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So you add those two, you get minus 4, 12 minus 16 is minus 4, x to the sixth, all of that over x to the fourth plus 27 squared. And that is our second derivative. Now, we've done all of the derivatives, and this was actually a pretty hairy problem, and now we can solve for when the first and the second derivatives equ... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
So first let's see where our first derivative is equal to 0 and get our critical points, or at least maybe, also maybe where it's undefined. So this is equal to 0, if we want to set, the only place that this can equal to 0 is if this numerator is equal to 0. This denominator actually, if we are assuming we're dealing w... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
So there's no undefined critical points here, but we can set the numerator equal to 0 pretty easily. If we wanted to set this equal to 0, we just say 4x to the third is equal to 0, and we know what x value will make that equal to 0, x has to be equal to 0. 4 times something to the third is equal to 0, that something ha... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
So we can write f prime of 0 is equal to 0. So 0 is a critical point. The slope at 0 is 0, we don't know if it's a maximum or a minimum or an inflection point yet, or it could be, you know, we'll explore it a little bit more. And actually just so we get the coordinate, what's the coordinate? The coordinate x is 0, and ... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
And actually just so we get the coordinate, what's the coordinate? The coordinate x is 0, and then y is the natural log of x is 0, this just turns out, so it's the natural log of 27. So it's the natural log of, let me figure out what that is and get the calculator out. I said I wouldn't use a graphing calculator, but I... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
I said I wouldn't use a graphing calculator, but I could use a regular calculator. 27, if I were to take the natural log of that, it's like 3 point, well for our purpose this is called 3.3, we're just trying to get the general shape of the graph. So 3.3, 3 point, well we could just say 2.9 and it kept going. So this is... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
So this is a critical point right here, the slope is 0 here. Slope is equal to 0 at x is equal to 0. So this is one thing we want to block off. And let's see if we can find any candidate inflection points. And remember, candidate inflection points are where the second derivative equals 0. Now if the second derivative e... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
And let's see if we can find any candidate inflection points. And remember, candidate inflection points are where the second derivative equals 0. Now if the second derivative equals 0, that doesn't tell us that those are definitely inflection points. Let me make this very clear. If x is inflection, then the second deri... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
Let me make this very clear. If x is inflection, then the second derivative at x is going to be equal to 0. Because you're having a change in the concavity, I always say con-ca-ti-vity, but it's the con-cavity. You have a change in the slope goes from either increasing to decreasing or from decreasing to increasing. Bu... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
You have a change in the slope goes from either increasing to decreasing or from decreasing to increasing. But if the derivative is equal to 0, you cannot assume that it's an inflection point. So what we're going to do is, we're going to find all of the points at which this is true, and then see if we actually do have ... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
And only if you have a sign change, then you can say it's an inflection point. So let's see if we can do that. So just because the second derivative is 0, that by itself does not tell you it's an inflection point. It has to have a second derivative of 0, and when you go below that x, the second derivative has to actual... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
It has to have a second derivative of 0, and when you go below that x, the second derivative has to actually change signs. Only then. So we can say if f' changes signs around x, then we can say that x is an inflection. And if it's changing signs around x, then it's definitely going to be 0 right at x. But you have to a... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
And if it's changing signs around x, then it's definitely going to be 0 right at x. But you have to actually see that if it's negative before x, it has to be positive after x, or if it's positive before x, it has to be negative after x. So let's test that out. So the first thing we need to do is find these candidate po... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
So the first thing we need to do is find these candidate points. Remember, the candidate points are where the second derivative is equal to 0. We're going to find those points and then see if this is true, that the sign actually changes. So we want to find where this thing over here is equal to 0. And once again, for t... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
So we want to find where this thing over here is equal to 0. And once again, for this to be equal to 0, the numerator has to be equal to 0. This denominator can never be equal to 0 if we're dealing with real numbers, which I think is a fair assumption. So let's see where our numerator can be equal to 0 for the second d... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
So let's see where our numerator can be equal to 0 for the second derivative. So let's set the numerator of the second derivative. 27 times 12 x squared minus 4x to the 6th is equal to 0. Remember, that's just the numerator of our second derivative. Any x that makes the numerator 0 is making the second derivative 0. So... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
Remember, that's just the numerator of our second derivative. Any x that makes the numerator 0 is making the second derivative 0. So let's factor out a 4x squared. Now we'll have 27 times, if we factor a 4 out of the 12, we'll just get a 3, and we factored out the x squared, minus, we factored out the 4, we factored ou... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
Now we'll have 27 times, if we factor a 4 out of the 12, we'll just get a 3, and we factored out the x squared, minus, we factored out the 4, we factored out an x squared, so we have x to the 4th is equal to 0. So the x's that will make this equal to 0 will satisfy either 4x squared is equal to 0, or, now 27 times 3, I... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
Any x that satisfies either of these will make this entire expression equal to 0, so if this thing is 0, the whole thing is going to be equal to 0. If this thing is 0, the whole thing is going to be equal to 0. Let me be clear, this is 81 right there. So let's solve this. This is going to be 0 when x is equal to 0 itse... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
So let's solve this. This is going to be 0 when x is equal to 0 itself. This is going to be equal to 0 when, let's see, if we add x to the 4th to both sides, you get x to the 4th is equal to 81. If we take the square root of both sides of this, you get x squared is equal to 9, or so you get x is plus or minus 3. x is e... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
If we take the square root of both sides of this, you get x squared is equal to 9, or so you get x is plus or minus 3. x is equal to plus or minus 3. These are our candidate inflection points. x is equal to 0, x is equal to plus 3, or x is equal to minus 3. What we have to do now is to see whether the second derivative... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
What we have to do now is to see whether the second derivative changes signs around these points in order to be able to label them inflection points. What happens when x is slightly below 0? Let's take the situation, let's do all the scenarios. What happens when x is slightly below 0? Not all of them necessarily, but i... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
What happens when x is slightly below 0? Not all of them necessarily, but if x is like 0.1, what is the second derivative going to be doing? If x is minus 0.1, this term right here is going to be positive, and then this is going to be 81 minus 0.1 to the 4th. That's going to be a very small number. It's going to be som... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
That's going to be a very small number. It's going to be some positive number times 81 minus a small number, so it's going to be a positive number. When x is less than 0, or just slightly less than 0, our second derivative is positive. What happens when x is slightly larger? When I write this notation, I want to be car... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
What happens when x is slightly larger? When I write this notation, I want to be careful. I mean really just right below 0. When x is right above 0, what happens? Let's say x was 0.01, or 0.1, positive 0.1. It's going to be the same thing, because in both cases we're squaring and we're taking the 4th, so you're kind of... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
When x is right above 0, what happens? Let's say x was 0.01, or 0.1, positive 0.1. It's going to be the same thing, because in both cases we're squaring and we're taking the 4th, so you're kind of losing your sign information. If x is 0.1, this thing is going to be a small positive number. You're going to be subtractin... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
If x is 0.1, this thing is going to be a small positive number. You're going to be subtracting a very small positive number from 81, but 81 minus a small number is still going to be positive. You're going to have a positive times a positive, so your second derivative is still going to be greater than 0. Something inter... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
Something interesting here. Your second derivative is 0 when x is equal to 0, but it is not an inflection point, because notice the concavity did not change around 0. Our second derivative is positive as we approach 0 from the left, and it's positive as we approach 0 from the right. In general, at 0, as we're near 0 fr... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
In general, at 0, as we're near 0 from either direction, we're going to be concave upwards. The fact that 0 is a critical point, and that we're always concave upward as we approach 0 from either side, this tells us that this is a minimum point. Because we're concave upwards all around 0. 0 is not an inflection point. L... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
0 is not an inflection point. Let's see if positive and negative 3 are inflection points. If you study this equation, let me write our... Actually, I just want to be clear. I've just been using the numerator of the second derivative. The whole second derivative is this thing right here, but I've been ignoring the denom... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
I've just been using the numerator of the second derivative. The whole second derivative is this thing right here, but I've been ignoring the denominator, because the denominator is always positive. If we're trying to understand whether things are positive or negative, we just really have to determine whether the numer... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
It's something to the second power. Let's test whether we have a change in concavity around x is equal to positive or negative 3. Remember, the numerator of our... Let me just rewrite our second derivative, just so you see it here. f prime prime of x. The numerator is this thing right here. It's 4x squared times 81 min... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
f prime prime of x. The numerator is this thing right here. It's 4x squared times 81 minus x to the fourth. The denominator was up here, x to the fourth plus 27 squared. x to the fourth plus 27 squared. That was our second derivative. Let's see if this changes signs around positive or negative 3. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
The denominator was up here, x to the fourth plus 27 squared. x to the fourth plus 27 squared. That was our second derivative. Let's see if this changes signs around positive or negative 3. Actually, we should get the same answer, because regardless of whether we put positive or negative 3 here, you lose all your sign ... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
Let's see if this changes signs around positive or negative 3. Actually, we should get the same answer, because regardless of whether we put positive or negative 3 here, you lose all your sign information, because you're taking it to the fourth power. You're taking it to the second power. Obviously, anything to the fou... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
Obviously, anything to the fourth power is always going to be positive. Anything to the second power is always going to be negative. When we do our test, if it's true for positive 3, it's probably going to be true for negative 3 as well. Let's just try it out. When x is just a little bit less than positive 3, what's th... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
Let's just try it out. When x is just a little bit less than positive 3, what's the sign of f prime prime of x? It's going to be 4 times 9. It's going to be 4 times a positive number. It might be like 2.999, but this is still going to be positive. This is going to be positive when x is approaching 3. Then this is going... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
It's going to be 4 times a positive number. It might be like 2.999, but this is still going to be positive. This is going to be positive when x is approaching 3. Then this is going to be, well, if x is 3, this is 0. If x is a little bit less than 3, if it's like 2.9999, this number is going to be less than 81. This is ... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
Then this is going to be, well, if x is 3, this is 0. If x is a little bit less than 3, if it's like 2.9999, this number is going to be less than 81. This is also going to be positive. Of course, the denominator is always positive. As x is less than 3, as it's approaching from the left, we are concave upwards. This thi... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
Of course, the denominator is always positive. As x is less than 3, as it's approaching from the left, we are concave upwards. This thing is going to be a positive. f prime prime is greater than 0. We are upwards, concave upwards. When x is just larger than 3, what's going to happen? This first term is still going to b... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
f prime prime is greater than 0. We are upwards, concave upwards. When x is just larger than 3, what's going to happen? This first term is still going to be positive, but if x is just larger than 3, x to the fourth is going to be just larger than 81. This second term is going to be negative in that situation. It's goin... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
This first term is still going to be positive, but if x is just larger than 3, x to the fourth is going to be just larger than 81. This second term is going to be negative in that situation. It's going to be negative. Let me do it in a new color. It's going to be negative when x is larger than 3 because this is going t... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
Let me do it in a new color. It's going to be negative when x is larger than 3 because this is going to be larger than 81. If this is negative and this is positive, then the whole thing is going to be negative because this denominator is still going to be positive. Then f prime prime is going to be less than 0. We're g... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
Then f prime prime is going to be less than 0. We're going to be concave downwards. One last one. What happens when x is just greater than minus 3? Just being greater than minus 3, that's like minus 2.9999. When you take minus 2.99 and square it, you're going to get a positive number. This is going to be positive. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
What happens when x is just greater than minus 3? Just being greater than minus 3, that's like minus 2.9999. When you take minus 2.99 and square it, you're going to get a positive number. This is going to be positive. If you take minus 2.99 to the fourth, that's going to be a little bit less than 81 because 2.99 to the... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
This is going to be positive. If you take minus 2.99 to the fourth, that's going to be a little bit less than 81 because 2.99 to the fourth is a little bit less than 81. This is still going to be positive. You have positive times a positive divided by a positive. You're going to be concave upwards because your second d... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
You have positive times a positive divided by a positive. You're going to be concave upwards because your second derivative is going to be greater than 0. Concave upwards. Finally, when x is just less than negative 3, remember when I write this, I don't mean for all x's larger than negative 3 or all x's smaller than ne... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
Finally, when x is just less than negative 3, remember when I write this, I don't mean for all x's larger than negative 3 or all x's smaller than negative 3. There's actually no notation that would say just as we just approach 3, in this case, from the left. What happens if we just go to minus 3.11 or 3.01? This term r... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
This term right here is going to be positive. If we take minus 3.1 to the fourth, that's going to be larger than positive 81. The sign will become positive. It will be larger than 81, so this will become negative. In that case as well, we'll have a positive times a negative divided by a positive. Then our second deriva... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
It will be larger than 81, so this will become negative. In that case as well, we'll have a positive times a negative divided by a positive. Then our second derivative is going to be negative. We're going to be downwards. I think we're ready to plot. First of all, is x plus or minus 3 inflection points? Sure. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
We're going to be downwards. I think we're ready to plot. First of all, is x plus or minus 3 inflection points? Sure. As we approach x is equal to 3 from the left, we are concave upwards. Then as we cross 3, the second derivative is 0. The second derivative is 0. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
Sure. As we approach x is equal to 3 from the left, we are concave upwards. Then as we cross 3, the second derivative is 0. The second derivative is 0. I lost it up here. The second derivative is 0. Then as we go to the right of 3, we become concave downwards. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
The second derivative is 0. I lost it up here. The second derivative is 0. Then as we go to the right of 3, we become concave downwards. We got our sign change in the second derivative. x is equal to 3. This 3 is definitely an inflection point. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
Then as we go to the right of 3, we become concave downwards. We got our sign change in the second derivative. x is equal to 3. This 3 is definitely an inflection point. The same argument can be made for negative 3. We switch signs as we cross 3. These definitely are inflection points. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
This 3 is definitely an inflection point. The same argument can be made for negative 3. We switch signs as we cross 3. These definitely are inflection points. Just so we get the exact coordinates, let's figure out what f of 3 is, or f of positive and negative 3. Then we're ready to graph. First of all, we know that the... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
These definitely are inflection points. Just so we get the exact coordinates, let's figure out what f of 3 is, or f of positive and negative 3. Then we're ready to graph. First of all, we know that the point 0,3.29, that this was a minimum. Because 0 is a critical point, the slope is 0 there. Because it's concave upwar... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
First of all, we know that the point 0,3.29, that this was a minimum. Because 0 is a critical point, the slope is 0 there. Because it's concave upwards, all around 0. So 0 is definitely not an inflection point. Then we know that the points positive 3 and minus 3 are inflection points. In order to figure out their y coo... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
So 0 is definitely not an inflection point. Then we know that the points positive 3 and minus 3 are inflection points. In order to figure out their y coordinates, we can just evaluate them. They're actually going to have the same y coordinates. Because if you put a minus 3 or positive 3 and take it to the fourth power,... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
They're actually going to have the same y coordinates. Because if you put a minus 3 or positive 3 and take it to the fourth power, you're going to get the same thing. Let's figure out what they are. If we take 3 to the fourth power, that's 81. 81 plus 27 is equal to 108. Then we want to take the natural log of it. We w... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
If we take 3 to the fourth power, that's 81. 81 plus 27 is equal to 108. Then we want to take the natural log of it. We want to take the natural log. It's like 4.7, just to get a rough idea. That's true of whether we do positive or negative 3, because we took to the fourth power. So it's 4.7. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
We want to take the natural log. It's like 4.7, just to get a rough idea. That's true of whether we do positive or negative 3, because we took to the fourth power. So it's 4.7. These are both inflection points. We should be ready to graph it. Let's graph it. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
So it's 4.7. These are both inflection points. We should be ready to graph it. Let's graph it. Let me draw my axis. Just like that. This is my y-axis. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
Let's graph it. Let me draw my axis. Just like that. This is my y-axis. This is my x-axis. This is y. You can even call it the f of x axis, if you like. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
This is my y-axis. This is my x-axis. This is y. You can even call it the f of x axis, if you like. This is x. The point 0, 3.29. Let's say this is 1, 2, 3, 4, 5. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
You can even call it the f of x axis, if you like. This is x. The point 0, 3.29. Let's say this is 1, 2, 3, 4, 5. The point 0, 3.29. That's 0, 1, 2, 3. A little bit above 3 is right there. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
Let's say this is 1, 2, 3, 4, 5. The point 0, 3.29. That's 0, 1, 2, 3. A little bit above 3 is right there. That's a minimum point. Then we're concave. The slope is 0 right there. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
A little bit above 3 is right there. That's a minimum point. Then we're concave. The slope is 0 right there. We figured that out, because the first derivative was 0 there. It's a critical point. It's concave upwards around there. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
The slope is 0 right there. We figured that out, because the first derivative was 0 there. It's a critical point. It's concave upwards around there. That told us we're at a minimum point right there. Then at positive 3. 1, 2, 3. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
It's concave upwards around there. That told us we're at a minimum point right there. Then at positive 3. 1, 2, 3. At positive 3, 4.7. 4.7 will look something like that. We have an inflection point. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
1, 2, 3. At positive 3, 4.7. 4.7 will look something like that. We have an inflection point. Before that, we're concave upwards. Then after that, we're concave downwards. It looks something like this. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
We have an inflection point. Before that, we're concave upwards. Then after that, we're concave downwards. It looks something like this. We're concave upwards up to that point. Maybe, actually, you should let me ignore that yellow thing I drew before. Let me get rid of that. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
It looks something like this. We're concave upwards up to that point. Maybe, actually, you should let me ignore that yellow thing I drew before. Let me get rid of that. Let me draw it like this. 1, 2, 3. 3, 4.7 looks like that. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
Let me get rid of that. Let me draw it like this. 1, 2, 3. 3, 4.7 looks like that. Minus 3, 4.7. 1, 2, 3. 4.7 looks like that. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
3, 4.7 looks like that. Minus 3, 4.7. 1, 2, 3. 4.7 looks like that. We know at 0, we have a slope of 0. We're concave upwards. We look like this. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
4.7 looks like that. We know at 0, we have a slope of 0. We're concave upwards. We look like this. We're concave upwards until x is equal to 3. Then at x equals 3, we become concave downwards. We become concave downwards. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
We look like this. We're concave upwards until x is equal to 3. Then at x equals 3, we become concave downwards. We become concave downwards. Let me try my best to draw it well. We go off like that. Then we're concave upwards around 0. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
We become concave downwards. Let me try my best to draw it well. We go off like that. Then we're concave upwards around 0. Until we get, we're concave upwards as long as x is greater than minus 3. Then at minus 3, we become concave downwards again. Maybe, I should do it in that color. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
Then we're concave upwards around 0. Until we get, we're concave upwards as long as x is greater than minus 3. Then at minus 3, we become concave downwards again. Maybe, I should do it in that color. This concave downwards right here, that's this right here. That's that right there. This concave downwards right here. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
Maybe, I should do it in that color. This concave downwards right here, that's this right here. That's that right there. This concave downwards right here. Sorry, I meant to do it in that red color. This concave downwards right here is this right there. Then the concave upwards around 0 is right there. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
This concave downwards right here. Sorry, I meant to do it in that red color. This concave downwards right here is this right there. Then the concave upwards around 0 is right there. You could even imagine this concave upwards that we measured. That's this concave upwards. This concave upwards is that. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
Then the concave upwards around 0 is right there. You could even imagine this concave upwards that we measured. That's this concave upwards. This concave upwards is that. Then around 0, we're always upwards. This is my sense of what the graph will look like. Maybe, it turns into, you could think about what it does. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
This concave upwards is that. Then around 0, we're always upwards. This is my sense of what the graph will look like. Maybe, it turns into, you could think about what it does. As x approaches positive or negative infinity, some of the terms. I won't go into that. Let's test whether we've graphed it correctly using a gr... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
Maybe, it turns into, you could think about what it does. As x approaches positive or negative infinity, some of the terms. I won't go into that. Let's test whether we've graphed it correctly using a graphing calculator. Let me get out my trusty TI-85 and let's graph this sucker. Press graph, y equals, it was the natur... | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
Let's test whether we've graphed it correctly using a graphing calculator. Let me get out my trusty TI-85 and let's graph this sucker. Press graph, y equals, it was the natural log of x to the fourth plus 27. I want to get that graph there. I do second graph. Let's cross our fingers. It looks pretty good. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
I want to get that graph there. I do second graph. Let's cross our fingers. It looks pretty good. It looks almost exactly like what we drew. I think our mathematics was correct. This was actually very satisfying. | Another example graphing with derivatives Differential Calculus Khan Academy.mp3 |
So the first thing, let's see if we can take the antiderivative of nine sine of x. So we could use some of our integration properties to simplify this a little bit. So this is going to be equal to, this is the same thing as nine times the integral from 11 pi over two to six pi of sine of x dx. And what's the antideriva... | Definite integral of trig function AP Calculus AB Khan Academy.mp3 |
And what's the antiderivative of sine of x? Well we know from our derivatives that the derivative with respect to x of cosine of x is equal to negative sine of x. Negative sine of x. So can we construct this in some way so this is a negative sine of x? Well what if I multiplied on the inside, what if I multiplied it by... | Definite integral of trig function AP Calculus AB Khan Academy.mp3 |
So can we construct this in some way so this is a negative sine of x? Well what if I multiplied on the inside, what if I multiplied it by negative one? Well I can't just multiply it only one place by negative one I need to multiply by negative one twice so I'm not changing its value. So what if I said negative nine tim... | Definite integral of trig function AP Calculus AB Khan Academy.mp3 |
So what if I said negative nine times negative sine of x? Well this is still gonna be nine sine of x. If you took negative nine times negative sine of x it is nine sine of x. And I did it this way because now negative sine of x it matches the derivative of cosine of x. So we could say that this is all going to be equal... | Definite integral of trig function AP Calculus AB Khan Academy.mp3 |
And I did it this way because now negative sine of x it matches the derivative of cosine of x. So we could say that this is all going to be equal to, it's all going to be equal to, you have your negative nine out front, negative nine times, times, and I'll put it in brackets, negative nine times the antiderivative of n... | Definite integral of trig function AP Calculus AB Khan Academy.mp3 |
And we're going to evaluate it at its bounds. We're going to evaluate it at six pi. Let me do that in a color I haven't used yet. We're gonna do that at six pi. And we're also going to do that at 11 pi over two. 11 pi over two. And so this is going to be equal to, this is equal to negative nine times, I'm gonna create ... | Definite integral of trig function AP Calculus AB Khan Academy.mp3 |
We're gonna do that at six pi. And we're also going to do that at 11 pi over two. 11 pi over two. And so this is going to be equal to, this is equal to negative nine times, I'm gonna create some space here. So, actually that's probably more space than I need. It's going to be cosine of six pi. Cosine of six, six pi. | Definite integral of trig function AP Calculus AB Khan Academy.mp3 |
And so this is going to be equal to, this is equal to negative nine times, I'm gonna create some space here. So, actually that's probably more space than I need. It's going to be cosine of six pi. Cosine of six, six pi. Cosine of six pi minus, minus cosine of 11 pi over two. Cosine of 11 pi over two. Well what is cosin... | Definite integral of trig function AP Calculus AB Khan Academy.mp3 |
Cosine of six, six pi. Cosine of six pi minus, minus cosine of 11 pi over two. Cosine of 11 pi over two. Well what is cosine of six pi going to be? Well, cosine of any multiple of two pi is going to be equal to one. You could use six pi as we're going around the unit circle three times. So, this is the same thing as co... | Definite integral of trig function AP Calculus AB Khan Academy.mp3 |
Well what is cosine of six pi going to be? Well, cosine of any multiple of two pi is going to be equal to one. You could use six pi as we're going around the unit circle three times. So, this is the same thing as cosine of two pi or the same thing as cosine of zero. So that is going to be equal to one. If that seems un... | Definite integral of trig function AP Calculus AB Khan Academy.mp3 |
So, this is the same thing as cosine of two pi or the same thing as cosine of zero. So that is going to be equal to one. If that seems unfamiliar to you, I encourage you to review the unit circle definition of cosine. And what is cosine of 11 pi over two? Let's see, let's subtract some, let's subtract some multiple of ... | Definite integral of trig function AP Calculus AB Khan Academy.mp3 |
And what is cosine of 11 pi over two? Let's see, let's subtract some, let's subtract some multiple of two pi here to put it in values that we can understand better. So this is, so let me write it here. Cosine of 11 pi over two. That is the same thing as, let's see, if we were to subtract, this is the same thing as cosi... | Definite integral of trig function AP Calculus AB Khan Academy.mp3 |
Cosine of 11 pi over two. That is the same thing as, let's see, if we were to subtract, this is the same thing as cosine of 11 pi over two minus, let's see, this is the same thing as five and 1 1⁄2 pi. Right? Yeah, so this is, so we could view this as, we could subtract, let's subtract four pi, which is going to be, we... | Definite integral of trig function AP Calculus AB Khan Academy.mp3 |
Yeah, so this is, so we could view this as, we could subtract, let's subtract four pi, which is going to be, we could write that as eight pi over two. In fact, no, let's subtract five. Let's subtract, no, let's subtract four pi, which is eight pi over two. So once again, I'm just subtracting a multiple of two pi, which... | Definite integral of trig function AP Calculus AB Khan Academy.mp3 |
So once again, I'm just subtracting a multiple of two pi, which isn't gonna change the value of cosine. And so this is going to be equal to cosine of three pi over two. And if we imagine the unit circle, let me draw the unit circle here. So it's my y-axis, my x-axis, and then I have the unit circle. So, whoops, all rig... | Definite integral of trig function AP Calculus AB Khan Academy.mp3 |
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