Sentence stringlengths 92 3.31k | video_title stringlengths 24 106 |
|---|---|
So five times e to the seventh power plus c is equal to 40, is equal to 40 plus 5e to the seventh power. And notice all I did is said okay, f of seven. Well if this is f of x, f of, let me write this down. So if this is f of seven, if this is f of x, I just replaced the x with a seven to find f of seven, and we know th... | Finding specific antiderivatives exponential function AP Calculus AB Khan Academy.mp3 |
So we've got the differential equation, the derivative of y with respect to x is equal to three times y. And we want to find the particular solution that gives us y being equal to two when x is equal to one. So I encourage you to pause this video and see if you can figure this out on your own. All right, now let's work... | Worked example exponential solution to differential equation AP Calculus AB Khan Academy.mp3 |
All right, now let's work through it together. So some of you might have immediately said, hey, this is the form of a differential equation where the solution is going to be an exponential and you just got right to it. But I'm not gonna go straight to that. I'm just gonna recognize that this is a separable differential... | Worked example exponential solution to differential equation AP Calculus AB Khan Academy.mp3 |
I'm just gonna recognize that this is a separable differential equation and then I'm gonna solve it that way. So when I say it's separable, that means we can separate all the y's, d y's on one side and all the x's, d x's on the other side. And so what I could do is if I divide both sides of this equation by y and multi... | Worked example exponential solution to differential equation AP Calculus AB Khan Academy.mp3 |
One, two, three, d x. Now on the left and right hand sides, I have these clean things that I can now integrate. That's what people talk about when they say separable differential equations. Now here on the left, if I wanted to write it in a fairly general form, I could write, well, the antiderivative of one over y is g... | Worked example exponential solution to differential equation AP Calculus AB Khan Academy.mp3 |
Now here on the left, if I wanted to write it in a fairly general form, I could write, well, the antiderivative of one over y is gonna be the natural log of the absolute value of y. I'm taking the antiderivative with respect to y here. Now I could add a constant, but I'm gonna add a constant on the right hand side so t... | Worked example exponential solution to differential equation AP Calculus AB Khan Academy.mp3 |
So that is going to be equal to, the antiderivative here is going to be three x and I'll add the promised constant plus c right over there. And now let's think about it a little bit. Well, we can rewrite this in exponential form. We could say, we could write that e to the three x plus c is equal to the natural log of y... | Worked example exponential solution to differential equation AP Calculus AB Khan Academy.mp3 |
We could say, we could write that e to the three x plus c is equal to the natural log of y. I could write the natural log of y is equal to e to the three x plus c. Now I could rewrite this as equal to e to the three x times e to the c. Now e to the c is just going to be some other arbitrary constant, which I could stil... | Worked example exponential solution to differential equation AP Calculus AB Khan Academy.mp3 |
We're trying to find a function solution to this differential equation. So this would tell us that either y is equal to c e to the three x or y is equal to negative c e to the three x. Well, we've kept it in general terms. I haven't put any, we don't know what c is, so what we could do instead is just pick this one, an... | Worked example exponential solution to differential equation AP Calculus AB Khan Academy.mp3 |
I haven't put any, we don't know what c is, so what we could do instead is just pick this one, and then we can solve for c, assuming this one right over here. And so we will see if we can meet these constraints using this, and it'll essentially take the other one into consideration, whether we're going positive or nega... | Worked example exponential solution to differential equation AP Calculus AB Khan Academy.mp3 |
So when y is equal to two, when y is equal to two, I'm not going to solve for c to find the particular solution, x is equal to one, or when x is equal to one, y is equal to two. So I could write it like that. And we get two is equal to c times e to the third power, three times one. And so to solve for c, I can just div... | Worked example exponential solution to differential equation AP Calculus AB Khan Academy.mp3 |
And so to solve for c, I can just divide both sides by e to the third, and so I could, or I could multiply both sides times e to the negative third, and I could get two e to the negative third power is equal to c. And so let's now substitute it back in. And our particular solution is going to be y is equal to c. C is t... | Worked example exponential solution to differential equation AP Calculus AB Khan Academy.mp3 |
So whenever you see something like this, it doesn't hurt to try to visualize it. You might want to draw it out or just visualize it in your head, but since you can't get in my head, I will draw it out. So let me draw the information that they are giving us. So that's x-axis, that is the y-axis. Let's see, the relevant ... | The derivative & tangent line equations Derivatives introduction AP Calculus AB Khan Academy.mp3 |
So that's x-axis, that is the y-axis. Let's see, the relevant points here are two comma three and seven comma six. So let me go one, two, three, four, five, six, seven along the x-axis. And I'm gonna go one, two, three, four, five, and six along the y-axis. And now this point, so we have the point two comma three. So l... | The derivative & tangent line equations Derivatives introduction AP Calculus AB Khan Academy.mp3 |
And I'm gonna go one, two, three, four, five, and six along the y-axis. And now this point, so we have the point two comma three. So let me mark that. So two comma three is right over there. So that's two comma three. And we also have the point seven comma six. Seven comma six is going to be right over there. | The derivative & tangent line equations Derivatives introduction AP Calculus AB Khan Academy.mp3 |
So two comma three is right over there. So that's two comma three. And we also have the point seven comma six. Seven comma six is going to be right over there. Seven comma six. Now let's remind ourselves what they're saying. They're saying the tangent line to the graph of function f at this point passes through the poi... | The derivative & tangent line equations Derivatives introduction AP Calculus AB Khan Academy.mp3 |
Seven comma six is going to be right over there. Seven comma six. Now let's remind ourselves what they're saying. They're saying the tangent line to the graph of function f at this point passes through the point seven comma six. So if it's the tangent line to the graph at that point, it must go through two comma three.... | The derivative & tangent line equations Derivatives introduction AP Calculus AB Khan Academy.mp3 |
They're saying the tangent line to the graph of function f at this point passes through the point seven comma six. So if it's the tangent line to the graph at that point, it must go through two comma three. That's the only place where it intersects our graph, and it goes through seven comma six. So you only need two po... | The derivative & tangent line equations Derivatives introduction AP Calculus AB Khan Academy.mp3 |
So you only need two points to define a line. And so the tangent line is going to look like, it's going to look like, let me see if I can, no, that's not right. Let me draw it. Like it's going to look, oh, that's not exactly right. Let me try one more time. Okay, there you go. So the tangent line is going to look like ... | The derivative & tangent line equations Derivatives introduction AP Calculus AB Khan Academy.mp3 |
Like it's going to look, oh, that's not exactly right. Let me try one more time. Okay, there you go. So the tangent line is going to look like that. It goes, it's tangent to f right at two comma three, and it goes through the point seven comma six. And so we don't know anything other than f, but we can imagine what f l... | The derivative & tangent line equations Derivatives introduction AP Calculus AB Khan Academy.mp3 |
So the tangent line is going to look like that. It goes, it's tangent to f right at two comma three, and it goes through the point seven comma six. And so we don't know anything other than f, but we can imagine what f looks like. Our function f could, so our function f, it could look something like this. It just has to... | The derivative & tangent line equations Derivatives introduction AP Calculus AB Khan Academy.mp3 |
Our function f could, so our function f, it could look something like this. It just has to be tangent. So that line has to be tangent to our function right at that point. So our function f could look something like that. So when they say find f prime of two, they're really saying what is the slope of the tangent line w... | The derivative & tangent line equations Derivatives introduction AP Calculus AB Khan Academy.mp3 |
So our function f could look something like that. So when they say find f prime of two, they're really saying what is the slope of the tangent line when x is equal to two? So when x is equal to two, well, the slope of the tangent line is the slope of this line. They gave us, they gave us the two points that sit on the ... | The derivative & tangent line equations Derivatives introduction AP Calculus AB Khan Academy.mp3 |
They gave us, they gave us the two points that sit on the tangent line. So we just have to figure out its slope, because that is going to be the rate of change of that function right over there. It's derivative. It's going to be the slope of the tangent line, because this is the tangent line. So let's do that. So as we... | The derivative & tangent line equations Derivatives introduction AP Calculus AB Khan Academy.mp3 |
It's going to be the slope of the tangent line, because this is the tangent line. So let's do that. So as we know, slope is change in y over change in x. So if we change our, to go from two comma three to seven comma six, our change in x, change in x, we go from x equals two to x equals seven, so our change in x is equ... | The derivative & tangent line equations Derivatives introduction AP Calculus AB Khan Academy.mp3 |
So if we change our, to go from two comma three to seven comma six, our change in x, change in x, we go from x equals two to x equals seven, so our change in x is equal to five. And our change in y, our change in y, we go from y equals three to y equals six, so our change in y is equal to three. So our change in y over... | The derivative & tangent line equations Derivatives introduction AP Calculus AB Khan Academy.mp3 |
Let's do another one of these. So, for a function g, we are given that g of negative one equals three, and g prime of negative one is equal to negative two. What is the equation of the tangent line to the graph of g at x equals negative one? All right, so once again, I think it will be helpful to graph this. So, we hav... | The derivative & tangent line equations Derivatives introduction AP Calculus AB Khan Academy.mp3 |
All right, so once again, I think it will be helpful to graph this. So, we have our y-axis. We have our x-axis. And let's see. We say for a function g, we are given that g of negative one is equal to three. So the point negative one comma three is on our function. So this is negative one, and then we have one, two, and... | The derivative & tangent line equations Derivatives introduction AP Calculus AB Khan Academy.mp3 |
And let's see. We say for a function g, we are given that g of negative one is equal to three. So the point negative one comma three is on our function. So this is negative one, and then we have one, two, and three. So that's that right over there. That is the point, that is the point negative one comma three. It's goi... | The derivative & tangent line equations Derivatives introduction AP Calculus AB Khan Academy.mp3 |
So this is negative one, and then we have one, two, and three. So that's that right over there. That is the point, that is the point negative one comma three. It's going to be on our function. And we also know that g prime of negative one is equal to negative two. So the slope of the tangent line right at that point on... | The derivative & tangent line equations Derivatives introduction AP Calculus AB Khan Academy.mp3 |
It's going to be on our function. And we also know that g prime of negative one is equal to negative two. So the slope of the tangent line right at that point on our function is going to be negative two. That's what that tells us. The slope of the tangent line when x is equal to negative one is equal to negative two. S... | The derivative & tangent line equations Derivatives introduction AP Calculus AB Khan Academy.mp3 |
That's what that tells us. The slope of the tangent line when x is equal to negative one is equal to negative two. So I could use that information to actually draw the tangent line. So let me see if I can, let me see if I can do this. So it will look, so I think it will, let me just draw it like this. So it's gonna go,... | The derivative & tangent line equations Derivatives introduction AP Calculus AB Khan Academy.mp3 |
So let me see if I can, let me see if I can do this. So it will look, so I think it will, let me just draw it like this. So it's gonna go, so it's a slope of negative two is going to look something like that. So as we can see, if we move positive one in the x direction, we go down two in the y direction. So that has a ... | The derivative & tangent line equations Derivatives introduction AP Calculus AB Khan Academy.mp3 |
So as we can see, if we move positive one in the x direction, we go down two in the y direction. So that has a slope of negative two. And so you might say, well where is g? Well we could draw what g could look like. G might look something like this. Might look something like that right over there where that is the tang... | The derivative & tangent line equations Derivatives introduction AP Calculus AB Khan Academy.mp3 |
Well we could draw what g could look like. G might look something like this. Might look something like that right over there where that is the tangent line. We could make g do all sorts of crazy things after that. But all we really care about is the equation for this green line. And there's a couple of ways that you co... | The derivative & tangent line equations Derivatives introduction AP Calculus AB Khan Academy.mp3 |
We could make g do all sorts of crazy things after that. But all we really care about is the equation for this green line. And there's a couple of ways that you could do this. You could say, well look, a line is generally, there's a bunch of different ways where you can define the equation for a line. You could say a l... | The derivative & tangent line equations Derivatives introduction AP Calculus AB Khan Academy.mp3 |
You could say, well look, a line is generally, there's a bunch of different ways where you can define the equation for a line. You could say a line has a form y is equal to mx plus b where m is the slope and b is the y intercept. Well we already know what the slope of this line is. It is negative two. So we could say y... | The derivative & tangent line equations Derivatives introduction AP Calculus AB Khan Academy.mp3 |
It is negative two. So we could say y is equal to negative two, negative two times x, times x plus b. And then to solve for b, we know that the point negative one comma three is on this line. And this goes back to some of your algebra one that you might have learned a few years ago. So let's substitute negative one and... | The derivative & tangent line equations Derivatives introduction AP Calculus AB Khan Academy.mp3 |
And this goes back to some of your algebra one that you might have learned a few years ago. So let's substitute negative one and three for x and y. So when y is equal to three, so three, three is equal to, is equal to negative two, negative two times x times negative one, times negative one plus b, plus b. And so let's... | The derivative & tangent line equations Derivatives introduction AP Calculus AB Khan Academy.mp3 |
And so let's see, this is negative two times negative one is positive two. And so if you subtract two from both sides, you get one is equal to b. And there you have it. That is the equation of our line. Y is equal to negative two x plus one. And there's other ways that you could have done this. You could have written t... | The derivative & tangent line equations Derivatives introduction AP Calculus AB Khan Academy.mp3 |
What we're going to do in this video is explore the relationship between local linearity at a point and differentiability at a point. So local linearity is this idea that if we zoom in sufficiently on a point, that even a nonlinear function that is differentiable at that point will actually look linear. So let me show ... | Local linearity and differentiability Derivatives introduction AP Calculus AB Khan Academy.mp3 |
So let's say we had y is equal to x squared. So that's that there, clearly a nonlinear function. But we can zoom in on a point and if we zoom sufficiently in, we will see that it looks roughly linear. So let's say we want to zoom in on the point one comma one. So let's do that. So zooming in on the point one comma one,... | Local linearity and differentiability Derivatives introduction AP Calculus AB Khan Academy.mp3 |
So let's say we want to zoom in on the point one comma one. So let's do that. So zooming in on the point one comma one, already it is looking roughly linear at that point. And this property of local linearity is very helpful when trying to approximate a function around a point. So for example, we could figure out, we c... | Local linearity and differentiability Derivatives introduction AP Calculus AB Khan Academy.mp3 |
And this property of local linearity is very helpful when trying to approximate a function around a point. So for example, we could figure out, we could take the derivative at the point one one, use that as the slope of our tangent line, find the equation of the tangent line, and use that equation to approximate values... | Local linearity and differentiability Derivatives introduction AP Calculus AB Khan Academy.mp3 |
But the big takeaway here, at the point one one, it is displaying this idea of local linearity. And it is also differentiable at that point. Now let's look at another example of a point on a function where we aren't differentiable and we also don't see the local linearity. So for example, let's do the absolute value of... | Local linearity and differentiability Derivatives introduction AP Calculus AB Khan Academy.mp3 |
So for example, let's do the absolute value of x. And let me shift it over a little bit just so that we don't overlap as much. All right. So the absolute value of x minus one, it actually is differentiable as long as we're not at this corner right over here, as long as we're not at the point one comma zero. For any oth... | Local linearity and differentiability Derivatives introduction AP Calculus AB Khan Academy.mp3 |
So the absolute value of x minus one, it actually is differentiable as long as we're not at this corner right over here, as long as we're not at the point one comma zero. For any other x value, it is differentiable. But right at x equals one, we've talked in other videos how we aren't differentiable there. And then we ... | Local linearity and differentiability Derivatives introduction AP Calculus AB Khan Academy.mp3 |
And then we can use this local linearity idea to test it as well. And once again, this is not rigorous mathematics, but it is to give you an intuition. No matter how far we zoom in, we still see this sharp corner. It would be hard to construct the only tangent line, a unique line that goes through this point one comma ... | Local linearity and differentiability Derivatives introduction AP Calculus AB Khan Academy.mp3 |
It would be hard to construct the only tangent line, a unique line that goes through this point one comma zero. I can construct an actual infinite number of lines that go through one comma zero, but that do not go through the rest of the curve. And so notice, wherever you see a hard corner, like we're seeing at one com... | Local linearity and differentiability Derivatives introduction AP Calculus AB Khan Academy.mp3 |
Now let's zoom out a little bit and let's take another function. Let's take a function where the differentiability or the lack of differentiability is not because of a corner, but it's because as we zoom in, it starts to look linear, but it starts to look like a vertical line. So a good example of that would be square ... | Local linearity and differentiability Derivatives introduction AP Calculus AB Khan Academy.mp3 |
So that's the top half of a circle of radius two. And let's focus on the point two comma zero because right over there, we actually are not differentiable. And if we zoom in far enough, we see right at two comma zero that we are approaching what looks like a vertical line. So once again, we would not be differentiable ... | Local linearity and differentiability Derivatives introduction AP Calculus AB Khan Academy.mp3 |
So once again, we would not be differentiable at two comma zero. Now another thing I wanna point out, all of these, you really didn't have to zoom in too much to appreciate that, hey, I got a corner here on this absolute value function, or at two comma zero or at negative two comma zero, something a little bit stranger... | Local linearity and differentiability Derivatives introduction AP Calculus AB Khan Academy.mp3 |
But as we zoom in, once again, we'll see the local linearity, and they're also differentiable at those points. So a good example of that, let me actually get rid of some of these just so we can really zoom in. Let's say y is equal to x to the, and I'm gonna make a very large exponent here. So x to the 10th power, it's ... | Local linearity and differentiability Derivatives introduction AP Calculus AB Khan Academy.mp3 |
So x to the 10th power, it's starting to look a little bit like a corner there. Let's make it to the 100th power. Well now, it's looking even more like a corner there. Let me go to the 1,000th power just for good measure. So at this scale, it looks like we have a corner at the point one comma zero. Now this curve actua... | Local linearity and differentiability Derivatives introduction AP Calculus AB Khan Academy.mp3 |
Let me go to the 1,000th power just for good measure. So at this scale, it looks like we have a corner at the point one comma zero. Now this curve actually does not go to the point one comma zero. If x is one, then y is going to be one. And we'll see that as we zoom in, this, what looks like a hard corner, is going to ... | Local linearity and differentiability Derivatives introduction AP Calculus AB Khan Academy.mp3 |
If x is one, then y is going to be one. And we'll see that as we zoom in, this, what looks like a hard corner, is going to soften. And that's good because this function is actually differentiable at every value of x. It's a little bit more exotic than what we typically see. But as we zoom in, we'll actually see that. L... | Local linearity and differentiability Derivatives introduction AP Calculus AB Khan Academy.mp3 |
It's a little bit more exotic than what we typically see. But as we zoom in, we'll actually see that. Let's just zoom in on what looks like a fairly hard corner. But if we zoom sufficiently enough, even at the part that looks like the hardest part of the corner, the real corner, we'll see that it starts to soften and i... | Local linearity and differentiability Derivatives introduction AP Calculus AB Khan Academy.mp3 |
But if we zoom sufficiently enough, even at the part that looks like the hardest part of the corner, the real corner, we'll see that it starts to soften and it curves. And if we zoom in sufficiently, it will actually look like a line. It's hard to believe when you're really zoomed out. And I'm going at the point that r... | Local linearity and differentiability Derivatives introduction AP Calculus AB Khan Academy.mp3 |
And I'm going at the point that really looked like a corner from a distance. But as we zoom in, we see once again this local linearity. And it's a non-vertical line. And so once again, this is true at any point on this curve, that we are going to be differentiable. So the whole point here is sometimes you might have to... | Local linearity and differentiability Derivatives introduction AP Calculus AB Khan Academy.mp3 |
And that is just a fancy way of saying does the function have a defined derivative at a point? So let's just remind ourselves a definition of a derivative. And there's multiple ways of writing this. For the sake of this video, I'll write it as the derivative of our function at point C, this is Lagrange notation with th... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
For the sake of this video, I'll write it as the derivative of our function at point C, this is Lagrange notation with this F prime, the derivative of our function F at C, is going to be equal to the limit as X approaches C of F of X minus F of C over X minus C. And at first when you see this formula, and we've seen it... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
And we're just trying to see, well, what is that slope as X gets closer and closer to C, as our change in X gets closer and closer to zero? And we talk about that in other videos. So I'm now going to make a few claims in this video. And I'm not going to prove them rigorously. There is another video that will go a littl... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
And I'm not going to prove them rigorously. There is another video that will go a little bit more into the proof direction. But this is more to get an intuition. And so the first claim that I'm going to make is if F is differentiable at X equals C, at X equals C, then F is continuous at X equals C. So I'm saying if we ... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
And so the first claim that I'm going to make is if F is differentiable at X equals C, at X equals C, then F is continuous at X equals C. So I'm saying if we know it's differentiable, if we can find this limit, if we can find this derivative at X equals C, then our function is also continuous at X equals C. It doesn't ... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
If F not continuous at X equals C, at X equals C, then F is not differentiable. Differentiable at X is equal to C. So let me give a few examples of a non-continuous function. And then think about would we be able to find this limit? So the first is where you have a discontinuity. Our function is defined at C. It's equa... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
So the first is where you have a discontinuity. Our function is defined at C. It's equal to this value. But you can see as X becomes larger than C, it just jumps down and shifts right over here. So what would happen if you were trying to find this limit? Well, remember, all this is is a slope of a line between when X i... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
So what would happen if you were trying to find this limit? Well, remember, all this is is a slope of a line between when X is some arbitrary value. Let's say it's out here. So that would be X. This would be the point X comma F of X. And then this is the point C comma F of C right over here. So this is C comma F of C. ... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
So that would be X. This would be the point X comma F of X. And then this is the point C comma F of C right over here. So this is C comma F of C. So if you find the left-sided limit right over here, you're essentially saying, okay, let's find this slope. And then let me get a little bit closer. And let's get X a little... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
So this is C comma F of C. So if you find the left-sided limit right over here, you're essentially saying, okay, let's find this slope. And then let me get a little bit closer. And let's get X a little bit closer, and then let's find this slope. And then let's get X even closer than that and find this slope. And in all... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
And then let's get X even closer than that and find this slope. And in all of those cases, it would be zero. The slope is zero. So one way to think about it, the derivative, or this limit as we approach from the left, seems to be approaching zero. But what about if we were to take Xs to the right? So instead of our Xs ... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
So one way to think about it, the derivative, or this limit as we approach from the left, seems to be approaching zero. But what about if we were to take Xs to the right? So instead of our Xs being there, what if we were to take Xs right over here? Well, for this point, X comma F of X, our slope, if we take F of X minu... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
Well, for this point, X comma F of X, our slope, if we take F of X minus F of C over X minus C, that would be the slope of this line. If we get X to be even closer, let's say right over here, then this would be the slope of this line. If we get even closer, then this expression would be the slope of this line. And so a... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
And so as we get closer and closer to X being equal to C, we see that our slope is actually approaching negative infinity. And most importantly, it's approaching a very different value from the right. This expression is approaching a very different value from the right as it is from the left. And so in this case, this ... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
And so in this case, this limit up here won't exist. So we can clearly say this is not differentiable. So once again, not a proof here. I'm just getting an intuition for, if something isn't continuous, it's pretty clear, at least in this case, that it's not going to be differentiable. Let's look at another case. Let's ... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
I'm just getting an intuition for, if something isn't continuous, it's pretty clear, at least in this case, that it's not going to be differentiable. Let's look at another case. Let's look at a case where we have what's sometimes called a removable discontinuity or a point discontinuity. So once again, let's say we're ... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
So once again, let's say we're approaching from the left. This is X, this is the point X comma F of X. Now what's interesting is, where as this expression is the slope of the line connecting X comma F of X and C comma F of C, which is this point, not that point. Remember, we have this removable discontinuity right over... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
Remember, we have this removable discontinuity right over here. And so this would be, this expression is calculating the slope of that line. And then if X gets even closer to C, well then we're gonna be calculating the slope of that line. If X gets even closer to C, we're gonna be calculating the slope of that line. An... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
If X gets even closer to C, we're gonna be calculating the slope of that line. And so as we approach from the left, as X approaches C from the left, we actually have a situation where this expression right over here is going to approach negative infinity. And if we approach from the right, if we approach with X as larg... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
So we have a positive slope. And then as we get closer, it gets more positive. More positive approaches positive infinity. But either way, it's not approaching a finite value. And one side is approaching positive infinity and the other side is approaching negative infinity. This, the limit of this expression is not goi... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
But either way, it's not approaching a finite value. And one side is approaching positive infinity and the other side is approaching negative infinity. This, the limit of this expression is not going to exist. So once again, I'm not doing a rigorous proof here. But try to construct a discontinuous function where you wi... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
So once again, I'm not doing a rigorous proof here. But try to construct a discontinuous function where you will be able to find this. It is very, very hard. And you might say, well what about the situations where F is not even defined at C? Which for sure, you're not gonna be continuous if F is not defined at C. Well ... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
And you might say, well what about the situations where F is not even defined at C? Which for sure, you're not gonna be continuous if F is not defined at C. Well if F is not defined at C, then this part of the expression wouldn't even make sense. So you definitely wouldn't be differentiable. But now let's ask another t... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
But now let's ask another thing. I've just given you good arguments for when you're not continuous, you're not going to be differentiable. But can we make another claim that if you are continuous, then you definitely will be differentiable? Well it turns out that there are for sure many functions, an infinite number of... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
Well it turns out that there are for sure many functions, an infinite number of functions, that can be continuous at C but not differentiable. So for example, this could be an absolute value function. It doesn't have to be an absolute value function, but this could be Y is equal to the absolute value of X minus C. And ... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
Well think about what's happening. Think about this expression. Remember, this expression, all it's doing is calculating the slope between the point X comma F of X and the point C comma F of C. So if X is say out here, this is X comma F of X, it's going to be calculated, and so as we take the limit as X approaches C fr... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
And then as we get closer, we'll be looking at this slope, which is actually going to be the same. In this case it would be a negative one. So as X approaches C from the left, this expression would be negative one. But as X approaches C from the right, this expression is going to be one. The slope of the line that conn... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
But as X approaches C from the right, this expression is going to be one. The slope of the line that connects these points is one. The slope of the line that connects these points is one. So the limit of this expression, or I would say the value of this expression, is approaching two different values as X approaches C ... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
So the limit of this expression, or I would say the value of this expression, is approaching two different values as X approaches C from the left or the right. From the left, it's approaching negative one, or it's constantly negative one, and so it's approaching negative one, you could say. And from the right, it's one... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
And so we know if you're approaching two different values from on the left-sided or the right-sided limit, then this limit will not exist. So here, this is not differentiable. And even intuitively, we think of the derivative as the slope of the tangent line, and you could actually draw an infinite number of tangent lin... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
It's one way to think about it. You could say, well, maybe this is the tangent line right over there, but then why can't I make something like this the tangent line? That only intersects at the point C comma zero, and then you could keep doing things like that. Why can't that be the tangent line? And you could go on an... | Differentiability and continuity Derivatives introduction AP Calculus AB Khan Academy.mp3 |
And it's a more constrained version of the general case we've been looking at, but it's still very powerful and very applicable. And the reason why we're going to go over this special case is because its proof is fairly straightforward and will give you an intuition for why L'Hospital's Rule works at all. So the specia... | Proof of special case of l'Hôpital's rule Differential Calculus Khan Academy.mp3 |
If these constraints are met, then the limit as x approaches a of f of x over g of x is going to be equal to f prime of a over g prime of a. So it's very similar to the general case, it's a little bit more constrained. We're assuming that f prime of a exists, we're not just taking the limit now. We're assuming f prime ... | Proof of special case of l'Hôpital's rule Differential Calculus Khan Academy.mp3 |
We're assuming f prime of a and g prime of a actually exist. But notice if we substitute a right over here, we get 0 over 0. But then if the derivatives exist, we can just evaluate the derivatives at a and then we get the limit. So this is very close to the general case of L'Hospital's Rule. Now let's actually prove it... | Proof of special case of l'Hôpital's rule Differential Calculus Khan Academy.mp3 |
So this is very close to the general case of L'Hospital's Rule. Now let's actually prove it. And to prove it, we're going to start with the right hand and then show that if we use the definition of derivatives, we get the left hand right over here. So let me do that. So I'll do it right over here. So f prime of a is eq... | Proof of special case of l'Hôpital's rule Differential Calculus Khan Academy.mp3 |
So let me do that. So I'll do it right over here. So f prime of a is equal to what by the definition of derivatives? Well, we could view that as the limit as x approaches a of f of x minus f of a over x minus a. So this is literally just the slope between two points. So if you have your function f of x like this, this ... | Proof of special case of l'Hôpital's rule Differential Calculus Khan Academy.mp3 |
Well, we could view that as the limit as x approaches a of f of x minus f of a over x minus a. So this is literally just the slope between two points. So if you have your function f of x like this, this is the point a, f of a right over here. This right over here is the point x, f of x. This expression right over here ... | Proof of special case of l'Hôpital's rule Differential Calculus Khan Academy.mp3 |
This right over here is the point x, f of x. This expression right over here is the slope between these two points. The change in our y value is f of x minus f of a. The change in our x value is x minus a. So this expression is just the slope of this line. And we're just taking the line that connects these two points. ... | Proof of special case of l'Hôpital's rule Differential Calculus Khan Academy.mp3 |
The change in our x value is x minus a. So this expression is just the slope of this line. And we're just taking the line that connects these two points. That's the slope of it. I'll do that in white. The slope of the line that connects those two points. And we're taking the limit as x gets closer and closer and closer... | Proof of special case of l'Hôpital's rule Differential Calculus Khan Academy.mp3 |
That's the slope of it. I'll do that in white. The slope of the line that connects those two points. And we're taking the limit as x gets closer and closer and closer to a. So this is just another way of writing the definition of the derivative. So that's fine. Let's do the same thing for g prime of a. | Proof of special case of l'Hôpital's rule Differential Calculus Khan Academy.mp3 |
And we're taking the limit as x gets closer and closer and closer to a. So this is just another way of writing the definition of the derivative. So that's fine. Let's do the same thing for g prime of a. So f prime of a over g prime of a is going to be this business, which is in orange, f prime of a over g prime of a, w... | Proof of special case of l'Hôpital's rule Differential Calculus Khan Academy.mp3 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.