Transcript

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What is the first derivative of that mysterious function s?

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And here it comes the amazing part ...

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... the function s is so mysterious. It's hard to even say something, well anything about it.

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But ...

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We are going to be able to compute what the derivative is ...

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... without even knowing the formula for s.

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Well the way we'll do it is by saying by definition the derivative is a limit ...

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... limit as delta t goes to zero ...

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... of the ratio ...

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... the change of the function s ...

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... s of t plus delta t minus s of t ...

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... divided by the change of ...

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... the function t [...] delta t.

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That's the definition.

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And now I'll do a little trick ...

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... rewriting that fraction in a little bit different way.

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And what I [...] is ...

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... this ...

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... this picture of how the cars are moving.

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So what I have there is ...

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... I have some time t.

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And at that time the first car is in the position f of t right here.

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And ...

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That position corresponds ...

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... to ...

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... the time s of t for the second car ...

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... when the second car was exactly at the same position ...

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... g of s of t.

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Right?

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So that was ...

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... a part of the definition of the function s of t.

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Now what happens when the time delta t passes?

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t becomes t plus delta t.

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And the first car shows up at ...

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... a different point. Somewhere there.

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What about that point? Well this is going exactly the point where the second car is ...

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... when t is equal to t plus delta t.

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So at that time the second car is exactly in the same point.

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So what is it that I see there in the fraction?

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What is going to be s of t plus delta t minus s of t?

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What's the meaning of the difference?

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Well that is the time ...

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... spent by the second car ...

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... as it moved from this point to that point.

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So this is delta s -- the time spent by the second car.

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And what I want to relate it to is I want to relate it ...

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... to the time spent by the first car ...

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... as it moved from the same first point to the same ending point.

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So two cars were travelling along the same segment of the road ...

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... and spent different times.

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Now if I relate ...

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... that time spent ...

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... to the length of the segment -- let's call it delta L -- the length of the segment of the road.

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And I multiply it by delta L back so that the fraction is kept.

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Then this fraction ...

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... has the perfect meaning.

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The distance travelled by the first car divided by the time spent.

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That's the speed of the first car.

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That speed of the first car is the magnitude ...

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... of the velocity.

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Now what is that fraction?

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Well that fraction is ...

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... the reciprocal of the speed of the second car ...

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... which is [...] to the magnitude of velocity.

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So the whole thing, the whole limit is going to be the fraction ...

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... f prime divided by g prime.

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It is simply the ratio of the speeds.

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Well it should be the speed, not the vector.

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Right?

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So that's the remaining thing.

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We found the formula for the derivative of s.

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An exact explicit formula in terms of f and g.

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Although we had no idea what the formula for s is.

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That's what I like about this. Er ...

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Well the use ... highly non-trivial use of the concept of a function.

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I didn't use any formula for the function.

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Just the idea of the existence of a function.

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And some definition of derivative helps me to differentiate.

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