WEBVTT
1
00:00:03.490 --> 00:00:06.459 A:middle L:90%
this problem is about high tides, and we have
2
00:00:06.459 --> 00:00:09.089 A:middle L:90%
this equation D of tea, which represents the depth
3
00:00:09.089 --> 00:00:12.429 A:middle L:90%
of the water and tea representative, represents the number
4
00:00:12.429 --> 00:00:15.800 A:middle L:90%
of hours since midnight, and we're asked for how
5
00:00:15.800 --> 00:00:19.420 A:middle L:90%
fast the tide is rising or falling at different times
6
00:00:19.420 --> 00:00:21.800 A:middle L:90%
. And when you see the phrase how fast you
7
00:00:21.800 --> 00:00:24.230 A:middle L:90%
want to think about rate of change and when you
8
00:00:24.230 --> 00:00:27.230 A:middle L:90%
think about rate of change, that means derivative in
9
00:00:27.230 --> 00:00:29.710 A:middle L:90%
calculus. And so we want to start by finding
10
00:00:29.710 --> 00:00:32.619 A:middle L:90%
the derivative of the function so deep prime of tea
11
00:00:33.500 --> 00:00:36.130 A:middle L:90%
, the derivative of 70 And then we move on
12
00:00:36.130 --> 00:00:38.850 A:middle L:90%
to the other term and we can leave the five
13
00:00:38.859 --> 00:00:40.840 A:middle L:90%
and take the derivative of co sign, which is
14
00:00:40.840 --> 00:00:43.710 A:middle L:90%
negative sign. So we have negative sign of the
15
00:00:43.710 --> 00:00:49.659 A:middle L:90%
entire inside. And then, according to the chain
16
00:00:49.659 --> 00:00:52.490 A:middle L:90%
rule, we're going to multiply by the derivative of
17
00:00:52.490 --> 00:00:55.560 A:middle L:90%
the inside. And so we would be multiplying by
18
00:00:56.039 --> 00:01:03.090 A:middle L:90%
0.503 Okay, so now that we have that derivative
19
00:01:03.090 --> 00:01:04.120 A:middle L:90%
, we could simplify it if we wanted to.
20
00:01:04.329 --> 00:01:07.480 A:middle L:90%
But chances are we're just going to be using a
21
00:01:07.480 --> 00:01:10.609 A:middle L:90%
calculator from this point. So the calculators not going
22
00:01:10.609 --> 00:01:12.120 A:middle L:90%
to mind if you don't simplify it for part A
23
00:01:12.120 --> 00:01:15.459 A:middle L:90%
. We want to find the speed of the tide
24
00:01:15.459 --> 00:01:19.480 A:middle L:90%
changing how fast the tide is rising or falling at
25
00:01:19.489 --> 00:01:22.540 A:middle L:90%
3 a.m. And 3 a.m. is going to be three
26
00:01:22.540 --> 00:01:25.420 A:middle L:90%
hours since midnight, so we're going to use T
27
00:01:25.420 --> 00:01:27.629 A:middle L:90%
equals three. We're going to substitute that into the
28
00:01:27.629 --> 00:01:33.549 A:middle L:90%
derivative, use a calculator and you get approximately 2.39
29
00:01:34.239 --> 00:01:40.469 A:middle L:90%
meters per hour for part B. Three more hours
30
00:01:40.469 --> 00:01:42.030 A:middle L:90%
have gone by, and now it's 6 a.m. So
31
00:01:42.030 --> 00:01:46.950 A:middle L:90%
T equals six. We're going to substitute six into
32
00:01:46.950 --> 00:01:49.150 A:middle L:90%
the derivative, use a calculator. Make sure your
33
00:01:49.150 --> 00:01:51.819 A:middle L:90%
calculators and radiance, by the way, and we
34
00:01:51.819 --> 00:01:59.069 A:middle L:90%
get 60.9 to 6 meters per hour. And for
35
00:01:59.069 --> 00:02:01.140 A:middle L:90%
part C, it's now 9 a.m. So we're going
36
00:02:01.140 --> 00:02:05.930 A:middle L:90%
to use T equals nine. Substitute that into the
37
00:02:05.939 --> 00:02:08.469 A:middle L:90%
derivative. Put it in your calculator on you get
38
00:02:08.479 --> 00:02:15.050 A:middle L:90%
approximately negative 2.28 Now, why is it negative?
39
00:02:15.840 --> 00:02:17.020 A:middle L:90%
If the derivative is negative, that means you're quantity
40
00:02:17.020 --> 00:02:20.300 A:middle L:90%
is decreasing. So that's telling us that the tide
41
00:02:20.300 --> 00:02:23.530 A:middle L:90%
is going down and then finally, for party,
42
00:02:23.530 --> 00:02:27.039 A:middle L:90%
it's noon, so T is 12 12 hours since
43
00:02:27.039 --> 00:02:30.939 A:middle L:90%
midnight, and the rate of change AT T equals
44
00:02:30.939 --> 00:02:34.889 A:middle L:90%
12 is approximately negative. 1.21 Again, the tide
45
00:02:34.889 --> 00:02:37.150 A:middle L:90%
must still be going down at this point in time
46
--> A:middle L:90%
.