the same types of properties and behavior , but also different . sweet ! now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death . they 're both mammals , and that means they share a lot in common , even if they 're also different . in that case , we want to use the idea of object inheritance . an object type could inherit properties and behavior from a parent object type , but then also have its own unique things about it . all the cats and dogs could inherit from mammal , so that they would n't have to invent eat ( ) ing from scratch . how would we do that in javascript ? let 's go back to our book example , and say that book is the `` parent '' object type , and we want to make two object types that inherit from it - paperback and ebook . a paperback is like a book , but it has one main thing different , at least for our program : it has a cover image . so , our constructor needs to take four arguments , to take in that extra info : var paperback = function ( title , author , numpages , cover ) { // ... } now , we do n't want to have to do all the work that we already did in the book constructor to remember those first three arguments - we want to take advantage of the fact that the code for that would be the same . so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages ! calvin.burn ( ) ; // omg , you burnt all 256 pages ! `` ` ( well , we 're not really going to burn it , because that 's an amazing book , but perhaps if we were stuck in a glacial desert and desperate for warmth and about to die . ) and now you can see how we can use object-oriented design principles in javascript to create more complex data for your programs and model your program worlds better .
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in that case , we want to use the idea of object inheritance . an object type could inherit properties and behavior from a parent object type , but then also have its own unique things about it . all the cats and dogs could inherit from mammal , so that they would n't have to invent eat ( ) ing from scratch .
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and what 'data type ' does the the object.create ( ) method returns ?
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this is a review of what we covered in this tutorial on object-oriented design . when we create programs , we often find that we want to create many different objects that all share similar properties - like many cats , that have slightly different fur color and size , or many buttons , with different labels and positions . we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages . it then sets the initial properties of the object based on those arguments , using the this keyword . when we use this in an object , we are referring to the current instance of an object , referring to itself . we need to store the properties on this to make sure we can remember them later . to create an instance of a book object , we declare a new variable to store it , then use the new keyword , followed by the constructor function name , and pass in the arguments that the constructor expects : var book = new book ( `` robot dreams '' , `` isaac asimov '' , 320 ) ; we can then access any properties that we stored in the object using dot notation : println ( `` i loved reading `` + book.title ) ; // i loved reading robot dreams println ( book.author + `` is my fav author '' ) ; // `` isaac asimov '' is my fav author let 's contrast this for a minute , and show what would have happened if we did n't set up our constructor function properly : var book = function ( title , author , numpages ) { } ; var book = new book ( `` little brother '' , `` cory doctorow '' , 380 ) ; println ( `` i loved reading `` + book.title ) ; // i loved reading undefined println ( book.author + `` is my fav author '' ) ; // undefined is my favorite author if we pass the arguments into the constructor function but do not explicitly store them on this , then we will not be able to access them later ! the object will have long forgotten about them . when we define object types , we often want to associate both properties and behavior with them - like all of our cat objects should be able to meow ( ) and eat ( ) . so we need to be able to attach functions to our object type definitions , and we can do that by defining them on what 's called the object prototype : book.prototype.readitall = function ( ) { this.currentpage = this.numpages ; println ( `` you read `` + this.numpages + `` pages ! `` ) ; } ; it 's like how we would define a function normally , except that we hang it off the book 's prototype instead of just defining it globally . that 's how javascript knows that this is a function that can be called on any book object , and that this function should have access to the this of the book that it 's called on . we can then call the function ( which we call a method , since it 's attached to an object ) , like so : var book = new book ( `` animal farm '' , `` george orwell '' , 112 ) ; book.readitall ( ) ; // you read 112 pages ! remember , the whole point of object-oriented design is that it makes it easy for us to make multiple related objects ( object instances ) . let 's see that in code : `` ` var pirate = new book ( `` pirate cinema '' , `` cory doctorow '' , 384 ) ; var giver = new book ( `` the giver '' , `` lois lowry '' , 179 ) ; var tuck = new book ( `` tuck everlasting '' , `` natalie babbit '' , 144 ) ; pirate.readitall ( ) ; // you read 384 pages ! giver.readitall ( ) ; // you read 179 pages ! tuck.readitall ( ) ; // you read 144 pages ! `` ` that code gives us three books that are similar - they all have the same types of properties and behavior , but also different . sweet ! now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death . they 're both mammals , and that means they share a lot in common , even if they 're also different . in that case , we want to use the idea of object inheritance . an object type could inherit properties and behavior from a parent object type , but then also have its own unique things about it . all the cats and dogs could inherit from mammal , so that they would n't have to invent eat ( ) ing from scratch . how would we do that in javascript ? let 's go back to our book example , and say that book is the `` parent '' object type , and we want to make two object types that inherit from it - paperback and ebook . a paperback is like a book , but it has one main thing different , at least for our program : it has a cover image . so , our constructor needs to take four arguments , to take in that extra info : var paperback = function ( title , author , numpages , cover ) { // ... } now , we do n't want to have to do all the work that we already did in the book constructor to remember those first three arguments - we want to take advantage of the fact that the code for that would be the same . so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages ! calvin.burn ( ) ; // omg , you burnt all 256 pages ! `` ` ( well , we 're not really going to burn it , because that 's an amazing book , but perhaps if we were stuck in a glacial desert and desperate for warmth and about to die . ) and now you can see how we can use object-oriented design principles in javascript to create more complex data for your programs and model your program worlds better .
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we need to store the properties on this to make sure we can remember them later . to create an instance of a book object , we declare a new variable to store it , then use the new keyword , followed by the constructor function name , and pass in the arguments that the constructor expects : var book = new book ( `` robot dreams '' , `` isaac asimov '' , 320 ) ; we can then access any properties that we stored in the object using dot notation : println ( `` i loved reading `` + book.title ) ; // i loved reading robot dreams println ( book.author + `` is my fav author '' ) ; // `` isaac asimov '' is my fav author let 's contrast this for a minute , and show what would have happened if we did n't set up our constructor function properly : var book = function ( title , author , numpages ) { } ; var book = new book ( `` little brother '' , `` cory doctorow '' , 380 ) ; println ( `` i loved reading `` + book.title ) ; // i loved reading undefined println ( book.author + `` is my fav author '' ) ; // undefined is my favorite author if we pass the arguments into the constructor function but do not explicitly store them on this , then we will not be able to access them later ! the object will have long forgotten about them .
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is there a way to create a new constructor function based on another constructor function without arguments ?
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this is a review of what we covered in this tutorial on object-oriented design . when we create programs , we often find that we want to create many different objects that all share similar properties - like many cats , that have slightly different fur color and size , or many buttons , with different labels and positions . we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages . it then sets the initial properties of the object based on those arguments , using the this keyword . when we use this in an object , we are referring to the current instance of an object , referring to itself . we need to store the properties on this to make sure we can remember them later . to create an instance of a book object , we declare a new variable to store it , then use the new keyword , followed by the constructor function name , and pass in the arguments that the constructor expects : var book = new book ( `` robot dreams '' , `` isaac asimov '' , 320 ) ; we can then access any properties that we stored in the object using dot notation : println ( `` i loved reading `` + book.title ) ; // i loved reading robot dreams println ( book.author + `` is my fav author '' ) ; // `` isaac asimov '' is my fav author let 's contrast this for a minute , and show what would have happened if we did n't set up our constructor function properly : var book = function ( title , author , numpages ) { } ; var book = new book ( `` little brother '' , `` cory doctorow '' , 380 ) ; println ( `` i loved reading `` + book.title ) ; // i loved reading undefined println ( book.author + `` is my fav author '' ) ; // undefined is my favorite author if we pass the arguments into the constructor function but do not explicitly store them on this , then we will not be able to access them later ! the object will have long forgotten about them . when we define object types , we often want to associate both properties and behavior with them - like all of our cat objects should be able to meow ( ) and eat ( ) . so we need to be able to attach functions to our object type definitions , and we can do that by defining them on what 's called the object prototype : book.prototype.readitall = function ( ) { this.currentpage = this.numpages ; println ( `` you read `` + this.numpages + `` pages ! `` ) ; } ; it 's like how we would define a function normally , except that we hang it off the book 's prototype instead of just defining it globally . that 's how javascript knows that this is a function that can be called on any book object , and that this function should have access to the this of the book that it 's called on . we can then call the function ( which we call a method , since it 's attached to an object ) , like so : var book = new book ( `` animal farm '' , `` george orwell '' , 112 ) ; book.readitall ( ) ; // you read 112 pages ! remember , the whole point of object-oriented design is that it makes it easy for us to make multiple related objects ( object instances ) . let 's see that in code : `` ` var pirate = new book ( `` pirate cinema '' , `` cory doctorow '' , 384 ) ; var giver = new book ( `` the giver '' , `` lois lowry '' , 179 ) ; var tuck = new book ( `` tuck everlasting '' , `` natalie babbit '' , 144 ) ; pirate.readitall ( ) ; // you read 384 pages ! giver.readitall ( ) ; // you read 179 pages ! tuck.readitall ( ) ; // you read 144 pages ! `` ` that code gives us three books that are similar - they all have the same types of properties and behavior , but also different . sweet ! now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death . they 're both mammals , and that means they share a lot in common , even if they 're also different . in that case , we want to use the idea of object inheritance . an object type could inherit properties and behavior from a parent object type , but then also have its own unique things about it . all the cats and dogs could inherit from mammal , so that they would n't have to invent eat ( ) ing from scratch . how would we do that in javascript ? let 's go back to our book example , and say that book is the `` parent '' object type , and we want to make two object types that inherit from it - paperback and ebook . a paperback is like a book , but it has one main thing different , at least for our program : it has a cover image . so , our constructor needs to take four arguments , to take in that extra info : var paperback = function ( title , author , numpages , cover ) { // ... } now , we do n't want to have to do all the work that we already did in the book constructor to remember those first three arguments - we want to take advantage of the fact that the code for that would be the same . so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages ! calvin.burn ( ) ; // omg , you burnt all 256 pages ! `` ` ( well , we 're not really going to burn it , because that 's an amazing book , but perhaps if we were stuck in a glacial desert and desperate for warmth and about to die . ) and now you can see how we can use object-oriented design principles in javascript to create more complex data for your programs and model your program worlds better .
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so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages !
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why could n't something like book.prototype.draw be var book.prototype.draw ?
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this is a review of what we covered in this tutorial on object-oriented design . when we create programs , we often find that we want to create many different objects that all share similar properties - like many cats , that have slightly different fur color and size , or many buttons , with different labels and positions . we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages . it then sets the initial properties of the object based on those arguments , using the this keyword . when we use this in an object , we are referring to the current instance of an object , referring to itself . we need to store the properties on this to make sure we can remember them later . to create an instance of a book object , we declare a new variable to store it , then use the new keyword , followed by the constructor function name , and pass in the arguments that the constructor expects : var book = new book ( `` robot dreams '' , `` isaac asimov '' , 320 ) ; we can then access any properties that we stored in the object using dot notation : println ( `` i loved reading `` + book.title ) ; // i loved reading robot dreams println ( book.author + `` is my fav author '' ) ; // `` isaac asimov '' is my fav author let 's contrast this for a minute , and show what would have happened if we did n't set up our constructor function properly : var book = function ( title , author , numpages ) { } ; var book = new book ( `` little brother '' , `` cory doctorow '' , 380 ) ; println ( `` i loved reading `` + book.title ) ; // i loved reading undefined println ( book.author + `` is my fav author '' ) ; // undefined is my favorite author if we pass the arguments into the constructor function but do not explicitly store them on this , then we will not be able to access them later ! the object will have long forgotten about them . when we define object types , we often want to associate both properties and behavior with them - like all of our cat objects should be able to meow ( ) and eat ( ) . so we need to be able to attach functions to our object type definitions , and we can do that by defining them on what 's called the object prototype : book.prototype.readitall = function ( ) { this.currentpage = this.numpages ; println ( `` you read `` + this.numpages + `` pages ! `` ) ; } ; it 's like how we would define a function normally , except that we hang it off the book 's prototype instead of just defining it globally . that 's how javascript knows that this is a function that can be called on any book object , and that this function should have access to the this of the book that it 's called on . we can then call the function ( which we call a method , since it 's attached to an object ) , like so : var book = new book ( `` animal farm '' , `` george orwell '' , 112 ) ; book.readitall ( ) ; // you read 112 pages ! remember , the whole point of object-oriented design is that it makes it easy for us to make multiple related objects ( object instances ) . let 's see that in code : `` ` var pirate = new book ( `` pirate cinema '' , `` cory doctorow '' , 384 ) ; var giver = new book ( `` the giver '' , `` lois lowry '' , 179 ) ; var tuck = new book ( `` tuck everlasting '' , `` natalie babbit '' , 144 ) ; pirate.readitall ( ) ; // you read 384 pages ! giver.readitall ( ) ; // you read 179 pages ! tuck.readitall ( ) ; // you read 144 pages ! `` ` that code gives us three books that are similar - they all have the same types of properties and behavior , but also different . sweet ! now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death . they 're both mammals , and that means they share a lot in common , even if they 're also different . in that case , we want to use the idea of object inheritance . an object type could inherit properties and behavior from a parent object type , but then also have its own unique things about it . all the cats and dogs could inherit from mammal , so that they would n't have to invent eat ( ) ing from scratch . how would we do that in javascript ? let 's go back to our book example , and say that book is the `` parent '' object type , and we want to make two object types that inherit from it - paperback and ebook . a paperback is like a book , but it has one main thing different , at least for our program : it has a cover image . so , our constructor needs to take four arguments , to take in that extra info : var paperback = function ( title , author , numpages , cover ) { // ... } now , we do n't want to have to do all the work that we already did in the book constructor to remember those first three arguments - we want to take advantage of the fact that the code for that would be the same . so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages ! calvin.burn ( ) ; // omg , you burnt all 256 pages ! `` ` ( well , we 're not really going to burn it , because that 's an amazing book , but perhaps if we were stuck in a glacial desert and desperate for warmth and about to die . ) and now you can see how we can use object-oriented design principles in javascript to create more complex data for your programs and model your program worlds better .
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we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' .
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how could i create and refer to randomized instances of an object ?
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this is a review of what we covered in this tutorial on object-oriented design . when we create programs , we often find that we want to create many different objects that all share similar properties - like many cats , that have slightly different fur color and size , or many buttons , with different labels and positions . we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages . it then sets the initial properties of the object based on those arguments , using the this keyword . when we use this in an object , we are referring to the current instance of an object , referring to itself . we need to store the properties on this to make sure we can remember them later . to create an instance of a book object , we declare a new variable to store it , then use the new keyword , followed by the constructor function name , and pass in the arguments that the constructor expects : var book = new book ( `` robot dreams '' , `` isaac asimov '' , 320 ) ; we can then access any properties that we stored in the object using dot notation : println ( `` i loved reading `` + book.title ) ; // i loved reading robot dreams println ( book.author + `` is my fav author '' ) ; // `` isaac asimov '' is my fav author let 's contrast this for a minute , and show what would have happened if we did n't set up our constructor function properly : var book = function ( title , author , numpages ) { } ; var book = new book ( `` little brother '' , `` cory doctorow '' , 380 ) ; println ( `` i loved reading `` + book.title ) ; // i loved reading undefined println ( book.author + `` is my fav author '' ) ; // undefined is my favorite author if we pass the arguments into the constructor function but do not explicitly store them on this , then we will not be able to access them later ! the object will have long forgotten about them . when we define object types , we often want to associate both properties and behavior with them - like all of our cat objects should be able to meow ( ) and eat ( ) . so we need to be able to attach functions to our object type definitions , and we can do that by defining them on what 's called the object prototype : book.prototype.readitall = function ( ) { this.currentpage = this.numpages ; println ( `` you read `` + this.numpages + `` pages ! `` ) ; } ; it 's like how we would define a function normally , except that we hang it off the book 's prototype instead of just defining it globally . that 's how javascript knows that this is a function that can be called on any book object , and that this function should have access to the this of the book that it 's called on . we can then call the function ( which we call a method , since it 's attached to an object ) , like so : var book = new book ( `` animal farm '' , `` george orwell '' , 112 ) ; book.readitall ( ) ; // you read 112 pages ! remember , the whole point of object-oriented design is that it makes it easy for us to make multiple related objects ( object instances ) . let 's see that in code : `` ` var pirate = new book ( `` pirate cinema '' , `` cory doctorow '' , 384 ) ; var giver = new book ( `` the giver '' , `` lois lowry '' , 179 ) ; var tuck = new book ( `` tuck everlasting '' , `` natalie babbit '' , 144 ) ; pirate.readitall ( ) ; // you read 384 pages ! giver.readitall ( ) ; // you read 179 pages ! tuck.readitall ( ) ; // you read 144 pages ! `` ` that code gives us three books that are similar - they all have the same types of properties and behavior , but also different . sweet ! now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death . they 're both mammals , and that means they share a lot in common , even if they 're also different . in that case , we want to use the idea of object inheritance . an object type could inherit properties and behavior from a parent object type , but then also have its own unique things about it . all the cats and dogs could inherit from mammal , so that they would n't have to invent eat ( ) ing from scratch . how would we do that in javascript ? let 's go back to our book example , and say that book is the `` parent '' object type , and we want to make two object types that inherit from it - paperback and ebook . a paperback is like a book , but it has one main thing different , at least for our program : it has a cover image . so , our constructor needs to take four arguments , to take in that extra info : var paperback = function ( title , author , numpages , cover ) { // ... } now , we do n't want to have to do all the work that we already did in the book constructor to remember those first three arguments - we want to take advantage of the fact that the code for that would be the same . so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages ! calvin.burn ( ) ; // omg , you burnt all 256 pages ! `` ` ( well , we 're not really going to burn it , because that 's an amazing book , but perhaps if we were stuck in a glacial desert and desperate for warmth and about to die . ) and now you can see how we can use object-oriented design principles in javascript to create more complex data for your programs and model your program worlds better .
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now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death .
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how would i create a new pokemon with randomized properties and manage that pokemon later in the code ?
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this is a review of what we covered in this tutorial on object-oriented design . when we create programs , we often find that we want to create many different objects that all share similar properties - like many cats , that have slightly different fur color and size , or many buttons , with different labels and positions . we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages . it then sets the initial properties of the object based on those arguments , using the this keyword . when we use this in an object , we are referring to the current instance of an object , referring to itself . we need to store the properties on this to make sure we can remember them later . to create an instance of a book object , we declare a new variable to store it , then use the new keyword , followed by the constructor function name , and pass in the arguments that the constructor expects : var book = new book ( `` robot dreams '' , `` isaac asimov '' , 320 ) ; we can then access any properties that we stored in the object using dot notation : println ( `` i loved reading `` + book.title ) ; // i loved reading robot dreams println ( book.author + `` is my fav author '' ) ; // `` isaac asimov '' is my fav author let 's contrast this for a minute , and show what would have happened if we did n't set up our constructor function properly : var book = function ( title , author , numpages ) { } ; var book = new book ( `` little brother '' , `` cory doctorow '' , 380 ) ; println ( `` i loved reading `` + book.title ) ; // i loved reading undefined println ( book.author + `` is my fav author '' ) ; // undefined is my favorite author if we pass the arguments into the constructor function but do not explicitly store them on this , then we will not be able to access them later ! the object will have long forgotten about them . when we define object types , we often want to associate both properties and behavior with them - like all of our cat objects should be able to meow ( ) and eat ( ) . so we need to be able to attach functions to our object type definitions , and we can do that by defining them on what 's called the object prototype : book.prototype.readitall = function ( ) { this.currentpage = this.numpages ; println ( `` you read `` + this.numpages + `` pages ! `` ) ; } ; it 's like how we would define a function normally , except that we hang it off the book 's prototype instead of just defining it globally . that 's how javascript knows that this is a function that can be called on any book object , and that this function should have access to the this of the book that it 's called on . we can then call the function ( which we call a method , since it 's attached to an object ) , like so : var book = new book ( `` animal farm '' , `` george orwell '' , 112 ) ; book.readitall ( ) ; // you read 112 pages ! remember , the whole point of object-oriented design is that it makes it easy for us to make multiple related objects ( object instances ) . let 's see that in code : `` ` var pirate = new book ( `` pirate cinema '' , `` cory doctorow '' , 384 ) ; var giver = new book ( `` the giver '' , `` lois lowry '' , 179 ) ; var tuck = new book ( `` tuck everlasting '' , `` natalie babbit '' , 144 ) ; pirate.readitall ( ) ; // you read 384 pages ! giver.readitall ( ) ; // you read 179 pages ! tuck.readitall ( ) ; // you read 144 pages ! `` ` that code gives us three books that are similar - they all have the same types of properties and behavior , but also different . sweet ! now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death . they 're both mammals , and that means they share a lot in common , even if they 're also different . in that case , we want to use the idea of object inheritance . an object type could inherit properties and behavior from a parent object type , but then also have its own unique things about it . all the cats and dogs could inherit from mammal , so that they would n't have to invent eat ( ) ing from scratch . how would we do that in javascript ? let 's go back to our book example , and say that book is the `` parent '' object type , and we want to make two object types that inherit from it - paperback and ebook . a paperback is like a book , but it has one main thing different , at least for our program : it has a cover image . so , our constructor needs to take four arguments , to take in that extra info : var paperback = function ( title , author , numpages , cover ) { // ... } now , we do n't want to have to do all the work that we already did in the book constructor to remember those first three arguments - we want to take advantage of the fact that the code for that would be the same . so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages ! calvin.burn ( ) ; // omg , you burnt all 256 pages ! `` ` ( well , we 're not really going to burn it , because that 's an amazing book , but perhaps if we were stuck in a glacial desert and desperate for warmth and about to die . ) and now you can see how we can use object-oriented design principles in javascript to create more complex data for your programs and model your program worlds better .
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so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages !
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what is oop and what is the thing with prototype called ?
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this is a review of what we covered in this tutorial on object-oriented design . when we create programs , we often find that we want to create many different objects that all share similar properties - like many cats , that have slightly different fur color and size , or many buttons , with different labels and positions . we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages . it then sets the initial properties of the object based on those arguments , using the this keyword . when we use this in an object , we are referring to the current instance of an object , referring to itself . we need to store the properties on this to make sure we can remember them later . to create an instance of a book object , we declare a new variable to store it , then use the new keyword , followed by the constructor function name , and pass in the arguments that the constructor expects : var book = new book ( `` robot dreams '' , `` isaac asimov '' , 320 ) ; we can then access any properties that we stored in the object using dot notation : println ( `` i loved reading `` + book.title ) ; // i loved reading robot dreams println ( book.author + `` is my fav author '' ) ; // `` isaac asimov '' is my fav author let 's contrast this for a minute , and show what would have happened if we did n't set up our constructor function properly : var book = function ( title , author , numpages ) { } ; var book = new book ( `` little brother '' , `` cory doctorow '' , 380 ) ; println ( `` i loved reading `` + book.title ) ; // i loved reading undefined println ( book.author + `` is my fav author '' ) ; // undefined is my favorite author if we pass the arguments into the constructor function but do not explicitly store them on this , then we will not be able to access them later ! the object will have long forgotten about them . when we define object types , we often want to associate both properties and behavior with them - like all of our cat objects should be able to meow ( ) and eat ( ) . so we need to be able to attach functions to our object type definitions , and we can do that by defining them on what 's called the object prototype : book.prototype.readitall = function ( ) { this.currentpage = this.numpages ; println ( `` you read `` + this.numpages + `` pages ! `` ) ; } ; it 's like how we would define a function normally , except that we hang it off the book 's prototype instead of just defining it globally . that 's how javascript knows that this is a function that can be called on any book object , and that this function should have access to the this of the book that it 's called on . we can then call the function ( which we call a method , since it 's attached to an object ) , like so : var book = new book ( `` animal farm '' , `` george orwell '' , 112 ) ; book.readitall ( ) ; // you read 112 pages ! remember , the whole point of object-oriented design is that it makes it easy for us to make multiple related objects ( object instances ) . let 's see that in code : `` ` var pirate = new book ( `` pirate cinema '' , `` cory doctorow '' , 384 ) ; var giver = new book ( `` the giver '' , `` lois lowry '' , 179 ) ; var tuck = new book ( `` tuck everlasting '' , `` natalie babbit '' , 144 ) ; pirate.readitall ( ) ; // you read 384 pages ! giver.readitall ( ) ; // you read 179 pages ! tuck.readitall ( ) ; // you read 144 pages ! `` ` that code gives us three books that are similar - they all have the same types of properties and behavior , but also different . sweet ! now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death . they 're both mammals , and that means they share a lot in common , even if they 're also different . in that case , we want to use the idea of object inheritance . an object type could inherit properties and behavior from a parent object type , but then also have its own unique things about it . all the cats and dogs could inherit from mammal , so that they would n't have to invent eat ( ) ing from scratch . how would we do that in javascript ? let 's go back to our book example , and say that book is the `` parent '' object type , and we want to make two object types that inherit from it - paperback and ebook . a paperback is like a book , but it has one main thing different , at least for our program : it has a cover image . so , our constructor needs to take four arguments , to take in that extra info : var paperback = function ( title , author , numpages , cover ) { // ... } now , we do n't want to have to do all the work that we already did in the book constructor to remember those first three arguments - we want to take advantage of the fact that the code for that would be the same . so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages ! calvin.burn ( ) ; // omg , you burnt all 256 pages ! `` ` ( well , we 're not really going to burn it , because that 's an amazing book , but perhaps if we were stuck in a glacial desert and desperate for warmth and about to die . ) and now you can see how we can use object-oriented design principles in javascript to create more complex data for your programs and model your program worlds better .
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in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages . it then sets the initial properties of the object based on those arguments , using the this keyword .
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what is the most efficient way to store color in a constructer function ?
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this is a review of what we covered in this tutorial on object-oriented design . when we create programs , we often find that we want to create many different objects that all share similar properties - like many cats , that have slightly different fur color and size , or many buttons , with different labels and positions . we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages . it then sets the initial properties of the object based on those arguments , using the this keyword . when we use this in an object , we are referring to the current instance of an object , referring to itself . we need to store the properties on this to make sure we can remember them later . to create an instance of a book object , we declare a new variable to store it , then use the new keyword , followed by the constructor function name , and pass in the arguments that the constructor expects : var book = new book ( `` robot dreams '' , `` isaac asimov '' , 320 ) ; we can then access any properties that we stored in the object using dot notation : println ( `` i loved reading `` + book.title ) ; // i loved reading robot dreams println ( book.author + `` is my fav author '' ) ; // `` isaac asimov '' is my fav author let 's contrast this for a minute , and show what would have happened if we did n't set up our constructor function properly : var book = function ( title , author , numpages ) { } ; var book = new book ( `` little brother '' , `` cory doctorow '' , 380 ) ; println ( `` i loved reading `` + book.title ) ; // i loved reading undefined println ( book.author + `` is my fav author '' ) ; // undefined is my favorite author if we pass the arguments into the constructor function but do not explicitly store them on this , then we will not be able to access them later ! the object will have long forgotten about them . when we define object types , we often want to associate both properties and behavior with them - like all of our cat objects should be able to meow ( ) and eat ( ) . so we need to be able to attach functions to our object type definitions , and we can do that by defining them on what 's called the object prototype : book.prototype.readitall = function ( ) { this.currentpage = this.numpages ; println ( `` you read `` + this.numpages + `` pages ! `` ) ; } ; it 's like how we would define a function normally , except that we hang it off the book 's prototype instead of just defining it globally . that 's how javascript knows that this is a function that can be called on any book object , and that this function should have access to the this of the book that it 's called on . we can then call the function ( which we call a method , since it 's attached to an object ) , like so : var book = new book ( `` animal farm '' , `` george orwell '' , 112 ) ; book.readitall ( ) ; // you read 112 pages ! remember , the whole point of object-oriented design is that it makes it easy for us to make multiple related objects ( object instances ) . let 's see that in code : `` ` var pirate = new book ( `` pirate cinema '' , `` cory doctorow '' , 384 ) ; var giver = new book ( `` the giver '' , `` lois lowry '' , 179 ) ; var tuck = new book ( `` tuck everlasting '' , `` natalie babbit '' , 144 ) ; pirate.readitall ( ) ; // you read 384 pages ! giver.readitall ( ) ; // you read 179 pages ! tuck.readitall ( ) ; // you read 144 pages ! `` ` that code gives us three books that are similar - they all have the same types of properties and behavior , but also different . sweet ! now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death . they 're both mammals , and that means they share a lot in common , even if they 're also different . in that case , we want to use the idea of object inheritance . an object type could inherit properties and behavior from a parent object type , but then also have its own unique things about it . all the cats and dogs could inherit from mammal , so that they would n't have to invent eat ( ) ing from scratch . how would we do that in javascript ? let 's go back to our book example , and say that book is the `` parent '' object type , and we want to make two object types that inherit from it - paperback and ebook . a paperback is like a book , but it has one main thing different , at least for our program : it has a cover image . so , our constructor needs to take four arguments , to take in that extra info : var paperback = function ( title , author , numpages , cover ) { // ... } now , we do n't want to have to do all the work that we already did in the book constructor to remember those first three arguments - we want to take advantage of the fact that the code for that would be the same . so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages ! calvin.burn ( ) ; // omg , you burnt all 256 pages ! `` ` ( well , we 're not really going to burn it , because that 's an amazing book , but perhaps if we were stuck in a glacial desert and desperate for warmth and about to die . ) and now you can see how we can use object-oriented design principles in javascript to create more complex data for your programs and model your program worlds better .
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now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death .
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would it be to make different arguments for all three color values ?
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this is a review of what we covered in this tutorial on object-oriented design . when we create programs , we often find that we want to create many different objects that all share similar properties - like many cats , that have slightly different fur color and size , or many buttons , with different labels and positions . we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages . it then sets the initial properties of the object based on those arguments , using the this keyword . when we use this in an object , we are referring to the current instance of an object , referring to itself . we need to store the properties on this to make sure we can remember them later . to create an instance of a book object , we declare a new variable to store it , then use the new keyword , followed by the constructor function name , and pass in the arguments that the constructor expects : var book = new book ( `` robot dreams '' , `` isaac asimov '' , 320 ) ; we can then access any properties that we stored in the object using dot notation : println ( `` i loved reading `` + book.title ) ; // i loved reading robot dreams println ( book.author + `` is my fav author '' ) ; // `` isaac asimov '' is my fav author let 's contrast this for a minute , and show what would have happened if we did n't set up our constructor function properly : var book = function ( title , author , numpages ) { } ; var book = new book ( `` little brother '' , `` cory doctorow '' , 380 ) ; println ( `` i loved reading `` + book.title ) ; // i loved reading undefined println ( book.author + `` is my fav author '' ) ; // undefined is my favorite author if we pass the arguments into the constructor function but do not explicitly store them on this , then we will not be able to access them later ! the object will have long forgotten about them . when we define object types , we often want to associate both properties and behavior with them - like all of our cat objects should be able to meow ( ) and eat ( ) . so we need to be able to attach functions to our object type definitions , and we can do that by defining them on what 's called the object prototype : book.prototype.readitall = function ( ) { this.currentpage = this.numpages ; println ( `` you read `` + this.numpages + `` pages ! `` ) ; } ; it 's like how we would define a function normally , except that we hang it off the book 's prototype instead of just defining it globally . that 's how javascript knows that this is a function that can be called on any book object , and that this function should have access to the this of the book that it 's called on . we can then call the function ( which we call a method , since it 's attached to an object ) , like so : var book = new book ( `` animal farm '' , `` george orwell '' , 112 ) ; book.readitall ( ) ; // you read 112 pages ! remember , the whole point of object-oriented design is that it makes it easy for us to make multiple related objects ( object instances ) . let 's see that in code : `` ` var pirate = new book ( `` pirate cinema '' , `` cory doctorow '' , 384 ) ; var giver = new book ( `` the giver '' , `` lois lowry '' , 179 ) ; var tuck = new book ( `` tuck everlasting '' , `` natalie babbit '' , 144 ) ; pirate.readitall ( ) ; // you read 384 pages ! giver.readitall ( ) ; // you read 179 pages ! tuck.readitall ( ) ; // you read 144 pages ! `` ` that code gives us three books that are similar - they all have the same types of properties and behavior , but also different . sweet ! now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death . they 're both mammals , and that means they share a lot in common , even if they 're also different . in that case , we want to use the idea of object inheritance . an object type could inherit properties and behavior from a parent object type , but then also have its own unique things about it . all the cats and dogs could inherit from mammal , so that they would n't have to invent eat ( ) ing from scratch . how would we do that in javascript ? let 's go back to our book example , and say that book is the `` parent '' object type , and we want to make two object types that inherit from it - paperback and ebook . a paperback is like a book , but it has one main thing different , at least for our program : it has a cover image . so , our constructor needs to take four arguments , to take in that extra info : var paperback = function ( title , author , numpages , cover ) { // ... } now , we do n't want to have to do all the work that we already did in the book constructor to remember those first three arguments - we want to take advantage of the fact that the code for that would be the same . so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages ! calvin.burn ( ) ; // omg , you burnt all 256 pages ! `` ` ( well , we 're not really going to burn it , because that 's an amazing book , but perhaps if we were stuck in a glacial desert and desperate for warmth and about to die . ) and now you can see how we can use object-oriented design principles in javascript to create more complex data for your programs and model your program worlds better .
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in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type .
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can i define an `` eat '' function to my animal object that is later overriden by the specific code for each subtype of animal ?
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this is a review of what we covered in this tutorial on object-oriented design . when we create programs , we often find that we want to create many different objects that all share similar properties - like many cats , that have slightly different fur color and size , or many buttons , with different labels and positions . we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages . it then sets the initial properties of the object based on those arguments , using the this keyword . when we use this in an object , we are referring to the current instance of an object , referring to itself . we need to store the properties on this to make sure we can remember them later . to create an instance of a book object , we declare a new variable to store it , then use the new keyword , followed by the constructor function name , and pass in the arguments that the constructor expects : var book = new book ( `` robot dreams '' , `` isaac asimov '' , 320 ) ; we can then access any properties that we stored in the object using dot notation : println ( `` i loved reading `` + book.title ) ; // i loved reading robot dreams println ( book.author + `` is my fav author '' ) ; // `` isaac asimov '' is my fav author let 's contrast this for a minute , and show what would have happened if we did n't set up our constructor function properly : var book = function ( title , author , numpages ) { } ; var book = new book ( `` little brother '' , `` cory doctorow '' , 380 ) ; println ( `` i loved reading `` + book.title ) ; // i loved reading undefined println ( book.author + `` is my fav author '' ) ; // undefined is my favorite author if we pass the arguments into the constructor function but do not explicitly store them on this , then we will not be able to access them later ! the object will have long forgotten about them . when we define object types , we often want to associate both properties and behavior with them - like all of our cat objects should be able to meow ( ) and eat ( ) . so we need to be able to attach functions to our object type definitions , and we can do that by defining them on what 's called the object prototype : book.prototype.readitall = function ( ) { this.currentpage = this.numpages ; println ( `` you read `` + this.numpages + `` pages ! `` ) ; } ; it 's like how we would define a function normally , except that we hang it off the book 's prototype instead of just defining it globally . that 's how javascript knows that this is a function that can be called on any book object , and that this function should have access to the this of the book that it 's called on . we can then call the function ( which we call a method , since it 's attached to an object ) , like so : var book = new book ( `` animal farm '' , `` george orwell '' , 112 ) ; book.readitall ( ) ; // you read 112 pages ! remember , the whole point of object-oriented design is that it makes it easy for us to make multiple related objects ( object instances ) . let 's see that in code : `` ` var pirate = new book ( `` pirate cinema '' , `` cory doctorow '' , 384 ) ; var giver = new book ( `` the giver '' , `` lois lowry '' , 179 ) ; var tuck = new book ( `` tuck everlasting '' , `` natalie babbit '' , 144 ) ; pirate.readitall ( ) ; // you read 384 pages ! giver.readitall ( ) ; // you read 179 pages ! tuck.readitall ( ) ; // you read 144 pages ! `` ` that code gives us three books that are similar - they all have the same types of properties and behavior , but also different . sweet ! now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death . they 're both mammals , and that means they share a lot in common , even if they 're also different . in that case , we want to use the idea of object inheritance . an object type could inherit properties and behavior from a parent object type , but then also have its own unique things about it . all the cats and dogs could inherit from mammal , so that they would n't have to invent eat ( ) ing from scratch . how would we do that in javascript ? let 's go back to our book example , and say that book is the `` parent '' object type , and we want to make two object types that inherit from it - paperback and ebook . a paperback is like a book , but it has one main thing different , at least for our program : it has a cover image . so , our constructor needs to take four arguments , to take in that extra info : var paperback = function ( title , author , numpages , cover ) { // ... } now , we do n't want to have to do all the work that we already did in the book constructor to remember those first three arguments - we want to take advantage of the fact that the code for that would be the same . so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages ! calvin.burn ( ) ; // omg , you burnt all 256 pages ! `` ` ( well , we 're not really going to burn it , because that 's an amazing book , but perhaps if we were stuck in a glacial desert and desperate for warmth and about to die . ) and now you can see how we can use object-oriented design principles in javascript to create more complex data for your programs and model your program worlds better .
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this is a review of what we covered in this tutorial on object-oriented design . when we create programs , we often find that we want to create many different objects that all share similar properties - like many cats , that have slightly different fur color and size , or many buttons , with different labels and positions .
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what conditions must be met to be awarded the `` intro to js : drawing & animation mastery badge '' ?
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this is a review of what we covered in this tutorial on object-oriented design . when we create programs , we often find that we want to create many different objects that all share similar properties - like many cats , that have slightly different fur color and size , or many buttons , with different labels and positions . we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages . it then sets the initial properties of the object based on those arguments , using the this keyword . when we use this in an object , we are referring to the current instance of an object , referring to itself . we need to store the properties on this to make sure we can remember them later . to create an instance of a book object , we declare a new variable to store it , then use the new keyword , followed by the constructor function name , and pass in the arguments that the constructor expects : var book = new book ( `` robot dreams '' , `` isaac asimov '' , 320 ) ; we can then access any properties that we stored in the object using dot notation : println ( `` i loved reading `` + book.title ) ; // i loved reading robot dreams println ( book.author + `` is my fav author '' ) ; // `` isaac asimov '' is my fav author let 's contrast this for a minute , and show what would have happened if we did n't set up our constructor function properly : var book = function ( title , author , numpages ) { } ; var book = new book ( `` little brother '' , `` cory doctorow '' , 380 ) ; println ( `` i loved reading `` + book.title ) ; // i loved reading undefined println ( book.author + `` is my fav author '' ) ; // undefined is my favorite author if we pass the arguments into the constructor function but do not explicitly store them on this , then we will not be able to access them later ! the object will have long forgotten about them . when we define object types , we often want to associate both properties and behavior with them - like all of our cat objects should be able to meow ( ) and eat ( ) . so we need to be able to attach functions to our object type definitions , and we can do that by defining them on what 's called the object prototype : book.prototype.readitall = function ( ) { this.currentpage = this.numpages ; println ( `` you read `` + this.numpages + `` pages ! `` ) ; } ; it 's like how we would define a function normally , except that we hang it off the book 's prototype instead of just defining it globally . that 's how javascript knows that this is a function that can be called on any book object , and that this function should have access to the this of the book that it 's called on . we can then call the function ( which we call a method , since it 's attached to an object ) , like so : var book = new book ( `` animal farm '' , `` george orwell '' , 112 ) ; book.readitall ( ) ; // you read 112 pages ! remember , the whole point of object-oriented design is that it makes it easy for us to make multiple related objects ( object instances ) . let 's see that in code : `` ` var pirate = new book ( `` pirate cinema '' , `` cory doctorow '' , 384 ) ; var giver = new book ( `` the giver '' , `` lois lowry '' , 179 ) ; var tuck = new book ( `` tuck everlasting '' , `` natalie babbit '' , 144 ) ; pirate.readitall ( ) ; // you read 384 pages ! giver.readitall ( ) ; // you read 179 pages ! tuck.readitall ( ) ; // you read 144 pages ! `` ` that code gives us three books that are similar - they all have the same types of properties and behavior , but also different . sweet ! now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death . they 're both mammals , and that means they share a lot in common , even if they 're also different . in that case , we want to use the idea of object inheritance . an object type could inherit properties and behavior from a parent object type , but then also have its own unique things about it . all the cats and dogs could inherit from mammal , so that they would n't have to invent eat ( ) ing from scratch . how would we do that in javascript ? let 's go back to our book example , and say that book is the `` parent '' object type , and we want to make two object types that inherit from it - paperback and ebook . a paperback is like a book , but it has one main thing different , at least for our program : it has a cover image . so , our constructor needs to take four arguments , to take in that extra info : var paperback = function ( title , author , numpages , cover ) { // ... } now , we do n't want to have to do all the work that we already did in the book constructor to remember those first three arguments - we want to take advantage of the fact that the code for that would be the same . so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages ! calvin.burn ( ) ; // omg , you burnt all 256 pages ! `` ` ( well , we 're not really going to burn it , because that 's an amazing book , but perhaps if we were stuck in a glacial desert and desperate for warmth and about to die . ) and now you can see how we can use object-oriented design principles in javascript to create more complex data for your programs and model your program worlds better .
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to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages .
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for instance , if i create three cats using a constructor , how can i animate one of them so that it runs across the screen ?
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this is a review of what we covered in this tutorial on object-oriented design . when we create programs , we often find that we want to create many different objects that all share similar properties - like many cats , that have slightly different fur color and size , or many buttons , with different labels and positions . we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages . it then sets the initial properties of the object based on those arguments , using the this keyword . when we use this in an object , we are referring to the current instance of an object , referring to itself . we need to store the properties on this to make sure we can remember them later . to create an instance of a book object , we declare a new variable to store it , then use the new keyword , followed by the constructor function name , and pass in the arguments that the constructor expects : var book = new book ( `` robot dreams '' , `` isaac asimov '' , 320 ) ; we can then access any properties that we stored in the object using dot notation : println ( `` i loved reading `` + book.title ) ; // i loved reading robot dreams println ( book.author + `` is my fav author '' ) ; // `` isaac asimov '' is my fav author let 's contrast this for a minute , and show what would have happened if we did n't set up our constructor function properly : var book = function ( title , author , numpages ) { } ; var book = new book ( `` little brother '' , `` cory doctorow '' , 380 ) ; println ( `` i loved reading `` + book.title ) ; // i loved reading undefined println ( book.author + `` is my fav author '' ) ; // undefined is my favorite author if we pass the arguments into the constructor function but do not explicitly store them on this , then we will not be able to access them later ! the object will have long forgotten about them . when we define object types , we often want to associate both properties and behavior with them - like all of our cat objects should be able to meow ( ) and eat ( ) . so we need to be able to attach functions to our object type definitions , and we can do that by defining them on what 's called the object prototype : book.prototype.readitall = function ( ) { this.currentpage = this.numpages ; println ( `` you read `` + this.numpages + `` pages ! `` ) ; } ; it 's like how we would define a function normally , except that we hang it off the book 's prototype instead of just defining it globally . that 's how javascript knows that this is a function that can be called on any book object , and that this function should have access to the this of the book that it 's called on . we can then call the function ( which we call a method , since it 's attached to an object ) , like so : var book = new book ( `` animal farm '' , `` george orwell '' , 112 ) ; book.readitall ( ) ; // you read 112 pages ! remember , the whole point of object-oriented design is that it makes it easy for us to make multiple related objects ( object instances ) . let 's see that in code : `` ` var pirate = new book ( `` pirate cinema '' , `` cory doctorow '' , 384 ) ; var giver = new book ( `` the giver '' , `` lois lowry '' , 179 ) ; var tuck = new book ( `` tuck everlasting '' , `` natalie babbit '' , 144 ) ; pirate.readitall ( ) ; // you read 384 pages ! giver.readitall ( ) ; // you read 179 pages ! tuck.readitall ( ) ; // you read 144 pages ! `` ` that code gives us three books that are similar - they all have the same types of properties and behavior , but also different . sweet ! now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death . they 're both mammals , and that means they share a lot in common , even if they 're also different . in that case , we want to use the idea of object inheritance . an object type could inherit properties and behavior from a parent object type , but then also have its own unique things about it . all the cats and dogs could inherit from mammal , so that they would n't have to invent eat ( ) ing from scratch . how would we do that in javascript ? let 's go back to our book example , and say that book is the `` parent '' object type , and we want to make two object types that inherit from it - paperback and ebook . a paperback is like a book , but it has one main thing different , at least for our program : it has a cover image . so , our constructor needs to take four arguments , to take in that extra info : var paperback = function ( title , author , numpages , cover ) { // ... } now , we do n't want to have to do all the work that we already did in the book constructor to remember those first three arguments - we want to take advantage of the fact that the code for that would be the same . so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages ! calvin.burn ( ) ; // omg , you burnt all 256 pages ! `` ` ( well , we 're not really going to burn it , because that 's an amazing book , but perhaps if we were stuck in a glacial desert and desperate for warmth and about to die . ) and now you can see how we can use object-oriented design principles in javascript to create more complex data for your programs and model your program worlds better .
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how would we do that in javascript ? let 's go back to our book example , and say that book is the `` parent '' object type , and we want to make two object types that inherit from it - paperback and ebook . a paperback is like a book , but it has one main thing different , at least for our program : it has a cover image .
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does the two parent object types , one we call in our constructor function and the other we inherit the methods from , need to be the same ?
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this is a review of what we covered in this tutorial on object-oriented design . when we create programs , we often find that we want to create many different objects that all share similar properties - like many cats , that have slightly different fur color and size , or many buttons , with different labels and positions . we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages . it then sets the initial properties of the object based on those arguments , using the this keyword . when we use this in an object , we are referring to the current instance of an object , referring to itself . we need to store the properties on this to make sure we can remember them later . to create an instance of a book object , we declare a new variable to store it , then use the new keyword , followed by the constructor function name , and pass in the arguments that the constructor expects : var book = new book ( `` robot dreams '' , `` isaac asimov '' , 320 ) ; we can then access any properties that we stored in the object using dot notation : println ( `` i loved reading `` + book.title ) ; // i loved reading robot dreams println ( book.author + `` is my fav author '' ) ; // `` isaac asimov '' is my fav author let 's contrast this for a minute , and show what would have happened if we did n't set up our constructor function properly : var book = function ( title , author , numpages ) { } ; var book = new book ( `` little brother '' , `` cory doctorow '' , 380 ) ; println ( `` i loved reading `` + book.title ) ; // i loved reading undefined println ( book.author + `` is my fav author '' ) ; // undefined is my favorite author if we pass the arguments into the constructor function but do not explicitly store them on this , then we will not be able to access them later ! the object will have long forgotten about them . when we define object types , we often want to associate both properties and behavior with them - like all of our cat objects should be able to meow ( ) and eat ( ) . so we need to be able to attach functions to our object type definitions , and we can do that by defining them on what 's called the object prototype : book.prototype.readitall = function ( ) { this.currentpage = this.numpages ; println ( `` you read `` + this.numpages + `` pages ! `` ) ; } ; it 's like how we would define a function normally , except that we hang it off the book 's prototype instead of just defining it globally . that 's how javascript knows that this is a function that can be called on any book object , and that this function should have access to the this of the book that it 's called on . we can then call the function ( which we call a method , since it 's attached to an object ) , like so : var book = new book ( `` animal farm '' , `` george orwell '' , 112 ) ; book.readitall ( ) ; // you read 112 pages ! remember , the whole point of object-oriented design is that it makes it easy for us to make multiple related objects ( object instances ) . let 's see that in code : `` ` var pirate = new book ( `` pirate cinema '' , `` cory doctorow '' , 384 ) ; var giver = new book ( `` the giver '' , `` lois lowry '' , 179 ) ; var tuck = new book ( `` tuck everlasting '' , `` natalie babbit '' , 144 ) ; pirate.readitall ( ) ; // you read 384 pages ! giver.readitall ( ) ; // you read 179 pages ! tuck.readitall ( ) ; // you read 144 pages ! `` ` that code gives us three books that are similar - they all have the same types of properties and behavior , but also different . sweet ! now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death . they 're both mammals , and that means they share a lot in common , even if they 're also different . in that case , we want to use the idea of object inheritance . an object type could inherit properties and behavior from a parent object type , but then also have its own unique things about it . all the cats and dogs could inherit from mammal , so that they would n't have to invent eat ( ) ing from scratch . how would we do that in javascript ? let 's go back to our book example , and say that book is the `` parent '' object type , and we want to make two object types that inherit from it - paperback and ebook . a paperback is like a book , but it has one main thing different , at least for our program : it has a cover image . so , our constructor needs to take four arguments , to take in that extra info : var paperback = function ( title , author , numpages , cover ) { // ... } now , we do n't want to have to do all the work that we already did in the book constructor to remember those first three arguments - we want to take advantage of the fact that the code for that would be the same . so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages ! calvin.burn ( ) ; // omg , you burnt all 256 pages ! `` ` ( well , we 're not really going to burn it , because that 's an amazing book , but perhaps if we were stuck in a glacial desert and desperate for warmth and about to die . ) and now you can see how we can use object-oriented design principles in javascript to create more complex data for your programs and model your program worlds better .
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we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' .
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is there a way to add if statements inside of objects ?
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this is a review of what we covered in this tutorial on object-oriented design . when we create programs , we often find that we want to create many different objects that all share similar properties - like many cats , that have slightly different fur color and size , or many buttons , with different labels and positions . we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages . it then sets the initial properties of the object based on those arguments , using the this keyword . when we use this in an object , we are referring to the current instance of an object , referring to itself . we need to store the properties on this to make sure we can remember them later . to create an instance of a book object , we declare a new variable to store it , then use the new keyword , followed by the constructor function name , and pass in the arguments that the constructor expects : var book = new book ( `` robot dreams '' , `` isaac asimov '' , 320 ) ; we can then access any properties that we stored in the object using dot notation : println ( `` i loved reading `` + book.title ) ; // i loved reading robot dreams println ( book.author + `` is my fav author '' ) ; // `` isaac asimov '' is my fav author let 's contrast this for a minute , and show what would have happened if we did n't set up our constructor function properly : var book = function ( title , author , numpages ) { } ; var book = new book ( `` little brother '' , `` cory doctorow '' , 380 ) ; println ( `` i loved reading `` + book.title ) ; // i loved reading undefined println ( book.author + `` is my fav author '' ) ; // undefined is my favorite author if we pass the arguments into the constructor function but do not explicitly store them on this , then we will not be able to access them later ! the object will have long forgotten about them . when we define object types , we often want to associate both properties and behavior with them - like all of our cat objects should be able to meow ( ) and eat ( ) . so we need to be able to attach functions to our object type definitions , and we can do that by defining them on what 's called the object prototype : book.prototype.readitall = function ( ) { this.currentpage = this.numpages ; println ( `` you read `` + this.numpages + `` pages ! `` ) ; } ; it 's like how we would define a function normally , except that we hang it off the book 's prototype instead of just defining it globally . that 's how javascript knows that this is a function that can be called on any book object , and that this function should have access to the this of the book that it 's called on . we can then call the function ( which we call a method , since it 's attached to an object ) , like so : var book = new book ( `` animal farm '' , `` george orwell '' , 112 ) ; book.readitall ( ) ; // you read 112 pages ! remember , the whole point of object-oriented design is that it makes it easy for us to make multiple related objects ( object instances ) . let 's see that in code : `` ` var pirate = new book ( `` pirate cinema '' , `` cory doctorow '' , 384 ) ; var giver = new book ( `` the giver '' , `` lois lowry '' , 179 ) ; var tuck = new book ( `` tuck everlasting '' , `` natalie babbit '' , 144 ) ; pirate.readitall ( ) ; // you read 384 pages ! giver.readitall ( ) ; // you read 179 pages ! tuck.readitall ( ) ; // you read 144 pages ! `` ` that code gives us three books that are similar - they all have the same types of properties and behavior , but also different . sweet ! now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death . they 're both mammals , and that means they share a lot in common , even if they 're also different . in that case , we want to use the idea of object inheritance . an object type could inherit properties and behavior from a parent object type , but then also have its own unique things about it . all the cats and dogs could inherit from mammal , so that they would n't have to invent eat ( ) ing from scratch . how would we do that in javascript ? let 's go back to our book example , and say that book is the `` parent '' object type , and we want to make two object types that inherit from it - paperback and ebook . a paperback is like a book , but it has one main thing different , at least for our program : it has a cover image . so , our constructor needs to take four arguments , to take in that extra info : var paperback = function ( title , author , numpages , cover ) { // ... } now , we do n't want to have to do all the work that we already did in the book constructor to remember those first three arguments - we want to take advantage of the fact that the code for that would be the same . so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages ! calvin.burn ( ) ; // omg , you burnt all 256 pages ! `` ` ( well , we 're not really going to burn it , because that 's an amazing book , but perhaps if we were stuck in a glacial desert and desperate for warmth and about to die . ) and now you can see how we can use object-oriented design principles in javascript to create more complex data for your programs and model your program worlds better .
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we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' .
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so here comes my question : where is the object , exactly ?
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this is a review of what we covered in this tutorial on object-oriented design . when we create programs , we often find that we want to create many different objects that all share similar properties - like many cats , that have slightly different fur color and size , or many buttons , with different labels and positions . we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages . it then sets the initial properties of the object based on those arguments , using the this keyword . when we use this in an object , we are referring to the current instance of an object , referring to itself . we need to store the properties on this to make sure we can remember them later . to create an instance of a book object , we declare a new variable to store it , then use the new keyword , followed by the constructor function name , and pass in the arguments that the constructor expects : var book = new book ( `` robot dreams '' , `` isaac asimov '' , 320 ) ; we can then access any properties that we stored in the object using dot notation : println ( `` i loved reading `` + book.title ) ; // i loved reading robot dreams println ( book.author + `` is my fav author '' ) ; // `` isaac asimov '' is my fav author let 's contrast this for a minute , and show what would have happened if we did n't set up our constructor function properly : var book = function ( title , author , numpages ) { } ; var book = new book ( `` little brother '' , `` cory doctorow '' , 380 ) ; println ( `` i loved reading `` + book.title ) ; // i loved reading undefined println ( book.author + `` is my fav author '' ) ; // undefined is my favorite author if we pass the arguments into the constructor function but do not explicitly store them on this , then we will not be able to access them later ! the object will have long forgotten about them . when we define object types , we often want to associate both properties and behavior with them - like all of our cat objects should be able to meow ( ) and eat ( ) . so we need to be able to attach functions to our object type definitions , and we can do that by defining them on what 's called the object prototype : book.prototype.readitall = function ( ) { this.currentpage = this.numpages ; println ( `` you read `` + this.numpages + `` pages ! `` ) ; } ; it 's like how we would define a function normally , except that we hang it off the book 's prototype instead of just defining it globally . that 's how javascript knows that this is a function that can be called on any book object , and that this function should have access to the this of the book that it 's called on . we can then call the function ( which we call a method , since it 's attached to an object ) , like so : var book = new book ( `` animal farm '' , `` george orwell '' , 112 ) ; book.readitall ( ) ; // you read 112 pages ! remember , the whole point of object-oriented design is that it makes it easy for us to make multiple related objects ( object instances ) . let 's see that in code : `` ` var pirate = new book ( `` pirate cinema '' , `` cory doctorow '' , 384 ) ; var giver = new book ( `` the giver '' , `` lois lowry '' , 179 ) ; var tuck = new book ( `` tuck everlasting '' , `` natalie babbit '' , 144 ) ; pirate.readitall ( ) ; // you read 384 pages ! giver.readitall ( ) ; // you read 179 pages ! tuck.readitall ( ) ; // you read 144 pages ! `` ` that code gives us three books that are similar - they all have the same types of properties and behavior , but also different . sweet ! now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death . they 're both mammals , and that means they share a lot in common , even if they 're also different . in that case , we want to use the idea of object inheritance . an object type could inherit properties and behavior from a parent object type , but then also have its own unique things about it . all the cats and dogs could inherit from mammal , so that they would n't have to invent eat ( ) ing from scratch . how would we do that in javascript ? let 's go back to our book example , and say that book is the `` parent '' object type , and we want to make two object types that inherit from it - paperback and ebook . a paperback is like a book , but it has one main thing different , at least for our program : it has a cover image . so , our constructor needs to take four arguments , to take in that extra info : var paperback = function ( title , author , numpages , cover ) { // ... } now , we do n't want to have to do all the work that we already did in the book constructor to remember those first three arguments - we want to take advantage of the fact that the code for that would be the same . so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages ! calvin.burn ( ) ; // omg , you burnt all 256 pages ! `` ` ( well , we 're not really going to burn it , because that 's an amazing book , but perhaps if we were stuck in a glacial desert and desperate for warmth and about to die . ) and now you can see how we can use object-oriented design principles in javascript to create more complex data for your programs and model your program worlds better .
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this is a review of what we covered in this tutorial on object-oriented design . when we create programs , we often find that we want to create many different objects that all share similar properties - like many cats , that have slightly different fur color and size , or many buttons , with different labels and positions .
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how to do challenge : flower grower ?
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this is a review of what we covered in this tutorial on object-oriented design . when we create programs , we often find that we want to create many different objects that all share similar properties - like many cats , that have slightly different fur color and size , or many buttons , with different labels and positions . we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages . it then sets the initial properties of the object based on those arguments , using the this keyword . when we use this in an object , we are referring to the current instance of an object , referring to itself . we need to store the properties on this to make sure we can remember them later . to create an instance of a book object , we declare a new variable to store it , then use the new keyword , followed by the constructor function name , and pass in the arguments that the constructor expects : var book = new book ( `` robot dreams '' , `` isaac asimov '' , 320 ) ; we can then access any properties that we stored in the object using dot notation : println ( `` i loved reading `` + book.title ) ; // i loved reading robot dreams println ( book.author + `` is my fav author '' ) ; // `` isaac asimov '' is my fav author let 's contrast this for a minute , and show what would have happened if we did n't set up our constructor function properly : var book = function ( title , author , numpages ) { } ; var book = new book ( `` little brother '' , `` cory doctorow '' , 380 ) ; println ( `` i loved reading `` + book.title ) ; // i loved reading undefined println ( book.author + `` is my fav author '' ) ; // undefined is my favorite author if we pass the arguments into the constructor function but do not explicitly store them on this , then we will not be able to access them later ! the object will have long forgotten about them . when we define object types , we often want to associate both properties and behavior with them - like all of our cat objects should be able to meow ( ) and eat ( ) . so we need to be able to attach functions to our object type definitions , and we can do that by defining them on what 's called the object prototype : book.prototype.readitall = function ( ) { this.currentpage = this.numpages ; println ( `` you read `` + this.numpages + `` pages ! `` ) ; } ; it 's like how we would define a function normally , except that we hang it off the book 's prototype instead of just defining it globally . that 's how javascript knows that this is a function that can be called on any book object , and that this function should have access to the this of the book that it 's called on . we can then call the function ( which we call a method , since it 's attached to an object ) , like so : var book = new book ( `` animal farm '' , `` george orwell '' , 112 ) ; book.readitall ( ) ; // you read 112 pages ! remember , the whole point of object-oriented design is that it makes it easy for us to make multiple related objects ( object instances ) . let 's see that in code : `` ` var pirate = new book ( `` pirate cinema '' , `` cory doctorow '' , 384 ) ; var giver = new book ( `` the giver '' , `` lois lowry '' , 179 ) ; var tuck = new book ( `` tuck everlasting '' , `` natalie babbit '' , 144 ) ; pirate.readitall ( ) ; // you read 384 pages ! giver.readitall ( ) ; // you read 179 pages ! tuck.readitall ( ) ; // you read 144 pages ! `` ` that code gives us three books that are similar - they all have the same types of properties and behavior , but also different . sweet ! now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death . they 're both mammals , and that means they share a lot in common , even if they 're also different . in that case , we want to use the idea of object inheritance . an object type could inherit properties and behavior from a parent object type , but then also have its own unique things about it . all the cats and dogs could inherit from mammal , so that they would n't have to invent eat ( ) ing from scratch . how would we do that in javascript ? let 's go back to our book example , and say that book is the `` parent '' object type , and we want to make two object types that inherit from it - paperback and ebook . a paperback is like a book , but it has one main thing different , at least for our program : it has a cover image . so , our constructor needs to take four arguments , to take in that extra info : var paperback = function ( title , author , numpages , cover ) { // ... } now , we do n't want to have to do all the work that we already did in the book constructor to remember those first three arguments - we want to take advantage of the fact that the code for that would be the same . so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages ! calvin.burn ( ) ; // omg , you burnt all 256 pages ! `` ` ( well , we 're not really going to burn it , because that 's an amazing book , but perhaps if we were stuck in a glacial desert and desperate for warmth and about to die . ) and now you can see how we can use object-oriented design principles in javascript to create more complex data for your programs and model your program worlds better .
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that 's how javascript knows that this is a function that can be called on any book object , and that this function should have access to the this of the book that it 's called on . we can then call the function ( which we call a method , since it 's attached to an object ) , like so : var book = new book ( `` animal farm '' , `` george orwell '' , 112 ) ; book.readitall ( ) ; // you read 112 pages ! remember , the whole point of object-oriented design is that it makes it easy for us to make multiple related objects ( object instances ) . let 's see that in code : `` ` var pirate = new book ( `` pirate cinema '' , `` cory doctorow '' , 384 ) ; var giver = new book ( `` the giver '' , `` lois lowry '' , 179 ) ; var tuck = new book ( `` tuck everlasting '' , `` natalie babbit '' , 144 ) ; pirate.readitall ( ) ; // you read 384 pages !
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how do you make a prototype function in an object global , so you can call an object 's method from inside another object 's method ?
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this is a review of what we covered in this tutorial on object-oriented design . when we create programs , we often find that we want to create many different objects that all share similar properties - like many cats , that have slightly different fur color and size , or many buttons , with different labels and positions . we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages . it then sets the initial properties of the object based on those arguments , using the this keyword . when we use this in an object , we are referring to the current instance of an object , referring to itself . we need to store the properties on this to make sure we can remember them later . to create an instance of a book object , we declare a new variable to store it , then use the new keyword , followed by the constructor function name , and pass in the arguments that the constructor expects : var book = new book ( `` robot dreams '' , `` isaac asimov '' , 320 ) ; we can then access any properties that we stored in the object using dot notation : println ( `` i loved reading `` + book.title ) ; // i loved reading robot dreams println ( book.author + `` is my fav author '' ) ; // `` isaac asimov '' is my fav author let 's contrast this for a minute , and show what would have happened if we did n't set up our constructor function properly : var book = function ( title , author , numpages ) { } ; var book = new book ( `` little brother '' , `` cory doctorow '' , 380 ) ; println ( `` i loved reading `` + book.title ) ; // i loved reading undefined println ( book.author + `` is my fav author '' ) ; // undefined is my favorite author if we pass the arguments into the constructor function but do not explicitly store them on this , then we will not be able to access them later ! the object will have long forgotten about them . when we define object types , we often want to associate both properties and behavior with them - like all of our cat objects should be able to meow ( ) and eat ( ) . so we need to be able to attach functions to our object type definitions , and we can do that by defining them on what 's called the object prototype : book.prototype.readitall = function ( ) { this.currentpage = this.numpages ; println ( `` you read `` + this.numpages + `` pages ! `` ) ; } ; it 's like how we would define a function normally , except that we hang it off the book 's prototype instead of just defining it globally . that 's how javascript knows that this is a function that can be called on any book object , and that this function should have access to the this of the book that it 's called on . we can then call the function ( which we call a method , since it 's attached to an object ) , like so : var book = new book ( `` animal farm '' , `` george orwell '' , 112 ) ; book.readitall ( ) ; // you read 112 pages ! remember , the whole point of object-oriented design is that it makes it easy for us to make multiple related objects ( object instances ) . let 's see that in code : `` ` var pirate = new book ( `` pirate cinema '' , `` cory doctorow '' , 384 ) ; var giver = new book ( `` the giver '' , `` lois lowry '' , 179 ) ; var tuck = new book ( `` tuck everlasting '' , `` natalie babbit '' , 144 ) ; pirate.readitall ( ) ; // you read 384 pages ! giver.readitall ( ) ; // you read 179 pages ! tuck.readitall ( ) ; // you read 144 pages ! `` ` that code gives us three books that are similar - they all have the same types of properties and behavior , but also different . sweet ! now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death . they 're both mammals , and that means they share a lot in common , even if they 're also different . in that case , we want to use the idea of object inheritance . an object type could inherit properties and behavior from a parent object type , but then also have its own unique things about it . all the cats and dogs could inherit from mammal , so that they would n't have to invent eat ( ) ing from scratch . how would we do that in javascript ? let 's go back to our book example , and say that book is the `` parent '' object type , and we want to make two object types that inherit from it - paperback and ebook . a paperback is like a book , but it has one main thing different , at least for our program : it has a cover image . so , our constructor needs to take four arguments , to take in that extra info : var paperback = function ( title , author , numpages , cover ) { // ... } now , we do n't want to have to do all the work that we already did in the book constructor to remember those first three arguments - we want to take advantage of the fact that the code for that would be the same . so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages ! calvin.burn ( ) ; // omg , you burnt all 256 pages ! `` ` ( well , we 're not really going to burn it , because that 's an amazing book , but perhaps if we were stuck in a glacial desert and desperate for warmth and about to die . ) and now you can see how we can use object-oriented design principles in javascript to create more complex data for your programs and model your program worlds better .
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we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages .
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what is the function of object.create here ?
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this is a review of what we covered in this tutorial on object-oriented design . when we create programs , we often find that we want to create many different objects that all share similar properties - like many cats , that have slightly different fur color and size , or many buttons , with different labels and positions . we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages . it then sets the initial properties of the object based on those arguments , using the this keyword . when we use this in an object , we are referring to the current instance of an object , referring to itself . we need to store the properties on this to make sure we can remember them later . to create an instance of a book object , we declare a new variable to store it , then use the new keyword , followed by the constructor function name , and pass in the arguments that the constructor expects : var book = new book ( `` robot dreams '' , `` isaac asimov '' , 320 ) ; we can then access any properties that we stored in the object using dot notation : println ( `` i loved reading `` + book.title ) ; // i loved reading robot dreams println ( book.author + `` is my fav author '' ) ; // `` isaac asimov '' is my fav author let 's contrast this for a minute , and show what would have happened if we did n't set up our constructor function properly : var book = function ( title , author , numpages ) { } ; var book = new book ( `` little brother '' , `` cory doctorow '' , 380 ) ; println ( `` i loved reading `` + book.title ) ; // i loved reading undefined println ( book.author + `` is my fav author '' ) ; // undefined is my favorite author if we pass the arguments into the constructor function but do not explicitly store them on this , then we will not be able to access them later ! the object will have long forgotten about them . when we define object types , we often want to associate both properties and behavior with them - like all of our cat objects should be able to meow ( ) and eat ( ) . so we need to be able to attach functions to our object type definitions , and we can do that by defining them on what 's called the object prototype : book.prototype.readitall = function ( ) { this.currentpage = this.numpages ; println ( `` you read `` + this.numpages + `` pages ! `` ) ; } ; it 's like how we would define a function normally , except that we hang it off the book 's prototype instead of just defining it globally . that 's how javascript knows that this is a function that can be called on any book object , and that this function should have access to the this of the book that it 's called on . we can then call the function ( which we call a method , since it 's attached to an object ) , like so : var book = new book ( `` animal farm '' , `` george orwell '' , 112 ) ; book.readitall ( ) ; // you read 112 pages ! remember , the whole point of object-oriented design is that it makes it easy for us to make multiple related objects ( object instances ) . let 's see that in code : `` ` var pirate = new book ( `` pirate cinema '' , `` cory doctorow '' , 384 ) ; var giver = new book ( `` the giver '' , `` lois lowry '' , 179 ) ; var tuck = new book ( `` tuck everlasting '' , `` natalie babbit '' , 144 ) ; pirate.readitall ( ) ; // you read 384 pages ! giver.readitall ( ) ; // you read 179 pages ! tuck.readitall ( ) ; // you read 144 pages ! `` ` that code gives us three books that are similar - they all have the same types of properties and behavior , but also different . sweet ! now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death . they 're both mammals , and that means they share a lot in common , even if they 're also different . in that case , we want to use the idea of object inheritance . an object type could inherit properties and behavior from a parent object type , but then also have its own unique things about it . all the cats and dogs could inherit from mammal , so that they would n't have to invent eat ( ) ing from scratch . how would we do that in javascript ? let 's go back to our book example , and say that book is the `` parent '' object type , and we want to make two object types that inherit from it - paperback and ebook . a paperback is like a book , but it has one main thing different , at least for our program : it has a cover image . so , our constructor needs to take four arguments , to take in that extra info : var paperback = function ( title , author , numpages , cover ) { // ... } now , we do n't want to have to do all the work that we already did in the book constructor to remember those first three arguments - we want to take advantage of the fact that the code for that would be the same . so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages ! calvin.burn ( ) ; // omg , you burnt all 256 pages ! `` ` ( well , we 're not really going to burn it , because that 's an amazing book , but perhaps if we were stuck in a glacial desert and desperate for warmth and about to die . ) and now you can see how we can use object-oriented design principles in javascript to create more complex data for your programs and model your program worlds better .
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let 's see that in code : `` ` var pirate = new book ( `` pirate cinema '' , `` cory doctorow '' , 384 ) ; var giver = new book ( `` the giver '' , `` lois lowry '' , 179 ) ; var tuck = new book ( `` tuck everlasting '' , `` natalie babbit '' , 144 ) ; pirate.readitall ( ) ; // you read 384 pages ! giver.readitall ( ) ; // you read 179 pages ! tuck.readitall ( ) ; // you read 144 pages !
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is `` draw '' a universally inherited trait for all objects , or is the computer trying to read it a a draw loop ?
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this is a review of what we covered in this tutorial on object-oriented design . when we create programs , we often find that we want to create many different objects that all share similar properties - like many cats , that have slightly different fur color and size , or many buttons , with different labels and positions . we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages . it then sets the initial properties of the object based on those arguments , using the this keyword . when we use this in an object , we are referring to the current instance of an object , referring to itself . we need to store the properties on this to make sure we can remember them later . to create an instance of a book object , we declare a new variable to store it , then use the new keyword , followed by the constructor function name , and pass in the arguments that the constructor expects : var book = new book ( `` robot dreams '' , `` isaac asimov '' , 320 ) ; we can then access any properties that we stored in the object using dot notation : println ( `` i loved reading `` + book.title ) ; // i loved reading robot dreams println ( book.author + `` is my fav author '' ) ; // `` isaac asimov '' is my fav author let 's contrast this for a minute , and show what would have happened if we did n't set up our constructor function properly : var book = function ( title , author , numpages ) { } ; var book = new book ( `` little brother '' , `` cory doctorow '' , 380 ) ; println ( `` i loved reading `` + book.title ) ; // i loved reading undefined println ( book.author + `` is my fav author '' ) ; // undefined is my favorite author if we pass the arguments into the constructor function but do not explicitly store them on this , then we will not be able to access them later ! the object will have long forgotten about them . when we define object types , we often want to associate both properties and behavior with them - like all of our cat objects should be able to meow ( ) and eat ( ) . so we need to be able to attach functions to our object type definitions , and we can do that by defining them on what 's called the object prototype : book.prototype.readitall = function ( ) { this.currentpage = this.numpages ; println ( `` you read `` + this.numpages + `` pages ! `` ) ; } ; it 's like how we would define a function normally , except that we hang it off the book 's prototype instead of just defining it globally . that 's how javascript knows that this is a function that can be called on any book object , and that this function should have access to the this of the book that it 's called on . we can then call the function ( which we call a method , since it 's attached to an object ) , like so : var book = new book ( `` animal farm '' , `` george orwell '' , 112 ) ; book.readitall ( ) ; // you read 112 pages ! remember , the whole point of object-oriented design is that it makes it easy for us to make multiple related objects ( object instances ) . let 's see that in code : `` ` var pirate = new book ( `` pirate cinema '' , `` cory doctorow '' , 384 ) ; var giver = new book ( `` the giver '' , `` lois lowry '' , 179 ) ; var tuck = new book ( `` tuck everlasting '' , `` natalie babbit '' , 144 ) ; pirate.readitall ( ) ; // you read 384 pages ! giver.readitall ( ) ; // you read 179 pages ! tuck.readitall ( ) ; // you read 144 pages ! `` ` that code gives us three books that are similar - they all have the same types of properties and behavior , but also different . sweet ! now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death . they 're both mammals , and that means they share a lot in common , even if they 're also different . in that case , we want to use the idea of object inheritance . an object type could inherit properties and behavior from a parent object type , but then also have its own unique things about it . all the cats and dogs could inherit from mammal , so that they would n't have to invent eat ( ) ing from scratch . how would we do that in javascript ? let 's go back to our book example , and say that book is the `` parent '' object type , and we want to make two object types that inherit from it - paperback and ebook . a paperback is like a book , but it has one main thing different , at least for our program : it has a cover image . so , our constructor needs to take four arguments , to take in that extra info : var paperback = function ( title , author , numpages , cover ) { // ... } now , we do n't want to have to do all the work that we already did in the book constructor to remember those first three arguments - we want to take advantage of the fact that the code for that would be the same . so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages ! calvin.burn ( ) ; // omg , you burnt all 256 pages ! `` ` ( well , we 're not really going to burn it , because that 's an amazing book , but perhaps if we were stuck in a glacial desert and desperate for warmth and about to die . ) and now you can see how we can use object-oriented design principles in javascript to create more complex data for your programs and model your program worlds better .
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to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages .
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why do you make the value of this.numpages = 0 if you 're going to create a new number of pages for each object instance ?
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this is a review of what we covered in this tutorial on object-oriented design . when we create programs , we often find that we want to create many different objects that all share similar properties - like many cats , that have slightly different fur color and size , or many buttons , with different labels and positions . we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages . it then sets the initial properties of the object based on those arguments , using the this keyword . when we use this in an object , we are referring to the current instance of an object , referring to itself . we need to store the properties on this to make sure we can remember them later . to create an instance of a book object , we declare a new variable to store it , then use the new keyword , followed by the constructor function name , and pass in the arguments that the constructor expects : var book = new book ( `` robot dreams '' , `` isaac asimov '' , 320 ) ; we can then access any properties that we stored in the object using dot notation : println ( `` i loved reading `` + book.title ) ; // i loved reading robot dreams println ( book.author + `` is my fav author '' ) ; // `` isaac asimov '' is my fav author let 's contrast this for a minute , and show what would have happened if we did n't set up our constructor function properly : var book = function ( title , author , numpages ) { } ; var book = new book ( `` little brother '' , `` cory doctorow '' , 380 ) ; println ( `` i loved reading `` + book.title ) ; // i loved reading undefined println ( book.author + `` is my fav author '' ) ; // undefined is my favorite author if we pass the arguments into the constructor function but do not explicitly store them on this , then we will not be able to access them later ! the object will have long forgotten about them . when we define object types , we often want to associate both properties and behavior with them - like all of our cat objects should be able to meow ( ) and eat ( ) . so we need to be able to attach functions to our object type definitions , and we can do that by defining them on what 's called the object prototype : book.prototype.readitall = function ( ) { this.currentpage = this.numpages ; println ( `` you read `` + this.numpages + `` pages ! `` ) ; } ; it 's like how we would define a function normally , except that we hang it off the book 's prototype instead of just defining it globally . that 's how javascript knows that this is a function that can be called on any book object , and that this function should have access to the this of the book that it 's called on . we can then call the function ( which we call a method , since it 's attached to an object ) , like so : var book = new book ( `` animal farm '' , `` george orwell '' , 112 ) ; book.readitall ( ) ; // you read 112 pages ! remember , the whole point of object-oriented design is that it makes it easy for us to make multiple related objects ( object instances ) . let 's see that in code : `` ` var pirate = new book ( `` pirate cinema '' , `` cory doctorow '' , 384 ) ; var giver = new book ( `` the giver '' , `` lois lowry '' , 179 ) ; var tuck = new book ( `` tuck everlasting '' , `` natalie babbit '' , 144 ) ; pirate.readitall ( ) ; // you read 384 pages ! giver.readitall ( ) ; // you read 179 pages ! tuck.readitall ( ) ; // you read 144 pages ! `` ` that code gives us three books that are similar - they all have the same types of properties and behavior , but also different . sweet ! now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death . they 're both mammals , and that means they share a lot in common , even if they 're also different . in that case , we want to use the idea of object inheritance . an object type could inherit properties and behavior from a parent object type , but then also have its own unique things about it . all the cats and dogs could inherit from mammal , so that they would n't have to invent eat ( ) ing from scratch . how would we do that in javascript ? let 's go back to our book example , and say that book is the `` parent '' object type , and we want to make two object types that inherit from it - paperback and ebook . a paperback is like a book , but it has one main thing different , at least for our program : it has a cover image . so , our constructor needs to take four arguments , to take in that extra info : var paperback = function ( title , author , numpages , cover ) { // ... } now , we do n't want to have to do all the work that we already did in the book constructor to remember those first three arguments - we want to take advantage of the fact that the code for that would be the same . so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages ! calvin.burn ( ) ; // omg , you burnt all 256 pages ! `` ` ( well , we 're not really going to burn it , because that 's an amazing book , but perhaps if we were stuck in a glacial desert and desperate for warmth and about to die . ) and now you can see how we can use object-oriented design principles in javascript to create more complex data for your programs and model your program worlds better .
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so , our constructor needs to take four arguments , to take in that extra info : var paperback = function ( title , author , numpages , cover ) { // ... } now , we do n't want to have to do all the work that we already did in the book constructor to remember those first three arguments - we want to take advantage of the fact that the code for that would be the same . so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages !
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why ca n't you just use paperback.prototype = ( new book ( title , author , numpages ) ) .prototype ?
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this is a review of what we covered in this tutorial on object-oriented design . when we create programs , we often find that we want to create many different objects that all share similar properties - like many cats , that have slightly different fur color and size , or many buttons , with different labels and positions . we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages . it then sets the initial properties of the object based on those arguments , using the this keyword . when we use this in an object , we are referring to the current instance of an object , referring to itself . we need to store the properties on this to make sure we can remember them later . to create an instance of a book object , we declare a new variable to store it , then use the new keyword , followed by the constructor function name , and pass in the arguments that the constructor expects : var book = new book ( `` robot dreams '' , `` isaac asimov '' , 320 ) ; we can then access any properties that we stored in the object using dot notation : println ( `` i loved reading `` + book.title ) ; // i loved reading robot dreams println ( book.author + `` is my fav author '' ) ; // `` isaac asimov '' is my fav author let 's contrast this for a minute , and show what would have happened if we did n't set up our constructor function properly : var book = function ( title , author , numpages ) { } ; var book = new book ( `` little brother '' , `` cory doctorow '' , 380 ) ; println ( `` i loved reading `` + book.title ) ; // i loved reading undefined println ( book.author + `` is my fav author '' ) ; // undefined is my favorite author if we pass the arguments into the constructor function but do not explicitly store them on this , then we will not be able to access them later ! the object will have long forgotten about them . when we define object types , we often want to associate both properties and behavior with them - like all of our cat objects should be able to meow ( ) and eat ( ) . so we need to be able to attach functions to our object type definitions , and we can do that by defining them on what 's called the object prototype : book.prototype.readitall = function ( ) { this.currentpage = this.numpages ; println ( `` you read `` + this.numpages + `` pages ! `` ) ; } ; it 's like how we would define a function normally , except that we hang it off the book 's prototype instead of just defining it globally . that 's how javascript knows that this is a function that can be called on any book object , and that this function should have access to the this of the book that it 's called on . we can then call the function ( which we call a method , since it 's attached to an object ) , like so : var book = new book ( `` animal farm '' , `` george orwell '' , 112 ) ; book.readitall ( ) ; // you read 112 pages ! remember , the whole point of object-oriented design is that it makes it easy for us to make multiple related objects ( object instances ) . let 's see that in code : `` ` var pirate = new book ( `` pirate cinema '' , `` cory doctorow '' , 384 ) ; var giver = new book ( `` the giver '' , `` lois lowry '' , 179 ) ; var tuck = new book ( `` tuck everlasting '' , `` natalie babbit '' , 144 ) ; pirate.readitall ( ) ; // you read 384 pages ! giver.readitall ( ) ; // you read 179 pages ! tuck.readitall ( ) ; // you read 144 pages ! `` ` that code gives us three books that are similar - they all have the same types of properties and behavior , but also different . sweet ! now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death . they 're both mammals , and that means they share a lot in common , even if they 're also different . in that case , we want to use the idea of object inheritance . an object type could inherit properties and behavior from a parent object type , but then also have its own unique things about it . all the cats and dogs could inherit from mammal , so that they would n't have to invent eat ( ) ing from scratch . how would we do that in javascript ? let 's go back to our book example , and say that book is the `` parent '' object type , and we want to make two object types that inherit from it - paperback and ebook . a paperback is like a book , but it has one main thing different , at least for our program : it has a cover image . so , our constructor needs to take four arguments , to take in that extra info : var paperback = function ( title , author , numpages , cover ) { // ... } now , we do n't want to have to do all the work that we already did in the book constructor to remember those first three arguments - we want to take advantage of the fact that the code for that would be the same . so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages ! calvin.burn ( ) ; // omg , you burnt all 256 pages ! `` ` ( well , we 're not really going to burn it , because that 's an amazing book , but perhaps if we were stuck in a glacial desert and desperate for warmth and about to die . ) and now you can see how we can use object-oriented design principles in javascript to create more complex data for your programs and model your program worlds better .
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a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death . they 're both mammals , and that means they share a lot in common , even if they 're also different .
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example : var animal = function ( birth , death ) { //atributes this.birth = birth ; this.death = death ; //methods this.calculateage = function ( ) { return this.death - this.death ; } ; } ; and then just use animal.call ( this , birth , death ) ; instead of having to use both that and object.create ( ) ?
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this is a review of what we covered in this tutorial on object-oriented design . when we create programs , we often find that we want to create many different objects that all share similar properties - like many cats , that have slightly different fur color and size , or many buttons , with different labels and positions . we want to be able to say `` this is generally what a cat is like '' and then say `` let 's make this specific cat , and this other cat , and they 'll be similar in some ways and different in a few ways as well . '' in that case , we want to use object-oriented design to define object types and create new instances of those objects . to define an object type in javascript , we first have to define a `` constructor function '' . this is the function that we 'll use whenever we want to create a new instance of that object type . here 's a constructor function for a book object type : var book = function ( title , author , numpages ) { this.title = title ; this.author = author ; this.numpages = numpages ; this.currentpage = 0 ; } ; the function takes in arguments for the aspects that will be different about each book - the title , author , and number of pages . it then sets the initial properties of the object based on those arguments , using the this keyword . when we use this in an object , we are referring to the current instance of an object , referring to itself . we need to store the properties on this to make sure we can remember them later . to create an instance of a book object , we declare a new variable to store it , then use the new keyword , followed by the constructor function name , and pass in the arguments that the constructor expects : var book = new book ( `` robot dreams '' , `` isaac asimov '' , 320 ) ; we can then access any properties that we stored in the object using dot notation : println ( `` i loved reading `` + book.title ) ; // i loved reading robot dreams println ( book.author + `` is my fav author '' ) ; // `` isaac asimov '' is my fav author let 's contrast this for a minute , and show what would have happened if we did n't set up our constructor function properly : var book = function ( title , author , numpages ) { } ; var book = new book ( `` little brother '' , `` cory doctorow '' , 380 ) ; println ( `` i loved reading `` + book.title ) ; // i loved reading undefined println ( book.author + `` is my fav author '' ) ; // undefined is my favorite author if we pass the arguments into the constructor function but do not explicitly store them on this , then we will not be able to access them later ! the object will have long forgotten about them . when we define object types , we often want to associate both properties and behavior with them - like all of our cat objects should be able to meow ( ) and eat ( ) . so we need to be able to attach functions to our object type definitions , and we can do that by defining them on what 's called the object prototype : book.prototype.readitall = function ( ) { this.currentpage = this.numpages ; println ( `` you read `` + this.numpages + `` pages ! `` ) ; } ; it 's like how we would define a function normally , except that we hang it off the book 's prototype instead of just defining it globally . that 's how javascript knows that this is a function that can be called on any book object , and that this function should have access to the this of the book that it 's called on . we can then call the function ( which we call a method , since it 's attached to an object ) , like so : var book = new book ( `` animal farm '' , `` george orwell '' , 112 ) ; book.readitall ( ) ; // you read 112 pages ! remember , the whole point of object-oriented design is that it makes it easy for us to make multiple related objects ( object instances ) . let 's see that in code : `` ` var pirate = new book ( `` pirate cinema '' , `` cory doctorow '' , 384 ) ; var giver = new book ( `` the giver '' , `` lois lowry '' , 179 ) ; var tuck = new book ( `` tuck everlasting '' , `` natalie babbit '' , 144 ) ; pirate.readitall ( ) ; // you read 384 pages ! giver.readitall ( ) ; // you read 179 pages ! tuck.readitall ( ) ; // you read 144 pages ! `` ` that code gives us three books that are similar - they all have the same types of properties and behavior , but also different . sweet ! now , if you think about the world , cats and dogs are different types of objects , so you 'd probably create different object types for them if you were programming a cat and a dog . a cat would meow ( ) , a dog would bark ( ) . but they 're also similar- both a cat and dog would eat ( ) , they both have an age , and a birth , and a death . they 're both mammals , and that means they share a lot in common , even if they 're also different . in that case , we want to use the idea of object inheritance . an object type could inherit properties and behavior from a parent object type , but then also have its own unique things about it . all the cats and dogs could inherit from mammal , so that they would n't have to invent eat ( ) ing from scratch . how would we do that in javascript ? let 's go back to our book example , and say that book is the `` parent '' object type , and we want to make two object types that inherit from it - paperback and ebook . a paperback is like a book , but it has one main thing different , at least for our program : it has a cover image . so , our constructor needs to take four arguments , to take in that extra info : var paperback = function ( title , author , numpages , cover ) { // ... } now , we do n't want to have to do all the work that we already did in the book constructor to remember those first three arguments - we want to take advantage of the fact that the code for that would be the same . so we can actually call the book constructor from the paperback constructor , and pass in those arguments : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; // ... } ; we still need to store the cover property in the object though , so we need one more line to take care of that : var paperback = function ( title , author , numpages , cover ) { book.call ( this , title , author , numpages ) ; this.cover = cover ; } ; now , we have a constructor for our paperback , which helps it share the same properties as books , but we also want our paperback to inherit its methods . here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages ! calvin.burn ( ) ; // omg , you burnt all 256 pages ! `` ` ( well , we 're not really going to burn it , because that 's an amazing book , but perhaps if we were stuck in a glacial desert and desperate for warmth and about to die . ) and now you can see how we can use object-oriented design principles in javascript to create more complex data for your programs and model your program worlds better .
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here 's how we do that , by telling the program that the paperback prototype should be based on the book prototype : paperback.prototype = object.create ( book.prototype ) ; we might also want to attach paperback-specific behavior , like being able to burn it , and we can do that by defining functions on the prototype , after that line above : paperback.prototype.burn = function ( ) { println ( `` omg , you burnt all `` + this.numpages + `` pages '' ) ; this.numpages = 0 ; } ; and now we can create a new paperback , read it all , and burn it ! `` ` var calvin = new paperback ( `` the essential calvin & amp ; hobbes '' , `` bill watterson '' , 256 , `` http : //ecx.images-amazon.com/images/i/61m41hxr0zl.jpg '' ) ; calvin.readitall ( ) ; // you read 256 pages ! calvin.burn ( ) ; // omg , you burnt all 256 pages ! `` ` ( well , we 're not really going to burn it , because that 's an amazing book , but perhaps if we were stuck in a glacial desert and desperate for warmth and about to die . )
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why would anyone want to burn a calvin and hobbes book ?
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overview the dawes act of 1887 authorized the federal government to break up tribal lands by partitioning them into individual plots . only those native american indians who accepted the individual allotments were allowed to become us citizens . the objective of the dawes act was to assimilate native american indians into mainstream us society by annihilating their cultural and social traditions . over ninety million acres of tribal land were stripped from native american indians and sold to non-natives . background to the “ indian problem ” although violent conflict had plagued relations between white settlers and native american indians from the very beginning of european colonization of the new world , such violence increased in the mid-nineteenth century as european settlers moved ever further west across the american continent . most white americans believed there was no way to live in peace and harmony with native americans , whom they regarded as “ backwards ” and “ primitive. ” as a result of this widespread belief , the federal government created the reservation system in 1851 to provide land to native americans and thereby keep them off of the lands that european-americans wished to settle . many tribes resisted their confinement to the reservations , resulting in a series of conflicts between various indian tribes and the us army known as the indian wars . ultimately , the army subdued the indians and forced them onto reservations , where they were allowed to govern themselves and maintain some of their traditions and culture. $ ^1 $ but as white americans pushed ever westward , they came into conflict with native americans on their tribal lands . many of these white settlers viewed the continued practice of native traditions as barbaric and intolerable . they believed that assimilation ( being completely absorbed ) into mainstream white american society was the only acceptable fate for native americans . this belief was often couched in religious terms ; many white christians argued that only by abandoning their spiritual traditions and accepting christian dogma could the indians be “ saved ” from the fires of hell . the forced assimilation of native americans was thus justified as being better for the indians themselves . in the late nineteenth century , a political consensus formed around these ideas , and the result was the 1887 passage of the dawes act . provisions and effects of the dawes act the dawes act of 1887 , sometimes referred to as the dawes severalty act of 1887 or the general allotment act , was signed into law on january 8 , 1887 , by us president grover cleveland . the act authorized the president to confiscate and redistribute tribal lands in the american west . it explicitly sought to destroy the social cohesion of indian tribes and to thereby eliminate the remaining vestiges of indian culture and society . only by disavowing their own traditions , it was believed , could the indians ever become truly “ american. ” $ ^2 $ as a result of the dawes act , tribal lands were parceled out into individual plots . only those native americans who accepted the individual plots of land were allowed to become us citizens . the remainder of the land was then sold off to white settlers . amendments to the dawes act initially , the dawes act did not apply to the so-called “ five civilized tribes ” ( cherokee , chickasaw , choctaw , creek , and seminole ) . these tribes had already adopted many elements of white european society and culture , which is why they were characterized as “ civilized. ” moreover , they were protected by treaties that had guaranteed that their tribal lands would remain free of white settlers . however , after these tribes had proven unwilling to voluntarily accept individual allotments of land , the curtis act of 1898 amended the dawes act to apply to the five civilized tribes . their tribal governments were obliterated , their tribal courts were destroyed , and over ninety million acres of their tribal lands were sold off to white americans. $ ^3 $ during the great depression , the administration of president franklin d. roosevelt supported the us indian reorganization act , which authorized a “ new deal ” for native american indians , allowing them to organize and form their own tribal governments , and ending the land allotments created by dawes act. $ ^4 $ what do you think ? why do you think white americans viewed native american indians as such a threat ? do you think the dawes act was intended to help or harm native americans ? what was the effect of the dawes act on native american cultural beliefs and traditions ? what do you see as the primary difference between native american and european american conceptions of land and ownership ?
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in the late nineteenth century , a political consensus formed around these ideas , and the result was the 1887 passage of the dawes act . provisions and effects of the dawes act the dawes act of 1887 , sometimes referred to as the dawes severalty act of 1887 or the general allotment act , was signed into law on january 8 , 1887 , by us president grover cleveland . the act authorized the president to confiscate and redistribute tribal lands in the american west .
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can someone explain the difference between the dawes act and the dawes plan ?
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overview the dawes act of 1887 authorized the federal government to break up tribal lands by partitioning them into individual plots . only those native american indians who accepted the individual allotments were allowed to become us citizens . the objective of the dawes act was to assimilate native american indians into mainstream us society by annihilating their cultural and social traditions . over ninety million acres of tribal land were stripped from native american indians and sold to non-natives . background to the “ indian problem ” although violent conflict had plagued relations between white settlers and native american indians from the very beginning of european colonization of the new world , such violence increased in the mid-nineteenth century as european settlers moved ever further west across the american continent . most white americans believed there was no way to live in peace and harmony with native americans , whom they regarded as “ backwards ” and “ primitive. ” as a result of this widespread belief , the federal government created the reservation system in 1851 to provide land to native americans and thereby keep them off of the lands that european-americans wished to settle . many tribes resisted their confinement to the reservations , resulting in a series of conflicts between various indian tribes and the us army known as the indian wars . ultimately , the army subdued the indians and forced them onto reservations , where they were allowed to govern themselves and maintain some of their traditions and culture. $ ^1 $ but as white americans pushed ever westward , they came into conflict with native americans on their tribal lands . many of these white settlers viewed the continued practice of native traditions as barbaric and intolerable . they believed that assimilation ( being completely absorbed ) into mainstream white american society was the only acceptable fate for native americans . this belief was often couched in religious terms ; many white christians argued that only by abandoning their spiritual traditions and accepting christian dogma could the indians be “ saved ” from the fires of hell . the forced assimilation of native americans was thus justified as being better for the indians themselves . in the late nineteenth century , a political consensus formed around these ideas , and the result was the 1887 passage of the dawes act . provisions and effects of the dawes act the dawes act of 1887 , sometimes referred to as the dawes severalty act of 1887 or the general allotment act , was signed into law on january 8 , 1887 , by us president grover cleveland . the act authorized the president to confiscate and redistribute tribal lands in the american west . it explicitly sought to destroy the social cohesion of indian tribes and to thereby eliminate the remaining vestiges of indian culture and society . only by disavowing their own traditions , it was believed , could the indians ever become truly “ american. ” $ ^2 $ as a result of the dawes act , tribal lands were parceled out into individual plots . only those native americans who accepted the individual plots of land were allowed to become us citizens . the remainder of the land was then sold off to white settlers . amendments to the dawes act initially , the dawes act did not apply to the so-called “ five civilized tribes ” ( cherokee , chickasaw , choctaw , creek , and seminole ) . these tribes had already adopted many elements of white european society and culture , which is why they were characterized as “ civilized. ” moreover , they were protected by treaties that had guaranteed that their tribal lands would remain free of white settlers . however , after these tribes had proven unwilling to voluntarily accept individual allotments of land , the curtis act of 1898 amended the dawes act to apply to the five civilized tribes . their tribal governments were obliterated , their tribal courts were destroyed , and over ninety million acres of their tribal lands were sold off to white americans. $ ^3 $ during the great depression , the administration of president franklin d. roosevelt supported the us indian reorganization act , which authorized a “ new deal ” for native american indians , allowing them to organize and form their own tribal governments , and ending the land allotments created by dawes act. $ ^4 $ what do you think ? why do you think white americans viewed native american indians as such a threat ? do you think the dawes act was intended to help or harm native americans ? what was the effect of the dawes act on native american cultural beliefs and traditions ? what do you see as the primary difference between native american and european american conceptions of land and ownership ?
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their tribal governments were obliterated , their tribal courts were destroyed , and over ninety million acres of their tribal lands were sold off to white americans. $ ^3 $ during the great depression , the administration of president franklin d. roosevelt supported the us indian reorganization act , which authorized a “ new deal ” for native american indians , allowing them to organize and form their own tribal governments , and ending the land allotments created by dawes act. $ ^4 $ what do you think ? why do you think white americans viewed native american indians as such a threat ? do you think the dawes act was intended to help or harm native americans ? what was the effect of the dawes act on native american cultural beliefs and traditions ? what do you see as the primary difference between native american and european american conceptions of land and ownership ?
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assimilation `` was the daws act successfully in achieving its goal relation to native american ?
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overview geographic living patterns in the united states changed during the postwar era as more americans moved to western and southern states . suburban living promoted the use of automobiles for transportation , which led to a vast expansion of america 's highway system . suburbs ' emphasis on conformity had negative effects on both white women and minorities . many white women began to feel trapped in the role of housewife , while restrictive covenants barred most african american and asian american families from living in suburban neighborhoods at all . suburbia , usa levitt and sons went on to build two more highly-successful suburbs in pennsylvania and new jersey ( both of which they also named levittown ) , and other developers quickly adopted their formula for suburban housing . between 1948 and 1958 , 85 % of the new homes built in the united states were located in suburbs . suburban construction across the country also meant that regional differences of architecture and urban planning began to erode in favor of identical housing across the united states . this suburban trend has endured : today , four out of five americans live in suburbs. $ ^1 $ living in suburbia meant that residents had to own cars in order to go to work or purchase groceries . by 1955 american automobile companies were producing eight million cars per year , more than three times as many as in 1945 . likewise , the system of roads had to expand in order to meet the demand of an increasingly car-oriented society : states and the federal government invested heavily in an interstate highway system in the late 1940s and 1950s . suburbia helped to promote a `` car culture '' in the united states that made it easier to drive than to take public transportation . the war and its aftermath also changed american living patterns on a large scale . defense plants in the southern and western united states drew workers during the war , and in the following decades more americans moved to the warmer states of the sunbelt in search of jobs . the population of california doubled between 1940 and 1960 . florida 's population nearly tripled in the same period . in general , people , jobs , and money began to move away from the industrial states of the northeast and the upper midwest and into the south and west. $ ^2 $ race , gender and suburbia with its cookie-cutter houses and firm emphasis on material comforts , from shiny new cars to washing machines , suburbia received its share of criticism . what most appalled critics was suburbia 's emphasis on sameness and conformity . on one hand , this `` sameness '' heralded a kind of democratic progress : suburban families made about the same amount of money , lived in identical or nearly identical houses , and generally were at about the same stage in life . class divisions narrowed as barriers to homeownership fell and the postwar economic boom elevated many families into the middle class . even longstanding prejudices based on religion and ethnicity eroded in the suburb , as the shared experiences of gis during the war trumped differences between italian-americans and german-americans , or catholics and jews. $ ^3 $ but this conformity also had a dark side . for white women , the charms of suburban life began to wear thin after a few years . although it should not be forgotten that more than 30 % of women did work outside the home in some capacity during the 1950s , popular culture was replete with messages counseling women that their greatest satisfaction in life would come from raising children , tending to their husbands ' needs , and owning all of the labor-saving household appliances that money could buy . but many began to identify a creeping sense that there ought to be more to life than childcare and housework. $ ^4 $ minority women did not experience the ennui of suburban life because , by and large , they were barred from suburbia altogether . william levitt was an unapologetic segregationist , declaring openly that his subdivisions were for whites only . in 1960 , not a single resident of levittown , new york was black . suburbs throughout the nation enacted restrictive covenants that prevented homeowners from selling their houses to african americans or asian americans , upon the pretense that their presence would lower property values . although the supreme court ruled in 1948 that such covenants were unenforceable , de facto segregation continued and was frequently enforced by violence and intimidation. $ ^5 $ banks also refused to loan money for new homes or improvements in the inner city neighborhoods where minorities lived in a practice known as redlining ( a term derived from mortgage security maps that shaded minority neighborhoods in red , signifying they were 'risky ' investments ) . $ ^6 $ thus , government subsidies for suburban home building and prejudice against lending to minorities combined to increase the distance -- both physically and economically -- between whites and african americans . what do you think ? what are the effects of american `` car culture '' ? consider its impact on americans ' ability to get to work and to the services they need , as well as its impact on the environment and the oil industry . do you think the `` sameness '' of the suburbs was an improvement on the `` ethnic enclaves '' found in the prewar period ( little italy in new york , for example ) , or was the emphasis on conformity stifling ? what was the overall impact of housing policies on african americans during this period ? do you think housing discrimination was a major factor in the emergence of the civil rights movement ?
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but many began to identify a creeping sense that there ought to be more to life than childcare and housework. $ ^4 $ minority women did not experience the ennui of suburban life because , by and large , they were barred from suburbia altogether . william levitt was an unapologetic segregationist , declaring openly that his subdivisions were for whites only . in 1960 , not a single resident of levittown , new york was black .
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was it true that william levitt was an `` unapologetic segregationist '' ?
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overview geographic living patterns in the united states changed during the postwar era as more americans moved to western and southern states . suburban living promoted the use of automobiles for transportation , which led to a vast expansion of america 's highway system . suburbs ' emphasis on conformity had negative effects on both white women and minorities . many white women began to feel trapped in the role of housewife , while restrictive covenants barred most african american and asian american families from living in suburban neighborhoods at all . suburbia , usa levitt and sons went on to build two more highly-successful suburbs in pennsylvania and new jersey ( both of which they also named levittown ) , and other developers quickly adopted their formula for suburban housing . between 1948 and 1958 , 85 % of the new homes built in the united states were located in suburbs . suburban construction across the country also meant that regional differences of architecture and urban planning began to erode in favor of identical housing across the united states . this suburban trend has endured : today , four out of five americans live in suburbs. $ ^1 $ living in suburbia meant that residents had to own cars in order to go to work or purchase groceries . by 1955 american automobile companies were producing eight million cars per year , more than three times as many as in 1945 . likewise , the system of roads had to expand in order to meet the demand of an increasingly car-oriented society : states and the federal government invested heavily in an interstate highway system in the late 1940s and 1950s . suburbia helped to promote a `` car culture '' in the united states that made it easier to drive than to take public transportation . the war and its aftermath also changed american living patterns on a large scale . defense plants in the southern and western united states drew workers during the war , and in the following decades more americans moved to the warmer states of the sunbelt in search of jobs . the population of california doubled between 1940 and 1960 . florida 's population nearly tripled in the same period . in general , people , jobs , and money began to move away from the industrial states of the northeast and the upper midwest and into the south and west. $ ^2 $ race , gender and suburbia with its cookie-cutter houses and firm emphasis on material comforts , from shiny new cars to washing machines , suburbia received its share of criticism . what most appalled critics was suburbia 's emphasis on sameness and conformity . on one hand , this `` sameness '' heralded a kind of democratic progress : suburban families made about the same amount of money , lived in identical or nearly identical houses , and generally were at about the same stage in life . class divisions narrowed as barriers to homeownership fell and the postwar economic boom elevated many families into the middle class . even longstanding prejudices based on religion and ethnicity eroded in the suburb , as the shared experiences of gis during the war trumped differences between italian-americans and german-americans , or catholics and jews. $ ^3 $ but this conformity also had a dark side . for white women , the charms of suburban life began to wear thin after a few years . although it should not be forgotten that more than 30 % of women did work outside the home in some capacity during the 1950s , popular culture was replete with messages counseling women that their greatest satisfaction in life would come from raising children , tending to their husbands ' needs , and owning all of the labor-saving household appliances that money could buy . but many began to identify a creeping sense that there ought to be more to life than childcare and housework. $ ^4 $ minority women did not experience the ennui of suburban life because , by and large , they were barred from suburbia altogether . william levitt was an unapologetic segregationist , declaring openly that his subdivisions were for whites only . in 1960 , not a single resident of levittown , new york was black . suburbs throughout the nation enacted restrictive covenants that prevented homeowners from selling their houses to african americans or asian americans , upon the pretense that their presence would lower property values . although the supreme court ruled in 1948 that such covenants were unenforceable , de facto segregation continued and was frequently enforced by violence and intimidation. $ ^5 $ banks also refused to loan money for new homes or improvements in the inner city neighborhoods where minorities lived in a practice known as redlining ( a term derived from mortgage security maps that shaded minority neighborhoods in red , signifying they were 'risky ' investments ) . $ ^6 $ thus , government subsidies for suburban home building and prejudice against lending to minorities combined to increase the distance -- both physically and economically -- between whites and african americans . what do you think ? what are the effects of american `` car culture '' ? consider its impact on americans ' ability to get to work and to the services they need , as well as its impact on the environment and the oil industry . do you think the `` sameness '' of the suburbs was an improvement on the `` ethnic enclaves '' found in the prewar period ( little italy in new york , for example ) , or was the emphasis on conformity stifling ? what was the overall impact of housing policies on african americans during this period ? do you think housing discrimination was a major factor in the emergence of the civil rights movement ?
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what do you think ? what are the effects of american `` car culture '' ? consider its impact on americans ' ability to get to work and to the services they need , as well as its impact on the environment and the oil industry .
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what are the effects of american `` car culture '' ?
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overview geographic living patterns in the united states changed during the postwar era as more americans moved to western and southern states . suburban living promoted the use of automobiles for transportation , which led to a vast expansion of america 's highway system . suburbs ' emphasis on conformity had negative effects on both white women and minorities . many white women began to feel trapped in the role of housewife , while restrictive covenants barred most african american and asian american families from living in suburban neighborhoods at all . suburbia , usa levitt and sons went on to build two more highly-successful suburbs in pennsylvania and new jersey ( both of which they also named levittown ) , and other developers quickly adopted their formula for suburban housing . between 1948 and 1958 , 85 % of the new homes built in the united states were located in suburbs . suburban construction across the country also meant that regional differences of architecture and urban planning began to erode in favor of identical housing across the united states . this suburban trend has endured : today , four out of five americans live in suburbs. $ ^1 $ living in suburbia meant that residents had to own cars in order to go to work or purchase groceries . by 1955 american automobile companies were producing eight million cars per year , more than three times as many as in 1945 . likewise , the system of roads had to expand in order to meet the demand of an increasingly car-oriented society : states and the federal government invested heavily in an interstate highway system in the late 1940s and 1950s . suburbia helped to promote a `` car culture '' in the united states that made it easier to drive than to take public transportation . the war and its aftermath also changed american living patterns on a large scale . defense plants in the southern and western united states drew workers during the war , and in the following decades more americans moved to the warmer states of the sunbelt in search of jobs . the population of california doubled between 1940 and 1960 . florida 's population nearly tripled in the same period . in general , people , jobs , and money began to move away from the industrial states of the northeast and the upper midwest and into the south and west. $ ^2 $ race , gender and suburbia with its cookie-cutter houses and firm emphasis on material comforts , from shiny new cars to washing machines , suburbia received its share of criticism . what most appalled critics was suburbia 's emphasis on sameness and conformity . on one hand , this `` sameness '' heralded a kind of democratic progress : suburban families made about the same amount of money , lived in identical or nearly identical houses , and generally were at about the same stage in life . class divisions narrowed as barriers to homeownership fell and the postwar economic boom elevated many families into the middle class . even longstanding prejudices based on religion and ethnicity eroded in the suburb , as the shared experiences of gis during the war trumped differences between italian-americans and german-americans , or catholics and jews. $ ^3 $ but this conformity also had a dark side . for white women , the charms of suburban life began to wear thin after a few years . although it should not be forgotten that more than 30 % of women did work outside the home in some capacity during the 1950s , popular culture was replete with messages counseling women that their greatest satisfaction in life would come from raising children , tending to their husbands ' needs , and owning all of the labor-saving household appliances that money could buy . but many began to identify a creeping sense that there ought to be more to life than childcare and housework. $ ^4 $ minority women did not experience the ennui of suburban life because , by and large , they were barred from suburbia altogether . william levitt was an unapologetic segregationist , declaring openly that his subdivisions were for whites only . in 1960 , not a single resident of levittown , new york was black . suburbs throughout the nation enacted restrictive covenants that prevented homeowners from selling their houses to african americans or asian americans , upon the pretense that their presence would lower property values . although the supreme court ruled in 1948 that such covenants were unenforceable , de facto segregation continued and was frequently enforced by violence and intimidation. $ ^5 $ banks also refused to loan money for new homes or improvements in the inner city neighborhoods where minorities lived in a practice known as redlining ( a term derived from mortgage security maps that shaded minority neighborhoods in red , signifying they were 'risky ' investments ) . $ ^6 $ thus , government subsidies for suburban home building and prejudice against lending to minorities combined to increase the distance -- both physically and economically -- between whites and african americans . what do you think ? what are the effects of american `` car culture '' ? consider its impact on americans ' ability to get to work and to the services they need , as well as its impact on the environment and the oil industry . do you think the `` sameness '' of the suburbs was an improvement on the `` ethnic enclaves '' found in the prewar period ( little italy in new york , for example ) , or was the emphasis on conformity stifling ? what was the overall impact of housing policies on african americans during this period ? do you think housing discrimination was a major factor in the emergence of the civil rights movement ?
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the war and its aftermath also changed american living patterns on a large scale . defense plants in the southern and western united states drew workers during the war , and in the following decades more americans moved to the warmer states of the sunbelt in search of jobs . the population of california doubled between 1940 and 1960 .
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did the president soon allow more freedom of jobs ?
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overview geographic living patterns in the united states changed during the postwar era as more americans moved to western and southern states . suburban living promoted the use of automobiles for transportation , which led to a vast expansion of america 's highway system . suburbs ' emphasis on conformity had negative effects on both white women and minorities . many white women began to feel trapped in the role of housewife , while restrictive covenants barred most african american and asian american families from living in suburban neighborhoods at all . suburbia , usa levitt and sons went on to build two more highly-successful suburbs in pennsylvania and new jersey ( both of which they also named levittown ) , and other developers quickly adopted their formula for suburban housing . between 1948 and 1958 , 85 % of the new homes built in the united states were located in suburbs . suburban construction across the country also meant that regional differences of architecture and urban planning began to erode in favor of identical housing across the united states . this suburban trend has endured : today , four out of five americans live in suburbs. $ ^1 $ living in suburbia meant that residents had to own cars in order to go to work or purchase groceries . by 1955 american automobile companies were producing eight million cars per year , more than three times as many as in 1945 . likewise , the system of roads had to expand in order to meet the demand of an increasingly car-oriented society : states and the federal government invested heavily in an interstate highway system in the late 1940s and 1950s . suburbia helped to promote a `` car culture '' in the united states that made it easier to drive than to take public transportation . the war and its aftermath also changed american living patterns on a large scale . defense plants in the southern and western united states drew workers during the war , and in the following decades more americans moved to the warmer states of the sunbelt in search of jobs . the population of california doubled between 1940 and 1960 . florida 's population nearly tripled in the same period . in general , people , jobs , and money began to move away from the industrial states of the northeast and the upper midwest and into the south and west. $ ^2 $ race , gender and suburbia with its cookie-cutter houses and firm emphasis on material comforts , from shiny new cars to washing machines , suburbia received its share of criticism . what most appalled critics was suburbia 's emphasis on sameness and conformity . on one hand , this `` sameness '' heralded a kind of democratic progress : suburban families made about the same amount of money , lived in identical or nearly identical houses , and generally were at about the same stage in life . class divisions narrowed as barriers to homeownership fell and the postwar economic boom elevated many families into the middle class . even longstanding prejudices based on religion and ethnicity eroded in the suburb , as the shared experiences of gis during the war trumped differences between italian-americans and german-americans , or catholics and jews. $ ^3 $ but this conformity also had a dark side . for white women , the charms of suburban life began to wear thin after a few years . although it should not be forgotten that more than 30 % of women did work outside the home in some capacity during the 1950s , popular culture was replete with messages counseling women that their greatest satisfaction in life would come from raising children , tending to their husbands ' needs , and owning all of the labor-saving household appliances that money could buy . but many began to identify a creeping sense that there ought to be more to life than childcare and housework. $ ^4 $ minority women did not experience the ennui of suburban life because , by and large , they were barred from suburbia altogether . william levitt was an unapologetic segregationist , declaring openly that his subdivisions were for whites only . in 1960 , not a single resident of levittown , new york was black . suburbs throughout the nation enacted restrictive covenants that prevented homeowners from selling their houses to african americans or asian americans , upon the pretense that their presence would lower property values . although the supreme court ruled in 1948 that such covenants were unenforceable , de facto segregation continued and was frequently enforced by violence and intimidation. $ ^5 $ banks also refused to loan money for new homes or improvements in the inner city neighborhoods where minorities lived in a practice known as redlining ( a term derived from mortgage security maps that shaded minority neighborhoods in red , signifying they were 'risky ' investments ) . $ ^6 $ thus , government subsidies for suburban home building and prejudice against lending to minorities combined to increase the distance -- both physically and economically -- between whites and african americans . what do you think ? what are the effects of american `` car culture '' ? consider its impact on americans ' ability to get to work and to the services they need , as well as its impact on the environment and the oil industry . do you think the `` sameness '' of the suburbs was an improvement on the `` ethnic enclaves '' found in the prewar period ( little italy in new york , for example ) , or was the emphasis on conformity stifling ? what was the overall impact of housing policies on african americans during this period ? do you think housing discrimination was a major factor in the emergence of the civil rights movement ?
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suburban living promoted the use of automobiles for transportation , which led to a vast expansion of america 's highway system . suburbs ' emphasis on conformity had negative effects on both white women and minorities . many white women began to feel trapped in the role of housewife , while restrictive covenants barred most african american and asian american families from living in suburban neighborhoods at all .
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is levvittoun still 100 % white ?
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iznik ceramics give us a wonderful opportunity to glimpse into both the thriving iznik ceramic industry and a 16th century ottoman home . the ewer shown on the left is a type of jug that is shaped like a vase . it was a common utensil used daily for carrying water from the kitchen to the dining area , and for serving family and guests . this particular ewer was made in iznik , the ottoman center of ceramic production . iznik ware the town of iznik was an important production center during the fifteenth , sixteenth and part of the seventeenth centuries that manufactured ceramics for both the court in the capital , istanbul , and for the open market . the ewer was only one form of ceramic kitchenware that was produced in iznik workshops . other products included dishes , bowls , tankards , and bottles . due to the absence of petuntse ( a variety of feldspar rock ) , used in chinese porcelain , true porcelain did not come to the middle east until the modern era . nevertheless , twelfth-century islamic potters in persia were able to produce a strong white clay that in many respects resembled chinese porcelain . this mixture of potter ’ s clay , ground quartz , and glassy frit , is called fritware . this porcelain substitute , together with the under-glaze painting technique used to decorate the ceramics , was used for centuries . it is not only seen in surviving kitchenware , but also in beautiful tiles covering the interior and exterior of important ottoman buildings . color , iconography , shape iznik ceramic production initially used blue-and-white decoration . however , by the second half of the sixteenth century , iznik pottery saw the gradual addition of new colors as pigments were developed . the brooklyn ewer is painted in black , cobalt blue , green , and red under a transparent glaze . the ewer ’ s round body shape with the narrow neck and the round handle is similar to the shape of metal jugs from the islamic world of the fifteenth and sixteenth centuries . it was also influenced by blue-and-white ceramic pot-bellied jugs produced by chinese potters of the ming period ( 1368-1644 ) . this form became especially popular in central and western asia after the mongol conquests and under the timurid and safavid dynasties . metalsmiths of this period often used bands to separate areas of their vessels—the base from the body and the body from the neck . similarly , the potter that created the ewer above employed decorative patterns to give an effect of visual separation similar to the metal bands . the simplicity of these bands , as opposed to the more complex motifs found on the metal jug , is repeated in the decorative floral imagery found on the larger surfaces—the body and neck of the ceramic ewer . the patterns that form the “ bands ” are common in iznik ewers from the second half of the sixteenth century ( see image above , `` similar iznik ewer ” ) . the base is decorated with black hatched lines and at the neck there are leaves alternating blue and green . an additional band of black lines separate this surface from the neck of the vessel . a black spiral called “ the snail ” is repeated just below the mouth of the ewer . the body is decorated in the saz style . this style was introduced to the ottoman court by the sixteenth century iranian painter , shah qulu , who moved to istanbul at the beginning of the century . he and his followers created the saz style , a name that derives both from the ottoman term for the marsh reed out of which the artists ’ pens were crafted and from the enchanted forest of turkic mythology . another name for this style , hatayi , recognizes the chinese origins of many of its elements , such as the lotus flower and the chinese-style dragon . the saz style was adapted to many different media for the ottoman court . tiles with saz style drawings , for example , were used for the restorations to istanbul ’ s topkapi palace after 1574 ( see “ tile ” image ) . note how the long , serrated , saz style leaf on this tile resembles the leaf on the ewer in both shape and color—blue with a narrow red line in the center . one of shah qulu ’ s pupils in the nakkashane ( royal atelier ) , the anatolian kara memi , developed another popular style seen on the ewer . shortly after the middle of the sixteenth century , kara memi introduced a set of motifs that is a virtual garden of stylized flowers , but most emblematic are the long , elegant tulips . these flowers constitute one of the most distinctive and familiar aspects of ottoman style art and we can identify them as well as leaves and additional garden flowers on the body and neck of the ewer . adopting the aesthetics of the aristocracy the fading , bleeding colors on the vessel indicate that it was not made for the royal court though it was produced from fine materials and its drawing imitates the court style . it was most likely made for the merchant class , subjects of the empire who were not aristocracy but were not slaves . the owners of the ewer adopted the aesthetics of the aristocracy and purchased their own ceramics in a style similar to that found in the royal court . the ottomans themselves accepted the influence of various cultures in to their art . their artistic strength comes from adaptation of influences such as chinese and iranian , together with local innovations to create what was eventually identified as an ottoman style . essay by ortal bensky additional resources introduction to iznik and ottoman ceramics , musée du louvre the art of the ottomans before 1600 , heilbrunn timeline of art history , the metropolitan museum of art akar , azade , treasury of turkish designs : 670 motifs from iznik pottery , new york : dover publications , 1988 carswell , john , iznik pottery , london : british museum press , 1998 denny , walter b , “ dating ottoman turkish works in the saz style , ” muqarnas 1 ( 1983 ) : 103-122 nurhan , atasoy and julian raby , iznik : the pottery of ottoman turkey , london : alexandria press , 1994 queiroz ribeiro , maria , iznik pottery and tiles , lisbon : fundacao calouste gulbenkian , 2009
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this style was introduced to the ottoman court by the sixteenth century iranian painter , shah qulu , who moved to istanbul at the beginning of the century . he and his followers created the saz style , a name that derives both from the ottoman term for the marsh reed out of which the artists ’ pens were crafted and from the enchanted forest of turkic mythology . another name for this style , hatayi , recognizes the chinese origins of many of its elements , such as the lotus flower and the chinese-style dragon .
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when you say that the term 'saz ' comes from the enchanted forest from the turkish mythology what are you referring to ?
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iznik ceramics give us a wonderful opportunity to glimpse into both the thriving iznik ceramic industry and a 16th century ottoman home . the ewer shown on the left is a type of jug that is shaped like a vase . it was a common utensil used daily for carrying water from the kitchen to the dining area , and for serving family and guests . this particular ewer was made in iznik , the ottoman center of ceramic production . iznik ware the town of iznik was an important production center during the fifteenth , sixteenth and part of the seventeenth centuries that manufactured ceramics for both the court in the capital , istanbul , and for the open market . the ewer was only one form of ceramic kitchenware that was produced in iznik workshops . other products included dishes , bowls , tankards , and bottles . due to the absence of petuntse ( a variety of feldspar rock ) , used in chinese porcelain , true porcelain did not come to the middle east until the modern era . nevertheless , twelfth-century islamic potters in persia were able to produce a strong white clay that in many respects resembled chinese porcelain . this mixture of potter ’ s clay , ground quartz , and glassy frit , is called fritware . this porcelain substitute , together with the under-glaze painting technique used to decorate the ceramics , was used for centuries . it is not only seen in surviving kitchenware , but also in beautiful tiles covering the interior and exterior of important ottoman buildings . color , iconography , shape iznik ceramic production initially used blue-and-white decoration . however , by the second half of the sixteenth century , iznik pottery saw the gradual addition of new colors as pigments were developed . the brooklyn ewer is painted in black , cobalt blue , green , and red under a transparent glaze . the ewer ’ s round body shape with the narrow neck and the round handle is similar to the shape of metal jugs from the islamic world of the fifteenth and sixteenth centuries . it was also influenced by blue-and-white ceramic pot-bellied jugs produced by chinese potters of the ming period ( 1368-1644 ) . this form became especially popular in central and western asia after the mongol conquests and under the timurid and safavid dynasties . metalsmiths of this period often used bands to separate areas of their vessels—the base from the body and the body from the neck . similarly , the potter that created the ewer above employed decorative patterns to give an effect of visual separation similar to the metal bands . the simplicity of these bands , as opposed to the more complex motifs found on the metal jug , is repeated in the decorative floral imagery found on the larger surfaces—the body and neck of the ceramic ewer . the patterns that form the “ bands ” are common in iznik ewers from the second half of the sixteenth century ( see image above , `` similar iznik ewer ” ) . the base is decorated with black hatched lines and at the neck there are leaves alternating blue and green . an additional band of black lines separate this surface from the neck of the vessel . a black spiral called “ the snail ” is repeated just below the mouth of the ewer . the body is decorated in the saz style . this style was introduced to the ottoman court by the sixteenth century iranian painter , shah qulu , who moved to istanbul at the beginning of the century . he and his followers created the saz style , a name that derives both from the ottoman term for the marsh reed out of which the artists ’ pens were crafted and from the enchanted forest of turkic mythology . another name for this style , hatayi , recognizes the chinese origins of many of its elements , such as the lotus flower and the chinese-style dragon . the saz style was adapted to many different media for the ottoman court . tiles with saz style drawings , for example , were used for the restorations to istanbul ’ s topkapi palace after 1574 ( see “ tile ” image ) . note how the long , serrated , saz style leaf on this tile resembles the leaf on the ewer in both shape and color—blue with a narrow red line in the center . one of shah qulu ’ s pupils in the nakkashane ( royal atelier ) , the anatolian kara memi , developed another popular style seen on the ewer . shortly after the middle of the sixteenth century , kara memi introduced a set of motifs that is a virtual garden of stylized flowers , but most emblematic are the long , elegant tulips . these flowers constitute one of the most distinctive and familiar aspects of ottoman style art and we can identify them as well as leaves and additional garden flowers on the body and neck of the ewer . adopting the aesthetics of the aristocracy the fading , bleeding colors on the vessel indicate that it was not made for the royal court though it was produced from fine materials and its drawing imitates the court style . it was most likely made for the merchant class , subjects of the empire who were not aristocracy but were not slaves . the owners of the ewer adopted the aesthetics of the aristocracy and purchased their own ceramics in a style similar to that found in the royal court . the ottomans themselves accepted the influence of various cultures in to their art . their artistic strength comes from adaptation of influences such as chinese and iranian , together with local innovations to create what was eventually identified as an ottoman style . essay by ortal bensky additional resources introduction to iznik and ottoman ceramics , musée du louvre the art of the ottomans before 1600 , heilbrunn timeline of art history , the metropolitan museum of art akar , azade , treasury of turkish designs : 670 motifs from iznik pottery , new york : dover publications , 1988 carswell , john , iznik pottery , london : british museum press , 1998 denny , walter b , “ dating ottoman turkish works in the saz style , ” muqarnas 1 ( 1983 ) : 103-122 nurhan , atasoy and julian raby , iznik : the pottery of ottoman turkey , london : alexandria press , 1994 queiroz ribeiro , maria , iznik pottery and tiles , lisbon : fundacao calouste gulbenkian , 2009
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color , iconography , shape iznik ceramic production initially used blue-and-white decoration . however , by the second half of the sixteenth century , iznik pottery saw the gradual addition of new colors as pigments were developed . the brooklyn ewer is painted in black , cobalt blue , green , and red under a transparent glaze .
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do you know about other archaeological sites in the new world ?
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an enormous triptych the elevation of the cross altarpiece is a masterpiece of baroque painting by the flemish painter peter paul rubens . the work was originally installed on the high altar of the church of st. walburga in antwerp ( since destroyed ) , and is now located in the cathedral of our lady in antwerp . this triptych ( a painting—usually an altarpiece—comprised of two outer “ wings ” and a central panel ) is impressive in its size , measuring 15 feet in height and 21 feet wide when open . the original frame , unfortunately lost , would have made the painting even more impressive in size ! due to its very size , rubens actually painted it on-site behind a curtain . four saints associated with the church of st. walburga can be found on the exterior of the wings ( visible when the altarpiece is closed ) : saints amandus and walburga on the left and saints catherine of alexandria and eligius on the right . baroque dynamism rubens was one of the most prolific and sought after painters of the baroque period , generally ( although not always ) defined in painting and sculpture by the representation of action and emotion in ways meant to inspire the catholic faithful ( this triptych was painted less than a century after martin luther 's challenge to the authority of the catholic church ) . in the central panel , we see the dramatic moment when the cross of christ ’ s crucifixion is being raised to its upright position . rubens created a strong diagonal emphasis by placing the base of the cross at the far lower right of the composition and the top of the cross in the upper left—making christ ’ s body the focal point . this strong diagonal reinforces the notion that this is an event unfolding before the viewer , as the men struggle to lift the weight of their burden . adding to this dynamic tension is the visual sensation that the two men in the lower right are about to burst into the viewer ’ s space as they work to pull the cross upward ( see image above ) . the viewer is caught in a moment of anxiety , waiting for the action to be complete . in the left panel ( below , left ) are st. john the evangelist and the virgin mary , who , standing in the shadow of the rocky outcrop above them , look to their left at what unfolds before their eyes . shown in quiet resignation and grief over the fate of christ , the group of women below is a stark contrast of overwrought emotion . here too rubens uses a diagonal along the line of the women from the lower right to the mid-left , setting john and mary apart , allowing the viewer to focus on their reaction . the right panel ( above , right ) continues the narrative event as roman soldiers prepare the two thieves for their fate as they will be crucified alongside christ . one thief—already being nailed to the cross on the ground—is foreshortened back into space , while the other—just behind him with his hands bound—is being forcefully led away by his hair . the diagonal rubens created here runs the opposite direction as that in the left panel , moving from the lower left to the upper right along the line created by the leg and neck of the gray horse . these opposing diagonals further create tension across the composition , heightening the viewer ’ s sense of drama and chaotic action . a unified narrative and biblical accuracy in addition to the powerful figural composition , the three panels are visually unified through the landscape and sky . the left and central panels share a rocky outcropping covered with oak trees and vines ( both of which have christological significance ) . notice that st. john , the virgin mary and the roman soldiers just to the left of the cross are standing on the same ground-line . the unification of the central and right panels is accomplished through the sky , which begins to darken in the central panel , moving to the impending eclipse of the sun on the right , an event recounted in the gospel of matthew ( 27:45 ) : “ from noon on , darkness came over the whole land ... . ” this attention to biblical accuracy is also seen in the text on the scroll at the top of the cross , which reads : “ jesus of nazareth , king of the jews , ” written in greek , latin , and aramaic , as told in the gospel of john ( 19:19-21 ) . in both cases , rubens was adhering to one of the primary mandates of the council of trent ( 1545-63 ) , which called for historical accuracy in the representation of sacred events ( at the council of trent , church authorities essentially decided theological questions raised by martin luther and the protestants , the period following the council is known as the counter-reformation—the catholic church ’ s response to martin luther ’ s protestant reformation ) . rubens and a reflection of italy the elevation of the cross altarpiece was the first commission rubens received after returning to antwerp from his italian sojourn from 1600 to 1608/9 where he worked in the cities of mantua , genoa , and rome . given his extended time in italy , it is not surprising that we see a number of italian influences in this work . the richness of the coloration ( notice the blues and reds throughout the composition ) and rubens ’ painterly technique recalls that of the venetian master titian , while the dramatic contrasts of light and dark bring to mind caravaggio ’ s tenebrism ( darkness ) in his roman compositions , such as the crucifixion of st. peter ( above ) . and indeed , we can clearly see rubens ’ interest in his italian counterpart in the sense of physical exertion , the use of foreshortening—where figures push past the boundaries of the picture plane into the space of the viewer , and in the use of the diagonal . in terms of the muscularity and physicality of ruben ’ s male figures , a clear connection can be drawn to michelangelo ’ s nude males ( the ignudi ) on the sistine chapel ceiling . in addition to looking at the works of past and contemporary masters , we know rubens was also interested in the study of classical antiquity ( ancient greece and rome ) . in fact , the figure of christ seems to have been based on one of the most famous works of antiquity , the laocoön , which rubens made drawings of during his time in rome . elevation : altarpiece and high altar when the elevation of the cross altarpiece was placed on the high altar , there was a specific connection being forged between the subject of the painting and the function of the altar . the act of raising an object up is known in latin as elevatio . during the mass performed by the priest at the high altar , there is a moment when the eucharistic wafer ( miraculously transformed into the body of christ ) is elevated . thus , when the congregation faced the high altar , they not only saw the elevatio of christ ’ s cross but the elevation of the wafer , and thus the altarpiece and the ritual of the mass performed in front of it visually reinforced the message of christ ’ s sacrifice on behalf of mankind . essay by dr. shannon pritchard additional resources : photographs of the triptych a study for the figure of christ in the harvard art museums biography of the artist from the national gallery rubens on the google art project
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the work was originally installed on the high altar of the church of st. walburga in antwerp ( since destroyed ) , and is now located in the cathedral of our lady in antwerp . this triptych ( a painting—usually an altarpiece—comprised of two outer “ wings ” and a central panel ) is impressive in its size , measuring 15 feet in height and 21 feet wide when open . the original frame , unfortunately lost , would have made the painting even more impressive in size ! due to its very size , rubens actually painted it on-site behind a curtain .
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how long did it take to make this painting ?
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an enormous triptych the elevation of the cross altarpiece is a masterpiece of baroque painting by the flemish painter peter paul rubens . the work was originally installed on the high altar of the church of st. walburga in antwerp ( since destroyed ) , and is now located in the cathedral of our lady in antwerp . this triptych ( a painting—usually an altarpiece—comprised of two outer “ wings ” and a central panel ) is impressive in its size , measuring 15 feet in height and 21 feet wide when open . the original frame , unfortunately lost , would have made the painting even more impressive in size ! due to its very size , rubens actually painted it on-site behind a curtain . four saints associated with the church of st. walburga can be found on the exterior of the wings ( visible when the altarpiece is closed ) : saints amandus and walburga on the left and saints catherine of alexandria and eligius on the right . baroque dynamism rubens was one of the most prolific and sought after painters of the baroque period , generally ( although not always ) defined in painting and sculpture by the representation of action and emotion in ways meant to inspire the catholic faithful ( this triptych was painted less than a century after martin luther 's challenge to the authority of the catholic church ) . in the central panel , we see the dramatic moment when the cross of christ ’ s crucifixion is being raised to its upright position . rubens created a strong diagonal emphasis by placing the base of the cross at the far lower right of the composition and the top of the cross in the upper left—making christ ’ s body the focal point . this strong diagonal reinforces the notion that this is an event unfolding before the viewer , as the men struggle to lift the weight of their burden . adding to this dynamic tension is the visual sensation that the two men in the lower right are about to burst into the viewer ’ s space as they work to pull the cross upward ( see image above ) . the viewer is caught in a moment of anxiety , waiting for the action to be complete . in the left panel ( below , left ) are st. john the evangelist and the virgin mary , who , standing in the shadow of the rocky outcrop above them , look to their left at what unfolds before their eyes . shown in quiet resignation and grief over the fate of christ , the group of women below is a stark contrast of overwrought emotion . here too rubens uses a diagonal along the line of the women from the lower right to the mid-left , setting john and mary apart , allowing the viewer to focus on their reaction . the right panel ( above , right ) continues the narrative event as roman soldiers prepare the two thieves for their fate as they will be crucified alongside christ . one thief—already being nailed to the cross on the ground—is foreshortened back into space , while the other—just behind him with his hands bound—is being forcefully led away by his hair . the diagonal rubens created here runs the opposite direction as that in the left panel , moving from the lower left to the upper right along the line created by the leg and neck of the gray horse . these opposing diagonals further create tension across the composition , heightening the viewer ’ s sense of drama and chaotic action . a unified narrative and biblical accuracy in addition to the powerful figural composition , the three panels are visually unified through the landscape and sky . the left and central panels share a rocky outcropping covered with oak trees and vines ( both of which have christological significance ) . notice that st. john , the virgin mary and the roman soldiers just to the left of the cross are standing on the same ground-line . the unification of the central and right panels is accomplished through the sky , which begins to darken in the central panel , moving to the impending eclipse of the sun on the right , an event recounted in the gospel of matthew ( 27:45 ) : “ from noon on , darkness came over the whole land ... . ” this attention to biblical accuracy is also seen in the text on the scroll at the top of the cross , which reads : “ jesus of nazareth , king of the jews , ” written in greek , latin , and aramaic , as told in the gospel of john ( 19:19-21 ) . in both cases , rubens was adhering to one of the primary mandates of the council of trent ( 1545-63 ) , which called for historical accuracy in the representation of sacred events ( at the council of trent , church authorities essentially decided theological questions raised by martin luther and the protestants , the period following the council is known as the counter-reformation—the catholic church ’ s response to martin luther ’ s protestant reformation ) . rubens and a reflection of italy the elevation of the cross altarpiece was the first commission rubens received after returning to antwerp from his italian sojourn from 1600 to 1608/9 where he worked in the cities of mantua , genoa , and rome . given his extended time in italy , it is not surprising that we see a number of italian influences in this work . the richness of the coloration ( notice the blues and reds throughout the composition ) and rubens ’ painterly technique recalls that of the venetian master titian , while the dramatic contrasts of light and dark bring to mind caravaggio ’ s tenebrism ( darkness ) in his roman compositions , such as the crucifixion of st. peter ( above ) . and indeed , we can clearly see rubens ’ interest in his italian counterpart in the sense of physical exertion , the use of foreshortening—where figures push past the boundaries of the picture plane into the space of the viewer , and in the use of the diagonal . in terms of the muscularity and physicality of ruben ’ s male figures , a clear connection can be drawn to michelangelo ’ s nude males ( the ignudi ) on the sistine chapel ceiling . in addition to looking at the works of past and contemporary masters , we know rubens was also interested in the study of classical antiquity ( ancient greece and rome ) . in fact , the figure of christ seems to have been based on one of the most famous works of antiquity , the laocoön , which rubens made drawings of during his time in rome . elevation : altarpiece and high altar when the elevation of the cross altarpiece was placed on the high altar , there was a specific connection being forged between the subject of the painting and the function of the altar . the act of raising an object up is known in latin as elevatio . during the mass performed by the priest at the high altar , there is a moment when the eucharistic wafer ( miraculously transformed into the body of christ ) is elevated . thus , when the congregation faced the high altar , they not only saw the elevatio of christ ’ s cross but the elevation of the wafer , and thus the altarpiece and the ritual of the mass performed in front of it visually reinforced the message of christ ’ s sacrifice on behalf of mankind . essay by dr. shannon pritchard additional resources : photographs of the triptych a study for the figure of christ in the harvard art museums biography of the artist from the national gallery rubens on the google art project
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in fact , the figure of christ seems to have been based on one of the most famous works of antiquity , the laocoön , which rubens made drawings of during his time in rome . elevation : altarpiece and high altar when the elevation of the cross altarpiece was placed on the high altar , there was a specific connection being forged between the subject of the painting and the function of the altar . the act of raising an object up is known in latin as elevatio .
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pm the woman with the child at her breast , is there a reason why she is feeding her child , or a specific meaning of symbolism ?
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in differential calculus we reasoned about the properties of a function $ f $ based on information given about its derivative $ f ' $ . in integral calculus , instead of talking about functions and their derivatives , we will talk about functions and their antiderivatives . reasoning about $ g $ from the graph of $ g'=f $ this is the graph of function $ f $ . let $ g ( x ) =\displaystyle\int_0^x f ( t ) \ , dt $ . defined this way , $ g $ is an antiderivative of $ f $ . in differential calculus we would write this as $ g'=f $ . since $ f $ is the derivative of $ g $ , we can reason about properties of $ g $ in similar to what we did in differential calculus . for example , $ f $ is positive on the interval $ [ 0,10 ] $ , so $ g $ must be increasing on this interval . furthermore , $ f $ changes its sign at $ x=10 $ , so $ g $ must have an extremum there . since $ f $ goes from positive to negative , that point must be a maximum point . the above examples showed how we can reason about the intervals where $ g $ increases or decreases and about its relative extrema . we can also reason about the concavity of $ g $ . since $ f $ is increasing on the interval $ [ -2,5 ] $ , we know $ g $ is concave up on that interval . and since $ f $ is decreasing on the interval $ [ 5,13 ] $ , we know $ g $ is concave down on that interval . $ g $ changes concavity at $ x=5 $ , so it has an inflection point there . want more practice ? try this exercise . it 's important not to confuse which properties of the function are related to which properties of its antiderivative . many students get confused and make all kinds of wrong inferences , like saying that an antiderivative is positive because the function is increasing ( in fact , it 's the other way around ) . this table summarizes all the relationships between the properties of a function and its antiderivative . when the function $ f $ is ... | the antiderivative $ g=\displaystyle\int_a^x f ( t ) \ , dt $ is ... : - : | : - : positive $ + $ | increasing $ \nearrow $ negative $ - $ | decreasing $ \searrow $ increasing $ \nearrow $ | concave up $ \cup $ decreasing $ \searrow $ | concave down $ \cap $ changes sign / crosses the $ x $ -axis | extremum point extremum point | inflection point
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reasoning about $ g $ from the graph of $ g'=f $ this is the graph of function $ f $ . let $ g ( x ) =\displaystyle\int_0^x f ( t ) \ , dt $ . defined this way , $ g $ is an antiderivative of $ f $ .
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i guess what im saying is wouldnt the constant make a huge difference in assessment of f ( x ) ?
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introduction the position of a car driving down the street , the value of currency adjusted for inflation , the number of bacteria in a culture , and the ac voltage of an electric signal are all examples of quantities that change with time . in this section , we will study the rate of change of a quantity and how is it related geometrically to secant and tangent lines . secant and tangent lines if two distinct points $ p ( x_0 , y_0 ) $ and $ q ( x_1 , y_1 ) $ lie on the curve $ y = f ( x ) $ , the slope of the secant line connecting the two points is $ \displaystyle m_ { \sec } = \frac { y_1 - y_0 } { x_1 - x_0 } = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ , $ . if we let the point $ x_1 $ approach $ x_0 $ , then $ q $ will approach $ p $ along the graph $ f $ . the slope of the secant line through points $ p $ and $ q $ will gradually approach the slope of the tangent line through $ p $ as $ x_1 $ approaches $ x_0 $ . in the limit , the previous equation becomes $ \displaystyle m_ { \tan } =\lim_ { x_1 \to x_0 } \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ , $ . if we let $ h = x_1 - x_0 $ , then $ x_1 = x_0 + h $ and $ h \rightarrow 0 $ as $ x_1 \rightarrow x_0 $ . we can rewrite the limit as $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ . when the limit exists , its value $ m_ { \tan } $ is the slope of the tangent line to the graph of $ f $ at the point $ p ( x_0 , y_0 ) $ . example 1 find the slope of the tangent line to the graph of the function $ f ( x ) = x^3 $ at the point $ ( 2 , 8 ) $ . solution since $ ( x_0 , y_0 ) = ( 2 , 8 ) $ , using the slope of the tangent line formula $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ we get $ \begin { align } m_ { \tan } & amp ; =\lim_ { h \to 0 } \frac { f ( 2 + h ) - f ( 2 ) } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { \left ( h^3 + 6h^2 + 12h + 8\right ) - 8 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { h^3 + 6h^2 + 12h } { h } \ \ & amp ; =\lim_ { h \to 0 } \left ( h^2 + 6h + 12\right ) \ \ & amp ; = 12 . \end { align } $ thus , the slope of the tangent line is $ 12 $ . recall from algebra that the point-slope form of the equation of the tangent line is $ \displaystyle y - y_0 = m_ { \tan } \cdot ( x - x_0 ) $ . the point-slope formula gives us the equation $ \displaystyle y - 8 = 12\cdot ( x - 2 ) $ which we can rewrite as $ \displaystyle y = 12x - 16 $ . finding the slope at any point next we are interested in finding a formula for the slope of the tangent line at any point on the graph of $ f $ . such a formula would be the same formula that we are using except we replace the constant $ x_0 $ by the variable $ x $ . this yields $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ . we denote this formula by $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ , where $ f\ , ' ( x ) $ is read `` $ f $ prime of $ x $ . '' the next example illustrates its usefulness . example 2 if $ f ( x ) = x^2 - 3 $ , find $ f\ , ' ( x ) $ and use the result to find the slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ . solution since $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ , then $ \begin { align } f\ , ' ( x ) & amp ; =\lim_ { h \to 0 } \frac { \left [ ( x + h ) ^2 - 3\right ] - \left [ x^2 - 3\right ] } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { x^2 + 2xh + h^2 - 3 - x^2 + 3 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { 2xh + h^2 } { h } \ \ & amp ; =\lim_ { h \to 0 } ( 2x + h ) \ \ & amp ; = 2x . \end { align } $ to find the slope , we substitute $ x = 2 $ and $ x = -1 $ into the result $ f\ , ' ( x ) $ , we get $ \displaystyle f\ , ' ( 2 ) = 2 ( 2 ) = 4 $ and $ \displaystyle f\ , ' ( -1 ) = 2 ( -1 ) = -2 $ . thus , slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ are $ 4 $ and $ -2 $ , respectively . example 3 find the slope of the tangent line to the curve $ y = 1/x $ at the point $ ( 1 , 1 ) $ . solution using the slope of the tangent line formula $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ and substituting $ y = 1/x $ gives us $ \begin { align } y\ , ' & amp ; =\lim_ { h \to 0 } \dfrac { \left ( \dfrac { 1 } { x+h } \right ) - \dfrac { 1 } { x } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { \dfrac { x - ( x + h ) } { x ( x + h ) } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { x - x - h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -1 } { x ( x + h ) } \ \ & amp ; = \frac { -1 } { x^2 } . \end { align } $ substituting $ x = 1 $ yields $ y\ , ' = \dfrac { -1 } { ( 1 ) ^2 } = -1 $ . thus , the slope of the tangent line at $ x = 1 $ for the curve $ y = 1/x $ is $ m = -1 $ . to find the equation of the tangent line , we use the point-slope formula , $ \displaystyle y - y_0 = m\cdot ( x - x_0 ) $ , where $ ( x_ { 0 } , y_ { 0 } ) = ( 1 , 1 ) $ . the equation of the tangent line is $ \begin { align } y - 1 & amp ; = -1 \cdot ( x - 1 ) \ y & amp ; = -x + 1 + 1\ y & amp ; = -x + 2 . \end { align } $ average speed the primary concept of differential is calculating the rate of change of one quantity with respect to another . for example , speed is defined as the rate of change of the distance traveled with respect to time . if a car travels $ 120 $ miles in $ 4 $ hours , his speed is $ \dfrac { 120\text { miles } } { 4\text { hours } } = 30 \text { mi/hr } $ . this speed is called the average speed or the average rate of change of distance with respect to time . of course , a car that travels $ 120 $ miles at an average rate of $ 30 $ miles per hour for $ 4 $ hours does not necessarily do so at constant speed . it may have slowed down or sped up during the $ 4 $ hour period . however , if the car hits a tree , it would not be its average speed that determines the resulting damage but its speed at the instant of the collision . so here we have two distinct kinds of speeds , average speed and instantaneous speed . the average speed of an object is defined as the object ’ s displacement $ \triangle x $ divided by the time interval $ \triangle t $ during which the displacement occurs : $ \displaystyle v = \frac { \triangle x } { \triangle t } = \frac { x_1 - x_0 } { t_1 - t_0 } $ . the average speed is also the expression for the slope of a secant line connecting the two points . figure 1 shows the $ \purple { \text { secant line } } $ through the points $ \pink { ( t_0 , x_0 ) } $ and $ \pink { ( t_1 , x_1 ) } $ on the $ \blue { \text { position-versus-time curve } } $ . thus we conclude that the average speed of an object between time $ t_0 $ and $ t_1 $ is represented geometrically by the slope of the secant line connecting the two points $ ( t_0 , x_0 ) $ and $ ( t_1 , x_1 ) $ . if we choose $ t_1 $ close to $ t_0 $ , then the average speed will closely approximate the instantaneous speed at time $ t_0 $ . rates of change the average rate of change of an arbitrary function $ f $ on an interval is represented geometrically by the slope of the secant line to the graph of $ f $ . the instantaneous rate of change of $ f $ at a particular point is represented by the slope of the tangent line to the graph of $ f $ at that point . let 's consider each case in more detail . average rate of change the average rate of change of the function $ f $ over the interval $ [ x_0 , x_1 ] $ is $ \displaystyle m_ { \sec } = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } $ . figure 2 shows the $ \purple { \text { secant line } } $ through the points $ \pink { ( x_0 , f ( x_0 ) ) } $ and $ \pink { ( x_1 , f ( x_1 ) ) } $ on the $ \blue { \text { graph of } \ , f } $ . the slope of the secant line is the average rate of change $ m_ { \sec } $ . instantaneous rate of change the instantaneous rate of change of the function $ f $ at the point $ x_0 $ is $ \displaystyle m_ { \tan } = f\ , ' ( x_0 ) = \lim_ { x_1 \to x_0 } \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } $ . figure 3 shows $ \purple { \text { tangent line } } $ through the point $ \pink { ( x_0 , f ( x_0 ) ) } $ on the $ \blue { \text { graph of } \ , f } $ . the slope of the tangent line is the instantaneous rate of change $ m_ { \tan } $ . example 4 suppose that $ y = x^2 - 3 $ . ( a ) find the average rate of change of $ y $ with respect to $ x $ over the interval $ [ 0 , 2 ] $ . ( b ) find the instantaneous rate of change of $ y $ with respect to $ x $ at the point $ x = -1 $ . solution ( a ) applying the formula for average rate of change with $ f ( x ) = x^2 - 3 $ and $ x_0 = 0 $ and $ x_1 = 2 $ yields $ \begin { align } m_ { \sec } & amp ; = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ \ & amp ; = \frac { f ( 2 ) - f ( 0 ) } { 2 - 0 } \ \ & amp ; = \frac { 1 - ( -3 ) } { 2 } \ \ & amp ; = 2 \end { align } $ this means the average rate of change over the interval $ [ 0 , 2 ] $ is 2 units of increase in $ y $ for each unit of increase in $ x $ . ( b ) from example 2 above , we found that $ f\ , ' ( x ) = 2x $ , so $ \begin { align } m_ { \tan } & amp ; = f\ , ' ( x_0 ) \ & amp ; = f\ , ' ( -1 ) \ & amp ; = 2 ( -1 ) \ & amp ; = -2 . \end { align } $ this means that the instantaneous rate of change is negative . that is , $ y $ is decreasing at $ x = -1 $ . it is decreasing at a rate of $ 2 $ units in $ y $ for each unit of increase in $ x $ . source ck-12 : tangent lines and rates of change license creative commons attribution-noncommercial
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solution since $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ , then $ \begin { align } f\ , ' ( x ) & amp ; =\lim_ { h \to 0 } \frac { \left [ ( x + h ) ^2 - 3\right ] - \left [ x^2 - 3\right ] } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { x^2 + 2xh + h^2 - 3 - x^2 + 3 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { 2xh + h^2 } { h } \ \ & amp ; =\lim_ { h \to 0 } ( 2x + h ) \ \ & amp ; = 2x . \end { align } $ to find the slope , we substitute $ x = 2 $ and $ x = -1 $ into the result $ f\ , ' ( x ) $ , we get $ \displaystyle f\ , ' ( 2 ) = 2 ( 2 ) = 4 $ and $ \displaystyle f\ , ' ( -1 ) = 2 ( -1 ) = -2 $ . thus , slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ are $ 4 $ and $ -2 $ , respectively .
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what is the purpose of the function that appears at the very end of example 2 ?
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introduction the position of a car driving down the street , the value of currency adjusted for inflation , the number of bacteria in a culture , and the ac voltage of an electric signal are all examples of quantities that change with time . in this section , we will study the rate of change of a quantity and how is it related geometrically to secant and tangent lines . secant and tangent lines if two distinct points $ p ( x_0 , y_0 ) $ and $ q ( x_1 , y_1 ) $ lie on the curve $ y = f ( x ) $ , the slope of the secant line connecting the two points is $ \displaystyle m_ { \sec } = \frac { y_1 - y_0 } { x_1 - x_0 } = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ , $ . if we let the point $ x_1 $ approach $ x_0 $ , then $ q $ will approach $ p $ along the graph $ f $ . the slope of the secant line through points $ p $ and $ q $ will gradually approach the slope of the tangent line through $ p $ as $ x_1 $ approaches $ x_0 $ . in the limit , the previous equation becomes $ \displaystyle m_ { \tan } =\lim_ { x_1 \to x_0 } \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ , $ . if we let $ h = x_1 - x_0 $ , then $ x_1 = x_0 + h $ and $ h \rightarrow 0 $ as $ x_1 \rightarrow x_0 $ . we can rewrite the limit as $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ . when the limit exists , its value $ m_ { \tan } $ is the slope of the tangent line to the graph of $ f $ at the point $ p ( x_0 , y_0 ) $ . example 1 find the slope of the tangent line to the graph of the function $ f ( x ) = x^3 $ at the point $ ( 2 , 8 ) $ . solution since $ ( x_0 , y_0 ) = ( 2 , 8 ) $ , using the slope of the tangent line formula $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ we get $ \begin { align } m_ { \tan } & amp ; =\lim_ { h \to 0 } \frac { f ( 2 + h ) - f ( 2 ) } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { \left ( h^3 + 6h^2 + 12h + 8\right ) - 8 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { h^3 + 6h^2 + 12h } { h } \ \ & amp ; =\lim_ { h \to 0 } \left ( h^2 + 6h + 12\right ) \ \ & amp ; = 12 . \end { align } $ thus , the slope of the tangent line is $ 12 $ . recall from algebra that the point-slope form of the equation of the tangent line is $ \displaystyle y - y_0 = m_ { \tan } \cdot ( x - x_0 ) $ . the point-slope formula gives us the equation $ \displaystyle y - 8 = 12\cdot ( x - 2 ) $ which we can rewrite as $ \displaystyle y = 12x - 16 $ . finding the slope at any point next we are interested in finding a formula for the slope of the tangent line at any point on the graph of $ f $ . such a formula would be the same formula that we are using except we replace the constant $ x_0 $ by the variable $ x $ . this yields $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ . we denote this formula by $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ , where $ f\ , ' ( x ) $ is read `` $ f $ prime of $ x $ . '' the next example illustrates its usefulness . example 2 if $ f ( x ) = x^2 - 3 $ , find $ f\ , ' ( x ) $ and use the result to find the slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ . solution since $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ , then $ \begin { align } f\ , ' ( x ) & amp ; =\lim_ { h \to 0 } \frac { \left [ ( x + h ) ^2 - 3\right ] - \left [ x^2 - 3\right ] } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { x^2 + 2xh + h^2 - 3 - x^2 + 3 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { 2xh + h^2 } { h } \ \ & amp ; =\lim_ { h \to 0 } ( 2x + h ) \ \ & amp ; = 2x . \end { align } $ to find the slope , we substitute $ x = 2 $ and $ x = -1 $ into the result $ f\ , ' ( x ) $ , we get $ \displaystyle f\ , ' ( 2 ) = 2 ( 2 ) = 4 $ and $ \displaystyle f\ , ' ( -1 ) = 2 ( -1 ) = -2 $ . thus , slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ are $ 4 $ and $ -2 $ , respectively . example 3 find the slope of the tangent line to the curve $ y = 1/x $ at the point $ ( 1 , 1 ) $ . solution using the slope of the tangent line formula $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ and substituting $ y = 1/x $ gives us $ \begin { align } y\ , ' & amp ; =\lim_ { h \to 0 } \dfrac { \left ( \dfrac { 1 } { x+h } \right ) - \dfrac { 1 } { x } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { \dfrac { x - ( x + h ) } { x ( x + h ) } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { x - x - h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -1 } { x ( x + h ) } \ \ & amp ; = \frac { -1 } { x^2 } . \end { align } $ substituting $ x = 1 $ yields $ y\ , ' = \dfrac { -1 } { ( 1 ) ^2 } = -1 $ . thus , the slope of the tangent line at $ x = 1 $ for the curve $ y = 1/x $ is $ m = -1 $ . to find the equation of the tangent line , we use the point-slope formula , $ \displaystyle y - y_0 = m\cdot ( x - x_0 ) $ , where $ ( x_ { 0 } , y_ { 0 } ) = ( 1 , 1 ) $ . the equation of the tangent line is $ \begin { align } y - 1 & amp ; = -1 \cdot ( x - 1 ) \ y & amp ; = -x + 1 + 1\ y & amp ; = -x + 2 . \end { align } $ average speed the primary concept of differential is calculating the rate of change of one quantity with respect to another . for example , speed is defined as the rate of change of the distance traveled with respect to time . if a car travels $ 120 $ miles in $ 4 $ hours , his speed is $ \dfrac { 120\text { miles } } { 4\text { hours } } = 30 \text { mi/hr } $ . this speed is called the average speed or the average rate of change of distance with respect to time . of course , a car that travels $ 120 $ miles at an average rate of $ 30 $ miles per hour for $ 4 $ hours does not necessarily do so at constant speed . it may have slowed down or sped up during the $ 4 $ hour period . however , if the car hits a tree , it would not be its average speed that determines the resulting damage but its speed at the instant of the collision . so here we have two distinct kinds of speeds , average speed and instantaneous speed . the average speed of an object is defined as the object ’ s displacement $ \triangle x $ divided by the time interval $ \triangle t $ during which the displacement occurs : $ \displaystyle v = \frac { \triangle x } { \triangle t } = \frac { x_1 - x_0 } { t_1 - t_0 } $ . the average speed is also the expression for the slope of a secant line connecting the two points . figure 1 shows the $ \purple { \text { secant line } } $ through the points $ \pink { ( t_0 , x_0 ) } $ and $ \pink { ( t_1 , x_1 ) } $ on the $ \blue { \text { position-versus-time curve } } $ . thus we conclude that the average speed of an object between time $ t_0 $ and $ t_1 $ is represented geometrically by the slope of the secant line connecting the two points $ ( t_0 , x_0 ) $ and $ ( t_1 , x_1 ) $ . if we choose $ t_1 $ close to $ t_0 $ , then the average speed will closely approximate the instantaneous speed at time $ t_0 $ . rates of change the average rate of change of an arbitrary function $ f $ on an interval is represented geometrically by the slope of the secant line to the graph of $ f $ . the instantaneous rate of change of $ f $ at a particular point is represented by the slope of the tangent line to the graph of $ f $ at that point . let 's consider each case in more detail . average rate of change the average rate of change of the function $ f $ over the interval $ [ x_0 , x_1 ] $ is $ \displaystyle m_ { \sec } = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } $ . figure 2 shows the $ \purple { \text { secant line } } $ through the points $ \pink { ( x_0 , f ( x_0 ) ) } $ and $ \pink { ( x_1 , f ( x_1 ) ) } $ on the $ \blue { \text { graph of } \ , f } $ . the slope of the secant line is the average rate of change $ m_ { \sec } $ . instantaneous rate of change the instantaneous rate of change of the function $ f $ at the point $ x_0 $ is $ \displaystyle m_ { \tan } = f\ , ' ( x_0 ) = \lim_ { x_1 \to x_0 } \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } $ . figure 3 shows $ \purple { \text { tangent line } } $ through the point $ \pink { ( x_0 , f ( x_0 ) ) } $ on the $ \blue { \text { graph of } \ , f } $ . the slope of the tangent line is the instantaneous rate of change $ m_ { \tan } $ . example 4 suppose that $ y = x^2 - 3 $ . ( a ) find the average rate of change of $ y $ with respect to $ x $ over the interval $ [ 0 , 2 ] $ . ( b ) find the instantaneous rate of change of $ y $ with respect to $ x $ at the point $ x = -1 $ . solution ( a ) applying the formula for average rate of change with $ f ( x ) = x^2 - 3 $ and $ x_0 = 0 $ and $ x_1 = 2 $ yields $ \begin { align } m_ { \sec } & amp ; = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ \ & amp ; = \frac { f ( 2 ) - f ( 0 ) } { 2 - 0 } \ \ & amp ; = \frac { 1 - ( -3 ) } { 2 } \ \ & amp ; = 2 \end { align } $ this means the average rate of change over the interval $ [ 0 , 2 ] $ is 2 units of increase in $ y $ for each unit of increase in $ x $ . ( b ) from example 2 above , we found that $ f\ , ' ( x ) = 2x $ , so $ \begin { align } m_ { \tan } & amp ; = f\ , ' ( x_0 ) \ & amp ; = f\ , ' ( -1 ) \ & amp ; = 2 ( -1 ) \ & amp ; = -2 . \end { align } $ this means that the instantaneous rate of change is negative . that is , $ y $ is decreasing at $ x = -1 $ . it is decreasing at a rate of $ 2 $ units in $ y $ for each unit of increase in $ x $ . source ck-12 : tangent lines and rates of change license creative commons attribution-noncommercial
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if we let $ h = x_1 - x_0 $ , then $ x_1 = x_0 + h $ and $ h \rightarrow 0 $ as $ x_1 \rightarrow x_0 $ . we can rewrite the limit as $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ . when the limit exists , its value $ m_ { \tan } $ is the slope of the tangent line to the graph of $ f $ at the point $ p ( x_0 , y_0 ) $ . example 1 find the slope of the tangent line to the graph of the function $ f ( x ) = x^3 $ at the point $ ( 2 , 8 ) $ .
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do i understand correctly that the growth `` h '' is at the end of reduction of limit omited because its value is so small it has practically no effect on value of slope ?
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introduction the position of a car driving down the street , the value of currency adjusted for inflation , the number of bacteria in a culture , and the ac voltage of an electric signal are all examples of quantities that change with time . in this section , we will study the rate of change of a quantity and how is it related geometrically to secant and tangent lines . secant and tangent lines if two distinct points $ p ( x_0 , y_0 ) $ and $ q ( x_1 , y_1 ) $ lie on the curve $ y = f ( x ) $ , the slope of the secant line connecting the two points is $ \displaystyle m_ { \sec } = \frac { y_1 - y_0 } { x_1 - x_0 } = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ , $ . if we let the point $ x_1 $ approach $ x_0 $ , then $ q $ will approach $ p $ along the graph $ f $ . the slope of the secant line through points $ p $ and $ q $ will gradually approach the slope of the tangent line through $ p $ as $ x_1 $ approaches $ x_0 $ . in the limit , the previous equation becomes $ \displaystyle m_ { \tan } =\lim_ { x_1 \to x_0 } \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ , $ . if we let $ h = x_1 - x_0 $ , then $ x_1 = x_0 + h $ and $ h \rightarrow 0 $ as $ x_1 \rightarrow x_0 $ . we can rewrite the limit as $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ . when the limit exists , its value $ m_ { \tan } $ is the slope of the tangent line to the graph of $ f $ at the point $ p ( x_0 , y_0 ) $ . example 1 find the slope of the tangent line to the graph of the function $ f ( x ) = x^3 $ at the point $ ( 2 , 8 ) $ . solution since $ ( x_0 , y_0 ) = ( 2 , 8 ) $ , using the slope of the tangent line formula $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ we get $ \begin { align } m_ { \tan } & amp ; =\lim_ { h \to 0 } \frac { f ( 2 + h ) - f ( 2 ) } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { \left ( h^3 + 6h^2 + 12h + 8\right ) - 8 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { h^3 + 6h^2 + 12h } { h } \ \ & amp ; =\lim_ { h \to 0 } \left ( h^2 + 6h + 12\right ) \ \ & amp ; = 12 . \end { align } $ thus , the slope of the tangent line is $ 12 $ . recall from algebra that the point-slope form of the equation of the tangent line is $ \displaystyle y - y_0 = m_ { \tan } \cdot ( x - x_0 ) $ . the point-slope formula gives us the equation $ \displaystyle y - 8 = 12\cdot ( x - 2 ) $ which we can rewrite as $ \displaystyle y = 12x - 16 $ . finding the slope at any point next we are interested in finding a formula for the slope of the tangent line at any point on the graph of $ f $ . such a formula would be the same formula that we are using except we replace the constant $ x_0 $ by the variable $ x $ . this yields $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ . we denote this formula by $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ , where $ f\ , ' ( x ) $ is read `` $ f $ prime of $ x $ . '' the next example illustrates its usefulness . example 2 if $ f ( x ) = x^2 - 3 $ , find $ f\ , ' ( x ) $ and use the result to find the slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ . solution since $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ , then $ \begin { align } f\ , ' ( x ) & amp ; =\lim_ { h \to 0 } \frac { \left [ ( x + h ) ^2 - 3\right ] - \left [ x^2 - 3\right ] } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { x^2 + 2xh + h^2 - 3 - x^2 + 3 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { 2xh + h^2 } { h } \ \ & amp ; =\lim_ { h \to 0 } ( 2x + h ) \ \ & amp ; = 2x . \end { align } $ to find the slope , we substitute $ x = 2 $ and $ x = -1 $ into the result $ f\ , ' ( x ) $ , we get $ \displaystyle f\ , ' ( 2 ) = 2 ( 2 ) = 4 $ and $ \displaystyle f\ , ' ( -1 ) = 2 ( -1 ) = -2 $ . thus , slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ are $ 4 $ and $ -2 $ , respectively . example 3 find the slope of the tangent line to the curve $ y = 1/x $ at the point $ ( 1 , 1 ) $ . solution using the slope of the tangent line formula $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ and substituting $ y = 1/x $ gives us $ \begin { align } y\ , ' & amp ; =\lim_ { h \to 0 } \dfrac { \left ( \dfrac { 1 } { x+h } \right ) - \dfrac { 1 } { x } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { \dfrac { x - ( x + h ) } { x ( x + h ) } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { x - x - h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -1 } { x ( x + h ) } \ \ & amp ; = \frac { -1 } { x^2 } . \end { align } $ substituting $ x = 1 $ yields $ y\ , ' = \dfrac { -1 } { ( 1 ) ^2 } = -1 $ . thus , the slope of the tangent line at $ x = 1 $ for the curve $ y = 1/x $ is $ m = -1 $ . to find the equation of the tangent line , we use the point-slope formula , $ \displaystyle y - y_0 = m\cdot ( x - x_0 ) $ , where $ ( x_ { 0 } , y_ { 0 } ) = ( 1 , 1 ) $ . the equation of the tangent line is $ \begin { align } y - 1 & amp ; = -1 \cdot ( x - 1 ) \ y & amp ; = -x + 1 + 1\ y & amp ; = -x + 2 . \end { align } $ average speed the primary concept of differential is calculating the rate of change of one quantity with respect to another . for example , speed is defined as the rate of change of the distance traveled with respect to time . if a car travels $ 120 $ miles in $ 4 $ hours , his speed is $ \dfrac { 120\text { miles } } { 4\text { hours } } = 30 \text { mi/hr } $ . this speed is called the average speed or the average rate of change of distance with respect to time . of course , a car that travels $ 120 $ miles at an average rate of $ 30 $ miles per hour for $ 4 $ hours does not necessarily do so at constant speed . it may have slowed down or sped up during the $ 4 $ hour period . however , if the car hits a tree , it would not be its average speed that determines the resulting damage but its speed at the instant of the collision . so here we have two distinct kinds of speeds , average speed and instantaneous speed . the average speed of an object is defined as the object ’ s displacement $ \triangle x $ divided by the time interval $ \triangle t $ during which the displacement occurs : $ \displaystyle v = \frac { \triangle x } { \triangle t } = \frac { x_1 - x_0 } { t_1 - t_0 } $ . the average speed is also the expression for the slope of a secant line connecting the two points . figure 1 shows the $ \purple { \text { secant line } } $ through the points $ \pink { ( t_0 , x_0 ) } $ and $ \pink { ( t_1 , x_1 ) } $ on the $ \blue { \text { position-versus-time curve } } $ . thus we conclude that the average speed of an object between time $ t_0 $ and $ t_1 $ is represented geometrically by the slope of the secant line connecting the two points $ ( t_0 , x_0 ) $ and $ ( t_1 , x_1 ) $ . if we choose $ t_1 $ close to $ t_0 $ , then the average speed will closely approximate the instantaneous speed at time $ t_0 $ . rates of change the average rate of change of an arbitrary function $ f $ on an interval is represented geometrically by the slope of the secant line to the graph of $ f $ . the instantaneous rate of change of $ f $ at a particular point is represented by the slope of the tangent line to the graph of $ f $ at that point . let 's consider each case in more detail . average rate of change the average rate of change of the function $ f $ over the interval $ [ x_0 , x_1 ] $ is $ \displaystyle m_ { \sec } = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } $ . figure 2 shows the $ \purple { \text { secant line } } $ through the points $ \pink { ( x_0 , f ( x_0 ) ) } $ and $ \pink { ( x_1 , f ( x_1 ) ) } $ on the $ \blue { \text { graph of } \ , f } $ . the slope of the secant line is the average rate of change $ m_ { \sec } $ . instantaneous rate of change the instantaneous rate of change of the function $ f $ at the point $ x_0 $ is $ \displaystyle m_ { \tan } = f\ , ' ( x_0 ) = \lim_ { x_1 \to x_0 } \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } $ . figure 3 shows $ \purple { \text { tangent line } } $ through the point $ \pink { ( x_0 , f ( x_0 ) ) } $ on the $ \blue { \text { graph of } \ , f } $ . the slope of the tangent line is the instantaneous rate of change $ m_ { \tan } $ . example 4 suppose that $ y = x^2 - 3 $ . ( a ) find the average rate of change of $ y $ with respect to $ x $ over the interval $ [ 0 , 2 ] $ . ( b ) find the instantaneous rate of change of $ y $ with respect to $ x $ at the point $ x = -1 $ . solution ( a ) applying the formula for average rate of change with $ f ( x ) = x^2 - 3 $ and $ x_0 = 0 $ and $ x_1 = 2 $ yields $ \begin { align } m_ { \sec } & amp ; = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ \ & amp ; = \frac { f ( 2 ) - f ( 0 ) } { 2 - 0 } \ \ & amp ; = \frac { 1 - ( -3 ) } { 2 } \ \ & amp ; = 2 \end { align } $ this means the average rate of change over the interval $ [ 0 , 2 ] $ is 2 units of increase in $ y $ for each unit of increase in $ x $ . ( b ) from example 2 above , we found that $ f\ , ' ( x ) = 2x $ , so $ \begin { align } m_ { \tan } & amp ; = f\ , ' ( x_0 ) \ & amp ; = f\ , ' ( -1 ) \ & amp ; = 2 ( -1 ) \ & amp ; = -2 . \end { align } $ this means that the instantaneous rate of change is negative . that is , $ y $ is decreasing at $ x = -1 $ . it is decreasing at a rate of $ 2 $ units in $ y $ for each unit of increase in $ x $ . source ck-12 : tangent lines and rates of change license creative commons attribution-noncommercial
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solution since $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ , then $ \begin { align } f\ , ' ( x ) & amp ; =\lim_ { h \to 0 } \frac { \left [ ( x + h ) ^2 - 3\right ] - \left [ x^2 - 3\right ] } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { x^2 + 2xh + h^2 - 3 - x^2 + 3 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { 2xh + h^2 } { h } \ \ & amp ; =\lim_ { h \to 0 } ( 2x + h ) \ \ & amp ; = 2x . \end { align } $ to find the slope , we substitute $ x = 2 $ and $ x = -1 $ into the result $ f\ , ' ( x ) $ , we get $ \displaystyle f\ , ' ( 2 ) = 2 ( 2 ) = 4 $ and $ \displaystyle f\ , ' ( -1 ) = 2 ( -1 ) = -2 $ . thus , slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ are $ 4 $ and $ -2 $ , respectively .
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how did we find the function at the end of example 2 ?
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introduction the position of a car driving down the street , the value of currency adjusted for inflation , the number of bacteria in a culture , and the ac voltage of an electric signal are all examples of quantities that change with time . in this section , we will study the rate of change of a quantity and how is it related geometrically to secant and tangent lines . secant and tangent lines if two distinct points $ p ( x_0 , y_0 ) $ and $ q ( x_1 , y_1 ) $ lie on the curve $ y = f ( x ) $ , the slope of the secant line connecting the two points is $ \displaystyle m_ { \sec } = \frac { y_1 - y_0 } { x_1 - x_0 } = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ , $ . if we let the point $ x_1 $ approach $ x_0 $ , then $ q $ will approach $ p $ along the graph $ f $ . the slope of the secant line through points $ p $ and $ q $ will gradually approach the slope of the tangent line through $ p $ as $ x_1 $ approaches $ x_0 $ . in the limit , the previous equation becomes $ \displaystyle m_ { \tan } =\lim_ { x_1 \to x_0 } \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ , $ . if we let $ h = x_1 - x_0 $ , then $ x_1 = x_0 + h $ and $ h \rightarrow 0 $ as $ x_1 \rightarrow x_0 $ . we can rewrite the limit as $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ . when the limit exists , its value $ m_ { \tan } $ is the slope of the tangent line to the graph of $ f $ at the point $ p ( x_0 , y_0 ) $ . example 1 find the slope of the tangent line to the graph of the function $ f ( x ) = x^3 $ at the point $ ( 2 , 8 ) $ . solution since $ ( x_0 , y_0 ) = ( 2 , 8 ) $ , using the slope of the tangent line formula $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ we get $ \begin { align } m_ { \tan } & amp ; =\lim_ { h \to 0 } \frac { f ( 2 + h ) - f ( 2 ) } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { \left ( h^3 + 6h^2 + 12h + 8\right ) - 8 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { h^3 + 6h^2 + 12h } { h } \ \ & amp ; =\lim_ { h \to 0 } \left ( h^2 + 6h + 12\right ) \ \ & amp ; = 12 . \end { align } $ thus , the slope of the tangent line is $ 12 $ . recall from algebra that the point-slope form of the equation of the tangent line is $ \displaystyle y - y_0 = m_ { \tan } \cdot ( x - x_0 ) $ . the point-slope formula gives us the equation $ \displaystyle y - 8 = 12\cdot ( x - 2 ) $ which we can rewrite as $ \displaystyle y = 12x - 16 $ . finding the slope at any point next we are interested in finding a formula for the slope of the tangent line at any point on the graph of $ f $ . such a formula would be the same formula that we are using except we replace the constant $ x_0 $ by the variable $ x $ . this yields $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ . we denote this formula by $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ , where $ f\ , ' ( x ) $ is read `` $ f $ prime of $ x $ . '' the next example illustrates its usefulness . example 2 if $ f ( x ) = x^2 - 3 $ , find $ f\ , ' ( x ) $ and use the result to find the slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ . solution since $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ , then $ \begin { align } f\ , ' ( x ) & amp ; =\lim_ { h \to 0 } \frac { \left [ ( x + h ) ^2 - 3\right ] - \left [ x^2 - 3\right ] } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { x^2 + 2xh + h^2 - 3 - x^2 + 3 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { 2xh + h^2 } { h } \ \ & amp ; =\lim_ { h \to 0 } ( 2x + h ) \ \ & amp ; = 2x . \end { align } $ to find the slope , we substitute $ x = 2 $ and $ x = -1 $ into the result $ f\ , ' ( x ) $ , we get $ \displaystyle f\ , ' ( 2 ) = 2 ( 2 ) = 4 $ and $ \displaystyle f\ , ' ( -1 ) = 2 ( -1 ) = -2 $ . thus , slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ are $ 4 $ and $ -2 $ , respectively . example 3 find the slope of the tangent line to the curve $ y = 1/x $ at the point $ ( 1 , 1 ) $ . solution using the slope of the tangent line formula $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ and substituting $ y = 1/x $ gives us $ \begin { align } y\ , ' & amp ; =\lim_ { h \to 0 } \dfrac { \left ( \dfrac { 1 } { x+h } \right ) - \dfrac { 1 } { x } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { \dfrac { x - ( x + h ) } { x ( x + h ) } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { x - x - h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -1 } { x ( x + h ) } \ \ & amp ; = \frac { -1 } { x^2 } . \end { align } $ substituting $ x = 1 $ yields $ y\ , ' = \dfrac { -1 } { ( 1 ) ^2 } = -1 $ . thus , the slope of the tangent line at $ x = 1 $ for the curve $ y = 1/x $ is $ m = -1 $ . to find the equation of the tangent line , we use the point-slope formula , $ \displaystyle y - y_0 = m\cdot ( x - x_0 ) $ , where $ ( x_ { 0 } , y_ { 0 } ) = ( 1 , 1 ) $ . the equation of the tangent line is $ \begin { align } y - 1 & amp ; = -1 \cdot ( x - 1 ) \ y & amp ; = -x + 1 + 1\ y & amp ; = -x + 2 . \end { align } $ average speed the primary concept of differential is calculating the rate of change of one quantity with respect to another . for example , speed is defined as the rate of change of the distance traveled with respect to time . if a car travels $ 120 $ miles in $ 4 $ hours , his speed is $ \dfrac { 120\text { miles } } { 4\text { hours } } = 30 \text { mi/hr } $ . this speed is called the average speed or the average rate of change of distance with respect to time . of course , a car that travels $ 120 $ miles at an average rate of $ 30 $ miles per hour for $ 4 $ hours does not necessarily do so at constant speed . it may have slowed down or sped up during the $ 4 $ hour period . however , if the car hits a tree , it would not be its average speed that determines the resulting damage but its speed at the instant of the collision . so here we have two distinct kinds of speeds , average speed and instantaneous speed . the average speed of an object is defined as the object ’ s displacement $ \triangle x $ divided by the time interval $ \triangle t $ during which the displacement occurs : $ \displaystyle v = \frac { \triangle x } { \triangle t } = \frac { x_1 - x_0 } { t_1 - t_0 } $ . the average speed is also the expression for the slope of a secant line connecting the two points . figure 1 shows the $ \purple { \text { secant line } } $ through the points $ \pink { ( t_0 , x_0 ) } $ and $ \pink { ( t_1 , x_1 ) } $ on the $ \blue { \text { position-versus-time curve } } $ . thus we conclude that the average speed of an object between time $ t_0 $ and $ t_1 $ is represented geometrically by the slope of the secant line connecting the two points $ ( t_0 , x_0 ) $ and $ ( t_1 , x_1 ) $ . if we choose $ t_1 $ close to $ t_0 $ , then the average speed will closely approximate the instantaneous speed at time $ t_0 $ . rates of change the average rate of change of an arbitrary function $ f $ on an interval is represented geometrically by the slope of the secant line to the graph of $ f $ . the instantaneous rate of change of $ f $ at a particular point is represented by the slope of the tangent line to the graph of $ f $ at that point . let 's consider each case in more detail . average rate of change the average rate of change of the function $ f $ over the interval $ [ x_0 , x_1 ] $ is $ \displaystyle m_ { \sec } = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } $ . figure 2 shows the $ \purple { \text { secant line } } $ through the points $ \pink { ( x_0 , f ( x_0 ) ) } $ and $ \pink { ( x_1 , f ( x_1 ) ) } $ on the $ \blue { \text { graph of } \ , f } $ . the slope of the secant line is the average rate of change $ m_ { \sec } $ . instantaneous rate of change the instantaneous rate of change of the function $ f $ at the point $ x_0 $ is $ \displaystyle m_ { \tan } = f\ , ' ( x_0 ) = \lim_ { x_1 \to x_0 } \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } $ . figure 3 shows $ \purple { \text { tangent line } } $ through the point $ \pink { ( x_0 , f ( x_0 ) ) } $ on the $ \blue { \text { graph of } \ , f } $ . the slope of the tangent line is the instantaneous rate of change $ m_ { \tan } $ . example 4 suppose that $ y = x^2 - 3 $ . ( a ) find the average rate of change of $ y $ with respect to $ x $ over the interval $ [ 0 , 2 ] $ . ( b ) find the instantaneous rate of change of $ y $ with respect to $ x $ at the point $ x = -1 $ . solution ( a ) applying the formula for average rate of change with $ f ( x ) = x^2 - 3 $ and $ x_0 = 0 $ and $ x_1 = 2 $ yields $ \begin { align } m_ { \sec } & amp ; = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ \ & amp ; = \frac { f ( 2 ) - f ( 0 ) } { 2 - 0 } \ \ & amp ; = \frac { 1 - ( -3 ) } { 2 } \ \ & amp ; = 2 \end { align } $ this means the average rate of change over the interval $ [ 0 , 2 ] $ is 2 units of increase in $ y $ for each unit of increase in $ x $ . ( b ) from example 2 above , we found that $ f\ , ' ( x ) = 2x $ , so $ \begin { align } m_ { \tan } & amp ; = f\ , ' ( x_0 ) \ & amp ; = f\ , ' ( -1 ) \ & amp ; = 2 ( -1 ) \ & amp ; = -2 . \end { align } $ this means that the instantaneous rate of change is negative . that is , $ y $ is decreasing at $ x = -1 $ . it is decreasing at a rate of $ 2 $ units in $ y $ for each unit of increase in $ x $ . source ck-12 : tangent lines and rates of change license creative commons attribution-noncommercial
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we denote this formula by $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ , where $ f\ , ' ( x ) $ is read `` $ f $ prime of $ x $ . '' the next example illustrates its usefulness . example 2 if $ f ( x ) = x^2 - 3 $ , find $ f\ , ' ( x ) $ and use the result to find the slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ . solution since $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ , then $ \begin { align } f\ , ' ( x ) & amp ; =\lim_ { h \to 0 } \frac { \left [ ( x + h ) ^2 - 3\right ] - \left [ x^2 - 3\right ] } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { x^2 + 2xh + h^2 - 3 - x^2 + 3 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { 2xh + h^2 } { h } \ \ & amp ; =\lim_ { h \to 0 } ( 2x + h ) \ \ & amp ; = 2x .
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in example 3 , what if x is given but we 'll have to find 'm ' and 'c ' with a given function ?
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introduction the position of a car driving down the street , the value of currency adjusted for inflation , the number of bacteria in a culture , and the ac voltage of an electric signal are all examples of quantities that change with time . in this section , we will study the rate of change of a quantity and how is it related geometrically to secant and tangent lines . secant and tangent lines if two distinct points $ p ( x_0 , y_0 ) $ and $ q ( x_1 , y_1 ) $ lie on the curve $ y = f ( x ) $ , the slope of the secant line connecting the two points is $ \displaystyle m_ { \sec } = \frac { y_1 - y_0 } { x_1 - x_0 } = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ , $ . if we let the point $ x_1 $ approach $ x_0 $ , then $ q $ will approach $ p $ along the graph $ f $ . the slope of the secant line through points $ p $ and $ q $ will gradually approach the slope of the tangent line through $ p $ as $ x_1 $ approaches $ x_0 $ . in the limit , the previous equation becomes $ \displaystyle m_ { \tan } =\lim_ { x_1 \to x_0 } \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ , $ . if we let $ h = x_1 - x_0 $ , then $ x_1 = x_0 + h $ and $ h \rightarrow 0 $ as $ x_1 \rightarrow x_0 $ . we can rewrite the limit as $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ . when the limit exists , its value $ m_ { \tan } $ is the slope of the tangent line to the graph of $ f $ at the point $ p ( x_0 , y_0 ) $ . example 1 find the slope of the tangent line to the graph of the function $ f ( x ) = x^3 $ at the point $ ( 2 , 8 ) $ . solution since $ ( x_0 , y_0 ) = ( 2 , 8 ) $ , using the slope of the tangent line formula $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ we get $ \begin { align } m_ { \tan } & amp ; =\lim_ { h \to 0 } \frac { f ( 2 + h ) - f ( 2 ) } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { \left ( h^3 + 6h^2 + 12h + 8\right ) - 8 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { h^3 + 6h^2 + 12h } { h } \ \ & amp ; =\lim_ { h \to 0 } \left ( h^2 + 6h + 12\right ) \ \ & amp ; = 12 . \end { align } $ thus , the slope of the tangent line is $ 12 $ . recall from algebra that the point-slope form of the equation of the tangent line is $ \displaystyle y - y_0 = m_ { \tan } \cdot ( x - x_0 ) $ . the point-slope formula gives us the equation $ \displaystyle y - 8 = 12\cdot ( x - 2 ) $ which we can rewrite as $ \displaystyle y = 12x - 16 $ . finding the slope at any point next we are interested in finding a formula for the slope of the tangent line at any point on the graph of $ f $ . such a formula would be the same formula that we are using except we replace the constant $ x_0 $ by the variable $ x $ . this yields $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ . we denote this formula by $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ , where $ f\ , ' ( x ) $ is read `` $ f $ prime of $ x $ . '' the next example illustrates its usefulness . example 2 if $ f ( x ) = x^2 - 3 $ , find $ f\ , ' ( x ) $ and use the result to find the slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ . solution since $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ , then $ \begin { align } f\ , ' ( x ) & amp ; =\lim_ { h \to 0 } \frac { \left [ ( x + h ) ^2 - 3\right ] - \left [ x^2 - 3\right ] } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { x^2 + 2xh + h^2 - 3 - x^2 + 3 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { 2xh + h^2 } { h } \ \ & amp ; =\lim_ { h \to 0 } ( 2x + h ) \ \ & amp ; = 2x . \end { align } $ to find the slope , we substitute $ x = 2 $ and $ x = -1 $ into the result $ f\ , ' ( x ) $ , we get $ \displaystyle f\ , ' ( 2 ) = 2 ( 2 ) = 4 $ and $ \displaystyle f\ , ' ( -1 ) = 2 ( -1 ) = -2 $ . thus , slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ are $ 4 $ and $ -2 $ , respectively . example 3 find the slope of the tangent line to the curve $ y = 1/x $ at the point $ ( 1 , 1 ) $ . solution using the slope of the tangent line formula $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ and substituting $ y = 1/x $ gives us $ \begin { align } y\ , ' & amp ; =\lim_ { h \to 0 } \dfrac { \left ( \dfrac { 1 } { x+h } \right ) - \dfrac { 1 } { x } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { \dfrac { x - ( x + h ) } { x ( x + h ) } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { x - x - h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -1 } { x ( x + h ) } \ \ & amp ; = \frac { -1 } { x^2 } . \end { align } $ substituting $ x = 1 $ yields $ y\ , ' = \dfrac { -1 } { ( 1 ) ^2 } = -1 $ . thus , the slope of the tangent line at $ x = 1 $ for the curve $ y = 1/x $ is $ m = -1 $ . to find the equation of the tangent line , we use the point-slope formula , $ \displaystyle y - y_0 = m\cdot ( x - x_0 ) $ , where $ ( x_ { 0 } , y_ { 0 } ) = ( 1 , 1 ) $ . the equation of the tangent line is $ \begin { align } y - 1 & amp ; = -1 \cdot ( x - 1 ) \ y & amp ; = -x + 1 + 1\ y & amp ; = -x + 2 . \end { align } $ average speed the primary concept of differential is calculating the rate of change of one quantity with respect to another . for example , speed is defined as the rate of change of the distance traveled with respect to time . if a car travels $ 120 $ miles in $ 4 $ hours , his speed is $ \dfrac { 120\text { miles } } { 4\text { hours } } = 30 \text { mi/hr } $ . this speed is called the average speed or the average rate of change of distance with respect to time . of course , a car that travels $ 120 $ miles at an average rate of $ 30 $ miles per hour for $ 4 $ hours does not necessarily do so at constant speed . it may have slowed down or sped up during the $ 4 $ hour period . however , if the car hits a tree , it would not be its average speed that determines the resulting damage but its speed at the instant of the collision . so here we have two distinct kinds of speeds , average speed and instantaneous speed . the average speed of an object is defined as the object ’ s displacement $ \triangle x $ divided by the time interval $ \triangle t $ during which the displacement occurs : $ \displaystyle v = \frac { \triangle x } { \triangle t } = \frac { x_1 - x_0 } { t_1 - t_0 } $ . the average speed is also the expression for the slope of a secant line connecting the two points . figure 1 shows the $ \purple { \text { secant line } } $ through the points $ \pink { ( t_0 , x_0 ) } $ and $ \pink { ( t_1 , x_1 ) } $ on the $ \blue { \text { position-versus-time curve } } $ . thus we conclude that the average speed of an object between time $ t_0 $ and $ t_1 $ is represented geometrically by the slope of the secant line connecting the two points $ ( t_0 , x_0 ) $ and $ ( t_1 , x_1 ) $ . if we choose $ t_1 $ close to $ t_0 $ , then the average speed will closely approximate the instantaneous speed at time $ t_0 $ . rates of change the average rate of change of an arbitrary function $ f $ on an interval is represented geometrically by the slope of the secant line to the graph of $ f $ . the instantaneous rate of change of $ f $ at a particular point is represented by the slope of the tangent line to the graph of $ f $ at that point . let 's consider each case in more detail . average rate of change the average rate of change of the function $ f $ over the interval $ [ x_0 , x_1 ] $ is $ \displaystyle m_ { \sec } = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } $ . figure 2 shows the $ \purple { \text { secant line } } $ through the points $ \pink { ( x_0 , f ( x_0 ) ) } $ and $ \pink { ( x_1 , f ( x_1 ) ) } $ on the $ \blue { \text { graph of } \ , f } $ . the slope of the secant line is the average rate of change $ m_ { \sec } $ . instantaneous rate of change the instantaneous rate of change of the function $ f $ at the point $ x_0 $ is $ \displaystyle m_ { \tan } = f\ , ' ( x_0 ) = \lim_ { x_1 \to x_0 } \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } $ . figure 3 shows $ \purple { \text { tangent line } } $ through the point $ \pink { ( x_0 , f ( x_0 ) ) } $ on the $ \blue { \text { graph of } \ , f } $ . the slope of the tangent line is the instantaneous rate of change $ m_ { \tan } $ . example 4 suppose that $ y = x^2 - 3 $ . ( a ) find the average rate of change of $ y $ with respect to $ x $ over the interval $ [ 0 , 2 ] $ . ( b ) find the instantaneous rate of change of $ y $ with respect to $ x $ at the point $ x = -1 $ . solution ( a ) applying the formula for average rate of change with $ f ( x ) = x^2 - 3 $ and $ x_0 = 0 $ and $ x_1 = 2 $ yields $ \begin { align } m_ { \sec } & amp ; = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ \ & amp ; = \frac { f ( 2 ) - f ( 0 ) } { 2 - 0 } \ \ & amp ; = \frac { 1 - ( -3 ) } { 2 } \ \ & amp ; = 2 \end { align } $ this means the average rate of change over the interval $ [ 0 , 2 ] $ is 2 units of increase in $ y $ for each unit of increase in $ x $ . ( b ) from example 2 above , we found that $ f\ , ' ( x ) = 2x $ , so $ \begin { align } m_ { \tan } & amp ; = f\ , ' ( x_0 ) \ & amp ; = f\ , ' ( -1 ) \ & amp ; = 2 ( -1 ) \ & amp ; = -2 . \end { align } $ this means that the instantaneous rate of change is negative . that is , $ y $ is decreasing at $ x = -1 $ . it is decreasing at a rate of $ 2 $ units in $ y $ for each unit of increase in $ x $ . source ck-12 : tangent lines and rates of change license creative commons attribution-noncommercial
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example 3 find the slope of the tangent line to the curve $ y = 1/x $ at the point $ ( 1 , 1 ) $ . solution using the slope of the tangent line formula $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ and substituting $ y = 1/x $ gives us $ \begin { align } y\ , ' & amp ; =\lim_ { h \to 0 } \dfrac { \left ( \dfrac { 1 } { x+h } \right ) - \dfrac { 1 } { x } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { \dfrac { x - ( x + h ) } { x ( x + h ) } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { x - x - h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -1 } { x ( x + h ) } \ \ & amp ; = \frac { -1 } { x^2 } . \end { align } $ substituting $ x = 1 $ yields $ y\ , ' = \dfrac { -1 } { ( 1 ) ^2 } = -1 $ .
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on example 3 , when taking the derivative of 1/x , at the third step while using the formula - `` h '' is somehow transferred from the lower denominator to the denominator above it : x- ( x+h ) /x ( x+h ) /h -- > x- ( x+h ) /xh ( x+h ) which algebric rule relates to that ?
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introduction the position of a car driving down the street , the value of currency adjusted for inflation , the number of bacteria in a culture , and the ac voltage of an electric signal are all examples of quantities that change with time . in this section , we will study the rate of change of a quantity and how is it related geometrically to secant and tangent lines . secant and tangent lines if two distinct points $ p ( x_0 , y_0 ) $ and $ q ( x_1 , y_1 ) $ lie on the curve $ y = f ( x ) $ , the slope of the secant line connecting the two points is $ \displaystyle m_ { \sec } = \frac { y_1 - y_0 } { x_1 - x_0 } = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ , $ . if we let the point $ x_1 $ approach $ x_0 $ , then $ q $ will approach $ p $ along the graph $ f $ . the slope of the secant line through points $ p $ and $ q $ will gradually approach the slope of the tangent line through $ p $ as $ x_1 $ approaches $ x_0 $ . in the limit , the previous equation becomes $ \displaystyle m_ { \tan } =\lim_ { x_1 \to x_0 } \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ , $ . if we let $ h = x_1 - x_0 $ , then $ x_1 = x_0 + h $ and $ h \rightarrow 0 $ as $ x_1 \rightarrow x_0 $ . we can rewrite the limit as $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ . when the limit exists , its value $ m_ { \tan } $ is the slope of the tangent line to the graph of $ f $ at the point $ p ( x_0 , y_0 ) $ . example 1 find the slope of the tangent line to the graph of the function $ f ( x ) = x^3 $ at the point $ ( 2 , 8 ) $ . solution since $ ( x_0 , y_0 ) = ( 2 , 8 ) $ , using the slope of the tangent line formula $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ we get $ \begin { align } m_ { \tan } & amp ; =\lim_ { h \to 0 } \frac { f ( 2 + h ) - f ( 2 ) } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { \left ( h^3 + 6h^2 + 12h + 8\right ) - 8 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { h^3 + 6h^2 + 12h } { h } \ \ & amp ; =\lim_ { h \to 0 } \left ( h^2 + 6h + 12\right ) \ \ & amp ; = 12 . \end { align } $ thus , the slope of the tangent line is $ 12 $ . recall from algebra that the point-slope form of the equation of the tangent line is $ \displaystyle y - y_0 = m_ { \tan } \cdot ( x - x_0 ) $ . the point-slope formula gives us the equation $ \displaystyle y - 8 = 12\cdot ( x - 2 ) $ which we can rewrite as $ \displaystyle y = 12x - 16 $ . finding the slope at any point next we are interested in finding a formula for the slope of the tangent line at any point on the graph of $ f $ . such a formula would be the same formula that we are using except we replace the constant $ x_0 $ by the variable $ x $ . this yields $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ . we denote this formula by $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ , where $ f\ , ' ( x ) $ is read `` $ f $ prime of $ x $ . '' the next example illustrates its usefulness . example 2 if $ f ( x ) = x^2 - 3 $ , find $ f\ , ' ( x ) $ and use the result to find the slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ . solution since $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ , then $ \begin { align } f\ , ' ( x ) & amp ; =\lim_ { h \to 0 } \frac { \left [ ( x + h ) ^2 - 3\right ] - \left [ x^2 - 3\right ] } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { x^2 + 2xh + h^2 - 3 - x^2 + 3 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { 2xh + h^2 } { h } \ \ & amp ; =\lim_ { h \to 0 } ( 2x + h ) \ \ & amp ; = 2x . \end { align } $ to find the slope , we substitute $ x = 2 $ and $ x = -1 $ into the result $ f\ , ' ( x ) $ , we get $ \displaystyle f\ , ' ( 2 ) = 2 ( 2 ) = 4 $ and $ \displaystyle f\ , ' ( -1 ) = 2 ( -1 ) = -2 $ . thus , slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ are $ 4 $ and $ -2 $ , respectively . example 3 find the slope of the tangent line to the curve $ y = 1/x $ at the point $ ( 1 , 1 ) $ . solution using the slope of the tangent line formula $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ and substituting $ y = 1/x $ gives us $ \begin { align } y\ , ' & amp ; =\lim_ { h \to 0 } \dfrac { \left ( \dfrac { 1 } { x+h } \right ) - \dfrac { 1 } { x } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { \dfrac { x - ( x + h ) } { x ( x + h ) } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { x - x - h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -1 } { x ( x + h ) } \ \ & amp ; = \frac { -1 } { x^2 } . \end { align } $ substituting $ x = 1 $ yields $ y\ , ' = \dfrac { -1 } { ( 1 ) ^2 } = -1 $ . thus , the slope of the tangent line at $ x = 1 $ for the curve $ y = 1/x $ is $ m = -1 $ . to find the equation of the tangent line , we use the point-slope formula , $ \displaystyle y - y_0 = m\cdot ( x - x_0 ) $ , where $ ( x_ { 0 } , y_ { 0 } ) = ( 1 , 1 ) $ . the equation of the tangent line is $ \begin { align } y - 1 & amp ; = -1 \cdot ( x - 1 ) \ y & amp ; = -x + 1 + 1\ y & amp ; = -x + 2 . \end { align } $ average speed the primary concept of differential is calculating the rate of change of one quantity with respect to another . for example , speed is defined as the rate of change of the distance traveled with respect to time . if a car travels $ 120 $ miles in $ 4 $ hours , his speed is $ \dfrac { 120\text { miles } } { 4\text { hours } } = 30 \text { mi/hr } $ . this speed is called the average speed or the average rate of change of distance with respect to time . of course , a car that travels $ 120 $ miles at an average rate of $ 30 $ miles per hour for $ 4 $ hours does not necessarily do so at constant speed . it may have slowed down or sped up during the $ 4 $ hour period . however , if the car hits a tree , it would not be its average speed that determines the resulting damage but its speed at the instant of the collision . so here we have two distinct kinds of speeds , average speed and instantaneous speed . the average speed of an object is defined as the object ’ s displacement $ \triangle x $ divided by the time interval $ \triangle t $ during which the displacement occurs : $ \displaystyle v = \frac { \triangle x } { \triangle t } = \frac { x_1 - x_0 } { t_1 - t_0 } $ . the average speed is also the expression for the slope of a secant line connecting the two points . figure 1 shows the $ \purple { \text { secant line } } $ through the points $ \pink { ( t_0 , x_0 ) } $ and $ \pink { ( t_1 , x_1 ) } $ on the $ \blue { \text { position-versus-time curve } } $ . thus we conclude that the average speed of an object between time $ t_0 $ and $ t_1 $ is represented geometrically by the slope of the secant line connecting the two points $ ( t_0 , x_0 ) $ and $ ( t_1 , x_1 ) $ . if we choose $ t_1 $ close to $ t_0 $ , then the average speed will closely approximate the instantaneous speed at time $ t_0 $ . rates of change the average rate of change of an arbitrary function $ f $ on an interval is represented geometrically by the slope of the secant line to the graph of $ f $ . the instantaneous rate of change of $ f $ at a particular point is represented by the slope of the tangent line to the graph of $ f $ at that point . let 's consider each case in more detail . average rate of change the average rate of change of the function $ f $ over the interval $ [ x_0 , x_1 ] $ is $ \displaystyle m_ { \sec } = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } $ . figure 2 shows the $ \purple { \text { secant line } } $ through the points $ \pink { ( x_0 , f ( x_0 ) ) } $ and $ \pink { ( x_1 , f ( x_1 ) ) } $ on the $ \blue { \text { graph of } \ , f } $ . the slope of the secant line is the average rate of change $ m_ { \sec } $ . instantaneous rate of change the instantaneous rate of change of the function $ f $ at the point $ x_0 $ is $ \displaystyle m_ { \tan } = f\ , ' ( x_0 ) = \lim_ { x_1 \to x_0 } \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } $ . figure 3 shows $ \purple { \text { tangent line } } $ through the point $ \pink { ( x_0 , f ( x_0 ) ) } $ on the $ \blue { \text { graph of } \ , f } $ . the slope of the tangent line is the instantaneous rate of change $ m_ { \tan } $ . example 4 suppose that $ y = x^2 - 3 $ . ( a ) find the average rate of change of $ y $ with respect to $ x $ over the interval $ [ 0 , 2 ] $ . ( b ) find the instantaneous rate of change of $ y $ with respect to $ x $ at the point $ x = -1 $ . solution ( a ) applying the formula for average rate of change with $ f ( x ) = x^2 - 3 $ and $ x_0 = 0 $ and $ x_1 = 2 $ yields $ \begin { align } m_ { \sec } & amp ; = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ \ & amp ; = \frac { f ( 2 ) - f ( 0 ) } { 2 - 0 } \ \ & amp ; = \frac { 1 - ( -3 ) } { 2 } \ \ & amp ; = 2 \end { align } $ this means the average rate of change over the interval $ [ 0 , 2 ] $ is 2 units of increase in $ y $ for each unit of increase in $ x $ . ( b ) from example 2 above , we found that $ f\ , ' ( x ) = 2x $ , so $ \begin { align } m_ { \tan } & amp ; = f\ , ' ( x_0 ) \ & amp ; = f\ , ' ( -1 ) \ & amp ; = 2 ( -1 ) \ & amp ; = -2 . \end { align } $ this means that the instantaneous rate of change is negative . that is , $ y $ is decreasing at $ x = -1 $ . it is decreasing at a rate of $ 2 $ units in $ y $ for each unit of increase in $ x $ . source ck-12 : tangent lines and rates of change license creative commons attribution-noncommercial
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this yields $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ . we denote this formula by $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ , where $ f\ , ' ( x ) $ is read `` $ f $ prime of $ x $ . '' the next example illustrates its usefulness . example 2 if $ f ( x ) = x^2 - 3 $ , find $ f\ , ' ( x ) $ and use the result to find the slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ . solution since $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ , then $ \begin { align } f\ , ' ( x ) & amp ; =\lim_ { h \to 0 } \frac { \left [ ( x + h ) ^2 - 3\right ] - \left [ x^2 - 3\right ] } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { x^2 + 2xh + h^2 - 3 - x^2 + 3 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { 2xh + h^2 } { h } \ \ & amp ; =\lim_ { h \to 0 } ( 2x + h ) \ \ & amp ; = 2x . \end { align } $ to find the slope , we substitute $ x = 2 $ and $ x = -1 $ into the result $ f\ , ' ( x ) $ , we get $ \displaystyle f\ , ' ( 2 ) = 2 ( 2 ) = 4 $ and $ \displaystyle f\ , ' ( -1 ) = 2 ( -1 ) = -2 $ . thus , slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ are $ 4 $ and $ -2 $ , respectively .
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in example 1 , how did you go from f ( 2+h ) to f ( x^3 + 6x^2 + 12x + 8 ) ?
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introduction the position of a car driving down the street , the value of currency adjusted for inflation , the number of bacteria in a culture , and the ac voltage of an electric signal are all examples of quantities that change with time . in this section , we will study the rate of change of a quantity and how is it related geometrically to secant and tangent lines . secant and tangent lines if two distinct points $ p ( x_0 , y_0 ) $ and $ q ( x_1 , y_1 ) $ lie on the curve $ y = f ( x ) $ , the slope of the secant line connecting the two points is $ \displaystyle m_ { \sec } = \frac { y_1 - y_0 } { x_1 - x_0 } = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ , $ . if we let the point $ x_1 $ approach $ x_0 $ , then $ q $ will approach $ p $ along the graph $ f $ . the slope of the secant line through points $ p $ and $ q $ will gradually approach the slope of the tangent line through $ p $ as $ x_1 $ approaches $ x_0 $ . in the limit , the previous equation becomes $ \displaystyle m_ { \tan } =\lim_ { x_1 \to x_0 } \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ , $ . if we let $ h = x_1 - x_0 $ , then $ x_1 = x_0 + h $ and $ h \rightarrow 0 $ as $ x_1 \rightarrow x_0 $ . we can rewrite the limit as $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ . when the limit exists , its value $ m_ { \tan } $ is the slope of the tangent line to the graph of $ f $ at the point $ p ( x_0 , y_0 ) $ . example 1 find the slope of the tangent line to the graph of the function $ f ( x ) = x^3 $ at the point $ ( 2 , 8 ) $ . solution since $ ( x_0 , y_0 ) = ( 2 , 8 ) $ , using the slope of the tangent line formula $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ we get $ \begin { align } m_ { \tan } & amp ; =\lim_ { h \to 0 } \frac { f ( 2 + h ) - f ( 2 ) } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { \left ( h^3 + 6h^2 + 12h + 8\right ) - 8 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { h^3 + 6h^2 + 12h } { h } \ \ & amp ; =\lim_ { h \to 0 } \left ( h^2 + 6h + 12\right ) \ \ & amp ; = 12 . \end { align } $ thus , the slope of the tangent line is $ 12 $ . recall from algebra that the point-slope form of the equation of the tangent line is $ \displaystyle y - y_0 = m_ { \tan } \cdot ( x - x_0 ) $ . the point-slope formula gives us the equation $ \displaystyle y - 8 = 12\cdot ( x - 2 ) $ which we can rewrite as $ \displaystyle y = 12x - 16 $ . finding the slope at any point next we are interested in finding a formula for the slope of the tangent line at any point on the graph of $ f $ . such a formula would be the same formula that we are using except we replace the constant $ x_0 $ by the variable $ x $ . this yields $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ . we denote this formula by $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ , where $ f\ , ' ( x ) $ is read `` $ f $ prime of $ x $ . '' the next example illustrates its usefulness . example 2 if $ f ( x ) = x^2 - 3 $ , find $ f\ , ' ( x ) $ and use the result to find the slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ . solution since $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ , then $ \begin { align } f\ , ' ( x ) & amp ; =\lim_ { h \to 0 } \frac { \left [ ( x + h ) ^2 - 3\right ] - \left [ x^2 - 3\right ] } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { x^2 + 2xh + h^2 - 3 - x^2 + 3 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { 2xh + h^2 } { h } \ \ & amp ; =\lim_ { h \to 0 } ( 2x + h ) \ \ & amp ; = 2x . \end { align } $ to find the slope , we substitute $ x = 2 $ and $ x = -1 $ into the result $ f\ , ' ( x ) $ , we get $ \displaystyle f\ , ' ( 2 ) = 2 ( 2 ) = 4 $ and $ \displaystyle f\ , ' ( -1 ) = 2 ( -1 ) = -2 $ . thus , slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ are $ 4 $ and $ -2 $ , respectively . example 3 find the slope of the tangent line to the curve $ y = 1/x $ at the point $ ( 1 , 1 ) $ . solution using the slope of the tangent line formula $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ and substituting $ y = 1/x $ gives us $ \begin { align } y\ , ' & amp ; =\lim_ { h \to 0 } \dfrac { \left ( \dfrac { 1 } { x+h } \right ) - \dfrac { 1 } { x } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { \dfrac { x - ( x + h ) } { x ( x + h ) } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { x - x - h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -1 } { x ( x + h ) } \ \ & amp ; = \frac { -1 } { x^2 } . \end { align } $ substituting $ x = 1 $ yields $ y\ , ' = \dfrac { -1 } { ( 1 ) ^2 } = -1 $ . thus , the slope of the tangent line at $ x = 1 $ for the curve $ y = 1/x $ is $ m = -1 $ . to find the equation of the tangent line , we use the point-slope formula , $ \displaystyle y - y_0 = m\cdot ( x - x_0 ) $ , where $ ( x_ { 0 } , y_ { 0 } ) = ( 1 , 1 ) $ . the equation of the tangent line is $ \begin { align } y - 1 & amp ; = -1 \cdot ( x - 1 ) \ y & amp ; = -x + 1 + 1\ y & amp ; = -x + 2 . \end { align } $ average speed the primary concept of differential is calculating the rate of change of one quantity with respect to another . for example , speed is defined as the rate of change of the distance traveled with respect to time . if a car travels $ 120 $ miles in $ 4 $ hours , his speed is $ \dfrac { 120\text { miles } } { 4\text { hours } } = 30 \text { mi/hr } $ . this speed is called the average speed or the average rate of change of distance with respect to time . of course , a car that travels $ 120 $ miles at an average rate of $ 30 $ miles per hour for $ 4 $ hours does not necessarily do so at constant speed . it may have slowed down or sped up during the $ 4 $ hour period . however , if the car hits a tree , it would not be its average speed that determines the resulting damage but its speed at the instant of the collision . so here we have two distinct kinds of speeds , average speed and instantaneous speed . the average speed of an object is defined as the object ’ s displacement $ \triangle x $ divided by the time interval $ \triangle t $ during which the displacement occurs : $ \displaystyle v = \frac { \triangle x } { \triangle t } = \frac { x_1 - x_0 } { t_1 - t_0 } $ . the average speed is also the expression for the slope of a secant line connecting the two points . figure 1 shows the $ \purple { \text { secant line } } $ through the points $ \pink { ( t_0 , x_0 ) } $ and $ \pink { ( t_1 , x_1 ) } $ on the $ \blue { \text { position-versus-time curve } } $ . thus we conclude that the average speed of an object between time $ t_0 $ and $ t_1 $ is represented geometrically by the slope of the secant line connecting the two points $ ( t_0 , x_0 ) $ and $ ( t_1 , x_1 ) $ . if we choose $ t_1 $ close to $ t_0 $ , then the average speed will closely approximate the instantaneous speed at time $ t_0 $ . rates of change the average rate of change of an arbitrary function $ f $ on an interval is represented geometrically by the slope of the secant line to the graph of $ f $ . the instantaneous rate of change of $ f $ at a particular point is represented by the slope of the tangent line to the graph of $ f $ at that point . let 's consider each case in more detail . average rate of change the average rate of change of the function $ f $ over the interval $ [ x_0 , x_1 ] $ is $ \displaystyle m_ { \sec } = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } $ . figure 2 shows the $ \purple { \text { secant line } } $ through the points $ \pink { ( x_0 , f ( x_0 ) ) } $ and $ \pink { ( x_1 , f ( x_1 ) ) } $ on the $ \blue { \text { graph of } \ , f } $ . the slope of the secant line is the average rate of change $ m_ { \sec } $ . instantaneous rate of change the instantaneous rate of change of the function $ f $ at the point $ x_0 $ is $ \displaystyle m_ { \tan } = f\ , ' ( x_0 ) = \lim_ { x_1 \to x_0 } \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } $ . figure 3 shows $ \purple { \text { tangent line } } $ through the point $ \pink { ( x_0 , f ( x_0 ) ) } $ on the $ \blue { \text { graph of } \ , f } $ . the slope of the tangent line is the instantaneous rate of change $ m_ { \tan } $ . example 4 suppose that $ y = x^2 - 3 $ . ( a ) find the average rate of change of $ y $ with respect to $ x $ over the interval $ [ 0 , 2 ] $ . ( b ) find the instantaneous rate of change of $ y $ with respect to $ x $ at the point $ x = -1 $ . solution ( a ) applying the formula for average rate of change with $ f ( x ) = x^2 - 3 $ and $ x_0 = 0 $ and $ x_1 = 2 $ yields $ \begin { align } m_ { \sec } & amp ; = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ \ & amp ; = \frac { f ( 2 ) - f ( 0 ) } { 2 - 0 } \ \ & amp ; = \frac { 1 - ( -3 ) } { 2 } \ \ & amp ; = 2 \end { align } $ this means the average rate of change over the interval $ [ 0 , 2 ] $ is 2 units of increase in $ y $ for each unit of increase in $ x $ . ( b ) from example 2 above , we found that $ f\ , ' ( x ) = 2x $ , so $ \begin { align } m_ { \tan } & amp ; = f\ , ' ( x_0 ) \ & amp ; = f\ , ' ( -1 ) \ & amp ; = 2 ( -1 ) \ & amp ; = -2 . \end { align } $ this means that the instantaneous rate of change is negative . that is , $ y $ is decreasing at $ x = -1 $ . it is decreasing at a rate of $ 2 $ units in $ y $ for each unit of increase in $ x $ . source ck-12 : tangent lines and rates of change license creative commons attribution-noncommercial
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solution using the slope of the tangent line formula $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ and substituting $ y = 1/x $ gives us $ \begin { align } y\ , ' & amp ; =\lim_ { h \to 0 } \dfrac { \left ( \dfrac { 1 } { x+h } \right ) - \dfrac { 1 } { x } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { \dfrac { x - ( x + h ) } { x ( x + h ) } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { x - x - h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -1 } { x ( x + h ) } \ \ & amp ; = \frac { -1 } { x^2 } . \end { align } $ substituting $ x = 1 $ yields $ y\ , ' = \dfrac { -1 } { ( 1 ) ^2 } = -1 $ . thus , the slope of the tangent line at $ x = 1 $ for the curve $ y = 1/x $ is $ m = -1 $ .
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in what condition we can use substitution to find the derivative like example 1 ?
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introduction the position of a car driving down the street , the value of currency adjusted for inflation , the number of bacteria in a culture , and the ac voltage of an electric signal are all examples of quantities that change with time . in this section , we will study the rate of change of a quantity and how is it related geometrically to secant and tangent lines . secant and tangent lines if two distinct points $ p ( x_0 , y_0 ) $ and $ q ( x_1 , y_1 ) $ lie on the curve $ y = f ( x ) $ , the slope of the secant line connecting the two points is $ \displaystyle m_ { \sec } = \frac { y_1 - y_0 } { x_1 - x_0 } = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ , $ . if we let the point $ x_1 $ approach $ x_0 $ , then $ q $ will approach $ p $ along the graph $ f $ . the slope of the secant line through points $ p $ and $ q $ will gradually approach the slope of the tangent line through $ p $ as $ x_1 $ approaches $ x_0 $ . in the limit , the previous equation becomes $ \displaystyle m_ { \tan } =\lim_ { x_1 \to x_0 } \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ , $ . if we let $ h = x_1 - x_0 $ , then $ x_1 = x_0 + h $ and $ h \rightarrow 0 $ as $ x_1 \rightarrow x_0 $ . we can rewrite the limit as $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ . when the limit exists , its value $ m_ { \tan } $ is the slope of the tangent line to the graph of $ f $ at the point $ p ( x_0 , y_0 ) $ . example 1 find the slope of the tangent line to the graph of the function $ f ( x ) = x^3 $ at the point $ ( 2 , 8 ) $ . solution since $ ( x_0 , y_0 ) = ( 2 , 8 ) $ , using the slope of the tangent line formula $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ we get $ \begin { align } m_ { \tan } & amp ; =\lim_ { h \to 0 } \frac { f ( 2 + h ) - f ( 2 ) } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { \left ( h^3 + 6h^2 + 12h + 8\right ) - 8 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { h^3 + 6h^2 + 12h } { h } \ \ & amp ; =\lim_ { h \to 0 } \left ( h^2 + 6h + 12\right ) \ \ & amp ; = 12 . \end { align } $ thus , the slope of the tangent line is $ 12 $ . recall from algebra that the point-slope form of the equation of the tangent line is $ \displaystyle y - y_0 = m_ { \tan } \cdot ( x - x_0 ) $ . the point-slope formula gives us the equation $ \displaystyle y - 8 = 12\cdot ( x - 2 ) $ which we can rewrite as $ \displaystyle y = 12x - 16 $ . finding the slope at any point next we are interested in finding a formula for the slope of the tangent line at any point on the graph of $ f $ . such a formula would be the same formula that we are using except we replace the constant $ x_0 $ by the variable $ x $ . this yields $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ . we denote this formula by $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ , where $ f\ , ' ( x ) $ is read `` $ f $ prime of $ x $ . '' the next example illustrates its usefulness . example 2 if $ f ( x ) = x^2 - 3 $ , find $ f\ , ' ( x ) $ and use the result to find the slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ . solution since $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ , then $ \begin { align } f\ , ' ( x ) & amp ; =\lim_ { h \to 0 } \frac { \left [ ( x + h ) ^2 - 3\right ] - \left [ x^2 - 3\right ] } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { x^2 + 2xh + h^2 - 3 - x^2 + 3 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { 2xh + h^2 } { h } \ \ & amp ; =\lim_ { h \to 0 } ( 2x + h ) \ \ & amp ; = 2x . \end { align } $ to find the slope , we substitute $ x = 2 $ and $ x = -1 $ into the result $ f\ , ' ( x ) $ , we get $ \displaystyle f\ , ' ( 2 ) = 2 ( 2 ) = 4 $ and $ \displaystyle f\ , ' ( -1 ) = 2 ( -1 ) = -2 $ . thus , slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ are $ 4 $ and $ -2 $ , respectively . example 3 find the slope of the tangent line to the curve $ y = 1/x $ at the point $ ( 1 , 1 ) $ . solution using the slope of the tangent line formula $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ and substituting $ y = 1/x $ gives us $ \begin { align } y\ , ' & amp ; =\lim_ { h \to 0 } \dfrac { \left ( \dfrac { 1 } { x+h } \right ) - \dfrac { 1 } { x } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { \dfrac { x - ( x + h ) } { x ( x + h ) } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { x - x - h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -1 } { x ( x + h ) } \ \ & amp ; = \frac { -1 } { x^2 } . \end { align } $ substituting $ x = 1 $ yields $ y\ , ' = \dfrac { -1 } { ( 1 ) ^2 } = -1 $ . thus , the slope of the tangent line at $ x = 1 $ for the curve $ y = 1/x $ is $ m = -1 $ . to find the equation of the tangent line , we use the point-slope formula , $ \displaystyle y - y_0 = m\cdot ( x - x_0 ) $ , where $ ( x_ { 0 } , y_ { 0 } ) = ( 1 , 1 ) $ . the equation of the tangent line is $ \begin { align } y - 1 & amp ; = -1 \cdot ( x - 1 ) \ y & amp ; = -x + 1 + 1\ y & amp ; = -x + 2 . \end { align } $ average speed the primary concept of differential is calculating the rate of change of one quantity with respect to another . for example , speed is defined as the rate of change of the distance traveled with respect to time . if a car travels $ 120 $ miles in $ 4 $ hours , his speed is $ \dfrac { 120\text { miles } } { 4\text { hours } } = 30 \text { mi/hr } $ . this speed is called the average speed or the average rate of change of distance with respect to time . of course , a car that travels $ 120 $ miles at an average rate of $ 30 $ miles per hour for $ 4 $ hours does not necessarily do so at constant speed . it may have slowed down or sped up during the $ 4 $ hour period . however , if the car hits a tree , it would not be its average speed that determines the resulting damage but its speed at the instant of the collision . so here we have two distinct kinds of speeds , average speed and instantaneous speed . the average speed of an object is defined as the object ’ s displacement $ \triangle x $ divided by the time interval $ \triangle t $ during which the displacement occurs : $ \displaystyle v = \frac { \triangle x } { \triangle t } = \frac { x_1 - x_0 } { t_1 - t_0 } $ . the average speed is also the expression for the slope of a secant line connecting the two points . figure 1 shows the $ \purple { \text { secant line } } $ through the points $ \pink { ( t_0 , x_0 ) } $ and $ \pink { ( t_1 , x_1 ) } $ on the $ \blue { \text { position-versus-time curve } } $ . thus we conclude that the average speed of an object between time $ t_0 $ and $ t_1 $ is represented geometrically by the slope of the secant line connecting the two points $ ( t_0 , x_0 ) $ and $ ( t_1 , x_1 ) $ . if we choose $ t_1 $ close to $ t_0 $ , then the average speed will closely approximate the instantaneous speed at time $ t_0 $ . rates of change the average rate of change of an arbitrary function $ f $ on an interval is represented geometrically by the slope of the secant line to the graph of $ f $ . the instantaneous rate of change of $ f $ at a particular point is represented by the slope of the tangent line to the graph of $ f $ at that point . let 's consider each case in more detail . average rate of change the average rate of change of the function $ f $ over the interval $ [ x_0 , x_1 ] $ is $ \displaystyle m_ { \sec } = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } $ . figure 2 shows the $ \purple { \text { secant line } } $ through the points $ \pink { ( x_0 , f ( x_0 ) ) } $ and $ \pink { ( x_1 , f ( x_1 ) ) } $ on the $ \blue { \text { graph of } \ , f } $ . the slope of the secant line is the average rate of change $ m_ { \sec } $ . instantaneous rate of change the instantaneous rate of change of the function $ f $ at the point $ x_0 $ is $ \displaystyle m_ { \tan } = f\ , ' ( x_0 ) = \lim_ { x_1 \to x_0 } \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } $ . figure 3 shows $ \purple { \text { tangent line } } $ through the point $ \pink { ( x_0 , f ( x_0 ) ) } $ on the $ \blue { \text { graph of } \ , f } $ . the slope of the tangent line is the instantaneous rate of change $ m_ { \tan } $ . example 4 suppose that $ y = x^2 - 3 $ . ( a ) find the average rate of change of $ y $ with respect to $ x $ over the interval $ [ 0 , 2 ] $ . ( b ) find the instantaneous rate of change of $ y $ with respect to $ x $ at the point $ x = -1 $ . solution ( a ) applying the formula for average rate of change with $ f ( x ) = x^2 - 3 $ and $ x_0 = 0 $ and $ x_1 = 2 $ yields $ \begin { align } m_ { \sec } & amp ; = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ \ & amp ; = \frac { f ( 2 ) - f ( 0 ) } { 2 - 0 } \ \ & amp ; = \frac { 1 - ( -3 ) } { 2 } \ \ & amp ; = 2 \end { align } $ this means the average rate of change over the interval $ [ 0 , 2 ] $ is 2 units of increase in $ y $ for each unit of increase in $ x $ . ( b ) from example 2 above , we found that $ f\ , ' ( x ) = 2x $ , so $ \begin { align } m_ { \tan } & amp ; = f\ , ' ( x_0 ) \ & amp ; = f\ , ' ( -1 ) \ & amp ; = 2 ( -1 ) \ & amp ; = -2 . \end { align } $ this means that the instantaneous rate of change is negative . that is , $ y $ is decreasing at $ x = -1 $ . it is decreasing at a rate of $ 2 $ units in $ y $ for each unit of increase in $ x $ . source ck-12 : tangent lines and rates of change license creative commons attribution-noncommercial
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the next example illustrates its usefulness . example 2 if $ f ( x ) = x^2 - 3 $ , find $ f\ , ' ( x ) $ and use the result to find the slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ . solution since $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ , then $ \begin { align } f\ , ' ( x ) & amp ; =\lim_ { h \to 0 } \frac { \left [ ( x + h ) ^2 - 3\right ] - \left [ x^2 - 3\right ] } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { x^2 + 2xh + h^2 - 3 - x^2 + 3 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { 2xh + h^2 } { h } \ \ & amp ; =\lim_ { h \to 0 } ( 2x + h ) \ \ & amp ; = 2x .
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what would happen if you defined the tangent line using a limit that drew the second y-value closer to the first , rather than drawing the second x-value closer to the first ?
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introduction the position of a car driving down the street , the value of currency adjusted for inflation , the number of bacteria in a culture , and the ac voltage of an electric signal are all examples of quantities that change with time . in this section , we will study the rate of change of a quantity and how is it related geometrically to secant and tangent lines . secant and tangent lines if two distinct points $ p ( x_0 , y_0 ) $ and $ q ( x_1 , y_1 ) $ lie on the curve $ y = f ( x ) $ , the slope of the secant line connecting the two points is $ \displaystyle m_ { \sec } = \frac { y_1 - y_0 } { x_1 - x_0 } = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ , $ . if we let the point $ x_1 $ approach $ x_0 $ , then $ q $ will approach $ p $ along the graph $ f $ . the slope of the secant line through points $ p $ and $ q $ will gradually approach the slope of the tangent line through $ p $ as $ x_1 $ approaches $ x_0 $ . in the limit , the previous equation becomes $ \displaystyle m_ { \tan } =\lim_ { x_1 \to x_0 } \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ , $ . if we let $ h = x_1 - x_0 $ , then $ x_1 = x_0 + h $ and $ h \rightarrow 0 $ as $ x_1 \rightarrow x_0 $ . we can rewrite the limit as $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ . when the limit exists , its value $ m_ { \tan } $ is the slope of the tangent line to the graph of $ f $ at the point $ p ( x_0 , y_0 ) $ . example 1 find the slope of the tangent line to the graph of the function $ f ( x ) = x^3 $ at the point $ ( 2 , 8 ) $ . solution since $ ( x_0 , y_0 ) = ( 2 , 8 ) $ , using the slope of the tangent line formula $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ we get $ \begin { align } m_ { \tan } & amp ; =\lim_ { h \to 0 } \frac { f ( 2 + h ) - f ( 2 ) } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { \left ( h^3 + 6h^2 + 12h + 8\right ) - 8 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { h^3 + 6h^2 + 12h } { h } \ \ & amp ; =\lim_ { h \to 0 } \left ( h^2 + 6h + 12\right ) \ \ & amp ; = 12 . \end { align } $ thus , the slope of the tangent line is $ 12 $ . recall from algebra that the point-slope form of the equation of the tangent line is $ \displaystyle y - y_0 = m_ { \tan } \cdot ( x - x_0 ) $ . the point-slope formula gives us the equation $ \displaystyle y - 8 = 12\cdot ( x - 2 ) $ which we can rewrite as $ \displaystyle y = 12x - 16 $ . finding the slope at any point next we are interested in finding a formula for the slope of the tangent line at any point on the graph of $ f $ . such a formula would be the same formula that we are using except we replace the constant $ x_0 $ by the variable $ x $ . this yields $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ . we denote this formula by $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ , where $ f\ , ' ( x ) $ is read `` $ f $ prime of $ x $ . '' the next example illustrates its usefulness . example 2 if $ f ( x ) = x^2 - 3 $ , find $ f\ , ' ( x ) $ and use the result to find the slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ . solution since $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ , then $ \begin { align } f\ , ' ( x ) & amp ; =\lim_ { h \to 0 } \frac { \left [ ( x + h ) ^2 - 3\right ] - \left [ x^2 - 3\right ] } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { x^2 + 2xh + h^2 - 3 - x^2 + 3 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { 2xh + h^2 } { h } \ \ & amp ; =\lim_ { h \to 0 } ( 2x + h ) \ \ & amp ; = 2x . \end { align } $ to find the slope , we substitute $ x = 2 $ and $ x = -1 $ into the result $ f\ , ' ( x ) $ , we get $ \displaystyle f\ , ' ( 2 ) = 2 ( 2 ) = 4 $ and $ \displaystyle f\ , ' ( -1 ) = 2 ( -1 ) = -2 $ . thus , slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ are $ 4 $ and $ -2 $ , respectively . example 3 find the slope of the tangent line to the curve $ y = 1/x $ at the point $ ( 1 , 1 ) $ . solution using the slope of the tangent line formula $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ and substituting $ y = 1/x $ gives us $ \begin { align } y\ , ' & amp ; =\lim_ { h \to 0 } \dfrac { \left ( \dfrac { 1 } { x+h } \right ) - \dfrac { 1 } { x } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { \dfrac { x - ( x + h ) } { x ( x + h ) } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { x - x - h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -1 } { x ( x + h ) } \ \ & amp ; = \frac { -1 } { x^2 } . \end { align } $ substituting $ x = 1 $ yields $ y\ , ' = \dfrac { -1 } { ( 1 ) ^2 } = -1 $ . thus , the slope of the tangent line at $ x = 1 $ for the curve $ y = 1/x $ is $ m = -1 $ . to find the equation of the tangent line , we use the point-slope formula , $ \displaystyle y - y_0 = m\cdot ( x - x_0 ) $ , where $ ( x_ { 0 } , y_ { 0 } ) = ( 1 , 1 ) $ . the equation of the tangent line is $ \begin { align } y - 1 & amp ; = -1 \cdot ( x - 1 ) \ y & amp ; = -x + 1 + 1\ y & amp ; = -x + 2 . \end { align } $ average speed the primary concept of differential is calculating the rate of change of one quantity with respect to another . for example , speed is defined as the rate of change of the distance traveled with respect to time . if a car travels $ 120 $ miles in $ 4 $ hours , his speed is $ \dfrac { 120\text { miles } } { 4\text { hours } } = 30 \text { mi/hr } $ . this speed is called the average speed or the average rate of change of distance with respect to time . of course , a car that travels $ 120 $ miles at an average rate of $ 30 $ miles per hour for $ 4 $ hours does not necessarily do so at constant speed . it may have slowed down or sped up during the $ 4 $ hour period . however , if the car hits a tree , it would not be its average speed that determines the resulting damage but its speed at the instant of the collision . so here we have two distinct kinds of speeds , average speed and instantaneous speed . the average speed of an object is defined as the object ’ s displacement $ \triangle x $ divided by the time interval $ \triangle t $ during which the displacement occurs : $ \displaystyle v = \frac { \triangle x } { \triangle t } = \frac { x_1 - x_0 } { t_1 - t_0 } $ . the average speed is also the expression for the slope of a secant line connecting the two points . figure 1 shows the $ \purple { \text { secant line } } $ through the points $ \pink { ( t_0 , x_0 ) } $ and $ \pink { ( t_1 , x_1 ) } $ on the $ \blue { \text { position-versus-time curve } } $ . thus we conclude that the average speed of an object between time $ t_0 $ and $ t_1 $ is represented geometrically by the slope of the secant line connecting the two points $ ( t_0 , x_0 ) $ and $ ( t_1 , x_1 ) $ . if we choose $ t_1 $ close to $ t_0 $ , then the average speed will closely approximate the instantaneous speed at time $ t_0 $ . rates of change the average rate of change of an arbitrary function $ f $ on an interval is represented geometrically by the slope of the secant line to the graph of $ f $ . the instantaneous rate of change of $ f $ at a particular point is represented by the slope of the tangent line to the graph of $ f $ at that point . let 's consider each case in more detail . average rate of change the average rate of change of the function $ f $ over the interval $ [ x_0 , x_1 ] $ is $ \displaystyle m_ { \sec } = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } $ . figure 2 shows the $ \purple { \text { secant line } } $ through the points $ \pink { ( x_0 , f ( x_0 ) ) } $ and $ \pink { ( x_1 , f ( x_1 ) ) } $ on the $ \blue { \text { graph of } \ , f } $ . the slope of the secant line is the average rate of change $ m_ { \sec } $ . instantaneous rate of change the instantaneous rate of change of the function $ f $ at the point $ x_0 $ is $ \displaystyle m_ { \tan } = f\ , ' ( x_0 ) = \lim_ { x_1 \to x_0 } \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } $ . figure 3 shows $ \purple { \text { tangent line } } $ through the point $ \pink { ( x_0 , f ( x_0 ) ) } $ on the $ \blue { \text { graph of } \ , f } $ . the slope of the tangent line is the instantaneous rate of change $ m_ { \tan } $ . example 4 suppose that $ y = x^2 - 3 $ . ( a ) find the average rate of change of $ y $ with respect to $ x $ over the interval $ [ 0 , 2 ] $ . ( b ) find the instantaneous rate of change of $ y $ with respect to $ x $ at the point $ x = -1 $ . solution ( a ) applying the formula for average rate of change with $ f ( x ) = x^2 - 3 $ and $ x_0 = 0 $ and $ x_1 = 2 $ yields $ \begin { align } m_ { \sec } & amp ; = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ \ & amp ; = \frac { f ( 2 ) - f ( 0 ) } { 2 - 0 } \ \ & amp ; = \frac { 1 - ( -3 ) } { 2 } \ \ & amp ; = 2 \end { align } $ this means the average rate of change over the interval $ [ 0 , 2 ] $ is 2 units of increase in $ y $ for each unit of increase in $ x $ . ( b ) from example 2 above , we found that $ f\ , ' ( x ) = 2x $ , so $ \begin { align } m_ { \tan } & amp ; = f\ , ' ( x_0 ) \ & amp ; = f\ , ' ( -1 ) \ & amp ; = 2 ( -1 ) \ & amp ; = -2 . \end { align } $ this means that the instantaneous rate of change is negative . that is , $ y $ is decreasing at $ x = -1 $ . it is decreasing at a rate of $ 2 $ units in $ y $ for each unit of increase in $ x $ . source ck-12 : tangent lines and rates of change license creative commons attribution-noncommercial
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the next example illustrates its usefulness . example 2 if $ f ( x ) = x^2 - 3 $ , find $ f\ , ' ( x ) $ and use the result to find the slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ . solution since $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ , then $ \begin { align } f\ , ' ( x ) & amp ; =\lim_ { h \to 0 } \frac { \left [ ( x + h ) ^2 - 3\right ] - \left [ x^2 - 3\right ] } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { x^2 + 2xh + h^2 - 3 - x^2 + 3 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { 2xh + h^2 } { h } \ \ & amp ; =\lim_ { h \to 0 } ( 2x + h ) \ \ & amp ; = 2x .
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is there a way that the x-a form can be used to find the derivative at any point ?
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introduction the position of a car driving down the street , the value of currency adjusted for inflation , the number of bacteria in a culture , and the ac voltage of an electric signal are all examples of quantities that change with time . in this section , we will study the rate of change of a quantity and how is it related geometrically to secant and tangent lines . secant and tangent lines if two distinct points $ p ( x_0 , y_0 ) $ and $ q ( x_1 , y_1 ) $ lie on the curve $ y = f ( x ) $ , the slope of the secant line connecting the two points is $ \displaystyle m_ { \sec } = \frac { y_1 - y_0 } { x_1 - x_0 } = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ , $ . if we let the point $ x_1 $ approach $ x_0 $ , then $ q $ will approach $ p $ along the graph $ f $ . the slope of the secant line through points $ p $ and $ q $ will gradually approach the slope of the tangent line through $ p $ as $ x_1 $ approaches $ x_0 $ . in the limit , the previous equation becomes $ \displaystyle m_ { \tan } =\lim_ { x_1 \to x_0 } \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ , $ . if we let $ h = x_1 - x_0 $ , then $ x_1 = x_0 + h $ and $ h \rightarrow 0 $ as $ x_1 \rightarrow x_0 $ . we can rewrite the limit as $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ . when the limit exists , its value $ m_ { \tan } $ is the slope of the tangent line to the graph of $ f $ at the point $ p ( x_0 , y_0 ) $ . example 1 find the slope of the tangent line to the graph of the function $ f ( x ) = x^3 $ at the point $ ( 2 , 8 ) $ . solution since $ ( x_0 , y_0 ) = ( 2 , 8 ) $ , using the slope of the tangent line formula $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ we get $ \begin { align } m_ { \tan } & amp ; =\lim_ { h \to 0 } \frac { f ( 2 + h ) - f ( 2 ) } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { \left ( h^3 + 6h^2 + 12h + 8\right ) - 8 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { h^3 + 6h^2 + 12h } { h } \ \ & amp ; =\lim_ { h \to 0 } \left ( h^2 + 6h + 12\right ) \ \ & amp ; = 12 . \end { align } $ thus , the slope of the tangent line is $ 12 $ . recall from algebra that the point-slope form of the equation of the tangent line is $ \displaystyle y - y_0 = m_ { \tan } \cdot ( x - x_0 ) $ . the point-slope formula gives us the equation $ \displaystyle y - 8 = 12\cdot ( x - 2 ) $ which we can rewrite as $ \displaystyle y = 12x - 16 $ . finding the slope at any point next we are interested in finding a formula for the slope of the tangent line at any point on the graph of $ f $ . such a formula would be the same formula that we are using except we replace the constant $ x_0 $ by the variable $ x $ . this yields $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ . we denote this formula by $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ , where $ f\ , ' ( x ) $ is read `` $ f $ prime of $ x $ . '' the next example illustrates its usefulness . example 2 if $ f ( x ) = x^2 - 3 $ , find $ f\ , ' ( x ) $ and use the result to find the slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ . solution since $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ , then $ \begin { align } f\ , ' ( x ) & amp ; =\lim_ { h \to 0 } \frac { \left [ ( x + h ) ^2 - 3\right ] - \left [ x^2 - 3\right ] } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { x^2 + 2xh + h^2 - 3 - x^2 + 3 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { 2xh + h^2 } { h } \ \ & amp ; =\lim_ { h \to 0 } ( 2x + h ) \ \ & amp ; = 2x . \end { align } $ to find the slope , we substitute $ x = 2 $ and $ x = -1 $ into the result $ f\ , ' ( x ) $ , we get $ \displaystyle f\ , ' ( 2 ) = 2 ( 2 ) = 4 $ and $ \displaystyle f\ , ' ( -1 ) = 2 ( -1 ) = -2 $ . thus , slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ are $ 4 $ and $ -2 $ , respectively . example 3 find the slope of the tangent line to the curve $ y = 1/x $ at the point $ ( 1 , 1 ) $ . solution using the slope of the tangent line formula $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ and substituting $ y = 1/x $ gives us $ \begin { align } y\ , ' & amp ; =\lim_ { h \to 0 } \dfrac { \left ( \dfrac { 1 } { x+h } \right ) - \dfrac { 1 } { x } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { \dfrac { x - ( x + h ) } { x ( x + h ) } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { x - x - h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -1 } { x ( x + h ) } \ \ & amp ; = \frac { -1 } { x^2 } . \end { align } $ substituting $ x = 1 $ yields $ y\ , ' = \dfrac { -1 } { ( 1 ) ^2 } = -1 $ . thus , the slope of the tangent line at $ x = 1 $ for the curve $ y = 1/x $ is $ m = -1 $ . to find the equation of the tangent line , we use the point-slope formula , $ \displaystyle y - y_0 = m\cdot ( x - x_0 ) $ , where $ ( x_ { 0 } , y_ { 0 } ) = ( 1 , 1 ) $ . the equation of the tangent line is $ \begin { align } y - 1 & amp ; = -1 \cdot ( x - 1 ) \ y & amp ; = -x + 1 + 1\ y & amp ; = -x + 2 . \end { align } $ average speed the primary concept of differential is calculating the rate of change of one quantity with respect to another . for example , speed is defined as the rate of change of the distance traveled with respect to time . if a car travels $ 120 $ miles in $ 4 $ hours , his speed is $ \dfrac { 120\text { miles } } { 4\text { hours } } = 30 \text { mi/hr } $ . this speed is called the average speed or the average rate of change of distance with respect to time . of course , a car that travels $ 120 $ miles at an average rate of $ 30 $ miles per hour for $ 4 $ hours does not necessarily do so at constant speed . it may have slowed down or sped up during the $ 4 $ hour period . however , if the car hits a tree , it would not be its average speed that determines the resulting damage but its speed at the instant of the collision . so here we have two distinct kinds of speeds , average speed and instantaneous speed . the average speed of an object is defined as the object ’ s displacement $ \triangle x $ divided by the time interval $ \triangle t $ during which the displacement occurs : $ \displaystyle v = \frac { \triangle x } { \triangle t } = \frac { x_1 - x_0 } { t_1 - t_0 } $ . the average speed is also the expression for the slope of a secant line connecting the two points . figure 1 shows the $ \purple { \text { secant line } } $ through the points $ \pink { ( t_0 , x_0 ) } $ and $ \pink { ( t_1 , x_1 ) } $ on the $ \blue { \text { position-versus-time curve } } $ . thus we conclude that the average speed of an object between time $ t_0 $ and $ t_1 $ is represented geometrically by the slope of the secant line connecting the two points $ ( t_0 , x_0 ) $ and $ ( t_1 , x_1 ) $ . if we choose $ t_1 $ close to $ t_0 $ , then the average speed will closely approximate the instantaneous speed at time $ t_0 $ . rates of change the average rate of change of an arbitrary function $ f $ on an interval is represented geometrically by the slope of the secant line to the graph of $ f $ . the instantaneous rate of change of $ f $ at a particular point is represented by the slope of the tangent line to the graph of $ f $ at that point . let 's consider each case in more detail . average rate of change the average rate of change of the function $ f $ over the interval $ [ x_0 , x_1 ] $ is $ \displaystyle m_ { \sec } = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } $ . figure 2 shows the $ \purple { \text { secant line } } $ through the points $ \pink { ( x_0 , f ( x_0 ) ) } $ and $ \pink { ( x_1 , f ( x_1 ) ) } $ on the $ \blue { \text { graph of } \ , f } $ . the slope of the secant line is the average rate of change $ m_ { \sec } $ . instantaneous rate of change the instantaneous rate of change of the function $ f $ at the point $ x_0 $ is $ \displaystyle m_ { \tan } = f\ , ' ( x_0 ) = \lim_ { x_1 \to x_0 } \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } $ . figure 3 shows $ \purple { \text { tangent line } } $ through the point $ \pink { ( x_0 , f ( x_0 ) ) } $ on the $ \blue { \text { graph of } \ , f } $ . the slope of the tangent line is the instantaneous rate of change $ m_ { \tan } $ . example 4 suppose that $ y = x^2 - 3 $ . ( a ) find the average rate of change of $ y $ with respect to $ x $ over the interval $ [ 0 , 2 ] $ . ( b ) find the instantaneous rate of change of $ y $ with respect to $ x $ at the point $ x = -1 $ . solution ( a ) applying the formula for average rate of change with $ f ( x ) = x^2 - 3 $ and $ x_0 = 0 $ and $ x_1 = 2 $ yields $ \begin { align } m_ { \sec } & amp ; = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ \ & amp ; = \frac { f ( 2 ) - f ( 0 ) } { 2 - 0 } \ \ & amp ; = \frac { 1 - ( -3 ) } { 2 } \ \ & amp ; = 2 \end { align } $ this means the average rate of change over the interval $ [ 0 , 2 ] $ is 2 units of increase in $ y $ for each unit of increase in $ x $ . ( b ) from example 2 above , we found that $ f\ , ' ( x ) = 2x $ , so $ \begin { align } m_ { \tan } & amp ; = f\ , ' ( x_0 ) \ & amp ; = f\ , ' ( -1 ) \ & amp ; = 2 ( -1 ) \ & amp ; = -2 . \end { align } $ this means that the instantaneous rate of change is negative . that is , $ y $ is decreasing at $ x = -1 $ . it is decreasing at a rate of $ 2 $ units in $ y $ for each unit of increase in $ x $ . source ck-12 : tangent lines and rates of change license creative commons attribution-noncommercial
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example 1 find the slope of the tangent line to the graph of the function $ f ( x ) = x^3 $ at the point $ ( 2 , 8 ) $ . solution since $ ( x_0 , y_0 ) = ( 2 , 8 ) $ , using the slope of the tangent line formula $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ we get $ \begin { align } m_ { \tan } & amp ; =\lim_ { h \to 0 } \frac { f ( 2 + h ) - f ( 2 ) } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { \left ( h^3 + 6h^2 + 12h + 8\right ) - 8 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { h^3 + 6h^2 + 12h } { h } \ \ & amp ; =\lim_ { h \to 0 } \left ( h^2 + 6h + 12\right ) \ \ & amp ; = 12 . \end { align } $ thus , the slope of the tangent line is $ 12 $ .
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what does the h in the formula represents ?
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introduction the position of a car driving down the street , the value of currency adjusted for inflation , the number of bacteria in a culture , and the ac voltage of an electric signal are all examples of quantities that change with time . in this section , we will study the rate of change of a quantity and how is it related geometrically to secant and tangent lines . secant and tangent lines if two distinct points $ p ( x_0 , y_0 ) $ and $ q ( x_1 , y_1 ) $ lie on the curve $ y = f ( x ) $ , the slope of the secant line connecting the two points is $ \displaystyle m_ { \sec } = \frac { y_1 - y_0 } { x_1 - x_0 } = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ , $ . if we let the point $ x_1 $ approach $ x_0 $ , then $ q $ will approach $ p $ along the graph $ f $ . the slope of the secant line through points $ p $ and $ q $ will gradually approach the slope of the tangent line through $ p $ as $ x_1 $ approaches $ x_0 $ . in the limit , the previous equation becomes $ \displaystyle m_ { \tan } =\lim_ { x_1 \to x_0 } \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ , $ . if we let $ h = x_1 - x_0 $ , then $ x_1 = x_0 + h $ and $ h \rightarrow 0 $ as $ x_1 \rightarrow x_0 $ . we can rewrite the limit as $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ . when the limit exists , its value $ m_ { \tan } $ is the slope of the tangent line to the graph of $ f $ at the point $ p ( x_0 , y_0 ) $ . example 1 find the slope of the tangent line to the graph of the function $ f ( x ) = x^3 $ at the point $ ( 2 , 8 ) $ . solution since $ ( x_0 , y_0 ) = ( 2 , 8 ) $ , using the slope of the tangent line formula $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ we get $ \begin { align } m_ { \tan } & amp ; =\lim_ { h \to 0 } \frac { f ( 2 + h ) - f ( 2 ) } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { \left ( h^3 + 6h^2 + 12h + 8\right ) - 8 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { h^3 + 6h^2 + 12h } { h } \ \ & amp ; =\lim_ { h \to 0 } \left ( h^2 + 6h + 12\right ) \ \ & amp ; = 12 . \end { align } $ thus , the slope of the tangent line is $ 12 $ . recall from algebra that the point-slope form of the equation of the tangent line is $ \displaystyle y - y_0 = m_ { \tan } \cdot ( x - x_0 ) $ . the point-slope formula gives us the equation $ \displaystyle y - 8 = 12\cdot ( x - 2 ) $ which we can rewrite as $ \displaystyle y = 12x - 16 $ . finding the slope at any point next we are interested in finding a formula for the slope of the tangent line at any point on the graph of $ f $ . such a formula would be the same formula that we are using except we replace the constant $ x_0 $ by the variable $ x $ . this yields $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ . we denote this formula by $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ , where $ f\ , ' ( x ) $ is read `` $ f $ prime of $ x $ . '' the next example illustrates its usefulness . example 2 if $ f ( x ) = x^2 - 3 $ , find $ f\ , ' ( x ) $ and use the result to find the slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ . solution since $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ , then $ \begin { align } f\ , ' ( x ) & amp ; =\lim_ { h \to 0 } \frac { \left [ ( x + h ) ^2 - 3\right ] - \left [ x^2 - 3\right ] } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { x^2 + 2xh + h^2 - 3 - x^2 + 3 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { 2xh + h^2 } { h } \ \ & amp ; =\lim_ { h \to 0 } ( 2x + h ) \ \ & amp ; = 2x . \end { align } $ to find the slope , we substitute $ x = 2 $ and $ x = -1 $ into the result $ f\ , ' ( x ) $ , we get $ \displaystyle f\ , ' ( 2 ) = 2 ( 2 ) = 4 $ and $ \displaystyle f\ , ' ( -1 ) = 2 ( -1 ) = -2 $ . thus , slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ are $ 4 $ and $ -2 $ , respectively . example 3 find the slope of the tangent line to the curve $ y = 1/x $ at the point $ ( 1 , 1 ) $ . solution using the slope of the tangent line formula $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ and substituting $ y = 1/x $ gives us $ \begin { align } y\ , ' & amp ; =\lim_ { h \to 0 } \dfrac { \left ( \dfrac { 1 } { x+h } \right ) - \dfrac { 1 } { x } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { \dfrac { x - ( x + h ) } { x ( x + h ) } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { x - x - h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -1 } { x ( x + h ) } \ \ & amp ; = \frac { -1 } { x^2 } . \end { align } $ substituting $ x = 1 $ yields $ y\ , ' = \dfrac { -1 } { ( 1 ) ^2 } = -1 $ . thus , the slope of the tangent line at $ x = 1 $ for the curve $ y = 1/x $ is $ m = -1 $ . to find the equation of the tangent line , we use the point-slope formula , $ \displaystyle y - y_0 = m\cdot ( x - x_0 ) $ , where $ ( x_ { 0 } , y_ { 0 } ) = ( 1 , 1 ) $ . the equation of the tangent line is $ \begin { align } y - 1 & amp ; = -1 \cdot ( x - 1 ) \ y & amp ; = -x + 1 + 1\ y & amp ; = -x + 2 . \end { align } $ average speed the primary concept of differential is calculating the rate of change of one quantity with respect to another . for example , speed is defined as the rate of change of the distance traveled with respect to time . if a car travels $ 120 $ miles in $ 4 $ hours , his speed is $ \dfrac { 120\text { miles } } { 4\text { hours } } = 30 \text { mi/hr } $ . this speed is called the average speed or the average rate of change of distance with respect to time . of course , a car that travels $ 120 $ miles at an average rate of $ 30 $ miles per hour for $ 4 $ hours does not necessarily do so at constant speed . it may have slowed down or sped up during the $ 4 $ hour period . however , if the car hits a tree , it would not be its average speed that determines the resulting damage but its speed at the instant of the collision . so here we have two distinct kinds of speeds , average speed and instantaneous speed . the average speed of an object is defined as the object ’ s displacement $ \triangle x $ divided by the time interval $ \triangle t $ during which the displacement occurs : $ \displaystyle v = \frac { \triangle x } { \triangle t } = \frac { x_1 - x_0 } { t_1 - t_0 } $ . the average speed is also the expression for the slope of a secant line connecting the two points . figure 1 shows the $ \purple { \text { secant line } } $ through the points $ \pink { ( t_0 , x_0 ) } $ and $ \pink { ( t_1 , x_1 ) } $ on the $ \blue { \text { position-versus-time curve } } $ . thus we conclude that the average speed of an object between time $ t_0 $ and $ t_1 $ is represented geometrically by the slope of the secant line connecting the two points $ ( t_0 , x_0 ) $ and $ ( t_1 , x_1 ) $ . if we choose $ t_1 $ close to $ t_0 $ , then the average speed will closely approximate the instantaneous speed at time $ t_0 $ . rates of change the average rate of change of an arbitrary function $ f $ on an interval is represented geometrically by the slope of the secant line to the graph of $ f $ . the instantaneous rate of change of $ f $ at a particular point is represented by the slope of the tangent line to the graph of $ f $ at that point . let 's consider each case in more detail . average rate of change the average rate of change of the function $ f $ over the interval $ [ x_0 , x_1 ] $ is $ \displaystyle m_ { \sec } = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } $ . figure 2 shows the $ \purple { \text { secant line } } $ through the points $ \pink { ( x_0 , f ( x_0 ) ) } $ and $ \pink { ( x_1 , f ( x_1 ) ) } $ on the $ \blue { \text { graph of } \ , f } $ . the slope of the secant line is the average rate of change $ m_ { \sec } $ . instantaneous rate of change the instantaneous rate of change of the function $ f $ at the point $ x_0 $ is $ \displaystyle m_ { \tan } = f\ , ' ( x_0 ) = \lim_ { x_1 \to x_0 } \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } $ . figure 3 shows $ \purple { \text { tangent line } } $ through the point $ \pink { ( x_0 , f ( x_0 ) ) } $ on the $ \blue { \text { graph of } \ , f } $ . the slope of the tangent line is the instantaneous rate of change $ m_ { \tan } $ . example 4 suppose that $ y = x^2 - 3 $ . ( a ) find the average rate of change of $ y $ with respect to $ x $ over the interval $ [ 0 , 2 ] $ . ( b ) find the instantaneous rate of change of $ y $ with respect to $ x $ at the point $ x = -1 $ . solution ( a ) applying the formula for average rate of change with $ f ( x ) = x^2 - 3 $ and $ x_0 = 0 $ and $ x_1 = 2 $ yields $ \begin { align } m_ { \sec } & amp ; = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ \ & amp ; = \frac { f ( 2 ) - f ( 0 ) } { 2 - 0 } \ \ & amp ; = \frac { 1 - ( -3 ) } { 2 } \ \ & amp ; = 2 \end { align } $ this means the average rate of change over the interval $ [ 0 , 2 ] $ is 2 units of increase in $ y $ for each unit of increase in $ x $ . ( b ) from example 2 above , we found that $ f\ , ' ( x ) = 2x $ , so $ \begin { align } m_ { \tan } & amp ; = f\ , ' ( x_0 ) \ & amp ; = f\ , ' ( -1 ) \ & amp ; = 2 ( -1 ) \ & amp ; = -2 . \end { align } $ this means that the instantaneous rate of change is negative . that is , $ y $ is decreasing at $ x = -1 $ . it is decreasing at a rate of $ 2 $ units in $ y $ for each unit of increase in $ x $ . source ck-12 : tangent lines and rates of change license creative commons attribution-noncommercial
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( b ) from example 2 above , we found that $ f\ , ' ( x ) = 2x $ , so $ \begin { align } m_ { \tan } & amp ; = f\ , ' ( x_0 ) \ & amp ; = f\ , ' ( -1 ) \ & amp ; = 2 ( -1 ) \ & amp ; = -2 . \end { align } $ this means that the instantaneous rate of change is negative . that is , $ y $ is decreasing at $ x = -1 $ .
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why does the negative change ?
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introduction the position of a car driving down the street , the value of currency adjusted for inflation , the number of bacteria in a culture , and the ac voltage of an electric signal are all examples of quantities that change with time . in this section , we will study the rate of change of a quantity and how is it related geometrically to secant and tangent lines . secant and tangent lines if two distinct points $ p ( x_0 , y_0 ) $ and $ q ( x_1 , y_1 ) $ lie on the curve $ y = f ( x ) $ , the slope of the secant line connecting the two points is $ \displaystyle m_ { \sec } = \frac { y_1 - y_0 } { x_1 - x_0 } = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ , $ . if we let the point $ x_1 $ approach $ x_0 $ , then $ q $ will approach $ p $ along the graph $ f $ . the slope of the secant line through points $ p $ and $ q $ will gradually approach the slope of the tangent line through $ p $ as $ x_1 $ approaches $ x_0 $ . in the limit , the previous equation becomes $ \displaystyle m_ { \tan } =\lim_ { x_1 \to x_0 } \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ , $ . if we let $ h = x_1 - x_0 $ , then $ x_1 = x_0 + h $ and $ h \rightarrow 0 $ as $ x_1 \rightarrow x_0 $ . we can rewrite the limit as $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ . when the limit exists , its value $ m_ { \tan } $ is the slope of the tangent line to the graph of $ f $ at the point $ p ( x_0 , y_0 ) $ . example 1 find the slope of the tangent line to the graph of the function $ f ( x ) = x^3 $ at the point $ ( 2 , 8 ) $ . solution since $ ( x_0 , y_0 ) = ( 2 , 8 ) $ , using the slope of the tangent line formula $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ we get $ \begin { align } m_ { \tan } & amp ; =\lim_ { h \to 0 } \frac { f ( 2 + h ) - f ( 2 ) } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { \left ( h^3 + 6h^2 + 12h + 8\right ) - 8 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { h^3 + 6h^2 + 12h } { h } \ \ & amp ; =\lim_ { h \to 0 } \left ( h^2 + 6h + 12\right ) \ \ & amp ; = 12 . \end { align } $ thus , the slope of the tangent line is $ 12 $ . recall from algebra that the point-slope form of the equation of the tangent line is $ \displaystyle y - y_0 = m_ { \tan } \cdot ( x - x_0 ) $ . the point-slope formula gives us the equation $ \displaystyle y - 8 = 12\cdot ( x - 2 ) $ which we can rewrite as $ \displaystyle y = 12x - 16 $ . finding the slope at any point next we are interested in finding a formula for the slope of the tangent line at any point on the graph of $ f $ . such a formula would be the same formula that we are using except we replace the constant $ x_0 $ by the variable $ x $ . this yields $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ . we denote this formula by $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ , where $ f\ , ' ( x ) $ is read `` $ f $ prime of $ x $ . '' the next example illustrates its usefulness . example 2 if $ f ( x ) = x^2 - 3 $ , find $ f\ , ' ( x ) $ and use the result to find the slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ . solution since $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ , then $ \begin { align } f\ , ' ( x ) & amp ; =\lim_ { h \to 0 } \frac { \left [ ( x + h ) ^2 - 3\right ] - \left [ x^2 - 3\right ] } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { x^2 + 2xh + h^2 - 3 - x^2 + 3 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { 2xh + h^2 } { h } \ \ & amp ; =\lim_ { h \to 0 } ( 2x + h ) \ \ & amp ; = 2x . \end { align } $ to find the slope , we substitute $ x = 2 $ and $ x = -1 $ into the result $ f\ , ' ( x ) $ , we get $ \displaystyle f\ , ' ( 2 ) = 2 ( 2 ) = 4 $ and $ \displaystyle f\ , ' ( -1 ) = 2 ( -1 ) = -2 $ . thus , slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ are $ 4 $ and $ -2 $ , respectively . example 3 find the slope of the tangent line to the curve $ y = 1/x $ at the point $ ( 1 , 1 ) $ . solution using the slope of the tangent line formula $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ and substituting $ y = 1/x $ gives us $ \begin { align } y\ , ' & amp ; =\lim_ { h \to 0 } \dfrac { \left ( \dfrac { 1 } { x+h } \right ) - \dfrac { 1 } { x } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { \dfrac { x - ( x + h ) } { x ( x + h ) } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { x - x - h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -1 } { x ( x + h ) } \ \ & amp ; = \frac { -1 } { x^2 } . \end { align } $ substituting $ x = 1 $ yields $ y\ , ' = \dfrac { -1 } { ( 1 ) ^2 } = -1 $ . thus , the slope of the tangent line at $ x = 1 $ for the curve $ y = 1/x $ is $ m = -1 $ . to find the equation of the tangent line , we use the point-slope formula , $ \displaystyle y - y_0 = m\cdot ( x - x_0 ) $ , where $ ( x_ { 0 } , y_ { 0 } ) = ( 1 , 1 ) $ . the equation of the tangent line is $ \begin { align } y - 1 & amp ; = -1 \cdot ( x - 1 ) \ y & amp ; = -x + 1 + 1\ y & amp ; = -x + 2 . \end { align } $ average speed the primary concept of differential is calculating the rate of change of one quantity with respect to another . for example , speed is defined as the rate of change of the distance traveled with respect to time . if a car travels $ 120 $ miles in $ 4 $ hours , his speed is $ \dfrac { 120\text { miles } } { 4\text { hours } } = 30 \text { mi/hr } $ . this speed is called the average speed or the average rate of change of distance with respect to time . of course , a car that travels $ 120 $ miles at an average rate of $ 30 $ miles per hour for $ 4 $ hours does not necessarily do so at constant speed . it may have slowed down or sped up during the $ 4 $ hour period . however , if the car hits a tree , it would not be its average speed that determines the resulting damage but its speed at the instant of the collision . so here we have two distinct kinds of speeds , average speed and instantaneous speed . the average speed of an object is defined as the object ’ s displacement $ \triangle x $ divided by the time interval $ \triangle t $ during which the displacement occurs : $ \displaystyle v = \frac { \triangle x } { \triangle t } = \frac { x_1 - x_0 } { t_1 - t_0 } $ . the average speed is also the expression for the slope of a secant line connecting the two points . figure 1 shows the $ \purple { \text { secant line } } $ through the points $ \pink { ( t_0 , x_0 ) } $ and $ \pink { ( t_1 , x_1 ) } $ on the $ \blue { \text { position-versus-time curve } } $ . thus we conclude that the average speed of an object between time $ t_0 $ and $ t_1 $ is represented geometrically by the slope of the secant line connecting the two points $ ( t_0 , x_0 ) $ and $ ( t_1 , x_1 ) $ . if we choose $ t_1 $ close to $ t_0 $ , then the average speed will closely approximate the instantaneous speed at time $ t_0 $ . rates of change the average rate of change of an arbitrary function $ f $ on an interval is represented geometrically by the slope of the secant line to the graph of $ f $ . the instantaneous rate of change of $ f $ at a particular point is represented by the slope of the tangent line to the graph of $ f $ at that point . let 's consider each case in more detail . average rate of change the average rate of change of the function $ f $ over the interval $ [ x_0 , x_1 ] $ is $ \displaystyle m_ { \sec } = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } $ . figure 2 shows the $ \purple { \text { secant line } } $ through the points $ \pink { ( x_0 , f ( x_0 ) ) } $ and $ \pink { ( x_1 , f ( x_1 ) ) } $ on the $ \blue { \text { graph of } \ , f } $ . the slope of the secant line is the average rate of change $ m_ { \sec } $ . instantaneous rate of change the instantaneous rate of change of the function $ f $ at the point $ x_0 $ is $ \displaystyle m_ { \tan } = f\ , ' ( x_0 ) = \lim_ { x_1 \to x_0 } \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } $ . figure 3 shows $ \purple { \text { tangent line } } $ through the point $ \pink { ( x_0 , f ( x_0 ) ) } $ on the $ \blue { \text { graph of } \ , f } $ . the slope of the tangent line is the instantaneous rate of change $ m_ { \tan } $ . example 4 suppose that $ y = x^2 - 3 $ . ( a ) find the average rate of change of $ y $ with respect to $ x $ over the interval $ [ 0 , 2 ] $ . ( b ) find the instantaneous rate of change of $ y $ with respect to $ x $ at the point $ x = -1 $ . solution ( a ) applying the formula for average rate of change with $ f ( x ) = x^2 - 3 $ and $ x_0 = 0 $ and $ x_1 = 2 $ yields $ \begin { align } m_ { \sec } & amp ; = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ \ & amp ; = \frac { f ( 2 ) - f ( 0 ) } { 2 - 0 } \ \ & amp ; = \frac { 1 - ( -3 ) } { 2 } \ \ & amp ; = 2 \end { align } $ this means the average rate of change over the interval $ [ 0 , 2 ] $ is 2 units of increase in $ y $ for each unit of increase in $ x $ . ( b ) from example 2 above , we found that $ f\ , ' ( x ) = 2x $ , so $ \begin { align } m_ { \tan } & amp ; = f\ , ' ( x_0 ) \ & amp ; = f\ , ' ( -1 ) \ & amp ; = 2 ( -1 ) \ & amp ; = -2 . \end { align } $ this means that the instantaneous rate of change is negative . that is , $ y $ is decreasing at $ x = -1 $ . it is decreasing at a rate of $ 2 $ units in $ y $ for each unit of increase in $ x $ . source ck-12 : tangent lines and rates of change license creative commons attribution-noncommercial
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the slope of the tangent line is the instantaneous rate of change $ m_ { \tan } $ . example 4 suppose that $ y = x^2 - 3 $ . ( a ) find the average rate of change of $ y $ with respect to $ x $ over the interval $ [ 0 , 2 ] $ .
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why in example 1 we have to evaluate x^3 ?
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introduction the position of a car driving down the street , the value of currency adjusted for inflation , the number of bacteria in a culture , and the ac voltage of an electric signal are all examples of quantities that change with time . in this section , we will study the rate of change of a quantity and how is it related geometrically to secant and tangent lines . secant and tangent lines if two distinct points $ p ( x_0 , y_0 ) $ and $ q ( x_1 , y_1 ) $ lie on the curve $ y = f ( x ) $ , the slope of the secant line connecting the two points is $ \displaystyle m_ { \sec } = \frac { y_1 - y_0 } { x_1 - x_0 } = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ , $ . if we let the point $ x_1 $ approach $ x_0 $ , then $ q $ will approach $ p $ along the graph $ f $ . the slope of the secant line through points $ p $ and $ q $ will gradually approach the slope of the tangent line through $ p $ as $ x_1 $ approaches $ x_0 $ . in the limit , the previous equation becomes $ \displaystyle m_ { \tan } =\lim_ { x_1 \to x_0 } \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ , $ . if we let $ h = x_1 - x_0 $ , then $ x_1 = x_0 + h $ and $ h \rightarrow 0 $ as $ x_1 \rightarrow x_0 $ . we can rewrite the limit as $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ . when the limit exists , its value $ m_ { \tan } $ is the slope of the tangent line to the graph of $ f $ at the point $ p ( x_0 , y_0 ) $ . example 1 find the slope of the tangent line to the graph of the function $ f ( x ) = x^3 $ at the point $ ( 2 , 8 ) $ . solution since $ ( x_0 , y_0 ) = ( 2 , 8 ) $ , using the slope of the tangent line formula $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ we get $ \begin { align } m_ { \tan } & amp ; =\lim_ { h \to 0 } \frac { f ( 2 + h ) - f ( 2 ) } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { \left ( h^3 + 6h^2 + 12h + 8\right ) - 8 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { h^3 + 6h^2 + 12h } { h } \ \ & amp ; =\lim_ { h \to 0 } \left ( h^2 + 6h + 12\right ) \ \ & amp ; = 12 . \end { align } $ thus , the slope of the tangent line is $ 12 $ . recall from algebra that the point-slope form of the equation of the tangent line is $ \displaystyle y - y_0 = m_ { \tan } \cdot ( x - x_0 ) $ . the point-slope formula gives us the equation $ \displaystyle y - 8 = 12\cdot ( x - 2 ) $ which we can rewrite as $ \displaystyle y = 12x - 16 $ . finding the slope at any point next we are interested in finding a formula for the slope of the tangent line at any point on the graph of $ f $ . such a formula would be the same formula that we are using except we replace the constant $ x_0 $ by the variable $ x $ . this yields $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ . we denote this formula by $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ , where $ f\ , ' ( x ) $ is read `` $ f $ prime of $ x $ . '' the next example illustrates its usefulness . example 2 if $ f ( x ) = x^2 - 3 $ , find $ f\ , ' ( x ) $ and use the result to find the slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ . solution since $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ , then $ \begin { align } f\ , ' ( x ) & amp ; =\lim_ { h \to 0 } \frac { \left [ ( x + h ) ^2 - 3\right ] - \left [ x^2 - 3\right ] } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { x^2 + 2xh + h^2 - 3 - x^2 + 3 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { 2xh + h^2 } { h } \ \ & amp ; =\lim_ { h \to 0 } ( 2x + h ) \ \ & amp ; = 2x . \end { align } $ to find the slope , we substitute $ x = 2 $ and $ x = -1 $ into the result $ f\ , ' ( x ) $ , we get $ \displaystyle f\ , ' ( 2 ) = 2 ( 2 ) = 4 $ and $ \displaystyle f\ , ' ( -1 ) = 2 ( -1 ) = -2 $ . thus , slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ are $ 4 $ and $ -2 $ , respectively . example 3 find the slope of the tangent line to the curve $ y = 1/x $ at the point $ ( 1 , 1 ) $ . solution using the slope of the tangent line formula $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ and substituting $ y = 1/x $ gives us $ \begin { align } y\ , ' & amp ; =\lim_ { h \to 0 } \dfrac { \left ( \dfrac { 1 } { x+h } \right ) - \dfrac { 1 } { x } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { \dfrac { x - ( x + h ) } { x ( x + h ) } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { x - x - h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -1 } { x ( x + h ) } \ \ & amp ; = \frac { -1 } { x^2 } . \end { align } $ substituting $ x = 1 $ yields $ y\ , ' = \dfrac { -1 } { ( 1 ) ^2 } = -1 $ . thus , the slope of the tangent line at $ x = 1 $ for the curve $ y = 1/x $ is $ m = -1 $ . to find the equation of the tangent line , we use the point-slope formula , $ \displaystyle y - y_0 = m\cdot ( x - x_0 ) $ , where $ ( x_ { 0 } , y_ { 0 } ) = ( 1 , 1 ) $ . the equation of the tangent line is $ \begin { align } y - 1 & amp ; = -1 \cdot ( x - 1 ) \ y & amp ; = -x + 1 + 1\ y & amp ; = -x + 2 . \end { align } $ average speed the primary concept of differential is calculating the rate of change of one quantity with respect to another . for example , speed is defined as the rate of change of the distance traveled with respect to time . if a car travels $ 120 $ miles in $ 4 $ hours , his speed is $ \dfrac { 120\text { miles } } { 4\text { hours } } = 30 \text { mi/hr } $ . this speed is called the average speed or the average rate of change of distance with respect to time . of course , a car that travels $ 120 $ miles at an average rate of $ 30 $ miles per hour for $ 4 $ hours does not necessarily do so at constant speed . it may have slowed down or sped up during the $ 4 $ hour period . however , if the car hits a tree , it would not be its average speed that determines the resulting damage but its speed at the instant of the collision . so here we have two distinct kinds of speeds , average speed and instantaneous speed . the average speed of an object is defined as the object ’ s displacement $ \triangle x $ divided by the time interval $ \triangle t $ during which the displacement occurs : $ \displaystyle v = \frac { \triangle x } { \triangle t } = \frac { x_1 - x_0 } { t_1 - t_0 } $ . the average speed is also the expression for the slope of a secant line connecting the two points . figure 1 shows the $ \purple { \text { secant line } } $ through the points $ \pink { ( t_0 , x_0 ) } $ and $ \pink { ( t_1 , x_1 ) } $ on the $ \blue { \text { position-versus-time curve } } $ . thus we conclude that the average speed of an object between time $ t_0 $ and $ t_1 $ is represented geometrically by the slope of the secant line connecting the two points $ ( t_0 , x_0 ) $ and $ ( t_1 , x_1 ) $ . if we choose $ t_1 $ close to $ t_0 $ , then the average speed will closely approximate the instantaneous speed at time $ t_0 $ . rates of change the average rate of change of an arbitrary function $ f $ on an interval is represented geometrically by the slope of the secant line to the graph of $ f $ . the instantaneous rate of change of $ f $ at a particular point is represented by the slope of the tangent line to the graph of $ f $ at that point . let 's consider each case in more detail . average rate of change the average rate of change of the function $ f $ over the interval $ [ x_0 , x_1 ] $ is $ \displaystyle m_ { \sec } = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } $ . figure 2 shows the $ \purple { \text { secant line } } $ through the points $ \pink { ( x_0 , f ( x_0 ) ) } $ and $ \pink { ( x_1 , f ( x_1 ) ) } $ on the $ \blue { \text { graph of } \ , f } $ . the slope of the secant line is the average rate of change $ m_ { \sec } $ . instantaneous rate of change the instantaneous rate of change of the function $ f $ at the point $ x_0 $ is $ \displaystyle m_ { \tan } = f\ , ' ( x_0 ) = \lim_ { x_1 \to x_0 } \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } $ . figure 3 shows $ \purple { \text { tangent line } } $ through the point $ \pink { ( x_0 , f ( x_0 ) ) } $ on the $ \blue { \text { graph of } \ , f } $ . the slope of the tangent line is the instantaneous rate of change $ m_ { \tan } $ . example 4 suppose that $ y = x^2 - 3 $ . ( a ) find the average rate of change of $ y $ with respect to $ x $ over the interval $ [ 0 , 2 ] $ . ( b ) find the instantaneous rate of change of $ y $ with respect to $ x $ at the point $ x = -1 $ . solution ( a ) applying the formula for average rate of change with $ f ( x ) = x^2 - 3 $ and $ x_0 = 0 $ and $ x_1 = 2 $ yields $ \begin { align } m_ { \sec } & amp ; = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ \ & amp ; = \frac { f ( 2 ) - f ( 0 ) } { 2 - 0 } \ \ & amp ; = \frac { 1 - ( -3 ) } { 2 } \ \ & amp ; = 2 \end { align } $ this means the average rate of change over the interval $ [ 0 , 2 ] $ is 2 units of increase in $ y $ for each unit of increase in $ x $ . ( b ) from example 2 above , we found that $ f\ , ' ( x ) = 2x $ , so $ \begin { align } m_ { \tan } & amp ; = f\ , ' ( x_0 ) \ & amp ; = f\ , ' ( -1 ) \ & amp ; = 2 ( -1 ) \ & amp ; = -2 . \end { align } $ this means that the instantaneous rate of change is negative . that is , $ y $ is decreasing at $ x = -1 $ . it is decreasing at a rate of $ 2 $ units in $ y $ for each unit of increase in $ x $ . source ck-12 : tangent lines and rates of change license creative commons attribution-noncommercial
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example 2 if $ f ( x ) = x^2 - 3 $ , find $ f\ , ' ( x ) $ and use the result to find the slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ . solution since $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ , then $ \begin { align } f\ , ' ( x ) & amp ; =\lim_ { h \to 0 } \frac { \left [ ( x + h ) ^2 - 3\right ] - \left [ x^2 - 3\right ] } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { x^2 + 2xh + h^2 - 3 - x^2 + 3 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { 2xh + h^2 } { h } \ \ & amp ; =\lim_ { h \to 0 } ( 2x + h ) \ \ & amp ; = 2x . \end { align } $ to find the slope , we substitute $ x = 2 $ and $ x = -1 $ into the result $ f\ , ' ( x ) $ , we get $ \displaystyle f\ , ' ( 2 ) = 2 ( 2 ) = 4 $ and $ \displaystyle f\ , ' ( -1 ) = 2 ( -1 ) = -2 $ .
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we get f ( xo+h ) ^3 since h- > 0 why ca n't we assume that the h inside the parenthesis is almost 0 and simply evaluate ( xo ) ^3 ?
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introduction the position of a car driving down the street , the value of currency adjusted for inflation , the number of bacteria in a culture , and the ac voltage of an electric signal are all examples of quantities that change with time . in this section , we will study the rate of change of a quantity and how is it related geometrically to secant and tangent lines . secant and tangent lines if two distinct points $ p ( x_0 , y_0 ) $ and $ q ( x_1 , y_1 ) $ lie on the curve $ y = f ( x ) $ , the slope of the secant line connecting the two points is $ \displaystyle m_ { \sec } = \frac { y_1 - y_0 } { x_1 - x_0 } = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ , $ . if we let the point $ x_1 $ approach $ x_0 $ , then $ q $ will approach $ p $ along the graph $ f $ . the slope of the secant line through points $ p $ and $ q $ will gradually approach the slope of the tangent line through $ p $ as $ x_1 $ approaches $ x_0 $ . in the limit , the previous equation becomes $ \displaystyle m_ { \tan } =\lim_ { x_1 \to x_0 } \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ , $ . if we let $ h = x_1 - x_0 $ , then $ x_1 = x_0 + h $ and $ h \rightarrow 0 $ as $ x_1 \rightarrow x_0 $ . we can rewrite the limit as $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ . when the limit exists , its value $ m_ { \tan } $ is the slope of the tangent line to the graph of $ f $ at the point $ p ( x_0 , y_0 ) $ . example 1 find the slope of the tangent line to the graph of the function $ f ( x ) = x^3 $ at the point $ ( 2 , 8 ) $ . solution since $ ( x_0 , y_0 ) = ( 2 , 8 ) $ , using the slope of the tangent line formula $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x_0 + h ) - f ( x_0 ) } { h } $ we get $ \begin { align } m_ { \tan } & amp ; =\lim_ { h \to 0 } \frac { f ( 2 + h ) - f ( 2 ) } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { \left ( h^3 + 6h^2 + 12h + 8\right ) - 8 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { h^3 + 6h^2 + 12h } { h } \ \ & amp ; =\lim_ { h \to 0 } \left ( h^2 + 6h + 12\right ) \ \ & amp ; = 12 . \end { align } $ thus , the slope of the tangent line is $ 12 $ . recall from algebra that the point-slope form of the equation of the tangent line is $ \displaystyle y - y_0 = m_ { \tan } \cdot ( x - x_0 ) $ . the point-slope formula gives us the equation $ \displaystyle y - 8 = 12\cdot ( x - 2 ) $ which we can rewrite as $ \displaystyle y = 12x - 16 $ . finding the slope at any point next we are interested in finding a formula for the slope of the tangent line at any point on the graph of $ f $ . such a formula would be the same formula that we are using except we replace the constant $ x_0 $ by the variable $ x $ . this yields $ \displaystyle m_ { \tan } =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ . we denote this formula by $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } \ , $ , where $ f\ , ' ( x ) $ is read `` $ f $ prime of $ x $ . '' the next example illustrates its usefulness . example 2 if $ f ( x ) = x^2 - 3 $ , find $ f\ , ' ( x ) $ and use the result to find the slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ . solution since $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ , then $ \begin { align } f\ , ' ( x ) & amp ; =\lim_ { h \to 0 } \frac { \left [ ( x + h ) ^2 - 3\right ] - \left [ x^2 - 3\right ] } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { x^2 + 2xh + h^2 - 3 - x^2 + 3 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { 2xh + h^2 } { h } \ \ & amp ; =\lim_ { h \to 0 } ( 2x + h ) \ \ & amp ; = 2x . \end { align } $ to find the slope , we substitute $ x = 2 $ and $ x = -1 $ into the result $ f\ , ' ( x ) $ , we get $ \displaystyle f\ , ' ( 2 ) = 2 ( 2 ) = 4 $ and $ \displaystyle f\ , ' ( -1 ) = 2 ( -1 ) = -2 $ . thus , slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ are $ 4 $ and $ -2 $ , respectively . example 3 find the slope of the tangent line to the curve $ y = 1/x $ at the point $ ( 1 , 1 ) $ . solution using the slope of the tangent line formula $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ and substituting $ y = 1/x $ gives us $ \begin { align } y\ , ' & amp ; =\lim_ { h \to 0 } \dfrac { \left ( \dfrac { 1 } { x+h } \right ) - \dfrac { 1 } { x } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { \dfrac { x - ( x + h ) } { x ( x + h ) } } { h } \ \ & amp ; = \lim_ { h \to 0 } \frac { x - x - h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -h } { hx ( x + h ) } \ \ & amp ; = \lim_ { h \to 0 } \frac { -1 } { x ( x + h ) } \ \ & amp ; = \frac { -1 } { x^2 } . \end { align } $ substituting $ x = 1 $ yields $ y\ , ' = \dfrac { -1 } { ( 1 ) ^2 } = -1 $ . thus , the slope of the tangent line at $ x = 1 $ for the curve $ y = 1/x $ is $ m = -1 $ . to find the equation of the tangent line , we use the point-slope formula , $ \displaystyle y - y_0 = m\cdot ( x - x_0 ) $ , where $ ( x_ { 0 } , y_ { 0 } ) = ( 1 , 1 ) $ . the equation of the tangent line is $ \begin { align } y - 1 & amp ; = -1 \cdot ( x - 1 ) \ y & amp ; = -x + 1 + 1\ y & amp ; = -x + 2 . \end { align } $ average speed the primary concept of differential is calculating the rate of change of one quantity with respect to another . for example , speed is defined as the rate of change of the distance traveled with respect to time . if a car travels $ 120 $ miles in $ 4 $ hours , his speed is $ \dfrac { 120\text { miles } } { 4\text { hours } } = 30 \text { mi/hr } $ . this speed is called the average speed or the average rate of change of distance with respect to time . of course , a car that travels $ 120 $ miles at an average rate of $ 30 $ miles per hour for $ 4 $ hours does not necessarily do so at constant speed . it may have slowed down or sped up during the $ 4 $ hour period . however , if the car hits a tree , it would not be its average speed that determines the resulting damage but its speed at the instant of the collision . so here we have two distinct kinds of speeds , average speed and instantaneous speed . the average speed of an object is defined as the object ’ s displacement $ \triangle x $ divided by the time interval $ \triangle t $ during which the displacement occurs : $ \displaystyle v = \frac { \triangle x } { \triangle t } = \frac { x_1 - x_0 } { t_1 - t_0 } $ . the average speed is also the expression for the slope of a secant line connecting the two points . figure 1 shows the $ \purple { \text { secant line } } $ through the points $ \pink { ( t_0 , x_0 ) } $ and $ \pink { ( t_1 , x_1 ) } $ on the $ \blue { \text { position-versus-time curve } } $ . thus we conclude that the average speed of an object between time $ t_0 $ and $ t_1 $ is represented geometrically by the slope of the secant line connecting the two points $ ( t_0 , x_0 ) $ and $ ( t_1 , x_1 ) $ . if we choose $ t_1 $ close to $ t_0 $ , then the average speed will closely approximate the instantaneous speed at time $ t_0 $ . rates of change the average rate of change of an arbitrary function $ f $ on an interval is represented geometrically by the slope of the secant line to the graph of $ f $ . the instantaneous rate of change of $ f $ at a particular point is represented by the slope of the tangent line to the graph of $ f $ at that point . let 's consider each case in more detail . average rate of change the average rate of change of the function $ f $ over the interval $ [ x_0 , x_1 ] $ is $ \displaystyle m_ { \sec } = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } $ . figure 2 shows the $ \purple { \text { secant line } } $ through the points $ \pink { ( x_0 , f ( x_0 ) ) } $ and $ \pink { ( x_1 , f ( x_1 ) ) } $ on the $ \blue { \text { graph of } \ , f } $ . the slope of the secant line is the average rate of change $ m_ { \sec } $ . instantaneous rate of change the instantaneous rate of change of the function $ f $ at the point $ x_0 $ is $ \displaystyle m_ { \tan } = f\ , ' ( x_0 ) = \lim_ { x_1 \to x_0 } \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } $ . figure 3 shows $ \purple { \text { tangent line } } $ through the point $ \pink { ( x_0 , f ( x_0 ) ) } $ on the $ \blue { \text { graph of } \ , f } $ . the slope of the tangent line is the instantaneous rate of change $ m_ { \tan } $ . example 4 suppose that $ y = x^2 - 3 $ . ( a ) find the average rate of change of $ y $ with respect to $ x $ over the interval $ [ 0 , 2 ] $ . ( b ) find the instantaneous rate of change of $ y $ with respect to $ x $ at the point $ x = -1 $ . solution ( a ) applying the formula for average rate of change with $ f ( x ) = x^2 - 3 $ and $ x_0 = 0 $ and $ x_1 = 2 $ yields $ \begin { align } m_ { \sec } & amp ; = \frac { f ( x_1 ) - f ( x_0 ) } { x_1 - x_0 } \ \ & amp ; = \frac { f ( 2 ) - f ( 0 ) } { 2 - 0 } \ \ & amp ; = \frac { 1 - ( -3 ) } { 2 } \ \ & amp ; = 2 \end { align } $ this means the average rate of change over the interval $ [ 0 , 2 ] $ is 2 units of increase in $ y $ for each unit of increase in $ x $ . ( b ) from example 2 above , we found that $ f\ , ' ( x ) = 2x $ , so $ \begin { align } m_ { \tan } & amp ; = f\ , ' ( x_0 ) \ & amp ; = f\ , ' ( -1 ) \ & amp ; = 2 ( -1 ) \ & amp ; = -2 . \end { align } $ this means that the instantaneous rate of change is negative . that is , $ y $ is decreasing at $ x = -1 $ . it is decreasing at a rate of $ 2 $ units in $ y $ for each unit of increase in $ x $ . source ck-12 : tangent lines and rates of change license creative commons attribution-noncommercial
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the next example illustrates its usefulness . example 2 if $ f ( x ) = x^2 - 3 $ , find $ f\ , ' ( x ) $ and use the result to find the slopes of the tangent lines at $ x = 2 $ and $ x = -1 $ . solution since $ \displaystyle f\ , ' ( x ) =\lim_ { h \to 0 } \frac { f ( x + h ) - f ( x ) } { h } $ , then $ \begin { align } f\ , ' ( x ) & amp ; =\lim_ { h \to 0 } \frac { \left [ ( x + h ) ^2 - 3\right ] - \left [ x^2 - 3\right ] } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { x^2 + 2xh + h^2 - 3 - x^2 + 3 } { h } \ \ & amp ; =\lim_ { h \to 0 } \frac { 2xh + h^2 } { h } \ \ & amp ; =\lim_ { h \to 0 } ( 2x + h ) \ \ & amp ; = 2x .
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what is the derivative for the function x^3 ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area .
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why we use cos ( theta ) instead of sin ( theta ) in magnetic flux density 's equation ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ .
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solution to how do we measure magnetic flux ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important .
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where do we get the magnetic field vector ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area .
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is it the magnetic flux density ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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what would the magnetic field in the wire be or in the magnet or whatever is generating the magnetic field ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading .
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because if we use the same formula , i suppose we will have to divide by 0 and that would be infinity or undefined , right ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux .
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for exercise 1 why ca n't you calculate b at 25mm and b at 75mm and then average the values to get the final b ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important .
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for exercise 1 , can someone please tell us how to calculate the area under x=25 and x=75 for magnetic field * distance using calculus ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area .
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why does flux has a unit of tm^2 ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area .
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is n't magnetic flux , how much component of magnetic field that is normal to the surface per given area and time.. ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil .
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i was wondering , what would happen if we had another wire with current going in the vertical direction , next to our loop ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area .
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how has the y axis been ploted in the last graph ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important .
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for the last question , after we found b , can we not find the average magnitude of magnetic field going through the loop and then use the formula to calculate the flux instead of using calculus method ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil .
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please explain more in detail of how exercise 1 in magnetic flux around a current-carrying wire is calculated ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area .
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how is the total area calculated ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area .
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what 's the meaning of glancing angle ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area .
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why does n't fieldline passing through at a glancing angle make striking result ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area .
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is there a range of glancing angle ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important .
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does iron core increase the strength of magnetic field ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area .
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what is the magnitude of the magnetic flux as a function of area and the magnetic field ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important .
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why in dynamo , whenever the coil reaches a position perpendicular to the magnetic field and the direction of motion of the longitudinal sides becomes parallel to the the magnetic field , the time rate in magnetic flux which cuts across the longitudinal sides = zero ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area .
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is the magnetic flux related to the number of coils , or is it only induction ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ?
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so if i have a 200 loop coil , when do i include the number of loops ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area .
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or do i solve the magnetic flux for one coil and then when i solve for the induced voltage include the 200 loops ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ .
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solution to how do we measure magnetic flux ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ?
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where did the y coordinate graph value come from in exercise 1 ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area .
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but why in the solution for exercise 2 use mt ( meter tesla ) for magnetic field and mwb ( meter weber ) for magnetic flux ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field .
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what would be the flux density ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism .
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when do we use sin and when do we use cos ?
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area . it is a useful tool for helping describe the effects of the magnetic force on something occupying a given area . the measurement of magnetic flux is tied to the particular area chosen . we can choose to make the area any size we want and orient it in any way relative to the magnetic field . if we use the field-line picture of a magnetic field then every field line passing through the given area contributes some magnetic flux . the angle at which the field line intersects the area is also important . a field line passing through at a glancing angle will only contribute a small component of the field to the magnetic flux . when calculating the magnetic flux we include only the component of the magnetic field vector which is normal to our test area . if we choose a simple flat surface with area $ a $ as our test area and there is an angle $ \theta $ between the normal to the surface and a magnetic field vector ( magnitude $ b $ ) then the magnetic flux is , $ \phi = b a \cos { \theta } $ in the case that the surface is perpendicular to the field then the angle is zero and the magnetic flux is simply $ b a $ . figure 1 shows an example of a flat test area at two different angles to a magnetic field and the resulting magnetic flux . exercise 1 : if the blue surfaces shown in figure 1 both have equal area and the angle $ \theta $ is $ 25^\circ $ , how much smaller is the flux through the area in figure 1-left vs figure 1-right ? how do we measure magnetic flux ? the si unit of magnetic flux is the weber ( named after german physicist and co-inventor of the telegraph wilhelm weber ) and the unit has the symbol $ \mathrm { wb } $ . because the magnetic flux is just a way of expressing the magnetic field in a given area , it can be measured with a magnetometer in the same way as the magnetic field . for example , suppose a small magnetometer probe is moved around ( without rotating ) inside a $ 0.5~\mathrm { m^2 } $ area near a large sheet of magnetic material and indicates a constant reading of $ 5~\mathrm { mt } $ . the magnetic flux through the area is then $ ( 5\cdot 10^ { -3 } ~\mathrm { t } ) \cdot ( 0.5~\mathrm { m^2 } ) = 0.0025~\mathrm { wb } $ . in the event that the magnetic field reading changes with position , it would be necessary to find the average reading . a related term that you may come across is the magnetic flux density . this is measured in $ \mathrm { wb/m^2 } $ . because we are dividing flux by area we could also directly state the units of flux density in tesla . in fact , the term magnetic flux density is often used synonymously with the magnitude of the magnetic field . exercise 2 : figure 2 shows a map of a non-uniform magnetic field measured near a sheet of magnetic material . if the green line represents a loop of wire , what is the magnetic flux through the loop ? why is this useful ? there are a couple of reasons why the description of magnetic flux can be more useful than that of a magnetic field directly . when a coil of wire is moved through a magnetic field a voltage is generated which depends on the magnetic flux through the area of the coil . this is described by faraday 's law and is explored in our article on faraday 's law . electric motors and generators apply faraday 's law to coils which rotate in a magnetic field as depicted in figure 3 . in this example the flux changes as the coil rotates . the description of magnetic flux allows engineers to easily calculate the voltage generated by an electric generator even when the magnetic field is complicated . although we have thus-far only concerned ourselves with magnetic flux measured for a simple flat test-area , we can make our test-area a surface of any shape we like . in-fact , we can use a closed surface such as a sphere which encloses a region of interest . closed surfaces are particularly interesting to physicists because of gauss 's law for magnetism . because magnets always have two poles there is no possibility ( as far as we know ) that there is a magnetic monopole inside a closed surface . this means that the net magnetic flux through such a closed surface is always zero and therefore all the magnetic field lines going into the closed surface are exactly balanced by field lines coming out . this fact is useful for simplifying magnetic field problems . magnetic flux around a current-carrying wire exercise 1 : figure 4 shows a square loop of wire placed near a current carrying wire . using the dimensions shown in the figure , find the magnetic flux through a coil . if you do n't know how to calculate the magnetic field around a wire , review our article on the magnetic field . hint : it may be useful to plot the magnetic field vs vertical distance from the wire .
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what is magnetic flux ? magnetic flux is a measurement of the total magnetic field which passes through a given area .
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why do we characterise the magnetic flux as a dot product ?
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key points the pressure exerted by an individual gas in a mixture is known as its partial pressure . assuming we have a mixture of ideal gases , we can use the ideal gas law to solve problems involving gases in a mixture . dalton 's law of partial pressures states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the component gases : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ dalton 's law can also be expressed using the mole fraction of a gas , $ x $ : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ introduction in day-to-day life , we measure gas pressure when we use a barometer to check the atmospheric pressure outside or a tire gauge to measure the pressure in a bike tube . when we do this , we are measuring a macroscopic physical property of a large number of gas molecules that are invisible to the naked eye . on the molecular level , the pressure we are measuring comes from the force of individual gas molecules colliding with other objects , such as the walls of their container . let 's take a closer look at pressure from a molecular perspective and learn how dalton 's law helps us calculate total and partial pressures for mixtures of gases . ideal gases and partial pressure in this article , we will be assuming the gases in our mixtures can be approximated as ideal gases . this assumption is generally reasonable as long as the temperature of the gas is not super low ( close to $ 0\ , \text k $ ) , and the pressure is around $ 1\ , \text { atm } $ . this means we are making some assumptions about our gas molecules : we assume that the gas molecules take up no volume . we assume that the molecules have no intermolecular attractions , which means they act independently of other gas molecules . based on these assumptions , we can calculate the contribution of different gases in a mixture to the total pressure . we refer to the pressure exerted by a specific gas in a mixture as its partial pressure . the partial pressure of a gas can be calculated using the ideal gas law , which we will cover in the next section , as well as using dalton 's law of partial pressures . example 1 : calculating the partial pressure of a gas let 's say we have a mixture of hydrogen gas , $ \text h_2 ( g ) $ , and oxygen gas , $ \text o_2 ( g ) $ . the mixture contains $ 6.7\ , \text { mol } $ hydrogen gas and $ 3.3\ , \text { mol } $ oxygen gas . the mixture is in a $ 300\ , \text l $ container at $ 273\ , \text k $ , and the total pressure of the gas mixture is $ 0.75\ , \text { atm } $ . the contribution of hydrogen gas to the total pressure is its partial pressure . since the gas molecules in an ideal gas behave independently of other gases in the mixture , the partial pressure of hydrogen is the same pressure as if there were no other gases in the container . therefore , if we want to know the partial pressure of hydrogen gas in the mixture , $ \text p_ { \text h_2 } $ , we can completely ignore the oxygen gas and use the ideal gas law : $ \text p_ { \text h_2 } \text v = \text { n } _ { \text h_2 } \text { rt } $ rearranging the ideal gas equation to solve for $ \text p_ { \text h_2 } $ , we get : $ \begin { align } \text p_ { \text h_2 } & amp ; = \dfrac { \text { n } _ { \text h_2 } \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 6.7\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text l ) } { 300\ , \text l } =0.50\ , \text { atm } \end { align } $ thus , the ideal gas law tells us that the partial pressure of hydrogen in the mixture is $ 0.50\ , \text { atm } $ . we can also calculate the partial pressure of hydrogen in this problem using dalton 's law of partial pressures , which will be discussed in the next section . dalton 's law of partial pressures dalton 's law of partial pressures states that the total pressure of a mixture of gases is the sum of the partial pressures of its components : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ where the partial pressure of each gas is the pressure that the gas would exert if it was the only gas in the container . that is because we assume there are no attractive forces between the gases . dalton 's law of partial pressure can also be expressed in terms of the mole fraction of a gas in the mixture . the mole fraction of a gas is the number of moles of that gas divided by the total moles of gas in the mixture , and it is often abbreviated as $ x $ : $ x_1=\text { mole fraction of gas 1 } =\dfrac { \text { moles of gas 1 } } { \text { total moles of gas } } $ dalton 's law for can be rearranged to give the partial pressure of gas 1 in a mixture in terms of the mole fraction of gas 1 : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ both forms of dalton 's law are extremely useful in solving different kinds of problems including : calculating the partial pressure of a gas when you know the mole ratio and total pressure calculating moles of an individual gas if you know the partial pressure and total pressure calculating the total pressure if you know the partial pressures of the components example 2 : calculating partial pressures and total pressure let 's say that we have one container with $ 24.0\ , \text l $ of nitrogen gas at $ 2.00 \ , \text { atm } $ , and another container with $ 12.0\ , \text l $ of oxygen gas at $ 2.00\ , \text { atm } $ . the temperature of both gases is $ 273\ , \text k $ . if both gases are mixed in a $ 10.0\ , \text l $ container , what are the partial pressures of nitrogen and oxygen in the resulting mixture ? what is the total pressure ? step 1 : calculate moles of oxygen and nitrogen gas since we know $ \text p $ , $ \text v $ , and $ \text t $ for each of the gases before they 're combined , we can find the number of moles of nitrogen gas and oxygen gas using the ideal gas law : $ \text n = \dfrac { \text { pv } } { \text { rt } } $ solving for nitrogen and oxygen , we get : $ \text n_ { \text { n } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 24.0\ , \text { l } ) } { ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 k ) } = 2.14\ , \text { mol nitrogen } $ $ \text n_ { \text { o } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 12.0\ , \text { l } ) } { ( 0.08206 \ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 \ , \text { k } ) } = 1.07\ , \text { mol oxygen } $ step 2 ( method 1 ) : calculate partial pressures and use dalton 's law to get $ \text p_\text { total } $ once we know the number of moles for each gas in our mixture , we can now use the ideal gas law to find the partial pressure of each component in the $ 10.0\ , \text l $ container : $ \text p = \dfrac { \text { nrt } } { \text v } $ $ \text p_ { \text { n } _2 } =\dfrac { ( 2.14\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 4.79\ , \text { atm } $ $ \text p_ { \text { o } _2 } =\dfrac { ( 1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 2.40\ , \text { atm } $ notice that the partial pressure for each of the gases increased compared to the pressure of the gas in the original container . this makes sense since the volume of both gases decreased , and pressure is inversely proportional to volume . we can now get the total pressure of the mixture by adding the partial pressures together using dalton 's law : $ \begin { align } \text p_\text { total } & amp ; =\text p_ { \text { n } 2 } + \text p { \text { o } _2 } \ \ & amp ; =4.79\ , \text { atm } + 2.40\ , \text { atm } = 7.19\ , \text { atm } \end { align } $ step 2 ( method 2 ) : use ideal gas law to calculate $ \text p_\text { total } $ without partial pressures since the pressure of an ideal gas mixture only depends on the number of gas molecules in the container ( and not the identity of the gas molecules ) , we can use the total moles of gas to calculate the total pressure using the ideal gas law : $ \begin { align } \text p_ { \text { total } } & amp ; = \dfrac { ( \text { n } { \text n_2 } +\text n { \text { o } _2 } ) \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 2.14\ , \text { mol } +1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =\dfrac { ( 3.21\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =7.19\ , \text { atm } \end { align } $ once we know the total pressure , we can use the mole fraction version of dalton 's law to calculate the partial pressures : $ \text p_ { \text { n } 2 } = x { \text n_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 2.14\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =4.79\ , \text { atm } $ $ \text p_ { \text { o } 2 } = x { \text o_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 1.07\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =2.40\ , \text { atm } $ luckily , both methods give the same answers ! you might be wondering when you might want to use each method . it mostly depends on which one you prefer , and partly on what you are solving for . for instance , if all you need to know is the total pressure , it might be better to use the second method to save a couple calculation steps . summary the pressure exerted by an individual gas in a mixture is known as its partial pressure . assuming we have a mixture of ideal gases , we can use the ideal gas law to solve problems involving gases in a mixture . dalton 's law of partial pressures states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the component gases : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ dalton 's law can also be expressed using the mole fraction of a gas , $ x $ : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ try it : evaporation in a closed system part 1 part 2
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the mixture is in a $ 300\ , \text l $ container at $ 273\ , \text k $ , and the total pressure of the gas mixture is $ 0.75\ , \text { atm } $ . the contribution of hydrogen gas to the total pressure is its partial pressure . since the gas molecules in an ideal gas behave independently of other gases in the mixture , the partial pressure of hydrogen is the same pressure as if there were no other gases in the container .
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in the very first example , where they are solving for the pressure of h2 , why does the equation say 273l , not 273k ?
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key points the pressure exerted by an individual gas in a mixture is known as its partial pressure . assuming we have a mixture of ideal gases , we can use the ideal gas law to solve problems involving gases in a mixture . dalton 's law of partial pressures states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the component gases : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ dalton 's law can also be expressed using the mole fraction of a gas , $ x $ : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ introduction in day-to-day life , we measure gas pressure when we use a barometer to check the atmospheric pressure outside or a tire gauge to measure the pressure in a bike tube . when we do this , we are measuring a macroscopic physical property of a large number of gas molecules that are invisible to the naked eye . on the molecular level , the pressure we are measuring comes from the force of individual gas molecules colliding with other objects , such as the walls of their container . let 's take a closer look at pressure from a molecular perspective and learn how dalton 's law helps us calculate total and partial pressures for mixtures of gases . ideal gases and partial pressure in this article , we will be assuming the gases in our mixtures can be approximated as ideal gases . this assumption is generally reasonable as long as the temperature of the gas is not super low ( close to $ 0\ , \text k $ ) , and the pressure is around $ 1\ , \text { atm } $ . this means we are making some assumptions about our gas molecules : we assume that the gas molecules take up no volume . we assume that the molecules have no intermolecular attractions , which means they act independently of other gas molecules . based on these assumptions , we can calculate the contribution of different gases in a mixture to the total pressure . we refer to the pressure exerted by a specific gas in a mixture as its partial pressure . the partial pressure of a gas can be calculated using the ideal gas law , which we will cover in the next section , as well as using dalton 's law of partial pressures . example 1 : calculating the partial pressure of a gas let 's say we have a mixture of hydrogen gas , $ \text h_2 ( g ) $ , and oxygen gas , $ \text o_2 ( g ) $ . the mixture contains $ 6.7\ , \text { mol } $ hydrogen gas and $ 3.3\ , \text { mol } $ oxygen gas . the mixture is in a $ 300\ , \text l $ container at $ 273\ , \text k $ , and the total pressure of the gas mixture is $ 0.75\ , \text { atm } $ . the contribution of hydrogen gas to the total pressure is its partial pressure . since the gas molecules in an ideal gas behave independently of other gases in the mixture , the partial pressure of hydrogen is the same pressure as if there were no other gases in the container . therefore , if we want to know the partial pressure of hydrogen gas in the mixture , $ \text p_ { \text h_2 } $ , we can completely ignore the oxygen gas and use the ideal gas law : $ \text p_ { \text h_2 } \text v = \text { n } _ { \text h_2 } \text { rt } $ rearranging the ideal gas equation to solve for $ \text p_ { \text h_2 } $ , we get : $ \begin { align } \text p_ { \text h_2 } & amp ; = \dfrac { \text { n } _ { \text h_2 } \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 6.7\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text l ) } { 300\ , \text l } =0.50\ , \text { atm } \end { align } $ thus , the ideal gas law tells us that the partial pressure of hydrogen in the mixture is $ 0.50\ , \text { atm } $ . we can also calculate the partial pressure of hydrogen in this problem using dalton 's law of partial pressures , which will be discussed in the next section . dalton 's law of partial pressures dalton 's law of partial pressures states that the total pressure of a mixture of gases is the sum of the partial pressures of its components : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ where the partial pressure of each gas is the pressure that the gas would exert if it was the only gas in the container . that is because we assume there are no attractive forces between the gases . dalton 's law of partial pressure can also be expressed in terms of the mole fraction of a gas in the mixture . the mole fraction of a gas is the number of moles of that gas divided by the total moles of gas in the mixture , and it is often abbreviated as $ x $ : $ x_1=\text { mole fraction of gas 1 } =\dfrac { \text { moles of gas 1 } } { \text { total moles of gas } } $ dalton 's law for can be rearranged to give the partial pressure of gas 1 in a mixture in terms of the mole fraction of gas 1 : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ both forms of dalton 's law are extremely useful in solving different kinds of problems including : calculating the partial pressure of a gas when you know the mole ratio and total pressure calculating moles of an individual gas if you know the partial pressure and total pressure calculating the total pressure if you know the partial pressures of the components example 2 : calculating partial pressures and total pressure let 's say that we have one container with $ 24.0\ , \text l $ of nitrogen gas at $ 2.00 \ , \text { atm } $ , and another container with $ 12.0\ , \text l $ of oxygen gas at $ 2.00\ , \text { atm } $ . the temperature of both gases is $ 273\ , \text k $ . if both gases are mixed in a $ 10.0\ , \text l $ container , what are the partial pressures of nitrogen and oxygen in the resulting mixture ? what is the total pressure ? step 1 : calculate moles of oxygen and nitrogen gas since we know $ \text p $ , $ \text v $ , and $ \text t $ for each of the gases before they 're combined , we can find the number of moles of nitrogen gas and oxygen gas using the ideal gas law : $ \text n = \dfrac { \text { pv } } { \text { rt } } $ solving for nitrogen and oxygen , we get : $ \text n_ { \text { n } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 24.0\ , \text { l } ) } { ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 k ) } = 2.14\ , \text { mol nitrogen } $ $ \text n_ { \text { o } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 12.0\ , \text { l } ) } { ( 0.08206 \ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 \ , \text { k } ) } = 1.07\ , \text { mol oxygen } $ step 2 ( method 1 ) : calculate partial pressures and use dalton 's law to get $ \text p_\text { total } $ once we know the number of moles for each gas in our mixture , we can now use the ideal gas law to find the partial pressure of each component in the $ 10.0\ , \text l $ container : $ \text p = \dfrac { \text { nrt } } { \text v } $ $ \text p_ { \text { n } _2 } =\dfrac { ( 2.14\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 4.79\ , \text { atm } $ $ \text p_ { \text { o } _2 } =\dfrac { ( 1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 2.40\ , \text { atm } $ notice that the partial pressure for each of the gases increased compared to the pressure of the gas in the original container . this makes sense since the volume of both gases decreased , and pressure is inversely proportional to volume . we can now get the total pressure of the mixture by adding the partial pressures together using dalton 's law : $ \begin { align } \text p_\text { total } & amp ; =\text p_ { \text { n } 2 } + \text p { \text { o } _2 } \ \ & amp ; =4.79\ , \text { atm } + 2.40\ , \text { atm } = 7.19\ , \text { atm } \end { align } $ step 2 ( method 2 ) : use ideal gas law to calculate $ \text p_\text { total } $ without partial pressures since the pressure of an ideal gas mixture only depends on the number of gas molecules in the container ( and not the identity of the gas molecules ) , we can use the total moles of gas to calculate the total pressure using the ideal gas law : $ \begin { align } \text p_ { \text { total } } & amp ; = \dfrac { ( \text { n } { \text n_2 } +\text n { \text { o } _2 } ) \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 2.14\ , \text { mol } +1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =\dfrac { ( 3.21\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =7.19\ , \text { atm } \end { align } $ once we know the total pressure , we can use the mole fraction version of dalton 's law to calculate the partial pressures : $ \text p_ { \text { n } 2 } = x { \text n_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 2.14\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =4.79\ , \text { atm } $ $ \text p_ { \text { o } 2 } = x { \text o_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 1.07\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =2.40\ , \text { atm } $ luckily , both methods give the same answers ! you might be wondering when you might want to use each method . it mostly depends on which one you prefer , and partly on what you are solving for . for instance , if all you need to know is the total pressure , it might be better to use the second method to save a couple calculation steps . summary the pressure exerted by an individual gas in a mixture is known as its partial pressure . assuming we have a mixture of ideal gases , we can use the ideal gas law to solve problems involving gases in a mixture . dalton 's law of partial pressures states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the component gases : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ dalton 's law can also be expressed using the mole fraction of a gas , $ x $ : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ try it : evaporation in a closed system part 1 part 2
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example 1 : calculating the partial pressure of a gas let 's say we have a mixture of hydrogen gas , $ \text h_2 ( g ) $ , and oxygen gas , $ \text o_2 ( g ) $ . the mixture contains $ 6.7\ , \text { mol } $ hydrogen gas and $ 3.3\ , \text { mol } $ oxygen gas . the mixture is in a $ 300\ , \text l $ container at $ 273\ , \text k $ , and the total pressure of the gas mixture is $ 0.75\ , \text { atm } $ .
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what is the rate of effusion for a gas that has a molar mass three times that of a gas that effuses at a rate of 5.0 mol/hour ?
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key points the pressure exerted by an individual gas in a mixture is known as its partial pressure . assuming we have a mixture of ideal gases , we can use the ideal gas law to solve problems involving gases in a mixture . dalton 's law of partial pressures states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the component gases : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ dalton 's law can also be expressed using the mole fraction of a gas , $ x $ : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ introduction in day-to-day life , we measure gas pressure when we use a barometer to check the atmospheric pressure outside or a tire gauge to measure the pressure in a bike tube . when we do this , we are measuring a macroscopic physical property of a large number of gas molecules that are invisible to the naked eye . on the molecular level , the pressure we are measuring comes from the force of individual gas molecules colliding with other objects , such as the walls of their container . let 's take a closer look at pressure from a molecular perspective and learn how dalton 's law helps us calculate total and partial pressures for mixtures of gases . ideal gases and partial pressure in this article , we will be assuming the gases in our mixtures can be approximated as ideal gases . this assumption is generally reasonable as long as the temperature of the gas is not super low ( close to $ 0\ , \text k $ ) , and the pressure is around $ 1\ , \text { atm } $ . this means we are making some assumptions about our gas molecules : we assume that the gas molecules take up no volume . we assume that the molecules have no intermolecular attractions , which means they act independently of other gas molecules . based on these assumptions , we can calculate the contribution of different gases in a mixture to the total pressure . we refer to the pressure exerted by a specific gas in a mixture as its partial pressure . the partial pressure of a gas can be calculated using the ideal gas law , which we will cover in the next section , as well as using dalton 's law of partial pressures . example 1 : calculating the partial pressure of a gas let 's say we have a mixture of hydrogen gas , $ \text h_2 ( g ) $ , and oxygen gas , $ \text o_2 ( g ) $ . the mixture contains $ 6.7\ , \text { mol } $ hydrogen gas and $ 3.3\ , \text { mol } $ oxygen gas . the mixture is in a $ 300\ , \text l $ container at $ 273\ , \text k $ , and the total pressure of the gas mixture is $ 0.75\ , \text { atm } $ . the contribution of hydrogen gas to the total pressure is its partial pressure . since the gas molecules in an ideal gas behave independently of other gases in the mixture , the partial pressure of hydrogen is the same pressure as if there were no other gases in the container . therefore , if we want to know the partial pressure of hydrogen gas in the mixture , $ \text p_ { \text h_2 } $ , we can completely ignore the oxygen gas and use the ideal gas law : $ \text p_ { \text h_2 } \text v = \text { n } _ { \text h_2 } \text { rt } $ rearranging the ideal gas equation to solve for $ \text p_ { \text h_2 } $ , we get : $ \begin { align } \text p_ { \text h_2 } & amp ; = \dfrac { \text { n } _ { \text h_2 } \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 6.7\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text l ) } { 300\ , \text l } =0.50\ , \text { atm } \end { align } $ thus , the ideal gas law tells us that the partial pressure of hydrogen in the mixture is $ 0.50\ , \text { atm } $ . we can also calculate the partial pressure of hydrogen in this problem using dalton 's law of partial pressures , which will be discussed in the next section . dalton 's law of partial pressures dalton 's law of partial pressures states that the total pressure of a mixture of gases is the sum of the partial pressures of its components : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ where the partial pressure of each gas is the pressure that the gas would exert if it was the only gas in the container . that is because we assume there are no attractive forces between the gases . dalton 's law of partial pressure can also be expressed in terms of the mole fraction of a gas in the mixture . the mole fraction of a gas is the number of moles of that gas divided by the total moles of gas in the mixture , and it is often abbreviated as $ x $ : $ x_1=\text { mole fraction of gas 1 } =\dfrac { \text { moles of gas 1 } } { \text { total moles of gas } } $ dalton 's law for can be rearranged to give the partial pressure of gas 1 in a mixture in terms of the mole fraction of gas 1 : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ both forms of dalton 's law are extremely useful in solving different kinds of problems including : calculating the partial pressure of a gas when you know the mole ratio and total pressure calculating moles of an individual gas if you know the partial pressure and total pressure calculating the total pressure if you know the partial pressures of the components example 2 : calculating partial pressures and total pressure let 's say that we have one container with $ 24.0\ , \text l $ of nitrogen gas at $ 2.00 \ , \text { atm } $ , and another container with $ 12.0\ , \text l $ of oxygen gas at $ 2.00\ , \text { atm } $ . the temperature of both gases is $ 273\ , \text k $ . if both gases are mixed in a $ 10.0\ , \text l $ container , what are the partial pressures of nitrogen and oxygen in the resulting mixture ? what is the total pressure ? step 1 : calculate moles of oxygen and nitrogen gas since we know $ \text p $ , $ \text v $ , and $ \text t $ for each of the gases before they 're combined , we can find the number of moles of nitrogen gas and oxygen gas using the ideal gas law : $ \text n = \dfrac { \text { pv } } { \text { rt } } $ solving for nitrogen and oxygen , we get : $ \text n_ { \text { n } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 24.0\ , \text { l } ) } { ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 k ) } = 2.14\ , \text { mol nitrogen } $ $ \text n_ { \text { o } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 12.0\ , \text { l } ) } { ( 0.08206 \ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 \ , \text { k } ) } = 1.07\ , \text { mol oxygen } $ step 2 ( method 1 ) : calculate partial pressures and use dalton 's law to get $ \text p_\text { total } $ once we know the number of moles for each gas in our mixture , we can now use the ideal gas law to find the partial pressure of each component in the $ 10.0\ , \text l $ container : $ \text p = \dfrac { \text { nrt } } { \text v } $ $ \text p_ { \text { n } _2 } =\dfrac { ( 2.14\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 4.79\ , \text { atm } $ $ \text p_ { \text { o } _2 } =\dfrac { ( 1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 2.40\ , \text { atm } $ notice that the partial pressure for each of the gases increased compared to the pressure of the gas in the original container . this makes sense since the volume of both gases decreased , and pressure is inversely proportional to volume . we can now get the total pressure of the mixture by adding the partial pressures together using dalton 's law : $ \begin { align } \text p_\text { total } & amp ; =\text p_ { \text { n } 2 } + \text p { \text { o } _2 } \ \ & amp ; =4.79\ , \text { atm } + 2.40\ , \text { atm } = 7.19\ , \text { atm } \end { align } $ step 2 ( method 2 ) : use ideal gas law to calculate $ \text p_\text { total } $ without partial pressures since the pressure of an ideal gas mixture only depends on the number of gas molecules in the container ( and not the identity of the gas molecules ) , we can use the total moles of gas to calculate the total pressure using the ideal gas law : $ \begin { align } \text p_ { \text { total } } & amp ; = \dfrac { ( \text { n } { \text n_2 } +\text n { \text { o } _2 } ) \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 2.14\ , \text { mol } +1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =\dfrac { ( 3.21\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =7.19\ , \text { atm } \end { align } $ once we know the total pressure , we can use the mole fraction version of dalton 's law to calculate the partial pressures : $ \text p_ { \text { n } 2 } = x { \text n_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 2.14\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =4.79\ , \text { atm } $ $ \text p_ { \text { o } 2 } = x { \text o_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 1.07\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =2.40\ , \text { atm } $ luckily , both methods give the same answers ! you might be wondering when you might want to use each method . it mostly depends on which one you prefer , and partly on what you are solving for . for instance , if all you need to know is the total pressure , it might be better to use the second method to save a couple calculation steps . summary the pressure exerted by an individual gas in a mixture is known as its partial pressure . assuming we have a mixture of ideal gases , we can use the ideal gas law to solve problems involving gases in a mixture . dalton 's law of partial pressures states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the component gases : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ dalton 's law can also be expressed using the mole fraction of a gas , $ x $ : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ try it : evaporation in a closed system part 1 part 2
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step 1 : calculate moles of oxygen and nitrogen gas since we know $ \text p $ , $ \text v $ , and $ \text t $ for each of the gases before they 're combined , we can find the number of moles of nitrogen gas and oxygen gas using the ideal gas law : $ \text n = \dfrac { \text { pv } } { \text { rt } } $ solving for nitrogen and oxygen , we get : $ \text n_ { \text { n } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 24.0\ , \text { l } ) } { ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 k ) } = 2.14\ , \text { mol nitrogen } $ $ \text n_ { \text { o } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 12.0\ , \text { l } ) } { ( 0.08206 \ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 \ , \text { k } ) } = 1.07\ , \text { mol oxygen } $ step 2 ( method 1 ) : calculate partial pressures and use dalton 's law to get $ \text p_\text { total } $ once we know the number of moles for each gas in our mixture , we can now use the ideal gas law to find the partial pressure of each component in the $ 10.0\ , \text l $ container : $ \text p = \dfrac { \text { nrt } } { \text v } $ $ \text p_ { \text { n } _2 } =\dfrac { ( 2.14\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 4.79\ , \text { atm } $ $ \text p_ { \text { o } _2 } =\dfrac { ( 1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 2.40\ , \text { atm } $ notice that the partial pressure for each of the gases increased compared to the pressure of the gas in the original container . this makes sense since the volume of both gases decreased , and pressure is inversely proportional to volume . we can now get the total pressure of the mixture by adding the partial pressures together using dalton 's law : $ \begin { align } \text p_\text { total } & amp ; =\text p_ { \text { n } 2 } + \text p { \text { o } _2 } \ \ & amp ; =4.79\ , \text { atm } + 2.40\ , \text { atm } = 7.19\ , \text { atm } \end { align } $ step 2 ( method 2 ) : use ideal gas law to calculate $ \text p_\text { total } $ without partial pressures since the pressure of an ideal gas mixture only depends on the number of gas molecules in the container ( and not the identity of the gas molecules ) , we can use the total moles of gas to calculate the total pressure using the ideal gas law : $ \begin { align } \text p_ { \text { total } } & amp ; = \dfrac { ( \text { n } { \text n_2 } +\text n { \text { o } _2 } ) \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 2.14\ , \text { mol } +1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =\dfrac { ( 3.21\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =7.19\ , \text { atm } \end { align } $ once we know the total pressure , we can use the mole fraction version of dalton 's law to calculate the partial pressures : $ \text p_ { \text { n } 2 } = x { \text n_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 2.14\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =4.79\ , \text { atm } $ $ \text p_ { \text { o } 2 } = x { \text o_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 1.07\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =2.40\ , \text { atm } $ luckily , both methods give the same answers !
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for example 1 above when we calculated for h2 's pressure , why did we use 300l as volume ?
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key points the pressure exerted by an individual gas in a mixture is known as its partial pressure . assuming we have a mixture of ideal gases , we can use the ideal gas law to solve problems involving gases in a mixture . dalton 's law of partial pressures states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the component gases : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ dalton 's law can also be expressed using the mole fraction of a gas , $ x $ : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ introduction in day-to-day life , we measure gas pressure when we use a barometer to check the atmospheric pressure outside or a tire gauge to measure the pressure in a bike tube . when we do this , we are measuring a macroscopic physical property of a large number of gas molecules that are invisible to the naked eye . on the molecular level , the pressure we are measuring comes from the force of individual gas molecules colliding with other objects , such as the walls of their container . let 's take a closer look at pressure from a molecular perspective and learn how dalton 's law helps us calculate total and partial pressures for mixtures of gases . ideal gases and partial pressure in this article , we will be assuming the gases in our mixtures can be approximated as ideal gases . this assumption is generally reasonable as long as the temperature of the gas is not super low ( close to $ 0\ , \text k $ ) , and the pressure is around $ 1\ , \text { atm } $ . this means we are making some assumptions about our gas molecules : we assume that the gas molecules take up no volume . we assume that the molecules have no intermolecular attractions , which means they act independently of other gas molecules . based on these assumptions , we can calculate the contribution of different gases in a mixture to the total pressure . we refer to the pressure exerted by a specific gas in a mixture as its partial pressure . the partial pressure of a gas can be calculated using the ideal gas law , which we will cover in the next section , as well as using dalton 's law of partial pressures . example 1 : calculating the partial pressure of a gas let 's say we have a mixture of hydrogen gas , $ \text h_2 ( g ) $ , and oxygen gas , $ \text o_2 ( g ) $ . the mixture contains $ 6.7\ , \text { mol } $ hydrogen gas and $ 3.3\ , \text { mol } $ oxygen gas . the mixture is in a $ 300\ , \text l $ container at $ 273\ , \text k $ , and the total pressure of the gas mixture is $ 0.75\ , \text { atm } $ . the contribution of hydrogen gas to the total pressure is its partial pressure . since the gas molecules in an ideal gas behave independently of other gases in the mixture , the partial pressure of hydrogen is the same pressure as if there were no other gases in the container . therefore , if we want to know the partial pressure of hydrogen gas in the mixture , $ \text p_ { \text h_2 } $ , we can completely ignore the oxygen gas and use the ideal gas law : $ \text p_ { \text h_2 } \text v = \text { n } _ { \text h_2 } \text { rt } $ rearranging the ideal gas equation to solve for $ \text p_ { \text h_2 } $ , we get : $ \begin { align } \text p_ { \text h_2 } & amp ; = \dfrac { \text { n } _ { \text h_2 } \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 6.7\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text l ) } { 300\ , \text l } =0.50\ , \text { atm } \end { align } $ thus , the ideal gas law tells us that the partial pressure of hydrogen in the mixture is $ 0.50\ , \text { atm } $ . we can also calculate the partial pressure of hydrogen in this problem using dalton 's law of partial pressures , which will be discussed in the next section . dalton 's law of partial pressures dalton 's law of partial pressures states that the total pressure of a mixture of gases is the sum of the partial pressures of its components : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ where the partial pressure of each gas is the pressure that the gas would exert if it was the only gas in the container . that is because we assume there are no attractive forces between the gases . dalton 's law of partial pressure can also be expressed in terms of the mole fraction of a gas in the mixture . the mole fraction of a gas is the number of moles of that gas divided by the total moles of gas in the mixture , and it is often abbreviated as $ x $ : $ x_1=\text { mole fraction of gas 1 } =\dfrac { \text { moles of gas 1 } } { \text { total moles of gas } } $ dalton 's law for can be rearranged to give the partial pressure of gas 1 in a mixture in terms of the mole fraction of gas 1 : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ both forms of dalton 's law are extremely useful in solving different kinds of problems including : calculating the partial pressure of a gas when you know the mole ratio and total pressure calculating moles of an individual gas if you know the partial pressure and total pressure calculating the total pressure if you know the partial pressures of the components example 2 : calculating partial pressures and total pressure let 's say that we have one container with $ 24.0\ , \text l $ of nitrogen gas at $ 2.00 \ , \text { atm } $ , and another container with $ 12.0\ , \text l $ of oxygen gas at $ 2.00\ , \text { atm } $ . the temperature of both gases is $ 273\ , \text k $ . if both gases are mixed in a $ 10.0\ , \text l $ container , what are the partial pressures of nitrogen and oxygen in the resulting mixture ? what is the total pressure ? step 1 : calculate moles of oxygen and nitrogen gas since we know $ \text p $ , $ \text v $ , and $ \text t $ for each of the gases before they 're combined , we can find the number of moles of nitrogen gas and oxygen gas using the ideal gas law : $ \text n = \dfrac { \text { pv } } { \text { rt } } $ solving for nitrogen and oxygen , we get : $ \text n_ { \text { n } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 24.0\ , \text { l } ) } { ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 k ) } = 2.14\ , \text { mol nitrogen } $ $ \text n_ { \text { o } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 12.0\ , \text { l } ) } { ( 0.08206 \ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 \ , \text { k } ) } = 1.07\ , \text { mol oxygen } $ step 2 ( method 1 ) : calculate partial pressures and use dalton 's law to get $ \text p_\text { total } $ once we know the number of moles for each gas in our mixture , we can now use the ideal gas law to find the partial pressure of each component in the $ 10.0\ , \text l $ container : $ \text p = \dfrac { \text { nrt } } { \text v } $ $ \text p_ { \text { n } _2 } =\dfrac { ( 2.14\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 4.79\ , \text { atm } $ $ \text p_ { \text { o } _2 } =\dfrac { ( 1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 2.40\ , \text { atm } $ notice that the partial pressure for each of the gases increased compared to the pressure of the gas in the original container . this makes sense since the volume of both gases decreased , and pressure is inversely proportional to volume . we can now get the total pressure of the mixture by adding the partial pressures together using dalton 's law : $ \begin { align } \text p_\text { total } & amp ; =\text p_ { \text { n } 2 } + \text p { \text { o } _2 } \ \ & amp ; =4.79\ , \text { atm } + 2.40\ , \text { atm } = 7.19\ , \text { atm } \end { align } $ step 2 ( method 2 ) : use ideal gas law to calculate $ \text p_\text { total } $ without partial pressures since the pressure of an ideal gas mixture only depends on the number of gas molecules in the container ( and not the identity of the gas molecules ) , we can use the total moles of gas to calculate the total pressure using the ideal gas law : $ \begin { align } \text p_ { \text { total } } & amp ; = \dfrac { ( \text { n } { \text n_2 } +\text n { \text { o } _2 } ) \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 2.14\ , \text { mol } +1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =\dfrac { ( 3.21\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =7.19\ , \text { atm } \end { align } $ once we know the total pressure , we can use the mole fraction version of dalton 's law to calculate the partial pressures : $ \text p_ { \text { n } 2 } = x { \text n_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 2.14\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =4.79\ , \text { atm } $ $ \text p_ { \text { o } 2 } = x { \text o_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 1.07\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =2.40\ , \text { atm } $ luckily , both methods give the same answers ! you might be wondering when you might want to use each method . it mostly depends on which one you prefer , and partly on what you are solving for . for instance , if all you need to know is the total pressure , it might be better to use the second method to save a couple calculation steps . summary the pressure exerted by an individual gas in a mixture is known as its partial pressure . assuming we have a mixture of ideal gases , we can use the ideal gas law to solve problems involving gases in a mixture . dalton 's law of partial pressures states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the component gases : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ dalton 's law can also be expressed using the mole fraction of a gas , $ x $ : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ try it : evaporation in a closed system part 1 part 2
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step 1 : calculate moles of oxygen and nitrogen gas since we know $ \text p $ , $ \text v $ , and $ \text t $ for each of the gases before they 're combined , we can find the number of moles of nitrogen gas and oxygen gas using the ideal gas law : $ \text n = \dfrac { \text { pv } } { \text { rt } } $ solving for nitrogen and oxygen , we get : $ \text n_ { \text { n } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 24.0\ , \text { l } ) } { ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 k ) } = 2.14\ , \text { mol nitrogen } $ $ \text n_ { \text { o } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 12.0\ , \text { l } ) } { ( 0.08206 \ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 \ , \text { k } ) } = 1.07\ , \text { mol oxygen } $ step 2 ( method 1 ) : calculate partial pressures and use dalton 's law to get $ \text p_\text { total } $ once we know the number of moles for each gas in our mixture , we can now use the ideal gas law to find the partial pressure of each component in the $ 10.0\ , \text l $ container : $ \text p = \dfrac { \text { nrt } } { \text v } $ $ \text p_ { \text { n } _2 } =\dfrac { ( 2.14\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 4.79\ , \text { atm } $ $ \text p_ { \text { o } _2 } =\dfrac { ( 1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 2.40\ , \text { atm } $ notice that the partial pressure for each of the gases increased compared to the pressure of the gas in the original container . this makes sense since the volume of both gases decreased , and pressure is inversely proportional to volume . we can now get the total pressure of the mixture by adding the partial pressures together using dalton 's law : $ \begin { align } \text p_\text { total } & amp ; =\text p_ { \text { n } 2 } + \text p { \text { o } _2 } \ \ & amp ; =4.79\ , \text { atm } + 2.40\ , \text { atm } = 7.19\ , \text { atm } \end { align } $ step 2 ( method 2 ) : use ideal gas law to calculate $ \text p_\text { total } $ without partial pressures since the pressure of an ideal gas mixture only depends on the number of gas molecules in the container ( and not the identity of the gas molecules ) , we can use the total moles of gas to calculate the total pressure using the ideal gas law : $ \begin { align } \text p_ { \text { total } } & amp ; = \dfrac { ( \text { n } { \text n_2 } +\text n { \text { o } _2 } ) \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 2.14\ , \text { mol } +1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =\dfrac { ( 3.21\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =7.19\ , \text { atm } \end { align } $ once we know the total pressure , we can use the mole fraction version of dalton 's law to calculate the partial pressures : $ \text p_ { \text { n } 2 } = x { \text n_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 2.14\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =4.79\ , \text { atm } $ $ \text p_ { \text { o } 2 } = x { \text o_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 1.07\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =2.40\ , \text { atm } $ luckily , both methods give the same answers !
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is n't that the volume of `` both '' gases ?
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key points the pressure exerted by an individual gas in a mixture is known as its partial pressure . assuming we have a mixture of ideal gases , we can use the ideal gas law to solve problems involving gases in a mixture . dalton 's law of partial pressures states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the component gases : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ dalton 's law can also be expressed using the mole fraction of a gas , $ x $ : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ introduction in day-to-day life , we measure gas pressure when we use a barometer to check the atmospheric pressure outside or a tire gauge to measure the pressure in a bike tube . when we do this , we are measuring a macroscopic physical property of a large number of gas molecules that are invisible to the naked eye . on the molecular level , the pressure we are measuring comes from the force of individual gas molecules colliding with other objects , such as the walls of their container . let 's take a closer look at pressure from a molecular perspective and learn how dalton 's law helps us calculate total and partial pressures for mixtures of gases . ideal gases and partial pressure in this article , we will be assuming the gases in our mixtures can be approximated as ideal gases . this assumption is generally reasonable as long as the temperature of the gas is not super low ( close to $ 0\ , \text k $ ) , and the pressure is around $ 1\ , \text { atm } $ . this means we are making some assumptions about our gas molecules : we assume that the gas molecules take up no volume . we assume that the molecules have no intermolecular attractions , which means they act independently of other gas molecules . based on these assumptions , we can calculate the contribution of different gases in a mixture to the total pressure . we refer to the pressure exerted by a specific gas in a mixture as its partial pressure . the partial pressure of a gas can be calculated using the ideal gas law , which we will cover in the next section , as well as using dalton 's law of partial pressures . example 1 : calculating the partial pressure of a gas let 's say we have a mixture of hydrogen gas , $ \text h_2 ( g ) $ , and oxygen gas , $ \text o_2 ( g ) $ . the mixture contains $ 6.7\ , \text { mol } $ hydrogen gas and $ 3.3\ , \text { mol } $ oxygen gas . the mixture is in a $ 300\ , \text l $ container at $ 273\ , \text k $ , and the total pressure of the gas mixture is $ 0.75\ , \text { atm } $ . the contribution of hydrogen gas to the total pressure is its partial pressure . since the gas molecules in an ideal gas behave independently of other gases in the mixture , the partial pressure of hydrogen is the same pressure as if there were no other gases in the container . therefore , if we want to know the partial pressure of hydrogen gas in the mixture , $ \text p_ { \text h_2 } $ , we can completely ignore the oxygen gas and use the ideal gas law : $ \text p_ { \text h_2 } \text v = \text { n } _ { \text h_2 } \text { rt } $ rearranging the ideal gas equation to solve for $ \text p_ { \text h_2 } $ , we get : $ \begin { align } \text p_ { \text h_2 } & amp ; = \dfrac { \text { n } _ { \text h_2 } \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 6.7\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text l ) } { 300\ , \text l } =0.50\ , \text { atm } \end { align } $ thus , the ideal gas law tells us that the partial pressure of hydrogen in the mixture is $ 0.50\ , \text { atm } $ . we can also calculate the partial pressure of hydrogen in this problem using dalton 's law of partial pressures , which will be discussed in the next section . dalton 's law of partial pressures dalton 's law of partial pressures states that the total pressure of a mixture of gases is the sum of the partial pressures of its components : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ where the partial pressure of each gas is the pressure that the gas would exert if it was the only gas in the container . that is because we assume there are no attractive forces between the gases . dalton 's law of partial pressure can also be expressed in terms of the mole fraction of a gas in the mixture . the mole fraction of a gas is the number of moles of that gas divided by the total moles of gas in the mixture , and it is often abbreviated as $ x $ : $ x_1=\text { mole fraction of gas 1 } =\dfrac { \text { moles of gas 1 } } { \text { total moles of gas } } $ dalton 's law for can be rearranged to give the partial pressure of gas 1 in a mixture in terms of the mole fraction of gas 1 : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ both forms of dalton 's law are extremely useful in solving different kinds of problems including : calculating the partial pressure of a gas when you know the mole ratio and total pressure calculating moles of an individual gas if you know the partial pressure and total pressure calculating the total pressure if you know the partial pressures of the components example 2 : calculating partial pressures and total pressure let 's say that we have one container with $ 24.0\ , \text l $ of nitrogen gas at $ 2.00 \ , \text { atm } $ , and another container with $ 12.0\ , \text l $ of oxygen gas at $ 2.00\ , \text { atm } $ . the temperature of both gases is $ 273\ , \text k $ . if both gases are mixed in a $ 10.0\ , \text l $ container , what are the partial pressures of nitrogen and oxygen in the resulting mixture ? what is the total pressure ? step 1 : calculate moles of oxygen and nitrogen gas since we know $ \text p $ , $ \text v $ , and $ \text t $ for each of the gases before they 're combined , we can find the number of moles of nitrogen gas and oxygen gas using the ideal gas law : $ \text n = \dfrac { \text { pv } } { \text { rt } } $ solving for nitrogen and oxygen , we get : $ \text n_ { \text { n } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 24.0\ , \text { l } ) } { ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 k ) } = 2.14\ , \text { mol nitrogen } $ $ \text n_ { \text { o } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 12.0\ , \text { l } ) } { ( 0.08206 \ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 \ , \text { k } ) } = 1.07\ , \text { mol oxygen } $ step 2 ( method 1 ) : calculate partial pressures and use dalton 's law to get $ \text p_\text { total } $ once we know the number of moles for each gas in our mixture , we can now use the ideal gas law to find the partial pressure of each component in the $ 10.0\ , \text l $ container : $ \text p = \dfrac { \text { nrt } } { \text v } $ $ \text p_ { \text { n } _2 } =\dfrac { ( 2.14\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 4.79\ , \text { atm } $ $ \text p_ { \text { o } _2 } =\dfrac { ( 1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 2.40\ , \text { atm } $ notice that the partial pressure for each of the gases increased compared to the pressure of the gas in the original container . this makes sense since the volume of both gases decreased , and pressure is inversely proportional to volume . we can now get the total pressure of the mixture by adding the partial pressures together using dalton 's law : $ \begin { align } \text p_\text { total } & amp ; =\text p_ { \text { n } 2 } + \text p { \text { o } _2 } \ \ & amp ; =4.79\ , \text { atm } + 2.40\ , \text { atm } = 7.19\ , \text { atm } \end { align } $ step 2 ( method 2 ) : use ideal gas law to calculate $ \text p_\text { total } $ without partial pressures since the pressure of an ideal gas mixture only depends on the number of gas molecules in the container ( and not the identity of the gas molecules ) , we can use the total moles of gas to calculate the total pressure using the ideal gas law : $ \begin { align } \text p_ { \text { total } } & amp ; = \dfrac { ( \text { n } { \text n_2 } +\text n { \text { o } _2 } ) \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 2.14\ , \text { mol } +1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =\dfrac { ( 3.21\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =7.19\ , \text { atm } \end { align } $ once we know the total pressure , we can use the mole fraction version of dalton 's law to calculate the partial pressures : $ \text p_ { \text { n } 2 } = x { \text n_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 2.14\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =4.79\ , \text { atm } $ $ \text p_ { \text { o } 2 } = x { \text o_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 1.07\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =2.40\ , \text { atm } $ luckily , both methods give the same answers ! you might be wondering when you might want to use each method . it mostly depends on which one you prefer , and partly on what you are solving for . for instance , if all you need to know is the total pressure , it might be better to use the second method to save a couple calculation steps . summary the pressure exerted by an individual gas in a mixture is known as its partial pressure . assuming we have a mixture of ideal gases , we can use the ideal gas law to solve problems involving gases in a mixture . dalton 's law of partial pressures states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the component gases : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ dalton 's law can also be expressed using the mole fraction of a gas , $ x $ : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ try it : evaporation in a closed system part 1 part 2
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step 1 : calculate moles of oxygen and nitrogen gas since we know $ \text p $ , $ \text v $ , and $ \text t $ for each of the gases before they 're combined , we can find the number of moles of nitrogen gas and oxygen gas using the ideal gas law : $ \text n = \dfrac { \text { pv } } { \text { rt } } $ solving for nitrogen and oxygen , we get : $ \text n_ { \text { n } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 24.0\ , \text { l } ) } { ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 k ) } = 2.14\ , \text { mol nitrogen } $ $ \text n_ { \text { o } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 12.0\ , \text { l } ) } { ( 0.08206 \ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 \ , \text { k } ) } = 1.07\ , \text { mol oxygen } $ step 2 ( method 1 ) : calculate partial pressures and use dalton 's law to get $ \text p_\text { total } $ once we know the number of moles for each gas in our mixture , we can now use the ideal gas law to find the partial pressure of each component in the $ 10.0\ , \text l $ container : $ \text p = \dfrac { \text { nrt } } { \text v } $ $ \text p_ { \text { n } _2 } =\dfrac { ( 2.14\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 4.79\ , \text { atm } $ $ \text p_ { \text { o } _2 } =\dfrac { ( 1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 2.40\ , \text { atm } $ notice that the partial pressure for each of the gases increased compared to the pressure of the gas in the original container . this makes sense since the volume of both gases decreased , and pressure is inversely proportional to volume . we can now get the total pressure of the mixture by adding the partial pressures together using dalton 's law : $ \begin { align } \text p_\text { total } & amp ; =\text p_ { \text { n } 2 } + \text p { \text { o } _2 } \ \ & amp ; =4.79\ , \text { atm } + 2.40\ , \text { atm } = 7.19\ , \text { atm } \end { align } $ step 2 ( method 2 ) : use ideal gas law to calculate $ \text p_\text { total } $ without partial pressures since the pressure of an ideal gas mixture only depends on the number of gas molecules in the container ( and not the identity of the gas molecules ) , we can use the total moles of gas to calculate the total pressure using the ideal gas law : $ \begin { align } \text p_ { \text { total } } & amp ; = \dfrac { ( \text { n } { \text n_2 } +\text n { \text { o } _2 } ) \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 2.14\ , \text { mol } +1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =\dfrac { ( 3.21\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =7.19\ , \text { atm } \end { align } $ once we know the total pressure , we can use the mole fraction version of dalton 's law to calculate the partial pressures : $ \text p_ { \text { n } 2 } = x { \text n_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 2.14\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =4.79\ , \text { atm } $ $ \text p_ { \text { o } 2 } = x { \text o_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 1.07\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =2.40\ , \text { atm } $ luckily , both methods give the same answers !
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why did n't we use the volume that is due to h2 alone ?
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key points the pressure exerted by an individual gas in a mixture is known as its partial pressure . assuming we have a mixture of ideal gases , we can use the ideal gas law to solve problems involving gases in a mixture . dalton 's law of partial pressures states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the component gases : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ dalton 's law can also be expressed using the mole fraction of a gas , $ x $ : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ introduction in day-to-day life , we measure gas pressure when we use a barometer to check the atmospheric pressure outside or a tire gauge to measure the pressure in a bike tube . when we do this , we are measuring a macroscopic physical property of a large number of gas molecules that are invisible to the naked eye . on the molecular level , the pressure we are measuring comes from the force of individual gas molecules colliding with other objects , such as the walls of their container . let 's take a closer look at pressure from a molecular perspective and learn how dalton 's law helps us calculate total and partial pressures for mixtures of gases . ideal gases and partial pressure in this article , we will be assuming the gases in our mixtures can be approximated as ideal gases . this assumption is generally reasonable as long as the temperature of the gas is not super low ( close to $ 0\ , \text k $ ) , and the pressure is around $ 1\ , \text { atm } $ . this means we are making some assumptions about our gas molecules : we assume that the gas molecules take up no volume . we assume that the molecules have no intermolecular attractions , which means they act independently of other gas molecules . based on these assumptions , we can calculate the contribution of different gases in a mixture to the total pressure . we refer to the pressure exerted by a specific gas in a mixture as its partial pressure . the partial pressure of a gas can be calculated using the ideal gas law , which we will cover in the next section , as well as using dalton 's law of partial pressures . example 1 : calculating the partial pressure of a gas let 's say we have a mixture of hydrogen gas , $ \text h_2 ( g ) $ , and oxygen gas , $ \text o_2 ( g ) $ . the mixture contains $ 6.7\ , \text { mol } $ hydrogen gas and $ 3.3\ , \text { mol } $ oxygen gas . the mixture is in a $ 300\ , \text l $ container at $ 273\ , \text k $ , and the total pressure of the gas mixture is $ 0.75\ , \text { atm } $ . the contribution of hydrogen gas to the total pressure is its partial pressure . since the gas molecules in an ideal gas behave independently of other gases in the mixture , the partial pressure of hydrogen is the same pressure as if there were no other gases in the container . therefore , if we want to know the partial pressure of hydrogen gas in the mixture , $ \text p_ { \text h_2 } $ , we can completely ignore the oxygen gas and use the ideal gas law : $ \text p_ { \text h_2 } \text v = \text { n } _ { \text h_2 } \text { rt } $ rearranging the ideal gas equation to solve for $ \text p_ { \text h_2 } $ , we get : $ \begin { align } \text p_ { \text h_2 } & amp ; = \dfrac { \text { n } _ { \text h_2 } \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 6.7\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text l ) } { 300\ , \text l } =0.50\ , \text { atm } \end { align } $ thus , the ideal gas law tells us that the partial pressure of hydrogen in the mixture is $ 0.50\ , \text { atm } $ . we can also calculate the partial pressure of hydrogen in this problem using dalton 's law of partial pressures , which will be discussed in the next section . dalton 's law of partial pressures dalton 's law of partial pressures states that the total pressure of a mixture of gases is the sum of the partial pressures of its components : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ where the partial pressure of each gas is the pressure that the gas would exert if it was the only gas in the container . that is because we assume there are no attractive forces between the gases . dalton 's law of partial pressure can also be expressed in terms of the mole fraction of a gas in the mixture . the mole fraction of a gas is the number of moles of that gas divided by the total moles of gas in the mixture , and it is often abbreviated as $ x $ : $ x_1=\text { mole fraction of gas 1 } =\dfrac { \text { moles of gas 1 } } { \text { total moles of gas } } $ dalton 's law for can be rearranged to give the partial pressure of gas 1 in a mixture in terms of the mole fraction of gas 1 : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ both forms of dalton 's law are extremely useful in solving different kinds of problems including : calculating the partial pressure of a gas when you know the mole ratio and total pressure calculating moles of an individual gas if you know the partial pressure and total pressure calculating the total pressure if you know the partial pressures of the components example 2 : calculating partial pressures and total pressure let 's say that we have one container with $ 24.0\ , \text l $ of nitrogen gas at $ 2.00 \ , \text { atm } $ , and another container with $ 12.0\ , \text l $ of oxygen gas at $ 2.00\ , \text { atm } $ . the temperature of both gases is $ 273\ , \text k $ . if both gases are mixed in a $ 10.0\ , \text l $ container , what are the partial pressures of nitrogen and oxygen in the resulting mixture ? what is the total pressure ? step 1 : calculate moles of oxygen and nitrogen gas since we know $ \text p $ , $ \text v $ , and $ \text t $ for each of the gases before they 're combined , we can find the number of moles of nitrogen gas and oxygen gas using the ideal gas law : $ \text n = \dfrac { \text { pv } } { \text { rt } } $ solving for nitrogen and oxygen , we get : $ \text n_ { \text { n } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 24.0\ , \text { l } ) } { ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 k ) } = 2.14\ , \text { mol nitrogen } $ $ \text n_ { \text { o } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 12.0\ , \text { l } ) } { ( 0.08206 \ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 \ , \text { k } ) } = 1.07\ , \text { mol oxygen } $ step 2 ( method 1 ) : calculate partial pressures and use dalton 's law to get $ \text p_\text { total } $ once we know the number of moles for each gas in our mixture , we can now use the ideal gas law to find the partial pressure of each component in the $ 10.0\ , \text l $ container : $ \text p = \dfrac { \text { nrt } } { \text v } $ $ \text p_ { \text { n } _2 } =\dfrac { ( 2.14\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 4.79\ , \text { atm } $ $ \text p_ { \text { o } _2 } =\dfrac { ( 1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 2.40\ , \text { atm } $ notice that the partial pressure for each of the gases increased compared to the pressure of the gas in the original container . this makes sense since the volume of both gases decreased , and pressure is inversely proportional to volume . we can now get the total pressure of the mixture by adding the partial pressures together using dalton 's law : $ \begin { align } \text p_\text { total } & amp ; =\text p_ { \text { n } 2 } + \text p { \text { o } _2 } \ \ & amp ; =4.79\ , \text { atm } + 2.40\ , \text { atm } = 7.19\ , \text { atm } \end { align } $ step 2 ( method 2 ) : use ideal gas law to calculate $ \text p_\text { total } $ without partial pressures since the pressure of an ideal gas mixture only depends on the number of gas molecules in the container ( and not the identity of the gas molecules ) , we can use the total moles of gas to calculate the total pressure using the ideal gas law : $ \begin { align } \text p_ { \text { total } } & amp ; = \dfrac { ( \text { n } { \text n_2 } +\text n { \text { o } _2 } ) \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 2.14\ , \text { mol } +1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =\dfrac { ( 3.21\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =7.19\ , \text { atm } \end { align } $ once we know the total pressure , we can use the mole fraction version of dalton 's law to calculate the partial pressures : $ \text p_ { \text { n } 2 } = x { \text n_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 2.14\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =4.79\ , \text { atm } $ $ \text p_ { \text { o } 2 } = x { \text o_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 1.07\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =2.40\ , \text { atm } $ luckily , both methods give the same answers ! you might be wondering when you might want to use each method . it mostly depends on which one you prefer , and partly on what you are solving for . for instance , if all you need to know is the total pressure , it might be better to use the second method to save a couple calculation steps . summary the pressure exerted by an individual gas in a mixture is known as its partial pressure . assuming we have a mixture of ideal gases , we can use the ideal gas law to solve problems involving gases in a mixture . dalton 's law of partial pressures states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the component gases : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ dalton 's law can also be expressed using the mole fraction of a gas , $ x $ : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ try it : evaporation in a closed system part 1 part 2
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the mixture is in a $ 300\ , \text l $ container at $ 273\ , \text k $ , and the total pressure of the gas mixture is $ 0.75\ , \text { atm } $ . the contribution of hydrogen gas to the total pressure is its partial pressure . since the gas molecules in an ideal gas behave independently of other gases in the mixture , the partial pressure of hydrogen is the same pressure as if there were no other gases in the container .
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how can i find partial pressure without being given volume ?
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key points the pressure exerted by an individual gas in a mixture is known as its partial pressure . assuming we have a mixture of ideal gases , we can use the ideal gas law to solve problems involving gases in a mixture . dalton 's law of partial pressures states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the component gases : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ dalton 's law can also be expressed using the mole fraction of a gas , $ x $ : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ introduction in day-to-day life , we measure gas pressure when we use a barometer to check the atmospheric pressure outside or a tire gauge to measure the pressure in a bike tube . when we do this , we are measuring a macroscopic physical property of a large number of gas molecules that are invisible to the naked eye . on the molecular level , the pressure we are measuring comes from the force of individual gas molecules colliding with other objects , such as the walls of their container . let 's take a closer look at pressure from a molecular perspective and learn how dalton 's law helps us calculate total and partial pressures for mixtures of gases . ideal gases and partial pressure in this article , we will be assuming the gases in our mixtures can be approximated as ideal gases . this assumption is generally reasonable as long as the temperature of the gas is not super low ( close to $ 0\ , \text k $ ) , and the pressure is around $ 1\ , \text { atm } $ . this means we are making some assumptions about our gas molecules : we assume that the gas molecules take up no volume . we assume that the molecules have no intermolecular attractions , which means they act independently of other gas molecules . based on these assumptions , we can calculate the contribution of different gases in a mixture to the total pressure . we refer to the pressure exerted by a specific gas in a mixture as its partial pressure . the partial pressure of a gas can be calculated using the ideal gas law , which we will cover in the next section , as well as using dalton 's law of partial pressures . example 1 : calculating the partial pressure of a gas let 's say we have a mixture of hydrogen gas , $ \text h_2 ( g ) $ , and oxygen gas , $ \text o_2 ( g ) $ . the mixture contains $ 6.7\ , \text { mol } $ hydrogen gas and $ 3.3\ , \text { mol } $ oxygen gas . the mixture is in a $ 300\ , \text l $ container at $ 273\ , \text k $ , and the total pressure of the gas mixture is $ 0.75\ , \text { atm } $ . the contribution of hydrogen gas to the total pressure is its partial pressure . since the gas molecules in an ideal gas behave independently of other gases in the mixture , the partial pressure of hydrogen is the same pressure as if there were no other gases in the container . therefore , if we want to know the partial pressure of hydrogen gas in the mixture , $ \text p_ { \text h_2 } $ , we can completely ignore the oxygen gas and use the ideal gas law : $ \text p_ { \text h_2 } \text v = \text { n } _ { \text h_2 } \text { rt } $ rearranging the ideal gas equation to solve for $ \text p_ { \text h_2 } $ , we get : $ \begin { align } \text p_ { \text h_2 } & amp ; = \dfrac { \text { n } _ { \text h_2 } \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 6.7\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text l ) } { 300\ , \text l } =0.50\ , \text { atm } \end { align } $ thus , the ideal gas law tells us that the partial pressure of hydrogen in the mixture is $ 0.50\ , \text { atm } $ . we can also calculate the partial pressure of hydrogen in this problem using dalton 's law of partial pressures , which will be discussed in the next section . dalton 's law of partial pressures dalton 's law of partial pressures states that the total pressure of a mixture of gases is the sum of the partial pressures of its components : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ where the partial pressure of each gas is the pressure that the gas would exert if it was the only gas in the container . that is because we assume there are no attractive forces between the gases . dalton 's law of partial pressure can also be expressed in terms of the mole fraction of a gas in the mixture . the mole fraction of a gas is the number of moles of that gas divided by the total moles of gas in the mixture , and it is often abbreviated as $ x $ : $ x_1=\text { mole fraction of gas 1 } =\dfrac { \text { moles of gas 1 } } { \text { total moles of gas } } $ dalton 's law for can be rearranged to give the partial pressure of gas 1 in a mixture in terms of the mole fraction of gas 1 : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ both forms of dalton 's law are extremely useful in solving different kinds of problems including : calculating the partial pressure of a gas when you know the mole ratio and total pressure calculating moles of an individual gas if you know the partial pressure and total pressure calculating the total pressure if you know the partial pressures of the components example 2 : calculating partial pressures and total pressure let 's say that we have one container with $ 24.0\ , \text l $ of nitrogen gas at $ 2.00 \ , \text { atm } $ , and another container with $ 12.0\ , \text l $ of oxygen gas at $ 2.00\ , \text { atm } $ . the temperature of both gases is $ 273\ , \text k $ . if both gases are mixed in a $ 10.0\ , \text l $ container , what are the partial pressures of nitrogen and oxygen in the resulting mixture ? what is the total pressure ? step 1 : calculate moles of oxygen and nitrogen gas since we know $ \text p $ , $ \text v $ , and $ \text t $ for each of the gases before they 're combined , we can find the number of moles of nitrogen gas and oxygen gas using the ideal gas law : $ \text n = \dfrac { \text { pv } } { \text { rt } } $ solving for nitrogen and oxygen , we get : $ \text n_ { \text { n } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 24.0\ , \text { l } ) } { ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 k ) } = 2.14\ , \text { mol nitrogen } $ $ \text n_ { \text { o } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 12.0\ , \text { l } ) } { ( 0.08206 \ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 \ , \text { k } ) } = 1.07\ , \text { mol oxygen } $ step 2 ( method 1 ) : calculate partial pressures and use dalton 's law to get $ \text p_\text { total } $ once we know the number of moles for each gas in our mixture , we can now use the ideal gas law to find the partial pressure of each component in the $ 10.0\ , \text l $ container : $ \text p = \dfrac { \text { nrt } } { \text v } $ $ \text p_ { \text { n } _2 } =\dfrac { ( 2.14\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 4.79\ , \text { atm } $ $ \text p_ { \text { o } _2 } =\dfrac { ( 1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 2.40\ , \text { atm } $ notice that the partial pressure for each of the gases increased compared to the pressure of the gas in the original container . this makes sense since the volume of both gases decreased , and pressure is inversely proportional to volume . we can now get the total pressure of the mixture by adding the partial pressures together using dalton 's law : $ \begin { align } \text p_\text { total } & amp ; =\text p_ { \text { n } 2 } + \text p { \text { o } _2 } \ \ & amp ; =4.79\ , \text { atm } + 2.40\ , \text { atm } = 7.19\ , \text { atm } \end { align } $ step 2 ( method 2 ) : use ideal gas law to calculate $ \text p_\text { total } $ without partial pressures since the pressure of an ideal gas mixture only depends on the number of gas molecules in the container ( and not the identity of the gas molecules ) , we can use the total moles of gas to calculate the total pressure using the ideal gas law : $ \begin { align } \text p_ { \text { total } } & amp ; = \dfrac { ( \text { n } { \text n_2 } +\text n { \text { o } _2 } ) \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 2.14\ , \text { mol } +1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =\dfrac { ( 3.21\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =7.19\ , \text { atm } \end { align } $ once we know the total pressure , we can use the mole fraction version of dalton 's law to calculate the partial pressures : $ \text p_ { \text { n } 2 } = x { \text n_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 2.14\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =4.79\ , \text { atm } $ $ \text p_ { \text { o } 2 } = x { \text o_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 1.07\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =2.40\ , \text { atm } $ luckily , both methods give the same answers ! you might be wondering when you might want to use each method . it mostly depends on which one you prefer , and partly on what you are solving for . for instance , if all you need to know is the total pressure , it might be better to use the second method to save a couple calculation steps . summary the pressure exerted by an individual gas in a mixture is known as its partial pressure . assuming we have a mixture of ideal gases , we can use the ideal gas law to solve problems involving gases in a mixture . dalton 's law of partial pressures states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the component gases : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ dalton 's law can also be expressed using the mole fraction of a gas , $ x $ : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ try it : evaporation in a closed system part 1 part 2
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the mixture is in a $ 300\ , \text l $ container at $ 273\ , \text k $ , and the total pressure of the gas mixture is $ 0.75\ , \text { atm } $ . the contribution of hydrogen gas to the total pressure is its partial pressure . since the gas molecules in an ideal gas behave independently of other gases in the mixture , the partial pressure of hydrogen is the same pressure as if there were no other gases in the container .
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in question 2 why did n't the addition of helium gas not affect the partial pressure of radon ?
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key points the pressure exerted by an individual gas in a mixture is known as its partial pressure . assuming we have a mixture of ideal gases , we can use the ideal gas law to solve problems involving gases in a mixture . dalton 's law of partial pressures states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the component gases : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ dalton 's law can also be expressed using the mole fraction of a gas , $ x $ : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ introduction in day-to-day life , we measure gas pressure when we use a barometer to check the atmospheric pressure outside or a tire gauge to measure the pressure in a bike tube . when we do this , we are measuring a macroscopic physical property of a large number of gas molecules that are invisible to the naked eye . on the molecular level , the pressure we are measuring comes from the force of individual gas molecules colliding with other objects , such as the walls of their container . let 's take a closer look at pressure from a molecular perspective and learn how dalton 's law helps us calculate total and partial pressures for mixtures of gases . ideal gases and partial pressure in this article , we will be assuming the gases in our mixtures can be approximated as ideal gases . this assumption is generally reasonable as long as the temperature of the gas is not super low ( close to $ 0\ , \text k $ ) , and the pressure is around $ 1\ , \text { atm } $ . this means we are making some assumptions about our gas molecules : we assume that the gas molecules take up no volume . we assume that the molecules have no intermolecular attractions , which means they act independently of other gas molecules . based on these assumptions , we can calculate the contribution of different gases in a mixture to the total pressure . we refer to the pressure exerted by a specific gas in a mixture as its partial pressure . the partial pressure of a gas can be calculated using the ideal gas law , which we will cover in the next section , as well as using dalton 's law of partial pressures . example 1 : calculating the partial pressure of a gas let 's say we have a mixture of hydrogen gas , $ \text h_2 ( g ) $ , and oxygen gas , $ \text o_2 ( g ) $ . the mixture contains $ 6.7\ , \text { mol } $ hydrogen gas and $ 3.3\ , \text { mol } $ oxygen gas . the mixture is in a $ 300\ , \text l $ container at $ 273\ , \text k $ , and the total pressure of the gas mixture is $ 0.75\ , \text { atm } $ . the contribution of hydrogen gas to the total pressure is its partial pressure . since the gas molecules in an ideal gas behave independently of other gases in the mixture , the partial pressure of hydrogen is the same pressure as if there were no other gases in the container . therefore , if we want to know the partial pressure of hydrogen gas in the mixture , $ \text p_ { \text h_2 } $ , we can completely ignore the oxygen gas and use the ideal gas law : $ \text p_ { \text h_2 } \text v = \text { n } _ { \text h_2 } \text { rt } $ rearranging the ideal gas equation to solve for $ \text p_ { \text h_2 } $ , we get : $ \begin { align } \text p_ { \text h_2 } & amp ; = \dfrac { \text { n } _ { \text h_2 } \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 6.7\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text l ) } { 300\ , \text l } =0.50\ , \text { atm } \end { align } $ thus , the ideal gas law tells us that the partial pressure of hydrogen in the mixture is $ 0.50\ , \text { atm } $ . we can also calculate the partial pressure of hydrogen in this problem using dalton 's law of partial pressures , which will be discussed in the next section . dalton 's law of partial pressures dalton 's law of partial pressures states that the total pressure of a mixture of gases is the sum of the partial pressures of its components : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ where the partial pressure of each gas is the pressure that the gas would exert if it was the only gas in the container . that is because we assume there are no attractive forces between the gases . dalton 's law of partial pressure can also be expressed in terms of the mole fraction of a gas in the mixture . the mole fraction of a gas is the number of moles of that gas divided by the total moles of gas in the mixture , and it is often abbreviated as $ x $ : $ x_1=\text { mole fraction of gas 1 } =\dfrac { \text { moles of gas 1 } } { \text { total moles of gas } } $ dalton 's law for can be rearranged to give the partial pressure of gas 1 in a mixture in terms of the mole fraction of gas 1 : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ both forms of dalton 's law are extremely useful in solving different kinds of problems including : calculating the partial pressure of a gas when you know the mole ratio and total pressure calculating moles of an individual gas if you know the partial pressure and total pressure calculating the total pressure if you know the partial pressures of the components example 2 : calculating partial pressures and total pressure let 's say that we have one container with $ 24.0\ , \text l $ of nitrogen gas at $ 2.00 \ , \text { atm } $ , and another container with $ 12.0\ , \text l $ of oxygen gas at $ 2.00\ , \text { atm } $ . the temperature of both gases is $ 273\ , \text k $ . if both gases are mixed in a $ 10.0\ , \text l $ container , what are the partial pressures of nitrogen and oxygen in the resulting mixture ? what is the total pressure ? step 1 : calculate moles of oxygen and nitrogen gas since we know $ \text p $ , $ \text v $ , and $ \text t $ for each of the gases before they 're combined , we can find the number of moles of nitrogen gas and oxygen gas using the ideal gas law : $ \text n = \dfrac { \text { pv } } { \text { rt } } $ solving for nitrogen and oxygen , we get : $ \text n_ { \text { n } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 24.0\ , \text { l } ) } { ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 k ) } = 2.14\ , \text { mol nitrogen } $ $ \text n_ { \text { o } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 12.0\ , \text { l } ) } { ( 0.08206 \ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 \ , \text { k } ) } = 1.07\ , \text { mol oxygen } $ step 2 ( method 1 ) : calculate partial pressures and use dalton 's law to get $ \text p_\text { total } $ once we know the number of moles for each gas in our mixture , we can now use the ideal gas law to find the partial pressure of each component in the $ 10.0\ , \text l $ container : $ \text p = \dfrac { \text { nrt } } { \text v } $ $ \text p_ { \text { n } _2 } =\dfrac { ( 2.14\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 4.79\ , \text { atm } $ $ \text p_ { \text { o } _2 } =\dfrac { ( 1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 2.40\ , \text { atm } $ notice that the partial pressure for each of the gases increased compared to the pressure of the gas in the original container . this makes sense since the volume of both gases decreased , and pressure is inversely proportional to volume . we can now get the total pressure of the mixture by adding the partial pressures together using dalton 's law : $ \begin { align } \text p_\text { total } & amp ; =\text p_ { \text { n } 2 } + \text p { \text { o } _2 } \ \ & amp ; =4.79\ , \text { atm } + 2.40\ , \text { atm } = 7.19\ , \text { atm } \end { align } $ step 2 ( method 2 ) : use ideal gas law to calculate $ \text p_\text { total } $ without partial pressures since the pressure of an ideal gas mixture only depends on the number of gas molecules in the container ( and not the identity of the gas molecules ) , we can use the total moles of gas to calculate the total pressure using the ideal gas law : $ \begin { align } \text p_ { \text { total } } & amp ; = \dfrac { ( \text { n } { \text n_2 } +\text n { \text { o } _2 } ) \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 2.14\ , \text { mol } +1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =\dfrac { ( 3.21\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =7.19\ , \text { atm } \end { align } $ once we know the total pressure , we can use the mole fraction version of dalton 's law to calculate the partial pressures : $ \text p_ { \text { n } 2 } = x { \text n_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 2.14\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =4.79\ , \text { atm } $ $ \text p_ { \text { o } 2 } = x { \text o_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 1.07\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =2.40\ , \text { atm } $ luckily , both methods give the same answers ! you might be wondering when you might want to use each method . it mostly depends on which one you prefer , and partly on what you are solving for . for instance , if all you need to know is the total pressure , it might be better to use the second method to save a couple calculation steps . summary the pressure exerted by an individual gas in a mixture is known as its partial pressure . assuming we have a mixture of ideal gases , we can use the ideal gas law to solve problems involving gases in a mixture . dalton 's law of partial pressures states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the component gases : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ dalton 's law can also be expressed using the mole fraction of a gas , $ x $ : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ try it : evaporation in a closed system part 1 part 2
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the mole fraction of a gas is the number of moles of that gas divided by the total moles of gas in the mixture , and it is often abbreviated as $ x $ : $ x_1=\text { mole fraction of gas 1 } =\dfrac { \text { moles of gas 1 } } { \text { total moles of gas } } $ dalton 's law for can be rearranged to give the partial pressure of gas 1 in a mixture in terms of the mole fraction of gas 1 : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ both forms of dalton 's law are extremely useful in solving different kinds of problems including : calculating the partial pressure of a gas when you know the mole ratio and total pressure calculating moles of an individual gas if you know the partial pressure and total pressure calculating the total pressure if you know the partial pressures of the components example 2 : calculating partial pressures and total pressure let 's say that we have one container with $ 24.0\ , \text l $ of nitrogen gas at $ 2.00 \ , \text { atm } $ , and another container with $ 12.0\ , \text l $ of oxygen gas at $ 2.00\ , \text { atm } $ . the temperature of both gases is $ 273\ , \text k $ . if both gases are mixed in a $ 10.0\ , \text l $ container , what are the partial pressures of nitrogen and oxygen in the resulting mixture ?
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what would you do if you are not given a temperature ?
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key points the pressure exerted by an individual gas in a mixture is known as its partial pressure . assuming we have a mixture of ideal gases , we can use the ideal gas law to solve problems involving gases in a mixture . dalton 's law of partial pressures states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the component gases : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ dalton 's law can also be expressed using the mole fraction of a gas , $ x $ : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ introduction in day-to-day life , we measure gas pressure when we use a barometer to check the atmospheric pressure outside or a tire gauge to measure the pressure in a bike tube . when we do this , we are measuring a macroscopic physical property of a large number of gas molecules that are invisible to the naked eye . on the molecular level , the pressure we are measuring comes from the force of individual gas molecules colliding with other objects , such as the walls of their container . let 's take a closer look at pressure from a molecular perspective and learn how dalton 's law helps us calculate total and partial pressures for mixtures of gases . ideal gases and partial pressure in this article , we will be assuming the gases in our mixtures can be approximated as ideal gases . this assumption is generally reasonable as long as the temperature of the gas is not super low ( close to $ 0\ , \text k $ ) , and the pressure is around $ 1\ , \text { atm } $ . this means we are making some assumptions about our gas molecules : we assume that the gas molecules take up no volume . we assume that the molecules have no intermolecular attractions , which means they act independently of other gas molecules . based on these assumptions , we can calculate the contribution of different gases in a mixture to the total pressure . we refer to the pressure exerted by a specific gas in a mixture as its partial pressure . the partial pressure of a gas can be calculated using the ideal gas law , which we will cover in the next section , as well as using dalton 's law of partial pressures . example 1 : calculating the partial pressure of a gas let 's say we have a mixture of hydrogen gas , $ \text h_2 ( g ) $ , and oxygen gas , $ \text o_2 ( g ) $ . the mixture contains $ 6.7\ , \text { mol } $ hydrogen gas and $ 3.3\ , \text { mol } $ oxygen gas . the mixture is in a $ 300\ , \text l $ container at $ 273\ , \text k $ , and the total pressure of the gas mixture is $ 0.75\ , \text { atm } $ . the contribution of hydrogen gas to the total pressure is its partial pressure . since the gas molecules in an ideal gas behave independently of other gases in the mixture , the partial pressure of hydrogen is the same pressure as if there were no other gases in the container . therefore , if we want to know the partial pressure of hydrogen gas in the mixture , $ \text p_ { \text h_2 } $ , we can completely ignore the oxygen gas and use the ideal gas law : $ \text p_ { \text h_2 } \text v = \text { n } _ { \text h_2 } \text { rt } $ rearranging the ideal gas equation to solve for $ \text p_ { \text h_2 } $ , we get : $ \begin { align } \text p_ { \text h_2 } & amp ; = \dfrac { \text { n } _ { \text h_2 } \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 6.7\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text l ) } { 300\ , \text l } =0.50\ , \text { atm } \end { align } $ thus , the ideal gas law tells us that the partial pressure of hydrogen in the mixture is $ 0.50\ , \text { atm } $ . we can also calculate the partial pressure of hydrogen in this problem using dalton 's law of partial pressures , which will be discussed in the next section . dalton 's law of partial pressures dalton 's law of partial pressures states that the total pressure of a mixture of gases is the sum of the partial pressures of its components : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ where the partial pressure of each gas is the pressure that the gas would exert if it was the only gas in the container . that is because we assume there are no attractive forces between the gases . dalton 's law of partial pressure can also be expressed in terms of the mole fraction of a gas in the mixture . the mole fraction of a gas is the number of moles of that gas divided by the total moles of gas in the mixture , and it is often abbreviated as $ x $ : $ x_1=\text { mole fraction of gas 1 } =\dfrac { \text { moles of gas 1 } } { \text { total moles of gas } } $ dalton 's law for can be rearranged to give the partial pressure of gas 1 in a mixture in terms of the mole fraction of gas 1 : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ both forms of dalton 's law are extremely useful in solving different kinds of problems including : calculating the partial pressure of a gas when you know the mole ratio and total pressure calculating moles of an individual gas if you know the partial pressure and total pressure calculating the total pressure if you know the partial pressures of the components example 2 : calculating partial pressures and total pressure let 's say that we have one container with $ 24.0\ , \text l $ of nitrogen gas at $ 2.00 \ , \text { atm } $ , and another container with $ 12.0\ , \text l $ of oxygen gas at $ 2.00\ , \text { atm } $ . the temperature of both gases is $ 273\ , \text k $ . if both gases are mixed in a $ 10.0\ , \text l $ container , what are the partial pressures of nitrogen and oxygen in the resulting mixture ? what is the total pressure ? step 1 : calculate moles of oxygen and nitrogen gas since we know $ \text p $ , $ \text v $ , and $ \text t $ for each of the gases before they 're combined , we can find the number of moles of nitrogen gas and oxygen gas using the ideal gas law : $ \text n = \dfrac { \text { pv } } { \text { rt } } $ solving for nitrogen and oxygen , we get : $ \text n_ { \text { n } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 24.0\ , \text { l } ) } { ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 k ) } = 2.14\ , \text { mol nitrogen } $ $ \text n_ { \text { o } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 12.0\ , \text { l } ) } { ( 0.08206 \ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 \ , \text { k } ) } = 1.07\ , \text { mol oxygen } $ step 2 ( method 1 ) : calculate partial pressures and use dalton 's law to get $ \text p_\text { total } $ once we know the number of moles for each gas in our mixture , we can now use the ideal gas law to find the partial pressure of each component in the $ 10.0\ , \text l $ container : $ \text p = \dfrac { \text { nrt } } { \text v } $ $ \text p_ { \text { n } _2 } =\dfrac { ( 2.14\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 4.79\ , \text { atm } $ $ \text p_ { \text { o } _2 } =\dfrac { ( 1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 2.40\ , \text { atm } $ notice that the partial pressure for each of the gases increased compared to the pressure of the gas in the original container . this makes sense since the volume of both gases decreased , and pressure is inversely proportional to volume . we can now get the total pressure of the mixture by adding the partial pressures together using dalton 's law : $ \begin { align } \text p_\text { total } & amp ; =\text p_ { \text { n } 2 } + \text p { \text { o } _2 } \ \ & amp ; =4.79\ , \text { atm } + 2.40\ , \text { atm } = 7.19\ , \text { atm } \end { align } $ step 2 ( method 2 ) : use ideal gas law to calculate $ \text p_\text { total } $ without partial pressures since the pressure of an ideal gas mixture only depends on the number of gas molecules in the container ( and not the identity of the gas molecules ) , we can use the total moles of gas to calculate the total pressure using the ideal gas law : $ \begin { align } \text p_ { \text { total } } & amp ; = \dfrac { ( \text { n } { \text n_2 } +\text n { \text { o } _2 } ) \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 2.14\ , \text { mol } +1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =\dfrac { ( 3.21\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =7.19\ , \text { atm } \end { align } $ once we know the total pressure , we can use the mole fraction version of dalton 's law to calculate the partial pressures : $ \text p_ { \text { n } 2 } = x { \text n_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 2.14\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =4.79\ , \text { atm } $ $ \text p_ { \text { o } 2 } = x { \text o_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 1.07\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =2.40\ , \text { atm } $ luckily , both methods give the same answers ! you might be wondering when you might want to use each method . it mostly depends on which one you prefer , and partly on what you are solving for . for instance , if all you need to know is the total pressure , it might be better to use the second method to save a couple calculation steps . summary the pressure exerted by an individual gas in a mixture is known as its partial pressure . assuming we have a mixture of ideal gases , we can use the ideal gas law to solve problems involving gases in a mixture . dalton 's law of partial pressures states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the component gases : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ dalton 's law can also be expressed using the mole fraction of a gas , $ x $ : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ try it : evaporation in a closed system part 1 part 2
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the mixture is in a $ 300\ , \text l $ container at $ 273\ , \text k $ , and the total pressure of the gas mixture is $ 0.75\ , \text { atm } $ . the contribution of hydrogen gas to the total pressure is its partial pressure . since the gas molecules in an ideal gas behave independently of other gases in the mixture , the partial pressure of hydrogen is the same pressure as if there were no other gases in the container .
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what will be the final pressure in the vessel ?
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key points the pressure exerted by an individual gas in a mixture is known as its partial pressure . assuming we have a mixture of ideal gases , we can use the ideal gas law to solve problems involving gases in a mixture . dalton 's law of partial pressures states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the component gases : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ dalton 's law can also be expressed using the mole fraction of a gas , $ x $ : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ introduction in day-to-day life , we measure gas pressure when we use a barometer to check the atmospheric pressure outside or a tire gauge to measure the pressure in a bike tube . when we do this , we are measuring a macroscopic physical property of a large number of gas molecules that are invisible to the naked eye . on the molecular level , the pressure we are measuring comes from the force of individual gas molecules colliding with other objects , such as the walls of their container . let 's take a closer look at pressure from a molecular perspective and learn how dalton 's law helps us calculate total and partial pressures for mixtures of gases . ideal gases and partial pressure in this article , we will be assuming the gases in our mixtures can be approximated as ideal gases . this assumption is generally reasonable as long as the temperature of the gas is not super low ( close to $ 0\ , \text k $ ) , and the pressure is around $ 1\ , \text { atm } $ . this means we are making some assumptions about our gas molecules : we assume that the gas molecules take up no volume . we assume that the molecules have no intermolecular attractions , which means they act independently of other gas molecules . based on these assumptions , we can calculate the contribution of different gases in a mixture to the total pressure . we refer to the pressure exerted by a specific gas in a mixture as its partial pressure . the partial pressure of a gas can be calculated using the ideal gas law , which we will cover in the next section , as well as using dalton 's law of partial pressures . example 1 : calculating the partial pressure of a gas let 's say we have a mixture of hydrogen gas , $ \text h_2 ( g ) $ , and oxygen gas , $ \text o_2 ( g ) $ . the mixture contains $ 6.7\ , \text { mol } $ hydrogen gas and $ 3.3\ , \text { mol } $ oxygen gas . the mixture is in a $ 300\ , \text l $ container at $ 273\ , \text k $ , and the total pressure of the gas mixture is $ 0.75\ , \text { atm } $ . the contribution of hydrogen gas to the total pressure is its partial pressure . since the gas molecules in an ideal gas behave independently of other gases in the mixture , the partial pressure of hydrogen is the same pressure as if there were no other gases in the container . therefore , if we want to know the partial pressure of hydrogen gas in the mixture , $ \text p_ { \text h_2 } $ , we can completely ignore the oxygen gas and use the ideal gas law : $ \text p_ { \text h_2 } \text v = \text { n } _ { \text h_2 } \text { rt } $ rearranging the ideal gas equation to solve for $ \text p_ { \text h_2 } $ , we get : $ \begin { align } \text p_ { \text h_2 } & amp ; = \dfrac { \text { n } _ { \text h_2 } \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 6.7\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text l ) } { 300\ , \text l } =0.50\ , \text { atm } \end { align } $ thus , the ideal gas law tells us that the partial pressure of hydrogen in the mixture is $ 0.50\ , \text { atm } $ . we can also calculate the partial pressure of hydrogen in this problem using dalton 's law of partial pressures , which will be discussed in the next section . dalton 's law of partial pressures dalton 's law of partial pressures states that the total pressure of a mixture of gases is the sum of the partial pressures of its components : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ where the partial pressure of each gas is the pressure that the gas would exert if it was the only gas in the container . that is because we assume there are no attractive forces between the gases . dalton 's law of partial pressure can also be expressed in terms of the mole fraction of a gas in the mixture . the mole fraction of a gas is the number of moles of that gas divided by the total moles of gas in the mixture , and it is often abbreviated as $ x $ : $ x_1=\text { mole fraction of gas 1 } =\dfrac { \text { moles of gas 1 } } { \text { total moles of gas } } $ dalton 's law for can be rearranged to give the partial pressure of gas 1 in a mixture in terms of the mole fraction of gas 1 : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ both forms of dalton 's law are extremely useful in solving different kinds of problems including : calculating the partial pressure of a gas when you know the mole ratio and total pressure calculating moles of an individual gas if you know the partial pressure and total pressure calculating the total pressure if you know the partial pressures of the components example 2 : calculating partial pressures and total pressure let 's say that we have one container with $ 24.0\ , \text l $ of nitrogen gas at $ 2.00 \ , \text { atm } $ , and another container with $ 12.0\ , \text l $ of oxygen gas at $ 2.00\ , \text { atm } $ . the temperature of both gases is $ 273\ , \text k $ . if both gases are mixed in a $ 10.0\ , \text l $ container , what are the partial pressures of nitrogen and oxygen in the resulting mixture ? what is the total pressure ? step 1 : calculate moles of oxygen and nitrogen gas since we know $ \text p $ , $ \text v $ , and $ \text t $ for each of the gases before they 're combined , we can find the number of moles of nitrogen gas and oxygen gas using the ideal gas law : $ \text n = \dfrac { \text { pv } } { \text { rt } } $ solving for nitrogen and oxygen , we get : $ \text n_ { \text { n } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 24.0\ , \text { l } ) } { ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 k ) } = 2.14\ , \text { mol nitrogen } $ $ \text n_ { \text { o } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 12.0\ , \text { l } ) } { ( 0.08206 \ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 \ , \text { k } ) } = 1.07\ , \text { mol oxygen } $ step 2 ( method 1 ) : calculate partial pressures and use dalton 's law to get $ \text p_\text { total } $ once we know the number of moles for each gas in our mixture , we can now use the ideal gas law to find the partial pressure of each component in the $ 10.0\ , \text l $ container : $ \text p = \dfrac { \text { nrt } } { \text v } $ $ \text p_ { \text { n } _2 } =\dfrac { ( 2.14\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 4.79\ , \text { atm } $ $ \text p_ { \text { o } _2 } =\dfrac { ( 1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 2.40\ , \text { atm } $ notice that the partial pressure for each of the gases increased compared to the pressure of the gas in the original container . this makes sense since the volume of both gases decreased , and pressure is inversely proportional to volume . we can now get the total pressure of the mixture by adding the partial pressures together using dalton 's law : $ \begin { align } \text p_\text { total } & amp ; =\text p_ { \text { n } 2 } + \text p { \text { o } _2 } \ \ & amp ; =4.79\ , \text { atm } + 2.40\ , \text { atm } = 7.19\ , \text { atm } \end { align } $ step 2 ( method 2 ) : use ideal gas law to calculate $ \text p_\text { total } $ without partial pressures since the pressure of an ideal gas mixture only depends on the number of gas molecules in the container ( and not the identity of the gas molecules ) , we can use the total moles of gas to calculate the total pressure using the ideal gas law : $ \begin { align } \text p_ { \text { total } } & amp ; = \dfrac { ( \text { n } { \text n_2 } +\text n { \text { o } _2 } ) \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 2.14\ , \text { mol } +1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =\dfrac { ( 3.21\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =7.19\ , \text { atm } \end { align } $ once we know the total pressure , we can use the mole fraction version of dalton 's law to calculate the partial pressures : $ \text p_ { \text { n } 2 } = x { \text n_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 2.14\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =4.79\ , \text { atm } $ $ \text p_ { \text { o } 2 } = x { \text o_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 1.07\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =2.40\ , \text { atm } $ luckily , both methods give the same answers ! you might be wondering when you might want to use each method . it mostly depends on which one you prefer , and partly on what you are solving for . for instance , if all you need to know is the total pressure , it might be better to use the second method to save a couple calculation steps . summary the pressure exerted by an individual gas in a mixture is known as its partial pressure . assuming we have a mixture of ideal gases , we can use the ideal gas law to solve problems involving gases in a mixture . dalton 's law of partial pressures states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the component gases : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ dalton 's law can also be expressed using the mole fraction of a gas , $ x $ : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ try it : evaporation in a closed system part 1 part 2
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that is because we assume there are no attractive forces between the gases . dalton 's law of partial pressure can also be expressed in terms of the mole fraction of a gas in the mixture . the mole fraction of a gas is the number of moles of that gas divided by the total moles of gas in the mixture , and it is often abbreviated as $ x $ : $ x_1=\text { mole fraction of gas 1 } =\dfrac { \text { moles of gas 1 } } { \text { total moles of gas } } $ dalton 's law for can be rearranged to give the partial pressure of gas 1 in a mixture in terms of the mole fraction of gas 1 : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ both forms of dalton 's law are extremely useful in solving different kinds of problems including : calculating the partial pressure of a gas when you know the mole ratio and total pressure calculating moles of an individual gas if you know the partial pressure and total pressure calculating the total pressure if you know the partial pressures of the components example 2 : calculating partial pressures and total pressure let 's say that we have one container with $ 24.0\ , \text l $ of nitrogen gas at $ 2.00 \ , \text { atm } $ , and another container with $ 12.0\ , \text l $ of oxygen gas at $ 2.00\ , \text { atm } $ . the temperature of both gases is $ 273\ , \text k $ .
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can someone explain what is meant by , 'mole fraction ' of the gas ?
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key points the pressure exerted by an individual gas in a mixture is known as its partial pressure . assuming we have a mixture of ideal gases , we can use the ideal gas law to solve problems involving gases in a mixture . dalton 's law of partial pressures states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the component gases : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ dalton 's law can also be expressed using the mole fraction of a gas , $ x $ : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ introduction in day-to-day life , we measure gas pressure when we use a barometer to check the atmospheric pressure outside or a tire gauge to measure the pressure in a bike tube . when we do this , we are measuring a macroscopic physical property of a large number of gas molecules that are invisible to the naked eye . on the molecular level , the pressure we are measuring comes from the force of individual gas molecules colliding with other objects , such as the walls of their container . let 's take a closer look at pressure from a molecular perspective and learn how dalton 's law helps us calculate total and partial pressures for mixtures of gases . ideal gases and partial pressure in this article , we will be assuming the gases in our mixtures can be approximated as ideal gases . this assumption is generally reasonable as long as the temperature of the gas is not super low ( close to $ 0\ , \text k $ ) , and the pressure is around $ 1\ , \text { atm } $ . this means we are making some assumptions about our gas molecules : we assume that the gas molecules take up no volume . we assume that the molecules have no intermolecular attractions , which means they act independently of other gas molecules . based on these assumptions , we can calculate the contribution of different gases in a mixture to the total pressure . we refer to the pressure exerted by a specific gas in a mixture as its partial pressure . the partial pressure of a gas can be calculated using the ideal gas law , which we will cover in the next section , as well as using dalton 's law of partial pressures . example 1 : calculating the partial pressure of a gas let 's say we have a mixture of hydrogen gas , $ \text h_2 ( g ) $ , and oxygen gas , $ \text o_2 ( g ) $ . the mixture contains $ 6.7\ , \text { mol } $ hydrogen gas and $ 3.3\ , \text { mol } $ oxygen gas . the mixture is in a $ 300\ , \text l $ container at $ 273\ , \text k $ , and the total pressure of the gas mixture is $ 0.75\ , \text { atm } $ . the contribution of hydrogen gas to the total pressure is its partial pressure . since the gas molecules in an ideal gas behave independently of other gases in the mixture , the partial pressure of hydrogen is the same pressure as if there were no other gases in the container . therefore , if we want to know the partial pressure of hydrogen gas in the mixture , $ \text p_ { \text h_2 } $ , we can completely ignore the oxygen gas and use the ideal gas law : $ \text p_ { \text h_2 } \text v = \text { n } _ { \text h_2 } \text { rt } $ rearranging the ideal gas equation to solve for $ \text p_ { \text h_2 } $ , we get : $ \begin { align } \text p_ { \text h_2 } & amp ; = \dfrac { \text { n } _ { \text h_2 } \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 6.7\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text l ) } { 300\ , \text l } =0.50\ , \text { atm } \end { align } $ thus , the ideal gas law tells us that the partial pressure of hydrogen in the mixture is $ 0.50\ , \text { atm } $ . we can also calculate the partial pressure of hydrogen in this problem using dalton 's law of partial pressures , which will be discussed in the next section . dalton 's law of partial pressures dalton 's law of partial pressures states that the total pressure of a mixture of gases is the sum of the partial pressures of its components : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ where the partial pressure of each gas is the pressure that the gas would exert if it was the only gas in the container . that is because we assume there are no attractive forces between the gases . dalton 's law of partial pressure can also be expressed in terms of the mole fraction of a gas in the mixture . the mole fraction of a gas is the number of moles of that gas divided by the total moles of gas in the mixture , and it is often abbreviated as $ x $ : $ x_1=\text { mole fraction of gas 1 } =\dfrac { \text { moles of gas 1 } } { \text { total moles of gas } } $ dalton 's law for can be rearranged to give the partial pressure of gas 1 in a mixture in terms of the mole fraction of gas 1 : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ both forms of dalton 's law are extremely useful in solving different kinds of problems including : calculating the partial pressure of a gas when you know the mole ratio and total pressure calculating moles of an individual gas if you know the partial pressure and total pressure calculating the total pressure if you know the partial pressures of the components example 2 : calculating partial pressures and total pressure let 's say that we have one container with $ 24.0\ , \text l $ of nitrogen gas at $ 2.00 \ , \text { atm } $ , and another container with $ 12.0\ , \text l $ of oxygen gas at $ 2.00\ , \text { atm } $ . the temperature of both gases is $ 273\ , \text k $ . if both gases are mixed in a $ 10.0\ , \text l $ container , what are the partial pressures of nitrogen and oxygen in the resulting mixture ? what is the total pressure ? step 1 : calculate moles of oxygen and nitrogen gas since we know $ \text p $ , $ \text v $ , and $ \text t $ for each of the gases before they 're combined , we can find the number of moles of nitrogen gas and oxygen gas using the ideal gas law : $ \text n = \dfrac { \text { pv } } { \text { rt } } $ solving for nitrogen and oxygen , we get : $ \text n_ { \text { n } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 24.0\ , \text { l } ) } { ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 k ) } = 2.14\ , \text { mol nitrogen } $ $ \text n_ { \text { o } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 12.0\ , \text { l } ) } { ( 0.08206 \ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 \ , \text { k } ) } = 1.07\ , \text { mol oxygen } $ step 2 ( method 1 ) : calculate partial pressures and use dalton 's law to get $ \text p_\text { total } $ once we know the number of moles for each gas in our mixture , we can now use the ideal gas law to find the partial pressure of each component in the $ 10.0\ , \text l $ container : $ \text p = \dfrac { \text { nrt } } { \text v } $ $ \text p_ { \text { n } _2 } =\dfrac { ( 2.14\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 4.79\ , \text { atm } $ $ \text p_ { \text { o } _2 } =\dfrac { ( 1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 2.40\ , \text { atm } $ notice that the partial pressure for each of the gases increased compared to the pressure of the gas in the original container . this makes sense since the volume of both gases decreased , and pressure is inversely proportional to volume . we can now get the total pressure of the mixture by adding the partial pressures together using dalton 's law : $ \begin { align } \text p_\text { total } & amp ; =\text p_ { \text { n } 2 } + \text p { \text { o } _2 } \ \ & amp ; =4.79\ , \text { atm } + 2.40\ , \text { atm } = 7.19\ , \text { atm } \end { align } $ step 2 ( method 2 ) : use ideal gas law to calculate $ \text p_\text { total } $ without partial pressures since the pressure of an ideal gas mixture only depends on the number of gas molecules in the container ( and not the identity of the gas molecules ) , we can use the total moles of gas to calculate the total pressure using the ideal gas law : $ \begin { align } \text p_ { \text { total } } & amp ; = \dfrac { ( \text { n } { \text n_2 } +\text n { \text { o } _2 } ) \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 2.14\ , \text { mol } +1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =\dfrac { ( 3.21\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =7.19\ , \text { atm } \end { align } $ once we know the total pressure , we can use the mole fraction version of dalton 's law to calculate the partial pressures : $ \text p_ { \text { n } 2 } = x { \text n_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 2.14\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =4.79\ , \text { atm } $ $ \text p_ { \text { o } 2 } = x { \text o_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 1.07\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =2.40\ , \text { atm } $ luckily , both methods give the same answers ! you might be wondering when you might want to use each method . it mostly depends on which one you prefer , and partly on what you are solving for . for instance , if all you need to know is the total pressure , it might be better to use the second method to save a couple calculation steps . summary the pressure exerted by an individual gas in a mixture is known as its partial pressure . assuming we have a mixture of ideal gases , we can use the ideal gas law to solve problems involving gases in a mixture . dalton 's law of partial pressures states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the component gases : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ dalton 's law can also be expressed using the mole fraction of a gas , $ x $ : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ try it : evaporation in a closed system part 1 part 2
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step 1 : calculate moles of oxygen and nitrogen gas since we know $ \text p $ , $ \text v $ , and $ \text t $ for each of the gases before they 're combined , we can find the number of moles of nitrogen gas and oxygen gas using the ideal gas law : $ \text n = \dfrac { \text { pv } } { \text { rt } } $ solving for nitrogen and oxygen , we get : $ \text n_ { \text { n } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 24.0\ , \text { l } ) } { ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 k ) } = 2.14\ , \text { mol nitrogen } $ $ \text n_ { \text { o } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 12.0\ , \text { l } ) } { ( 0.08206 \ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 \ , \text { k } ) } = 1.07\ , \text { mol oxygen } $ step 2 ( method 1 ) : calculate partial pressures and use dalton 's law to get $ \text p_\text { total } $ once we know the number of moles for each gas in our mixture , we can now use the ideal gas law to find the partial pressure of each component in the $ 10.0\ , \text l $ container : $ \text p = \dfrac { \text { nrt } } { \text v } $ $ \text p_ { \text { n } _2 } =\dfrac { ( 2.14\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 4.79\ , \text { atm } $ $ \text p_ { \text { o } _2 } =\dfrac { ( 1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 2.40\ , \text { atm } $ notice that the partial pressure for each of the gases increased compared to the pressure of the gas in the original container . this makes sense since the volume of both gases decreased , and pressure is inversely proportional to volume . we can now get the total pressure of the mixture by adding the partial pressures together using dalton 's law : $ \begin { align } \text p_\text { total } & amp ; =\text p_ { \text { n } 2 } + \text p { \text { o } _2 } \ \ & amp ; =4.79\ , \text { atm } + 2.40\ , \text { atm } = 7.19\ , \text { atm } \end { align } $ step 2 ( method 2 ) : use ideal gas law to calculate $ \text p_\text { total } $ without partial pressures since the pressure of an ideal gas mixture only depends on the number of gas molecules in the container ( and not the identity of the gas molecules ) , we can use the total moles of gas to calculate the total pressure using the ideal gas law : $ \begin { align } \text p_ { \text { total } } & amp ; = \dfrac { ( \text { n } { \text n_2 } +\text n { \text { o } _2 } ) \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 2.14\ , \text { mol } +1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =\dfrac { ( 3.21\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =7.19\ , \text { atm } \end { align } $ once we know the total pressure , we can use the mole fraction version of dalton 's law to calculate the partial pressures : $ \text p_ { \text { n } 2 } = x { \text n_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 2.14\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =4.79\ , \text { atm } $ $ \text p_ { \text { o } 2 } = x { \text o_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 1.07\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =2.40\ , \text { atm } $ luckily , both methods give the same answers !
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why in the first question did we not use the given temperature and volume ?
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key points the pressure exerted by an individual gas in a mixture is known as its partial pressure . assuming we have a mixture of ideal gases , we can use the ideal gas law to solve problems involving gases in a mixture . dalton 's law of partial pressures states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the component gases : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ dalton 's law can also be expressed using the mole fraction of a gas , $ x $ : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ introduction in day-to-day life , we measure gas pressure when we use a barometer to check the atmospheric pressure outside or a tire gauge to measure the pressure in a bike tube . when we do this , we are measuring a macroscopic physical property of a large number of gas molecules that are invisible to the naked eye . on the molecular level , the pressure we are measuring comes from the force of individual gas molecules colliding with other objects , such as the walls of their container . let 's take a closer look at pressure from a molecular perspective and learn how dalton 's law helps us calculate total and partial pressures for mixtures of gases . ideal gases and partial pressure in this article , we will be assuming the gases in our mixtures can be approximated as ideal gases . this assumption is generally reasonable as long as the temperature of the gas is not super low ( close to $ 0\ , \text k $ ) , and the pressure is around $ 1\ , \text { atm } $ . this means we are making some assumptions about our gas molecules : we assume that the gas molecules take up no volume . we assume that the molecules have no intermolecular attractions , which means they act independently of other gas molecules . based on these assumptions , we can calculate the contribution of different gases in a mixture to the total pressure . we refer to the pressure exerted by a specific gas in a mixture as its partial pressure . the partial pressure of a gas can be calculated using the ideal gas law , which we will cover in the next section , as well as using dalton 's law of partial pressures . example 1 : calculating the partial pressure of a gas let 's say we have a mixture of hydrogen gas , $ \text h_2 ( g ) $ , and oxygen gas , $ \text o_2 ( g ) $ . the mixture contains $ 6.7\ , \text { mol } $ hydrogen gas and $ 3.3\ , \text { mol } $ oxygen gas . the mixture is in a $ 300\ , \text l $ container at $ 273\ , \text k $ , and the total pressure of the gas mixture is $ 0.75\ , \text { atm } $ . the contribution of hydrogen gas to the total pressure is its partial pressure . since the gas molecules in an ideal gas behave independently of other gases in the mixture , the partial pressure of hydrogen is the same pressure as if there were no other gases in the container . therefore , if we want to know the partial pressure of hydrogen gas in the mixture , $ \text p_ { \text h_2 } $ , we can completely ignore the oxygen gas and use the ideal gas law : $ \text p_ { \text h_2 } \text v = \text { n } _ { \text h_2 } \text { rt } $ rearranging the ideal gas equation to solve for $ \text p_ { \text h_2 } $ , we get : $ \begin { align } \text p_ { \text h_2 } & amp ; = \dfrac { \text { n } _ { \text h_2 } \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 6.7\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text l ) } { 300\ , \text l } =0.50\ , \text { atm } \end { align } $ thus , the ideal gas law tells us that the partial pressure of hydrogen in the mixture is $ 0.50\ , \text { atm } $ . we can also calculate the partial pressure of hydrogen in this problem using dalton 's law of partial pressures , which will be discussed in the next section . dalton 's law of partial pressures dalton 's law of partial pressures states that the total pressure of a mixture of gases is the sum of the partial pressures of its components : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ where the partial pressure of each gas is the pressure that the gas would exert if it was the only gas in the container . that is because we assume there are no attractive forces between the gases . dalton 's law of partial pressure can also be expressed in terms of the mole fraction of a gas in the mixture . the mole fraction of a gas is the number of moles of that gas divided by the total moles of gas in the mixture , and it is often abbreviated as $ x $ : $ x_1=\text { mole fraction of gas 1 } =\dfrac { \text { moles of gas 1 } } { \text { total moles of gas } } $ dalton 's law for can be rearranged to give the partial pressure of gas 1 in a mixture in terms of the mole fraction of gas 1 : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ both forms of dalton 's law are extremely useful in solving different kinds of problems including : calculating the partial pressure of a gas when you know the mole ratio and total pressure calculating moles of an individual gas if you know the partial pressure and total pressure calculating the total pressure if you know the partial pressures of the components example 2 : calculating partial pressures and total pressure let 's say that we have one container with $ 24.0\ , \text l $ of nitrogen gas at $ 2.00 \ , \text { atm } $ , and another container with $ 12.0\ , \text l $ of oxygen gas at $ 2.00\ , \text { atm } $ . the temperature of both gases is $ 273\ , \text k $ . if both gases are mixed in a $ 10.0\ , \text l $ container , what are the partial pressures of nitrogen and oxygen in the resulting mixture ? what is the total pressure ? step 1 : calculate moles of oxygen and nitrogen gas since we know $ \text p $ , $ \text v $ , and $ \text t $ for each of the gases before they 're combined , we can find the number of moles of nitrogen gas and oxygen gas using the ideal gas law : $ \text n = \dfrac { \text { pv } } { \text { rt } } $ solving for nitrogen and oxygen , we get : $ \text n_ { \text { n } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 24.0\ , \text { l } ) } { ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 k ) } = 2.14\ , \text { mol nitrogen } $ $ \text n_ { \text { o } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 12.0\ , \text { l } ) } { ( 0.08206 \ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 \ , \text { k } ) } = 1.07\ , \text { mol oxygen } $ step 2 ( method 1 ) : calculate partial pressures and use dalton 's law to get $ \text p_\text { total } $ once we know the number of moles for each gas in our mixture , we can now use the ideal gas law to find the partial pressure of each component in the $ 10.0\ , \text l $ container : $ \text p = \dfrac { \text { nrt } } { \text v } $ $ \text p_ { \text { n } _2 } =\dfrac { ( 2.14\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 4.79\ , \text { atm } $ $ \text p_ { \text { o } _2 } =\dfrac { ( 1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 2.40\ , \text { atm } $ notice that the partial pressure for each of the gases increased compared to the pressure of the gas in the original container . this makes sense since the volume of both gases decreased , and pressure is inversely proportional to volume . we can now get the total pressure of the mixture by adding the partial pressures together using dalton 's law : $ \begin { align } \text p_\text { total } & amp ; =\text p_ { \text { n } 2 } + \text p { \text { o } _2 } \ \ & amp ; =4.79\ , \text { atm } + 2.40\ , \text { atm } = 7.19\ , \text { atm } \end { align } $ step 2 ( method 2 ) : use ideal gas law to calculate $ \text p_\text { total } $ without partial pressures since the pressure of an ideal gas mixture only depends on the number of gas molecules in the container ( and not the identity of the gas molecules ) , we can use the total moles of gas to calculate the total pressure using the ideal gas law : $ \begin { align } \text p_ { \text { total } } & amp ; = \dfrac { ( \text { n } { \text n_2 } +\text n { \text { o } _2 } ) \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 2.14\ , \text { mol } +1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =\dfrac { ( 3.21\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =7.19\ , \text { atm } \end { align } $ once we know the total pressure , we can use the mole fraction version of dalton 's law to calculate the partial pressures : $ \text p_ { \text { n } 2 } = x { \text n_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 2.14\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =4.79\ , \text { atm } $ $ \text p_ { \text { o } 2 } = x { \text o_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 1.07\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =2.40\ , \text { atm } $ luckily , both methods give the same answers ! you might be wondering when you might want to use each method . it mostly depends on which one you prefer , and partly on what you are solving for . for instance , if all you need to know is the total pressure , it might be better to use the second method to save a couple calculation steps . summary the pressure exerted by an individual gas in a mixture is known as its partial pressure . assuming we have a mixture of ideal gases , we can use the ideal gas law to solve problems involving gases in a mixture . dalton 's law of partial pressures states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the component gases : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ dalton 's law can also be expressed using the mole fraction of a gas , $ x $ : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ try it : evaporation in a closed system part 1 part 2
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dalton 's law of partial pressure can also be expressed in terms of the mole fraction of a gas in the mixture . the mole fraction of a gas is the number of moles of that gas divided by the total moles of gas in the mixture , and it is often abbreviated as $ x $ : $ x_1=\text { mole fraction of gas 1 } =\dfrac { \text { moles of gas 1 } } { \text { total moles of gas } } $ dalton 's law for can be rearranged to give the partial pressure of gas 1 in a mixture in terms of the mole fraction of gas 1 : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ both forms of dalton 's law are extremely useful in solving different kinds of problems including : calculating the partial pressure of a gas when you know the mole ratio and total pressure calculating moles of an individual gas if you know the partial pressure and total pressure calculating the total pressure if you know the partial pressures of the components example 2 : calculating partial pressures and total pressure let 's say that we have one container with $ 24.0\ , \text l $ of nitrogen gas at $ 2.00 \ , \text { atm } $ , and another container with $ 12.0\ , \text l $ of oxygen gas at $ 2.00\ , \text { atm } $ . the temperature of both gases is $ 273\ , \text k $ .
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is there a way to calculate the partial pressures of different reactants and products in a reaction when you only have the total pressure of the all gases and the number of moles of each gas but no volume ?
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key points the pressure exerted by an individual gas in a mixture is known as its partial pressure . assuming we have a mixture of ideal gases , we can use the ideal gas law to solve problems involving gases in a mixture . dalton 's law of partial pressures states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the component gases : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ dalton 's law can also be expressed using the mole fraction of a gas , $ x $ : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ introduction in day-to-day life , we measure gas pressure when we use a barometer to check the atmospheric pressure outside or a tire gauge to measure the pressure in a bike tube . when we do this , we are measuring a macroscopic physical property of a large number of gas molecules that are invisible to the naked eye . on the molecular level , the pressure we are measuring comes from the force of individual gas molecules colliding with other objects , such as the walls of their container . let 's take a closer look at pressure from a molecular perspective and learn how dalton 's law helps us calculate total and partial pressures for mixtures of gases . ideal gases and partial pressure in this article , we will be assuming the gases in our mixtures can be approximated as ideal gases . this assumption is generally reasonable as long as the temperature of the gas is not super low ( close to $ 0\ , \text k $ ) , and the pressure is around $ 1\ , \text { atm } $ . this means we are making some assumptions about our gas molecules : we assume that the gas molecules take up no volume . we assume that the molecules have no intermolecular attractions , which means they act independently of other gas molecules . based on these assumptions , we can calculate the contribution of different gases in a mixture to the total pressure . we refer to the pressure exerted by a specific gas in a mixture as its partial pressure . the partial pressure of a gas can be calculated using the ideal gas law , which we will cover in the next section , as well as using dalton 's law of partial pressures . example 1 : calculating the partial pressure of a gas let 's say we have a mixture of hydrogen gas , $ \text h_2 ( g ) $ , and oxygen gas , $ \text o_2 ( g ) $ . the mixture contains $ 6.7\ , \text { mol } $ hydrogen gas and $ 3.3\ , \text { mol } $ oxygen gas . the mixture is in a $ 300\ , \text l $ container at $ 273\ , \text k $ , and the total pressure of the gas mixture is $ 0.75\ , \text { atm } $ . the contribution of hydrogen gas to the total pressure is its partial pressure . since the gas molecules in an ideal gas behave independently of other gases in the mixture , the partial pressure of hydrogen is the same pressure as if there were no other gases in the container . therefore , if we want to know the partial pressure of hydrogen gas in the mixture , $ \text p_ { \text h_2 } $ , we can completely ignore the oxygen gas and use the ideal gas law : $ \text p_ { \text h_2 } \text v = \text { n } _ { \text h_2 } \text { rt } $ rearranging the ideal gas equation to solve for $ \text p_ { \text h_2 } $ , we get : $ \begin { align } \text p_ { \text h_2 } & amp ; = \dfrac { \text { n } _ { \text h_2 } \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 6.7\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text l ) } { 300\ , \text l } =0.50\ , \text { atm } \end { align } $ thus , the ideal gas law tells us that the partial pressure of hydrogen in the mixture is $ 0.50\ , \text { atm } $ . we can also calculate the partial pressure of hydrogen in this problem using dalton 's law of partial pressures , which will be discussed in the next section . dalton 's law of partial pressures dalton 's law of partial pressures states that the total pressure of a mixture of gases is the sum of the partial pressures of its components : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ where the partial pressure of each gas is the pressure that the gas would exert if it was the only gas in the container . that is because we assume there are no attractive forces between the gases . dalton 's law of partial pressure can also be expressed in terms of the mole fraction of a gas in the mixture . the mole fraction of a gas is the number of moles of that gas divided by the total moles of gas in the mixture , and it is often abbreviated as $ x $ : $ x_1=\text { mole fraction of gas 1 } =\dfrac { \text { moles of gas 1 } } { \text { total moles of gas } } $ dalton 's law for can be rearranged to give the partial pressure of gas 1 in a mixture in terms of the mole fraction of gas 1 : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ both forms of dalton 's law are extremely useful in solving different kinds of problems including : calculating the partial pressure of a gas when you know the mole ratio and total pressure calculating moles of an individual gas if you know the partial pressure and total pressure calculating the total pressure if you know the partial pressures of the components example 2 : calculating partial pressures and total pressure let 's say that we have one container with $ 24.0\ , \text l $ of nitrogen gas at $ 2.00 \ , \text { atm } $ , and another container with $ 12.0\ , \text l $ of oxygen gas at $ 2.00\ , \text { atm } $ . the temperature of both gases is $ 273\ , \text k $ . if both gases are mixed in a $ 10.0\ , \text l $ container , what are the partial pressures of nitrogen and oxygen in the resulting mixture ? what is the total pressure ? step 1 : calculate moles of oxygen and nitrogen gas since we know $ \text p $ , $ \text v $ , and $ \text t $ for each of the gases before they 're combined , we can find the number of moles of nitrogen gas and oxygen gas using the ideal gas law : $ \text n = \dfrac { \text { pv } } { \text { rt } } $ solving for nitrogen and oxygen , we get : $ \text n_ { \text { n } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 24.0\ , \text { l } ) } { ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 k ) } = 2.14\ , \text { mol nitrogen } $ $ \text n_ { \text { o } _2 } = \dfrac { ( 2\ , \text { atm } ) ( 12.0\ , \text { l } ) } { ( 0.08206 \ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273 \ , \text { k } ) } = 1.07\ , \text { mol oxygen } $ step 2 ( method 1 ) : calculate partial pressures and use dalton 's law to get $ \text p_\text { total } $ once we know the number of moles for each gas in our mixture , we can now use the ideal gas law to find the partial pressure of each component in the $ 10.0\ , \text l $ container : $ \text p = \dfrac { \text { nrt } } { \text v } $ $ \text p_ { \text { n } _2 } =\dfrac { ( 2.14\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 4.79\ , \text { atm } $ $ \text p_ { \text { o } _2 } =\dfrac { ( 1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } = 2.40\ , \text { atm } $ notice that the partial pressure for each of the gases increased compared to the pressure of the gas in the original container . this makes sense since the volume of both gases decreased , and pressure is inversely proportional to volume . we can now get the total pressure of the mixture by adding the partial pressures together using dalton 's law : $ \begin { align } \text p_\text { total } & amp ; =\text p_ { \text { n } 2 } + \text p { \text { o } _2 } \ \ & amp ; =4.79\ , \text { atm } + 2.40\ , \text { atm } = 7.19\ , \text { atm } \end { align } $ step 2 ( method 2 ) : use ideal gas law to calculate $ \text p_\text { total } $ without partial pressures since the pressure of an ideal gas mixture only depends on the number of gas molecules in the container ( and not the identity of the gas molecules ) , we can use the total moles of gas to calculate the total pressure using the ideal gas law : $ \begin { align } \text p_ { \text { total } } & amp ; = \dfrac { ( \text { n } { \text n_2 } +\text n { \text { o } _2 } ) \text { rt } } { \text v } \ \ & amp ; =\dfrac { ( 2.14\ , \text { mol } +1.07\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =\dfrac { ( 3.21\ , \text { mol } ) ( 0.08206\ , \dfrac { \text { atm } \cdot \text l } { \text { mol } \cdot \text k } ) ( 273\ , \text k ) } { 10\ , \text l } \ \ & amp ; =7.19\ , \text { atm } \end { align } $ once we know the total pressure , we can use the mole fraction version of dalton 's law to calculate the partial pressures : $ \text p_ { \text { n } 2 } = x { \text n_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 2.14\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =4.79\ , \text { atm } $ $ \text p_ { \text { o } 2 } = x { \text o_2 } \text { p } _ { \text { total } } =\left ( \dfrac { 1.07\ , \text { mol } } { 3.21\ , \text { mol } } \right ) ( 7.19\ , \text { atm } ) =2.40\ , \text { atm } $ luckily , both methods give the same answers ! you might be wondering when you might want to use each method . it mostly depends on which one you prefer , and partly on what you are solving for . for instance , if all you need to know is the total pressure , it might be better to use the second method to save a couple calculation steps . summary the pressure exerted by an individual gas in a mixture is known as its partial pressure . assuming we have a mixture of ideal gases , we can use the ideal gas law to solve problems involving gases in a mixture . dalton 's law of partial pressures states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the component gases : $ \text { p } { \text { total } } = \text p { \text { gas 1 } } + \text p_ { \text { gas 2 } } + \text p_ { \text { gas 3 } } ... $ dalton 's law can also be expressed using the mole fraction of a gas , $ x $ : $ \text p_ { \text { gas 1 } } = x_1 \text { p } _ { \text { total } } $ try it : evaporation in a closed system part 1 part 2
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the mixture is in a $ 300\ , \text l $ container at $ 273\ , \text k $ , and the total pressure of the gas mixture is $ 0.75\ , \text { atm } $ . the contribution of hydrogen gas to the total pressure is its partial pressure . since the gas molecules in an ideal gas behave independently of other gases in the mixture , the partial pressure of hydrogen is the same pressure as if there were no other gases in the container .
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what is the partial pressure of each component of this gas ?
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from the physical to spiritual world ( the shaman ) an important visual theme in many ancient american art styles is that of transformation : one thing changing into another . often the physical change seen in the art is meant to convey a spiritual transmutation , as inhabitants of this physical world communicate with , and often temporarily become , the inhabitants of the spirit world . communication with the spirit world is essential . the inhabitants of that world hold power over this one , and they can aid humans in making crops grow , healing the sick , or helping the dead pass into the afterlife . however , there is also a dark side to the spirit world and its inhabitants . some spirit beings are malevolent , causing disease , crop failure , droughts , floods , and other disasters . in order to negotiate with the perilous spirit world , a person with special gifts and training must be able to travel there and represent their people . the term “ shaman ” initially referred solely to the specific kinds of healers and holy people of siberia ( the word is tungusk in origin ) , but today it is often used as a broad category designating ritual specialists who undergo spiritual transformations and trances that place them into communication with the spirit world , and allow them to journey into that world if the need arises . usually , if the local term for such a practitioner is known , that is what they are called in scholarly literature , but in the case of long-passed cultures with no written language , “ shaman ” is frequently the catch-all designation . most shamans have spirit alter egos who help them in this world and in the spirit world , and these alter egos are usually animals . the qualities of particular animals , such as swiftness , strength , and heightened senses , all aid the shaman as they travel in the spirit world , where they may have to fight evil spirits or rival shamans as well as find and communicate with more benevolent forces . doe / shaman the doe shaman ( above ) is such a figure . she is from the greater nicoya area , comprised of parts of modern-day costa rica and nicaragua . unlike mesoamerica and the central andes , there was no empire-building in the area we now call central america . instead , small egalitarian bands and larger chiefdoms were the primary political units . a characteristic of such small societies was their desire to clearly distinguish themselves from their neighbors , with whom they often fought , competed for resources , married , and traded . a product of this competition was remarkable innovation in material culture styles , meant to create distinct wares that announce and confirm social identity . the red and black slip painting burnished to a high sheen and fine incised lines all characterize the particular local style ( called rosales zoned engraved ) that the doe shaman ’ s people used . however , it is what the artistic style depicts that was of central importance . a woman ... the doe shaman depicts a woman , identifiable by her protruding breasts and rounded , pregnant belly , seated with her legs crossed and her hands placed on her thighs . she wears a necklace , perhaps made of stone beads , around her neck , and her body is painted or tattooed with numerous swirling and straight lines . her head is considerably larger in proportion to her body than a normal human ’ s would be , and her shoulders are exceptionally broad . the swirling forms that cross her body at the armpits , navel , and buttocks may reference the shapes modern shamans report seeing as they fall into trance and prepare to enter the spirit world . she is seated in a cross-legged pose that is commonly used to show shamans in a trance state . her crossed legs ( below ) are rendered in a combination of three-and two-dimensional forms : the thighs down to the knees are rounded , three-dimensional forms , while the lower legs and feet are painted on the surface of the thighs they cross . the ankles cross between the legs , both marking and hiding her genital area . in general her body is depicted in an abstract manner , with rounded forms and a lack of interest in musculature . but also a white-tailed deer ... upon closer inspection , it also becomes clear that this woman does not just have human features . instead she is being portrayed as someone in between the human and animal worlds : a shaman transforming into her animal self in preparation for a journey to the spirit world . her features are a mixture of those of a human and those of a white-tailed deer ( odocoileus virginianus ) . several parts of her body reveal this subtle transformation , most especially the knobby protrusions on the sides of her head , the split in her lower lip , and the fact that her hands lack fingers and instead look like hooves . the protrusions on her head mimic the pedicels , or antler stumps , of deer , from which antlers grow and then are shed . the pedicels reference the regenerative nature of antlers , endowing the shaman with their property of death and rebirth . her split lower lip , viewed against her protruding chin , looks somewhat like a deer ’ s sensitive muzzle . her hoof-hands , viewed in contrast to her nearby human feet , emphasize her ability to transform into the deer in her travels to the spirit world , and to use the swiftness of the deer in her dangerous journey . this beautifully-made piece would have been placed in a grave , where the vessel would embody the shaman ’ s power that would help the deceased make their safe passage from life to death , being reborn into the spirit world through the regenerative and maternal power of the shaman as well as her protection in the journey from life to afterlife . [ 1 ] essay by dr. sarahh scher [ 1 ] rebecca rollins stone , the jaguar within : shamanic visions in ancient central and south american art ( austin : university of texas press , 2011 ) , pp . 94-104 . additional resources : this work at the michael c. carlos museum , emory university—with zoomable image rebecca rollins stone , *the jaguar within : shamanic visions in ancient central and south american art . austin : university of texas press , 2011 . further reading : rebecca stone-miller , seeing with new eyes : highlights of the michael c. carlos museum collection of the art of the ancient americas . atlanta : michael c. carlos museum , 2002 .
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in general her body is depicted in an abstract manner , with rounded forms and a lack of interest in musculature . but also a white-tailed deer ... upon closer inspection , it also becomes clear that this woman does not just have human features .
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; ) also , why was nudity so common in this kind of art ?
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what is an inverse ? recall that a number multiplied by its inverse equals 1 . from basic arithmetic we know that : the inverse of a number a is 1/a since a * 1/a = 1 e.g . the inverse of 5 is 1/5 * all real numbers other than 0 have an inverse multiplying a number by the inverse of a is equivalent to dividing by a e.g . 10/5 is the same as 10* 1/5 what is a modular inverse ? in modular arithmetic we do not have a division operation . however , we do have modular inverses . the modular inverse of a ( mod c ) is a^-1 ( a * a^-1 ) ≡ 1 ( mod c ) or equivalently ( a * a^-1 ) mod c = 1 only the numbers coprime to c ( numbers that share no prime factors with c ) have a modular inverse ( mod c ) how to find a modular inverse a naive method of finding a modular inverse for a ( mod c ) is : step 1 . calculate a * b mod c for b values 0 through c-1 step 2 . the modular inverse of a mod c is the b value that makes a * b mod c = 1 note that the term b mod c can only have an integer value 0 through c-1 , so testing larger values for b is redundant . example : a=3 c=7 step 1 . calculate a * b mod c for b values 0 through c-1**** 3 * 0 ≡ 0 ( mod 7 ) 3 * 1 ≡ 3 ( mod 7 ) 3 * 2 ≡ 6 ( mod 7 ) 3 * 3 ≡ 9 ≡ 2 ( mod 7 ) 3 * 4 ≡ 12 ≡ 5 ( mod 7 ) 3 * 5 ≡ 15 ( mod 7 ) ≡ 1 ( mod 7 ) & lt ; -- -- -- found inverse ! 3 * 6 ≡ 18 ( mod 7 ) ≡ 4 ( mod 7 ) step 2 . the modular inverse of a mod c is the b value that makes __a * b mod c = 1__ 5 is the modular inverse of 3 mod 7 since 5*3 mod 7 = 1 simple ! let 's do one more example where we do n't find an inverse . example : a=2 c=6 step 1 . calculate a * b mod c for b values 0 through c-1 2 * 0 ≡ 0 ( mod 6 ) 2 * 1 ≡ 2 ( mod 6 ) 2 * 2 ≡ 4 ( mod 6 ) 2 * 3 ≡ 6 ≡ 0 ( mod 6 ) 2 * 4 ≡ 8 ≡ 2 ( mod 6 ) 2 * 5 ≡ 10 ≡ 4 ( mod 6 ) step 2 . the modular inverse of a mod c is the b value that makes a * b mod c = 1 no value of b makes a * b mod c = 1 . therefore , a has no modular inverse ( mod 6 ) . this is because 2 is not coprime to 6 ( they share the prime factor 2 ) . this method seems slow ... there is a much faster method for finding the inverse of a ( mod c ) that we will discuss in the next articles on the extended euclidean algorithm . first , let 's do some exercises !
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in modular arithmetic we do not have a division operation . however , we do have modular inverses . the modular inverse of a ( mod c ) is a^-1 ( a * a^-1 ) ≡ 1 ( mod c ) or equivalently ( a * a^-1 ) mod c = 1 only the numbers coprime to c ( numbers that share no prime factors with c ) have a modular inverse ( mod c ) how to find a modular inverse a naive method of finding a modular inverse for a ( mod c ) is : step 1 . calculate a * b mod c for b values 0 through c-1 step 2 .
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why is it that a has to be coprime to c to have a modular inverse ?
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what is an inverse ? recall that a number multiplied by its inverse equals 1 . from basic arithmetic we know that : the inverse of a number a is 1/a since a * 1/a = 1 e.g . the inverse of 5 is 1/5 * all real numbers other than 0 have an inverse multiplying a number by the inverse of a is equivalent to dividing by a e.g . 10/5 is the same as 10* 1/5 what is a modular inverse ? in modular arithmetic we do not have a division operation . however , we do have modular inverses . the modular inverse of a ( mod c ) is a^-1 ( a * a^-1 ) ≡ 1 ( mod c ) or equivalently ( a * a^-1 ) mod c = 1 only the numbers coprime to c ( numbers that share no prime factors with c ) have a modular inverse ( mod c ) how to find a modular inverse a naive method of finding a modular inverse for a ( mod c ) is : step 1 . calculate a * b mod c for b values 0 through c-1 step 2 . the modular inverse of a mod c is the b value that makes a * b mod c = 1 note that the term b mod c can only have an integer value 0 through c-1 , so testing larger values for b is redundant . example : a=3 c=7 step 1 . calculate a * b mod c for b values 0 through c-1**** 3 * 0 ≡ 0 ( mod 7 ) 3 * 1 ≡ 3 ( mod 7 ) 3 * 2 ≡ 6 ( mod 7 ) 3 * 3 ≡ 9 ≡ 2 ( mod 7 ) 3 * 4 ≡ 12 ≡ 5 ( mod 7 ) 3 * 5 ≡ 15 ( mod 7 ) ≡ 1 ( mod 7 ) & lt ; -- -- -- found inverse ! 3 * 6 ≡ 18 ( mod 7 ) ≡ 4 ( mod 7 ) step 2 . the modular inverse of a mod c is the b value that makes __a * b mod c = 1__ 5 is the modular inverse of 3 mod 7 since 5*3 mod 7 = 1 simple ! let 's do one more example where we do n't find an inverse . example : a=2 c=6 step 1 . calculate a * b mod c for b values 0 through c-1 2 * 0 ≡ 0 ( mod 6 ) 2 * 1 ≡ 2 ( mod 6 ) 2 * 2 ≡ 4 ( mod 6 ) 2 * 3 ≡ 6 ≡ 0 ( mod 6 ) 2 * 4 ≡ 8 ≡ 2 ( mod 6 ) 2 * 5 ≡ 10 ≡ 4 ( mod 6 ) step 2 . the modular inverse of a mod c is the b value that makes a * b mod c = 1 no value of b makes a * b mod c = 1 . therefore , a has no modular inverse ( mod 6 ) . this is because 2 is not coprime to 6 ( they share the prime factor 2 ) . this method seems slow ... there is a much faster method for finding the inverse of a ( mod c ) that we will discuss in the next articles on the extended euclidean algorithm . first , let 's do some exercises !
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in modular arithmetic we do not have a division operation . however , we do have modular inverses . the modular inverse of a ( mod c ) is a^-1 ( a * a^-1 ) ≡ 1 ( mod c ) or equivalently ( a * a^-1 ) mod c = 1 only the numbers coprime to c ( numbers that share no prime factors with c ) have a modular inverse ( mod c ) how to find a modular inverse a naive method of finding a modular inverse for a ( mod c ) is : step 1 .
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do you always have to use modular inverses ?
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what is an inverse ? recall that a number multiplied by its inverse equals 1 . from basic arithmetic we know that : the inverse of a number a is 1/a since a * 1/a = 1 e.g . the inverse of 5 is 1/5 * all real numbers other than 0 have an inverse multiplying a number by the inverse of a is equivalent to dividing by a e.g . 10/5 is the same as 10* 1/5 what is a modular inverse ? in modular arithmetic we do not have a division operation . however , we do have modular inverses . the modular inverse of a ( mod c ) is a^-1 ( a * a^-1 ) ≡ 1 ( mod c ) or equivalently ( a * a^-1 ) mod c = 1 only the numbers coprime to c ( numbers that share no prime factors with c ) have a modular inverse ( mod c ) how to find a modular inverse a naive method of finding a modular inverse for a ( mod c ) is : step 1 . calculate a * b mod c for b values 0 through c-1 step 2 . the modular inverse of a mod c is the b value that makes a * b mod c = 1 note that the term b mod c can only have an integer value 0 through c-1 , so testing larger values for b is redundant . example : a=3 c=7 step 1 . calculate a * b mod c for b values 0 through c-1**** 3 * 0 ≡ 0 ( mod 7 ) 3 * 1 ≡ 3 ( mod 7 ) 3 * 2 ≡ 6 ( mod 7 ) 3 * 3 ≡ 9 ≡ 2 ( mod 7 ) 3 * 4 ≡ 12 ≡ 5 ( mod 7 ) 3 * 5 ≡ 15 ( mod 7 ) ≡ 1 ( mod 7 ) & lt ; -- -- -- found inverse ! 3 * 6 ≡ 18 ( mod 7 ) ≡ 4 ( mod 7 ) step 2 . the modular inverse of a mod c is the b value that makes __a * b mod c = 1__ 5 is the modular inverse of 3 mod 7 since 5*3 mod 7 = 1 simple ! let 's do one more example where we do n't find an inverse . example : a=2 c=6 step 1 . calculate a * b mod c for b values 0 through c-1 2 * 0 ≡ 0 ( mod 6 ) 2 * 1 ≡ 2 ( mod 6 ) 2 * 2 ≡ 4 ( mod 6 ) 2 * 3 ≡ 6 ≡ 0 ( mod 6 ) 2 * 4 ≡ 8 ≡ 2 ( mod 6 ) 2 * 5 ≡ 10 ≡ 4 ( mod 6 ) step 2 . the modular inverse of a mod c is the b value that makes a * b mod c = 1 no value of b makes a * b mod c = 1 . therefore , a has no modular inverse ( mod 6 ) . this is because 2 is not coprime to 6 ( they share the prime factor 2 ) . this method seems slow ... there is a much faster method for finding the inverse of a ( mod c ) that we will discuss in the next articles on the extended euclidean algorithm . first , let 's do some exercises !
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this method seems slow ... there is a much faster method for finding the inverse of a ( mod c ) that we will discuss in the next articles on the extended euclidean algorithm . first , let 's do some exercises !
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did the extended euclidean algorithm articles ever get published ?
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what is an inverse ? recall that a number multiplied by its inverse equals 1 . from basic arithmetic we know that : the inverse of a number a is 1/a since a * 1/a = 1 e.g . the inverse of 5 is 1/5 * all real numbers other than 0 have an inverse multiplying a number by the inverse of a is equivalent to dividing by a e.g . 10/5 is the same as 10* 1/5 what is a modular inverse ? in modular arithmetic we do not have a division operation . however , we do have modular inverses . the modular inverse of a ( mod c ) is a^-1 ( a * a^-1 ) ≡ 1 ( mod c ) or equivalently ( a * a^-1 ) mod c = 1 only the numbers coprime to c ( numbers that share no prime factors with c ) have a modular inverse ( mod c ) how to find a modular inverse a naive method of finding a modular inverse for a ( mod c ) is : step 1 . calculate a * b mod c for b values 0 through c-1 step 2 . the modular inverse of a mod c is the b value that makes a * b mod c = 1 note that the term b mod c can only have an integer value 0 through c-1 , so testing larger values for b is redundant . example : a=3 c=7 step 1 . calculate a * b mod c for b values 0 through c-1**** 3 * 0 ≡ 0 ( mod 7 ) 3 * 1 ≡ 3 ( mod 7 ) 3 * 2 ≡ 6 ( mod 7 ) 3 * 3 ≡ 9 ≡ 2 ( mod 7 ) 3 * 4 ≡ 12 ≡ 5 ( mod 7 ) 3 * 5 ≡ 15 ( mod 7 ) ≡ 1 ( mod 7 ) & lt ; -- -- -- found inverse ! 3 * 6 ≡ 18 ( mod 7 ) ≡ 4 ( mod 7 ) step 2 . the modular inverse of a mod c is the b value that makes __a * b mod c = 1__ 5 is the modular inverse of 3 mod 7 since 5*3 mod 7 = 1 simple ! let 's do one more example where we do n't find an inverse . example : a=2 c=6 step 1 . calculate a * b mod c for b values 0 through c-1 2 * 0 ≡ 0 ( mod 6 ) 2 * 1 ≡ 2 ( mod 6 ) 2 * 2 ≡ 4 ( mod 6 ) 2 * 3 ≡ 6 ≡ 0 ( mod 6 ) 2 * 4 ≡ 8 ≡ 2 ( mod 6 ) 2 * 5 ≡ 10 ≡ 4 ( mod 6 ) step 2 . the modular inverse of a mod c is the b value that makes a * b mod c = 1 no value of b makes a * b mod c = 1 . therefore , a has no modular inverse ( mod 6 ) . this is because 2 is not coprime to 6 ( they share the prime factor 2 ) . this method seems slow ... there is a much faster method for finding the inverse of a ( mod c ) that we will discuss in the next articles on the extended euclidean algorithm . first , let 's do some exercises !
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this method seems slow ... there is a much faster method for finding the inverse of a ( mod c ) that we will discuss in the next articles on the extended euclidean algorithm . first , let 's do some exercises !
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when will the articles on the extended euclidean algorithm be posted ?
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