쉐도잉 연습: Derivatives: Crash Course Physics #2 - 영상으로 영어 말하기 배우기

Every discipline of science has its very own special language, the way it communicates the ideas that it investigates.

For example, biology finds order in the world by giving every living thing a name in Latin.

Chemistry has a system of prefixes, suffixes, and numerals to tell you, in a word or two, the exact composition of an atom or a compound.

Physics has to communicate its ideas differently.

The language of physics is mathematics.

Because if you're trying to describe how the world works, you really have to know how things relate to each other in a mathematical way.

For example, we've been talking a lot about position, velocity, and acceleration, and how they're all connected.

Velocity is a measure of your change in position, and acceleration is a measure of your change in velocity.

They're connected.

One quality will describe how the other is changing.

And the way we describe change in mathematics is through calculus.

Calculus explains how and why things change using derivatives, which help you to determine how an equation is changing.

as well as with integrals, which you can use to calculate the area under a curve.

Derivatives and integrals themselves are closely connected, but let's start with derivatives.

You probably won't be able to go straight from this lesson to your calculus final, but hopefully in about 10 minutes you will be able to understand some of the maths

that scientists have been using to think about physics for the last 400 years or so.

And you'll also have a new way to fight speeding tickets.

You know, just in case.

Last time we talked about that unfortunate incident where you got a speeding ticket.

Your speedometer was broken, but because we knew your acceleration, we were able to calculate how fast you were going when the cops pulled you over.

So now, let's talk about what happens next.

Say the police drive off.

You're ready to get back on the road, so you hit the gas and zoom forward, moving faster and faster.

But in this scenario, we don't know your acceleration we only know how much your position is changing over time.

In this instance, your position happens to be equal to the amount of time you've been driving squared.

So we'd write that as the equation x equals t squared.

20 seconds in, you pass a detector with a sign that tells you your speed.

You keep driving, foot still on the gas, before you realize what number you saw on the sign.

And oh no!

You just got a speeding ticket in the last episode for

doing 126 km an hour in a 100 km an hour zone.

And now the detector says you're going even faster!

Now, you want to know if the number on the detector is accurate.

In other words, you want to find your velocity at the exact moment you passed it.

That velocity is just a measure of your changing position, its derivative.

So to find your velocity, we'll need to find the derivative of your position.

And in order to determine that, we first need to talk about limits.

Not speed limits, I mean the derivatives kind.

I'll explain.

Limits are based on the idea that if you have an equation on a graph, you can often predict what it's going to look like at one point, just by knowing what it looks like at the surrounding points.

For example, let's say you have a graph of x equals t squared from our speeding scenario above,

and you want to find out how your position is changing at the exact moment that time is equal to zero.

That is what we call the limit as t approaches zero.

So you take a look around at what's happening around t equals zero.

At t equals one, x is one.

At t equals zero point five, x is zero point two five.

And at t equals 0.1, x is 0.01.

You can probably tell that as we get closer and closer to t equals zero, your value of x is getting closer to zero, too.

That's what mathematicians mean when they talk about a limit.

Limits are useful because they can help predict what happens as you make intervals smaller.

An interval is just a range on a graph.

It's the space between two points on the horizontal axis.

So the first thing we can try is calculating your average velocity over the interval from 15 to 20 seconds.

To do that, we use an equation that we talked about last time – your average velocity, which is equal to the change in your position divided by the change in time.

That turns out to be 35 meters per second.

Problem is, it's just an average.

It's not exactly how fast you were going after 20 seconds of acceleration when you passed the detector.

Because of limits, we know

that you could get a little closer to the right number by calculating your average over smaller and smaller intervals.

Then you'd see that the number seems to be getting closer and closer to 40 meters per second.

Which means you're going to need to slow way down

if you don't want to get your second speeding ticket of the day.

But that's the idea of derivatives.

You can use infinitely tiny intervals to figure out exactly how an equation is changing at any moment.

You can even come up with an equation to describe the change.

That's exactly what velocity is, an equation that describes change in position.

An acceleration describes change in velocity.

So we call velocity the derivative of position, and acceleration the derivative of velocity.

Now, when it comes to how you can express a derivative in writing, mathematicians have come up with shortcuts.

Like what's known as the power rule.

As the name suggests, it's used for equations with variables raised to powers or exponents, as long as the exponent is a number.

For example, x equals t squared would work with the power rule, because t is raised to the power of 2.

The power rule says that, for these kinds of equations, to calculate the derivative, all you need is one weird trick.

Take the number of that exponent, in this case 2, and stick it in front of the variable.

Then you subtract 1 from the exponent.

And that's your derivative!

So the derivative of x equals t squared is just 2t, Which means that no matter how many seconds you've got your foot on the gas, your velocity will be 2t.

So double the number of seconds.

After 5 seconds, you were going a modest 10 meters per second.

But after 20 seconds, you were going a full 40 meters per second, which is not good.

We'd write that like this, where dx over dt is just a way of saying

that we're taking the derivative of the part of the equation that involves t.

Or, as a mathematician would put it, we're taking the derivative of x with respect to t.

You'll also sometimes see this written in a different way.

If f is equal to t squared, then f prime of t is equal to 2t.

Now let's try to find a couple more derivatives using the power rule.

X equals 7t to the power 6 is another power-style equation.

It has a variable t raised to the power 6, with a number in front of it, 7.

The first thing we do is take the exponent and stick it in front of the variable.

But there's already a number in front of t, 7.

So we end up multiplying them.

7 times 6 is 42.

Then we subtract 1 from the power that t is raised to.

So we end up with 42t to the power 5.

Same goes for equations where the exponents are fractions or decimals.

So the derivative of t to the power half is half t to the negative one-half.

It works for negative exponents, too.

The derivative of t to the power minus 2 is just negative 2t to the negative third.

Now, there are a few more equations whose derivatives you should understand.

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Trigonometry, which we use to calculate the angles and size of triangles, is going to This is going to come up a lot in physics, because we'll be using right-angled triangles all the time.

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So it's a good idea to know how to find the derivatives of sine x and cosine x.

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Sine tells you that if you have a right-angled triangle, and x is the angle in that triangle, then sine x will be the length of the side opposite that angle, divided by the hypotenuse.

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Cosine does the same thing, just with the side next to the angle, divided by the hypotenuse.

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So their graphs will tell you what those ratios will be, depending on the angle.

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We can actually try to guess the derivative of sine x, just by looking at its graph.

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We can see that the curve has turning points every so often, at x equals minus 90 degrees, x equals 90 degrees, and so on, repeating every 180 degrees.

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Meaning at those points, the equations aren't changing at all.

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So the derivative at these turning points is also going to be exactly zero.

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Let's pull up another graph where we'll plot the derivative and put little dots where we'll know it'll be zero.

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Now, what's happening between those turning points?

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Well, from minus 270 to minus 90 degrees, sine x is decreasing.

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In other words, its change, and therefore its derivative, must be negative.

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Then, from minus 90 to 90 degrees, sine x is increasing, so it'll have a positive derivative, and so on.

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There are actually a lot more clues in this graph to help us find the derivative, but we already know enough to make a decent guess.

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If we smoothly connect the dots on the graph of our derivative, keeping in mind where the curve should be positive and where it should be negative,

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hey, this derivative is looking a whole lot like the graph of cosine x.

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That's because it is!

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The derivative of sine is just cosine, and that's going to come up a lot.

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And so will these, which you can work out on your own by repeating what we just did with the graphs of sine x

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and cosine x.

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Another important derivative that comes up a lot is a very special case, and that's e to the x.

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The derivative of e x is just...e x.

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Yep, and that's it, no matter what.

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In fact, that's one way to define e, which is kind of like pi, in the sense that it's a simple letter, representing a very specific, irrational number,

125

about 2.718, with more digits after the decimal point that go on forever.

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It has all sorts of uses in calculus, but it also shows up when you're studying things like finance and probability.

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Armed with all these ways to find derivatives, you could pretty much take any equation of your position and calculate its derivative, and therefore, your velocity.

128

In the same way, you could take the derivative of your velocity and find your acceleration.

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But there's a whole other part of calculus that we haven't even talked about yet.

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Integrals, which will let you do this backwards.

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With integrals, you can use your acceleration to find your velocity, and your velocity to find your position.

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But we'll save that for next time.

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Today you learned about limits and that derivatives use them to describe how an equation is changing.

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We also talked about a few different kinds of derivatives – powers, constants, trigonometry, and e to the power x.

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Crash Course Physics is produced in association with PBS Digital Studios.

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You can head over to their channel to check out amazing shows like Deep Look, The Good Stuff, and PBS Space Time.

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This episode of Crash Course is filmed in the Dr. Cheryl C.

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Kinney Crash Course Studio, the help of these amazing people and our graphics team is Thought Cafe.

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영상의 맥락과 배경

이 비디오는 물리학에서 수학이 어떻게 사용되는지 설명하는 내용으로, 위치, 속도, 가속도의 관계와 미분의 기초를 다룹니다. 말투는 친절하고 쉽게 풀어내어, 복잡한 개념도 일상적인 예시(과속 티켓 사건)를 통해 이해하기 쉽게 만들었습니다. 영상 속 대화는 학술적이지만 생생한 예시가 많아, 영어 듣기와 말하기 연습에 좋은 자료입니다.

일상 대화에 쓸 수 있는 상위 5개 표현

find order in the world: 세상에서 규칙을 찾다. "생물학은 라틴어 이름을 통해 세상에서 규칙을 찾습니다."처럼 사용합니다.
connected to: ~와 연결되다. "속도는 위치 변화와 연결되어 있습니다."
figure out: 알아내다, 파악하다. "정확한 속도를 알아내기 위해 미분을 사용합니다."
slow down: 느리게 하다, 감속하다. "과속을 피하려면 감속해야 합니다."
come up with: 생각해 내다, 고안하다. "수학자들은 미분 기호를 생각해 냈습니다."

스텝별 샤도잉 가이드 (shadow speak)

이 영상은 과학 용어가 많아 조금 어렵지만, 천천히 따라하면 실력이 늘어날 것입니다. 다음 단계를 따라해 보세요:

영상을 2-3번 듣고 전체 흐름 파악하기: 속도와 억양을 먼저 느껴보세요. "derivatives"나 "limits" 같은 단어의 발음에 집중하지 말고, 전체 내용을 이해하세요.
짧은 구절로 나누어 반복하기: "Velocity is a measure of your change in position"처럼 짧은 문장을 골라, 영상이 끝나는 동시에 따라 말해보세요. shadowing site에서도 이 방법을 자주 사용해요.
억양과 강세에 주의하기: "OH NO!"처럼 감정이 담긴 부분은 얼마나 높게 말하는지, "exactly"처럼 강조하는 단어는 어떻게 발음하는지 관찰하세요. shadow speech 연습에 중요합니다.
자신의 목소리를 녹음하고 비교하기: 영상과 자신의 녹음을 비교해서, 발음이 다른 부분이나 속도가 맞지 않는 부분을 찾아 고치세요. 영상 영어 공부에 효과적입니다!

천천히 연습하면, 과학 용어도 자연스럽게 사용할 수 있고, 영어 듣기와 말하기 실력이 크게 향상될 거예요. 파이팅!

이 영상의 문법

화자가 가장 많이 쓰는 문형을 영상 속 실제 표현과 함께 정리했습니다.

문형	영상 속 표현
현재완료진행 have/has been + -ing — 전에 시작해 지금도 계속되는 동작	we've been talking · have been using · you've been driving
수동태 be + 과거분사 — 누가 하는지보다 무슨 일이 일어나는지에 초점	was broken · are based · is raised
관계절 who / which + 절 — 사람이나 사물에 대한 추가 정보	velocity, which is · second, which is · Integrals, which will
현재완료 have/has + 과거분사 — 과거의 일이 지금도 관련이 있을 때	have come · haven't even talked

쉐도잉이란? 영어 실력을 빠르게 키우는 과학적 방법

쉐도잉(Shadowing)은 원래 전문 통역사 훈련을 위해 개발된 언어 학습 기법으로, 다언어 학자인 Dr. Alexander Arguelles에 의해 대중화된 방법입니다. 핵심 원리는 간단하지만 매우 강력합니다: 원어민의 영어를 들으면서 1~2초의 짧은 지연으로 즉시 소리 내어 따라 말하는 것——마치 '그림자(shadow)'처럼 화자를 따라가는 것입니다. 문법 공부나 수동적인 청취와 달리, 쉐도잉은 뇌와 입 근육이 동시에 실시간으로 영어를 처리하고 재현하도록 훈련합니다. 연구에 따르면 이 방법은 발음 정확도, 억양, 리듬, 연음, 청취력, 말하기 유창성을 크게 향상시킵니다. IELTS 스피킹 준비와 자연스러운 영어 소통을 원하는 분들에게 특히 효과적입니다.

섀도잉 방법: 단계별 전체 가이드 읽기 →

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영상의 맥락과 배경

일상 대화에 쓸 수 있는 상위 5개 표현

스텝별 샤도잉 가이드 (shadow speak)

이 영상의 문법

쉐도잉이란? 영어 실력을 빠르게 키우는 과학적 방법