1
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Let me generalize this idea of average speed.
2
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So, average speed from time 1 (t₁) to time 2 (t₂),
3
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some time interval along your journey,
4
00:00:10,108 --> 00:00:15,194
is just the distance you traveled during that time interval,
5
00:00:15,194 --> 00:00:19,117
from time 1 (t₁) to time 2 (t₂), divided by how long that time took,
6
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this final time minus the initial time (t₂ - t₁).
7
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It's still a distance over a time.
8
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And this distance is how far you traveled during this time.
9
00:00:31,529 --> 00:00:37,270
Okay, so the problem with this is we still haven't answered the question:
10
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How fast were you going exactly 10 minutes into the ride?
11
00:00:48,834 --> 00:00:55,202
So the question is: how fast were you going exactly 10 minutes into your ride...
12
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not "What's your average speed from 9 to 11?"
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but, "At exactly t=10, what is your speed?"
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Well, we know how to calculate the speed using this formula.
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But this formula requires two times and we're really only given one.
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So this is over some time interval from t₁ to t₂,
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but I want to know the time at exactly 10.
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So, we're in a little bit of a bind, and calculus
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and this idea of a derivative is going to get us out of the bind.
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So here's what we do: We take a time interval
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maybe from t=10 to t=11
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and then we calculate the average speed for that.
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So we might say that the speed at exactly t=10
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is the average speed from t=10 to t=11
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But you might object and say, "This can't
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be right because the speed might be changing.
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Maybe you're speeding up or slowing down a lot
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as you go from minute 10 to minute 11.
29
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So this really isn't your speed at exactly t=10, this could be changing
30
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and this is not an accurate way of looking at it."
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So that's a reasonable objection, actually.
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And here's one way we could address it.
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You could say, okay, yeah, you're right
34
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the speed might be changing
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from 10 to 11, so what if I consider a smaller interval?
36
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I'll go from 10 to 10.1
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That's just a tenth of a minute.
38
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Surely the speed isn't changing much.
39
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So the average speed is a good approximation
40
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to the speed exactly at t=10.
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And you might object again:
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"Well, how do you know? Maybe you're speeding up or slowing down
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a lot and maybe you stopped for just an instant in
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here so that this average speed is not
45
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a good representation, is not a good approximation to this exact speed."
46
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Okay, that's again a reasonable objection.
47
00:03:06,946 --> 00:03:12,681
So all right, fine, I'll try this.
48
00:03:12,681 --> 00:03:14,609
Maybe I'll just calculate the average speed
49
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from t=10 to t=10.01
50
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And again you might object
51
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"Well, yes, this is a really small time interval
52
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but still, your speed could be changing in
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this time interval, so it's not a good approximation or
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representation of the speed at this particular
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instant." So again I could meet this objection
56
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and we could continue arguing back and forth
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And so on and so on and so on
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considering a smaller and smaller and smaller
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time interval here. And so the way out of this bind
60
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(it seems we could play this game forever)
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is to agree that if the right hand side is getting closer and closer
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to something, then we say that's the speed exactly at t=10.