WEBVTT
1
00:00:01.780 --> 00:00:03.509 A:middle L:90%
this problem. Number thirty four of the story.
2
00:00:03.509 --> 00:00:05.950 A:middle L:90%
Calculus. In addition, section two point eight.
3
00:00:07.040 --> 00:00:10.099 A:middle L:90%
Party if ever Vex equals X plus one Rex,
4
00:00:10.580 --> 00:00:18.449 A:middle L:90%
find F Primex. So let's use the definition one
5
00:00:19.539 --> 00:00:25.579 A:middle L:90%
before the derivative. This is the limit. And
6
00:00:25.589 --> 00:00:36.240 A:middle L:90%
experts, eh? Of the function ethnics minus every
7
00:00:40.340 --> 00:00:46.960 A:middle L:90%
guided by X minus eighty. Okay, our limit
8
00:00:46.960 --> 00:00:59.859 A:middle L:90%
will be FX X plus one of rex minus the
9
00:00:59.859 --> 00:01:03.929 A:middle L:90%
function evaluated, eh? A plus one over,
10
00:01:03.939 --> 00:01:11.060 A:middle L:90%
eh? All divided by X minus a. Okay
11
00:01:11.150 --> 00:01:14.810 A:middle L:90%
, Right Next step will be to multiply by and
12
00:01:14.810 --> 00:01:19.620 A:middle L:90%
the lowest common denominator to eliminate the fractions. So
13
00:01:19.620 --> 00:01:29.799 A:middle L:90%
an axe over eggs limited express his aim. Um
14
00:01:30.180 --> 00:01:37.040 A:middle L:90%
X times x x squared plus one over x Times
15
00:01:37.040 --> 00:01:42.769 A:middle L:90%
X A minus eight times x His critics minus one
16
00:01:42.769 --> 00:01:49.650 A:middle L:90%
toe Ray Tim's a X minus X all over X
17
00:01:49.650 --> 00:01:56.129 A:middle L:90%
minus. Hey, time's a X Now, awaken
18
00:01:56.129 --> 00:02:00.769 A:middle L:90%
groups some of the terms in the numerator. That's
19
00:02:00.769 --> 00:02:07.010 A:middle L:90%
group the X squared and the cortex term. Let's
20
00:02:07.010 --> 00:02:09.930 A:middle L:90%
factor out that X and we're just up with explaining
21
00:02:09.930 --> 00:02:17.430 A:middle L:90%
saying on their many terms are minus the quantity expensing
22
00:02:21.819 --> 00:02:23.889 A:middle L:90%
and all this divided by, uh, explain.
23
00:02:23.889 --> 00:02:30.830 A:middle L:90%
It's a in the pine by X Now, each
24
00:02:30.830 --> 00:02:32.090 A:middle L:90%
turn in. The Raider has an expense. A
25
00:02:32.090 --> 00:02:36.629 A:middle L:90%
turn I can cancel with e expectancy in the denominator
26
00:02:37.620 --> 00:02:46.229 A:middle L:90%
. And that leaves us with a X minus one
27
00:02:47.259 --> 00:02:54.430 A:middle L:90%
surrounded by a ex as expertise. Ain't this becomes
28
00:02:55.819 --> 00:03:01.919 A:middle L:90%
he squared minus one threated by a squared or one
29
00:03:01.919 --> 00:03:07.610 A:middle L:90%
minus one over a squared. And if we generalize
30
00:03:07.610 --> 00:03:09.479 A:middle L:90%
is for any a value in the domain of the
31
00:03:09.479 --> 00:03:15.229 A:middle L:90%
function Given its that I mean being any number X
32
00:03:15.319 --> 00:03:19.469 A:middle L:90%
, then our dirt is one minus one over x
33
00:03:19.469 --> 00:03:23.900 A:middle L:90%
squared. So a scored executed first to the same
34
00:03:24.189 --> 00:03:28.509 A:middle L:90%
derivative function. So we've concluded the dirt over this
35
00:03:28.509 --> 00:03:34.599 A:middle L:90%
function is one minus one over X Quit now Bernie's
36
00:03:34.599 --> 00:03:38.150 A:middle L:90%
ah, the grafts of infinite Pregnant To confirm that
37
00:03:38.229 --> 00:03:43.210 A:middle L:90%
the derivative of dysfunction experts when Rex is indeed one
38
00:03:43.210 --> 00:03:46.430 A:middle L:90%
man is one of the X word. So here
39
00:03:46.430 --> 00:03:49.599 A:middle L:90%
the two functions planted on the same graph the blue
40
00:03:49.610 --> 00:03:53.789 A:middle L:90%
is the original function affects and oranges that order function
41
00:03:54.419 --> 00:03:55.669 A:middle L:90%
. If we look at the blue function, we
42
00:03:55.669 --> 00:04:00.000 A:middle L:90%
see that there are straight lines here to this straight
43
00:04:00.000 --> 00:04:03.203 A:middle L:90%
line or we'LL have a constant slope. Therefore a
44
00:04:03.203 --> 00:04:08.663 A:middle L:90%
concentrated because it slips that the tension lines or constant
45
00:04:09.052 --> 00:04:10.913 A:middle L:90%
. So that's what we see here on the dirt
46
00:04:10.913 --> 00:04:13.802 A:middle L:90%
of graph. It's a constant value exactly equal to
47
00:04:13.802 --> 00:04:16.403 A:middle L:90%
one. So it's constant until it reaches a point
48
00:04:16.413 --> 00:04:19.213 A:middle L:90%
where this slope against change a little bit and then
49
00:04:19.213 --> 00:04:21.163 A:middle L:90%
we have, ah, local movement, Maxima,
50
00:04:21.653 --> 00:04:24.192 A:middle L:90%
which has the slip of zero. That's what we
51
00:04:24.192 --> 00:04:28.562 A:middle L:90%
see here constant slope and then it decreases to zero
52
00:04:29.202 --> 00:04:30.552 A:middle L:90%
. And then afterward we have a very sharp decrease
53
00:04:30.552 --> 00:04:32.622 A:middle L:90%
in the slope of the tension line, so very
54
00:04:32.923 --> 00:04:38.353 A:middle L:90%
negative slips. And this derivative function shows very negative
55
00:04:38.353 --> 00:04:42.723 A:middle L:90%
soaps as we approach Sarah from the left. As
56
00:04:42.723 --> 00:04:44.833 A:middle L:90%
we approach zero from the right for this function.
57
00:04:44.833 --> 00:04:46.533 A:middle L:90%
After Becks, we see that the slopes of the
58
00:04:46.533 --> 00:04:47.942 A:middle L:90%
tension lines are very, very negative. So this
59
00:04:47.942 --> 00:04:51.593 A:middle L:90%
continues Armand to the right of zero very, very
60
00:04:51.593 --> 00:04:56.023 A:middle L:90%
negative slopes until we reach this local minimum, which
61
00:04:56.023 --> 00:04:59.362 A:middle L:90%
has a slip of zero that is consistent with this
62
00:04:59.362 --> 00:05:02.562 A:middle L:90%
derivative or function. And then afterwards the slopes increase
63
00:05:03.252 --> 00:05:08.723 A:middle L:90%
until the slopes become a concept once again. So
64
00:05:08.802 --> 00:05:14.762 A:middle L:90%
disturbing a function is definitely consistent with our original function
65
00:05:15.853 --> 00:05:18.362 A:middle L:90%
and confirms that we computed that our function correctly