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anytime you 're trying to come up with a mechanism for a reaction , it 's worthwhile to study a little bit of what you are starting with and then thinking about what you finish with and think about what is different . so what we 're starting with , we could call this one , two , three , four , five , so this is , let ... | because the higher order of carbon , the more stable it will be if it forms some type of cation . so if you look at this carbon right over here , our number two carbon , let me circle it , our number two carbon is bonded to one , two , three carbons . so this is a tertiary carbon . this one right over here , this carbo... | and by the same logic , is the right carbon not a tertiary carbon ? |
anytime you 're trying to come up with a mechanism for a reaction , it 's worthwhile to study a little bit of what you are starting with and then thinking about what you finish with and think about what is different . so what we 're starting with , we could call this one , two , three , four , five , so this is , let ... | so one way to think about this is , in the presence of an acid , it 's acid-catalyzed , we have gained two hydrogens and an oxygen , which is what we 've gained , what could be used to make a water . and this is actually called an acid-catalyzed addition of water . the water is n't sitting on one part of the molecule ,... | an electophilic addition or a neucleophilic addition ? |
anytime you 're trying to come up with a mechanism for a reaction , it 's worthwhile to study a little bit of what you are starting with and then thinking about what you finish with and think about what is different . so what we 're starting with , we could call this one , two , three , four , five , so this is , let ... | it now , it now forms an actual bond . and we 're really close to our final product , we have our hydrogen on the number three carbon , we have more than we want on our number two carbon , we just want a hydroxyl group , now we have a whole water bonded to the carbon . so somehow we have to get one of these other hydro... | what 's the name of the final product 2 methil , 2 penthanol ? |
now i 'm gon na show you what a circuit , that 's called a voltage divider . this is the name we give to a simple circuit of two series resistors . so , i 'm just gon na draw two series resistors here . and it 's a nickname , in the sense of , it 's just a pattern that we see when we look at circuits . and , i 'll sho... | and then , the midpoint of the two resistors , and typically the bottom , that 's called the 'out ' . so , we basically just have a pattern here with the series resistor , driven by some voltage from the ends of the two resistors . and we 're curious about the voltage across one of them . | what do the loose ends around the lower resistor mean ? |
now i 'm gon na show you what a circuit , that 's called a voltage divider . this is the name we give to a simple circuit of two series resistors . so , i 'm just gon na draw two series resistors here . and it 's a nickname , in the sense of , it 's just a pattern that we see when we look at circuits . and , i 'll sho... | which means that 'v out ' is always somewhat less than 'v in ' . and it 's adjustable , by adjusting the resistor values . it 's a really handy circuit to have . | conceptually , in this oversimplified case , why not just use 1 resistor ? |
now i 'm gon na show you what a circuit , that 's called a voltage divider . this is the name we give to a simple circuit of two series resistors . so , i 'm just gon na draw two series resistors here . and it 's a nickname , in the sense of , it 's just a pattern that we see when we look at circuits . and , i 'll sho... | and we get 'v out ' equals 'v1 ' . sorry , 'v in ' times 'r2 ' over 'r1 ' plus 'r2 ' . and this is called , this is called the voltage divider expression . | why we only take r2 for v out ? |
now i 'm gon na show you what a circuit , that 's called a voltage divider . this is the name we give to a simple circuit of two series resistors . so , i 'm just gon na draw two series resistors here . and it 's a nickname , in the sense of , it 's just a pattern that we see when we look at circuits . and , i 'll sho... | and if you remember at the beginning , if you remember at the beginning , we made an assumption that this current going out here , was appr -- about zero . if that current is really small , you can use this voltage divider expression . which as , we see up here , is the ratio of the bottom resistor to both resistors . | why is the voltage on the node between the resistors assumed to be very small or 0 ? |
now i 'm gon na show you what a circuit , that 's called a voltage divider . this is the name we give to a simple circuit of two series resistors . so , i 'm just gon na draw two series resistors here . and it 's a nickname , in the sense of , it 's just a pattern that we see when we look at circuits . and , i 'll sho... | and we 'll say that that is 'v out ' . so let 's solve this using the voltage divider expression . 'v out ' equals 'v in ' , which is 6 volts . | it looks to me as if vout is equivalent to the voltage that `` exists '' on the r2 resistor , and therefore could be figured out using a voltmeter ? |
now i 'm gon na show you what a circuit , that 's called a voltage divider . this is the name we give to a simple circuit of two series resistors . so , i 'm just gon na draw two series resistors here . and it 's a nickname , in the sense of , it 's just a pattern that we see when we look at circuits . and , i 'll sho... | and we assume that there 's a voltage over here . we hook up a voltage over here like this . so , that 's called an input voltage . we 'll call it 'vi ' , for 'v in ' . | also can this also be theoretically ( or practically ) used as a new `` battery or voltage source '' for `` new circuits '' that require lesser voltage input that vi ? |
now i 'm gon na show you what a circuit , that 's called a voltage divider . this is the name we give to a simple circuit of two series resistors . so , i 'm just gon na draw two series resistors here . and it 's a nickname , in the sense of , it 's just a pattern that we see when we look at circuits . and , i 'll sho... | now i 'm gon na show you what a circuit , that 's called a voltage divider . this is the name we give to a simple circuit of two series resistors . | why do the headlights on a car dim when the starter motor is operated ? |
now i 'm gon na show you what a circuit , that 's called a voltage divider . this is the name we give to a simple circuit of two series resistors . so , i 'm just gon na draw two series resistors here . and it 's a nickname , in the sense of , it 's just a pattern that we see when we look at circuits . and , i 'll sho... | and if i do my calculations right , 'v out ' is 4.5 volts . so that 's what a voltage divider is . and if you remember at the beginning , if you remember at the beginning , we made an assumption that this current going out here , was appr -- about zero . | so the voltage in the voltage divider , is that basically what a voltmeter would read if you connected it to the circuit at the second resistor ? |
now i 'm gon na show you what a circuit , that 's called a voltage divider . this is the name we give to a simple circuit of two series resistors . so , i 'm just gon na draw two series resistors here . and it 's a nickname , in the sense of , it 's just a pattern that we see when we look at circuits . and , i 'll sho... | and we 're gon na develop an expression for this . so , let 's first put a current through here . we 'll call that current 'i ' . | if the current in the first wire is to be accounted for as mentioned in how would the calculation continue and how would the current affect the output voltage ? |
now i 'm gon na show you what a circuit , that 's called a voltage divider . this is the name we give to a simple circuit of two series resistors . so , i 'm just gon na draw two series resistors here . and it 's a nickname , in the sense of , it 's just a pattern that we see when we look at circuits . and , i 'll sho... | and if i do my calculations right , 'v out ' is 4.5 volts . so that 's what a voltage divider is . and if you remember at the beginning , if you remember at the beginning , we made an assumption that this current going out here , was appr -- about zero . | what is a voltage divider , why is it important and what are some of its applications ? |
now i 'm gon na show you what a circuit , that 's called a voltage divider . this is the name we give to a simple circuit of two series resistors . so , i 'm just gon na draw two series resistors here . and it 's a nickname , in the sense of , it 's just a pattern that we see when we look at circuits . and , i 'll sho... | in a specific case here , 'v in ' equals 'i ' times what ? times the series combination of 'r1 ' and 'r2 ' . and the series combination is the sum : 'r1 ' plus 'r2 ' . | since both r1 and r2 are in series , why do n't we use simple ohms law to find the vo ? |
now i 'm gon na show you what a circuit , that 's called a voltage divider . this is the name we give to a simple circuit of two series resistors . so , i 'm just gon na draw two series resistors here . and it 's a nickname , in the sense of , it 's just a pattern that we see when we look at circuits . and , i 'll sho... | and we 'll take an output off of this . right here , is where the output of our voltage divider is . and we 'll say that that is 'v out ' . | does that mean that the output voltage ( vout ) in this case is the voltage going out after r2 only ? |
now i 'm gon na show you what a circuit , that 's called a voltage divider . this is the name we give to a simple circuit of two series resistors . so , i 'm just gon na draw two series resistors here . and it 's a nickname , in the sense of , it 's just a pattern that we see when we look at circuits . and , i 'll sho... | and we 'll hook it up to an input source that looks like , let 's say it 's 6 volts . like that . and we 'll take an output off of this . | the current in the extending wire is 0 amps as told , just like when a voltmeter is connected , which has high resistance and so current instead of going in voltmeter ( like it does in parallel resistors ) goes through resistors only ... right ? |
now i 'm gon na show you what a circuit , that 's called a voltage divider . this is the name we give to a simple circuit of two series resistors . so , i 'm just gon na draw two series resistors here . and it 's a nickname , in the sense of , it 's just a pattern that we see when we look at circuits . and , i 'll sho... | and if i do my calculations right , 'v out ' is 4.5 volts . so that 's what a voltage divider is . and if you remember at the beginning , if you remember at the beginning , we made an assumption that this current going out here , was appr -- about zero . | what exactly is the purpose of a voltage divider ? |
now i 'm gon na show you what a circuit , that 's called a voltage divider . this is the name we give to a simple circuit of two series resistors . so , i 'm just gon na draw two series resistors here . and it 's a nickname , in the sense of , it 's just a pattern that we see when we look at circuits . and , i 'll sho... | and we 'll hook it up to an input source that looks like , let 's say it 's 6 volts . like that . and we 'll take an output off of this . | in the end of the video sal poses a scenario where the current is not 0 like maybe it is connected to a bulb or something , how would the calculation proceed in that scenario ? |
we 're asked to determine whether each expression is equivalent to the seventh root of v to the third power . and , like always , pause the video and see if you can figure out which of these are equivalent to the seventh root of v to the third power . well , a good way to figure out if things are equivalent is to just... | and then , if you multiply these exponents , you get what we have right over there . but , either way , d is equal to negative 1/7 . | should n't the negative be put on the base ? |
we 're asked to determine whether each expression is equivalent to the seventh root of v to the third power . and , like always , pause the video and see if you can figure out which of these are equivalent to the seventh root of v to the third power . well , a good way to figure out if things are equivalent is to just... | and i forgot to tell you in the last one , but pause this video as well and see if you can work it out on ... or pause for this question as well and see if you can work it out . well , here , let 's just start rewriting the root as an exponent . so , i can rewrite the whole thing . | how do you rewrite the root if its a negitive decimal with a fraction as an exponent ? |
we 're asked to determine whether each expression is equivalent to the seventh root of v to the third power . and , like always , pause the video and see if you can figure out which of these are equivalent to the seventh root of v to the third power . well , a good way to figure out if things are equivalent is to just... | so , this is going to be the same thing as v to the three times 1/7 power , which , of course , is 3/7 . 3/7 . so , we 've written it in multiple forms now . | how does work , when x has a positive exponent , turn into a negative ? |
we 're asked to determine whether each expression is equivalent to the seventh root of v to the third power . and , like always , pause the video and see if you can figure out which of these are equivalent to the seventh root of v to the third power . well , a good way to figure out if things are equivalent is to just... | so , v to the third to the 1/7 power , well , that was the form that we have right over here , so that is equivalent . v to the 3/7 . that 's what we have right over here , so that one is definitely equivalent . | at about 0.50 , why do you multiply the two numbers for v to the 3rd to the 1/7 ? |
we 're asked to determine whether each expression is equivalent to the seventh root of v to the third power . and , like always , pause the video and see if you can figure out which of these are equivalent to the seventh root of v to the third power . well , a good way to figure out if things are equivalent is to just... | so , this is going to be the same thing as v to the three times 1/7 power , which , of course , is 3/7 . 3/7 . so , we 've written it in multiple forms now . | am i wrong in believing that x^1/7 * x^7 = x^1/7 * x^7/1 which is equal to x^7/7=x ? |
we 're asked to determine whether each expression is equivalent to the seventh root of v to the third power . and , like always , pause the video and see if you can figure out which of these are equivalent to the seventh root of v to the third power . well , a good way to figure out if things are equivalent is to just... | so , this is going to be the same thing as v to the three times 1/7 power , which , of course , is 3/7 . 3/7 . so , we 've written it in multiple forms now . | how do you raise a number to a fraction like 3/7 ? |
we 're asked to determine whether each expression is equivalent to the seventh root of v to the third power . and , like always , pause the video and see if you can figure out which of these are equivalent to the seventh root of v to the third power . well , a good way to figure out if things are equivalent is to just... | and then , if you multiply these exponents , you get what we have right over there . but , either way , d is equal to negative 1/7 . | if you raise a number to 1/7 , it 's the same as taking the 7th root , but how does it work with a faction with a numerator other than 1 ? |
we 're asked to determine whether each expression is equivalent to the seventh root of v to the third power . and , like always , pause the video and see if you can figure out which of these are equivalent to the seventh root of v to the third power . well , a good way to figure out if things are equivalent is to just... | another way of thinking about it is you could view this as , you could view it as , x to the 1/7 to the negative one power . and then , if you multiply these exponents , you get what we have right over there . but , either way , d is equal to negative 1/7 . | is it possible to change order which exponents multiplication takes place , since multiplication is commutative ? |
we 're asked to determine whether each expression is equivalent to the seventh root of v to the third power . and , like always , pause the video and see if you can figure out which of these are equivalent to the seventh root of v to the third power . well , a good way to figure out if things are equivalent is to just... | so , that 's the same thing as g to the 5/6 power . and so d is 5/6 . five over six . | for example , taken from practice rational exponents challenge : ( d^1/8 ) ^5 is equivalent to ( d^5 ) ^1/8 ? |
we 're asked to determine whether each expression is equivalent to the seventh root of v to the third power . and , like always , pause the video and see if you can figure out which of these are equivalent to the seventh root of v to the third power . well , a good way to figure out if things are equivalent is to just... | so , this is going to be the same thing as v to the three times 1/7 power , which , of course , is 3/7 . 3/7 . so , we 've written it in multiple forms now . | on number 3 , how come whenever you take the reciprocal of the left side , the right side does n't become a reciprocal ? |
we 're asked to determine whether each expression is equivalent to the seventh root of v to the third power . and , like always , pause the video and see if you can figure out which of these are equivalent to the seventh root of v to the third power . well , a good way to figure out if things are equivalent is to just... | let 's do a few more of these , or similar types of problems dealing with roots and fractional exponents . the following equation is true for g greater than or equal to zero , and d is a constant . what is the value of d ? | what does , `` g greater than or equal to zero '' mean ? |
we 're asked to determine whether each expression is equivalent to the seventh root of v to the third power . and , like always , pause the video and see if you can figure out which of these are equivalent to the seventh root of v to the third power . well , a good way to figure out if things are equivalent is to just... | let 's do one more of these . the following equation is true for x greater than zero , and d is a constant . what is the value of d ? | and what is a constant ? |
we 're asked to determine whether each expression is equivalent to the seventh root of v to the third power . and , like always , pause the video and see if you can figure out which of these are equivalent to the seventh root of v to the third power . well , a good way to figure out if things are equivalent is to just... | so , that 's the same thing as g to the 5/6 power . and so d is 5/6 . five over six . | why do you make n to the 6th into n to the 1/6 ? |
we 're asked to determine whether each expression is equivalent to the seventh root of v to the third power . and , like always , pause the video and see if you can figure out which of these are equivalent to the seventh root of v to the third power . well , a good way to figure out if things are equivalent is to just... | so , this is going to be the same thing as v to the three times 1/7 power , which , of course , is 3/7 . 3/7 . so , we 've written it in multiple forms now . | what is 15 ( ab/b^-3 ) ^-4 ? |
we 're asked to determine whether each expression is equivalent to the seventh root of v to the third power . and , like always , pause the video and see if you can figure out which of these are equivalent to the seventh root of v to the third power . well , a good way to figure out if things are equivalent is to just... | so , this is going to be the same thing as v to the three times 1/7 power , which , of course , is 3/7 . 3/7 . so , we 've written it in multiple forms now . | why dose 3 go on the top and not the bottom ? |
we 're asked to determine whether each expression is equivalent to the seventh root of v to the third power . and , like always , pause the video and see if you can figure out which of these are equivalent to the seventh root of v to the third power . well , a good way to figure out if things are equivalent is to just... | this is the same thing as one over , instead of writing the seventh root of x , i 'll write x to the 1/7 power is equal to x to the d. and if i have one over something to a power , that 's the same thing as that something raised to the negative of that power . so , that is the same thing as x to the negative 1/7 power ... | 3rd example question : if 1 / x ^ 1/7 is `` the same as '' x ^ -1/7 , why could n't the answer just be x ^ 1/7 ? |
let 's try to come up with a mathematically rigorous definition for what this statement means . the statement that the limit of f of x as x approaches c is equal to l. so let 's say that this means that you can get f of x as close to l as you want . i 'll put that in quotes right over here , because it 's kind of a lit... | let 's try to come up with a mathematically rigorous definition for what this statement means . the statement that the limit of f of x as x approaches c is equal to l. so let 's say that this means that you can get f of x as close to l as you want . i 'll put that in quotes right over here , because it 's kind of a lit... | what does y=f ( x ) mean ? |
let 's try to come up with a mathematically rigorous definition for what this statement means . the statement that the limit of f of x as x approaches c is equal to l. so let 's say that this means that you can get f of x as close to l as you want . i 'll put that in quotes right over here , because it 's kind of a lit... | i 'm going to draw a slightly different function , just so we can really focus on what 's going on around here . the range is around c , and the range is around l. so that 's x . this right over here is y . | why is it that the range of the l is 0.5 and the range around c was 0.25 , my point is that will the range for the x-axis always be less than that for the range for the y-axis ? |
let 's try to come up with a mathematically rigorous definition for what this statement means . the statement that the limit of f of x as x approaches c is equal to l. so let 's say that this means that you can get f of x as close to l as you want . i 'll put that in quotes right over here , because it 's kind of a lit... | let 's try to come up with a mathematically rigorous definition for what this statement means . the statement that the limit of f of x as x approaches c is equal to l. so let 's say that this means that you can get f of x as close to l as you want . i 'll put that in quotes right over here , because it 's kind of a lit... | are limits used only in situations when x approaches the value for which the function f ( x ) is undefined , and not for other values ? |
let 's try to come up with a mathematically rigorous definition for what this statement means . the statement that the limit of f of x as x approaches c is equal to l. so let 's say that this means that you can get f of x as close to l as you want . i 'll put that in quotes right over here , because it 's kind of a lit... | so in the next video , we will now generalize that . and that will really bring us to the famous epsilon delta definition of limits . | why were epsilon delta forms created ? |
let 's try to come up with a mathematically rigorous definition for what this statement means . the statement that the limit of f of x as x approaches c is equal to l. so let 's say that this means that you can get f of x as close to l as you want . i 'll put that in quotes right over here , because it 's kind of a lit... | so in the next video , we will now generalize that . and that will really bring us to the famous epsilon delta definition of limits . | in what kind of circumstances would people use this epsilon-delta definition out of the studies of mathematics ? |
let 's try to come up with a mathematically rigorous definition for what this statement means . the statement that the limit of f of x as x approaches c is equal to l. so let 's say that this means that you can get f of x as close to l as you want . i 'll put that in quotes right over here , because it 's kind of a lit... | so i want to get within 0.5 . i want to get f of x within 0.5 of l. so this right over here would be l plus 0.5 . and this right over here is l minus 0.5 . and then you say , fine . | your graph is between l-0.5 and l+0.5 at all times but what if at l-0.25 it all of the sudden dramatically spikes out of l-0.5 just for a quick dip and comes back ? |
let 's try to come up with a mathematically rigorous definition for what this statement means . the statement that the limit of f of x as x approaches c is equal to l. so let 's say that this means that you can get f of x as close to l as you want . i 'll put that in quotes right over here , because it 's kind of a lit... | i 'll put that in quotes right over here , because it 's kind of a little loosey goosey as how close is that . but as close as you want by getting x sufficiently close to c. so another way of saying this is , if you tell me , hey , i want to get my f of x to be within 0.5 of this limit . then you 're telling me if this... | how does this affect the `` close as you want '' concept ? |
let 's try to come up with a mathematically rigorous definition for what this statement means . the statement that the limit of f of x as x approaches c is equal to l. so let 's say that this means that you can get f of x as close to l as you want . i 'll put that in quotes right over here , because it 's kind of a lit... | i 'm going to draw a slightly different function , just so we can really focus on what 's going on around here . the range is around c , and the range is around l. so that 's x . this right over here is y . | is it possible for the range around c that x must be in ( for y to equal l ) to not be centered/symmetrical around c ? |
let 's try to come up with a mathematically rigorous definition for what this statement means . the statement that the limit of f of x as x approaches c is equal to l. so let 's say that this means that you can get f of x as close to l as you want . i 'll put that in quotes right over here , because it 's kind of a lit... | i 'm going to draw a slightly different function , just so we can really focus on what 's going on around here . the range is around c , and the range is around l. so that 's x . this right over here is y . | do the upper and lower boundaries of the range have to be equidistant from the given l or the given c ? |
let 's try to come up with a mathematically rigorous definition for what this statement means . the statement that the limit of f of x as x approaches c is equal to l. so let 's say that this means that you can get f of x as close to l as you want . i 'll put that in quotes right over here , because it 's kind of a lit... | there 's a lot to think about . but hopefully this made sense . we did it for the particular example of someone hands you the 0.5 , i want f of x within the 0.5 of l , and you say , well , as long as x is within 0.25 of c , you 're going to match it . | why limits were made in the first place ? |
let 's try to come up with a mathematically rigorous definition for what this statement means . the statement that the limit of f of x as x approaches c is equal to l. so let 's say that this means that you can get f of x as close to l as you want . i 'll put that in quotes right over here , because it 's kind of a lit... | you won that round . let me make it even tighter . maybe instead of saying within the 0.5 , i want to get within is 0.05 . and then you 'd have to do this exercise again and find another range . | why can we make coordinate planes where the `` origin '' is n't at ( 0,0 ) ? |
let 's try to come up with a mathematically rigorous definition for what this statement means . the statement that the limit of f of x as x approaches c is equal to l. so let 's say that this means that you can get f of x as close to l as you want . i 'll put that in quotes right over here , because it 's kind of a lit... | let 's try to come up with a mathematically rigorous definition for what this statement means . the statement that the limit of f of x as x approaches c is equal to l. so let 's say that this means that you can get f of x as close to l as you want . i 'll put that in quotes right over here , because it 's kind of a lit... | for example- will this definition fail for ( lim x tending to 0 , then f ( x ) = sin1/x ) ? |
let 's try to come up with a mathematically rigorous definition for what this statement means . the statement that the limit of f of x as x approaches c is equal to l. so let 's say that this means that you can get f of x as close to l as you want . i 'll put that in quotes right over here , because it 's kind of a lit... | so let 's say our function looks , is doing something like , let 's say it does something like , let 's see , i do n't want it to be defined at c. at least just for the -- it could be . you can always find a limit even where is defined . but let 's say our function looks something like that . | would such a second discontinuity render sal 's limit incorrect ? |
let 's try to come up with a mathematically rigorous definition for what this statement means . the statement that the limit of f of x as x approaches c is equal to l. so let 's say that this means that you can get f of x as close to l as you want . i 'll put that in quotes right over here , because it 's kind of a lit... | and in order for this to be true , you would have to be able to do this for any range that they give you . for any range around l that they give you , you have to be able to get f of x within that range by finding a range around c. that as long as x is that range around c , f of x is going to sit within that range . so... | in sal says that we need to be able to do `` this '' for any range from c. what if requested range maps to the part of the function that also is not defined ( for example if there is no `` x '' that would equal l + 12 ) ? |
let 's try to come up with a mathematically rigorous definition for what this statement means . the statement that the limit of f of x as x approaches c is equal to l. so let 's say that this means that you can get f of x as close to l as you want . i 'll put that in quotes right over here , because it 's kind of a lit... | i 'm going to draw a slightly different function , just so we can really focus on what 's going on around here . the range is around c , and the range is around l. so that 's x . this right over here is y . | after | so , basically when someone gives me a certain range around the limit , do i have to find the range of the function ? |
let 's try to come up with a mathematically rigorous definition for what this statement means . the statement that the limit of f of x as x approaches c is equal to l. so let 's say that this means that you can get f of x as close to l as you want . i 'll put that in quotes right over here , because it 's kind of a lit... | and in order for this to be true , you would have to be able to do this for any range that they give you . for any range around l that they give you , you have to be able to get f of x within that range by finding a range around c. that as long as x is that range around c , f of x is going to sit within that range . so... | so ... is the range inclusive ? |
let 's try to come up with a mathematically rigorous definition for what this statement means . the statement that the limit of f of x as x approaches c is equal to l. so let 's say that this means that you can get f of x as close to l as you want . i 'll put that in quotes right over here , because it 's kind of a lit... | so i want to get within 0.5 . i want to get f of x within 0.5 of l. so this right over here would be l plus 0.5 . and this right over here is l minus 0.5 . and then you say , fine . | are l-0.5 and l+0.5 also attainable if we define a range for c ? |
let 's try to come up with a mathematically rigorous definition for what this statement means . the statement that the limit of f of x as x approaches c is equal to l. so let 's say that this means that you can get f of x as close to l as you want . i 'll put that in quotes right over here , because it 's kind of a lit... | well , it 's saying that you can get f of x as close to l as you want . so if you tell someone , i want to get f of x within a certain range of l , then if this limit is actually true , if the limit of f of x as x approaches c really is equal to l , then they should be able to find a range around c. that as long as x i... | i mean to find instantaneous velocity , why should th limit of f ( x ) aproach 0 ? |
let 's try to come up with a mathematically rigorous definition for what this statement means . the statement that the limit of f of x as x approaches c is equal to l. so let 's say that this means that you can get f of x as close to l as you want . i 'll put that in quotes right over here , because it 's kind of a lit... | i 'm going to draw a slightly different function , just so we can really focus on what 's going on around here . the range is around c , and the range is around l. so that 's x . this right over here is y . | why is the range around c , c - .25 and c + .25 and the ranges around l are l + .5 and l - .5 ? |
let 's try to come up with a mathematically rigorous definition for what this statement means . the statement that the limit of f of x as x approaches c is equal to l. so let 's say that this means that you can get f of x as close to l as you want . i 'll put that in quotes right over here , because it 's kind of a lit... | let 's say that this is c. and let 's just zoom in on our function . so let 's say our function looks , is doing something like , let 's say it does something like , let 's see , i do n't want it to be defined at c. at least just for the -- it could be . you can always find a limit even where is defined . but let 's sa... | you can only take a real , defined limit of something if it is a point/removable discontinuity or a continuous function , right ? |
let 's try to come up with a mathematically rigorous definition for what this statement means . the statement that the limit of f of x as x approaches c is equal to l. so let 's say that this means that you can get f of x as close to l as you want . i 'll put that in quotes right over here , because it 's kind of a lit... | let me draw it a little bit different . so it is undefined when x is equal to c. so this is the point where there 's a hole . it is undefined when x is equal to c. so it even has a little kink in it , just like that . | so what if you had two point/removable discontinuities `` in a row '' on your graph ? |
let 's try to come up with a mathematically rigorous definition for what this statement means . the statement that the limit of f of x as x approaches c is equal to l. so let 's say that this means that you can get f of x as close to l as you want . i 'll put that in quotes right over here , because it 's kind of a lit... | and what we want to do is prove that the limit , as x , the limit of f of x -- and let me make it clear , this is the graph of y is equal to f of x -- we want to get an idea for what this definition is saying . if we 're claiming that the limit of f of x , as x approaches c , is l. so conceptually , we get the gist alr... | how do you decide what to chose as c- x to be ? |
let 's try to come up with a mathematically rigorous definition for what this statement means . the statement that the limit of f of x as x approaches c is equal to l. so let 's say that this means that you can get f of x as close to l as you want . i 'll put that in quotes right over here , because it 's kind of a lit... | i 'm going to draw a slightly different function , just so we can really focus on what 's going on around here . the range is around c , and the range is around l. so that 's x . this right over here is y . | for physical systems does the behavior of the l range versus the c range tell us anything meaningful about the system ? |
now , we 've talked about hypertension , and you know that it means that you have high blood pressure . so the next logical thing to think about is why is that bad ? why is it a problem to have high blood pressure ? and i like to think about high blood pressure from two different perspectives -- one would be the perspe... | ok , going back to the large and middle arteries , you also can have a situation -- i 'll draw it here -- where you have an artery , let 's say -- actually , let me write what it is first . you can have an aneurysm . and an aneurysm is where you have a vessel -- let 's say this is my vessel , and it 's taking blood thr... | 7 is the body able to repair an aneurysm or is surgery required to repair it ? |
now , we 've talked about hypertension , and you know that it means that you have high blood pressure . so the next logical thing to think about is why is that bad ? why is it a problem to have high blood pressure ? and i like to think about high blood pressure from two different perspectives -- one would be the perspe... | so we have the large and middle sized arteries -- and specifically i mean arteries that are between , let 's say , 25 millimeters in diameter all the way down to about one millimeter in diameter . so primarily these are the vessels that are going to get blood from the heart to the different organs that it needs to get ... | can people get aneurysms in the veins and if so are they called something different ? |
now , we 've talked about hypertension , and you know that it means that you have high blood pressure . so the next logical thing to think about is why is that bad ? why is it a problem to have high blood pressure ? and i like to think about high blood pressure from two different perspectives -- one would be the perspe... | so primarily these are the vessels that are going to get blood from the heart to the different organs that it needs to get to . and then you of course have the small arteries and arterioles . and these are going to be at the high end . | are the aneurysms in the the small arteries just as life threatening as the ones in the large arteries over time ? |
now , we 've talked about hypertension , and you know that it means that you have high blood pressure . so the next logical thing to think about is why is that bad ? why is it a problem to have high blood pressure ? and i like to think about high blood pressure from two different perspectives -- one would be the perspe... | now , we 've talked about hypertension , and you know that it means that you have high blood pressure . so the next logical thing to think about is why is that bad ? | what are the symptoms of hypertension ? |
now , we 've talked about hypertension , and you know that it means that you have high blood pressure . so the next logical thing to think about is why is that bad ? why is it a problem to have high blood pressure ? and i like to think about high blood pressure from two different perspectives -- one would be the perspe... | and actually that aneurysm , if it 's a sack of blood , can actually burst and break . and that blood can spill out , and we call that hemorrhage . so you can actually have an aneurysm because of a weak vessel wall . | can you have hemorrhage from an artery that is arteriosclerotic ? |
now , we 've talked about hypertension , and you know that it means that you have high blood pressure . so the next logical thing to think about is why is that bad ? why is it a problem to have high blood pressure ? and i like to think about high blood pressure from two different perspectives -- one would be the perspe... | so at one spot , it starts to get weak . let 's say right here instead of being like that , it starts to look like this . and you get this little area of weakness . | in the sense of an aneurism but i imagine it like a crack in a pipe ? |
now , we 've talked about hypertension , and you know that it means that you have high blood pressure . so the next logical thing to think about is why is that bad ? why is it a problem to have high blood pressure ? and i like to think about high blood pressure from two different perspectives -- one would be the perspe... | so the next logical thing to think about is why is that bad ? why is it a problem to have high blood pressure ? and i like to think about high blood pressure from two different perspectives -- one would be the perspective of the heart , and the other is the perspective of the blood vessels . | if constant high blood pressure in the arteries weakens them , would n't exercise be somewhat dangerous ? |
now , we 've talked about hypertension , and you know that it means that you have high blood pressure . so the next logical thing to think about is why is that bad ? why is it a problem to have high blood pressure ? and i like to think about high blood pressure from two different perspectives -- one would be the perspe... | now , we 've talked about hypertension , and you know that it means that you have high blood pressure . so the next logical thing to think about is why is that bad ? | how does htn cause a headache ? |
now , we 've talked about hypertension , and you know that it means that you have high blood pressure . so the next logical thing to think about is why is that bad ? why is it a problem to have high blood pressure ? and i like to think about high blood pressure from two different perspectives -- one would be the perspe... | now , we 've talked about hypertension , and you know that it means that you have high blood pressure . so the next logical thing to think about is why is that bad ? | what does hypertension do to the blood vassals and the body ? |
now , we 've talked about hypertension , and you know that it means that you have high blood pressure . so the next logical thing to think about is why is that bad ? why is it a problem to have high blood pressure ? and i like to think about high blood pressure from two different perspectives -- one would be the perspe... | ok , going back to the large and middle arteries , you also can have a situation -- i 'll draw it here -- where you have an artery , let 's say -- actually , let me write what it is first . you can have an aneurysm . and an aneurysm is where you have a vessel -- let 's say this is my vessel , and it 's taking blood thr... | if you get an aneurysm and it is untreated what can happen ? |
now , we 've talked about hypertension , and you know that it means that you have high blood pressure . so the next logical thing to think about is why is that bad ? why is it a problem to have high blood pressure ? and i like to think about high blood pressure from two different perspectives -- one would be the perspe... | and in fact , that change from being elastic to firm , we call that arteriosclerosis . i 'll write that in white -- arteriosclerosis . and in fact , a very similar thing happens on the other side with the small arteries and arterioles . | is arteriosclerosis and atherosclerosis the same thing ? |
now , we 've talked about hypertension , and you know that it means that you have high blood pressure . so the next logical thing to think about is why is that bad ? why is it a problem to have high blood pressure ? and i like to think about high blood pressure from two different perspectives -- one would be the perspe... | so this is the difference , right ? they 're very similar things , kind of similar processes , but one is in the smaller arteries and one is in the larger and middle sized arteries . so this is one of the things that can happen when you have lots of high blood pressure constantly exposed to the vessels . | i thought arteriosclerosis was the buildup of plaque in the arteries..does the term include both plaque buildup and he hardening/stiffening of arteries ? |
now , we 've talked about hypertension , and you know that it means that you have high blood pressure . so the next logical thing to think about is why is that bad ? why is it a problem to have high blood pressure ? and i like to think about high blood pressure from two different perspectives -- one would be the perspe... | ok , going back to the large and middle arteries , you also can have a situation -- i 'll draw it here -- where you have an artery , let 's say -- actually , let me write what it is first . you can have an aneurysm . and an aneurysm is where you have a vessel -- let 's say this is my vessel , and it 's taking blood thr... | can you feel aneurysm happening ? |
now , we 've talked about hypertension , and you know that it means that you have high blood pressure . so the next logical thing to think about is why is that bad ? why is it a problem to have high blood pressure ? and i like to think about high blood pressure from two different perspectives -- one would be the perspe... | or if it 's in your eye , you can get what we call retinopathy , basically meaning that the retina is not working properly . so you can have kidney damage or retinopathy . you can have aneurysms , arteriosclerosis , or arteriolosclerosis . | how do you cure retinopathy ? |
now , we 've talked about hypertension , and you know that it means that you have high blood pressure . so the next logical thing to think about is why is that bad ? why is it a problem to have high blood pressure ? and i like to think about high blood pressure from two different perspectives -- one would be the perspe... | so this could be because the vessel literally breaks right here and blood spills out . and it could also be because these tiny vessels are necessary to make the organ work . for example , the kidneys require that these small arteries and arterioles are working properly . | does arteriosclerosis happens in small vessels first or in the bigger vessels first ? |
now , we 've talked about hypertension , and you know that it means that you have high blood pressure . so the next logical thing to think about is why is that bad ? why is it a problem to have high blood pressure ? and i like to think about high blood pressure from two different perspectives -- one would be the perspe... | now , we 've talked about hypertension , and you know that it means that you have high blood pressure . so the next logical thing to think about is why is that bad ? | how does htn cause headaches ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | and it is true with water . the liquid form is definitely more dense than the gas form . in the gas form , the molecules are going to jump around , not touch each other . | where can you find the fifth form of matter which is bose-einstein condensate ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | you have some bonds to hydrogen . and then you have two extra pairs of valence electrons in the oxygen . and a couple of videos ago , we said oxygen is a lot more electronegative than the hydrogen . | could you explain me why the oxygen atoms in the molecular formula of water have 4 extra electrons on their outermost shell ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | and in the case of water , when you 're a solid , you 're ice . when you 're a liquid , some people would call ice water , but let 's call it liquid water . i think we know what that is . | what makes liquid water have more density than ice ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | and let me write this down . this is in the solid phase , or the solid state of matter . now something very interesting happens . | how will we come to know the state of matter in a chemical equation ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | well , it turns into a liquid . ice melts . not all solids , we 're talking in particular about water , about h2o . | so does that mean that there is some ice that can be colder or warmer that other ice ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | or a gas . so let 's just draw a water molecule . so you have oxygen there . | what is the difference between an atom and a molecule ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | and i 'll be frank , liquids are kind of fascinating because you can never nail them down , i guess is the best way to view them . or a gas . so let 's just draw a water molecule . | how come oxygen is a gas and hydrogen is a gas but water is a liquid ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | and this is the temperature . we 'll talk about the states of matter in a second . so heat is often denoted by q . | is there any other types of states ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . | what is the bose-einstein condensate ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | and what do we know is average kinetic energy ? well , that 's temperature . then this lattice structure will be solid . | is temperature a vector or scalar quantity ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . | why is it vector/scalar ( which ever 's the answer ) ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | the temperature was not increasing . but the ice was going from ice to water . so what was happening at that state , is that the kinetic energy , the heat , was being used to essentially break these bonds . | why does ice float on water ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | and let me write this down . this is in the solid phase , or the solid state of matter . now something very interesting happens . | but if you search it up at science websites , and many other places such as google and bing , they say there is a 4th state of matter ( plasma ) .so is plasma really a state of matter ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | i 'll do it in celsius because that 's what we 're familiar with . what happens ? that 's the temperature at which water will vaporize or which water will boil . | can anyone explain what happens at about 1 000 000 000 000 kelvin/celsius ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | now let 's say at low temperatures i 'm here and as i add heat my temperature will go up . temperature is average kinetic energy . let 's say i 'm in the solid state here . | does applying more heat not change the average kinetic energy or something while in its liquid state ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | and let me write this down . this is in the solid phase , or the solid state of matter . now something very interesting happens . | what state of matter would fire be in ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | the temperature was not increasing . but the ice was going from ice to water . so what was happening at that state , is that the kinetic energy , the heat , was being used to essentially break these bonds . | in what degrees does water turns to ice ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | and in the case of water , when you 're a solid , you 're ice . when you 're a liquid , some people would call ice water , but let 's call it liquid water . i think we know what that is . | how is the gases converted to liquid ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | because it 's the same amount of heat regardless how much direction we go in . when we go from solid to liquid , you view it as the heat of melting . it 's the head that you need to put in to melt the ice into liquid . | is ketchup a solid or a liquid ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | the temperature was not increasing . but the ice was going from ice to water . so what was happening at that state , is that the kinetic energy , the heat , was being used to essentially break these bonds . | why is ice less dense than water ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | and this is the temperature . we 'll talk about the states of matter in a second . so heat is often denoted by q . | how do we identify the states of the products which are formed from a reaction ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | that 's not very scientific . but they would vibrate around , they would buzz around a little bit . i 'm drawing arrows to show that they are vibrating . | what would be the state of sodium sulfate ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | and energy is the ability to do work . and what 's the unit for work ? well , it 's joules . | what is the smallest unit of matter ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | and it is true with water . the liquid form is definitely more dense than the gas form . in the gas form , the molecules are going to jump around , not touch each other . | well , since water can be solid , liquid and gas , does that mean that solid can be gas when its liquid form is heated to a certain temperature ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | at least , as far as my neural connections could make it . but enthalpy is closely related to heat . it 's heat content . | how are entropy and enthalpy related , if they even are ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | in the gas form , the molecules are going to jump around , not touch each other . and because of that , more light can get through the substance . now the question is , how do we measure the amount of heat that it takes to do this to water ? | what is a volatile and non-volatile substance ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | all solids are n't ice . although , you could think of a rock as solid magma . because that 's what it is . | could a rock become a gas ? |
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