1
00:00:01,660 --> 00:00:04,926
Let me generalize this idea of average speed.
2
00:00:04,926 --> 00:00:08,332
So, average speed from time 1 (t₁) to time 2 (t₂),
3
00:00:08,332 --> 00:00:10,686
some time interval along your journey,
4
00:00:10,686 --> 00:00:13,222
is just the distance you traveled
5
00:00:13,222 --> 00:00:17,023
during that time interval from time 1 (t₁) to time 2 (t₂),
6
00:00:17,023 --> 00:00:19,902
divided by how long that time took,
7
00:00:19,902 --> 00:00:22,660
this final time minus the initial time.
8
00:00:22,660 --> 00:00:25,977
It's still a distance over a time.
9
00:00:26,147 --> 00:00:31,628
And this distance is how far you traveled during this time.
10
00:00:32,689 --> 00:00:37,239
OK. So, the problem with this is we still haven't answered the question.
11
00:00:37,239 --> 00:00:41,232
How fast were you going exactly 10 minutes into the ride?
12
00:00:50,394 --> 00:00:55,223
So, the question is how fast were you going exactly 10 minutes into your ride?
13
00:00:55,223 --> 00:00:58,071
not what's your average speed from 9 to 11,
14
00:00:58,071 --> 00:01:01,180
but at exactly t = 10, what is your speed?
15
00:01:02,213 --> 00:01:05,557
Well, we know how to calculate the speed using this formula,
16
00:01:06,679 --> 00:01:11,857
but this formula requires two times, and we're really only given one.
17
00:01:12,803 --> 00:01:16,329
So this is over some time interval from t₁ to t₂,
18
00:01:16,329 --> 00:01:19,169
but I want to know the time at exactly 10.
19
00:01:19,748 --> 00:01:21,602
So, we're in a little bit of a bind,
20
00:01:21,602 --> 00:01:25,731
and calculus, and this idea of a derivative is going to get us out of the bind.
21
00:01:26,564 --> 00:01:28,341
So, here is what we do.
22
00:01:29,920 --> 00:01:34,613
We take a time interval, maybe from t = 10 to t = 11,
23
00:01:36,629 --> 00:01:39,241
and then we calculate the average speed for that.
24
00:01:48,573 --> 00:01:51,979
So, we might say that the speed at exactly t = 10,
25
00:01:51,979 --> 00:01:55,672
is the average speed from t = 10 to t = 11.
26
00:01:56,439 --> 00:01:59,251
But, you might object and say, well, this can't be right
27
00:01:59,251 --> 00:02:01,662
because speed might be changing.
28
00:02:01,662 --> 00:02:04,461
Maybe you're speeding up or slowing down a lot
29
00:02:04,461 --> 00:02:08,486
as you go from minute 10 to minute 11.
30
00:02:08,486 --> 00:02:11,678
So this really isn't your speed at exactly t = 10.
31
00:02:11,678 --> 00:02:16,576
This could be changing. This is not an accurate way of looking at it.
32
00:02:17,149 --> 00:02:20,064
So, that's a reasonable objection actually
33
00:02:20,064 --> 00:02:22,889
and here is one way we could address it.
34
00:02:22,889 --> 00:02:27,156
Say, OK, yeah, you're right, the speed might be changing from 10 to 11,
35
00:02:27,156 --> 00:02:30,069
so what if I insert a smaller interval?
36
00:02:30,069 --> 00:02:34,443
I'll go from 10 to 10.1.
37
00:02:34,937 --> 00:02:38,842
That's just a tenth of a minute, surely the speed isn't changing much.
38
00:02:40,712 --> 00:02:46,128
So the average speed is a good approximation to the speed exactly at t = 10.
39
00:02:47,434 --> 00:02:49,149
And you might object again.
40
00:02:49,149 --> 00:02:53,714
Well, how do you know? Maybe you're speeding up or slowing down a lot.
41
00:02:53,714 --> 00:02:56,521
Maybe you stopped for just an instant in here
42
00:02:56,521 --> 00:02:59,990
so that this average speed is not a good representation,
43
00:02:59,990 --> 00:03:02,789
it's not a good approximation to this exact speed.
44
00:03:03,192 --> 00:03:06,597
OK, well, that's again a reasonable objection.
45
00:03:06,597 --> 00:03:12,448
So, fine, I'll try this.
46
00:03:12,448 --> 00:03:18,942
Maybe I'll just calculate the average speed from t = 10 to 10.01.
47
00:03:19,527 --> 00:03:22,472
And again you might object, well,
48
00:03:25,012 --> 00:03:26,417
yes this is a really small time interval,
49
00:03:26,417 --> 00:03:29,612
but still your speed could be changing in this time interval,
50
00:03:29,612 --> 00:03:32,642
so it's not a good approximation or representation
51
00:03:32,642 --> 00:03:35,392
of the speed at this particular instant.
52
00:03:35,392 --> 00:03:38,054
So, again, I could meet this objection
53
00:03:44,313 --> 00:03:47,451
and we could continue arguing back and forth
54
00:03:53,249 --> 00:03:55,163
and so on and so on and so on,
55
00:03:55,163 --> 00:03:59,188
considering a smaller and smaller and smaller time interval here.
56
00:03:59,788 --> 00:04:04,309
And so, the way out of this bind
57
00:04:04,309 --> 00:04:06,400
-- so it seems we could play this game forever --
58
00:04:06,400 --> 00:04:11,342
is to agree that if the right hand side is getting closer and closer to something,
59
00:04:11,342 --> 00:04:15,599
then we say that's the speed exactly at t = 10.