Sentence stringlengths 92 3.31k | video_title stringlengths 24 106 |
|---|---|
So our original integral, once again, this is just algebra, everything we've done so far is algebra. We've just rewritten it using a little bit of algebraic long division. Our original integral has simplified to negative 1 1 2. And some might argue it's not simplified, but it's actually much more useful for finding the... | Dividing expressions to evaluate integral AP Calculus BC Khan Academy.mp3 |
And some might argue it's not simplified, but it's actually much more useful for finding the integral. Negative 1 1 2 plus two over x minus one dx. Now, how do we evaluate this? Well, the antiderivative of negative 1 1 2 is pretty straightforward. That's just going to be negative 1 1 2 x. Negative 1 1 2 x. What's the a... | Dividing expressions to evaluate integral AP Calculus BC Khan Academy.mp3 |
Well, the antiderivative of negative 1 1 2 is pretty straightforward. That's just going to be negative 1 1 2 x. Negative 1 1 2 x. What's the antiderivative of two over x minus one? And you might be able to do this in your head. The derivative of x minus one is just one. So you could say that the derivative is sitting t... | Dividing expressions to evaluate integral AP Calculus BC Khan Academy.mp3 |
What's the antiderivative of two over x minus one? And you might be able to do this in your head. The derivative of x minus one is just one. So you could say that the derivative is sitting there. And so we can essentially do u substitution in our heads and say, okay, let's just take the antiderivative, I guess you coul... | Dividing expressions to evaluate integral AP Calculus BC Khan Academy.mp3 |
So you could say that the derivative is sitting there. And so we can essentially do u substitution in our heads and say, okay, let's just take the antiderivative, I guess you could say with respect to x minus one, which would be the natural log of the absolute value of x minus one. If all of that sounds really confusin... | Dividing expressions to evaluate integral AP Calculus BC Khan Academy.mp3 |
So if I were just trying to evaluate, if I were just trying to evaluate the integral two over x minus one dx, I could see, okay, the derivative of x minus one is just one. So I could say u is equal to x minus one. And then du is going to be equal to dx. And so this is going to be, we can rewrite in terms of u as two, I... | Dividing expressions to evaluate integral AP Calculus BC Khan Academy.mp3 |
What we will talk about in this video is the product rule, which is one of the fundamental ways of evaluating derivatives. And we won't prove it in this video, but we will learn how to apply it. And all it tells us is that if we have a function that can be expressed as the product of two functions, so let's say it can ... | Product rule Derivative rules AP Calculus AB Khan Academy.mp3 |
So here we have two terms. In each term, we took the derivative of one of the functions and not the other. And we multiply the derivative of the first function times the second function plus just the first function times the derivative of the second function. Now let's see if we can actually apply this to actually find... | Product rule Derivative rules AP Calculus AB Khan Academy.mp3 |
Now let's see if we can actually apply this to actually find the derivative of something. So let's say we are dealing with x squared times cosine of x. Or let's say, well, yeah, sure. Let's do x squared times sine of x. We could have done it either way. And we are curious about taking the derivative of this. We are cur... | Product rule Derivative rules AP Calculus AB Khan Academy.mp3 |
Let's do x squared times sine of x. We could have done it either way. And we are curious about taking the derivative of this. We are curious about what its derivative is. Well, we might immediately recognize that this can be expressed as a product of two functions. We could set f of x is equal to x squared. So that is ... | Product rule Derivative rules AP Calculus AB Khan Academy.mp3 |
We are curious about what its derivative is. Well, we might immediately recognize that this can be expressed as a product of two functions. We could set f of x is equal to x squared. So that is f of x right over there. And we could set g of x to be equal to sine of x. And there we have it. We have our f of x times g of... | Product rule Derivative rules AP Calculus AB Khan Academy.mp3 |
So that is f of x right over there. And we could set g of x to be equal to sine of x. And there we have it. We have our f of x times g of x. And we could think about what these individual derivatives are. The derivative of f of x is just going to be equal to 2x by the power rule. And the derivative of g of x is just th... | Product rule Derivative rules AP Calculus AB Khan Academy.mp3 |
We have our f of x times g of x. And we could think about what these individual derivatives are. The derivative of f of x is just going to be equal to 2x by the power rule. And the derivative of g of x is just the derivative of sine of x. And we covered this when we just talked about common derivatives. Derivative of s... | Product rule Derivative rules AP Calculus AB Khan Academy.mp3 |
And the derivative of g of x is just the derivative of sine of x. And we covered this when we just talked about common derivatives. Derivative of sine of x is cosine of x. And so now we're ready to apply the product rule. This is going to be equal to f prime of x times g of x. So f prime of x, the derivative of f, is 2... | Product rule Derivative rules AP Calculus AB Khan Academy.mp3 |
Now at first this might seem daunting. I have this rational expression. I have x's in the numerators and x's in the denominators, but we just have to remember, we just have to do some algebraic manipulation and this is going to seem a lot more tractable. This is the same thing as the definite integral from negative one... | Definite integral of rational function AP Calculus AB Khan Academy.mp3 |
This is the same thing as the definite integral from negative one to negative two of 16 over x to the third minus x to the third over x to the third. Minus x to the third over x to the third dx. And now what is that going to be equal to? That is going to be equal to the definite integral from negative one to negative t... | Definite integral of rational function AP Calculus AB Khan Academy.mp3 |
That is going to be equal to the definite integral from negative one to negative two of, I could write this first term right over here. Let me do this in a different color. I could write this as 16x to the negative three. x to the negative three. And this second one, we have minus x to the third over x to the third. We... | Definite integral of rational function AP Calculus AB Khan Academy.mp3 |
x to the negative three. And this second one, we have minus x to the third over x to the third. Well, x to the third is just over, x to the third over x to the third is just going to be equal to one. So this is going to be minus one dx. So dx. And so what is this going to be equal to? Well, let's take the antiderivativ... | Definite integral of rational function AP Calculus AB Khan Academy.mp3 |
So this is going to be minus one dx. So dx. And so what is this going to be equal to? Well, let's take the antiderivative of each of these parts and then we're going to have to evaluate them at the different bounds. So let's see. The antiderivative of 16x to the negative three. We're just going to do the power rule for... | Definite integral of rational function AP Calculus AB Khan Academy.mp3 |
Well, let's take the antiderivative of each of these parts and then we're going to have to evaluate them at the different bounds. So let's see. The antiderivative of 16x to the negative three. We're just going to do the power rule for derivatives in reverse. You could view this as the power rule of integration or power... | Definite integral of rational function AP Calculus AB Khan Academy.mp3 |
We're just going to do the power rule for derivatives in reverse. You could view this as the power rule of integration or power rule of taking the antiderivative where what you do is you're going to increase our exponent by one. So you're going to go from negative three to negative two. And then you're going to divide ... | Definite integral of rational function AP Calculus AB Khan Academy.mp3 |
And then you're going to divide by that amount, by negative two. So it's going to be 16 divided by negative two times x to the negative two. All I did is I increased the exponent and I divided by that amount. So that's the antiderivative here. And 16 divided by negative two, that is just negative eight. So we have nega... | Definite integral of rational function AP Calculus AB Khan Academy.mp3 |
So that's the antiderivative here. And 16 divided by negative two, that is just negative eight. So we have negative eight x to the negative two. And then the antiderivative of negative one, well, that's just negative x. Negative, negative x. Negative x. And actually you could, you might just know that. | Definite integral of rational function AP Calculus AB Khan Academy.mp3 |
And then the antiderivative of negative one, well, that's just negative x. Negative, negative x. Negative x. And actually you could, you might just know that. And hey, if I take the derivative of negative x, I get negative one. Or if you viewed this as negative x to the zero power, because that's what one is, well, it'... | Definite integral of rational function AP Calculus AB Khan Academy.mp3 |
And actually you could, you might just know that. And hey, if I take the derivative of negative x, I get negative one. Or if you viewed this as negative x to the zero power, because that's what one is, well, it's the same thing. You increase the exponent by one to get x to the first power. And then you divide by one. A... | Definite integral of rational function AP Calculus AB Khan Academy.mp3 |
You increase the exponent by one to get x to the first power. And then you divide by one. And so, I mean, you could view it as that right over there. But either way, you get to negative or minus x. And so now we want to evaluate that. We're going to evaluate that at the bounds and take the difference. So we're going to... | Definite integral of rational function AP Calculus AB Khan Academy.mp3 |
But either way, you get to negative or minus x. And so now we want to evaluate that. We're going to evaluate that at the bounds and take the difference. So we're going to evaluate that at negative two and then subtract from that, this evaluated at negative one. And let me do those in two different colors so we can see ... | Definite integral of rational function AP Calculus AB Khan Academy.mp3 |
So we're going to evaluate that at negative two and then subtract from that, this evaluated at negative one. And let me do those in two different colors so we can see what's going on. So we're gonna evaluate it at negative two and we're going to evaluate it at negative one. So let's first evaluate it at negative two. S... | Definite integral of rational function AP Calculus AB Khan Academy.mp3 |
So let's first evaluate it at negative two. So this is going to be equal to, this is going to be equal to, when you evaluate it at negative two, it's going to be negative eight, negative eight times x to the negative two. So negative two to the negative two power minus negative two. And from that, we're going to subtra... | Definite integral of rational function AP Calculus AB Khan Academy.mp3 |
And from that, we're going to subtract it, evaluate it at negative one. So it's gonna be negative eight times negative one to the negative two power minus negative one. All right, so what is this going to be? So negative two to the negative two, so negative two to the negative two is equal to one over negative two squa... | Definite integral of rational function AP Calculus AB Khan Academy.mp3 |
So negative two to the negative two, so negative two to the negative two is equal to one over negative two squared, which is equal to 1 4th. So this is equal to positive 1 4th, but then negative eight times positive 1 4th is going to be equal to negative two. And then we have negative two minus negative two, so that's ... | Definite integral of rational function AP Calculus AB Khan Academy.mp3 |
We now have a lot of experience finding the areas under curves when we're dealing with things in rectangular coordinates. And we saw, we took the Riemann sums, a bunch of rectangles, we took the limit as we had an infinite number of infinitely thin rectangles, and we were able to find the area. But now let's move on to... | Area bounded by polar curves Applications of definite integrals AP Calculus BC Khan Academy.mp3 |
And polar coordinates, I won't say we're finding the area under a curve, but really, in this example right over here, we have a part of the graph of r is equal to f of theta, and we've graphed it between theta is equal to alpha and theta is equal to beta. What I wanna do in this video is come up with a general expressi... | Area bounded by polar curves Applications of definite integrals AP Calculus BC Khan Academy.mp3 |
When we did it in rectangular coordinates, we divided things into rectangles. Over here, rectangles don't seem as obvious, because they're all kind of coming to this point. But what if we could divide things into, if we could divide things into sector, I guess we could say, little pie pieces. Someone's doing some serio... | Area bounded by polar curves Applications of definite integrals AP Calculus BC Khan Academy.mp3 |
Someone's doing some serious drilling downstairs. I don't know if it's picking up on the microphone, but anyway, I will continue. So what would happen if we could divide this into a whole series of kind of pie pieces? If we could divide it into a whole series of pie pieces, and then take the limit as if we had an infin... | Area bounded by polar curves Applications of definite integrals AP Calculus BC Khan Academy.mp3 |
If we could divide it into a whole series of pie pieces, and then take the limit as if we had an infinite number, an infinite number of pie pieces. So we want to find the area of each of these pie pieces, and then take the limit as the pie pieces, I guess you could say, become infinitely thin, and we have an infinite n... | Area bounded by polar curves Applications of definite integrals AP Calculus BC Khan Academy.mp3 |
I'll give you one more hint. For thinking about the area of these pie, I guess you could say pie, the area of these pie wedges. I'll give you another hint. So if I have a circle, I'm doing my best attempt at a circle. Luckily the plumbing or whatever is going on downstairs has stopped for now, allowing me to focus more... | Area bounded by polar curves Applications of definite integrals AP Calculus BC Khan Academy.mp3 |
So if I have a circle, I'm doing my best attempt at a circle. Luckily the plumbing or whatever is going on downstairs has stopped for now, allowing me to focus more on the calculus, which is obviously more important. All right, so if I have a circle, that's my best attempt at a circle, and it's of radius r, it's of rad... | Area bounded by polar curves Applications of definite integrals AP Calculus BC Khan Academy.mp3 |
It's a sector of a circle, so that's obviously r as well. And if this angle right here is theta, what is going to be the area, what is going to be the area of this sector right over here? So that's my hint for you. Think about what this area is going to be, and we're assuming theta is in radians. Think about what this ... | Area bounded by polar curves Applications of definite integrals AP Calculus BC Khan Academy.mp3 |
Think about what this area is going to be, and we're assuming theta is in radians. Think about what this area is going to be, and then see if you can extend that to what we're trying to do here to figure out a, somehow, I'm giving you a hint again, using integration, finding an expression for this area. So I'm assuming... | Area bounded by polar curves Applications of definite integrals AP Calculus BC Khan Academy.mp3 |
So what's the area of the entire, the area of the entire circle? Well, we already know that. That's going to be pi r squared, formula for the area of a circle. And then what's going to be the area of this? Well, it's going to be a fraction of the circle. If this is pi, sorry, if this is theta, where if we went two pi r... | Area bounded by polar curves Applications of definite integrals AP Calculus BC Khan Academy.mp3 |
And then what's going to be the area of this? Well, it's going to be a fraction of the circle. If this is pi, sorry, if this is theta, where if we went two pi radians, that would be the whole circle. So this is going to be pi, sorry, this is going to be theta over two pi of the circle. So times theta over, over two pi,... | Area bounded by polar curves Applications of definite integrals AP Calculus BC Khan Academy.mp3 |
So this is going to be pi, sorry, this is going to be theta over two pi of the circle. So times theta over, over two pi, times theta over two pi, would be the area of this sector right over here, area of the whole circle times the proportion of the circle that we've kind of, we have defined, or that this sector is made... | Area bounded by polar curves Applications of definite integrals AP Calculus BC Khan Academy.mp3 |
Now, what happens if instead of theta, so let's look at each of these over here. So each of these things that I've drawn. So let's focus just on one of these, one of these wedges. I will highlight it in orange. I'll highlight it in orange. So instead of the angle being theta, let's just assume it's a really, really, re... | Area bounded by polar curves Applications of definite integrals AP Calculus BC Khan Academy.mp3 |
I will highlight it in orange. I'll highlight it in orange. So instead of the angle being theta, let's just assume it's a really, really, really small angle. We'll use a differential, although this is a little bit of loosey-goosey mathematics, but the important here is to give you the conceptual understanding. I could ... | Area bounded by polar curves Applications of definite integrals AP Calculus BC Khan Academy.mp3 |
We'll use a differential, although this is a little bit of loosey-goosey mathematics, but the important here is to give you the conceptual understanding. I could call it a delta theta, and then eventually take the limit as our delta thetas approach zero, but I'm just gonna, just for conceptual purposes, assume we have ... | Area bounded by polar curves Applications of definite integrals AP Calculus BC Khan Academy.mp3 |
It's going to be r, it's going to be r as a function of the thetas that were around right over here, but we're just gonna call that our r right over there. And so what is going to be the area of this little sector? Well, the area of this little sector is going to, instead of my angle being theta, I'm calling my angle d... | Area bounded by polar curves Applications of definite integrals AP Calculus BC Khan Academy.mp3 |
In the last video, we tried to figure out the slope of a point, or the slope of a curve at a certain point. And the way we did it, we said, OK, well, let's find the slope between that point and then another point that's not too far away from that point. And we got the slope of the secant line. And it looks all fancy, b... | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
And it looks all fancy, but this is just the y value of the point that's not too far away. And this is just the y value of the point in question. So this is just your change in y. And then you divide that by your change in x. So in the example we did, h was the difference between our two x values. This distance was h. ... | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
And then you divide that by your change in x. So in the example we did, h was the difference between our two x values. This distance was h. And that gave us the point, the slope of that line. We said, hey, what if we take the limit as this point right here gets closer and closer to this point? If this point essentially... | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
We said, hey, what if we take the limit as this point right here gets closer and closer to this point? If this point essentially almost becomes this point, then our slope is going to be the slope of our tangent line. And we define that as the derivative of our function. We said that's equal to f prime of x. So let's se... | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
We said that's equal to f prime of x. So let's see if we can apply this in this video to maybe make things a little bit more concrete in your head. So let me do one. First I'll do a particular case where I want to find the slope at exactly some point. So let's say, let me draw my axes again. Let's draw some axes right ... | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
First I'll do a particular case where I want to find the slope at exactly some point. So let's say, let me draw my axes again. Let's draw some axes right there. And let's say I have the curve. This is the curve. y is equal to x squared. So this is my y-axis. | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
And let's say I have the curve. This is the curve. y is equal to x squared. So this is my y-axis. This is my x-axis. And I want to know the slope at the point x is equal to 3. When I say the slope, you can imagine a tangent line here. | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
So this is my y-axis. This is my x-axis. And I want to know the slope at the point x is equal to 3. When I say the slope, you can imagine a tangent line here. Do it in a light yellow. You can imagine a tangent line that goes just like that. And it would just barely graze the curve at that point. | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
When I say the slope, you can imagine a tangent line here. Do it in a light yellow. You can imagine a tangent line that goes just like that. And it would just barely graze the curve at that point. But what is the slope of that tangent line? What is the slope of that tangent line? Which is the same as the slope of the c... | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
And it would just barely graze the curve at that point. But what is the slope of that tangent line? What is the slope of that tangent line? Which is the same as the slope of the curve right at that point. So to do it, I'm going to actually do this exact technique that we did before. And then we'll generalize it so you ... | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
Which is the same as the slope of the curve right at that point. So to do it, I'm going to actually do this exact technique that we did before. And then we'll generalize it so you don't have to do it every time for a particular number. So let's take some other point here. Let's call this 3 plus delta x. I'm changing th... | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
So let's take some other point here. Let's call this 3 plus delta x. I'm changing the notation because in some books you'll see an h, some books you'll see a delta x. Doesn't hurt to be exposed to both of them. So this is 3 plus delta x. So first of all, what is this point right here? This is a curve y is equal to x sq... | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
So this is 3 plus delta x. So first of all, what is this point right here? This is a curve y is equal to x squared. So f of x is 3 squared. This is the point 9. This is the point 3, 9 right here. Let me draw that out. | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
So f of x is 3 squared. This is the point 9. This is the point 3, 9 right here. Let me draw that out. Well, I'll draw it out in a second. And what is this point right here? So if we were to go all the way up here, what is that point? | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
Let me draw that out. Well, I'll draw it out in a second. And what is this point right here? So if we were to go all the way up here, what is that point? Well, here our x is now 3 plus delta x. It's the same thing as this one right here. It's x naught plus h. I could have called this 3 plus h just as easily. | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
So if we were to go all the way up here, what is that point? Well, here our x is now 3 plus delta x. It's the same thing as this one right here. It's x naught plus h. I could have called this 3 plus h just as easily. So it's 3 plus delta x up there. So what's the y value going to be? Well, whatever x value is, it's on ... | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
It's x naught plus h. I could have called this 3 plus h just as easily. So it's 3 plus delta x up there. So what's the y value going to be? Well, whatever x value is, it's on the curve, it's going to be that squared. So it's going to be, this is going to be the point 3 plus delta x squared. So let's figure out the slop... | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
Well, whatever x value is, it's on the curve, it's going to be that squared. So it's going to be, this is going to be the point 3 plus delta x squared. So let's figure out the slope of this secant line. Let me zoom in a little bit because that might help. So if I zoom in on just this part of the curve, it might look li... | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
Let me zoom in a little bit because that might help. So if I zoom in on just this part of the curve, it might look like that. And then I have one point here, and then I have the other point is up here. Let me see if I can draw that. That's the secant line. Just like that. This was the point over here. | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
Let me see if I can draw that. That's the secant line. Just like that. This was the point over here. This point is the point 3, 9. And then this point up here is the point 3 plus delta x. So just some larger number than 3. | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
This was the point over here. This point is the point 3, 9. And then this point up here is the point 3 plus delta x. So just some larger number than 3. And then it's going to be that number squared. So it's going to be 3 plus delta x squared. What is that? | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
So just some larger number than 3. And then it's going to be that number squared. So it's going to be 3 plus delta x squared. What is that? That's going to be 9. I'm just FOILing this out, or you do the distributive property twice. If a plus b squared is a squared plus 2ab plus b squared, so it's going to be 9 plus 2 t... | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
What is that? That's going to be 9. I'm just FOILing this out, or you do the distributive property twice. If a plus b squared is a squared plus 2ab plus b squared, so it's going to be 9 plus 2 times the product of these things. So plus 6 delta x, and then plus delta x squared. Plus delta x plus delta x squared. That's ... | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
If a plus b squared is a squared plus 2ab plus b squared, so it's going to be 9 plus 2 times the product of these things. So plus 6 delta x, and then plus delta x squared. Plus delta x plus delta x squared. That's the coordinate of the second line. This looks complicated, but I just took this x value and I squared it, ... | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
That's the coordinate of the second line. This looks complicated, but I just took this x value and I squared it, because it's on the line y is equal to x squared. So the slope of this line, the slope of the secant line is going to be the change in y divided by the change in x. So the change in y is just going to be thi... | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
So the change in y is just going to be this guy's y value, which is 9 plus 6 delta x plus delta x squared. That's this guy's y value. Minus this guy's y value. I'll do it in green. So minus 9. That's your change in y. And you want to divide that by your change in x. | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
I'll do it in green. So minus 9. That's your change in y. And you want to divide that by your change in x. Well, what is your change in x? This is actually going to be pretty convenient. This larger x value, we started with this point on the top. | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
And you want to divide that by your change in x. Well, what is your change in x? This is actually going to be pretty convenient. This larger x value, we started with this point on the top. So we have to start with this point on the bottom. So it's going to be 3 plus delta x. And then what's this x value? | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
This larger x value, we started with this point on the top. So we have to start with this point on the bottom. So it's going to be 3 plus delta x. And then what's this x value? Well, it's minus 3. That's his x value. So what does this simplify to? | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
And then what's this x value? Well, it's minus 3. That's his x value. So what does this simplify to? The numerator, this 9 and that 9 cancel out. We get a 9 minus 9. And the denominator, what happens? | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
So what does this simplify to? The numerator, this 9 and that 9 cancel out. We get a 9 minus 9. And the denominator, what happens? This 3 and minus 3 cancel out. So the change in x actually end up becoming this delta x, which makes sense, because this delta x is essentially how much more this guy is than that guy. So t... | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
And the denominator, what happens? This 3 and minus 3 cancel out. So the change in x actually end up becoming this delta x, which makes sense, because this delta x is essentially how much more this guy is than that guy. So that should be the change in x, delta x. So the slope of my secant line has simplified to 6 times... | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
So that should be the change in x, delta x. So the slope of my secant line has simplified to 6 times my change in x plus my change in x squared, all of that over my change in x. Now we can simplify this even more. Let's divide the numerator and the denominator by our change in x. And I will switch colors just to ease t... | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
Let's divide the numerator and the denominator by our change in x. And I will switch colors just to ease the monotony. So my slope of my tangent of my secant line, the one that goes through both of these, is going to be equal if you divide the numerator and denominator. This becomes 6. I'm just dividing numerator and d... | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
This becomes 6. I'm just dividing numerator and denominator by delta x. Plus 6 plus delta x. So that is the slope of the secant line. So slope is equal to 6 plus delta x. That's this one right here. That's this reddish line that I've drawn right there. | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
So that is the slope of the secant line. So slope is equal to 6 plus delta x. That's this one right here. That's this reddish line that I've drawn right there. So if this number right here, if the delta x was 1, if these were the points 3 and 4, then my slope would be 6 plus 1. Because I'm picking a point 4 where the d... | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
That's this reddish line that I've drawn right there. So if this number right here, if the delta x was 1, if these were the points 3 and 4, then my slope would be 6 plus 1. Because I'm picking a point 4 where the delta x here would have to be 1. So the slope would be 7. So we have a general formula for no matter what m... | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
So the slope would be 7. So we have a general formula for no matter what my delta x is, I can find the slope between 3 and 3 plus delta x, between those two points. Now, we wanted to find the slope at exactly that point right there. So let's see what happens when delta x gets smaller and smaller. This is what delta x i... | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
So let's see what happens when delta x gets smaller and smaller. This is what delta x is right now. It's this distance. But if delta x got a little bit smaller, then it would start, the secant line would look like that. Got even smaller, the secant line would look like that. It gets even smaller, then we're getting pre... | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
But if delta x got a little bit smaller, then it would start, the secant line would look like that. Got even smaller, the secant line would look like that. It gets even smaller, then we're getting pretty close to the slope of the tangent line. The tangent line is this thing right here that I want to find the slope of. ... | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
The tangent line is this thing right here that I want to find the slope of. So let's find the limit as our delta x approaches 0. So the limit as delta x approaches 0 of our slope of the secant line of 6 plus delta x is equal to what? This is pretty straightforward. You could just set this equal to 0 and it's equal to 6... | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
This is pretty straightforward. You could just set this equal to 0 and it's equal to 6. So the slope of our tangent line at the point x is equal to 3 right there is equal to 6. And another way we could write this, if we wrote that f of x is equal to x squared, we now know that the derivative, or the slope, of the tange... | Calculating slope of tangent line using derivative definition Differential Calculus Khan Academy.mp3 |
Let me underline that, the limit comparison test, in order to determine whether S converges. So let's just remind ourselves about the limit comparison test. If we say, if we say that we have two series, and I'll just use this notation, A sub n, and then another series, B sub n, and we know that A sub n and B sub n are ... | Worked example limit comparison test Series AP Calculus BC Khan Academy.mp3 |
And it really makes a lot of sense, because it's saying, look, as we get into our really large values of n, as we go really far out there in terms of the terms, if our behavior starts to look the same, then it makes sense that both these series would converge or diverge, and we have an introductory video on this in ano... | Worked example limit comparison test Series AP Calculus BC Khan Academy.mp3 |
This one doesn't look that similar. It has a three to the n minus one in the denominator, but the numerator doesn't behave the same. This one over here is interesting, because we could write this, this is the same thing as the sum n equals one to infinity. We could write this as two to the n over three to the n, and th... | Worked example limit comparison test Series AP Calculus BC Khan Academy.mp3 |
We could write this as two to the n over three to the n, and these are very similar. The only difference between this and this is that in the denominator here, or in the denominator up here, we have a minus one, and down here, we don't have that minus one. And so it makes sense, given that that's just a constant, that ... | Worked example limit comparison test Series AP Calculus BC Khan Academy.mp3 |
So let's try it out. Let's find the limit, and we also know that the A sub n's and the B sub n's, if we say that this right over here is B sub n, if we say that's B sub n, that this is going to be positive, or this is going to be greater than or equal to zero for n equals one, two, three, so for any values, this is goi... | Worked example limit comparison test Series AP Calculus BC Khan Academy.mp3 |
So we meet these first constraints, and so let's find the limit as n approaches infinity of A sub n, which is, I'll write it in that red color, which is two to the nth power over three to the n minus one over B sub n, over two to the nth, over three to the nth. So let me actually do a little algebraic manipulation righ... | Worked example limit comparison test Series AP Calculus BC Khan Academy.mp3 |
And so this will give us three to the n over three to the n minus one. We can divide the numerator and denominator by three to the n, and that'll give us one over one minus one over three to the n. So we could say this is the same thing as the limit as n approaches infinity of one over one minus one over three to the n... | Worked example limit comparison test Series AP Calculus BC Khan Academy.mp3 |
So this whole thing is just going to approach one. And one is clearly between zero and infinity. So the destinies of these two series are tied. They either both converge or they both diverge. And so this is a good one to use the limit comparison test with. And so let's think about it. Do they either both converge or do... | Worked example limit comparison test Series AP Calculus BC Khan Academy.mp3 |
They either both converge or they both diverge. And so this is a good one to use the limit comparison test with. And so let's think about it. Do they either both converge or do they both diverge? Well, this is a geometric series. Our common ratio here is less than one. So this is going to converge. | Worked example limit comparison test Series AP Calculus BC Khan Academy.mp3 |
Do they either both converge or do they both diverge? Well, this is a geometric series. Our common ratio here is less than one. So this is going to converge. This is going to converge. And because this one converges, by the limit comparison test, our original series S converges. Converges. | Worked example limit comparison test Series AP Calculus BC Khan Academy.mp3 |
So let's start with a little bit of a geometric or trigonometric construction that I have here. So this white circle, this is a unit circle. Let me label it as such. So it has radius one, unit circle. So what does the length of this salmon colored line represent? Well, the height of this line would be the y coordinate ... | Limit of sin(x) x as x approaches 0 Derivative rules AP Calculus AB Khan Academy (2).mp3 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.