Teaching Calculus

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  • Nikolay Brodskiy
  • Alexander Brodsky

Part 05 (Transcript)

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Er ... But ...

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There are two important things in the whole discussion that I wanted to make.

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One thing is that in reality, in real world ...

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... the concept of distance is never precise, has nothing in real world.

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And there is no concept of straight.

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There is nothing straight in real world.

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Unless you firmly believe that light follows straight path ...

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But then you know it's not true even in ...

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... small scale world.

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You know that if you have water ...

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Right? And you have light ...

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... hitting water under angle.

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It will go ...

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It will leave ... it will go underwater under other angle, right?

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So, that's what you were learning in school and you probably believe in that, don't you?

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So, it makes angle.

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And it doesn't go straight.

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And of course there is a drastic change of things happening here.

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But we can easily imaging the slow change of density.

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Right? And the ...

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... mirage, right?

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When you drive and you see ...

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... on the interstate ...

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... the piece of water that disappears after some time ...

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... so that's actually a reflection of the sky.

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And that happens sort of because of this.

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So, er ...

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There's nothing straight and ...

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The only straight thing can be there.

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So, come there.

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Across this fine line ...

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... between reality and abstraction.

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And ...

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Well, we can step back sometimes but ...

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That world is so beautiful.

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We can find straight things and not only straight lines but straight planes.

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Right? The ...

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... plane ...

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... you were thinking about on the homework er ...

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... is also an abstraction.

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Well, as there are no straight lines in the world, well, there is ...

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... no way we can have straight plane in the world.

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Only approximation of that. Er ...

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Yes, so, that was one point nothing straight in reality.

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But the other point is that er ...

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... what we do in reality is always an approximation.

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And that will be the main thing in this course though the whole semester.

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We will only do approximations.

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And we'll do approximations even there.

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But we'll do them for different reason.

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It is not because we don't have exact things ...

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... and we have to do approximations only like in reality.

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We do have exact things.

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But ...

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Despite of having exact things ...

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... we will do approximations because those exact things are too complicated.

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Because if you want to express algebraically the number Pi ...

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... you can not do it in finite time, right?

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So, you'll have to do approximation and ...

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It will happen all the time ...

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... at the level of numbers, at the level of functions, at the level of, well, anything over there.

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But the reason is only because the exact things are too complicated.

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All right. So what we created there ...

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... in that abstract world is a plane made up of ...

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... two lines, right? And those are called X-axis, Y-axis.

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Any point on the plane gets ...

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... two numbers assigned x and y ...

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... in the given coordinate system and ...

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... those two numbers together determine the point uniquely. And ...

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And then we measure distance by that formula square root of sum of squares.

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OK.

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So what else is there?

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