Quadratic approximation example Khan Academy English (auto-generated) English

Quadratic approximation exampleYouTube↗ IA↗(long shot)
Howto & Style 20.9K views 56 likes 3 dislikes 16 Jun 2016

Khan Academy 9.4M subscribers United States YouTube↗

Click to Play Video

when we last left off in the riveting

- [Voiceover] When we last left off in the riveting saga

saga of quadratic approximations of multivariable functions we were

of quadratic approximations of multi variable functions,

approximating a two variable function f

we were approximating a two variable function, f of xy,

of X Y and we ended up with this pretty monstrous expression and because it's

and we ended up with this pretty monstrous expression,

written in its full abstract form I

and because it's written its full abstract form, I almost

almost feel like it looks more monstrous

feel like it looks more monstrous than it needs to,

than it needs to so I'm going to go

so I'm going to go ahead and go through a specific

ahead and go through a specific example here and just to remind you of kind of

example here, and just to remind you of kind of what

what all these terms are how there's

all these terms are, how there's actually kind of a

actually kind of a pattern to what's going on this here represents you could

pattern to what's going on, this here represents,

think of it as the constant term where

you can think of it as the constant term, where you know

you know this is just going to evaluate to some kind of number these two terms

this is just going to evaluate to some kind of number.

are what you might call the linear term

These two terms are what you might call the linear term,

linear because if you actually look the

linear, because if you actually look, the only places

only places where the variable x and y comes up is here where it's just being

where the variable x and y comes up is here,

multiplied by a constant and here where

where it's just being multiplied by a constant,

it's just being multiplied by a constant

and here, where it's just being multiplied by a constant,

so it's just variables times constant in

so it's just variables times constant in there.

there and then all of this stuff at the

And then all of this stuff at the end, which is kind of

end which is kind of the whole essence of a quadratic approximation where you

the whole essence of a quadratic approximation,

start to have things like you get an x

where you start to have things like you get an x squared,

squared and you get like X gets to be multiplied by Y all of this stuff is the

and you get x gets to be multiplied by y,

quadratic term and though it seems like

all of this stuff is the quadratic term, and though

a lot now you'll see in the context of

it seems like a lot now, you'll see in the context

an actual example it's it's not

of an actual example, it's not necessarily as bad

necessarily as bad as it seems so let's

say we're looking at the function f of X Y and let's say it's going to be e to

So let's say we're looking at the function f of xy and let's say it's going to be

the X divided by 2 x sine of Y this is

e to the x divided by two, multiplied by sin of y.

our multiply very function and let's say

This is our multi variable function.

we want to approximate this near some

And let's say we want to approximate this near some

kind of point and I'm going to choose a

kind of point, and I'm going to choose a point that's

point that's you know something that we can actually evaluate these at so like X

something that we can actually evaluate these at,

it would be convenient if that was 0 and

so like x, it would be convenient if that was zero,

then Y I'll go with PI halves because

and then y, I'll go with pi halves, because that's

that's something where I'll know how to evaluate sine and where I'll know how to

something where I'll know how to evaluate sin,

evaluate its derivatives things like

and where I'll know how to evaluate its derivatives,

that so we're trying to approximate this

things like that.

function near this point now first

So we're trying to approximate this function near

things first we're just going to need to

this point. Now first things first.

get all of the different partial

We're just going to need to get all of the different

derivatives and second partial

partial derivatives and second partial derivatives.

derivatives we know we're going to need

We know we're going to need them, so let's just kind

them so let's just kind of start working it through and figuring out what all of

of start working it through, and figuring out what

them are so let's start with the partial derivative with respect to X so this is

all of them are. Let's start with the partial derivative with respect to x.

also a function of XY and we look up at

So this is also a function of xy,

the original function the only place

and we look up at the original function,

where X shows up is in this e to the x

the only place where x shows up is in this e to the x

over 2 the derivative of that is 1/2 we

over two, the derivative of that is one half,

bring down that 1/2

we bring down that one half,

times e to the X over 2 and this is

times e to the x over two,

being multiplied by something that looks

and this is being multiplied by something that looks

like a constant as far as X is concerned

like a constant, as far as x is concerned,

sine of Y now when we do the partial

derivative with respect to Y what we get

Now when we do the partial derivative with respect to y,

this first part just looks like a

constant so we kind of keep that

this first part just looks like a constant, so we kind

constant there as far as Y is concerned

of keep that constant there, as far as y is concerned,

and the derivative of sine is cosine

and the derivative of sin is cosin.

cosine of Y and then now we let's start

Cosin of y. And then now we

taking second partial derivatives so

let's start taking second partial derivatives,

I'll start by doing the one where we

so I'll start by doing the one where

take the partial derivative with respect to X twice now here I'll actually do

we take the partial derivative with respect to x twice.

this I'll do this in a different color

Now here I'll actually do this in a different color.

what let's do like yellow just to make

Let's do yellow.

clear which ones are the second partial

Just to make clear which ones are the second

derivatives so partial with respect to X

partial derivatives.

twice also a function of X Y like all of these guys and so let's look up the

So partial with respect to x twice, also, function of xy, like all of these guys.

original partial derivative with respect

So let's look up at the original partial derivative with

to X and we're now going to take its

respect to x, and we're now going to take its

derivative again with respect to X this

derivative, again with respect to x.

is the only place where X shows up that

This is the only place where x shows up,

1/2 kind of comes down again so now it's

that one half kind of comes down again,

going to be 1/4 times e to the x over 2

so now it's going to be one fourth times

and we just keep that sine of Y because

e to the x over two,

it looks it looks like we're just

and we just keep that sin of y,

multiplying by a constant sine of Y next

because it looks like we're just multiplying by a constant, sin of y.

we'll do the mixed partial derivative where you do first with respect to X

Next we'll do the mixed partial derivative, where you do

then with respect to Y or you could do

first with respect to x, then with respect to y,

it the other way because with almost all

or you could do it the other way, because with almost

functions it kind of doesn't matter

all functions, it kind of doesn't matter which

which order you take the two so I'll go

order you take the two.

ahead and just look at the one that was

So I'll go ahead and just look at the one that

with respect to X and now let's think of

was with respect to x, and now let's think of its

its derivative with respect to Y this

derivative with respect to y.

whole 1/2 e to the X halves looks like a

This whole one half e to the x halves

constant derivative of sine of Y is

looks like a constant, derivative of sin of y

cosine Y so we take that constant the

1/2 e to the e to the X halves and then

So we take that constant, the one half e to the x halves,

we multiply it by the derivative of sine

and then we multiply it by the derivative of sin of y,

of Y which is cosine of Y and then

which is cosin of y.

finally we take we take the second

And then finally we take the second derivative,

derivative second partial derivative

second partial derivative

with respect to Y twice in a row so F

with respect to y, twice in a row.

with respect to Y twice in a row and for

So f with respect to y, twice in a row.

this one let's take a look at the

And for this one, let's take a look at the partial

partial derivative with respect to Y

derivative with respect to y.

this part is the only part where Y shows

This part is the only part where y shows up,

up derivative of cosine is negative sine

derivative of cosin is negative sin,

and then e to the X halves just still looks like a constant so we bring that

and then e to the x halves just still looks like a constant,

negative out front that constant e to

so we'll bring that negative out front,

the X halves and it was

that constant e to the x halves, and it was negative sin,

sign so that negative one out front sign

so that negative went out front,

so that's that's all of the partial

So that's all of the partial differential information

differential information that we're

that we're going to need.

going to need and now we know we're

And now we know we're going to need to evaluate

going to need to evaluate all of these guys all of these partial derivatives at

all of these guys, all of these partial derivatives

the specific point because if we go up

at the specific point, because if we go up and

and look at the original function that

look at the original function that we have,

we have we're going to need to evaluate F at this point both of the partial

we're going to need to evaluate f at this point,

derivatives at this point the second

both of the partial derivatives at this point,

partial derivatives I'm realizing

the second partial derivatives.

actually that I made a little bit of a

Oh, I'm realizing, actually, that I made a little

mistake here this should be a one-half

bit of a mistake here.

out in front of each of these guys that

This should be a one half

should be plus one half of this second

out in front of each of these guys. That should be plus one half of this second

partial derivative and one half of this

partial derivative, and one half of this second

second partial derivative the mixed

partial derivative.

partial derivative it's still one but

The mixed partial derivative it's still one,

these guys should have a one-half that's

but these guys should have a one half.

a that was a mistake on my part another

That was a mistake on my part.

case that we're going to need to

In any case, though, we're going to need to evaluate

evaluate all of these guys so if we go

all of these guys, so if we go back down,

back down let's just start plugging in

let's just start plugging in the point zero and

the point 0 and PI halves to each one of

pi halves to each one of these.

these so the function itself when we

So the function itself, when we plug in zero,

plug in 0 e to the 0 is 1 and sine of PI

e to the zero is one, and sin of pi halves,

halves sine of PI halves is also 1 so

sin of pi halves is also one,

this entire thing just comes to 1 if we

so this entire thing just comes to one.

do this for the next one again e to the

If we do this for the next one, again

e to the 0 is going to be 1 sine of Y is

e to the zero is going to be one,

also going to be 1 but now we have that

sin of y is also going to be one,

1/2 sitting there so that'll end up as

but now we have that one half sitting there,

1/2 if we look at the partial derivative

so that'll end up as one half.

with respect to Y cosine of PI halves is

If we look at the partial derivative with respect to y, cosin of pi halves is zero,

0 so this entire thing is going to be 0

so this entire thing is going to be zero.

moving right along if we do let's take a

Moving right along, let's take a look at this second

look at the second partial derivative with respect to X again e to the 0 will

partial derivative with respect to x.

be 1 and sine of PI halves will be 1 so

Again, e to the zero will be one, and sin of pi halves will be one,

this ends up just being that 1/4 the

so this ends up just being that one fourth,

mixed partial derivative here if we have

the mixed partial derivative here,

1/2 by you know the pattern starting to

if we have one half by the pattern's starting to continue,

continue have got the 1 this one's

you've got the one, this one's actually zero,

actually zero so cosine of PI halves is

so cosin of pi halves is zero,

0 so the whole thing will be 0 and then

so the whole thing will be zero.

the last one it'll be negative 1 times

And then the last one it'll be negative

that one again for sine of PI halves as

one times that one again,

1 so all of that just comes out to be

for sin of pi halves is one,

negative 1 so this is I mean it kind of

so all of that just comes out to be negative one.

chose a convenient example right where

So this, I mean I kind of chose a convenient example,

all the derivatives look very similar to

where all the derivatives look very similar to the

the thing itself which is actually

thing itself, which is actually pretty common,

pretty common so we get to leverage a

so we get to leverage a lot of the work that

lot of the work that we did earlier so

we did earlier.

now we have these six different

So now we have these six different constants,

Stinson's kind of can't keep them all on the screen at the same time but we've

can't keep them all on the screen at the same time.

got this six different constants so now

But we've got these six different constants,

we just plug each one of these into the

so now we just plug each one of them in to the

quadratic approximation so if I make our

quadratic approximation.

quadratic approximation of our function

So if we make our quadratic approximation of our function,

the first term is that constant term so

the first term is that constant term,

we take a look up and we say where does

so we take a look up and we say where does f of xy

f of X Y go at this point and it'll just be one I'm going to have to do a lot of

go at this point, and it'll just be one.

scrolling back and forth here there's a

We're going to have to do a lot of scrolling back

lot of text to deal with and then the

and forth here. There's a lot of text to deal with.

next thing is going to be something

The next thing is going to be something

times X minus minus zero the kind of x

times x minus zero, the kind of x coordinate

coordinate of our specified point and

of our specified point,

that something is the first derivative

and that something is the first derivative with

with respect to X so that's going to be one half so coming back down here

respect to x, so that's going to be one half.

that one half and then similarly we're

So come back down here. That one half.

going to have something multiplied by y

And then similarly, we're going to have something

minus the y coordinate of the point

multiplied by y minus the y coordinate

about which we are approximating and for

of the point about which we are approximating,

that we take a look at the partial

and for that we take a look at the partial derivative

derivative with respect to Y which was

with respect to y, which was just zero,

just zero so that's pretty convenient

so that's pretty convenient.

that's just going to end up being zero

That's just going to end up being zero.

and then for the second partial

And then for the second partial derivative terms,

derivative terms maybe I'll actually be

maybe I'll actually be able to keep it on

able to keep it on the same screen here

the same screen here.

we're going to have something multiplied by X minus you know it's coordinate squared and that something is whatever

We're going to have something multiplied by x minus its coordinate squared,

the partial derivative with respect to X

and that something is whatever the partial derivative

twice's which is 1/4 so we go ahead and

with respect to x twice is, which is one fourth,

plug in 1/4 and then for the mixed

so we go ahead and plug in one fourth,

partial derivative I'll put it down here

and then for the mixed partial derivative,

it'll be something multiplied by you know X minus its constant and then Y

I'll put it down here, it'll be something multiplied by x, minus its constant,

minus that PI halves and that something

and then y minus that pi halves,

is the mixed partial derivative which in

and that something is the mixed partial derivative,

this case is zero what I'm realizing I

which in this case is zero.

made the same mistake again it's a it's

Oh and I'm realizing I made the same mistake again.

not 1/4 it's 1/2 we you know for the

It's not one fourth, it's one half.

same reason that I made a mistake up

For the same reason that I made a mistake up here earlier,

here earlier where it's actually 1/2

where it's actually one half multiplied by this

multiplied by this second partial

second partial derivative, and one half by the second

derivative and 1/2 by the second partial derivative there I guess I keep

partial derivative there, I guess I keep forgetting that.

forgetting that good lesson I suppose that that's a that's an easy thing to

Good lesson, I suppose, that that's an easy thing to forget,

forget if you find yourself computing

if you find yourself computing one of these,

one of these where I'll put it in here

where I'll put it in here,

multiply that guy by 1/2 it's similar to

multiply that guy by one half.

a Taylor expansion in single variable

It's similar to a Taylor expansion in single variable

calculus where you kind of have to

calculus, where you kind of have to remember what's

remember what's what that squared term would be has a 1/2 associated with it so

what that squared term would be, has a one half associated with it.

for that same reason now we're going to

So for that same reason, now we're going to have,

have and this time I won't forget it

and this time I won't forget it,

will be 1/2 multiplied by something

will be one half multiplied by something,

multiplied by the Y minus PI halves

multiplied by the y minus pi halves,

minus that y-coordinate of the point

minus that y coordinate of the point we're approximating

we're approximating there and this time

there, and this time that something

that something is is negative 1 so we

is negative one.

can kind of plug in here negative 1 and

So we can kind of plug in here negative one.

now this is something we can we can simplify quite a bit because that one

And now this is something we can simplify quite a bit, because that one stays there,

stays there 1/2 of X minus 0 that's just

one half of x minus zero,

X halves this whole part cancels out to

that's just x halves,

0 so there's nothing there over here we

this whole part cancels out to zero,

have 1/2 times 1/4 1/8 times x squared

so there's nothing there. Over here we have half times a fourth, one eighth

so that's x squared divided by 8 this

times x squared, so that's x squared divided by eight.

mixed partial derivative term is 0 so

This mixed partial derivative term is zero,

that's pretty nice and then this last

so that's pretty nice.

term here is just a negative 1/2 so

And then this last term here is just

let's see how I write it down as negative 1/2 by Y minus PI halves

negative one half, so let's see, I'll write it down as negative one half by y minus pi halves squared,

squared by Y minus PI halves squared so

by y minus pi halves squared.

that that is the quadratic approximation

So that is the quadratic approximation,

and you can see this is low this

and you can see this actually feels like a

actually feels like a quadratic function

quadratic function.

we've got you know up to x squared and up to Y squared and there's a sense in

We've got up to x squared, and up to y squared,

which this is a a simpler function I

and there's a sense in which this is a

mean it looks like it's got more terms

simpler function.

than the original one which was e to the

I mean, it looks like it's got more terms than the

X have sine of Y but if it's you know if

original one, which was e to the x halved sin of y,

it's a computer that needs to compute

but if it's a computer that needs to compute these

these things for example it's much

things, for example, it's much easier to deal with

easier to deal with polynomials that's that's a faster thing to do also for

polynomials, that's a faster thing to do.

Theoretical purposes it can be nice to

Also for theoretical purposes, it can be nice to

deal with just a quadratic polynomial to

deal with just a quadratic polynomial to make

make conclusions about things we'll see

conclusions about things.

that in the context of something called

We'll see that in the context of something called the

the second partial derivative test which

second partial derivative test.

is just to get a feel for what this what

But just to get a feel for what this means, let's pull

this means let's pull up the graph of the relevant functions so this here is

up the graph of the relevant functions.

the graph of the original function e to

So this here is the graph of the original function,

the X halves times sine of Y and the

e to the x halves times sin of y,

point that we're approximating near was

and the point that we're approximating near

where x equals 0 so let's see how we get

was where x equals zero,

oriented X is equal to 0 and then Y is

so let's see how we get oriented.

equal to PI halves so this is the point

X is equal to zero, and then y is equal to pi halves.

we're approximating near and the

So this is the point we're approximating near.

quadratic approximation when you plug

And the quadratic approximation, when you plug everything

everything in has a graph that looks like this white surface here so if I get

in, has a graph that looks like this white surface here.

rid of that original graph this is how

So if I get rid of that original graph, this is how

we're approximating the function near

we're approximating the function near that point.

that point and then that does a pretty

And that does a pretty good job, right?

good job right because even as you step pretty far away from

Because even as you step pretty far away from that

that point it's pretty closely hugging

point, it's pretty closely hugging the original surface.

the original surface if you go very far away you know it certainly doesn't get

If you go very far away, it certainly doesn't get

the oscillating nature of that sign

the oscillating nature of that sin component,

component and the the exponential

and the exponential component grows faster than

component grows faster than the quadratic one but nearby nearby this

the quadratic one, but nearby,

actually gives a very good feel for the

nearby this actually gives a very good feel for

the shape of the graph and again later

the shape of the graph.

on we'll see how this is a pretty useful

And again, later on we'll see how this is a pretty

theoretical tool for drawing conclusions

useful theoretical tool for drawing conclusions about

about qualitative features of the shape

qualitative features of the shape of the graph,

of the graph the fact that this looks kind of like a saddle is going to end up

the fact that this looks kind of like a saddle,

being kind of important in certain

is going to end up being kind of important

contexts but before we get to any of

in certain contexts.

that in the next couple videos I'm going

But before we get to any of that, in the next couple

to talk about a a simpler or rather a

videos I'm going to talk about a simpler,

more generalizable form of writing down

or rather a more generalizable form,

this quadratic approximation using

of writing down this quadratic approximation

vector notation because right now we're

using vector notation.

just limited to you know two variables

Because right now we're just limited to

and you can imagine how monstrous this

two variables, and you can imagine how monstrous this

might look if you were dealing even just

might look if you were dealing even just with a

with a three variable function right

three variable function,

where think of all the different

where think of all the different possible second

possible second partial derivatives of a

partial derivatives of a three variable function,

three variable function or a four available variable function it would

or a four variable function.

quickly get out of hand but there is

It would quickly get out of hand, but there is kind

kind of a nice general way to write all

of a nice general way to write all of these.

of these so with that I will see you next video

So with that, I will see you next video.

Similar videos:

Quadratic approximation formula, part 2
Khan Academy
21.7K views 111 likes 3 dislikes 16 Jun 201638% match

Second partial derivative test
Khan Academy
52.3K views 205 likes 17 dislikes 22 Jun 201632% match

Computing a tangent plane
Khan Academy
55.9K views 249 likes 8 dislikes 15 Jun 201631% match

Quadratic approximation formula, part 1
Khan Academy
42.4K views 126 likes 15 dislikes 16 Jun 201631% match

Symmetry of second partial derivatives
Khan Academy
105.9K views 397 likes 14 dislikes 11 May 201630% match

The Hessian matrix
Khan Academy
165.5K views 771 likes 24 dislikes 16 Jun 201630% match

Multivariable chain rule
Khan Academy
161.4K views 671 likes 24 dislikes 20 May 201630% match

Controlling a plane in space
Khan Academy
30.6K views 141 likes 3 dislikes 15 Jun 201629% match

Warm up to the second partial derivative test
Khan Academy
34.1K views 137 likes 1 dislikes 22 Jun 201629% match

Local linearization
Khan Academy
33K views 140 likes 4 dislikes 15 Jun 201629% match

Second partial derivative test intuition
Khan Academy
42.3K views 357 likes 7 dislikes 22 Jun 201629% match

Second partial derivative test example, part 2
Khan Academy
29.6K views 215 likes 2 dislikes 22 Jun 201629% match

Partial derivatives, introduction
Khan Academy
836.1K views 6.5K likes 294 dislikes 11 May 201628% match

The Lagrangian
Khan Academy
178.5K views 1.3K likes 30 dislikes 29 Nov 201628% match

Partial derivatives 2 | Multivariable Calculus | Khan Academy
Khan Academy
304.6K views 652 likes 10 dislikes 09 Aug 200828% match

Vector form of multivariable quadratic approximation
Khan Academy
40.9K views 305 likes 2 dislikes 16 Jun 201627% match

Second Derivative Test - Two Variables (KristaKingMath)
Krista King
31.6K views 271 likes 5 dislikes 10 Nov 201127% match

Taylor & Maclaurin polynomials intro (part 1) | Series | AP Calculus BC | Khan Academy
Khan Academy
1.1M views 3.3K likes 124 dislikes 17 May 201125% match

Settings Privacy About Contact Donate