Main content

## Multivariable calculus

### Course: Multivariable calculus > Unit 3

Lesson 2: Quadratic approximations- What do quadratic approximations look like
- Quadratic approximation formula, part 1
- Quadratic approximation formula, part 2
- Quadratic approximation example
- The Hessian matrix
- The Hessian matrix
- Expressing a quadratic form with a matrix
- Vector form of multivariable quadratic approximation
- The Hessian
- Quadratic approximation

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# 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.

## Want to join the conversation?

- Why the linear approximation does not have the constant term?(instead it's called the affine)(3 votes)
- I believe that for a function to be linear it must pass through the origin. The constant term shifts the function up or down, which causes it to no longer pass through the origin. For more complex answers: https://math.stackexchange.com/questions/275310/what-is-the-difference-between-linear-and-affine-function(8 votes)

- What is the tool Grant is using for math illustrations?(6 votes)
- Are we building up to Taylor Polynomials for n dimensions?(6 votes)
- As of we're dealing with approximations of functions, there's some kind of Taylor series for multivariable functions?(3 votes)
- Are we building up to Taylor polynomials?(3 votes)

## 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.