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So it's my y-axis, my x-axis, and then I have the unit circle. So, whoops, all right, the unit circle, just like that. So if we start at, this is zero, then you go to pi over two, then you go to pi, then you go to three pi over two. So that's this point on the unit circle. So the cosine is the x-coordinate, so this is ... | Definite integral of trig function AP Calculus AB Khan Academy.mp3 |
So that's this point on the unit circle. So the cosine is the x-coordinate, so this is going to be zero. This is zero, so this is zero. And so we get one minus zero, so everything in the brackets evaluates out to one. And so we are left with, so let me do that. So all of this is equal to one. And so you have negative n... | Definite integral of trig function AP Calculus AB Khan Academy.mp3 |
This is going to be the same thing as the limit as x approaches negative one of x plus one over, over the limit, the limit as x approaches negative one of square root of x plus five minus two. And then we could say, all right, this thing up here, x plus one, if we think about the graph y equals x plus one, it's continu... | Limits by rationalizing Limits and continuity AP Calculus AB Khan Academy.mp3 |
And then our denominator, square root of x plus five minus two isn't continuous everywhere, but it is continuous at x equals negative one, and so we can do the same thing. We can just substitute negative one for x. So this is going to be the square root of negative one plus five minus two. Now what does this evaluate t... | Limits by rationalizing Limits and continuity AP Calculus AB Khan Academy.mp3 |
Now what does this evaluate to? Well, in the numerator we get a zero, and in the denominator, negative one plus five is four, take the principal root is two minus two, we get zero again. So we get, we got zero over zero. Now when you see that, you might be tempted to give up. You say, oh look, there's a zero in the den... | Limits by rationalizing Limits and continuity AP Calculus AB Khan Academy.mp3 |
Now when you see that, you might be tempted to give up. You say, oh look, there's a zero in the denominator, maybe this limit doesn't exist, maybe I'm done here, what do I do? And if this was non-zero up here in the numerator, if you're taking a non-zero value and dividing it by zero, that is undefined, and your limit ... | Limits by rationalizing Limits and continuity AP Calculus AB Khan Academy.mp3 |
But when you have zero over zero, this is indeterminate form, and it doesn't mean necessarily that your limit does not exist. And as we'll see in this video and many future ones, there are tools at our disposal to address this, and we will look at one of them. Now the tool that we're gonna look at is, is there another ... | Limits by rationalizing Limits and continuity AP Calculus AB Khan Academy.mp3 |
Well let's just rewrite, let's just take this, let me give it, so let's take this thing right over here, and let's say this is g of x. So essentially what we're trying to do is find the limit of g of x as x approaches negative one. So we could write g of x is equal to x plus one, and the whole reason why I'm defining i... | Limits by rationalizing Limits and continuity AP Calculus AB Khan Academy.mp3 |
Now the technique we're gonna use is, when you get this indeterminate form, and if you have a square root in either the numerator or the denominator, it might help to get rid of that square root. And this is often called rationalizing expression. In this case you have a square root in the denominator, so it would be ra... | Limits by rationalizing Limits and continuity AP Calculus AB Khan Academy.mp3 |
And so this would be, the way we would do it, is we'd be leveraging our knowledge of difference of squares. We know, we know that a plus b times a minus b is equal to a squared minus b squared, you learned that in algebra a little while ago. Or, if we had the square root of a plus b, and we were to multiply that times ... | Limits by rationalizing Limits and continuity AP Calculus AB Khan Academy.mp3 |
So we can just leverage these ideas to get rid of this radical down here. The way we're going to do it, is we're gonna multiply the numerator and the denominator by the square root of x plus five plus two, right? We have the minus two, so we're gonna multiply it times the plus two. So let's do that. So we have square r... | Limits by rationalizing Limits and continuity AP Calculus AB Khan Academy.mp3 |
So let's do that. So we have square root of x plus five plus two, and we're gonna multiply the numerator times the same thing, because we don't want to change the value of the expression. This is one, so if we take the expression divided by the same expression, it's going to be one. So this is, so square root of x plus... | Limits by rationalizing Limits and continuity AP Calculus AB Khan Academy.mp3 |
So this is, so square root of x plus five plus two. And so this is going to be equal to, this is going to be equal to x plus one times the square root, times the square root of x plus five plus two. And then the denominator is going to be, well, it's going to be the square root of x plus five squared, which would be ju... | Limits by rationalizing Limits and continuity AP Calculus AB Khan Academy.mp3 |
And so this down here simplifies to x plus five minus four, it's just x plus one. So this is just, this is just x plus one. And it probably jumps out at you that both the numerator and the denominator have an x plus one in it, so maybe we can simplify. So we could simplify and just say, well, g of x is equal to the squ... | Limits by rationalizing Limits and continuity AP Calculus AB Khan Academy.mp3 |
So we could simplify and just say, well, g of x is equal to the square root of x plus five plus two. Now some of you might be feeling a little off here, and you would be correct. Your spider senses would be, say, is this, is this definitely the same thing as what we originally had before we canceled out the x plus ones... | Limits by rationalizing Limits and continuity AP Calculus AB Khan Academy.mp3 |
And the answer is the way I just wrote it, it is not the exact same thing. It is the exact same thing everywhere, except at x equals negative one. This thing right over here is defined at x equals negative one. This thing right over here is not defined at x equals negative one. And g of x was not, was not, so g of x ri... | Limits by rationalizing Limits and continuity AP Calculus AB Khan Academy.mp3 |
This thing right over here is not defined at x equals negative one. And g of x was not, was not, so g of x right over here, you don't get a good result when you try x equals negative one. And so in order for this to truly be the same thing as g of x, the same function, we have to say for x not equal to negative one. No... | Limits by rationalizing Limits and continuity AP Calculus AB Khan Academy.mp3 |
Now this is a simplified version of g of x. It is the same thing. For any input x that g of x is defined, this is going to give you the same output. And this has the exact same domain now, now that we've put this constraint in, as g of x. Now you might say, okay, well, how does this help us? Because we want to find the... | Limits by rationalizing Limits and continuity AP Calculus AB Khan Academy.mp3 |
And this has the exact same domain now, now that we've put this constraint in, as g of x. Now you might say, okay, well, how does this help us? Because we want to find the limit as x approaches negative one. And even here, I had to put this little constraint here that x cannot be equal to negative one. How do we think ... | Limits by rationalizing Limits and continuity AP Calculus AB Khan Academy.mp3 |
And even here, I had to put this little constraint here that x cannot be equal to negative one. How do we think about this limit? Well, lucky for us, we know, lucky for us, we know that if we just take another function, f of x, if we say f of x is equal to the square root of x plus five plus two, well, then we know tha... | Limits by rationalizing Limits and continuity AP Calculus AB Khan Academy.mp3 |
And we know if this is true of two, if this is true of two functions, then the limit as x approaches, the limit, let me write this down, is since we know this, because of this, we know that the limit of f of x as x approaches negative one is going to be equal to the limit of g of x as x approaches negative one. And thi... | Limits by rationalizing Limits and continuity AP Calculus AB Khan Academy.mp3 |
And if you were to graph g of x, it just has a point discontinuity, or a removable, or I should just say, yeah, a point discontinuity right over here at x equals negative one. And so what is the limit? And we are in the home stretch now. What is the limit of f of x, or we could say the limit of the square root of x plu... | Limits by rationalizing Limits and continuity AP Calculus AB Khan Academy.mp3 |
What is the limit of f of x, or we could say the limit of the square root of x plus five plus two as x approaches negative one? Well, this expression is continuous, or this function is continuous at x equals negative one, so we can just evaluate it at x equals negative one. So this is going to be the square root of neg... | Limits by rationalizing Limits and continuity AP Calculus AB Khan Academy.mp3 |
So this is four, square root, principle root of four is two. Two plus two is equal to four. So since the limit of f of x as x approaches negative one is four, the limit of g of x as x approaches negative one is also four. And if this little, this little, I guess you could say, leap that I just made right over here does... | Limits by rationalizing Limits and continuity AP Calculus AB Khan Academy.mp3 |
And if this little, this little, I guess you could say, leap that I just made right over here doesn't make sense to you, think about it visually. Think about it visually. So if this is my y-axis, and this is my x-axis, g of x looked something like this. The g of x, the g of x, let me draw it, g of x looked something, s... | Limits by rationalizing Limits and continuity AP Calculus AB Khan Academy.mp3 |
The g of x, the g of x, let me draw it, g of x looked something, something like this. And it had a gap at negative one. So it had a gap right over there. While f of x, f of x would have the same graph, except it wouldn't have, it wouldn't have the gap. And so if you're trying to find the limit, it seems completely reas... | Limits by rationalizing Limits and continuity AP Calculus AB Khan Academy.mp3 |
Here we have a function of t, and we're taking the antiderivative with respect to t. And so you would not write a dx here. That is not the notation. You'll see why when we focus on definite integrals. So what's the antiderivative of this business right over here? Well, it's going to be the same thing as the antiderivat... | Indefinite integrals of sin(x), cos(x), and e_ AP Calculus AB Khan Academy.mp3 |
So what's the antiderivative of this business right over here? Well, it's going to be the same thing as the antiderivative of sine of t. It's going to be the antiderivative of sine of t, or the indefinite integral of sine of t, plus the indefinite integral, or the antiderivative of cosine of t. Plus the antiderivative ... | Indefinite integrals of sin(x), cos(x), and e_ AP Calculus AB Khan Academy.mp3 |
We know that the derivative with respect to t of cosine of t is equal to negative sine of t. So if we want a sine of t here, we would just have to take the derivative of negative cosine t. If we take the derivative of negative cosine t, then we get positive sine of t. Derivative with respect to t of cosine t is negativ... | Indefinite integrals of sin(x), cos(x), and e_ AP Calculus AB Khan Academy.mp3 |
We found the antiderivative. Now let's tackle this. Now we don't have a t. We're taking the indefinite integral with respect to, actually, this is a mistake. This should be with respect to a. Let me clean this up. This should be a dA. If we were taking this with respect to t, then we would treat all of these things as ... | Indefinite integrals of sin(x), cos(x), and e_ AP Calculus AB Khan Academy.mp3 |
This should be with respect to a. Let me clean this up. This should be a dA. If we were taking this with respect to t, then we would treat all of these things as just constants. But I don't want to confuse you right now. Let me make it clear. This is going to be dA. | Indefinite integrals of sin(x), cos(x), and e_ AP Calculus AB Khan Academy.mp3 |
If we were taking this with respect to t, then we would treat all of these things as just constants. But I don't want to confuse you right now. Let me make it clear. This is going to be dA. That's what we are integrating. We're taking the antiderivative with respect to. So what is this going to be equal to? | Indefinite integrals of sin(x), cos(x), and e_ AP Calculus AB Khan Academy.mp3 |
This is going to be dA. That's what we are integrating. We're taking the antiderivative with respect to. So what is this going to be equal to? Well, once again, we can rewrite it as the sum of integrals. This is the indefinite integral of e to the a dA. So this one right over here. | Indefinite integrals of sin(x), cos(x), and e_ AP Calculus AB Khan Academy.mp3 |
So what is this going to be equal to? Well, once again, we can rewrite it as the sum of integrals. This is the indefinite integral of e to the a dA. So this one right over here. I'll do it in green. Plus the indefinite integral, or the antiderivative, of 1 over a dA. Now, what is the antiderivative of e to the a? | Indefinite integrals of sin(x), cos(x), and e_ AP Calculus AB Khan Academy.mp3 |
So this one right over here. I'll do it in green. Plus the indefinite integral, or the antiderivative, of 1 over a dA. Now, what is the antiderivative of e to the a? Well, we already know a little bit about exponentials. The derivative with respect to x of e to the x is equal to e to the x. That's one of the reasons wh... | Indefinite integrals of sin(x), cos(x), and e_ AP Calculus AB Khan Academy.mp3 |
Now, what is the antiderivative of e to the a? Well, we already know a little bit about exponentials. The derivative with respect to x of e to the x is equal to e to the x. That's one of the reasons why e and the exponential function in general is so amazing. And if we just replaced a with x, or x with a, you get the d... | Indefinite integrals of sin(x), cos(x), and e_ AP Calculus AB Khan Academy.mp3 |
That's one of the reasons why e and the exponential function in general is so amazing. And if we just replaced a with x, or x with a, you get the derivative with respect to a of e to the a is equal to e to the a. So the antiderivative here, the derivative of e to the a is e to the a. The antiderivative is going to be e... | Indefinite integrals of sin(x), cos(x), and e_ AP Calculus AB Khan Academy.mp3 |
The antiderivative is going to be e to the a. And maybe you could shift it by some type of a constant. Oh, and let me not forget, I have to put my constant right over here. I could have a constant factor. So let me, always important, remember the constant. So you have a constant factor right over here. Never forget tha... | Indefinite integrals of sin(x), cos(x), and e_ AP Calculus AB Khan Academy.mp3 |
I could have a constant factor. So let me, always important, remember the constant. So you have a constant factor right over here. Never forget that. I almost did. So once again, over here, what's the antiderivative of e to the a? It is e to the a. | Indefinite integrals of sin(x), cos(x), and e_ AP Calculus AB Khan Academy.mp3 |
Never forget that. I almost did. So once again, over here, what's the antiderivative of e to the a? It is e to the a. What's the antiderivative of 1 over a? Well, we've seen that in the last video. It is going to be the natural log of the absolute value of a. | Indefinite integrals of sin(x), cos(x), and e_ AP Calculus AB Khan Academy.mp3 |
It is e to the a. What's the antiderivative of 1 over a? Well, we've seen that in the last video. It is going to be the natural log of the absolute value of a. And then we want to have the most general antiderivative. So there could be a constant factor out here as well. And we are done. | Indefinite integrals of sin(x), cos(x), and e_ AP Calculus AB Khan Academy.mp3 |
And what I want to do is I want to figure out the limit of g of x as x approaches positive six from values that are less than positive six, or you could say from the left, from the, you could say the negative direction. So what is this going to be equal to? And if you have a sense of it, pause the video and give a go a... | One-sided limits from graphs asymptote Limits and continuity AP Calculus AB Khan Academy.mp3 |
Well, to think about this, let's just take different x values that approach six from the left and look at what the values of the function are. So g of two looks like it's a little bit more than one. G of three, it's a little bit more than that. G of four looks like it's a little under two. G of five, it looks like it's... | One-sided limits from graphs asymptote Limits and continuity AP Calculus AB Khan Academy.mp3 |
G of four looks like it's a little under two. G of five, it looks like it's around three. G of 5.5 looks like it's around five. G of, let's say 5.75, looks like it's like nine. And so as x gets closer and closer to six from the left, it looks like the value of our function just becomes unbounded. It's just getting infi... | One-sided limits from graphs asymptote Limits and continuity AP Calculus AB Khan Academy.mp3 |
G of, let's say 5.75, looks like it's like nine. And so as x gets closer and closer to six from the left, it looks like the value of our function just becomes unbounded. It's just getting infinitely large. And so in some context, you might see someone write that. Maybe this is equal to infinity. But infinity isn't, we'... | One-sided limits from graphs asymptote Limits and continuity AP Calculus AB Khan Academy.mp3 |
And so in some context, you might see someone write that. Maybe this is equal to infinity. But infinity isn't, we're not talking about a specific number. And if we're talking technically about limits, the way that we've looked at it, what is, you'll sometimes see this in some classes, but in this context, especially on... | One-sided limits from graphs asymptote Limits and continuity AP Calculus AB Khan Academy.mp3 |
And if we're talking technically about limits, the way that we've looked at it, what is, you'll sometimes see this in some classes, but in this context, especially on the exercises on Khan Academy, we'll say that this does not exist. Not exist. This thing right over here is unbounded. And this is interesting because th... | One-sided limits from graphs asymptote Limits and continuity AP Calculus AB Khan Academy.mp3 |
And once again, it reinforces the idea that e is really this somewhat magical number. So we're gonna do a little bit of an exploration. Let's just pick some points on this curve of y is equal to e to the x, and think about what the slope of the tangent line is, or what the derivative looks like. And so let's say when y... | Derivative of __ Advanced derivatives AP Calculus AB Khan Academy.mp3 |
And so let's say when y is equal to one, or when e to the x is equal to one, this is the case when x is equal to zero, well, the slope of the tangent line looks like it is one, which is curious, because that's exactly the value of the function at that point. What about when e to the x is equal to two, right over here? ... | Derivative of __ Advanced derivatives AP Calculus AB Khan Academy.mp3 |
What about, what about when e to the x is equal to 1 1⁄2? So that's happening right about here. Well, it sure looks like the slope of the tangent line is about 1 1⁄2. We could try, what happens when e to the x is equal to five? Well, the slope of the tangent line here sure does look pretty close, sure does look pretty ... | Derivative of __ Advanced derivatives AP Calculus AB Khan Academy.mp3 |
We could try, what happens when e to the x is equal to five? Well, the slope of the tangent line here sure does look pretty close, sure does look pretty close to five. And so, just eyeballing it, is it the case that the slope of the tangent line of e to the x is the same thing, is e to the x? And I will tell you, and t... | Derivative of __ Advanced derivatives AP Calculus AB Khan Academy.mp3 |
And I will tell you, and this is an amazing thing, that that is indeed true. That if I have some function, f of x, that is equal to e to the x, and if I were to take the derivative of this, this is going to be equal to e to the x as well. Or another way of saying it, the derivative with respect to x of e to the x is eq... | Derivative of __ Advanced derivatives AP Calculus AB Khan Academy.mp3 |
And that is an amazing thing. In previous lessons or courses, you've learned about ways to define e. And this could be a new one. E is the number that where if you take that number to the power x, if you define a function or expression as e to the x, it's that number where if you take the derivative of that, it's still... | Derivative of __ Advanced derivatives AP Calculus AB Khan Academy.mp3 |
And this time we're going to rotate our function. We're going to rotate it around a vertical line that is not the y-axis. And if we do that, so we're going to rotate y is equal to x squared minus 1, or at least this part of it, we're going to rotate it around the vertical line x is equal to negative 2. And if we do tha... | Disc method rotating around vertical line AP Calculus AB Khan Academy.mp3 |
And if we do that, we get this gumball shape that looks something like this. So what I want to do is I want to find the volume of this using the disk method. So what I want to do is construct some disks. So construct some disks. So let's take this one of the disks right over here. It's going to have some depth. And tha... | Disc method rotating around vertical line AP Calculus AB Khan Academy.mp3 |
So construct some disks. So let's take this one of the disks right over here. It's going to have some depth. And that depth is going to be dy. It's going to be dy right over there. And it's going to have some area on top of it that is a function of any given y that I have. So the volume of a given disk is going to be t... | Disc method rotating around vertical line AP Calculus AB Khan Academy.mp3 |
And that depth is going to be dy. It's going to be dy right over there. And it's going to have some area on top of it that is a function of any given y that I have. So the volume of a given disk is going to be the area as a function of y times the depth of the disk times dy. And then we just have to integrate it over t... | Disc method rotating around vertical line AP Calculus AB Khan Academy.mp3 |
So the volume of a given disk is going to be the area as a function of y times the depth of the disk times dy. And then we just have to integrate it over the interval that we care about. And we're doing it in terms of the volume of the disk. And then we're going to integrate it over the interval that we care about. And... | Disc method rotating around vertical line AP Calculus AB Khan Academy.mp3 |
And then we're going to integrate it over the interval that we care about. And we're doing it all in terms of y. And in this case, we're going to integrate from y is equal to... Well, this is going to hit this y intercept right over here. So y is equal to negative 1. And let's go all the way to y is equal to... Let's s... | Disc method rotating around vertical line AP Calculus AB Khan Academy.mp3 |
So y is equal to negative 1. And let's go all the way to y is equal to... Let's say y is equal to 3. y is equal to 3 right over here. So from y equals negative 1 to y equals 3. So y equals negative 1 to y is equal to 3. And that's going to give us the volume of our upside-down gumdrop-type-looking thing. So the key her... | Disc method rotating around vertical line AP Calculus AB Khan Academy.mp3 |
So y equals negative 1 to y is equal to 3. And that's going to give us the volume of our upside-down gumdrop-type-looking thing. So the key here is so that we can start evaluating the double integral is to just figure out what the area of each of these disks are as a function of y. And we know that area is just... Area... | Disc method rotating around vertical line AP Calculus AB Khan Academy.mp3 |
And we know that area is just... Area as a function of y is just going to be pi times radius as a function of y squared. So the real key is, what is the radius as a function of y for any one of these y's? The radius as a function of y. So let's think about that a little bit. What is this curve? Well, let's write it as ... | Disc method rotating around vertical line AP Calculus AB Khan Academy.mp3 |
So let's think about that a little bit. What is this curve? Well, let's write it as a function of y. If you add 1 to both sides, and I'm going to swap sides, you'll get x squared is equal to y plus 1. I just added 1 to both sides and then swapped sides. And then you get x is equal to the principal root of the square ro... | Disc method rotating around vertical line AP Calculus AB Khan Academy.mp3 |
If you add 1 to both sides, and I'm going to swap sides, you'll get x squared is equal to y plus 1. I just added 1 to both sides and then swapped sides. And then you get x is equal to the principal root of the square root of y plus 1. So this we can write as x or we can even write it as f of y if we want. f of y is equ... | Disc method rotating around vertical line AP Calculus AB Khan Academy.mp3 |
So this we can write as x or we can even write it as f of y if we want. f of y is equal to the square root of y plus 1. Or we could say x is equal to a function of y, which is the square root of y plus 1. So what's the distance? What's the distance here at any point? Well, this distance... Let me make it very clear. So... | Disc method rotating around vertical line AP Calculus AB Khan Academy.mp3 |
So what's the distance? What's the distance here at any point? Well, this distance... Let me make it very clear. So it's going to be our total distance in the horizontal direction. So this first part, as we're... I'm going to do it in a different color that you can't see. So this part right over here is just going to b... | Disc method rotating around vertical line AP Calculus AB Khan Academy.mp3 |
So it's going to be our total distance in the horizontal direction. So this first part, as we're... I'm going to do it in a different color that you can't see. So this part right over here is just going to be the value of the function. It's going to give you an x value. But then you have to add another 2 to go all the ... | Disc method rotating around vertical line AP Calculus AB Khan Academy.mp3 |
So this part right over here is just going to be the value of the function. It's going to give you an x value. But then you have to add another 2 to go all the way over here. So your entire radius is a function of y. Your radius as a function of y is going to be equal to the square root of y plus 1. This essentially wi... | Disc method rotating around vertical line AP Calculus AB Khan Academy.mp3 |
So your entire radius is a function of y. Your radius as a function of y is going to be equal to the square root of y plus 1. This essentially will give you one of these x values when you're sitting on this curve. It's x as a function of y. It'll give you one of these x values. And then from that you add another 2. So ... | Disc method rotating around vertical line AP Calculus AB Khan Academy.mp3 |
It's x as a function of y. It'll give you one of these x values. And then from that you add another 2. So plus 2. Another way of thinking about it, you get an x value here and from that x value you subtract out x is equal to negative 2. And when you subtract x is equal to negative 2 you're adding 2 here. But hopefully ... | Disc method rotating around vertical line AP Calculus AB Khan Academy.mp3 |
So plus 2. Another way of thinking about it, you get an x value here and from that x value you subtract out x is equal to negative 2. And when you subtract x is equal to negative 2 you're adding 2 here. But hopefully this makes intuitive sense. This is the x value... Let me do this in a better color. This right over he... | Disc method rotating around vertical line AP Calculus AB Khan Academy.mp3 |
But hopefully this makes intuitive sense. This is the x value... Let me do this in a better color. This right over here, this distance right over here is the x value you get when you just evaluate the function of y. But then if you wanted the full radius, you have to go another 2 to go to the center of our axis of rota... | Disc method rotating around vertical line AP Calculus AB Khan Academy.mp3 |
But then if you wanted the full radius, you have to go another 2 to go to the center of our axis of rotation. Once again, if you just take a given y right over there, you evaluate the y you get an x value. That x value will just give you this distance. If you want the full distance, you have to subtract negative 2 from... | Disc method rotating around vertical line AP Calculus AB Khan Academy.mp3 |
If you want the full distance, you have to subtract negative 2 from that x value, which is essentially the same thing as adding 2 to get our full radius. So our radius as a function of y is this thing right over here. So substituting back into this, we can now write our definite integral for our volume. The volume is g... | Disc method rotating around vertical line AP Calculus AB Khan Academy.mp3 |
The volume is going to be equal to the definite integral from negative 1 to 3 of pi times our radius squared dy. So I can write the pi out here. We've done this multiple times. Times radius squared, so it's going to be square root of y plus 1 plus 2 squared, that's our radius, times dy. So we've set up the definite int... | Disc method rotating around vertical line AP Calculus AB Khan Academy.mp3 |
And then later we are asked, is Robert's work correct? If not, what's his mistake? So pause this video and try to figure it out on your own. All right, now let's work through this together. So our original g of x is equal to the cube root of x, which is the same thing as x to the 1 3rd. So in step one, it looks like Ro... | Mistakes when finding inflection points second derivative undefined AP Calculus AB Khan Academy.mp3 |
All right, now let's work through this together. So our original g of x is equal to the cube root of x, which is the same thing as x to the 1 3rd. So in step one, it looks like Robert's trying to find the first and second derivative. So the first derivative, we just do the power rule. So it'll be 1 3rd x to the decreme... | Mistakes when finding inflection points second derivative undefined AP Calculus AB Khan Academy.mp3 |
So the first derivative, we just do the power rule. So it'll be 1 3rd x to the decrement the exponent. So this is looking good. Second derivative, we take this, multiply this times 1 3rd, which would be negative 2 9ths, and then decrement negative 2 3rds, which would indeed be negative 5 3rds. So that looks right. And ... | Mistakes when finding inflection points second derivative undefined AP Calculus AB Khan Academy.mp3 |
Second derivative, we take this, multiply this times 1 3rd, which would be negative 2 9ths, and then decrement negative 2 3rds, which would indeed be negative 5 3rds. So that looks right. And then it looks like Robert's trying to rewrite it. So we have the negative 2 9ths still. But then he recognized that this is the ... | Mistakes when finding inflection points second derivative undefined AP Calculus AB Khan Academy.mp3 |
So we have the negative 2 9ths still. But then he recognized that this is the same thing as x to the 5 3rds in the denominator. And x to the 5 3rds is the same thing as x, as the cube root of x to the 5th. So this is all looking good. Step one looks good. And then step two, it looks like he's trying to find the solutio... | Mistakes when finding inflection points second derivative undefined AP Calculus AB Khan Academy.mp3 |
So this is all looking good. Step one looks good. And then step two, it looks like he's trying to find the solution, or he's trying to find x values where the second derivative is equal to zero. And it is indeed true that this has no solution, that you can never make this second derivative equal to zero. In order to be... | Mistakes when finding inflection points second derivative undefined AP Calculus AB Khan Academy.mp3 |
And it is indeed true that this has no solution, that you can never make this second derivative equal to zero. In order to be zero, the numerator would have to be zero, and well, two is never going to be equal to zero. So this is correct. And then step three, he says g doesn't have any inflection points. Now this is a ... | Mistakes when finding inflection points second derivative undefined AP Calculus AB Khan Academy.mp3 |
And then step three, he says g doesn't have any inflection points. Now this is a little bit suspect. It is, in many cases, our inflection point is a situation where our second derivative is equal to zero. And even then, we don't know it's an inflection point. It would be a candidate inflection point. We would have to c... | Mistakes when finding inflection points second derivative undefined AP Calculus AB Khan Academy.mp3 |
And even then, we don't know it's an inflection point. It would be a candidate inflection point. We would have to confirm that our second derivative crosses signs, or switches signs, as we cross that x value. But here we can't find a situation where our second derivative is equal to zero. But we have to remind ourselve... | Mistakes when finding inflection points second derivative undefined AP Calculus AB Khan Academy.mp3 |
But here we can't find a situation where our second derivative is equal to zero. But we have to remind ourselves that other candidate inflection points are where our second derivative is undefined. And so he can't make this statement without seeing where our second derivative could be undefined. So for example, he coul... | Mistakes when finding inflection points second derivative undefined AP Calculus AB Khan Academy.mp3 |
So for example, he could say that g prime prime is undefined, undefined when what? Well, this is going to be undefined when x is equal to zero. X zero to the fifth, cube root of that, that's gonna be zero but then you're dividing by zero. So g prime prime undefined when x is equal to zero. So therefore, x equals, so we... | Mistakes when finding inflection points second derivative undefined AP Calculus AB Khan Academy.mp3 |
So g prime prime undefined when x is equal to zero. So therefore, x equals, so we could say, candidate, candidate, candidate inflection point when x equals zero. And so then we would want to test it. And we could set up a traditional table that you might have seen before where we have our interval or intervals. We coul... | Mistakes when finding inflection points second derivative undefined AP Calculus AB Khan Academy.mp3 |
And we could set up a traditional table that you might have seen before where we have our interval or intervals. We could have test values in our intervals. We have to be careful with those, make sure that they are indicative. And then we would say the sine of our second derivative of g prime prime. And then we would h... | Mistakes when finding inflection points second derivative undefined AP Calculus AB Khan Academy.mp3 |
And then we would say the sine of our second derivative of g prime prime. And then we would have our concavity, concavity of g. And in order for x equals zero to be an inflection point, we would have to switch sines as, or our second derivative would have to switch sines as we cross x equals zero, and which would mean ... | Mistakes when finding inflection points second derivative undefined AP Calculus AB Khan Academy.mp3 |
I could do test values, let's say, I'll use negative one and one. And you have to be careful when you use these, you have to make sure that we are close enough that nothing unusual happens between these test values up until we get to that candidate inflection point. And now what's the sine of our second derivative when... | Mistakes when finding inflection points second derivative undefined AP Calculus AB Khan Academy.mp3 |
When x equals negative one, so let's see, negative one to the fifth power is negative one cube root of negative one is negative one. And so we're gonna have negative 2 9ths divided by negative one is gonna be positive 2 9ths. So our sine right over here is gonna be positive. And when, and this is gonna be in general wh... | Mistakes when finding inflection points second derivative undefined AP Calculus AB Khan Academy.mp3 |
And when, and this is gonna be in general when we're dealing with any negative value, because if you take any negative value to the fifth power it's gonna be negative. And you take that, the cube root of that, you're gonna have negative, but then when you have a negative value divided by that, you're gonna get a positi... | Mistakes when finding inflection points second derivative undefined AP Calculus AB Khan Academy.mp3 |
And if you're dealing with a positive value, well, that to the fifth power is gonna be positive, cube root of that is still going to be positive, but then your gonna have negative 2 9ths divided by that positive value. So this is going to be negative. So it is indeed the case that our concavity of g switches as we cros... | Mistakes when finding inflection points second derivative undefined AP Calculus AB Khan Academy.mp3 |
We're concave upwards when x is less than zero, our second derivative is positive, and we are concave downwards when x is greater than zero. Let me write that a little bit. Downwards, downwards when x is greater than zero. So we are switching concavity as we cross x equals zero, and so this tells us that x, so let's se... | Mistakes when finding inflection points second derivative undefined AP Calculus AB Khan Academy.mp3 |
So we are switching concavity as we cross x equals zero, and so this tells us that x, so let's see, we are switching signs, switching, let me say g prime prime switching signs as we cross x equals zero, and our function is defined at x equals zero, and function defined at x equals zero. So we have an inflection point a... | Mistakes when finding inflection points second derivative undefined AP Calculus AB Khan Academy.mp3 |
We're told that f of seven is equal to 40 plus 5e to the seventh power. And f prime of x is equal to 5e to the x. What is f of zero? So to evaluate f of zero, let's take the antiderivative of f prime of x, and then we're going to have a constant of integration there, so we can use the information that they gave us up h... | Finding specific antiderivatives exponential function AP Calculus AB Khan Academy.mp3 |
So to evaluate f of zero, let's take the antiderivative of f prime of x, and then we're going to have a constant of integration there, so we can use the information that they gave us up here, that f of seven is equal to this. This might look like an expression, well it is an expression, but it's really just a number. T... | Finding specific antiderivatives exponential function AP Calculus AB Khan Academy.mp3 |
And so we can use that to solve for our constant of integration, and then we will have fully known what f of x is, and we can use that to evaluate f of zero. So let's just do it. So if f prime of x is equal to 5e to the x, then f of x is going to be equal to the antiderivative of f prime of x, so the antiderivative of ... | Finding specific antiderivatives exponential function AP Calculus AB Khan Academy.mp3 |
And this is the thing that I always find amazing about exponentials, and actually let me just take a step. I'll take that five out of the integral so it becomes a little bit more obvious. And so the antiderivative of e to the x, well that's just e to the x, because the derivative of e to the x is e to the x, which I fi... | Finding specific antiderivatives exponential function AP Calculus AB Khan Academy.mp3 |
So this is going to be 5e to the x plus c. And you can verify, take the derivative of 5e to the x plus c. The derivative of 5e to the x, well that's 5e to the x, so that works out, and the derivative of c is zero, so you wouldn't see it over here. So now let's use this information to figure out what c is so that we kno... | Finding specific antiderivatives exponential function AP Calculus AB Khan Academy.mp3 |
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