Teaching Calculus

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  • Nikolay Brodskiy
  • Alexander Brodsky

Part 02 (Transcript)

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So the ...

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The typical example of use is ...

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... [...] can be seen on Google Maps.

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If you go to Smoky Mountains ...

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And then by default you don't see all those contour maps.

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So you have to check the terrain.

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And if you check that and zoom in ...

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... you begin see those curves.

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And the way they do it on a map ... well these days ...

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... you see they show you the terrain by colors.

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Er ... in ...

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Well on the old maps you may see only those level curves.

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And each curve has a number sitting on it.

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And you will know what number is, right?

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It's the elevation of the curve.

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So it is that you can think of this terrain being a function ...

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... in three-dimensional space ...

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... where the domain is horizontal.

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Think of it as x, y plane.

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And the value of the function is equal to the height.

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And it goes vertically.

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And what we see in 3D is the terrain as a graph.

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But what you see on a map is only the contour map of all those ...

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Er ...

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And then of course you can do something about this map.

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You can analyze some details.

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Say er ... even without ...

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... that ...

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Well without that color ...

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... showing you what the hill looks like ...

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... you can still look at all those curves ...

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... and see where the top of the hill ...

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... is probably located, right?

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So if you look at that ...

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... that curve, right?

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Well the top of the hill is probably somewhere inside.

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And you see how those curves converge to something ...

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... like a point.

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And that's probably is the top of the hill.

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If you know that it is a hill ...

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Not ...

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... the other way around, right? So ...

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Those numbers help you to identify ...

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... whether the height increases or decreases.

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That's what we will never see by looking at curves.

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And OK. So this is going to be our ...

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... connection of a function of two variables to ...

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... maybe everyday experience.

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You may want to think about ...

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... walking in mountains.

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And thinking about those level curves ...

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... thinking about maps and thinking about what is it that you can learn about a function ...

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... by looking at the contour maps, by looking at the terrain.

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Er ... you are going back and forth from algebra to geometry.

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