1
00:00:00,012 --> 00:00:12,766
>> [music] Let's fill up this bowl with
this green water.

2
00:00:13,828 --> 00:00:22,608
We can graph the height of the water.
On the x-axis, I've plotted time in

3
00:00:22,608 --> 00:00:25,830
seconds.
And on the y-axis, I've plotted the

4
00:00:25,830 --> 00:00:29,247
water's height in pixels.
And here's the curve that I get.

5
00:00:29,247 --> 00:00:33,873
Each of these little crosses is one of the
measurements that I made at some given

6
00:00:33,873 --> 00:00:37,446
time, I measured the height of the water
and I get this curve.

7
00:00:37,446 --> 00:00:42,250
What do you notice about this graph?
So, one thing I notice right off the bat,

8
00:00:42,250 --> 00:00:47,266
is that the water level increases fairly
rapidly at first, but at the end of this

9
00:00:47,266 --> 00:00:51,750
process, the water level is still
increasing but it's increasing more

10
00:00:51,750 --> 00:00:54,850
slowly.
Of course, this makes sense since this

11
00:00:54,850 --> 00:00:59,855
bowl is narrower at the bottom than it is
at the top, and the water is coming in at

12
00:00:59,855 --> 00:01:04,222
a constant rate.
A bit more formally, alright, dv/dt, the

13
00:01:04,222 --> 00:01:08,317
change in volume of water over time,
that's constant.

14
00:01:08,317 --> 00:01:13,930
I'm pouring in water at a constant rate,
but dh/dt starts off very large, the

15
00:01:13,930 --> 00:01:17,990
water's height is increasing very rapidly
at first.

16
00:01:17,991 --> 00:01:21,213
And by the end of this process, dh/dt is
small.

17
00:01:21,213 --> 00:01:25,237
So, I'd like a function that relates water
height to volume.

18
00:01:25,237 --> 00:01:30,655
Well, if I take a look at the side of the
bowl, the bowl starts off fairly slanted,

19
00:01:30,655 --> 00:01:35,131
and then sort of flattens out, and that's
what I'm graphing here.

20
00:01:35,131 --> 00:01:40,224
I'm graphing the radius of the bowl.
On the x-axis, I'm plotting the height,

21
00:01:40,224 --> 00:01:43,062
that's how far I am from the bottom of the
bowl.

22
00:01:43,062 --> 00:01:47,820
And as I move up the bowl, the radius of
the bowl is increasing and that's what I'm

23
00:01:47,820 --> 00:01:51,914
plotting on the y-axis.
And if you look at the bowl, right, the

24
00:01:51,914 --> 00:01:56,901
bottom of the bowl starts off very slanted
and then it sort of flattens out.

25
00:01:56,901 --> 00:02:02,574
And that's exactly what I'm representing
here with, with this broken line, right?

26
00:02:02,574 --> 00:02:06,545
The bowl starts off fairly slanted and
then flattens out.

27
00:02:06,545 --> 00:02:12,029
Once I know what the radius of the bowl is
at a particular height, I can then think

28
00:02:12,029 --> 00:02:17,103
about the volume of water when then water
level is at some given height.

29
00:02:17,103 --> 00:02:22,002
So, once I know the radius of the bowl at
a particular height, from the bottom of

30
00:02:22,002 --> 00:02:26,516
the bowl, I can then make a graph that
shows the volume of water in the bowl,

31
00:02:26,516 --> 00:02:31,241
given a particular height of water.
So, here on the x-axis, I'm plotting the

32
00:02:31,241 --> 00:02:35,104
height of the water.
And on the y-axis, I'm plotting the volume

33
00:02:35,104 --> 00:02:38,825
of water in the bowl.
And you can see that when the water is

34
00:02:38,825 --> 00:02:43,708
deeper, there's more water in the bowl.
It's an increasing function.

35
00:02:43,708 --> 00:02:49,381
Now, let's think a little bit about the
slope of the tangent line to this graph.

36
00:02:49,381 --> 00:02:55,390
The tangent line initially has not very
large slope, compared to after a while,

37
00:02:55,390 --> 00:03:00,784
the tangent line has a much higher slope.
And what that really means is that

38
00:03:00,784 --> 00:03:06,460
initially, for a given change in volume,
there's quite a change in the height of

39
00:03:06,460 --> 00:03:09,992
the water.
But after a while, the same change in

40
00:03:09,992 --> 00:03:14,460
volume leads to a less dramatic change in
the height of the water.

41
00:03:14,460 --> 00:03:20,066
That really makes sense from our original
experience, we poured all that water into

42
00:03:20,066 --> 00:03:23,150
the bowl.
We think back to a graph of the water

43
00:03:23,150 --> 00:03:26,202
level in the bowl.
Here, I've got time.

44
00:03:26,202 --> 00:03:31,043
Here, I've got the height of the water.
The water is flowing in at a constant

45
00:03:31,043 --> 00:03:36,887
changing volume per unit time.
And initially, aIright, the slope of this

46
00:03:36,887 --> 00:03:43,347
tangent line is quite high, because the
water height is increasing very rapidly,

47
00:03:43,347 --> 00:03:49,617
but after a while, because there's the
same change in volume during this entire

48
00:03:49,617 --> 00:03:54,808
process, the change in the water height is
less dramatic, right?

49
00:03:54,808 --> 00:04:03,189
It's the same amount of water coming in,
but the water's moving up the sides of the

50
00:04:03,189 --> 00:04:05,606
bowl less quickly.