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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, which we can write as the limit as x approaches a of g of x minus g of a over x minus a. Well, in the numerator we're taking the limit as x approaches a. In the d... | Proof of special case of l'Hôpital's rule Differential Calculus Khan Academy.mp3 |
In the denominator we're taking the limit as x approaches a. So we can just rewrite this. This we can rewrite as the limit as x approaches a of all this business in orange, f of x minus f of a over x minus a over all the business in green, g of x minus g of a, all of that over x minus a. Now to simplify this, we can mu... | Proof of special case of l'Hôpital's rule Differential Calculus Khan Academy.mp3 |
Now to simplify this, we can multiply the numerator and the denominator by x minus a to get rid of these x minus a. So let's do that. Let's multiply by x minus a over x minus a. So the numerator x minus a and we're dividing by x minus a, those cancel out, and then these two cancel out, and we're left with this thing ov... | Proof of special case of l'Hôpital's rule Differential Calculus Khan Academy.mp3 |
So the numerator x minus a and we're dividing by x minus a, those cancel out, and then these two cancel out, and we're left with this thing over here is equal to the limit as x approaches a of, in the numerator we have f of x minus f of a, and in the denominator we have g of x minus g of a. I think you see where this i... | Proof of special case of l'Hôpital's rule Differential Calculus Khan Academy.mp3 |
When I say vector-valued, it means you give me a t, it's a function of t, and so you give me a t, I'm not just gonna give you a number, I'm gonna give you a vector. And as we'll see, you're gonna get a two-dimensional vector. You could view this as the x component of the vector and the y component of the vector. And yo... | Second derivatives (vector-valued functions) Advanced derivatives AP Calculus BC Khan Academy.mp3 |
And you are probably familiar by now that there's multiple notations for even a two-dimensional vector. For example, you could use what's often viewed as engineering notation here, where the x component is being multiplied by the horizontal unit vector. So you might see something like that, where that's the unit vector... | Second derivatives (vector-valued functions) Advanced derivatives AP Calculus BC Khan Academy.mp3 |
So these are both representing the same thing, it just has a different notation. And sometimes you'll see vector-valued functions with an arrow on top to make it explicit that this is a vector-valued function. Sometimes you'll just hear people say, well, let h be a vector-valued function, and they might not write that ... | Second derivatives (vector-valued functions) Advanced derivatives AP Calculus BC Khan Academy.mp3 |
So now that we have that out of the way, what we are interested in is, well, let's find the first and second derivatives of h with respect to t. So let's take the first derivative, h prime of t. Well, as you'll see, that's actually quite straightforward. You're just gonna take the respective components, take the deriva... | Second derivatives (vector-valued functions) Advanced derivatives AP Calculus BC Khan Academy.mp3 |
Five times the negative one, or times the negative, you're gonna get negative five, times t to the five minus one power, so t to the fourth power. The derivative with respect to t of negative six, well, that's just zero. So that's the rate of change of the x component with respect to t. And now we go to the y component... | Second derivatives (vector-valued functions) Advanced derivatives AP Calculus BC Khan Academy.mp3 |
So we're gonna do the same thing. Derivative with respect to t is going to be, and once again, we just use the power rule, four times four is 16t to the third power. Derivative of 2t is just two. And then derivative of a constant, well, that's zero, we've already seen that. So there you have it. So this is the rate of ... | Second derivatives (vector-valued functions) Advanced derivatives AP Calculus BC Khan Academy.mp3 |
And then derivative of a constant, well, that's zero, we've already seen that. So there you have it. So this is the rate of change of the x component with respect to t. This is the rate of change of the y component with respect to t. And one way to do it, and vectors can represent many, many, many different things, but... | Second derivatives (vector-valued functions) Advanced derivatives AP Calculus BC Khan Academy.mp3 |
And then if you're looking at the rate of change of position with respect to time, well, then this would be the velocity vector. And then if we were to take the derivative of this with respect to time, well, we're going to get the acceleration vector. So if we say h prime prime of t, what is that going to be equal to? ... | Second derivatives (vector-valued functions) Advanced derivatives AP Calculus BC Khan Academy.mp3 |
H prime prime of t? Well, we just apply the power rule again. So four times negative five is equal to negative 20 t to the four minus one, so t to the third power. And then we have three times 16 is 48t squared, and then the derivative of two is just zero. And so there you have it. For any, if you view t as time, for a... | Second derivatives (vector-valued functions) Advanced derivatives AP Calculus BC Khan Academy.mp3 |
It would look something like this. It would look something like that. So that is e to the x. And what I want to do is I want to approximate f of x is equal to e to the x using a Taylor series approximation, or Taylor series expansion. And I want to do it not around x is equal to 0. I want to do it around x is equal to ... | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
And what I want to do is I want to approximate f of x is equal to e to the x using a Taylor series approximation, or Taylor series expansion. And I want to do it not around x is equal to 0. I want to do it around x is equal to 3, just to pick another arbitrary value. So we're going to do it around x is equal to 3. This... | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
So we're going to do it around x is equal to 3. This is x is equal to 3. This right there, that is f of 3. f of 3 is e to the third power. So this is e to the third power right over there. So when we take the Taylor series expansion, if we have a 0 degree polynomial approximating it, the best we could probably do is ha... | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
So this is e to the third power right over there. So when we take the Taylor series expansion, if we have a 0 degree polynomial approximating it, the best we could probably do is have a constant function going straight through e to the third. If we do a first order approximation, so we have a first degree term, then it... | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
And as we add more and more degrees to it, we should hopefully be able to kind of contour or converge with the curve better and better and better. And in the future, we'll talk a little bit more about how we can test for convergences and how well are we converging and all of that type of thing. But with that said, let'... | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
So the Taylor series expansion for f of x is equal to e to the x will be the polynomial. So what's f of c? Well, if x is equal to 3, we're saying that c is 3 in this situation. So if c is 3, f of 3 is e to the third power. So it's e to the third power plus. What's f prime of c? Well, f prime of x is also going to be e ... | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
So if c is 3, f of 3 is e to the third power. So it's e to the third power plus. What's f prime of c? Well, f prime of x is also going to be e to the x. You take the derivative of e to the x, you get e to the x. That's one of the super cool things about e to the x. So this is also f prime of x. | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
Well, f prime of x is also going to be e to the x. You take the derivative of e to the x, you get e to the x. That's one of the super cool things about e to the x. So this is also f prime of x. Frankly, this is the same thing as f, the nth derivative of x. You could just keep taking the derivative of this and you'll ge... | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
So this is also f prime of x. Frankly, this is the same thing as f, the nth derivative of x. You could just keep taking the derivative of this and you'll get e to the x. So f prime of x is e to the x. You evaluate that at 3. You get e to the third power again times x minus 3. c is 3. Plus the second derivative of our f... | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
So f prime of x is e to the x. You evaluate that at 3. You get e to the third power again times x minus 3. c is 3. Plus the second derivative of our function is still e to the x. Evaluate that at 3. You get e to the third power over 2 factorial times x minus 3 to the second power. And then we could keep going. | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
Plus the second derivative of our function is still e to the x. Evaluate that at 3. You get e to the third power over 2 factorial times x minus 3 to the second power. And then we could keep going. The third derivative is still e to the x. Evaluate that at 3. c is 3 in this situation. So you get e to the third power ove... | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
And then we could keep going. The third derivative is still e to the x. Evaluate that at 3. c is 3 in this situation. So you get e to the third power over 3 factorial times x minus 3 to the third power. And we can keep going with this, but I think you get the general idea. But what's even more interesting than just kin... | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
So you get e to the third power over 3 factorial times x minus 3 to the third power. And we can keep going with this, but I think you get the general idea. But what's even more interesting than just kind of going through the mechanics of finding the expansion is seeing how, as we add more and more terms, it starts to a... | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
And our approximation gets good further and further away from x is equal to 3. And to do that, I used Wolfram Alpha, available at wolframalpha.com. And I think I typed in like Taylor series expansion e to the x and x equals 3. And I just knew what I wanted and gave me all of this business right over here. And I actuall... | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
And I just knew what I wanted and gave me all of this business right over here. And I actually calculated the expansion. You can see it's the exact same thing that we have over here. e to the third plus e to the third times x minus 3. We have e to the third plus e to the third times x minus 3 plus 1 half. They actually... | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
e to the third plus e to the third times x minus 3. We have e to the third plus e to the third times x minus 3 plus 1 half. They actually expanded out the factorial. So instead of 3 factorial, they wrote a 6 over here. And they did a bunch of terms up here. But what's even more interesting is that they actually graph e... | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
So instead of 3 factorial, they wrote a 6 over here. And they did a bunch of terms up here. But what's even more interesting is that they actually graph each of these polynomials with more and more terms. So in orange, we have e to the x. We have f of x is equal to e to the x. And then they tell us order n approximatio... | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
So in orange, we have e to the x. We have f of x is equal to e to the x. And then they tell us order n approximation shown with n dots. So the order 1 approximation, so that should be the situation where we have a first degree polynomial. So that's literally a first degree polynomial would be these two terms right over... | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
So the order 1 approximation, so that should be the situation where we have a first degree polynomial. So that's literally a first degree polynomial would be these two terms right over here. Because this is a 0th degree. This is a first degree. We just have x to the first power involved here. If we just were to plot th... | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
This is a first degree. We just have x to the first power involved here. If we just were to plot this, if this was our polynomial, that is plotted with one dot. And that is this one right over here with one dot. And they plot it right over here. And we can see that it's just a tangent line at x is equal to 3. That is x... | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
And that is this one right over here with one dot. And they plot it right over here. And we can see that it's just a tangent line at x is equal to 3. That is x is equal to 3 right over there. So this is the tangent line. If we add a term, now we're getting to a second degree polynomial. Because we're adding an x square... | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
That is x is equal to 3 right over there. So this is the tangent line. If we add a term, now we're getting to a second degree polynomial. Because we're adding an x squared. If you expand this out, you'll have an x squared term. And you'll have another x term. But the degree of the polynomial will now be a second degree... | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
Because we're adding an x squared. If you expand this out, you'll have an x squared term. And you'll have another x term. But the degree of the polynomial will now be a second degree. So let's look for two dots. So that's this one right over here. So let's see, two dots coming in. | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
But the degree of the polynomial will now be a second degree. So let's look for two dots. So that's this one right over here. So let's see, two dots coming in. So you'll notice one, two dots. So you have two dots. And it comes in. | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
So let's see, two dots coming in. So you'll notice one, two dots. So you have two dots. And it comes in. And this is a parabola. It's a second degree polynomial. And then it comes back like this. | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
And it comes in. And this is a parabola. It's a second degree polynomial. And then it comes back like this. But notice, it does a better job, especially around x equals 3, of approximating e to the x. It stays with the curve a little bit longer. You add another term. | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
And then it comes back like this. But notice, it does a better job, especially around x equals 3, of approximating e to the x. It stays with the curve a little bit longer. You add another term. Let me do this in a new color, a color that I have not used. You add another term. Now you have a third degree polynomial. | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
You add another term. Let me do this in a new color, a color that I have not used. You add another term. Now you have a third degree polynomial. If you have all of these combined, if this is your polynomial, and you were to graph that. So let's look for the three dots right over here. So one, two, three. | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
Now you have a third degree polynomial. If you have all of these combined, if this is your polynomial, and you were to graph that. So let's look for the three dots right over here. So one, two, three. So it's this curve. Third degree polynomial is this curve right over here. And notice, it starts contouring e to the x ... | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
So one, two, three. So it's this curve. Third degree polynomial is this curve right over here. And notice, it starts contouring e to the x a little bit sooner than the second degree version. And it stays with it a little bit longer. And so you have it just like that. You add another term to it. | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
And notice, it starts contouring e to the x a little bit sooner than the second degree version. And it stays with it a little bit longer. And so you have it just like that. You add another term to it. You add the fourth degree term to it. So now we have all of this plus all of this. If this is your polynomial, now you ... | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
You add another term to it. You add the fourth degree term to it. So now we have all of this plus all of this. If this is your polynomial, now you have this curve right over here. Notice, every time you add a term, it's getting better and better at approximating e to the x further and further away from x is equal to 3.... | Visualizing Taylor polynomial approximations AP Calculus BC Khan Academy.mp3 |
Position as a function of time. And let me graph a potential s of t right over here. We have a horizontal axis as the time axis. And let me just graph something. I'll draw it kind of parabola looking. Although I could have done it general, but just to make things a little bit simpler for me. So I'll draw it kind of par... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
And let me just graph something. I'll draw it kind of parabola looking. Although I could have done it general, but just to make things a little bit simpler for me. So I'll draw it kind of parabola looking. So that is, if we call this a y-axis, we could even call this y equals s of t as a reasonable way to graph our pos... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
So I'll draw it kind of parabola looking. So that is, if we call this a y-axis, we could even call this y equals s of t as a reasonable way to graph our position as a function of time function. And now let's think about what happens if we want to think about the change in position between two times. Let's say between t... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
Let's say between time a. Let's say that's time a right over there. And then this right over here is time b. So time b is right over here. So what would be the change in position between time a and between time b? Well, at time b, we are at s of b. We are at s of b position. | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
So time b is right over here. So what would be the change in position between time a and between time b? Well, at time b, we are at s of b. We are at s of b position. And at time a, we were at s of a position. So the change in position between time a and time b, let me write this down, the change in position between, a... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
We are at s of b position. And at time a, we were at s of a position. So the change in position between time a and time b, let me write this down, the change in position between, and this might be obvious to you, but I'll write it down, between times a and b is going to be equal to s of b, this position, s of b minus t... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
So nothing earth shattering so far. But now let's think about what happens if we take the derivative of this function right over here. So what happens when we take the derivative of a position as a function of time? So remember, the derivative gives us the slope of the tangent line at any point. So let's say we're look... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
So remember, the derivative gives us the slope of the tangent line at any point. So let's say we're looking at a point right over there. The slope of the tangent line, it tells us for a very small change in t, I'm exaggerating it visually, for a very, very small change in t, how much are we changing in position? How mu... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
How much are we changing in position? So we write that as ds dt is the derivative of our position function at any given time. So when we're talking about how the rate at which position changes with respect to time, what is that? Well, that is equal to velocity. So this is equal to velocity. But let me write this in dif... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
Well, that is equal to velocity. So this is equal to velocity. But let me write this in different notations. So this itself is going to be a function of time. So we could write this, this is equal to s prime of t. These are just two different ways of writing the derivative of s with respect to t. This makes it a little... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
So this itself is going to be a function of time. So we could write this, this is equal to s prime of t. These are just two different ways of writing the derivative of s with respect to t. This makes it a little bit clearer that this itself is a function of time. And we know that this is the exact same thing as velocit... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
Let's graph it. So let me put another axis down here that looks pretty close to the original. Give myself some real estate. So that looks pretty good. And then let me try to graph v of t. So once again, if this is my y-axis, this is my t-axis. And I'm going to graph y is equal to v of t. And if this really is a parabol... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
So that looks pretty good. And then let me try to graph v of t. So once again, if this is my y-axis, this is my t-axis. And I'm going to graph y is equal to v of t. And if this really is a parabola, then the slope over here is 0. The slope, the rate of change is 0. And then it keeps increasing. The slope gets steeper a... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
The slope, the rate of change is 0. And then it keeps increasing. The slope gets steeper and steeper and steeper. And so v of t might look something like this. So this is the graph of y is equal to v of t. Now, using this graph, let's think if we can conceptualize the distance or the change in position between time a a... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
And so v of t might look something like this. So this is the graph of y is equal to v of t. Now, using this graph, let's think if we can conceptualize the distance or the change in position between time a and between time b. Well, let's go back to our Riemann sums. Let's think about what an area of a very small rectang... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
Let's think about what an area of a very small rectangle would represent. So let's divide this into a bunch of rectangles. So I'll do fairly large rectangles so we have some space to work with. You can imagine much smaller ones. And I'm going to do a left Riemann sum here, just because we've done those a bunch. But we ... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
You can imagine much smaller ones. And I'm going to do a left Riemann sum here, just because we've done those a bunch. But we could do a right Riemann sum. We could do a trapezoidal sum. We could do anything we want. So then we could keep going all the way. Actually, let me just do 3 right now. | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
We could do a trapezoidal sum. We could do anything we want. So then we could keep going all the way. Actually, let me just do 3 right now. Let me just do 3 right over here. And so this is actually a very rough approximation. But you can imagine it might get closer. | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
Actually, let me just do 3 right now. Let me just do 3 right over here. And so this is actually a very rough approximation. But you can imagine it might get closer. But what is the area of each of these rectangles trying? What is it an approximation for? Well, this one right over here, you have f of a, or I should say ... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
But you can imagine it might get closer. But what is the area of each of these rectangles trying? What is it an approximation for? Well, this one right over here, you have f of a, or I should say v of a. So your velocity at time a is the height right over here. And then this distance right over here is a change in time... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
Well, this one right over here, you have f of a, or I should say v of a. So your velocity at time a is the height right over here. And then this distance right over here is a change in time, times delta t. So the area for that rectangle is your velocity at that moment times your change in time. What is the velocity at ... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
What is the velocity at that moment times your change in time? Well, that's going to be your change in position. So this will tell you this is an approximation of your change in position over this time. Then this rectangle, the area of this rectangle, is another approximation for your change in position over the next d... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
Then this rectangle, the area of this rectangle, is another approximation for your change in position over the next delta t. And then you can imagine this right over here is an approximation for your change in position for the next delta t. So if you really wanted to figure out your change in position between a and b, ... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
We could do trapezoids. We could do the right Riemann sum. But I'll just do a left one because that's what I depicted right here. v of t of i minus 1. So if this would be t0, it would be a. So this is the first rectangle. So the first rectangle, you use the function evaluated at t0. | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
v of t of i minus 1. So if this would be t0, it would be a. So this is the first rectangle. So the first rectangle, you use the function evaluated at t0. For the second rectangle, you use the function evaluated at t1. We've done this in multiple videos already. And then we multiply it times each of the changes in time. | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
So the first rectangle, you use the function evaluated at t0. For the second rectangle, you use the function evaluated at t1. We've done this in multiple videos already. And then we multiply it times each of the changes in time. This will be an approximation for our total. And let me make it clear. Where delta t is equ... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
And then we multiply it times each of the changes in time. This will be an approximation for our total. And let me make it clear. Where delta t is equal to b minus a over the number of intervals we have. We already know from many, many videos when we looked at Riemann sums that this will be an approximation. Well, it w... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
Where delta t is equal to b minus a over the number of intervals we have. We already know from many, many videos when we looked at Riemann sums that this will be an approximation. Well, it will be an approximation for two things. We just talked about it will be an approximation for our change in position. But it's also... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
We just talked about it will be an approximation for our change in position. But it's also an approximation for our area. So this right over here. So we're trying to approximate change in position. And this is also approximate of the area under the curve. So hopefully this satisfies you that if you are able to calculat... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
So we're trying to approximate change in position. And this is also approximate of the area under the curve. So hopefully this satisfies you that if you are able to calculate the area under the curve. And actually, this one's pretty easy because it's a trapezoid. But even if this was a function, if it was kind of a wac... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
And actually, this one's pretty easy because it's a trapezoid. But even if this was a function, if it was kind of a wacky function, it would still apply. That when you're calculating the area under the curve of the velocity function, you are actually figuring out the change in position. These are the two things. Well, ... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
These are the two things. Well, we already know what could we do to get the exact area under the curve or to get the exact change in position. Well, we just have a ton of rectangles. We take the limit as the number of rectangles we have approaches infinity. We take the limit as n approaches infinity. And as n approache... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
We take the limit as the number of rectangles we have approaches infinity. We take the limit as n approaches infinity. And as n approaches infinity, because delta t is b minus a divided by n, delta t is going to become infinitely small. It's going to turn into dt. This is one way to think about it. And we already have ... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
It's going to turn into dt. This is one way to think about it. And we already have notation for this. This is one way to think about a Riemann integral. We just use the left Riemann sum. Once again, we could use the right Riemann sum, et cetera, et cetera. We could have used a more general Riemann sum, but this one wil... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
This is one way to think about a Riemann integral. We just use the left Riemann sum. Once again, we could use the right Riemann sum, et cetera, et cetera. We could have used a more general Riemann sum, but this one will work. So this will be equal to the definite integral from a to b of v of t dt. So this right over he... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
We could have used a more general Riemann sum, but this one will work. So this will be equal to the definite integral from a to b of v of t dt. So this right over here is one way of saying, look, if we want the exact area under the curve of the velocity curve, which is going to be the exact change in position between a... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
It's the limit of this Riemann sum as n approaches infinity or the definite integral from a to b of v of t dt. But what did we just figure out? So remember, this is another. We could call this the exact change in position between times a and b. But we already figured out what the exact change in positions between times... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
We could call this the exact change in position between times a and b. But we already figured out what the exact change in positions between times a and b are. It's this thing right over here. And so this gets interesting. We now have a way of evaluating this definite integral. Conceptually, we knew that this is the ex... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
And so this gets interesting. We now have a way of evaluating this definite integral. Conceptually, we knew that this is the exact change in position between a and b, but we already figured out a way to figure out the exact change in position between a and b. So let me write all this down. We have that the definite int... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
So let me write all this down. We have that the definite integral between a and b of v of t dt is equal to s of b minus s of a, where s of t is the anti-derivative of v of t. And this notion, although I've written in a very nontraditional used position velocity, this is the second fundamental theorem of calculus. And y... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
We'll talk about that in another video. But this is a super useful way of evaluating definite integrals and finding the area under a curve. Second fundamental theorem of calculus, very closely tied to the first fundamental theorem, which we won't talk about now. So why is this such a big deal? Well, let me write it in ... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
So why is this such a big deal? Well, let me write it in a more general notation, the way that you might be used to seeing it in your calculus book. It's telling us that if we want the area under the curve between two points, a and b, between two x points, a and b, of f of x. And so this is how we would denote the area... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
And so this is how we would denote the area under the curve between those two intervals. So let me draw that just to make it clear what I'm talking about in general terms. So this right over here could be f of x. And we care about the area under the curve between a and b. If we want to find the exact area under the cur... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
And we care about the area under the curve between a and b. If we want to find the exact area under the curve, we can figure it out by taking the antiderivative of f. And let's just say that capital F of x is the antiderivative, or is an antiderivative, because you can have multiple that are shifted by constants, is an... | Intuition for second part of fundamental theorem of calculus AP Calculus AB Khan Academy.mp3 |
See if you can evaluate this integral right over here. So I'm assuming you've had a go at it, so let's work through this together. So you probably realize that some of the traditional techniques that we've already had in our toolkits don't seem to be directly applicable. You substitution and others. And the key here to... | Dividing expressions to evaluate integral AP Calculus BC Khan Academy.mp3 |
You substitution and others. And the key here to realize is we have a rational expression here where the numerator has the same degree or higher than the denominator. In this case, the numerator and the denominator have the same degree. And whenever you see something like that, it's probably a good idea to divide the d... | Dividing expressions to evaluate integral AP Calculus BC Khan Academy.mp3 |
And whenever you see something like that, it's probably a good idea to divide the denominator into the numerator. That's what this rational expression could be interpreted as, x minus five divided by negative two x plus two. So let's do a little bit of algebraic long division to actually divide negative two x plus two ... | Dividing expressions to evaluate integral AP Calculus BC Khan Academy.mp3 |
So let's do that. We're gonna take x minus five. So x minus five. And divide negative two x plus two into that. So negative two x plus two. So look at the highest degree terms. How many times does negative two x go into x? | Dividing expressions to evaluate integral AP Calculus BC Khan Academy.mp3 |
And divide negative two x plus two into that. So negative two x plus two. So look at the highest degree terms. How many times does negative two x go into x? Well, it's gonna go negative 1 1 2 times. Negative 1 1 2 times two is negative one. Negative 1 1 2 times negative two x is just going to be positive x, just like t... | Dividing expressions to evaluate integral AP Calculus BC Khan Academy.mp3 |
How many times does negative two x go into x? Well, it's gonna go negative 1 1 2 times. Negative 1 1 2 times two is negative one. Negative 1 1 2 times negative two x is just going to be positive x, just like that. And now we want to subtract this yellow expression from this blue expression. And so let's just, let me ju... | Dividing expressions to evaluate integral AP Calculus BC Khan Academy.mp3 |
Negative 1 1 2 times negative two x is just going to be positive x, just like that. And now we want to subtract this yellow expression from this blue expression. And so let's just, let me just take the negative of this and then add. So I'm just gonna take the negative of it and add. And so we are left with negative fiv... | Dividing expressions to evaluate integral AP Calculus BC Khan Academy.mp3 |
So I'm just gonna take the negative of it and add. And so we are left with negative five plus one is negative four. So you could say negative two x plus two goes into x minus five negative 1 1 2 times with negative four left over. And we can rewrite this integral, our original integral, as, we can rewrite it as negativ... | Dividing expressions to evaluate integral AP Calculus BC Khan Academy.mp3 |
And we can rewrite this integral, our original integral, as, we can rewrite it as negative 1 1 2 minus four over negative two x plus two dx. Now let's see, it looks like we can simplify this expression a little bit more. The numerator and the denominator, they're both divisible by two, all of these terms are divisible ... | Dividing expressions to evaluate integral AP Calculus BC Khan Academy.mp3 |
Actually we have all these negatives, that always unnecessarily complicates things. So let's actually divide the numerator and the denominator by negative two. So what are we gonna have then? So if we divide the numerator by negative two, if this is negative four, this is going to become positive two. Then this, if we ... | Dividing expressions to evaluate integral AP Calculus BC Khan Academy.mp3 |
So if we divide the numerator by negative two, if this is negative four, this is going to become positive two. Then this, if we divide negative two x by negative two, that's just going to become x. And then two divided by negative two is going to be minus one. So our original integral, once again, this is just algebra,... | Dividing expressions to evaluate integral AP Calculus BC Khan Academy.mp3 |
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