WEBVTT
1
00:00:01.740 --> 00:00:04.809 A:middle L:90%
Okay, let's bring in some devil's advocates to kind
2
00:00:04.809 --> 00:00:07.950 A:middle L:90%
of shake things up. We're talking about these derivatives
3
00:00:07.960 --> 00:00:10.869 A:middle L:90%
finding the slope of the tangent line. Everything's great
4
00:00:11.539 --> 00:00:14.890 A:middle L:90%
, but what about if a function is not differential
5
00:00:14.900 --> 00:00:17.620 A:middle L:90%
at the point? So here you have this function
6
00:00:17.620 --> 00:00:21.489 A:middle L:90%
GFT t to the two thirds. We want to
7
00:00:21.489 --> 00:00:25.660 A:middle L:90%
determine whether the function is differential at the point.
8
00:00:26.739 --> 00:00:32.259 A:middle L:90%
And so remember that differential. What does that mean
9
00:00:36.740 --> 00:00:40.079 A:middle L:90%
? It just means that f prime in this case
10
00:00:40.640 --> 00:00:50.030 A:middle L:90%
of zero exists. Yeah, so we need to
11
00:00:50.039 --> 00:00:53.979 A:middle L:90%
figure out whether or not f prime of zero exists
12
00:00:55.140 --> 00:00:58.450 A:middle L:90%
as a limit. Okay, I guess I should
13
00:00:58.450 --> 00:01:00.950 A:middle L:90%
say g here, right? Because it's actually G
14
00:01:03.340 --> 00:01:06.689 A:middle L:90%
, not F and you'll see kind of a We
15
00:01:06.689 --> 00:01:08.739 A:middle L:90%
go along, we'll start mixing up. The notation
16
00:01:08.739 --> 00:01:14.640 A:middle L:90%
will use different letters for the variables. Different letters
17
00:01:14.640 --> 00:01:18.659 A:middle L:90%
for independent variables, dependent variables, because in different
18
00:01:18.659 --> 00:01:21.739 A:middle L:90%
contexts you use different variables. So here we're using
19
00:01:21.739 --> 00:01:25.730 A:middle L:90%
GFT. So maybe t represents time or something like
20
00:01:25.730 --> 00:01:29.780 A:middle L:90%
that. So for GFT, we want to know
21
00:01:29.780 --> 00:01:33.310 A:middle L:90%
whether or not g prime of zero exists. So
22
00:01:33.310 --> 00:01:34.480 A:middle L:90%
let's just write down what g prime of zero.
23
00:01:34.489 --> 00:01:41.260 A:middle L:90%
It's so g prime of zero is going to be
24
00:01:42.239 --> 00:01:48.000 A:middle L:90%
H to the two thirds. It's a really zero
25
00:01:48.000 --> 00:01:53.950 A:middle L:90%
plus h to the two thirds minus zero all over
26
00:01:53.540 --> 00:01:57.349 A:middle L:90%
h. And so just a reminder. This is
27
00:01:57.359 --> 00:02:05.250 A:middle L:90%
g of a plus itch. This is G A
28
00:02:06.340 --> 00:02:08.849 A:middle L:90%
and in this case is equal to zero. So
29
00:02:08.849 --> 00:02:14.979 A:middle L:90%
let you verify that. And of course, very
30
00:02:14.979 --> 00:02:17.080 A:middle L:90%
easy to forget or taking the limit is h goes
31
00:02:17.080 --> 00:02:25.560 A:middle L:90%
to zero, and it really is important. As
32
00:02:25.560 --> 00:02:29.979 A:middle L:90%
you're taking these limits toe leave the limit. That's
33
00:02:29.979 --> 00:02:32.169 A:middle L:90%
just good notation. It's good practices, good mathematical
34
00:02:32.169 --> 00:02:36.770 A:middle L:90%
practice. It will also help you get full credit
35
00:02:37.139 --> 00:02:39.409 A:middle L:90%
on problems or partial credit on problems that maybe you
36
00:02:39.409 --> 00:02:43.319 A:middle L:90%
don't get exactly right. It will make your greater
37
00:02:43.319 --> 00:02:46.819 A:middle L:90%
, very happy if you use good notation. So
38
00:02:46.819 --> 00:02:50.349 A:middle L:90%
you lead this limit until you actually evaluate the limit
39
00:02:52.340 --> 00:02:54.199 A:middle L:90%
. So this is just the limit is h goes
40
00:02:54.199 --> 00:03:00.060 A:middle L:90%
to zero of H to the two thirds divided by
41
00:03:00.539 --> 00:03:07.949 A:middle L:90%
H or limit is h goes to zero. So
42
00:03:07.960 --> 00:03:09.800 A:middle L:90%
H to the two thirds divided by H to the
43
00:03:09.800 --> 00:03:14.960 A:middle L:90%
first is the same thing is one over age to
44
00:03:14.960 --> 00:03:17.710 A:middle L:90%
the one third. But this is something that we've
45
00:03:17.710 --> 00:03:22.060 A:middle L:90%
seen before. When we talked about infinite limits.
46
00:03:22.639 --> 00:03:25.680 A:middle L:90%
So we know that the limit is h goes to
47
00:03:25.680 --> 00:03:30.129 A:middle L:90%
zero from the right of one over age to the
48
00:03:30.129 --> 00:03:35.250 A:middle L:90%
one third is going to be infinity because we're dividing
49
00:03:35.250 --> 00:03:40.550 A:middle L:90%
by a small positive number. But the limit his
50
00:03:40.550 --> 00:03:45.610 A:middle L:90%
h purchase zero from the left. Now we're going
51
00:03:45.610 --> 00:03:47.889 A:middle L:90%
to be dividing by a small negative number, so
52
00:03:47.889 --> 00:03:52.580 A:middle L:90%
it's going to be approaching minus infinity. So what
53
00:03:52.580 --> 00:04:09.819 A:middle L:90%
does this mean? G is not differential At T
54
00:04:09.819 --> 00:04:14.490 A:middle L:90%
equals zero. And why is that? Because this
55
00:04:14.490 --> 00:04:19.870 A:middle L:90%
limit does not exist. So again, we see
56
00:04:19.870 --> 00:04:25.990 A:middle L:90%
this connection between does Geep Ryan of Zero exists and
57
00:04:25.990 --> 00:04:30.110 A:middle L:90%
she not being differential AT T equals zero. So
58
00:04:30.110 --> 00:04:33.540 A:middle L:90%
both of these are, of course, equivalent ways
59
00:04:33.540 --> 00:04:36.879 A:middle L:90%
to say the same thing. But the point is
60
00:04:38.339 --> 00:04:43.230 A:middle L:90%
, dysfunction is not differential. And actually, if
61
00:04:43.230 --> 00:04:46.990 A:middle L:90%
you remember our discussion on cusp sis dysfunction G has
62
00:04:46.990 --> 00:04:50.689 A:middle L:90%
a cusp. At T equals zero. So let's
63
00:04:50.689 --> 00:05:01.610 A:middle L:90%
see a picture. Okay, so here's a plot
64
00:05:01.620 --> 00:05:04.060 A:middle L:90%
of the function. G F T equals T to
65
00:05:04.060 --> 00:05:08.410 A:middle L:90%
the two thirds and we see we have a problem
66
00:05:08.420 --> 00:05:12.769 A:middle L:90%
. This function is not smooth at this point.
67
00:05:14.339 --> 00:05:15.949 A:middle L:90%
It has a nice sharp corner, but it's actually
68
00:05:15.949 --> 00:05:19.629 A:middle L:90%
stronger than a corner is a cusp and it's a
69
00:05:19.629 --> 00:05:23.149 A:middle L:90%
cuss. We actually showed that it was a custom
70
00:05:24.139 --> 00:05:27.389 A:middle L:90%
because the limit from the left we approach zero goes
71
00:05:27.389 --> 00:05:30.110 A:middle L:90%
to minus infinity Limit from the right goes to infinity
72
00:05:30.250 --> 00:05:33.949 A:middle L:90%
, so it's really, really sharp. It's like
73
00:05:34.439 --> 00:05:36.439 A:middle L:90%
if you put the ends of a piece of paper
74
00:05:36.439 --> 00:05:41.529 A:middle L:90%
together, Andi kind of bend it inwards. It
75
00:05:41.540 --> 00:05:46.269 A:middle L:90%
kind of goes like If here's the piece of paper
76
00:05:47.240 --> 00:05:49.589 A:middle L:90%
and I put the two ends of the piece of
77
00:05:49.589 --> 00:05:53.740 A:middle L:90%
paper like that, they're meeting at a really,
78
00:05:53.740 --> 00:05:58.959 A:middle L:90%
really sharp corner. That's kind of like a cusp
79
00:05:59.079 --> 00:06:00.180 A:middle L:90%
. If you want to kind of think about where
80
00:06:00.180 --> 00:06:04.810 A:middle L:90%
these occur naturally and we see that if we try
81
00:06:04.810 --> 00:06:10.750 A:middle L:90%
to find a tangent line through this point, we're
82
00:06:10.750 --> 00:06:13.959 A:middle L:90%
gonna be out of luck, right? So we
83
00:06:13.959 --> 00:06:17.670 A:middle L:90%
see, Really. Actually, we can almost argue
84
00:06:20.040 --> 00:06:29.860 A:middle L:90%
that any line passing through this point could almost be
85
00:06:29.860 --> 00:06:32.480 A:middle L:90%
a tangent line. And in fact, if you
86
00:06:32.480 --> 00:06:36.259 A:middle L:90%
go on and learn more math, it's actually the
87
00:06:36.269 --> 00:06:43.600 A:middle L:90%
case. It's not that a tangent line doesn't exist
88
00:06:43.600 --> 00:06:46.870 A:middle L:90%
here. It's actually that they're infinitely many tangent lines
89
00:06:46.339 --> 00:06:48.839 A:middle L:90%
at this point, so the tangent space really is
90
00:06:48.839 --> 00:06:50.980 A:middle L:90%
the whole space. Instead of being kind of a
91
00:06:50.980 --> 00:06:55.860 A:middle L:90%
one dimensional line. It's actually just the whole plane
92
00:06:56.839 --> 00:06:59.290 A:middle L:90%
, so I mean, this is probably more than
93
00:06:59.290 --> 00:07:00.189 A:middle L:90%
you signed up for. But the point is,
94
00:07:00.189 --> 00:07:03.490 A:middle L:90%
this function has what's called the cusp. You have
95
00:07:03.540 --> 00:07:05.740 A:middle L:90%
infinite limits from the left and the right, not
96
00:07:05.750 --> 00:07:11.860 A:middle L:90%
equally. And so it's not smooth. The derivative
97
00:07:11.860 --> 00:07:15.089 A:middle L:90%
doesn't exist. The tangent line doesn't exist or is
98
00:07:15.089 --> 00:07:19.480 A:middle L:90%
not well defined anyway. It's not differential, etcetera
99
00:07:19.490 --> 00:07:23.220 A:middle L:90%
, instantaneous rate of change. We don't know what's
100
00:07:23.220 --> 00:07:26.319 A:middle L:90%
happening, right? So we don't actually know how
101
00:07:26.319 --> 00:07:29.139 A:middle L:90%
the function is changing. At that point, it
102
00:07:29.139 --> 00:07:30.920 A:middle L:90%
really could be doing a lot of different things.
103
00:07:30.930 --> 00:07:33.800 A:middle L:90%
It could be just changing direction. Or it could
104
00:07:33.800 --> 00:07:36.399 A:middle L:90%
be. We really don't know. That's the point
105
00:07:36.399 --> 00:07:40.449 A:middle L:90%
. We don't know how the function is changing at
106
00:07:40.449 --> 00:07:51.759 A:middle L:90%
that point at that corner, her cusp. Okay
107
00:07:51.769 --> 00:07:55.329 A:middle L:90%
, so let's look at one more example. This
108
00:07:55.329 --> 00:07:58.050 A:middle L:90%
time we have h of our is equal to our
109
00:07:58.439 --> 00:08:03.449 A:middle L:90%
to the one third again R equals zero. So
110
00:08:03.449 --> 00:08:07.870 A:middle L:90%
let's take the limit now. This is gonna be
111
00:08:07.870 --> 00:08:09.649 A:middle L:90%
a little bit confusing, but not so confusing.
112
00:08:11.139 --> 00:08:13.139 A:middle L:90%
His h goes to zero. Of course, this
113
00:08:13.139 --> 00:08:16.220 A:middle L:90%
H is not the same as this H but we're
114
00:08:16.220 --> 00:08:20.089 A:middle L:90%
not going to explicitly use h here, but we're
115
00:08:20.089 --> 00:08:26.500 A:middle L:90%
going to plug in H to the one third minus
116
00:08:26.519 --> 00:08:31.870 A:middle L:90%
zero over H. And again, this is a
117
00:08:31.870 --> 00:08:33.230 A:middle L:90%
church of well, here it is going to get
118
00:08:33.230 --> 00:08:37.850 A:middle L:90%
confusing zero plus h. And this is a job
119
00:08:37.850 --> 00:08:41.830 A:middle L:90%
zero. And this h is actually the limit.
120
00:08:41.840 --> 00:08:45.129 A:middle L:90%
H And in this age is, of course,
121
00:08:45.129 --> 00:08:48.000 A:middle L:90%
the function H. But the point is, the
122
00:08:48.000 --> 00:08:52.080 A:middle L:90%
limit is just equal to that. And this is
123
00:08:54.240 --> 00:09:00.049 A:middle L:90%
h prime of zero. So this is really the
124
00:09:00.049 --> 00:09:03.600 A:middle L:90%
limit is h goes to zero of h two,
125
00:09:03.600 --> 00:09:09.139 A:middle L:90%
the one third over H and very similarly to the
126
00:09:09.139 --> 00:09:16.580 A:middle L:90%
last example we can rewrite. This is one over
127
00:09:16.590 --> 00:09:20.360 A:middle L:90%
H to the two thirds in this case. Okay
128
00:09:20.360 --> 00:09:22.740 A:middle L:90%
, so what does this do? Well, what
129
00:09:22.740 --> 00:09:28.360 A:middle L:90%
this does is because this age and the denominator is
130
00:09:28.360 --> 00:09:31.500 A:middle L:90%
being raised to an even power as well as being
131
00:09:31.500 --> 00:09:37.190 A:middle L:90%
cute bruited. We know that the limit is h
132
00:09:37.190 --> 00:09:41.240 A:middle L:90%
purchase zero from the right of one over H to
133
00:09:41.240 --> 00:09:43.919 A:middle L:90%
the two thirds is going to infinity. This is
134
00:09:43.919 --> 00:09:48.500 A:middle L:90%
one over a small, positive number, but actually
135
00:09:48.500 --> 00:09:50.539 A:middle L:90%
the same thing happens. Is agent purchase zero from
136
00:09:50.539 --> 00:09:54.629 A:middle L:90%
the left because even though we're plugging in negative,
137
00:09:54.629 --> 00:09:56.990 A:middle L:90%
small, negative numbers, they're becoming positive because we're
138
00:09:56.990 --> 00:10:00.570 A:middle L:90%
raising them to the second power. So this is
139
00:10:00.570 --> 00:10:03.450 A:middle L:90%
again one over a small positive number. So the
140
00:10:03.450 --> 00:10:09.070 A:middle L:90%
limit is infinity. So, actually, in this
141
00:10:09.070 --> 00:10:11.299 A:middle L:90%
case, now we're getting that. The slope of
142
00:10:11.299 --> 00:10:15.110 A:middle L:90%
the tangent line is undefined, but there's a consistency
143
00:10:15.120 --> 00:10:16.940 A:middle L:90%
from the left and the right, So the limit
144
00:10:16.940 --> 00:10:18.240 A:middle L:90%
from the left is infinity. The limit from the
145
00:10:18.240 --> 00:10:20.850 A:middle L:90%
right is infinity. And what we'll see when we
146
00:10:20.850 --> 00:10:24.580 A:middle L:90%
look at the picture is that this is a vertical
147
00:10:24.580 --> 00:10:35.519 A:middle L:90%
tangent. Okay, so here's the graph of the
148
00:10:35.519 --> 00:10:37.460 A:middle L:90%
function. H bar is our to the one third
149
00:10:39.240 --> 00:10:45.129 A:middle L:90%
, and here we see this vertical tangent. And
150
00:10:45.129 --> 00:10:50.250 A:middle L:90%
so it's a little bit interesting because on one hand
151
00:10:54.840 --> 00:10:56.659 A:middle L:90%
you can see that this function is smooth. I
152
00:10:56.659 --> 00:11:00.610 A:middle L:90%
mean, if I trace this with my finger,
153
00:11:00.620 --> 00:11:03.350 A:middle L:90%
I don't feel any sharp points. It is smooth
154
00:11:05.440 --> 00:11:07.350 A:middle L:90%
, but the problem is coming from the fact that
155
00:11:07.350 --> 00:11:11.389 A:middle L:90%
to be smooth, my slope actually has to become
156
00:11:11.399 --> 00:11:20.820 A:middle L:90%
undefined. So my slope at X equals well are
157
00:11:20.830 --> 00:11:26.529 A:middle L:90%
zero Sorry at Article zero is undefined. So even
158
00:11:26.529 --> 00:11:31.879 A:middle L:90%
though the function appears smooth, the tangent line is
159
00:11:31.879 --> 00:11:35.330 A:middle L:90%
vertical, so the slope is undefined, so the
160
00:11:35.330 --> 00:11:43.279 A:middle L:90%
function is not differential. So the answer are original
161
00:11:43.289 --> 00:11:50.159 A:middle L:90%
. Question. Its function is not differential at zero
162
00:11:52.840 --> 00:11:56.120 A:middle L:90%
, so it's a little tricky. But that's that's
163
00:11:56.120 --> 00:12:00.889 A:middle L:90%
how it goes. Undefined slope functions, not differential
164
00:12:01.159 --> 00:12:03.269 A:middle L:90%
, even though in some sense the function is smooth
165
00:12:03.039 --> 00:12:05.379 A:middle L:90%
. But what we'll see is what that actually means
166
00:12:05.379 --> 00:12:09.159 A:middle L:90%
is that the derivative function is not defined, so
167
00:12:09.159 --> 00:12:11.669 A:middle L:90%
that's why it's not differential.