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What do quadratic approximations look like

After learning about local linearizations of multivariable functions, the next step is to understand how to approximate a function even more closely with a quadratic approximation. Created by Grant Sanderson.

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Video transcript

- [Voiceover] In the last couple videos I talked about the local linearization of a function. And in terms of graphs, there's a nice interpretation here where if you imagine a graph of a function and you want to approximate it near a specific point, you picture that point somewhere on the graph, and it doesn't have to be there, you know I can choose to be anywhere else along the graph, but if you have some sort of point and you want to approximate the function near there you can have another function whose graph is just a flat plane, and specifically a plane which is tangent to your original graph at that point. And that's kind of visually how you think about the local linearization. And what I'm going to start doing here in this next video and in the ones following, is talking about quadratic approximations. So quadratic approximations, and these, these basically take these to the next level. And first I'll show what they look like graphically and then I'll show you what it actually means in formulas. But graphically instead of having a plane that's flat, you have a few more parameters to deal with, and you can give yourself some kind of surface that hugs the graph a little bit more closely. It's still going to be simpler in terms of formulas, it can still be notably simpler than the original function, but this actually hugs it closely. And as we move around the point that it's approximating here, the way that it hugs it can look pretty different. And if you want to think graphically what a quadratic approximation is, you can basically say if you slice this surface, this kind of ghostly white surface in any direction it'll look like a parabola of some kind. And notice that given that we're dealing in multiple dimensions that can make things look pretty complicated, like this right here, you know if you slice it kind of in this direction, whoa, if you look at it from this angle it kind of looks like a concave up parabola, but if you were looking at it from another direction it kind of looks concave down, and all-in-all you get a surface that actually has quite a bit of information carried within it. And you can see that by hugging the graph very closely this approximation is going to be, well, it's going to be even closer, because near the point where you're approximating you can go out, you know, you can take a couple steps away and the approximation is still going to be very close to what the graph is, and it's only when you step really far away from the original point that the approximation starts to deviate away from the graph itself. So this is going to be something that although it takes more information to describe than a local linearization it gives us a much closer approximation. So a linear function which, you know, one that just draws a plane like this, in terms of actual function what this means, so kind of a linear, this is going to be some kind of function of x and y, and what it looks like is some kind of constant, which I'll say a plus another constant times the variable x, plus another constant times the variable y, this is sort of the basic form of linear functions. And technically this isn't linear if one is going to be really pedantic and they would say that that's actually affine, because I'm strictly speaking linear functions shouldn't have this constant term, it should be purely x's and y's, but in the context of approximations people would usually call this a linear term. So quadratic term, what this is going to look like, quadratic, we are allowed to have all the same terms as that linear one, so you can have constant, you can have these two linear terms bx and cy, and then you're allowed to have anything that has two variables multiplied into it. So maybe I'll have d times x squared, and then you can also have something times xy, this is considered a quadratic term. Which is a little bit weird at first, because usually we think of quadratics as associated with that exponent two, but really it's just saying any time you have two variables multiplied in, and then we can add some other constants, say f times y squared. Where, you know, now we're multiplying two y's into it. So all of these guys, these are what you would call your quadratic terms. Things that either x squared, y squared, or x times y, anything that has two variables in it. So you can see this gives us a lot more control because previously, as we tweaked the constants a, b, and c, you're able to give yourself, you know, that gives you control over all sorts of planes in space, and if you choose the most optimal one you'll get one that's tangent to your curve at this specific point, and kind of, it'll depend on where that point is, you'll get different planes, but they're all tangent. So what we're going to do in the next couple of videos, is talk about how you tweak all of these six different constants so that you can get functions that really closely hug the curve, right? And as you, and they're all going to depend on the original point because as you move that point around, what it takes to hug the curve is going to be different. It's going to have to do with partial differential information about your original function, the function whose graph this is, and it's going to look pretty similar to the local linearization case, just you know, added complexity so we have to add a few more steps in there. And I'll see you next video talking about that.