WEBVTT
1
00:00:00.940 --> 00:00:02.339 A:middle L:90%
This is a problem. Number fifty four of the
2
00:00:02.339 --> 00:00:06.030 A:middle L:90%
Stuart Calculus eighth Edition Section two point eight. Use
3
00:00:06.030 --> 00:00:08.220 A:middle L:90%
the definition of a derivative to find at prime of
4
00:00:08.220 --> 00:00:11.150 A:middle L:90%
X and F double primary Kes, then Graff.
5
00:00:11.839 --> 00:00:13.980 A:middle L:90%
Time enough. Double prime on a common screen and
6
00:00:13.980 --> 00:00:16.960 A:middle L:90%
check to see if your answers are reasonable and the
7
00:00:16.960 --> 00:00:19.449 A:middle L:90%
function F it's given us at of X equal to
8
00:00:19.460 --> 00:00:25.739 A:middle L:90%
execute minus X minus three X name. So let's
9
00:00:25.739 --> 00:00:30.179 A:middle L:90%
start with and paramedics by using the derivative ah,
10
00:00:30.190 --> 00:00:33.250 A:middle L:90%
definition of a derivative. The prime of X is
11
00:00:33.250 --> 00:00:37.649 A:middle L:90%
given as the limit as each of purchase. Zero
12
00:00:39.670 --> 00:00:44.100 A:middle L:90%
of the function hearing Next Hugo Industry X Evaluated Expo
13
00:00:44.100 --> 00:00:49.149 A:middle L:90%
Sage. So the text plus age quantity cubed minus
14
00:00:49.149 --> 00:00:52.619 A:middle L:90%
three times his age. And then we're going to
15
00:00:52.619 --> 00:01:00.549 A:middle L:90%
subtract the function execute ministry and this on the right
16
00:01:00.549 --> 00:01:03.250 A:middle L:90%
of right teach. Okay. Our next step is
17
00:01:03.250 --> 00:01:07.409 A:middle L:90%
to expand the numerator. Separate all the terms X
18
00:01:07.409 --> 00:01:11.010 A:middle L:90%
plus h Cubed is binomial cubed will give us X
19
00:01:11.010 --> 00:01:18.549 A:middle L:90%
cubed plus three expert H plus three x h squared
20
00:01:19.540 --> 00:01:23.840 A:middle L:90%
not plus age cubed. Then we distribute the negative
21
00:01:23.840 --> 00:01:26.959 A:middle L:90%
three in the next term, making three expense three
22
00:01:26.969 --> 00:01:30.219 A:middle L:90%
each and then we should make it appear negative,
23
00:01:30.219 --> 00:01:37.150 A:middle L:90%
X cubed Class three X and they're all divided by
24
00:01:37.170 --> 00:01:41.090 A:middle L:90%
each. Hey, we have a next cute how
25
00:01:41.090 --> 00:01:44.049 A:middle L:90%
you hear minus and execute high here. Ah,
26
00:01:44.209 --> 00:01:49.140 A:middle L:90%
a positive three x here and a negative directs there
27
00:01:49.239 --> 00:01:53.629 A:middle L:90%
that were canceled and then each remaining term has it
28
00:01:53.739 --> 00:01:56.849 A:middle L:90%
one age values. So the agent the denominator counsels
29
00:01:56.849 --> 00:01:59.450 A:middle L:90%
with one h and each of the remaining terms.
30
00:02:01.719 --> 00:02:07.489 A:middle L:90%
And that leaves us with for Lim as each approaches
31
00:02:07.500 --> 00:02:14.330 A:middle L:90%
zero out of the function three x squared less three
32
00:02:14.330 --> 00:02:23.129 A:middle L:90%
x h pompous h squared minus three. And if
33
00:02:23.129 --> 00:02:27.370 A:middle L:90%
we take a look here each term, that has
34
00:02:27.370 --> 00:02:30.650 A:middle L:90%
an a jewel that being negligible since ages approaching zero
35
00:02:30.650 --> 00:02:31.580 A:middle L:90%
. So three x h and H squared will go
36
00:02:31.580 --> 00:02:36.379 A:middle L:90%
to zero and the resulting terms are are derivative three
37
00:02:36.379 --> 00:02:40.520 A:middle L:90%
X squared minus three. Now, for the second
38
00:02:40.520 --> 00:02:43.830 A:middle L:90%
derivative, we do the same procedure except we are
39
00:02:43.830 --> 00:02:49.180 A:middle L:90%
using the function have paramedics instead of ethics so that
40
00:02:49.189 --> 00:02:52.319 A:middle L:90%
the second rate of double prime is the limited.
41
00:02:52.319 --> 00:02:54.949 A:middle L:90%
H approaches zero of primal axe evaluated at X plus
42
00:02:54.949 --> 00:03:02.819 A:middle L:90%
H three times X plus H squared minus three and
43
00:03:02.819 --> 00:03:07.270 A:middle L:90%
there were subjecting the function minus three X squared minus
44
00:03:07.270 --> 00:03:15.469 A:middle L:90%
three all the better for age. Next step is
45
00:03:15.469 --> 00:03:21.599 A:middle L:90%
to simplify the numerator. Here's a binomial squared so
46
00:03:21.599 --> 00:03:24.620 A:middle L:90%
we'LL have X squared plus two Ex age plus eight
47
00:03:24.620 --> 00:03:29.020 A:middle L:90%
squared or not by three. Gives us three X
48
00:03:29.020 --> 00:03:38.810 A:middle L:90%
squared class six X h plus three age squared minus
49
00:03:38.810 --> 00:03:39.729 A:middle L:90%
three And then over here, we're going to subtract
50
00:03:40.620 --> 00:03:43.680 A:middle L:90%
Are we're going to This should be the negatives from
51
00:03:43.680 --> 00:03:50.250 A:middle L:90%
the ministry X squared plus three This all the h
52
00:03:51.819 --> 00:03:53.210 A:middle L:90%
ah, now we canceled three extorted with negative three
53
00:03:53.210 --> 00:03:55.889 A:middle L:90%
x squared And then? Then they got three and
54
00:03:55.889 --> 00:04:00.289 A:middle L:90%
the positive three. And then agent the denominator comes
55
00:04:00.289 --> 00:04:01.439 A:middle L:90%
around with one inch of each of the remaining terms
56
00:04:02.439 --> 00:04:09.909 A:middle L:90%
leaving us with the limit is H approaches zero of
57
00:04:09.909 --> 00:04:15.840 A:middle L:90%
six. Eight x plus three each and his age
58
00:04:15.840 --> 00:04:17.209 A:middle L:90%
approaches zero three to purchase zero, leaving us with
59
00:04:17.220 --> 00:04:19.970 A:middle L:90%
just six. X as our derivative, our secondary
60
00:04:19.970 --> 00:04:24.899 A:middle L:90%
of F double prime of acts. So we have
61
00:04:25.290 --> 00:04:28.879 A:middle L:90%
determined deaf prime and asked about crime I'm to recall
62
00:04:28.879 --> 00:04:31.120 A:middle L:90%
after vexes x cubed minus tree x f Prima vex
63
00:04:31.129 --> 00:04:33.500 A:middle L:90%
is at three x squared my street. Enough double
64
00:04:33.500 --> 00:04:36.279 A:middle L:90%
primer vexes six x So the next part is to
65
00:04:36.279 --> 00:04:38.990 A:middle L:90%
grab all three of these air to show that the
66
00:04:38.990 --> 00:04:43.699 A:middle L:90%
answers seem reasonable. So the original function of X
67
00:04:44.089 --> 00:04:46.459 A:middle L:90%
is shown in purple executed minus three x the derivative
68
00:04:46.459 --> 00:04:48.600 A:middle L:90%
, or that it's three expert ministrations in red,
69
00:04:49.089 --> 00:04:53.250 A:middle L:90%
and it threw her after that is six x,
70
00:04:53.350 --> 00:04:57.660 A:middle L:90%
which is shown in blue. So the original function
71
00:04:57.990 --> 00:05:00.069 A:middle L:90%
in purple is a cubic function. It ah is
72
00:05:00.069 --> 00:05:02.410 A:middle L:90%
increasing up until its maximum point here, meaning that
73
00:05:02.410 --> 00:05:04.500 A:middle L:90%
it's derivative which is shown in red must be all
74
00:05:04.500 --> 00:05:06.490 A:middle L:90%
positive, which is true, but then as the
75
00:05:06.490 --> 00:05:11.129 A:middle L:90%
derivative zero, meaning that service crosses the X axis
76
00:05:11.139 --> 00:05:13.980 A:middle L:90%
. Then afterwards it's decreasing until it's minimums. All
77
00:05:13.980 --> 00:05:17.209 A:middle L:90%
negative in that region followed a pie, and afterwards
78
00:05:17.209 --> 00:05:19.370 A:middle L:90%
it's increasing, which means that it's rude is all
79
00:05:19.370 --> 00:05:23.629 A:middle L:90%
positive, so that seems pretty consistent. On the
80
00:05:23.629 --> 00:05:26.370 A:middle L:90%
drill you have prime in red. It is always
81
00:05:26.370 --> 00:05:29.610 A:middle L:90%
decreasing up until its minimum. Here on DH.
82
00:05:29.610 --> 00:05:32.199 A:middle L:90%
It's derivative. Here in blue is showing just that
83
00:05:32.689 --> 00:05:36.819 A:middle L:90%
since function have prime is decreasing, double crime must
84
00:05:36.829 --> 00:05:41.569 A:middle L:90%
be negative. And then it's ah, derivative of
85
00:05:41.579 --> 00:05:44.120 A:middle L:90%
a derivative of brain eyes equal to zero here at
86
00:05:44.120 --> 00:05:45.790 A:middle L:90%
X equals zero, which is true. And then
87
00:05:45.790 --> 00:05:47.529 A:middle L:90%
afterwards it's all positive derivative, thought positive, meaning
88
00:05:47.529 --> 00:05:51.519 A:middle L:90%
that the function of peace increasing half their X equals
89
00:05:51.519 --> 00:05:55.449 A:middle L:90%
zero, which is true. So the graph confirms
90
00:05:55.449 --> 00:05:57.449 A:middle L:90%
that all of these functions are consistent with each other
91
00:05:57.449 --> 00:06:00.100 A:middle L:90%
as F f F prime and double prime.