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let 's do some practice problems dealing with variable expressions . so these first problems say write the following in a more condensed form by leaving out the multiplication symbol or leaving out a multiplication symbol . so here we have 2 times 11x , so if we have 11 x 's and then we 're going to have 2 times those ... | evaluate the following expressions for a is equal to 3 , b is equal to 2 , c is equal to 5 , and d is equal to minus 4 -- or , actually , i should say negative 4 is the correct terminology . negative 4 . so we just substitute . | how does the negative come into the question ? |
let 's do some practice problems dealing with variable expressions . so these first problems say write the following in a more condensed form by leaving out the multiplication symbol or leaving out a multiplication symbol . so here we have 2 times 11x , so if we have 11 x 's and then we 're going to have 2 times those ... | so that 's minus 5 , or negative 5 minus 4 . so negative 1 minus 5 is negative 6 , minus 4 is equal to negative 10 . and i 'll do these last two just to get a sample of all of the types of problems in this variable expression section . | how is negative one squared equal to one ? |
let 's do some practice problems dealing with variable expressions . so these first problems say write the following in a more condensed form by leaving out the multiplication symbol or leaving out a multiplication symbol . so here we have 2 times 11x , so if we have 11 x 's and then we 're going to have 2 times those ... | remember , this right here is a , that right there is b . how do i know ? they 're telling me up here . | how do you know to know to do the exponents or to simplify first ? |
let 's do some practice problems dealing with variable expressions . so these first problems say write the following in a more condensed form by leaving out the multiplication symbol or leaving out a multiplication symbol . so here we have 2 times 11x , so if we have 11 x 's and then we 're going to have 2 times those ... | 1 times 9 is 9 . 1 times 4 is 4 . and then we add the two . | what is the variable expression for 3x -4 ? |
let 's do some practice problems dealing with variable expressions . so these first problems say write the following in a more condensed form by leaving out the multiplication symbol or leaving out a multiplication symbol . so here we have 2 times 11x , so if we have 11 x 's and then we 're going to have 2 times those ... | let 's do some practice problems dealing with variable expressions . so these first problems say write the following in a more condensed form by leaving out the multiplication symbol or leaving out a multiplication symbol . | how can linear equations be easier and not so confusing ? |
let 's do some practice problems dealing with variable expressions . so these first problems say write the following in a more condensed form by leaving out the multiplication symbol or leaving out a multiplication symbol . so here we have 2 times 11x , so if we have 11 x 's and then we 're going to have 2 times those ... | so this whole expression simplifies to negative 1 plus 5 times negative 1 -- we do the multiplication first , of course . so that 's minus 5 , or negative 5 minus 4 . so negative 1 minus 5 is negative 6 , minus 4 is equal to negative 10 . and i 'll do these last two just to get a sample of all of the types of problems ... | how do i remember what to minus and what to add in equations on both sides of the equal sign ? |
let 's do some practice problems dealing with variable expressions . so these first problems say write the following in a more condensed form by leaving out the multiplication symbol or leaving out a multiplication symbol . so here we have 2 times 11x , so if we have 11 x 's and then we 're going to have 2 times those ... | multiply the denominators , 1 times 4 , you get 4 . so number 3 , i got 3/4 . and then finally , you have 1/4 times z . | examples of evaluating variable expressions 12/27 number 22 b ) , i do n't understand how you took 4 ( 3 ) ( 10 - 3 ) 2 and got 12 x 49 ? |
let 's do some practice problems dealing with variable expressions . so these first problems say write the following in a more condensed form by leaving out the multiplication symbol or leaving out a multiplication symbol . so here we have 2 times 11x , so if we have 11 x 's and then we 're going to have 2 times those ... | now we 're at problem 22 . the volume of a box without a lid is given by the formula , volume is equal to 4x times 10 minus x squared , where x is a length in inches , and v is the volume in cubic inches . what is the volume when x is equal to 2 ? | how far down the road in the study of mathematics does one go to be able to come up with the formula for the volume of a box without a lid as given in the review example ? |
let 's do some practice problems dealing with variable expressions . so these first problems say write the following in a more condensed form by leaving out the multiplication symbol or leaving out a multiplication symbol . so here we have 2 times 11x , so if we have 11 x 's and then we 're going to have 2 times those ... | let 's do some practice problems dealing with variable expressions . so these first problems say write the following in a more condensed form by leaving out the multiplication symbol or leaving out a multiplication symbol . | why does sal rephrase the equation ? |
let 's do some practice problems dealing with variable expressions . so these first problems say write the following in a more condensed form by leaving out the multiplication symbol or leaving out a multiplication symbol . so here we have 2 times 11x , so if we have 11 x 's and then we 're going to have 2 times those ... | in algebra we can just get rid of that dot symbol . if we have a variable following a number , we know that means 1.35 times that variable . so that , we could rewrite as just being equal to 1.35y . | how do you solve a expression if you dont know what the variable is ? |
let 's do some practice problems dealing with variable expressions . so these first problems say write the following in a more condensed form by leaving out the multiplication symbol or leaving out a multiplication symbol . so here we have 2 times 11x , so if we have 11 x 's and then we 're going to have 2 times those ... | put a 0 . 1 times 9 is 9 . 1 times 4 is 4 . | what is the algebraic expression for the quotient of 9 times a number and 6 ? |
let 's do some practice problems dealing with variable expressions . so these first problems say write the following in a more condensed form by leaving out the multiplication symbol or leaving out a multiplication symbol . so here we have 2 times 11x , so if we have 11 x 's and then we 're going to have 2 times those ... | so that 's minus 5 , or negative 5 minus 4 . so negative 1 minus 5 is negative 6 , minus 4 is equal to negative 10 . and i 'll do these last two just to get a sample of all of the types of problems in this variable expression section . | can a negative fraction like -6/-20 be simplified ? |
let 's do some practice problems dealing with variable expressions . so these first problems say write the following in a more condensed form by leaving out the multiplication symbol or leaving out a multiplication symbol . so here we have 2 times 11x , so if we have 11 x 's and then we 're going to have 2 times those ... | so you have 4 times c. 4 times -- now what 's c equal to ? they tell us c is equal to 5 . so 4 times 5 , that 's our c , plus d. d is minus or negative 4 . | how do you tell what the z is ? |
let 's do some practice problems dealing with variable expressions . so these first problems say write the following in a more condensed form by leaving out the multiplication symbol or leaving out a multiplication symbol . so here we have 2 times 11x , so if we have 11 x 's and then we 're going to have 2 times those ... | so 2x squared minus 3x squared , plus 5x minus 4 . ok , well , this was n't that hard . all of them are dealing with x . | what happens when you make n bigger ? |
let 's do some practice problems dealing with variable expressions . so these first problems say write the following in a more condensed form by leaving out the multiplication symbol or leaving out a multiplication symbol . so here we have 2 times 11x , so if we have 11 x 's and then we 're going to have 2 times those ... | so that 's part a . $ 5,000 . now part b . | so would it be 7w to the power of 5. ? |
let 's do some practice problems dealing with variable expressions . so these first problems say write the following in a more condensed form by leaving out the multiplication symbol or leaving out a multiplication symbol . so here we have 2 times 11x , so if we have 11 x 's and then we 're going to have 2 times those ... | so we have 4 times 5 is 20 , plus negative 4 -- that 's the same thing as minus 4 , so that is equal to 16 . problem 6 . now , let 's do one of the harder ones down here . | how is -6+6 equal to zero ? |
let 's do some practice problems dealing with variable expressions . so these first problems say write the following in a more condensed form by leaving out the multiplication symbol or leaving out a multiplication symbol . so here we have 2 times 11x , so if we have 11 x 's and then we 're going to have 2 times those ... | put a 0 . 1 times 9 is 9 . 1 times 4 is 4 . | for questions 13-20 , can someone please explain to me a ) what the variables are ( ca n't read ) and b ) for question 20 , would , at the end , when the problem comes to 3+ ( -1/9 ) would it become just ( -3^1/9 ) or ( 2^8/9 ) ? |
let 's do some practice problems dealing with variable expressions . so these first problems say write the following in a more condensed form by leaving out the multiplication symbol or leaving out a multiplication symbol . so here we have 2 times 11x , so if we have 11 x 's and then we 're going to have 2 times those ... | 1 times 9 is 9 . 1 times 4 is 4 . and then we add the two . | # 6 why is `` '' + -4 `` ' the same as `` '' - 4 `` '' ? |
let 's do some practice problems dealing with variable expressions . so these first problems say write the following in a more condensed form by leaving out the multiplication symbol or leaving out a multiplication symbol . so here we have 2 times 11x , so if we have 11 x 's and then we 're going to have 2 times those ... | i 'll do it here in pink . b , when is equal to 3 , then the volume is equal to 4 times 3 -- x is equal to 3 now -- times 10 minus 3 squared . 4 times 3 is 12 , times 10 minus 3 is 7 squared . | a is equal to -3 not to 3 ? |
so they 're asking us to find the least common multiple of these two different polynomials . so the first one 's three z to the third minus six z squared minus nine z and the second is seven z to the fourth plus 21 z to the third plus 14 z squared . now , if you 're saying , `` well , what is a , '' you 're familiar w... | one times negative three is negative three . one z minus three z is negative two z , so that looks good . so now let 's factor , now let 's factor this other character over here , this fourth degree polynomial . | i 'm probably missing something simply , but why would n't you have to include the original `` z '' and a `` z^2 '' ? |
milena 's town is built on a grid similar to the coordinate plane . she is riding her bicycle from her home at point negative 3 , 4 to the mall at point negative 3 , negative 7 . each unit on the graph denotes one city block . plot the two points , and find the distance between milena 's home and the mall . so let 's s... | so i 'll use this point right over here . so negative 3 is our x-coordinate . so we 're going to go 3 to the left of the origin 1 , 2 , 3 . | what would happen if the x and/or y is imaginary like 5i or -3.5i ? |
milena 's town is built on a grid similar to the coordinate plane . she is riding her bicycle from her home at point negative 3 , 4 to the mall at point negative 3 , negative 7 . each unit on the graph denotes one city block . plot the two points , and find the distance between milena 's home and the mall . so let 's s... | that 's 3 to the right , 3 to the left . so he wants to put the nail at the point x equals 0 , y is equal to 4 . so he wants to put it at x is equal to 0 , y is equal to 4 . | if the numbers on the x axis or y axis start at 0 and jump to a number like 40 and then starts counting by 20 's ( 0 , 40 , 60 , 80 ) , would you put a jagged line after the zero indicating that you are not labeling your numbers in a certain pattern ? |
milena 's town is built on a grid similar to the coordinate plane . she is riding her bicycle from her home at point negative 3 , 4 to the mall at point negative 3 , negative 7 . each unit on the graph denotes one city block . plot the two points , and find the distance between milena 's home and the mall . so let 's s... | so he wants to place a nail in the center of the blue line . the blue line is 6 units long . the center is right over here . | how long is each side of the park ? |
milena 's town is built on a grid similar to the coordinate plane . she is riding her bicycle from her home at point negative 3 , 4 to the mall at point negative 3 , negative 7 . each unit on the graph denotes one city block . plot the two points , and find the distance between milena 's home and the mall . so let 's s... | so i 'll use this point right over here . so negative 3 is our x-coordinate . so we 're going to go 3 to the left of the origin 1 , 2 , 3 . | what are the quadrants of the standard coordinate plane ( x , y ) that contain points on the graph of equation 4x - 2y = 8 ? |
milena 's town is built on a grid similar to the coordinate plane . she is riding her bicycle from her home at point negative 3 , 4 to the mall at point negative 3 , negative 7 . each unit on the graph denotes one city block . plot the two points , and find the distance between milena 's home and the mall . so let 's s... | she is riding her bicycle from her home at point negative 3 , 4 to the mall at point negative 3 , negative 7 . each unit on the graph denotes one city block . plot the two points , and find the distance between milena 's home and the mall . | how many axis in a problems can you fit into one graph ? |
milena 's town is built on a grid similar to the coordinate plane . she is riding her bicycle from her home at point negative 3 , 4 to the mall at point negative 3 , negative 7 . each unit on the graph denotes one city block . plot the two points , and find the distance between milena 's home and the mall . so let 's s... | and now we need to figure out the distance between her home and the mall . now , we could actually count it out , or we could just compute it . if we wanted to count it out , it 's 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 blocks . | why do we have to count the quadrants in that order ? |
milena 's town is built on a grid similar to the coordinate plane . she is riding her bicycle from her home at point negative 3 , 4 to the mall at point negative 3 , negative 7 . each unit on the graph denotes one city block . plot the two points , and find the distance between milena 's home and the mall . so let 's s... | milena 's town is built on a grid similar to the coordinate plane . she is riding her bicycle from her home at point negative 3 , 4 to the mall at point negative 3 , negative 7 . | poyo ... what does poyo mean ? |
milena 's town is built on a grid similar to the coordinate plane . she is riding her bicycle from her home at point negative 3 , 4 to the mall at point negative 3 , negative 7 . each unit on the graph denotes one city block . plot the two points , and find the distance between milena 's home and the mall . so let 's s... | this is at a negative 7 . positive 4 , negative 7 . so we 're really trying to find the distance between 4 and negative 7 . | are top boxes positive and are the bottom boxes negative ? |
milena 's town is built on a grid similar to the coordinate plane . she is riding her bicycle from her home at point negative 3 , 4 to the mall at point negative 3 , negative 7 . each unit on the graph denotes one city block . plot the two points , and find the distance between milena 's home and the mall . so let 's s... | and then to go from 8 to 3 , you 're going to go 5 more . so you could also not necessarily count it out , you can actually just think about the coordinates . but either way , you see that town a is closer . | also does the square boxes count with the distance ? |
milena 's town is built on a grid similar to the coordinate plane . she is riding her bicycle from her home at point negative 3 , 4 to the mall at point negative 3 , negative 7 . each unit on the graph denotes one city block . plot the two points , and find the distance between milena 's home and the mall . so let 's s... | and then to go from 8 to 3 , you 're going to go 5 more . so you could also not necessarily count it out , you can actually just think about the coordinates . but either way , you see that town a is closer . | what do the lines outside of the coordinates represent ? |
milena 's town is built on a grid similar to the coordinate plane . she is riding her bicycle from her home at point negative 3 , 4 to the mall at point negative 3 , negative 7 . each unit on the graph denotes one city block . plot the two points , and find the distance between milena 's home and the mall . so let 's s... | milena 's town is built on a grid similar to the coordinate plane . she is riding her bicycle from her home at point negative 3 , 4 to the mall at point negative 3 , negative 7 . | how does the coordinate grid represent her town ? |
milena 's town is built on a grid similar to the coordinate plane . she is riding her bicycle from her home at point negative 3 , 4 to the mall at point negative 3 , negative 7 . each unit on the graph denotes one city block . plot the two points , and find the distance between milena 's home and the mall . so let 's s... | so i 'll use this point right over here . so negative 3 is our x-coordinate . so we 're going to go 3 to the left of the origin 1 , 2 , 3 . | why do we start at the x-axis before the y-axis ? |
milena 's town is built on a grid similar to the coordinate plane . she is riding her bicycle from her home at point negative 3 , 4 to the mall at point negative 3 , negative 7 . each unit on the graph denotes one city block . plot the two points , and find the distance between milena 's home and the mall . so let 's s... | milena 's town is built on a grid similar to the coordinate plane . she is riding her bicycle from her home at point negative 3 , 4 to the mall at point negative 3 , negative 7 . | is there an end to a coordinate plane ? |
milena 's town is built on a grid similar to the coordinate plane . she is riding her bicycle from her home at point negative 3 , 4 to the mall at point negative 3 , negative 7 . each unit on the graph denotes one city block . plot the two points , and find the distance between milena 's home and the mall . so let 's s... | milena 's town is built on a grid similar to the coordinate plane . she is riding her bicycle from her home at point negative 3 , 4 to the mall at point negative 3 , negative 7 . | what is the difference between a 'scatter plot ' and a 'coordinate plane ' ? |
ocean sunfishes are well-known for rapidly gaining a lot of weight on a diet based on jellyfish . the relationship between the elapsed time , t , in days , since an ocean sunfish is born , and its mass , m of t , in milligrams , is modeled by the following function . all right . complete the following sentence about t... | the common ratio here is n't the way i 've written it . is n't 1.35 . it 's 1.35 to the 1/6 power . let me draw a little table here to make that really , really clear . | what rule allows one to transform 5^x into just 1 ? |
ocean sunfishes are well-known for rapidly gaining a lot of weight on a diet based on jellyfish . the relationship between the elapsed time , t , in days , since an ocean sunfish is born , and its mass , m of t , in milligrams , is modeled by the following function . all right . complete the following sentence about t... | the common ratio here is n't the way i 've written it . is n't 1.35 . it 's 1.35 to the 1/6 power . let me draw a little table here to make that really , really clear . | how and why does the expression have an initial value of 1.35^5 ? |
ocean sunfishes are well-known for rapidly gaining a lot of weight on a diet based on jellyfish . the relationship between the elapsed time , t , in days , since an ocean sunfish is born , and its mass , m of t , in milligrams , is modeled by the following function . all right . complete the following sentence about t... | so we can say this is approximately 1.35 times 1.051 to the t-th power . so every day , we are growing by a factor of 1.051 . well growing by a factor of 1.051 means that you are adding a little bit more than 5 % . you 're adding 0.51 every day of your mass , so you 're adding 5.1 % . | if the common ratio of the growth factor would be 1.051^t and it is growing since birth , where did the initial value come from ? |
ocean sunfishes are well-known for rapidly gaining a lot of weight on a diet based on jellyfish . the relationship between the elapsed time , t , in days , since an ocean sunfish is born , and its mass , m of t , in milligrams , is modeled by the following function . all right . complete the following sentence about t... | the common ratio here is n't the way i 've written it . is n't 1.35 . it 's 1.35 to the 1/6 power . let me draw a little table here to make that really , really clear . | did sal forget to write 1.35^5 ? |
( lighthearted music ) female : we 're in the palazzo barbarini , looking at beautiful painting by caravaggio of narcissus . male : narcissus at the source is their title . female : this is a story from ovid of a boy who falls in love with his own reflection in the water , so much so that he falls in and drowns . ma... | it 's especially interesting because you get the reality of the figure , and then the reflection . of course , paintings themselves are kinds of mirrors , or reflections , in a way . male : they certainly are , and the idea of the artist 's responsibility in terms of depiction , in terms of creating a faithfulness , a... | did caravaggio do a series of greek mythology based paintings or was this a one off ? |
( lighthearted music ) female : we 're in the palazzo barbarini , looking at beautiful painting by caravaggio of narcissus . male : narcissus at the source is their title . female : this is a story from ovid of a boy who falls in love with his own reflection in the water , so much so that he falls in and drowns . ma... | male : they certainly are , and the idea of the artist 's responsibility in terms of depiction , in terms of creating a faithfulness , and the dangers that are inherent in that . it 's interesting if you look at this painting that the reflection , in the sense , the painting within the painting is a dimmer image . fem... | why does the reflection look like a man with five-o-clock shadow ? |
( lighthearted music ) female : we 're in the palazzo barbarini , looking at beautiful painting by caravaggio of narcissus . male : narcissus at the source is their title . female : this is a story from ovid of a boy who falls in love with his own reflection in the water , so much so that he falls in and drowns . ma... | it 's especially interesting because you get the reality of the figure , and then the reflection . of course , paintings themselves are kinds of mirrors , or reflections , in a way . male : they certainly are , and the idea of the artist 's responsibility in terms of depiction , in terms of creating a faithfulness , a... | is it because caravaggio wanted to show how people in paintings can be idealized ? |
( lighthearted music ) female : we 're in the palazzo barbarini , looking at beautiful painting by caravaggio of narcissus . male : narcissus at the source is their title . female : this is a story from ovid of a boy who falls in love with his own reflection in the water , so much so that he falls in and drowns . ma... | ( lighthearted music ) female : we 're in the palazzo barbarini , looking at beautiful painting by caravaggio of narcissus . male : narcissus at the source is their title . | how big is the picture ? |
( lighthearted music ) female : we 're in the palazzo barbarini , looking at beautiful painting by caravaggio of narcissus . male : narcissus at the source is their title . female : this is a story from ovid of a boy who falls in love with his own reflection in the water , so much so that he falls in and drowns . ma... | female : right , it 's almost like he 's reaching out to embrace himself ; he 's fallen in love with himself , literally . male : but he 'll embrace only his reflection , which is , of course , intangible . ( lighthearted music ) | how is it possible for its reflection to be as portrayed ? |
which of the following functions are continuous at x equals three ? well , as we said in the previous video , in the previous example , in order to be continuous at a point , you at least have to be defined at that point . we saw our definition of continuity , f is continuous at a , if and only if , the limit of f as ... | so g of x , ln of x minus three is gon na look something like this . if you put three in it , it 's not defined , if you put four in it , ln of four , well , that 's gon na , sorry , ln of four minus one , so that 's gon na be ln of four minus three , is actually let me just draw a table here , i know i 'm confusing yo... | how did you know to put zero ? |
which of the following functions are continuous at x equals three ? well , as we said in the previous video , in the previous example , in order to be continuous at a point , you at least have to be defined at that point . we saw our definition of continuity , f is continuous at a , if and only if , the limit of f as ... | g of three , let me write it here , g of three is equal to the natural log of zero . three minus three . this is not defined . | are you saying that nothing raised to a power equals three therefore its zero ? |
which of the following functions are continuous at x equals three ? well , as we said in the previous video , in the previous example , in order to be continuous at a point , you at least have to be defined at that point . we saw our definition of continuity , f is continuous at a , if and only if , the limit of f as ... | now let 's look at this first function right over here . natural log of x minus three . well , try to evaluate it , and it 's not an f now , it 's g , try to evaluate g of three . | what is meant by the natural logarithm ? |
which of the following functions are continuous at x equals three ? well , as we said in the previous video , in the previous example , in order to be continuous at a point , you at least have to be defined at that point . we saw our definition of continuity , f is continuous at a , if and only if , the limit of f as ... | we saw our definition of continuity , f is continuous at a , if and only if , the limit of f as x approaches a is equal to f of a . so , over here , in this case , we could say that a function is continuous at x equals three , so f is continuous at x equals three , if and only if the limit as x approaches three of f of... | what if a specific x-value is defined but the point still not continuous ? |
which of the following functions are continuous at x equals three ? well , as we said in the previous video , in the previous example , in order to be continuous at a point , you at least have to be defined at that point . we saw our definition of continuity , f is continuous at a , if and only if , the limit of f as ... | now let 's look at this first function right over here . natural log of x minus three . well , try to evaluate it , and it 's not an f now , it 's g , try to evaluate g of three . | what is a natural log ? |
which of the following functions are continuous at x equals three ? well , as we said in the previous video , in the previous example , in order to be continuous at a point , you at least have to be defined at that point . we saw our definition of continuity , f is continuous at a , if and only if , the limit of f as ... | we saw our definition of continuity , f is continuous at a , if and only if , the limit of f as x approaches a is equal to f of a . so , over here , in this case , we could say that a function is continuous at x equals three , so f is continuous at x equals three , if and only if the limit as x approaches three of f of... | on the practice for this section , it asks us to know if a function is continuous for all real numbers.. one of the options is cubic root of ( x+1 ) and it is at f ( x ) at -2 ... how can you take a cubic root of -1 ? |
which of the following functions are continuous at x equals three ? well , as we said in the previous video , in the previous example , in order to be continuous at a point , you at least have to be defined at that point . we saw our definition of continuity , f is continuous at a , if and only if , the limit of f as ... | now let 's look at this first function right over here . natural log of x minus three . well , try to evaluate it , and it 's not an f now , it 's g , try to evaluate g of three . | are `` natural logs '' a concept we were supposed to learn from a previous section ? |
which of the following functions are continuous at x equals three ? well , as we said in the previous video , in the previous example , in order to be continuous at a point , you at least have to be defined at that point . we saw our definition of continuity , f is continuous at a , if and only if , the limit of f as ... | we saw our definition of continuity , f is continuous at a , if and only if , the limit of f as x approaches a is equal to f of a . so , over here , in this case , we could say that a function is continuous at x equals three , so f is continuous at x equals three , if and only if the limit as x approaches three of f of... | how this can be evaluated if a function is continuous or not by using a calculator ? |
which of the following functions are continuous at x equals three ? well , as we said in the previous video , in the previous example , in order to be continuous at a point , you at least have to be defined at that point . we saw our definition of continuity , f is continuous at a , if and only if , the limit of f as ... | well , as we said in the previous video , in the previous example , in order to be continuous at a point , you at least have to be defined at that point . we saw our definition of continuity , f is continuous at a , if and only if , the limit of f as x approaches a is equal to f of a . so , over here , in this case , w... | if you 're finding continuity algebraically without a graph , how would you know whether the function is continuous or not ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | pi over 2 is the same thing as to tau over 4 . pi is the same thing as to tau over 2 . 3 pi over 2 is 3 pi -- oh , sorry , 3 tau over 4 , 3/4 tau . and then one revolution is tau . | tau is not very common ... pi is much more common , but why ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . | why a full rotation is taken as 360 degree ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | they say , hey , would n't it be natural to define some number , the ratio of the circumference to the radius ? and as you see here , this pi is just one half times this over here , right ? circumference over 2 r , this the same thing as one half times circumference over r. so pi is just half of tau . or another way to... | is half a circle pi ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | it would be tau radii , or it would be to tau radians would be the angle subtended by that arc length . it would be tau radians . all the way around is tau radians . | how do you convert a time constant to a frequency in radians ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | it would be tau radians . all the way around is tau radians . so that by itself is intuitive . | how long has tau been around ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | pi over 2 is the same thing as to tau over 4 . pi is the same thing as to tau over 2 . 3 pi over 2 is 3 pi -- oh , sorry , 3 tau over 4 , 3/4 tau . and then one revolution is tau . | why was pi discovered as an idea before tau ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | pi over 2 is the same thing as to tau over 4 . pi is the same thing as to tau over 2 . 3 pi over 2 is 3 pi -- oh , sorry , 3 tau over 4 , 3/4 tau . and then one revolution is tau . | so , why do n't more teachers use tau rather than pi if it 's so much easier to use ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | well , now it is 1 . this is the unit circle . it has a radius 1 . | what is the area of a unit circle ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . | why a full rotation is taken as 360 degree ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | and as you see here , this pi is just one half times this over here , right ? circumference over 2 r , this the same thing as one half times circumference over r. so pi is just half of tau . or another way to think about it is that tau is just 2 times pi . or , and i 'm sure you probably do n't have this memorized , be... | is n't tau basically half of pi ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | and you get two times the radius times pi is equal to the circumference , or more familiarly , it would be circumference is equal to 2 pi r. this is one of those fundamental things that you learn early on in your career and you use it to find circumferences , usually , or figure out radiuses if you know circumference .... | why use theta as the unknown measure of angles ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | pi is the same thing as to tau over 2 . 3 pi over 2 is 3 pi -- oh , sorry , 3 tau over 4 , 3/4 tau . and then one revolution is tau . | 4 , is the plural form of radius really radiuseseseseses ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | radii , actually let me do that just so no one says , sal , you 're teaching people the wrong plural form of radius . radii . so this arc length is pi over 2 radii and it subtends an angle of pi over 2 radians . | what is the difference between radians and radii ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | pi over 2 is the same thing as to tau over 4 . pi is the same thing as to tau over 2 . 3 pi over 2 is 3 pi -- oh , sorry , 3 tau over 4 , 3/4 tau . and then one revolution is tau . | what is the point of tau versus pi ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | so 3 pi over 2 radians , sine of theta is the y-coordinate on the unit circle right over here . so it 's going to be negative 1 . so this is negative 1 . and then finally , when you go all the way around the circle , you 've gone 2 pi radians , and you 're back where you began . | for what i know up till now is that we do n't have any roots for negative numbers , so how can root -1 be explained as the variable `` i '' and further why is it done so ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | well , let 's just try it out and see what happens . so if we take e to the i tau , that will give us cosine of tau plus i sine of tau . and once again , let 's just think about what this is . | or we can even take the tangent of tau/4 , so , can it be considered tangent of 90 degrees ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | actually , i think it 's radii but it 's fun to try to say radiuses . radii , actually let me do that just so no one says , sal , you 're teaching people the wrong plural form of radius . radii . so this arc length is pi over 2 radii and it subtends an angle of pi over 2 radians . | little grammar question ... is the plural form of radius that sal is trying to say at around supposed to be radians or radii ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | and it is pretty profound . you have all of the fundamental numbers in one equation . e , i , pi , 1 , 0 , although for my aesthetic taste it would have been even more powerful if this was a 1 right over here . | and how can you create formulas with imaginary numbers ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | pi over 2 is the same thing as to tau over 4 . pi is the same thing as to tau over 2 . 3 pi over 2 is 3 pi -- oh , sorry , 3 tau over 4 , 3/4 tau . and then one revolution is tau . | so is tau better than pi ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | so that by itself is intuitive . one revolution is one tau radians . if you go only one fourth of that , it 's going to be tau over four radians . | if the area of a circle in terms of tau is the area of a triangle with both base and height equal to its radius multiplied by tau , can tau be used to quickly find the area of an ellipse given any right triangle with two vertexes on it and one at the origin ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | and as you see here , this pi is just one half times this over here , right ? circumference over 2 r , this the same thing as one half times circumference over r. so pi is just half of tau . or another way to think about it is that tau is just 2 times pi . | what about the circle formula , pi r ( r ) =area , you do n't want to say tau/2 r ( r ) ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | and so , out of this comes all of the things that we know about how to graph trigonometric functions or at least how we measure the graph on the x-axis . and i 'll also touch on euler 's formula , which is the most beautiful formula , i think , in all of mathematics . and let 's visit those right now , just to remind o... | why is eulers formula so significant ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | pi over 2 is the same thing as to tau over 4 . pi is the same thing as to tau over 2 . 3 pi over 2 is 3 pi -- oh , sorry , 3 tau over 4 , 3/4 tau . and then one revolution is tau . | does anybody else realize that pi and tau are both made up of capital t 's ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | but it sometimes looks even more mind boggling when you put pi in for theta because then , from euler 's formula , you would get e to the i pi is equal to -- well , what 's cosine of pi ? cosine of pi is negative 1 . and then sine of pi is 0 . | how ( e ) power i *pi give as a negative number ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | well now , all of a sudden the arc length is the entire circumference of the circle . it would be 2 pi r , which is the same thing as 2 pi radii . and we would say that the angle subtended by this arc length , the angle that we care about going all the way around the circle , is 2 pi radians . | if pi is wrong , would the computation of the circumference of the circle change with pi , or would the equation c=d ( pi ) be the same ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | let me make that clear . cosine of theta is the x value , sine of theta is the y value . and so if you were to graph one of these functions , and i 'll just do sine of theta for convenience . | i do n't understand sal , what was the value of the zero ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | pi over 2 is the same thing as to tau over 4 . pi is the same thing as to tau over 2 . 3 pi over 2 is 3 pi -- oh , sorry , 3 tau over 4 , 3/4 tau . and then one revolution is tau . | if tau is easier to look at than pi , then why do most people know about pi and not tau ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | well now , all of a sudden the arc length is the entire circumference of the circle . it would be 2 pi r , which is the same thing as 2 pi radii . and we would say that the angle subtended by this arc length , the angle that we care about going all the way around the circle , is 2 pi radians . | how do i type the symbol for pi and square root ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | pi over 2 is the same thing as to tau over 4 . pi is the same thing as to tau over 2 . 3 pi over 2 is 3 pi -- oh , sorry , 3 tau over 4 , 3/4 tau . and then one revolution is tau . | how did people come up with pi and tau ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | it comes from -- well , it 's inspired by -- many people are on this movement now , the tau movement , but these are kind of the people that gave me the thinking on this . and the first is robert palais on `` pi is wrong . '' and he does n't argue that pi is calculated wrong . he still agrees that it is the ratio of th... | how is pi still wrong , i dont get ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | well now , all of a sudden the arc length is the entire circumference of the circle . it would be 2 pi r , which is the same thing as 2 pi radii . and we would say that the angle subtended by this arc length , the angle that we care about going all the way around the circle , is 2 pi radians . | so for the area of a circle , would [ a = ( tau * r^2 ) / 2 ] be logical ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | it would be tau radii , or it would be to tau radians would be the angle subtended by that arc length . it would be tau radians . all the way around is tau radians . | if i am wrong , how would the area of a circle be expressed using tau instead of pi ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | pi over 2 is the same thing as to tau over 4 . pi is the same thing as to tau over 2 . 3 pi over 2 is 3 pi -- oh , sorry , 3 tau over 4 , 3/4 tau . and then one revolution is tau . | why ca n't pi and tau be friends ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | how does tau hold up to this ? well , let 's just try it out and see what happens . so if we take e to the i tau , that will give us cosine of tau plus i sine of tau . | so what happens when you compute the volume of a sphere ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | and it is pretty profound . you have all of the fundamental numbers in one equation . e , i , pi , 1 , 0 , although for my aesthetic taste it would have been even more powerful if this was a 1 right over here . | what does the letter i in the equation stand for ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | well , let 's just try it out and see what happens . so if we take e to the i tau , that will give us cosine of tau plus i sine of tau . and once again , let 's just think about what this is . | if 2pi=tau then what is the area of a circle in terms of tau ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | so you get this formula , which is pretty profound , and then you say , ok , if i want to put all of the fundamental numbers together in one formula , i can add 1 to both sides of this . and you get e to the i pi plus 1 is equal to 0 . sometimes this is called euler 's identity , the most beautiful formula or equation ... | how is e^ipi+1 equal to 0 ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | well , let 's just try it out and see what happens . so if we take e to the i tau , that will give us cosine of tau plus i sine of tau . and once again , let 's just think about what this is . | if tau is so effective and much more convenient , why do people still use pi , not tau ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | and you get e to the i pi plus 1 is equal to 0 . sometimes this is called euler 's identity , the most beautiful formula or equation in all of mathematics . and it is pretty profound . | please forgive my ignorance , but can someone summarize what euler 's formula/identity is ( ) and what it 's used for and what 's so profound and beautiful about it ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | pi over 2 is the same thing as to tau over 4 . pi is the same thing as to tau over 2 . 3 pi over 2 is 3 pi -- oh , sorry , 3 tau over 4 , 3/4 tau . and then one revolution is tau . | so your saying that tau is a simpler why to do math than pi ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | all of this is available online . and what they argue for is a number called tau , or what they call tau . and they define tau , and it 's a very simple change from pi . | what is the number tau ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | when theta is equal to pi , the y value of this point right here is once again 0 . so we go back to 0 . remember , we 're talking about sine of theta . | where did the `` i '' in the equation go ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | well , let 's just try it out and see what happens . so if we take e to the i tau , that will give us cosine of tau plus i sine of tau . and once again , let 's just think about what this is . | is tau frequently used or universally recognized ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | well , let 's just try it out and see what happens . so if we take e to the i tau , that will give us cosine of tau plus i sine of tau . and once again , let 's just think about what this is . | will the average mathematician understand what you mean when you say tau ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | so this is a nice review of all of that . you assume you have a unit circle , a circle of radius 1 . and then the trig functions are defined as , for any angle you have here , for any angle , theta , cosine of theta is how far you have to move in -- or the x-coordinate of the point along the arc that subtends this angl... | why a circle ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | so when the angle is 0 , we 're right here on the unit circle . the y value there is 0 . so sine of theta is going to be right like that . | if 0 is a point on the number line and i is i units away from 0 perpendicular to the number line , does n't it have a nonzero value ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | and then the trig functions are defined as , for any angle you have here , for any angle , theta , cosine of theta is how far you have to move in -- or the x-coordinate of the point along the arc that subtends this angle . so that 's cosine of theta . and then sine of theta is the y value of that point . | so , when we say that an angle approaches 90 degrees , can we say that the '' sin of theta '' is increasing and `` cosine of theta '' is decreasing ? |
what i want to do in this video is revisit a little bit of what we know about pi , and really how we measure angles in radians . and then think about whether pi is necessarily the best number to be paying attention to . so let 's think a little bit about what i just said . so pi , we know , is defined -- and i 'll writ... | well now , all of a sudden the arc length is the entire circumference of the circle . it would be 2 pi r , which is the same thing as 2 pi radii . and we would say that the angle subtended by this arc length , the angle that we care about going all the way around the circle , is 2 pi radians . | is there truly a final decimal digit to pi ? |
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