Gradient and graphs Khan Academy English Korean

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Education 189.1K views 1K likes 17 dislikes 11 May 2016

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- [Voiceover] So here I'd like to talk

이번 시간에는

about what the gradient means

함수의 그래프에 있어서

in the context of the graph of a function.

기울기의 뜻을 배워 보려 합니다

So in the last video, I defined the gradient,

지난 동영상에서 기울기의 정의를 내렸지요

but let me just take a function here.

여기 함수를 하나 예로 듭시다

And the one that I had graphed is x-squared plus y-squared,

제가 그래프로 나타낸 것은 x²+y²입니다

f of x, y, equals x-squared plus y-squared.

f(x, y) = x²+y²

So two-dimensional input, which we think about

우리가 좌표평면이라 생각하는

as being kind of the xy-plane,

이차원적 입력값이 있고

and then a one-dimensional output that's just the height

그 평면 위의 그래프의 높이인

of the graph above that plane.

일차원적 출력값이 있습니다

And I defined in the last video, the gradient,

지난 동영상에서 기울기를

to be a certain operator.

특정한 연산자라고 정의했습니다

An operator just means you've taken a function

연산자라 함은 함수를 가져다가

and you output another function,

또 다른 함수를 출력시킨다는 뜻인데

and we use this upside down triangle.

이렇게 거꾸로 된 삼각형을 사용합니다

So it gives you another function that's also of x and y,

그러면 x, y에 대한 또 다른 함수가 생기는데

but this time it has a vector valued output.

이 함수는 벡터 출력값을 가지고 있습니다

And the two components of its output

그 출력값의 두 요소는

are the partial derivatives, partial of f with respect to x,

x에 대한 f의 부분과 y에 대한 f의 부분인

and the partial of f with respect to y.

편도함수입니다

So for a function like this, we actually evaluated it.

이런 함수 때문에 그 값을 실제로 구해 봤습니다

Let's take a look.

한번 봅시다

The first one is taking the derivative

첫 번째 값이 x에 대한 도함수를 보고

with respect to x, so it looks at x and says,

x를 보고 하는 말이

"You look like a variable to me.

“넌 변수 같아”

"I'm gonna take your derivative, your 2x."

“네 도함수 2x를 가져와야겠어”

2x, but the y component just looks like a constant

하지만 부분적인 x가 관여된 한

as far as the partial x is concerned.

y성분은 그냥 상수처럼 보입니다

And the derivative of a constant is zero.

그리고 상수의 도함수는 0이죠

But when you take the partial derivative

하지만 편도함수를 y에 대해 계산하면

with respect to y, things reverse.

일이 뒤바뀝니다

It looks at the x component and says,

얘가 x성분을 보고

"You look like a constant.

“넌 상수 같아”

"Your derivative is zero."

“네 도함수는 0이야”

But it looks at the y component and says,

하지만 y성분을 보고는

"Ah, you look like a variable.

“넌 변수 같아”

"Your derivative is 2y."

“네 도함수는 2y야”

So this ultimate function we get, the gradient,

따라서 최종적으로 얻는 함수인 기울기는

which takes in a two variable input, xy,

이 좌표평면 어딘가에 있는

some point on this plane,

입력값 xy의 미지수가 두 개이고

but outputs a vector,

출력값은 벡터 값입니다

can nicely be visualized with a vector field.

벡터장으로 제대로 시각화할 수 있죠

And I have another video on vector fields

자신없다면 벡터장을 다룬 동영상도

if you're feeling unsure.

But I want you to just take a moment, pause if you need to,

지금 필요하시다면 동영상을 멈추고

and guess, or try to think about what vector field

이 함수가 벡터장에서 어떻게 보일지

this will look like.

생각해 보시기 바랍니다

I'm gonna show you in a moment,

잠시 뒤에 보여드리죠

but what's it gonna look like,

xy를 입력하면

the one that takes in xy and outputs 2x, 2y?

2x, 2y가 출력되는 함수가

Alright, have you done it,

어떻게 생겼을까요?

have you thought about what it's gonna look like?

생각해 보셨습니까?

Here's what we get.

이런 모양이 나옵니다

It's a bunch of vectors pointing away from the origin.

원점으로부터 바깥쪽을 가리키는 수많은 벡터값입니다

And the basic reason for that

그 기본적인 이유는

is that if you have any given input point,

x, y좌표를 지닌

and say it's got coordinates x, y,

어떤 입력점이 있을 경우

then the vector that that input point represents

그 입력점이 나타내는 벡터값이

would, you know, if it went from the origin here,

여기 원점으로부터 출발하면

that's what that vector looks like,

이렇게 보이겠지만

but the output is two times that vector.

출력값은 그 벡터값의 두 배이기 때문입니다

So when we attach that output to the original point,

그 출력값을 원점에 붙이면

we get something that's two times that original vector

같은 방향을 가리키면서

but pointing in the same direction,

원래 벡터값의 두 배인 값이 나옵니다

which is away from the origin.

원점 반대 방향입니다

We kind of drew it poorly here.

그림을 잘못 그렸습니다

And of course, when we draw vector fields,

물론 벡터장을 그릴 때

we don't usually draw them to scale.

대부분 비율에 맞게 그리지 않습니다

You scale them down

지저분해 보이지 않도록

just so that things don't look as cluttered.

축소해서 그리죠

That's why everything here, they all look the same length,

여기서는 모두 같은 길이로 보이지만

but color indicates length.

색깔은 길이를 나타냅니다

So you should think of these red guys as being really long,

따라서 빨간색은 매우 길고

the blue ones as being really short.

파란색은 매우 짧다고 봐야 합니다

So what does this have to do with the graph of the function?

이것과 함수 그래프가 무슨 상관이냐고요?

There's actually a really cool interpretation.

사실 아주 좋은 해석이 하나 있습니다

So imagine that you are just walking along this graph,

이 그래프를 따라 걷는다고 상상해 보세요

you know, you're a hiker and this is a mountain.

여러분이 등산객이고 이것이 산입니다

And you picture yourself at any old point on this graph,

어떤 점이든 상관없이 이 그래프 위에

let's say, what color should I use?

어떤 색을 쓸까요

Let's say you're sitting at a point like this.

이 점에 앉아 있다고 칩시다

And you say, "What direction should I walk

“어느 방향으로 가야

"to increase my altitude the fastest?"

제일 빨리 올라갈 수 있을까?”

You want to get uphill as quickly as possible.

여러분은 가능한 한 빨리 정상에 오르고 싶어합니다

And from that point,

그 점에서부터

you might walk what looks like straight up there.

일직선으로 쭉 올라갈 수도 있죠

You certainly wouldn't go around,

옆으로 돌아가지도 않고

and this way you wouldn't go down.

아래로 내려갈 일도 없을 겁니다

So you might go straight up there.

위로 바로 올라갈 수 있습니다

And if you project your point down onto the input space,

여러분의 위치를 입력 공간에 표시하면

so this is the point above which you are,

여러분이 이 점 위에 있습니다

that vector, the one that's gonna get you going

이 그래프에서

uphill the fastest, the direction you should walk.

가장 빨리 올라갈 수 있는 벡터의 방향은

For this graph, it should kind of make sense,

원점의 반대 방향으로

is directly away from the origin,

향하는 것이 맞습니다

'cause here, I'll erase this

이 표시를 지우겠습니다

'cause once I start moving things, that won't stick.

그래프를 움직이면 못 쓰게 되니까요

If you were to look at things from the very bottom,

밑바닥에서 올려다보면

any point that you are on the mountain on the graph here

이 그래프 산 위 당신이 어떤 점에 있든

and when you want to increase the fastest,

가장 빨리 올라가려면

you should just go directly away from the origin

원점 반대 방향으로 곧바로 가야 합니다

'cause that's when it's the steepest.

가장 가파르니까요

And all of these vectors are also pointing

이 벡터들 또한 모두

directly away from the origin.

원점 반대 방향으로 뻗어나가고 있습니다

So people will say the gradient points in the direction

그래서 기울기가 최대 경사의 방향을 가리킨다고하는 겁니다

of steepest ascent, that might even be worth writing down.

하는 겁니다, 써두어도 되겠네요

Direction of steepest ascent.

최대 경사의 방향

And let's just see what that looks like

그게 어떤 모습인지

in the context of another example.

또 다른 예시를 들어 봅시다

So I'll pull up another graph here,

새 그래프를 하나 만들겠습니다

pull up another graph and its vector field.

새 그래프와 그 벡터장을요

So this graph, it's all negative values,

이 그래프는 모두 음수값이고

it's all below the xy-plane,

xy 평면 밑에 있습니다

and it's got these two different peaks.

꼭지점이 두 개입니다

And I've also drawn the gradient field,

기울기장도 그렸습니다

which is the word for the vector field

기울기장이란 기울기를 위의 평면에 나타내는

representing the gradient on top.

벡터장을 뜻합니다

And you'll notice near the peak

보시다시피

all of the vectors are pointing

꼭지점 가까이의 벡터값들은 모두

kind of in the uphill direction,

위쪽을 가리키고 있습니다

sort of telling you to go towards that peak in some way.

그 꼭지점으로 가라고 하는 것처럼요

And as you get a feel around, you can see here,

살펴보다 보면 여기서

this very top one, like the point that it's stemming from

맨 꼭대기의 것이 위치한 점이

corresponds with something just a little bit shy

저쪽 꼭지점에서 약간 떨어진 이것과

of the peak there.

대응합니다

And everybody's telling you to go uphill.

다들 위쪽으로 가라고 하죠

Each vector is telling you which way to walk

각 벡터값마다 그래프 꼭대기로 가장 빨리 가는 방향을

to increase the altitude on the graph the fastest.

알려주고 있습니다

It's the direction of steepest ascent.

최대 경사의 방향입니다

And that's what the direction means,

방향은 그런 뜻이지만

but what does the length mean?

길이는 무슨 뜻일까요?

Well, if you take a look,

한번 보세요

take a look at these red vectors here.

여기 빨간색 벡터를 보세요

So red means that they should be considered very, very long.

빨간색은 아주 길다는 뜻입니다

And the graph itself,

그래프 자체는

the point they correspond to on the graph

이 점들에 대응하는 그래프 위 점은

is just way off screen for us

우리가 못 볼 정도로 화면 밖으로 나가 있습니다

because this graph gets really steep

그래프가 굉장히 가파르고

and really negative very fast.

음수 방향으로 빠르게 증가하거든요

So the points these correspond to

그래서 이것들에 대응하는 점들은

have really, really steep slopes

기울기가 많이 가파릅니다

whereas these blue ones over here,

한편 이쪽 파란 것들은

you know, it's kind of a relatively shallow slope.

비교적 완만한 기울기입니다

By the time you're getting to the peak,

꼭대기에 다다를 때쯤 되면

things start leveling off.

경사가 완만해지기 시작합니다

So the length of the gradient vector

기울기 벡터의 길이는

actually tells you the steepness

사실 그 최대 경사 방향의

of that direction of steepest ascent.

기울기를 알려 줍니다

But one thing I want to point out here,

여기서 짚고 넘어갈 것 한 가지는

it doesn't really make sense immediately looking at it,

보는 순간 이해가 되진 않는다는 것입니다

why just throwing the partial derivatives into a vector

왜 벡터값에 편도함수를 던져대면

is gonna give you this direction of steepest ascent.

최대 경사의 방향이 나오는지요

Ultimately it will.

나중엔 이해가 될 겁니다

We're gonna talk through that

말로 설명해 드리고

and I hope to make that connection pretty clear,

그 관련성을 명확하게 하고 싶지만

but unless you're some kind of intuitive genius,

타고난 천재가 아닌 이상

I don't think that connection is at all obvious at first.

처음엔 눈치채기 어려울 것 같습니다

But you will see it in due time.

시간이 지나면 보일 겁니다

It's gonna require something

방향도함수라 불리는 것이

called the directional derivative.

필요하게 됩니다

See you next video.

다음 동영상에서 만납시다

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