Computing a tangent plane Khan Academy English Korean

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Howto & Style 55.9K views 249 likes 8 dislikes 15 Jun 2016

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- [Voiceover] Hey guys, so in the last video

안녕하세요 여러분

I was talking about how you can define a function

지난번 비디오에서는

whose graph is a plane,

평면의 함수를 찾고

and moreover a plane that passes through a specified point

특정한 점을 지나고 방향이 정의되는

and whose orientation you can somehow specify.

평면의 함수를 찾았습니다

And we ended up seeing how specifying that orientation

지난 영상에서는

comes down to certain partial derivative information.

평면의 방향을 특정 편도함수 정보로 표현했습니다

And first let me just kind of repeat what the conclusion was

지난 영상의 결론을 간략하게

but I'll put it in more abstract terms

다시 한번 알려드리겠습니다

since I did it in a very specific example last time.

So basically if you want some kind of function

특정한 점을 지나는 평면의

which gives you a plane that

함수를 얻기 위해서는

passes through a certain point.

Well first let's see what that point is right.

우선, 그 평면이 지나는 점을 알아야 합니다

Let's say the point was x nought, y nought, and z nought.

그 점을 (x0,y0,z0) 라고 합시다

So these are constant values

이 값은 당연히 상숫값입니다

and this is my way of abstractly describing

(x0,y0,z0)의 표현은

a single point in space using

공간상의 한 정점에 대한

x nought to represent a constant x value,

저의 추상적인 표현입니다

y nought to represent constant y value, that kind of thing.

So what it is, is it's gonna be some sort of other constant

또 다른 상수 a 를 도입해서

a multiplied by x minus x nought.

a 곱하기 (x-x0) 라고 표시하겠습니다

So this white x here is the variable

여기 보이는 흰색 x가 변수이고

and then x nought is just a constant.

빨간색 x0는 상숫값입니다

Now let me go ahead and make that parentheses there.

Then we add to that b multiplied by

마찬가지의 방법으로 또 다른 상수 b 를 도입해서

and then b is just some other constant

just like a is some other constant

b 곱하기( y-y0) 라고 표시하겠습니다

multiplied by y minus y nought

and then all of that you add z nought.

마지막으로는 상수항 z0를 쓰겠습니다

And now if you were just presented this as it is

보시다시피 이 식에는

it's kind of a lot right,

다섯개의 서로 다른 상숫값이

there's five different constants going on.

포함되어 있습니다

But really what this is saying is

하지만 이 식이 정말로 의미하는 것은

you want something where the partial derivative

우리는 단지 함수L을 x에 대해 편미분 할 때

with respect to x is just some constant

상숫값이 나오는 경우만을

and you want to be able to specify what that constant is.

다루고 있다는 것입니다

And similarly, the partial derivative

마찬가지로 y에 대해 편미분할 경우에도

with respect to y is another constant.

다른 상숫값이 나옵니다

And you just want to ensure that this

우리는 이 식이

passes through this point x nought, y nought, z nought.

(x0, y0, z0) 를 지나는지 확인하고 싶습니다

And if you imagine plugging in x the variable equals x nought

자, 여기 보이는 이 식에 x 대신 x0를 대입하면

the constant this part goes to zero.

이 부분은 0이 됩니다

Similarly plugging in y nought the constant

같은 방법으로 y에 y0를 대입하면

makes this part go to zero.

이 부분도 0이 됩니다

So this is a way of specifying that when you

이 방법을 통해서라면

evaluate the function x nought y nought equals z nought,

(x0,y0,z0) 라는 점을 함수 L이 지나는지를

and that's what makes sure that the graph

확인하기가 수월합니다

passes through that point.

So with that said let's start thinking about

지금까지의 내용을 바탕으로

how you can find the tangent plane to a graph.

그래프의 접평면을 찾는 방법을 알아봅시다

And first of all let's think about what that point is,

우선 정점을 설정하는 방법을 알아봅시다

how you specify such a point.

Instead of specifying any three numbers in space,

앞서 공간상의 한 점을 임의로 잡은 것과는 달리

because you have to make sure the point

이제는 그래프 상에 존재하는 점을

is somewhere on the graph,

잡아야 하기 때문에

you instead only specify two.

두 상수만을 구체화합시다

You're basically gonna say what's the x coordinate

지금 보이는 이 평면의 경우

and in this case let's say the x coordinate was one,

x 좌표는 1이고

and then the y coordinate which

y 좌표는 -2입니다

looks about like negative two.

To make it easier I'm just gonna say it is negative two.

Then the z coordinate is specified

x 와 y 좌표만 정해주면

because this is a graph.

z 좌표는 그래프 상에서 자동적으로 결정됩니다

Z coordinate is forced to be whatever the output

of the function is at one negative two.

So this is gonna be whatever the output of our function is

입력값인 1과 -2 에 대한 출력값으로

at one negative two.

z 값이 결정됩니다

And f here, f is going to be whatever

함수 f 로 출력값 z 를

function gives us this graph.

표현하겠습니다

So maybe I should write down the actual function

이 그래프의 함수식을 정확하게 적어드려야 할 것 같습니다

that I'm using for this graph.

In this case f which is a function of x and y

x 와 y 를 변수로 갖는 함수 f 는

is equal to three minus one third

of x squared minus y squared.

곱하기 x² 빼기 y² 입니다

So this is the function that we're using

함수 f 를 이용하여

and you evaluate it at that point

공간상의 점을 정하고

and this will give you your point in three dimensional space

that our linear function,

선형함수로 표현되는

that our tangent plane has to pass through.

접평면이 이를 지나게 할 것입니다

So we can start writing out our linear function.

접평면의 함수L을 적어보도록 합시다

We can say okay so our linear function

has a function of x and y.

변수 x, y 의 함수입니다

It's gotta make sure it goes through

또한 (1,-2)를 지난다는 사실을

that one and that negative two,

까먹지 맙시다

so this is gonna be some constant a

다시 표현해보면

that we'll fill in in a moment,

상수 a 곱하기

multiplied by x minus that one,

(x-1) 더하기

plus and then b also a constant we'll specify in a moment,

상수 b 곱하기

times y minus that negative two,

so it's minus a negative two

and then the thing that we add to it

is f of one negative two.

더하기 f(1,-2) 입니다

Now let's just go ahead and evaluate that.

아까 파란색으로 적었던 함수 f 로 돌아가

Let's say we plug in one to negative two.

(1,-2)를 대입해 봅시다

So if we go up here and we plug in three minus one third of

계산해보면

if x equals one, one third of one squared,

so that's one third one squared,

1/3 곱하기 1²

and then y is negative two.

빼기 (-2)² 이고,

So that would be minus negative two squared.

정리해보면

So that's three minus a third minus four

3 -⅓ - 4 이므로

so the whole thing is equal to

three minus four is negative one minus another third

is negative four thirds.

- 4/3 이 됩니다

So that's what we add to this entire thing.

다시 L 로 돌아가

We add negative four thirds

상수항으로 -4/3 을 더해줍시다

or maybe I should just kind of

make clear the separation here.

So this is our function but we don't know what a and b are.

함수 L에서 아직 a 와 b 의 값을 모르는 상태입니다

Those are things that we need to plug in.

이제 이 값들을 구합시다

Now the whole idea of the tangent plane

접평면의 기본적인 아이디어로 돌아갑시다

is that the partial derivative with respect to x

접평면의 x 에 대한 편도함수는

should match that of the original function.

기존의 함수와 일치해야 합니다

So if we go over to the graph here

그림에 보이는 그래프를 살펴봅시다

and start thinking about partial derivative information.

If we want the partial derivative with respect to x

x 에 대한 편도함수를 알기 위해서는

then you imagine moving purely in the x direction here.

x 축 방향의 움직임을 알아야 합니다

This intersects the graph along some kind of curve

흰색 평면은 그래프와 일종의 곡선을 이루며 교차합니다

and what the partial derivative with respect to x

x 에 대한 편도함수가 주는 정보는

at this point tells you, is the slope of the tangent line,

빨간 점에서의 접선의 기울기입니다

in that direction of that point.

So that's what the partial derivative with respect to x

이 기울기가 x에 대한 편도함수가 주는 정보입니다

is telling you and what you want when you look

그래프의 접평면에서 알 수 있는 것은

at the tangent plane is that the tangent plane

접평면에서도 동일한 기울기를

also has that same slope.

가지리라는 것입니다

If I lined things up here,

you'd want it also to have that same slope.

So you can specify over here and say a.

이제 상수 a 값을 정할 수 있습니다

You want a to be equal to the partial derivative

of the function with respect to x

f의 x에 대한 편미분 값에

evaluated at this one negative two.

(1,-2)를 대입한 것입니다

Evaluated at that special point, one negative two.

And similarly b for pretty much the same reasons

같은 방법으로 상수 b 를 알 수 있습니다

and I'll draw it out here so let's erase this line.

So instead of intersecting it with that slice

x 축 방향의 움직임 대신에

let's see what movement in the y direction looks like.

y 축 방향의 움직임을 관찰해 봅시다

So in this case it looks like a very steep slope right

흰색 평면과 그래프가 이루는 곡선이 매우 경사져 보입니다

because in this case the tangent line in that direction

빨간 점에서의 접선의 기울기도

is a pretty steep slope and now when we bring in

매우 크다는 것을 알 수 있습니다

the tangent plane it should intersect with that

접평면을 그려보면

constant x value plane along that same slope.

역시나 빨간 점에서의 접선과 동일한 기울기를 갖습니다

Made it kind of messy there

약간은 지저분해 보이지만

but you can see the line formed

잘 보이실거라 믿습니다

by intersecting these two planes

should be that desired tangent,

and what that corresponds to in formulas

수식으로 표현하자면

is that this b which represents the partial derivative of l,

b 는 함수 L의 y 에 대한

l is the tangent plane function,

편미분 값이며

that should be the same as if we took the partial derivative

또한 앞서 한 것과 마찬가지로

of f with respect to y at that point,

f 를 y 에 대해서 편미분 한 후

at this point one negative two.

(1,-2)를 대입한 값입니다

And this is stuff that we can compute

and that we can figure out.

So let's start plugging that in.

그러면 지금부터 상숫값을 찾읍시다

First let me just copy this function

우선 파란색으로 써진

because we're gonna need it.

함수 f를 복사한 다음

Now let's go on down here.

아래에 붙여 넣겠습니다

I'm just gonna, let's paste it down here

[붙여넣는 중]

in the bottom because that's what we'll need.

[붙여넣는 중]

So let's compute the partial derivative of f

이제 f 를 x에 대해서

with respect to x.

편미분 하겠습니다

So we look down here, the only place where x shows up

함수 f 를 살펴보면 두 번째 항에서만 x가 등장합니다

is in this negative one third of x squared context

so the partial derivative of f with respect to x

따라서 함수 f 를 x 에 대해 편미분하는 것은

is gonna be just the derivative of this little guy

두 번째 항만 고려하면 되며

which is negative, we bring down the two,

편도함수는

negative two thirds of x.

-2/3 곱하기 x 입니다

So when we go ahead and plug in x equals one

(1,-2)를 x에 대한 편도함수에 대입하면

to see what it looks like when we evaluate at this point

that's just gonna be equal to negative two thirds.

-2/3이 됩니다

So that tells us that a is gonna

즉 상수 a는

have to be negative two thirds.

-2/3 이 됩니다

Now for similar reasons,

비슷한 방법으로

let's go ahead and compute the

y 에 대한 f의 편도함수를

partial derivative with respect to y.

계산합니다

We look down here, the only place that y shows up

함수 f 를 보면 y 가 등장하는 곳은

in the entire expression is this negative y squared.

맨 마지막 항이 유일합니다

So the partial derivative of f with respect to y

따라서 f 의 y 에 대한 편도함수는

is equal to negative two y, negative two y.

-2 곱하기 y 입니다

And now when we plug in y equals negative two

여기에 (1,-2)를 대입하면

what we get is negative two multiplied coincidentally

-2 곱하기 -2가 되어

by negative two, didn't have to be the case

that those were the same, and that whole thing equals four.

4가 됨을 알 수 있습니다

So the partial derivative of f with respect to y

즉 f 의 y 에 대한 편도함수를 구하고

evaluated at this point one negative two

(1,-2) 에서 이 값을 계산하면

is equal to four.

4가 나옵니다

So if we were to plug this information back up

아까 적었던

into our formula we would replace

함수 L의 식으로 돌아가서

a with negative two thirds.

a 대신 -2/3을

It would say negative two thirds.

And we would replace b with four, replace b with four.

b 대신에 4를 적어줍시다

And that would give us the full formula,

이제 완벽한 함수가 완성되었습니다

the full formula for the tangent plane.

접평면의 함수 말입니다

And this can be kind of a lot to look at at first

접평면을 구하는 것은 쉽지 않아 보일 수 있습니다

because we have to specify the input point one negative two.

입력값으로 (1,-2)를 넣어

And then we had to figure out where

그래프 상의 점을 찾고

the function evaluates at that point.

그래프 상의 점을 찾고

And then we had to figure out both of the

partial derivatives with respect to x and with respect to y.

f 의 x에 대한 편도함수와 f의 y에 대한 편도함수 를 구해야 합니다

But all in all, there's not actually

하지만 사실 기억할 것이

a lot to remember from how you go about computing this.

그리 많지는 않습니다

Looking at the graph actually makes

그림에 있는 그래프를 봅시다

things seem a lot more reasonable

각 항들은 고유하고 있기 때문에

because each of those terms has an actual meaning.

일련의 과정은 꽤 합리적으로 와 닿습니다

If we look at the one and negative two,

(1,-2)를 봅시다

that's just telling us the input,

the kind of x and y coordindates of the input

이는 x와 y의 입력값을 의미하며

and of course we have to evaluate that

간단한 계산을 거치면

because that tells us the z coordinate

그래프의 z 좌표를 알 수 있습니다

that will put us on the graph corresponding to that point

and then to get a tangent plane

그 점에서의 접평면을 알기 위해서는

you just need to specify the two bits of

그저 두 편도함수에 관한 정보를

partial differential information

구체적으로 얻으면 됩니다

and that will tell you how this graph

needs to be oriented,

and once you start thinking of things in that way,

한 번 이렇게 기하학적으로 접평면을 생각하기 시작하면

geometrically, even though there's a lot going on here,

다섯개의 상숫값을 정해야 하는

there's five different numbers you have to put in,

약간의 복잡한 과정이 기다리고 있지만

each one of them feels like of course you need that number

더 이상 이를 복잡하게 여기지 않고

otherwise you couldn't specify a tangent plane.

'당연히 계산되는 값' 으로 여기게 될 것 입니다

There's kind of a lot of information required

하지만 정점에서의 접평면을 구하기 위해

to put it on the appropriate spot.

많은 정보가 필요한 것은 사실입니다

So with that I will see you next video.

다음 영상에서 만납시다

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