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Let me draw one of those rectangles where you could view. And there's different ways to do this, but this is just a review. Where you could review, that's maybe one of the rectangles. The area of the rectangle is just base times height, right? We're going to make these rectangles really skinny and just sum up an infinite number of them. So we want to make them infinitely small, but let's just call the base of this rectangle dx. And then the height of this rectangle is going to be f of x at that point.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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The area of the rectangle is just base times height, right? We're going to make these rectangles really skinny and just sum up an infinite number of them. So we want to make them infinitely small, but let's just call the base of this rectangle dx. And then the height of this rectangle is going to be f of x at that point. It's going to be f of, if this is x naught or whatever, you could just call it f of x. That's the height of that rectangle. And if we wanted to take the sum of all of these rectangles, there's just going to be a bunch of them.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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And then the height of this rectangle is going to be f of x at that point. It's going to be f of, if this is x naught or whatever, you could just call it f of x. That's the height of that rectangle. And if we wanted to take the sum of all of these rectangles, there's just going to be a bunch of them. One there, one there. Then we'll get the area. And if we have infinite number of these rectangles and they're infinitely skinny, we have exactly the area under that curve.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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And if we wanted to take the sum of all of these rectangles, there's just going to be a bunch of them. One there, one there. Then we'll get the area. And if we have infinite number of these rectangles and they're infinitely skinny, we have exactly the area under that curve. That's the intuition behind the definite integral. And the way we write that, it's the definite integral. We're going to take the sums of these rectangles from x is equal to a to x is equal to b.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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And if we have infinite number of these rectangles and they're infinitely skinny, we have exactly the area under that curve. That's the intuition behind the definite integral. And the way we write that, it's the definite integral. We're going to take the sums of these rectangles from x is equal to a to x is equal to b. And the sum, or the areas that we're summing up, are going to be the height is f of x and the width is d of x. It's going to be f of x times d of x. This is equal to the area under the curve, f of x, y is equal to f of x, from x is equal to a to x is equal to b.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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We're going to take the sums of these rectangles from x is equal to a to x is equal to b. And the sum, or the areas that we're summing up, are going to be the height is f of x and the width is d of x. It's going to be f of x times d of x. This is equal to the area under the curve, f of x, y is equal to f of x, from x is equal to a to x is equal to b. And that's just a little bit of review. But hopefully you'll now see the parallel of how we extend this to taking the volume under a surface. First of all, what is a surface?
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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This is equal to the area under the curve, f of x, y is equal to f of x, from x is equal to a to x is equal to b. And that's just a little bit of review. But hopefully you'll now see the parallel of how we extend this to taking the volume under a surface. First of all, what is a surface? If we're thinking in three dimensions, a surface is going to be a function of x and y. So we can write a surface as, instead of saying that y is a function of x, we can write a surface as z is equal to a function of x and y. So you can kind of view it as the domain.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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First of all, what is a surface? If we're thinking in three dimensions, a surface is going to be a function of x and y. So we can write a surface as, instead of saying that y is a function of x, we can write a surface as z is equal to a function of x and y. So you can kind of view it as the domain. The domain is all of the set of valid things that you can input into a function. So now, before our domain was just, at least for most of what we dealt with, was just the x-axis, or kind of the real number line in the x direction. Now, our domain is the xy-plane.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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So you can kind of view it as the domain. The domain is all of the set of valid things that you can input into a function. So now, before our domain was just, at least for most of what we dealt with, was just the x-axis, or kind of the real number line in the x direction. Now, our domain is the xy-plane. We can give it any x and any y, and we'll just deal with reals right now, but I don't want to get too technical. And then it'll pop out another number, and if we wanted to graph it, it'll be our height. And so that could be the height of a surface.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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Now, our domain is the xy-plane. We can give it any x and any y, and we'll just deal with reals right now, but I don't want to get too technical. And then it'll pop out another number, and if we wanted to graph it, it'll be our height. And so that could be the height of a surface. So let me just show you what a surface looks like, in case you don't remember, and we'll actually figure out the volume under this surface. So this is a surface. I'll tell you its function in a second, but it's pretty neat to look at.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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And so that could be the height of a surface. So let me just show you what a surface looks like, in case you don't remember, and we'll actually figure out the volume under this surface. So this is a surface. I'll tell you its function in a second, but it's pretty neat to look at. But as you can see, it's a surface. It's like a piece of paper that's bent. Let me rotate it to its traditional form.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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I'll tell you its function in a second, but it's pretty neat to look at. But as you can see, it's a surface. It's like a piece of paper that's bent. Let me rotate it to its traditional form. So this is the x direction, this is the y direction, and the height is a function of where we are in the xy-plane. So how do we figure out the volume under a surface like this? How do we figure out the volume?
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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Let me rotate it to its traditional form. So this is the x direction, this is the y direction, and the height is a function of where we are in the xy-plane. So how do we figure out the volume under a surface like this? How do we figure out the volume? It seems like a bit of a stretch, given what we've learned from this. But what if, and I'm just going to draw an abstract surface here, let me draw some axes. Let's say that's my x-axis, that's my y-axis, that's my z-axis.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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How do we figure out the volume? It seems like a bit of a stretch, given what we've learned from this. But what if, and I'm just going to draw an abstract surface here, let me draw some axes. Let's say that's my x-axis, that's my y-axis, that's my z-axis. I practice these videos ahead of time, so I'm often wondering what I'm about to draw. So that's x, that's y, and that's z. And let's say I have some surface.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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Let's say that's my x-axis, that's my y-axis, that's my z-axis. I practice these videos ahead of time, so I'm often wondering what I'm about to draw. So that's x, that's y, and that's z. And let's say I have some surface. I'll just draw something. I don't know what it is. Some surface.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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And let's say I have some surface. I'll just draw something. I don't know what it is. Some surface. This is our surface. z is a function of x and y. So if you give me a coordinate in the xy-plane, say here, I'll put it into the function and it'll give us a z value.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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Some surface. This is our surface. z is a function of x and y. So if you give me a coordinate in the xy-plane, say here, I'll put it into the function and it'll give us a z value. And I'll plot it there and it'll be a point on the surface. So what we want to figure out is the volume under the surface, and we have to specify bounds. From here we said x is equal to a to x is equal to b.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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So if you give me a coordinate in the xy-plane, say here, I'll put it into the function and it'll give us a z value. And I'll plot it there and it'll be a point on the surface. So what we want to figure out is the volume under the surface, and we have to specify bounds. From here we said x is equal to a to x is equal to b. So let's make a square bound first, because this keeps it a lot simpler. So let's say that the region of the x and y region of this part of the surface under which we want to calculate the volume, let's say the shadow, if the sun was right above the surface, the shadow would be right there. Let me try my best to draw this neatly.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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From here we said x is equal to a to x is equal to b. So let's make a square bound first, because this keeps it a lot simpler. So let's say that the region of the x and y region of this part of the surface under which we want to calculate the volume, let's say the shadow, if the sun was right above the surface, the shadow would be right there. Let me try my best to draw this neatly. So this is what we're going to try to figure out the volume of. So if we wanted to draw it in the xy-plane, you can kind of view it as the projection of the surface in the xy-plane or the shadow of the surface in the xy-plane. What are the bounds?
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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Let me try my best to draw this neatly. So this is what we're going to try to figure out the volume of. So if we wanted to draw it in the xy-plane, you can kind of view it as the projection of the surface in the xy-plane or the shadow of the surface in the xy-plane. What are the bounds? You can almost view it as what are the bounds of the domain? Let's say that this right here, that's 0, 0 in the xy-plane. Let's say that this is y is equal to a.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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What are the bounds? You can almost view it as what are the bounds of the domain? Let's say that this right here, that's 0, 0 in the xy-plane. Let's say that this is y is equal to a. That's this line right here, y is equal to a. And let's say that this line right here is x is equal to b. I hope you get that. This is the xy-plane.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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Let's say that this is y is equal to a. That's this line right here, y is equal to a. And let's say that this line right here is x is equal to b. I hope you get that. This is the xy-plane. If we have a constant x, it would be a line like that, a constant y, a line like that, and then we have the area in between it. So how do we figure out the volume under this? Well, if I just wanted to figure out the area of, let's just say, this sliver.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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This is the xy-plane. If we have a constant x, it would be a line like that, a constant y, a line like that, and then we have the area in between it. So how do we figure out the volume under this? Well, if I just wanted to figure out the area of, let's just say, this sliver. Let's say we had a constant y. So let's say I had some sliver. I don't want to confuse you.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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Well, if I just wanted to figure out the area of, let's just say, this sliver. Let's say we had a constant y. So let's say I had some sliver. I don't want to confuse you. Let's say that I have some constant y. I just want to give you the intuition. Let's say, I don't know what that is, it's an arbitrary y. But for some constant y, what if I could just figure out the area under the curve there?
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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I don't want to confuse you. Let's say that I have some constant y. I just want to give you the intuition. Let's say, I don't know what that is, it's an arbitrary y. But for some constant y, what if I could just figure out the area under the curve there? How would I figure out just the area under that curve? It'll be a function of which y I pick, right? Because if I pick a y here, it'll be a different area.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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But for some constant y, what if I could just figure out the area under the curve there? How would I figure out just the area under that curve? It'll be a function of which y I pick, right? Because if I pick a y here, it'll be a different area. If I pick a y there, it'll be a different area. But I could view this now as a very similar problem to this one up here. I could have my dx's.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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Because if I pick a y here, it'll be a different area. If I pick a y there, it'll be a different area. But I could view this now as a very similar problem to this one up here. I could have my dx's. Let me pick a vibrant color so you can see it. Let's say that's a dx, right? That's a change in x.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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I could have my dx's. Let me pick a vibrant color so you can see it. Let's say that's a dx, right? That's a change in x. And then the height, this height, is going to be a function of the x I have and the y I've picked. Although I'm assuming to some degree that that's a constant y. So what would be the area of this sheet of paper?
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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That's a change in x. And then the height, this height, is going to be a function of the x I have and the y I've picked. Although I'm assuming to some degree that that's a constant y. So what would be the area of this sheet of paper? It's kind of a constant y. It's a sheet of paper within this volume. You can kind of view it.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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So what would be the area of this sheet of paper? It's kind of a constant y. It's a sheet of paper within this volume. You can kind of view it. Well, we said the height of each of these rectangles is f of xy. That's the height. It depends which x and y we pick down here.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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You can kind of view it. Well, we said the height of each of these rectangles is f of xy. That's the height. It depends which x and y we pick down here. And then its width is going to be dx. And then if we integrated it from x is equal to 0, which was back here, all the way to x is equal to b, what would it look like? It would look like that.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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It depends which x and y we pick down here. And then its width is going to be dx. And then if we integrated it from x is equal to 0, which was back here, all the way to x is equal to b, what would it look like? It would look like that. x is going from 0 to b. Fair enough. And this would actually give us a function of y.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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It would look like that. x is going from 0 to b. Fair enough. And this would actually give us a function of y. This would give us an expression so that if I would know the area of this kind of sliver of the volume for any given value of y. If you give me a y, I can tell you the area of the sliver that corresponds to that y. Now what could I do?
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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And this would actually give us a function of y. This would give us an expression so that if I would know the area of this kind of sliver of the volume for any given value of y. If you give me a y, I can tell you the area of the sliver that corresponds to that y. Now what could I do? If I know the area of any given sliver, what if I multiplied the area of that sliver times dy? This is a dy. Let me do it in a vibrant color.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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Now what could I do? If I know the area of any given sliver, what if I multiplied the area of that sliver times dy? This is a dy. Let me do it in a vibrant color. So dy, a very small change in y. If I multiply this area times a small dy, then all of a sudden I have a sliver of volume. Hopefully that makes some sense.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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Let me do it in a vibrant color. So dy, a very small change in y. If I multiply this area times a small dy, then all of a sudden I have a sliver of volume. Hopefully that makes some sense. I'm making that little cut that I took the area of, I'm making it three dimensional. So what would be the volume of that sliver? The volume of that sliver will be this function of y times dy or this whole thing times dy.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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Hopefully that makes some sense. I'm making that little cut that I took the area of, I'm making it three dimensional. So what would be the volume of that sliver? The volume of that sliver will be this function of y times dy or this whole thing times dy. So it would be the integral from 0 to b of f of x, y dx. That gives us the area of this blue sheet. Now if I multiply this whole thing times dy, I get this volume.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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The volume of that sliver will be this function of y times dy or this whole thing times dy. So it would be the integral from 0 to b of f of x, y dx. That gives us the area of this blue sheet. Now if I multiply this whole thing times dy, I get this volume. It gets some depth. This little area that I'm shading right here gives depth of that sheet. Now if I added all of those sheets that now have depth, if I took the infinite sum, so if I took the integral of this from my lower y bound, from 0 to my upper y bound, then, at least based on our intuition here, maybe I will have figured out the volume under this surface.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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Now if I multiply this whole thing times dy, I get this volume. It gets some depth. This little area that I'm shading right here gives depth of that sheet. Now if I added all of those sheets that now have depth, if I took the infinite sum, so if I took the integral of this from my lower y bound, from 0 to my upper y bound, then, at least based on our intuition here, maybe I will have figured out the volume under this surface. But anyway, I didn't want to confuse you, but that's the intuition of what we're going to do. I think you're going to find out that actually calculating the volumes are pretty straightforward, especially when you have fixed x and y bounds. That's what we're going to do in the next video.
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Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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You produce some little trinket that people enjoy buying. And the main costs that you have are labor, you know, the workers that you have creating these, and steel. And let's just say that your labor costs are $20 per hour, $20 each hour, and then your steel costs are $2,000. $2,000, keep the numbers kind of related to each other, $2,000 for every ton of steel. And then you've had your analysts work a little bit on trying to model the revenues you can make with your widgets as a function of hours of labor and tons of steel. Now let's say the revenue model that they've come up with, the revenue as a function of hours of labor, and then S for steel, let's say, is equal to about 100 times the hours of labor to the power 2 3rds multiplied by the tons of steel to the power 1 3rd. If you put in a given amount of labor and a given amount of steel, this is about how much money you're gonna expect to earn.
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Lagrange multiplier example, part 1.mp3
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$2,000, keep the numbers kind of related to each other, $2,000 for every ton of steel. And then you've had your analysts work a little bit on trying to model the revenues you can make with your widgets as a function of hours of labor and tons of steel. Now let's say the revenue model that they've come up with, the revenue as a function of hours of labor, and then S for steel, let's say, is equal to about 100 times the hours of labor to the power 2 3rds multiplied by the tons of steel to the power 1 3rd. If you put in a given amount of labor and a given amount of steel, this is about how much money you're gonna expect to earn. And of course you wanna earn as much as you can, but let's say you actually have a budget for how much you're able to spend on all these things, and your budget, the budget is $20,000. You're willing to spend $20,000 and you wanna make as much money as you can according to this model based on that. Now this is exactly the kind of problem that the Lagrange multiplier technique is made for.
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Lagrange multiplier example, part 1.mp3
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If you put in a given amount of labor and a given amount of steel, this is about how much money you're gonna expect to earn. And of course you wanna earn as much as you can, but let's say you actually have a budget for how much you're able to spend on all these things, and your budget, the budget is $20,000. You're willing to spend $20,000 and you wanna make as much money as you can according to this model based on that. Now this is exactly the kind of problem that the Lagrange multiplier technique is made for. We're trying to maximize some kind of function and we have a constraint. Now right now the constraint isn't written as a formula, but we can pretty easily write it as a formula because what makes up our budget? Well, it's gonna be the number of hours of labor multiplied by 20, so that's gonna be $20 per hour multiplied by the number of hours you put in plus $2,000 per tons of steel times the tons of steel that you put in.
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Lagrange multiplier example, part 1.mp3
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Now this is exactly the kind of problem that the Lagrange multiplier technique is made for. We're trying to maximize some kind of function and we have a constraint. Now right now the constraint isn't written as a formula, but we can pretty easily write it as a formula because what makes up our budget? Well, it's gonna be the number of hours of labor multiplied by 20, so that's gonna be $20 per hour multiplied by the number of hours you put in plus $2,000 per tons of steel times the tons of steel that you put in. So the constraint is basically that you have to have these values equal $20,000. I mean, you could say less than, right? You could say you're not willing to go any more than that, but intuitively and in reality, it's gonna be the case that in order to maximize your revenues, you should squeeze every dollar that you have available and actually hit this constraint.
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Lagrange multiplier example, part 1.mp3
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Well, it's gonna be the number of hours of labor multiplied by 20, so that's gonna be $20 per hour multiplied by the number of hours you put in plus $2,000 per tons of steel times the tons of steel that you put in. So the constraint is basically that you have to have these values equal $20,000. I mean, you could say less than, right? You could say you're not willing to go any more than that, but intuitively and in reality, it's gonna be the case that in order to maximize your revenues, you should squeeze every dollar that you have available and actually hit this constraint. So this right here is the constraint of our problem. And let's go ahead and give this guy a name, the function that we're dealing with a name. And I'm gonna call it G of HS, which is gonna be that guy.
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Lagrange multiplier example, part 1.mp3
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You could say you're not willing to go any more than that, but intuitively and in reality, it's gonna be the case that in order to maximize your revenues, you should squeeze every dollar that you have available and actually hit this constraint. So this right here is the constraint of our problem. And let's go ahead and give this guy a name, the function that we're dealing with a name. And I'm gonna call it G of HS, which is gonna be that guy. And now if you'll remember in the last few videos, the way we visualize something like this is to think about the set of all possible inputs. So in this case, you might be thinking about the HS plane, you know, the number of hours of labor on one axis, the number of tons of steel on another. And this constraint, well, in this case, it's a linear function.
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Lagrange multiplier example, part 1.mp3
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And I'm gonna call it G of HS, which is gonna be that guy. And now if you'll remember in the last few videos, the way we visualize something like this is to think about the set of all possible inputs. So in this case, you might be thinking about the HS plane, you know, the number of hours of labor on one axis, the number of tons of steel on another. And this constraint, well, in this case, it's a linear function. So this constraint is gonna give us some kind of line that tells us which pairs of S and H are gonna achieve that constraint. And then the revenue function that we're dealing with will have certain contours. You know, maybe revenues of $10,000 have a certain contour that looks like this, and revenues of $100,000 have a certain contour that looks like this.
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Lagrange multiplier example, part 1.mp3
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And this constraint, well, in this case, it's a linear function. So this constraint is gonna give us some kind of line that tells us which pairs of S and H are gonna achieve that constraint. And then the revenue function that we're dealing with will have certain contours. You know, maybe revenues of $10,000 have a certain contour that looks like this, and revenues of $100,000 have a certain contour that looks like this. But what we want is to find which value is barely touching the constraint curve, just tangent to it at a given point, because that's gonna be the contour line where if you up the value by just a little bit, it would no longer intersect with that curve. There would no longer be values of H and S that satisfy this constraint. And the way to think about finding that tangency is to consider the vector perpendicular to the tangent line to the curve at that point, which fortunately is represented by, let's see, let me make some room for myself here, represented by the gradient of our R function, the function whose contours this is, the revenue.
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Lagrange multiplier example, part 1.mp3
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You know, maybe revenues of $10,000 have a certain contour that looks like this, and revenues of $100,000 have a certain contour that looks like this. But what we want is to find which value is barely touching the constraint curve, just tangent to it at a given point, because that's gonna be the contour line where if you up the value by just a little bit, it would no longer intersect with that curve. There would no longer be values of H and S that satisfy this constraint. And the way to think about finding that tangency is to consider the vector perpendicular to the tangent line to the curve at that point, which fortunately is represented by, let's see, let me make some room for myself here, represented by the gradient of our R function, the function whose contours this is, the revenue. And what it means for this to be tangent to the constraint line is that there's gonna be another vector, the gradient of G, of our constraint function, that points in the same direction, that's proportional to that. And typically the way you write this is to say that the gradient of this function is proportional to the gradient of G, and this proportionality constant is called our Lagrange multiplier. It's called the Lagrange multiplier.
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Lagrange multiplier example, part 1.mp3
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And the way to think about finding that tangency is to consider the vector perpendicular to the tangent line to the curve at that point, which fortunately is represented by, let's see, let me make some room for myself here, represented by the gradient of our R function, the function whose contours this is, the revenue. And what it means for this to be tangent to the constraint line is that there's gonna be another vector, the gradient of G, of our constraint function, that points in the same direction, that's proportional to that. And typically the way you write this is to say that the gradient of this function is proportional to the gradient of G, and this proportionality constant is called our Lagrange multiplier. It's called the Lagrange multiplier. So let's go ahead and start working it out. Let's first compute the gradient of R. So the gradient of R is gonna be the partial derivative of R with respect to its first variable, which is H. So partial derivative with respect to H. And the second component is its partial derivative with respect to that second variable, S, with respect to S. And in this case, that first partial derivative, if we treat H as a variable and S as a constant, then that 2 3rds gets brought down. So that'll be 100 times 2 3rds times H, H to the power of, well, we've gotta subtract one from 2 3rds when we bring it down, so that'll be negative 1 3rd, multiplied by S to the 1 3rd.
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Lagrange multiplier example, part 1.mp3
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It's called the Lagrange multiplier. So let's go ahead and start working it out. Let's first compute the gradient of R. So the gradient of R is gonna be the partial derivative of R with respect to its first variable, which is H. So partial derivative with respect to H. And the second component is its partial derivative with respect to that second variable, S, with respect to S. And in this case, that first partial derivative, if we treat H as a variable and S as a constant, then that 2 3rds gets brought down. So that'll be 100 times 2 3rds times H, H to the power of, well, we've gotta subtract one from 2 3rds when we bring it down, so that'll be negative 1 3rd, multiplied by S to the 1 3rd. And then the second component here, the partial derivative with respect to S, is gonna be 100 times, well, now by treating S as the variable, we take down that 1 3rd, so that's 1 3rd, H to the 2 3rds just looks like a constant as far as S is concerned, and then we take S to the 1 3rd minus one, which is negative 2 3rds, negative 2 3rds. Great, so that's the gradient of R, and now we need the gradient of G, and that one's a lot easier, actually, because G is just a linear function. So when we take the gradient of G, which is its partial derivative with respect to H, partial H, and its partial derivative with respect to S, partial S, well, the partial with respect to H is just 20.
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Lagrange multiplier example, part 1.mp3
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So that'll be 100 times 2 3rds times H, H to the power of, well, we've gotta subtract one from 2 3rds when we bring it down, so that'll be negative 1 3rd, multiplied by S to the 1 3rd. And then the second component here, the partial derivative with respect to S, is gonna be 100 times, well, now by treating S as the variable, we take down that 1 3rd, so that's 1 3rd, H to the 2 3rds just looks like a constant as far as S is concerned, and then we take S to the 1 3rd minus one, which is negative 2 3rds, negative 2 3rds. Great, so that's the gradient of R, and now we need the gradient of G, and that one's a lot easier, actually, because G is just a linear function. So when we take the gradient of G, which is its partial derivative with respect to H, partial H, and its partial derivative with respect to S, partial S, well, the partial with respect to H is just 20. The function looks like 20 times H plus something that's a constant. So that ends up being 20, and then the partial with respect to S, likewise, it's just 2,000, because it's just some constant multiplied by S plus a bunch of other stuff that looks like constants. So that's great.
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Lagrange multiplier example, part 1.mp3
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So when we take the gradient of G, which is its partial derivative with respect to H, partial H, and its partial derivative with respect to S, partial S, well, the partial with respect to H is just 20. The function looks like 20 times H plus something that's a constant. So that ends up being 20, and then the partial with respect to S, likewise, it's just 2,000, because it's just some constant multiplied by S plus a bunch of other stuff that looks like constants. So that's great. And this means when we set the gradient of R equal to the gradient of G, the pair of equations that we get, and let me just write it all out again, is we have this top one, which I'll call 2 3rds times, and let's go ahead and do a little simplifying while I'm rewriting things here. So H to the 1 3rd is really one over, H to the negative 1 3rd, sorry, is one over H to the 1 3rd, and that's S to the 1 3rd. So all of this, that first component, is being set equal to the first component of the gradient of G, which is 20 times lambda, times this Lagrange multiplier.
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Lagrange multiplier example, part 1.mp3
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So that's great. And this means when we set the gradient of R equal to the gradient of G, the pair of equations that we get, and let me just write it all out again, is we have this top one, which I'll call 2 3rds times, and let's go ahead and do a little simplifying while I'm rewriting things here. So H to the 1 3rd is really one over, H to the negative 1 3rd, sorry, is one over H to the 1 3rd, and that's S to the 1 3rd. So all of this, that first component, is being set equal to the first component of the gradient of G, which is 20 times lambda, times this Lagrange multiplier. Because we're not setting the gradients equal to each other, we're just setting them proportional to each other. So that's the first equation. And then the second one, I'll go ahead and do some simplifying while I rewrite that one also.
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Lagrange multiplier example, part 1.mp3
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So all of this, that first component, is being set equal to the first component of the gradient of G, which is 20 times lambda, times this Lagrange multiplier. Because we're not setting the gradients equal to each other, we're just setting them proportional to each other. So that's the first equation. And then the second one, I'll go ahead and do some simplifying while I rewrite that one also. That's gonna be 100 3rds, and then H to the 2 3rds, so times H to the 2 3rds, divided by S to the 2 3rds, because S to the negative 2 3rds is the same as one over S to the 2 3rds. All of that is equal to 2,000. 2,000 times lambda.
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Lagrange multiplier example, part 1.mp3
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In the last couple videos, I showed how you can take a function, just a function with two inputs, and find the tangent plane to its graph. And the way that you think about this, you first find a point, some kind of input point, which is, I'll just write abstractly as x-naught and y-naught, and then you see where that point ends up on the graph, and you want to find a new function, a new function which we were calling L, and maybe you say L sub f, which also is a function of x and y, and you want the graph of that function to be a plane tangent to the graph. Now this often goes by another name. This will go under the name local linearization. Local linearization. It's kind of a long word. Zation.
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Local linearization.mp3
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This will go under the name local linearization. Local linearization. It's kind of a long word. Zation. And what this basically means, the word local means you're looking at a specific input point. So in this case, it's at a specific input point, x-naught, y-naught. And the idea of a linearization, a linearization means you're approximating the function with something simpler, with something that's actually linear.
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Local linearization.mp3
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Zation. And what this basically means, the word local means you're looking at a specific input point. So in this case, it's at a specific input point, x-naught, y-naught. And the idea of a linearization, a linearization means you're approximating the function with something simpler, with something that's actually linear. And I'll tell you what I mean by linear in just a moment. But the whole idea here is that we don't really care about, you know, tangent planes in an abstract 3D space to some kind of graph. The whole reason for doing this is that this is a really good way to approximate a function, which is potentially a very complicated function, with something that's much easier, something that has constant partial derivatives.
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Local linearization.mp3
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And the idea of a linearization, a linearization means you're approximating the function with something simpler, with something that's actually linear. And I'll tell you what I mean by linear in just a moment. But the whole idea here is that we don't really care about, you know, tangent planes in an abstract 3D space to some kind of graph. The whole reason for doing this is that this is a really good way to approximate a function, which is potentially a very complicated function, with something that's much easier, something that has constant partial derivatives. Now my goal of this video is going to be to show how we write this local linearization here in vector form, because it'll be both more compact and hopefully easier to remember, and also it's more general. It'll apply to things that have more than just two input variables, like this one does. So just to remind us of where we were and what we got to in the last couple videos, I'll write it a little bit more abstractly this time, rather than a specific example.
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Local linearization.mp3
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The whole reason for doing this is that this is a really good way to approximate a function, which is potentially a very complicated function, with something that's much easier, something that has constant partial derivatives. Now my goal of this video is going to be to show how we write this local linearization here in vector form, because it'll be both more compact and hopefully easier to remember, and also it's more general. It'll apply to things that have more than just two input variables, like this one does. So just to remind us of where we were and what we got to in the last couple videos, I'll write it a little bit more abstractly this time, rather than a specific example. The way you do this local linearization is first you find the partial derivative of f with respect to x, which I'll write with the subscript notation. And you evaluate that at x sub o, or x naught, y naught. You evaluate it at the point about which you're approximating.
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Local linearization.mp3
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So just to remind us of where we were and what we got to in the last couple videos, I'll write it a little bit more abstractly this time, rather than a specific example. The way you do this local linearization is first you find the partial derivative of f with respect to x, which I'll write with the subscript notation. And you evaluate that at x sub o, or x naught, y naught. You evaluate it at the point about which you're approximating. And then you multiply that by x minus that constant. So the only variable right here, everything is a constant, but the only variable part is that x. And then we add to that, basically doing the same thing with y.
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Local linearization.mp3
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You evaluate it at the point about which you're approximating. And then you multiply that by x minus that constant. So the only variable right here, everything is a constant, but the only variable part is that x. And then we add to that, basically doing the same thing with y. You take the partial derivative with respect to y, you evaluate it at the input point, the point about which you are linearizing, and then you multiply it by y minus y sub o. And then to this entire thing, because you want to make sure that when you evaluate this function at the input point itself, you see when you plug in x naught and y naught, this term goes to zero, because x naught minus x naught is zero, this term goes to zero, and this is the whole reason we kind of paired up these terms and organized the constants in this way. This way you can just think about adding whatever the function itself evaluates to at that point.
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Local linearization.mp3
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And then we add to that, basically doing the same thing with y. You take the partial derivative with respect to y, you evaluate it at the input point, the point about which you are linearizing, and then you multiply it by y minus y sub o. And then to this entire thing, because you want to make sure that when you evaluate this function at the input point itself, you see when you plug in x naught and y naught, this term goes to zero, because x naught minus x naught is zero, this term goes to zero, and this is the whole reason we kind of paired up these terms and organized the constants in this way. This way you can just think about adding whatever the function itself evaluates to at that point. And this will ensure that your linearization actually equals the function itself at the local point, because hopefully if you're approximating it near a point, then at that point it's actually equal. So what do I mean by this word linear? The word linear has a very precise formulation, especially in the context of linear algebra, and admittedly, this is not actually a linear function in the technical sense.
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Local linearization.mp3
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This way you can just think about adding whatever the function itself evaluates to at that point. And this will ensure that your linearization actually equals the function itself at the local point, because hopefully if you're approximating it near a point, then at that point it's actually equal. So what do I mean by this word linear? The word linear has a very precise formulation, especially in the context of linear algebra, and admittedly, this is not actually a linear function in the technical sense. But loosely what it means, and the reason people call it linear, is that this x term here, this variable term, doesn't have anything fancy going on with it. It's just being multiplied by a constant. And similarly, this y term, it's just being multiplied by a constant.
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Local linearization.mp3
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The word linear has a very precise formulation, especially in the context of linear algebra, and admittedly, this is not actually a linear function in the technical sense. But loosely what it means, and the reason people call it linear, is that this x term here, this variable term, doesn't have anything fancy going on with it. It's just being multiplied by a constant. And similarly, this y term, it's just being multiplied by a constant. It's not squared, there's no square root, it's not in an exponent or anything like that. And although there is a more technical meaning of the word linear, this is all it really means in this context. This is all you need to think about.
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Local linearization.mp3
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And similarly, this y term, it's just being multiplied by a constant. It's not squared, there's no square root, it's not in an exponent or anything like that. And although there is a more technical meaning of the word linear, this is all it really means in this context. This is all you need to think about. Each variable is just multiplied by a constant. Now you might see this in a more complicated form, or what's at first a more complicated form, using vectors. So first of all, let's think about how we would start describing everything going on here with vectors.
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Local linearization.mp3
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This is all you need to think about. Each variable is just multiplied by a constant. Now you might see this in a more complicated form, or what's at first a more complicated form, using vectors. So first of all, let's think about how we would start describing everything going on here with vectors. So the input, rather than talking about the input as being a pair of points, what I want to say is that there's some vector, some vector that has these as its components, and we just want to capture that all, and I want to give that a name. And kind of unfortunately, the name that we give this, it's very common to just call it x, and maybe a bold-faced x, and that would be easier to do typing than it is writing. So I'll just kind of try to emphasize bold-faced x equals this vector, and that's confusing, because x is already one of the input variables that's just a number.
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Local linearization.mp3
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So first of all, let's think about how we would start describing everything going on here with vectors. So the input, rather than talking about the input as being a pair of points, what I want to say is that there's some vector, some vector that has these as its components, and we just want to capture that all, and I want to give that a name. And kind of unfortunately, the name that we give this, it's very common to just call it x, and maybe a bold-faced x, and that would be easier to do typing than it is writing. So I'll just kind of try to emphasize bold-faced x equals this vector, and that's confusing, because x is already one of the input variables that's just a number. But I'll try to emphasize it, just making it bold. You'll see this in writing a lot. x is this input vector.
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Local linearization.mp3
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So I'll just kind of try to emphasize bold-faced x equals this vector, and that's confusing, because x is already one of the input variables that's just a number. But I'll try to emphasize it, just making it bold. You'll see this in writing a lot. x is this input vector. And then similarly, the specified input about which we are approximating, you would call, let's say I'll make it a nice bold-faced x, not. We'll do that not to just kind of indicate that it's a constant of some kind. And what that is, it's a vector containing the two numbers x-not, y-not.
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Local linearization.mp3
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x is this input vector. And then similarly, the specified input about which we are approximating, you would call, let's say I'll make it a nice bold-faced x, not. We'll do that not to just kind of indicate that it's a constant of some kind. And what that is, it's a vector containing the two numbers x-not, y-not. So this is just our starting to write things in a more vectorized way, and the convenience here is that then if you're dealing with a function with three input variables, or four, or 100, you could still just write it as this bold-faced x with the understanding that the vector has a lot more components. So now, let's take a look at these first two terms in our linearization. We can start thinking of this as a dot product, actually.
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Local linearization.mp3
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And what that is, it's a vector containing the two numbers x-not, y-not. So this is just our starting to write things in a more vectorized way, and the convenience here is that then if you're dealing with a function with three input variables, or four, or 100, you could still just write it as this bold-faced x with the understanding that the vector has a lot more components. So now, let's take a look at these first two terms in our linearization. We can start thinking of this as a dot product, actually. So let me first just kind of move this guy out of the way and give ourselves some room. So he's gonna just go up there, just the same guy. And now I want to think about writing this other term here as a dot product.
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Local linearization.mp3
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We can start thinking of this as a dot product, actually. So let me first just kind of move this guy out of the way and give ourselves some room. So he's gonna just go up there, just the same guy. And now I want to think about writing this other term here as a dot product. And what that looks like is we have the two partial derivatives, f sub x and f sub y, indicating the partial derivatives with respect to x and y, and each one of them is evaluated, let's see, I'll do it evaluating at our bold-faced x-not. And then this one is also evaluated at that bold-faced x-not. So really, you're thinking about this as being a vector that contains two different variables, you're just packing it into a single symbol.
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Local linearization.mp3
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And now I want to think about writing this other term here as a dot product. And what that looks like is we have the two partial derivatives, f sub x and f sub y, indicating the partial derivatives with respect to x and y, and each one of them is evaluated, let's see, I'll do it evaluating at our bold-faced x-not. And then this one is also evaluated at that bold-faced x-not. So really, you're thinking about this as being a vector that contains two different variables, you're just packing it into a single symbol. And the dot product here is against, you know, the first component is x minus x-not, so I'd write that as x minus x-not, the number, and then similarly, y minus, let's see, I'll do it in the same color, y-not, the number. But we can write each one of these in a more compact form, where this, the vector that has the partial derivatives, that's the gradient. And if that feels unfamiliar, maybe go back and check out the videos on the gradient, but this whole vector is basically just saying take the gradient and evaluate it at that vector input, you know, x-not.
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Local linearization.mp3
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So really, you're thinking about this as being a vector that contains two different variables, you're just packing it into a single symbol. And the dot product here is against, you know, the first component is x minus x-not, so I'd write that as x minus x-not, the number, and then similarly, y minus, let's see, I'll do it in the same color, y-not, the number. But we can write each one of these in a more compact form, where this, the vector that has the partial derivatives, that's the gradient. And if that feels unfamiliar, maybe go back and check out the videos on the gradient, but this whole vector is basically just saying take the gradient and evaluate it at that vector input, you know, x-not. And then the second component here, that's telling you you've got x and y minus x-not, y-not. So what you're basically doing is taking the, you know, bold-faced input, the variable vector, x, and then you're subtracting off, you know, x-not, where x-not is some kind of constant. So this right here, this is just vector terms where you're thinking of this as being a vector with two components, and this one is a vector with two components.
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Local linearization.mp3
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And if that feels unfamiliar, maybe go back and check out the videos on the gradient, but this whole vector is basically just saying take the gradient and evaluate it at that vector input, you know, x-not. And then the second component here, that's telling you you've got x and y minus x-not, y-not. So what you're basically doing is taking the, you know, bold-faced input, the variable vector, x, and then you're subtracting off, you know, x-not, where x-not is some kind of constant. So this right here, this is just vector terms where you're thinking of this as being a vector with two components, and this one is a vector with two components. But if your function happened to be something more complicated with, you know, 100 input variables, this would be the same thing you write down. You would just understand that when you expand this, there's gonna be 100 different components in the vector. And this is what a linear term looks like in vector terminology, because this dot product is telling you that all of the components of that bold-faced x vector, the, that expands into, you know, not bold-faced x, y, z, whatever else it expands to, all of those are just being multiplied by some kind of constant.
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Local linearization.mp3
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So this right here, this is just vector terms where you're thinking of this as being a vector with two components, and this one is a vector with two components. But if your function happened to be something more complicated with, you know, 100 input variables, this would be the same thing you write down. You would just understand that when you expand this, there's gonna be 100 different components in the vector. And this is what a linear term looks like in vector terminology, because this dot product is telling you that all of the components of that bold-faced x vector, the, that expands into, you know, not bold-faced x, y, z, whatever else it expands to, all of those are just being multiplied by some kind of constant. So we take that whole thing, that's how you simplify the first couple terms here. And then, of course, we just add on the value of the function itself. So you would take that as the linear term.
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Local linearization.mp3
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And this is what a linear term looks like in vector terminology, because this dot product is telling you that all of the components of that bold-faced x vector, the, that expands into, you know, not bold-faced x, y, z, whatever else it expands to, all of those are just being multiplied by some kind of constant. So we take that whole thing, that's how you simplify the first couple terms here. And then, of course, we just add on the value of the function itself. So you would take that as the linear term. And now, I kind of like to add it on to the front, actually, where you think about taking the function itself and evaluating it at that constant input, x-not. Because that way, you can kind of think this is your constant term, this is your constant term, and then the rest of the stuff here is your linear term. The rest of your stuff is your linear.
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Local linearization.mp3
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So you would take that as the linear term. And now, I kind of like to add it on to the front, actually, where you think about taking the function itself and evaluating it at that constant input, x-not. Because that way, you can kind of think this is your constant term, this is your constant term, and then the rest of the stuff here is your linear term. The rest of your stuff is your linear. Because later on, if we start adding other terms, like a quadratic term or more complicated things, you can kind of keep adding them on the end. So this right here is the expression that you will often see for the local linearization. And the only place where the actual variable shows up, the variable vector, is right here, is this guy.
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Local linearization.mp3
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The rest of your stuff is your linear. Because later on, if we start adding other terms, like a quadratic term or more complicated things, you can kind of keep adding them on the end. So this right here is the expression that you will often see for the local linearization. And the only place where the actual variable shows up, the variable vector, is right here, is this guy. Because, you know, when you evaluate the function f at a specified input, that's just a constant. When you evaluate the gradient at that input, it's just a constant. And we're subtracting off that specified input, that's just a constant.
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Local linearization.mp3
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And the only place where the actual variable shows up, the variable vector, is right here, is this guy. Because, you know, when you evaluate the function f at a specified input, that's just a constant. When you evaluate the gradient at that input, it's just a constant. And we're subtracting off that specified input, that's just a constant. So this is the only place where your variable shows up. So once all is said and done, and once you do your computations, this is a very simple function. And the important part is maybe this is much simpler than the function f itself, which allows you to, you know, maybe compute something more quickly if you're writing a program that needs to, you know, deal with some kind of complicated function, but runtime is an issue.
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Local linearization.mp3
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And we're subtracting off that specified input, that's just a constant. So this is the only place where your variable shows up. So once all is said and done, and once you do your computations, this is a very simple function. And the important part is maybe this is much simpler than the function f itself, which allows you to, you know, maybe compute something more quickly if you're writing a program that needs to, you know, deal with some kind of complicated function, but runtime is an issue. Or maybe it's a function that you never knew in the first place, but you were able to approximate its value at a point and approximate its gradient. So this is what lets you approximate the function as a whole near that point. So again, this might look very abstract, but if you just kind of unravel everything and think back to where it came from and look at the specific example of a, you know, tangent plane, hopefully it all makes a little bit of sense and you see that this is really just the simplest possible function that evaluates to the same value as f when you input this point, and whose partial derivatives all evaluate to the same values as those of f at that specified point.
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Local linearization.mp3
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And the important part is maybe this is much simpler than the function f itself, which allows you to, you know, maybe compute something more quickly if you're writing a program that needs to, you know, deal with some kind of complicated function, but runtime is an issue. Or maybe it's a function that you never knew in the first place, but you were able to approximate its value at a point and approximate its gradient. So this is what lets you approximate the function as a whole near that point. So again, this might look very abstract, but if you just kind of unravel everything and think back to where it came from and look at the specific example of a, you know, tangent plane, hopefully it all makes a little bit of sense and you see that this is really just the simplest possible function that evaluates to the same value as f when you input this point, and whose partial derivatives all evaluate to the same values as those of f at that specified point. And if you want to see more examples of this and what it looks like and maybe how you can use it to approximate certain functions, I have an article on that that you can go check out. And it would be particularly good to kind of go in with a piece of paper and sort of work through the examples yourself as you work through it. And with that said, I will see you next video.
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Local linearization.mp3
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So that's the plane that kind of slants down, slants down like that. It's the intersection of that plane and the cylinder, actually I shouldn't even call it a cylinder because if you just have x squared plus y squared is equal to one, it would essentially be like a pole, an infinite pole that keeps going up forever and keeps going down forever. So it would never have a top or a bottom. But we've sliced that pole with y plus z is equal to two to get and where they intersect, we get our path C. We also have this vector field defined in this way and we're asked to evaluate the line integral of F dot dr over this path with this orientation. And we could directly solve this line integral. We've done it before and actually we'll do it again to show that we're getting the same answer. But this is about Stokes' Theorem.
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Stokes example part 1 Multivariable Calculus Khan Academy.mp3
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But we've sliced that pole with y plus z is equal to two to get and where they intersect, we get our path C. We also have this vector field defined in this way and we're asked to evaluate the line integral of F dot dr over this path with this orientation. And we could directly solve this line integral. We've done it before and actually we'll do it again to show that we're getting the same answer. But this is about Stokes' Theorem. So let's see if we can apply Stokes' Theorem to this circumstance. So Stokes' Theorem tells us that this is going to be the same thing as the surface integral over a piecewise smooth surface bounded by this boundary right over here. And actually we'll pick the simplest of all the, of all of the surfaces bounded by this.
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Stokes example part 1 Multivariable Calculus Khan Academy.mp3
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But this is about Stokes' Theorem. So let's see if we can apply Stokes' Theorem to this circumstance. So Stokes' Theorem tells us that this is going to be the same thing as the surface integral over a piecewise smooth surface bounded by this boundary right over here. And actually we'll pick the simplest of all the, of all of the surfaces bounded by this. I'll pick the portion of the plane that is bounded by C. So S is going to be the portion, portion of the plane, of the plane Y plus Z is equal to two bounded, bounded by C. So it's gonna be the surface integral over this. We could have picked any other surface too that's piecewise smooth bounded by this, but this is gonna be the easiest for us to work with. We have information on it.
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Stokes example part 1 Multivariable Calculus Khan Academy.mp3
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And actually we'll pick the simplest of all the, of all of the surfaces bounded by this. I'll pick the portion of the plane that is bounded by C. So S is going to be the portion, portion of the plane, of the plane Y plus Z is equal to two bounded, bounded by C. So it's gonna be the surface integral over this. We could have picked any other surface too that's piecewise smooth bounded by this, but this is gonna be the easiest for us to work with. We have information on it. It's a nice flat, it's at an incline, but it's a nice flat surface that we're gonna be able to operate on analytically. And so we're gonna take the surface integral of the curl, the curl of our vector field dotted with, and we could say dot ds if we think of the vector differential, or we could say dotted with the normal vector, dotted with the normal vector at any point, ds times the little section of our surface, a little surface differential right over there. And we've talked already about orientation, so we wanna make sure that we get our orientation right.
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Stokes example part 1 Multivariable Calculus Khan Academy.mp3
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We have information on it. It's a nice flat, it's at an incline, but it's a nice flat surface that we're gonna be able to operate on analytically. And so we're gonna take the surface integral of the curl, the curl of our vector field dotted with, and we could say dot ds if we think of the vector differential, or we could say dotted with the normal vector, dotted with the normal vector at any point, ds times the little section of our surface, a little surface differential right over there. And we've talked already about orientation, so we wanna make sure that we get our orientation right. We are traversing it in this direction, and if we think about the little man analogy, if when he's walking in this direction, if he's walking with his head upwards, then the surface will be to his left, and so we want the normal vector that would also go upwards. So the normal vector would need to go, the unit normal vector would need to look something like that. The other analogy, if you're twisting a bottle cap like this then the bottle cap will move in that direction, in the direction of this normal vector.
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Stokes example part 1 Multivariable Calculus Khan Academy.mp3
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And we've talked already about orientation, so we wanna make sure that we get our orientation right. We are traversing it in this direction, and if we think about the little man analogy, if when he's walking in this direction, if he's walking with his head upwards, then the surface will be to his left, and so we want the normal vector that would also go upwards. So the normal vector would need to go, the unit normal vector would need to look something like that. The other analogy, if you're twisting a bottle cap like this then the bottle cap will move in that direction, in the direction of this normal vector. So we've set it all up. Now all we have to do is evaluate, now we have to do is evaluate this integral. To evaluate it, we have to do a couple of things.
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Stokes example part 1 Multivariable Calculus Khan Academy.mp3
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The other analogy, if you're twisting a bottle cap like this then the bottle cap will move in that direction, in the direction of this normal vector. So we've set it all up. Now all we have to do is evaluate, now we have to do is evaluate this integral. To evaluate it, we have to do a couple of things. We one, have to come up with a parameterization for our surface, which shouldn't be too difficult. We've done it before when we evaluated surface integrals. And we also have to calculate what the curl of f is going to be, and then we just need to evaluate the double integral after all of that.
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Stokes example part 1 Multivariable Calculus Khan Academy.mp3
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All right. So in the last video, I set up the scaffolding for the quadratic approximation, which I'm calling Q, of a function, an arbitrary two-variable function, which I'm calling F. And the form that we have right now looks like quite a lot, actually. We have six different terms. Now, the first three were just basically stolen from the local linearization formula and written in their full abstractness. It almost makes it seem a little bit more complicated than it is. And then these next three terms are basically the quadratic parts. We have what is basically x squared.
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Quadratic approximation formula, part 2.mp3
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Now, the first three were just basically stolen from the local linearization formula and written in their full abstractness. It almost makes it seem a little bit more complicated than it is. And then these next three terms are basically the quadratic parts. We have what is basically x squared. We take it as x minus x-naught squared so that we don't mess with anything previously once we plug in x equals x-naught. But basically, we think of this as x-squared. And then this here is basically x times y, but of course, we're matching each one of them with the corresponding x-naught, y-naught.
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Quadratic approximation formula, part 2.mp3
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We have what is basically x squared. We take it as x minus x-naught squared so that we don't mess with anything previously once we plug in x equals x-naught. But basically, we think of this as x-squared. And then this here is basically x times y, but of course, we're matching each one of them with the corresponding x-naught, y-naught. And then this term is the y-squared. And the question at hand is how do we fill in these constants, the coefficients in front of each one of these quadratic terms to make it so that this guy q hugs the graph of f as closely as possible? And I showed that in the very first video, kind of what that hugging means.
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Quadratic approximation formula, part 2.mp3
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And then this here is basically x times y, but of course, we're matching each one of them with the corresponding x-naught, y-naught. And then this term is the y-squared. And the question at hand is how do we fill in these constants, the coefficients in front of each one of these quadratic terms to make it so that this guy q hugs the graph of f as closely as possible? And I showed that in the very first video, kind of what that hugging means. Now, in formulas, the goal here, I should probably state what it is that we want is for the second partial derivatives of q. So for example, if we take the partial derivative with respect to x twice in a row, we want it to be the case that if you take that guy and you evaluate it at the point of interest, the point about which we are approximating, it should be the same as when you take the second partial derivative of f, or the corresponding second partial derivative, I should say, since there's multiple different second partial derivatives, and you evaluate it at that same point. And of course, we want this to be true not just with the second partial derivative with respect to x twice in a row, but if we did it with the other ones.
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Quadratic approximation formula, part 2.mp3
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And I showed that in the very first video, kind of what that hugging means. Now, in formulas, the goal here, I should probably state what it is that we want is for the second partial derivatives of q. So for example, if we take the partial derivative with respect to x twice in a row, we want it to be the case that if you take that guy and you evaluate it at the point of interest, the point about which we are approximating, it should be the same as when you take the second partial derivative of f, or the corresponding second partial derivative, I should say, since there's multiple different second partial derivatives, and you evaluate it at that same point. And of course, we want this to be true not just with the second partial derivative with respect to x twice in a row, but if we did it with the other ones. Like for example, let's say we took the partial derivative first with respect to x, and then with respect to y. This is called the mixed partial derivative. We want it to be the case that when we evaluate that at the point of interest, it's the same as taking the mixed partial derivative of f, with respect to x, and then with respect to y, and we evaluate it at that same point.
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Quadratic approximation formula, part 2.mp3
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And of course, we want this to be true not just with the second partial derivative with respect to x twice in a row, but if we did it with the other ones. Like for example, let's say we took the partial derivative first with respect to x, and then with respect to y. This is called the mixed partial derivative. We want it to be the case that when we evaluate that at the point of interest, it's the same as taking the mixed partial derivative of f, with respect to x, and then with respect to y, and we evaluate it at that same point. At that same point. And remember, for almost all functions that you deal with, when you take this second partial derivative where we mix two of the variables, it doesn't matter the order in which you take them. You could take it first with respect to x, then y, or you could do it first with respect to y, and then with respect to x.
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Quadratic approximation formula, part 2.mp3
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We want it to be the case that when we evaluate that at the point of interest, it's the same as taking the mixed partial derivative of f, with respect to x, and then with respect to y, and we evaluate it at that same point. At that same point. And remember, for almost all functions that you deal with, when you take this second partial derivative where we mix two of the variables, it doesn't matter the order in which you take them. You could take it first with respect to x, then y, or you could do it first with respect to y, and then with respect to x. Usually, these guys are equal. There are some functions for which this isn't true, but we're gonna basically assume that we're dealing with functions where this is. So, that's the only mixed partial derivative that we have to take into account.
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Quadratic approximation formula, part 2.mp3
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You could take it first with respect to x, then y, or you could do it first with respect to y, and then with respect to x. Usually, these guys are equal. There are some functions for which this isn't true, but we're gonna basically assume that we're dealing with functions where this is. So, that's the only mixed partial derivative that we have to take into account. And I'll just kinda get rid of that guy there. And then of course, the final one, final one, just to have it on record here, is that we want the partial derivative when we take it with respect to y two times in a row, and we evaluate that at the same point. There's kind of a lot of, this is, there's a lot of writing that goes on with these things, and that's just kind of par for the course when it comes to multivariable calculus, but you take the partial derivative with respect to y at both of them, and you want it to be the same value at this point.
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Quadratic approximation formula, part 2.mp3
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So, that's the only mixed partial derivative that we have to take into account. And I'll just kinda get rid of that guy there. And then of course, the final one, final one, just to have it on record here, is that we want the partial derivative when we take it with respect to y two times in a row, and we evaluate that at the same point. There's kind of a lot of, this is, there's a lot of writing that goes on with these things, and that's just kind of par for the course when it comes to multivariable calculus, but you take the partial derivative with respect to y at both of them, and you want it to be the same value at this point. So, even though there's a lot going on here, all I'm basically saying is all the second partial differential information should be the same for q as it is for f. So, let's actually go up and take a look at our function and start thinking about what its partial derivatives are, what its first and second partial derivatives are. And to do that, let me first just kind of clear up some of the board here, just to make it so we can actually start computing what this second partial derivative is. So, let's go ahead and do it.
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Quadratic approximation formula, part 2.mp3
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There's kind of a lot of, this is, there's a lot of writing that goes on with these things, and that's just kind of par for the course when it comes to multivariable calculus, but you take the partial derivative with respect to y at both of them, and you want it to be the same value at this point. So, even though there's a lot going on here, all I'm basically saying is all the second partial differential information should be the same for q as it is for f. So, let's actually go up and take a look at our function and start thinking about what its partial derivatives are, what its first and second partial derivatives are. And to do that, let me first just kind of clear up some of the board here, just to make it so we can actually start computing what this second partial derivative is. So, let's go ahead and do it. This first, this partial derivative with respect to x twice, what we'll do is I'll take one of those out and think partial derivative with respect to x, and then on the inside, I'm gonna put what the partial derivative of this entire expression with respect to x is. Well, we just take it one term at a time. This first term here is a constant, so that goes to zero.
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Quadratic approximation formula, part 2.mp3
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So, let's go ahead and do it. This first, this partial derivative with respect to x twice, what we'll do is I'll take one of those out and think partial derivative with respect to x, and then on the inside, I'm gonna put what the partial derivative of this entire expression with respect to x is. Well, we just take it one term at a time. This first term here is a constant, so that goes to zero. The second term here actually has the variable x in it. And when we take its partial derivative, since this is a linear term, it's just gonna be that constant sitting in front of it. So, it'll be that constant, which is the value of the partial derivative of f with respect to x, evaluated at the point of interest.
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Quadratic approximation formula, part 2.mp3
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This first term here is a constant, so that goes to zero. The second term here actually has the variable x in it. And when we take its partial derivative, since this is a linear term, it's just gonna be that constant sitting in front of it. So, it'll be that constant, which is the value of the partial derivative of f with respect to x, evaluated at the point of interest. And that's just a constant. All right, so that's there. This next term has no x's in it, so that's just gonna go to zero.
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Quadratic approximation formula, part 2.mp3
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