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So it has radius one, unit circle. So what does the length of this salmon colored line represent? Well, the height of this line would be the y coordinate of where this radius intersects the unit circle. And so by definition, by the unit circle definition of trig functions, the length of this line is going to be sine of... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
And so by definition, by the unit circle definition of trig functions, the length of this line is going to be sine of theta. If we wanted to make sure that it also worked for thetas that end up in the fourth quadrant, that will be useful, we can just ensure that it's the absolute value of the sine of theta. Now what ab... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
Can I express that in terms of a trigonometric function? Well, let's think about it. What would tangent of theta be? Let me write it over here. Tangent of theta is equal to opposite over adjacent. So if we look at this broader triangle right over here, this is our angle theta in radians. This is the opposite side. | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
Let me write it over here. Tangent of theta is equal to opposite over adjacent. So if we look at this broader triangle right over here, this is our angle theta in radians. This is the opposite side. The adjacent side down here, this just has length one. Remember, this is a unit circle. So this just has length one. | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
This is the opposite side. The adjacent side down here, this just has length one. Remember, this is a unit circle. So this just has length one. So the tangent of theta is the opposite side. The opposite side is equal to the tangent of theta. And just like before, this is going to be a positive value if we're sitting he... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
So this just has length one. So the tangent of theta is the opposite side. The opposite side is equal to the tangent of theta. And just like before, this is going to be a positive value if we're sitting here in the first quadrant, but I want things to work in both the first and the fourth quadrant for the sake of our p... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
And just like before, this is going to be a positive value if we're sitting here in the first quadrant, but I want things to work in both the first and the fourth quadrant for the sake of our proof, so I'm just gonna put an absolute value here. So now that we've done that, I'm gonna think about some triangles and their... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
So I can construct this triangle. And so let's think about the area of what I am shading in right over here. How can I express that area? Well, it's a triangle. We know that the area of a triangle is 1 1 2 base times height. We know the height is the absolute value of the sine of theta, and we know that the base is equ... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
Well, it's a triangle. We know that the area of a triangle is 1 1 2 base times height. We know the height is the absolute value of the sine of theta, and we know that the base is equal to one. So the area here is going to be equal to 1 1 2 times our base, which is one, times our height, which is the absolute value of t... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
So the area here is going to be equal to 1 1 2 times our base, which is one, times our height, which is the absolute value of the sine of theta. I'll rewrite it over here. I could just rewrite that as the absolute value of the sine of theta over two. Now let's think about the area of this wedge that I am highlighting i... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
Now let's think about the area of this wedge that I am highlighting in this yellow color. So what fraction of the entire circle is this going to be? If I were to go all the way around the circle, it would be two pi radians. So this is theta over two piths of the entire circle, and we know the area of the circle. This i... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
So this is theta over two piths of the entire circle, and we know the area of the circle. This is a unit circle. It has a radius one. So it would be times the area of the circle, which would be pi times the radius squared. The radius is one, so it's just gonna be times pi. And so the area of this wedge right over here,... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
So it would be times the area of the circle, which would be pi times the radius squared. The radius is one, so it's just gonna be times pi. And so the area of this wedge right over here, theta over two. And if we wanted to make this work for thetas in the fourth quadrant, we could just write an absolute value sign righ... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
And if we wanted to make this work for thetas in the fourth quadrant, we could just write an absolute value sign right over there because we're talking about positive area. And now let's think about this larger triangle in this blue color. And this is pretty straightforward. The area here is gonna be 1 1⁄2 times base t... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
The area here is gonna be 1 1⁄2 times base times height. So the area, and once again, this is this entire area, that's going to be 1 1⁄2 times our base, which is one, times our height, which is our absolute value of tangent of theta. And so I can just write that down as the absolute value of the tangent of theta over t... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
Now how would you compare the areas of this pink or this salmon-colored triangle, which sits inside of this wedge, and how would you compare that area of the wedge to the bigger triangle? Well, it's clear that the area of the salmon triangle is less than or equal to the area of the wedge, and the area of the wedge is l... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
And then the blue triangle includes the wedge plus it has this area right over here. So I think we can feel good visually that this statement right over here is true. And now I'm just going to do a little bit of algebraic manipulation. Let me multiply everything by two. So I can rewrite that the absolute value of sine ... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
Let me multiply everything by two. So I can rewrite that the absolute value of sine of theta is less than or equal to the absolute value of theta, which is less than or equal to the absolute value of tangent of theta. And let's see, actually, instead of writing the absolute value of tangent of theta, I'm gonna rewrite ... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
That's going to be the same thing as the absolute value of tangent of theta. And the reason why I did that is we can now divide everything by the absolute value of sine of theta. Since we're dividing by a positive quantity, it's not going to change the direction of the inequalities. So let's do that. I'm gonna divide t... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
So let's do that. I'm gonna divide this by an absolute value of sine of theta. I'm gonna divide this by an absolute value of the sine of theta. And then I'm gonna divide this by an absolute value of the sine of theta. And what do I get? Well, over here, I get a one. And on the right-hand side, I get a one over the abso... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
And then I'm gonna divide this by an absolute value of the sine of theta. And what do I get? Well, over here, I get a one. And on the right-hand side, I get a one over the absolute value of cosine theta. These two cancel out. So the next step I'm gonna do is take the reciprocal of everything. And so when I take the rec... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
And on the right-hand side, I get a one over the absolute value of cosine theta. These two cancel out. So the next step I'm gonna do is take the reciprocal of everything. And so when I take the reciprocal of everything, that actually will switch the inequalities. The reciprocal of one is still going to be one. But now,... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
And so when I take the reciprocal of everything, that actually will switch the inequalities. The reciprocal of one is still going to be one. But now, since I'm taking the reciprocal of this here, it's going to be greater than or equal to the absolute value of the sine of theta over the absolute value of theta. And that... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
And that's going to be greater than or equal to the reciprocal of one over the absolute value of cosine of theta is the absolute value of cosine of theta. We really just care about the first and fourth quadrants. You can think about this theta approaching zero from that direction or from that direction there. So that w... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
So that would be the first and fourth quadrants. So if we're in the first quadrant and theta is positive, sine of theta is gonna be positive as well. And if we're in the fourth quadrant and theta's negative, well, sine of theta's gonna have the same sign. It's going to be negative as well. And so these absolute value s... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
It's going to be negative as well. And so these absolute value signs aren't necessary. In the first quadrant, sine of theta and theta are both positive. In the fourth quadrant, they're both negative, but when you divide them, you're going to get a positive value. So I can erase those. If we're in the first or fourth qu... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
In the fourth quadrant, they're both negative, but when you divide them, you're going to get a positive value. So I can erase those. If we're in the first or fourth quadrant, our x value is not negative, and so cosine of theta, which is the x coordinate on our unit circle, is not going to be negative. And so we don't n... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
And so we don't need the absolute value signs over there. Now, we should pause a second because we're actually almost done. We have just set up three functions. You could think of this as f of x is equal to, you could view this as f of theta is equal to one, g of theta is equal to this, and h of theta is equal to that.... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
You could think of this as f of x is equal to, you could view this as f of theta is equal to one, g of theta is equal to this, and h of theta is equal to that. And over the interval that we care about, we could say four, negative pi over two is less than theta, is less than pi over two. But over this interval, this is ... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
Sine of theta over theta is defined over this interval except where theta is equal to zero. But since we're defined everywhere else, we can now find the limit. So what we can say is, well, by the squeeze theorem or by the sandwich theorem, if this is true over the interval, then we also know that the following is true.... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
And this we deserve a little bit of a drum roll. The limit as theta approaches zero of this is going to be greater than or equal to the limit as theta approaches zero of this, which is the one that we care about, sine of theta over theta, which is going to be greater than or equal to the limit as theta approaches zero ... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
In the last video, we tried to come up with a somewhat rigorous definition of what a limit is, where we say when you say that the limit of f of x as x approaches c is equal to l, you're really saying, and this is a somewhat rigorous definition, that you can get f of x as close as you want to l by making x sufficiently ... | Epsilon-delta definition of limits.mp3 |
So it really turns into a game. So you tell me how close you want. So this is the game. You tell me how close you want f of x to be to l. And you do this by giving me a positive number that we call epsilon, which is really how close you want f of x to be to l. So you give a positive number epsilon. And epsilon is how c... | Epsilon-delta definition of limits.mp3 |
You tell me how close you want f of x to be to l. And you do this by giving me a positive number that we call epsilon, which is really how close you want f of x to be to l. So you give a positive number epsilon. And epsilon is how close do you want to be. How close. So for example, if epsilon is 0.01, that says that yo... | Epsilon-delta definition of limits.mp3 |
So for example, if epsilon is 0.01, that says that you want f of x to be within 0.01 of epsilon. And so what I then do is I say, well, OK, you've given me that epsilon. I'm going to find you. I will find you another number, find another positive number, another number which we'll call delta, the lowercase delta, the Gr... | Epsilon-delta definition of limits.mp3 |
I will find you another number, find another positive number, another number which we'll call delta, the lowercase delta, the Greek letter delta, such that, so I'll say where, if x is within delta of c, then f of x will be within epsilon of our limit. So let's see if these are really saying the same thing. And this yel... | Epsilon-delta definition of limits.mp3 |
Someone is saying how close they want f of x to be to l. And the burden is then to find a delta where as long as x is within delta of c, then f of x will be within epsilon of the limit. So that is doing it. It's saying, look, if we were constraining x in such a way that if x is in that range to c, then f of x will be a... | Epsilon-delta definition of limits.mp3 |
So let's make this a little bit clearer by diagramming right over here. You show up and you say, well, I want f of x to be within epsilon of our limit. So this right over here, this point right over here is our limit plus epsilon. And this right over here might be our limit minus. This right over here is the limit minu... | Epsilon-delta definition of limits.mp3 |
And this right over here might be our limit minus. This right over here is the limit minus epsilon. And you say, OK, sure. I think I can get your f of x within this range of our limit. And I can do that by defining a range around c. And really, I could visually look at this boundary. But I could even go narrower than b... | Epsilon-delta definition of limits.mp3 |
I think I can get your f of x within this range of our limit. And I can do that by defining a range around c. And really, I could visually look at this boundary. But I could even go narrower than boundary. I can go right over here. Says, OK, I meet your challenge. I will find another number, delta. So this right over h... | Epsilon-delta definition of limits.mp3 |
I can go right over here. Says, OK, I meet your challenge. I will find another number, delta. So this right over here is c plus delta. This right over here is c minus. Let me write this down. Is c minus delta. | Epsilon-delta definition of limits.mp3 |
So this right over here is c plus delta. This right over here is c minus. Let me write this down. Is c minus delta. So I'll find you some delta so that if you take any x in the range c minus delta to c plus delta, and maybe the function's not even defined at c. So we think of ones that maybe aren't c, but are getting v... | Epsilon-delta definition of limits.mp3 |
Is c minus delta. So I'll find you some delta so that if you take any x in the range c minus delta to c plus delta, and maybe the function's not even defined at c. So we think of ones that maybe aren't c, but are getting very close. If you find any x in that range, f of those x's are going to be as close as you want to... | Epsilon-delta definition of limits.mp3 |
They're going to be within the range l plus epsilon or l minus epsilon. So what's another way of saying this? Another way of saying this is you give me an epsilon. Then I will find you a delta. So let me write this in a little bit more math notation. So I'll write the same exact statements with a little bit more math h... | Epsilon-delta definition of limits.mp3 |
Then I will find you a delta. So let me write this in a little bit more math notation. So I'll write the same exact statements with a little bit more math here, but it's the exact same thing. So you give me, or let me write it this way. Given an epsilon greater than 0, we can find, so that's kind of the first part of t... | Epsilon-delta definition of limits.mp3 |
So you give me, or let me write it this way. Given an epsilon greater than 0, we can find, so that's kind of the first part of the game, we can find a delta greater than 0 such that if x is within delta of c. So what's another way of saying that x is within delta of c? Well, one way you could say, well, what's the dist... | Epsilon-delta definition of limits.mp3 |
This statement is true for any x that's within delta of c. The difference between the two is going to be less than delta. So that if you pick an x that is in this range between c minus delta and c plus delta, and these are the x's that satisfy that right over here, then the distance between your f of x and your limit, ... | Epsilon-delta definition of limits.mp3 |
So we can define a range around c so that if we take any x value that is within delta of c, that's all this statement is saying, that the distance between x and c is less than delta. So it's within delta c. So that's these points right over here. That f of those x's, the function evaluated at those x's is going to be w... | Epsilon-delta definition of limits.mp3 |
So first, differentiability. Differentiability. So let's think about that first. It's always helpful to draw ourselves a function. So that's our y-axis. This is our x-axis. And let's just draw some function here. | Proof Differentiability implies continuity Derivative rules AP Calculus AB Khan Academy (2).mp3 |
It's always helpful to draw ourselves a function. So that's our y-axis. This is our x-axis. And let's just draw some function here. So let's say my function looks like this. And we care about the point x equals C, which is right over here. So that's the point x equals C. And then this value, of course, is going to be f... | Proof Differentiability implies continuity Derivative rules AP Calculus AB Khan Academy (2).mp3 |
And let's just draw some function here. So let's say my function looks like this. And we care about the point x equals C, which is right over here. So that's the point x equals C. And then this value, of course, is going to be f of C. And one way that we can find the derivative at x equals C, or the slope of the tangen... | Proof Differentiability implies continuity Derivative rules AP Calculus AB Khan Academy (2).mp3 |
So that's the point x equals C. And then this value, of course, is going to be f of C. And one way that we can find the derivative at x equals C, or the slope of the tangent line at x equals C, is we could start with some other point, say some arbitrary x out here. So let's say this is some arbitrary x out here. So the... | Proof Differentiability implies continuity Derivative rules AP Calculus AB Khan Academy (2).mp3 |
This graph, of course, is a graph of y equals f of x. And we can think about finding the slope of this line, this secant line between these two points. But then we can find the limit as x approaches C. And as x approaches C, this secant, the slope of the secant line is going to approach the slope of the tangent line, o... | Proof Differentiability implies continuity Derivative rules AP Calculus AB Khan Academy (2).mp3 |
And so we could take the limit as x approaches C, of the slope of this secant line. So what's the slope? Well, it's going to be change in y over change in x. The change in y is f of x minus f of C. That's our change in y right over here. And this is all a review. This is just one definition of the derivative, or one wa... | Proof Differentiability implies continuity Derivative rules AP Calculus AB Khan Academy (2).mp3 |
The change in y is f of x minus f of C. That's our change in y right over here. And this is all a review. This is just one definition of the derivative, or one way to think about the derivative. So it's going to be f of x minus f of C, that's our change in y, over our change in x, which is x minus C. It is x minus C. S... | Proof Differentiability implies continuity Derivative rules AP Calculus AB Khan Academy (2).mp3 |
So it's going to be f of x minus f of C, that's our change in y, over our change in x, which is x minus C. It is x minus C. So if this limit exists, then we're able to find the slope of the tangent line at this point. And we call that slope of the tangent line, we call that the derivative at x equals C. We say that thi... | Proof Differentiability implies continuity Derivative rules AP Calculus AB Khan Academy (2).mp3 |
And if this limit actually exists, we just call that value f prime of C. So that's just a review of differentiability. Now let's give ourselves a review of continuity. Con-ti-nuity. So the definition for continuity is if the limit as x approaches C of f of x is equal to f of C. Now this might seem a little bit, you kno... | Proof Differentiability implies continuity Derivative rules AP Calculus AB Khan Academy (2).mp3 |
So the definition for continuity is if the limit as x approaches C of f of x is equal to f of C. Now this might seem a little bit, you know, well, it might pop out to you as being intuitive, or it might seem a little, well, where did this come from? Well, let's visualize it, and then hopefully it'll make some intuitive... | Proof Differentiability implies continuity Derivative rules AP Calculus AB Khan Academy (2).mp3 |
And that actually might make it a little bit more clear. So if you had a point discontinuity at x equals C, so this is x equals C. So if you had a point discontinuity, so let me draw it like this actually. So you have a gap here, and x equals, when x equals C, f of C is actually way up here. So this is f of C, and then... | Proof Differentiability implies continuity Derivative rules AP Calculus AB Khan Academy (2).mp3 |
So this is f of C, and then the function continues like this. The limit as x approaches C of f of x is going to be this value, which is clearly different than f of C, this value right over here. If you take the limit, if you take the limit as x approaches C of f of x, you're approaching this value. This right over here... | Proof Differentiability implies continuity Derivative rules AP Calculus AB Khan Academy (2).mp3 |
This right over here is the limit as x approaches C of f of x, which is different than f of C. So this definition of continuity seems to be good at least for this case, because this is not a continuous function. You have a point discontinuity. So for at least in this case, this definition of continuity would properly i... | Proof Differentiability implies continuity Derivative rules AP Calculus AB Khan Academy (2).mp3 |
Now you could also think about a jump discontinuity. You could also think about a jump discontinuity. So let's look at this. And all of this is hopefully a little bit of review. So a jump discontinuity at C, at x equals C, might look like this. Might look like this. So this is at x equals C. So this is x equals C right... | Proof Differentiability implies continuity Derivative rules AP Calculus AB Khan Academy (2).mp3 |
And all of this is hopefully a little bit of review. So a jump discontinuity at C, at x equals C, might look like this. Might look like this. So this is at x equals C. So this is x equals C right over here. This would be f of C. But if you tried to evaluate the limit as x approaches C of f of x, you'd get a different v... | Proof Differentiability implies continuity Derivative rules AP Calculus AB Khan Academy (2).mp3 |
So this is at x equals C. So this is x equals C right over here. This would be f of C. But if you tried to evaluate the limit as x approaches C of f of x, you'd get a different value as you approach C from the negative side. You would approach this value. And as you approach C from the positive side, you would approach... | Proof Differentiability implies continuity Derivative rules AP Calculus AB Khan Academy (2).mp3 |
And as you approach C from the positive side, you would approach f of C. And so the limit wouldn't exist. So this limit right over here wouldn't exist in the case of this type of a jump discontinuity. So once again, this definition would properly say that this is not, this one right over here is not continuous. This li... | Proof Differentiability implies continuity Derivative rules AP Calculus AB Khan Academy (2).mp3 |
This limit actually would not even exist. And then you could even look at a, you could look at a function that is truly continuous. If you look at a function that is truly continuous, so something like this. Something like this. That is x equals C. Well, this is f of C. This is f of C. And if you were to take the limit... | Proof Differentiability implies continuity Derivative rules AP Calculus AB Khan Academy (2).mp3 |
Something like this. That is x equals C. Well, this is f of C. This is f of C. And if you were to take the limit as x approaches C, as x approaches C from either side of f of x, you're going to approach f of C. So here you have the limit as x approaches C of f of x indeed is equal to f of C. So it's what you would expe... | Proof Differentiability implies continuity Derivative rules AP Calculus AB Khan Academy (2).mp3 |
And I think it's important to kind of do this review just so that you can really visualize things. So differentiability implies this, this limit right over here exists. So let's start with a slightly different limit. Let me draw a line here actually. Let me draw a line just so we're doing something different. So let's ... | Proof Differentiability implies continuity Derivative rules AP Calculus AB Khan Academy (2).mp3 |
Let me draw a line here actually. Let me draw a line just so we're doing something different. So let's take, let us take the limit as x approaches C of f of x, of f of x minus f of C. Of f of x minus f of C. Well can we rewrite this? Well we could rewrite this as the limit as x approaches C. And we can essentially take... | Proof Differentiability implies continuity Derivative rules AP Calculus AB Khan Academy (2).mp3 |
Well we could rewrite this as the limit as x approaches C. And we can essentially take this expression and multiply and divide it by x minus C. So let's multiply it times x minus C. x minus C and divide it by x minus C. So we have f of x minus f of C. All of that over x minus C. So all I did is I multiplied and I divid... | Proof Differentiability implies continuity Derivative rules AP Calculus AB Khan Academy (2).mp3 |
So it's the limit as x approaches C of x minus C times the limit, let me write it this way, times the limit as x approaches C of f of x minus f of C. All of that over x minus C. Now what is this thing right over here? Well if we assume that f is differentiable at C, and we're going to do that, actually I should have st... | Proof Differentiability implies continuity Derivative rules AP Calculus AB Khan Academy (2).mp3 |
If we assume f differentiable, differentiable at C, well then this right over here is just going to be f prime of C. This right over here, we just saw it right over here, that's this exact same thing. This is f prime, f prime of C. And what is this thing right over here? The limit as x approaches C of x minus C? Well t... | Proof Differentiability implies continuity Derivative rules AP Calculus AB Khan Academy (2).mp3 |
Well that's just going to be zero. As x approaches C, it's going to approach C minus C, it's just going to be zero. So what's zero times f prime of C? Well f prime of C is just going to be some value, so zero times anything is just going to be zero. So I did all that work to get a zero. Now why is this interesting? Wel... | Proof Differentiability implies continuity Derivative rules AP Calculus AB Khan Academy (2).mp3 |
Well f prime of C is just going to be some value, so zero times anything is just going to be zero. So I did all that work to get a zero. Now why is this interesting? Well we just said, we just assumed that if f is differentiable at C, and we evaluate this limit, we get zero. So if we assume f is differentiable at C, we... | Proof Differentiability implies continuity Derivative rules AP Calculus AB Khan Academy (2).mp3 |
Well we just said, we just assumed that if f is differentiable at C, and we evaluate this limit, we get zero. So if we assume f is differentiable at C, we can write, we can write the limit, I'm just rewriting it, the limit as x approaches C of f of x minus f of C, and I could even put parentheses around it like that, w... | Proof Differentiability implies continuity Derivative rules AP Calculus AB Khan Academy (2).mp3 |
The limit as x approaches C of f of x minus the limit as x approaches C of f of C, of f of C, is equal to zero. The limit of the difference is the same thing as the difference of the limits. Well what's this thing over here going to be? Well f of C is just a number, it's not a function of x anymore, it's just f of C is... | Proof Differentiability implies continuity Derivative rules AP Calculus AB Khan Academy (2).mp3 |
Well f of C is just a number, it's not a function of x anymore, it's just f of C is going to evaluate to something. So this is just going to be f of C. This is just going to be f of C. So if the limit of f of x as x approaches C minus f of C is equal to zero. Well just add f of C to both sides and what do you get? Well... | Proof Differentiability implies continuity Derivative rules AP Calculus AB Khan Academy (2).mp3 |
And for the sake of this video, we can assume that the graph of this function just keeps getting lower and lower and lower as x becomes more and more negative, and lower and lower and lower as x goes beyond the interval that I've depicted right over here. So what is the maximum value that this function takes on? Well, ... | Critical points introduction AP Calculus AB Khan Academy.mp3 |
It looks like it's at this point. It looks like it's at that point right over there. So we would call this a global maximum. The function never takes on a value larger than this. So we could say that we have a global maximum at the point x naught, because f of x naught is greater than or equal to f of x for any other x... | Critical points introduction AP Calculus AB Khan Academy.mp3 |
The function never takes on a value larger than this. So we could say that we have a global maximum at the point x naught, because f of x naught is greater than or equal to f of x for any other x in the domain. And that's pretty obvious when you look at it like this. Now, do we have a global minimum point the way that ... | Critical points introduction AP Calculus AB Khan Academy.mp3 |
Now, do we have a global minimum point the way that I've drawn it? Well, no. This function can take on arbitrarily negative values. It approaches negative infinity as x approaches negative infinity. It approaches negative infinity as x approaches positive infinity. So we have, let me write this down, we have no global ... | Critical points introduction AP Calculus AB Khan Academy.mp3 |
It approaches negative infinity as x approaches negative infinity. It approaches negative infinity as x approaches positive infinity. So we have, let me write this down, we have no global minimum. Now, let me ask you a question. Do we have local minima or local maximum? When I say minima, it's just the plural of minimu... | Critical points introduction AP Calculus AB Khan Academy.mp3 |
Now, let me ask you a question. Do we have local minima or local maximum? When I say minima, it's just the plural of minimum. And maxima is just the plural of maximum. So do we have a local minima here or a local minimum here? Well, a local minimum, you could imagine, means that that value of the function at that point... | Critical points introduction AP Calculus AB Khan Academy.mp3 |
And maxima is just the plural of maximum. So do we have a local minima here or a local minimum here? Well, a local minimum, you could imagine, means that that value of the function at that point is lower than the points around it. So right over here, it looks like we have a local minimum. And I'm not giving you a very ... | Critical points introduction AP Calculus AB Khan Academy.mp3 |
So right over here, it looks like we have a local minimum. And I'm not giving you a very rigorous definition here, but one way to think about it is we can say that we have a local minimum point at x1 is if we have a region around x1 where f of x1 is less than an f of x for any x in this region right over here. And it's... | Critical points introduction AP Calculus AB Khan Academy.mp3 |
This is a low point for any of the values of f around it right over there. Now, do we have any other local minima? Well, it doesn't look like we do. Now, what about local maxima? Well, this one right over here, let me do it in purple. I don't want to get people confused, actually. Let me do it in this color. | Critical points introduction AP Calculus AB Khan Academy.mp3 |
Now, what about local maxima? Well, this one right over here, let me do it in purple. I don't want to get people confused, actually. Let me do it in this color. This point right over here looks like a local maximum. Local, not lox. That would have to deal with salmon. | Critical points introduction AP Calculus AB Khan Academy.mp3 |
Let me do it in this color. This point right over here looks like a local maximum. Local, not lox. That would have to deal with salmon. Local maximum right over there. So we could say at the point x2, we have a local maximum point at x2 because f of x2 is larger than f of x for any x around a neighborhood around x2. I'... | Critical points introduction AP Calculus AB Khan Academy.mp3 |
That would have to deal with salmon. Local maximum right over there. So we could say at the point x2, we have a local maximum point at x2 because f of x2 is larger than f of x for any x around a neighborhood around x2. I'm not being very rigorous, but you can see it just by looking at it. So that's fair enough. We've i... | Critical points introduction AP Calculus AB Khan Academy.mp3 |
I'm not being very rigorous, but you can see it just by looking at it. So that's fair enough. We've identified all of the maxima and minima, often called the extrema, for this function. Now, how can we identify those if we knew something about the derivative of the function? Well, let's look at the derivative at each o... | Critical points introduction AP Calculus AB Khan Academy.mp3 |
Now, how can we identify those if we knew something about the derivative of the function? Well, let's look at the derivative at each of these points. So at this first point right over here, if I were to try to visualize the tangent line, let me do that in a better color than brown. If I were to try to visualize the tan... | Critical points introduction AP Calculus AB Khan Academy.mp3 |
If I were to try to visualize the tangent line, it would look something like that. So the slope here is 0. So we would say that f prime of x0 is equal to 0. So the slope of the tangent line at this point is 0. What about over here? Well, once again, the tangent line would look something like that. So once again, we wou... | Critical points introduction AP Calculus AB Khan Academy.mp3 |
So the slope of the tangent line at this point is 0. What about over here? Well, once again, the tangent line would look something like that. So once again, we would say f prime at x1 is equal to 0. What about over here? Well, here, the tangent line is actually not well-defined. We have a positive slope going into it, ... | Critical points introduction AP Calculus AB Khan Academy.mp3 |
So once again, we would say f prime at x1 is equal to 0. What about over here? Well, here, the tangent line is actually not well-defined. We have a positive slope going into it, and then it immediately jumps to being a negative slope. So over here, f prime of x2 is not defined. Let me just write undefined. So we have a... | Critical points introduction AP Calculus AB Khan Academy.mp3 |
We have a positive slope going into it, and then it immediately jumps to being a negative slope. So over here, f prime of x2 is not defined. Let me just write undefined. So we have an interesting, and once again, I'm not rigorously proving it to you. I just want you to get the intuition here. We see that if we have som... | Critical points introduction AP Calculus AB Khan Academy.mp3 |
So we have an interesting, and once again, I'm not rigorously proving it to you. I just want you to get the intuition here. We see that if we have some type of an extrema, and we're not talking about when x is at an endpoint of an interval. Just to be clear what I'm talking about when I'm talking about x as an endpoint... | Critical points introduction AP Calculus AB Khan Academy.mp3 |
Just to be clear what I'm talking about when I'm talking about x as an endpoint of an interval. We're saying, let's say that the function is, let's say we have an interval from there. So let's say a function starts right over there and then keeps going. This would be a maximum point, but it would be an endpoint. We're ... | Critical points introduction AP Calculus AB Khan Academy.mp3 |
This would be a maximum point, but it would be an endpoint. We're not talking about endpoints right now. We're talking about when we have points in between, or when our interval is infinite. So we're not talking about points like that or points like this. We're talking about the points in between. So if you have a poin... | Critical points introduction AP Calculus AB Khan Academy.mp3 |
So we're not talking about points like that or points like this. We're talking about the points in between. So if you have a point inside of an interval, it's going to be a minimum or maximum, and we see the intuition here. If you have non-endpoint min or max at, let's say, x is equal to a. So if you know that you have... | Critical points introduction AP Calculus AB Khan Academy.mp3 |
If you have non-endpoint min or max at, let's say, x is equal to a. So if you know that you have a minimum or maximum point at some point x is equal to a and x isn't the endpoint of some interval, this tells you something interesting, or at least we have the intuition. We see that the derivative at x is equal to a is g... | Critical points introduction AP Calculus AB Khan Academy.mp3 |
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