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PROFESSOR: OK, we're ready to
start the eleventh lecture.
10
00:00:25,38 --> 00:00:29,39
We're still in the
middle of sketching.
11
00:00:29,39 --> 00:00:33,93
And, indeed, one of the reasons
why we did not talk about
12
00:00:33,93 --> 00:00:38,44
hyperbolic functions is this
that we're running just
13
00:00:38,44 --> 00:00:39,54
a little bit behind.
14
00:00:39,54 --> 00:00:41,72
And we'll catch up
a tiny bit today.
15
00:00:41,72 --> 00:00:46
And I hope all the way on
Tuesday of next week.
16
00:00:46 --> 00:00:58,64
So let me pick up where we
left off, with sketching.
17
00:00:58,64 --> 00:01:05,22
So this is a continuation.
18
00:01:05,22 --> 00:01:07,62
I want to give you one
more example of how
19
00:01:07,62 --> 00:01:08,39
to sketch things.
20
00:01:08,39 --> 00:01:10,38
And then we'll go through
it systematically.
21
00:01:10,38 --> 00:01:13,45
So the second example that
we did as one example
22
00:01:13,45 --> 00:01:18,06
last time, is this.
23
00:01:18,06 --> 00:01:19,76
The function is x +
24
00:01:19,76 --> 00:01:21,97
+ 1 / x + + 2.
25
00:01:21,97 --> 00:01:25,78
And I'm going to save
you the time right now.
26
00:01:25,78 --> 00:01:29,4
This is very typical of me,
especially if you're in a hurry
27
00:01:29,4 --> 00:01:32,92
on an exam, I'll just tell
you what the derivative is.
28
00:01:32,92 --> 00:01:37,76
So in this case,
it's 1 / (x + 2)^2.
29
00:01:37,76 --> 00:01:42,65
Now, the reason why I'm
bringing this example up, even
30
00:01:42,65 --> 00:01:44,72
though it'll turn out to be a
relatively simple one to
31
00:01:44,72 --> 00:01:51,69
sketch, is that it's easy to
fall into a black hole
32
00:01:51,69 --> 00:01:53,89
with this problem.
33
00:01:53,89 --> 00:01:57,47
So let me just show you.
34
00:01:57,47 --> 00:01:59,6
This is not equal to 0.
35
00:01:59,6 --> 00:02:01,18
It's never equal to 0.
36
00:02:01,18 --> 00:02:08,94
So that means there are
no critical points.
37
00:02:08,94 --> 00:02:15,46
At this point, students, many
students who have been trained
38
00:02:15,46 --> 00:02:19,23
like monkeys to do exactly what
they've been told, suddenly
39
00:02:19,23 --> 00:02:21,17
freeze and give up.
40
00:02:21,17 --> 00:02:23,97
Because there's nothing to do.
41
00:02:23,97 --> 00:02:28
So this is the one thing that
I have to train out of you.
42
00:02:28 --> 00:02:32,26
You can't just give
up at this point.
43
00:02:32,26 --> 00:02:35,06
So what would you suggest?
44
00:02:35,06 --> 00:02:38,1
Can anybody get us
out of this jam?
45
00:02:38,1 --> 00:02:38,78
Yeah.
46
00:02:38,78 --> 00:02:47,41
STUDENT: [INAUDIBLE]
47
00:02:47,41 --> 00:02:48,15
PROFESSOR: Right.
48
00:02:48,15 --> 00:02:53,93
So the suggestion was to
find the x values where
49
00:02:53,93 --> 00:02:55,8
f (x) is undefined.
50
00:02:55,8 --> 00:03:01,09
In fact, so now that's a fairly
sophisticated way of putting
51
00:03:01,09 --> 00:03:04,69
the point that I want to make,
which is that what we want to
52
00:03:04,69 --> 00:03:07,77
do is go back to our
precalculus skills.
53
00:03:07,77 --> 00:03:11,15
And just plot points.
54
00:03:11,15 --> 00:03:13,84
So instead, you go back
to precalculus and you
55
00:03:13,84 --> 00:03:17,22
just plot some points.
56
00:03:17,22 --> 00:03:18,35
It's a perfectly
reasonable thing.
57
00:03:18,35 --> 00:03:22,18
Now, it turns out that the most
important point to plot is
58
00:03:22,18 --> 00:03:24,73
the one that's not there.
59
00:03:24,73 --> 00:03:29,3
Namely, the value of x = - 2.
60
00:03:29,3 --> 00:03:31,78
Which is just what
was suggested.
61
00:03:31,78 --> 00:03:35,47
Namely, we plot the points
where the function
62
00:03:35,47 --> 00:03:38,71
is not defined.
63
00:03:38,71 --> 00:03:41,22
So how do we do that?
64
00:03:41,22 --> 00:03:43,77
Well, you have to think
about it for a second and
65
00:03:43,77 --> 00:03:46,47
I'll introduce some new
notation when I do it.
66
00:03:46,47 --> 00:03:49,29
If I evaluate 2 at this place,
actually I can't do it.
67
00:03:49,29 --> 00:03:51,25
I have to do it from the
left and the right.
68
00:03:51,25 --> 00:03:56,61
So if I plug in - 2 on the
positive side, from the
69
00:03:56,61 --> 00:04:01,15
right, that's going
to be equal to - 2 +
70
00:04:01,15 --> 00:04:05,3
1 / - 2, a little
bit more than - 2,
71
00:04:05,3 --> 00:04:06,59
+2.
72
00:04:06,59 --> 00:04:11,11
Which is - 1 divided by - now,
this denominator is - 2,
73
00:04:11,11 --> 00:04:12,5
a little more than that,
74
00:04:12,5 --> 00:04:12,73
+2.
75
00:04:12,73 --> 00:04:20,2
So it's a little more than 0.
76
00:04:20,2 --> 00:04:24,72
And that is, well we'll
fill that in in a second.
77
00:04:24,72 --> 00:04:25,62
Everybody's puzzled.
78
00:04:25,62 --> 00:04:26,04
Yes.
79
00:04:26,04 --> 00:04:30,67
STUDENT: [INAUDIBLE]
80
00:04:30,67 --> 00:04:37,66
PROFESSOR: No, that's
the function.
81
00:04:37,66 --> 00:04:37,98
I'm plotting points, I'm
not differentiating.
82
00:04:37,98 --> 00:04:38,25
I've already differentiated it.
83
00:04:38,25 --> 00:04:39,63
I've already got something
that's a little puzzling.
84
00:04:39,63 --> 00:04:41,25
Now I'm focusing on
the weird spot.
85
00:04:41,25 --> 00:04:41,59
Yes, another question.
86
00:04:41,59 --> 00:04:49,11
STUDENT: Wouldn't it be
a little less than 0?
87
00:04:49,11 --> 00:04:49,61
PROFESSOR: Wouldn't it be
a little less than 0?
88
00:04:49,61 --> 00:04:50,74
OK, that's a very good
point and this is a
89
00:04:50,74 --> 00:04:53,31
matter of notation here.
90
00:04:53,31 --> 00:04:55,69
And a matter of parentheses.
91
00:04:55,69 --> 00:04:57,43
So wouldn't this be a
little less than 2.
92
00:04:57,43 --> 00:05:01,45
Well, if the parentheses
were this way; that is, 2+
93
00:05:01,45 --> 00:05:04,595
, with a - after I did the 2+
94
00:05:04,595 --> 00:05:06,79
, then it would be less.
95
00:05:06,79 --> 00:05:09,83
But it's this way.
96
00:05:09,83 --> 00:05:10,67
OK.
97
00:05:10,67 --> 00:05:16,77
So the notation is, you have
a number and you take the
98
00:05:16,77 --> 00:05:17,19
part of it.
99
00:05:17,19 --> 00:05:21,55
That's the part which is a
little bit bigger than it.
100
00:05:21,55 --> 00:05:26,84
And so this is what I mean.
101
00:05:26,84 --> 00:05:29,87
And if you like, here I can
put in those parentheses too.
102
00:05:29,87 --> 00:05:31,27
Yeah, another question.
103
00:05:31,27 --> 00:05:34,73
STUDENT: [INAUDIBLE]
104
00:05:34,73 --> 00:05:39,19
PROFESSOR: Why doesn't
the top one have a plus?
105
00:05:39,19 --> 00:05:39,47
The only reason why the top one
doesn't have a plus is that I
106
00:05:39,47 --> 00:05:42,86
don't need it to evaluate this.
107
00:05:42,86 --> 00:05:45,41
And when I take the limit, I
can just plug in the value.
108
00:05:45,41 --> 00:05:48,25
Whereas here, I'm
still uncertain.
109
00:05:48,25 --> 00:05:49,43
Because it's going to be 0.
110
00:05:49,43 --> 00:05:51,74
And I want to know which
side of 0 it's on.
111
00:05:51,74 --> 00:05:55,19
Whether it's on the positive
side or the negative side.
112
00:05:55,19 --> 00:05:58,02
So this one, I could have
written here a parentheses
113
00:05:58,02 --> 00:06:01,41
2+, but then it would have
just simplified to - 1.
114
00:06:01,41 --> 00:06:04,41
In the limit.
115
00:06:04,41 --> 00:06:07,5
So now, I've got a negative
number divided by a
116
00:06:07,5 --> 00:06:10,1
tiny positive number.
117
00:06:10,1 --> 00:06:11,94
And so, somebody want to
tell me what that is?
118
00:06:11,94 --> 00:06:16,75
Negative infinity.
119
00:06:16,75 --> 00:06:21,46
So, we just evaluated this
function from one side.
120
00:06:21,46 --> 00:06:29,19
And if you follow through the
other side, so this one here,
121
00:06:29,19 --> 00:06:31,8
you get something very similar,
except that this should be --
122
00:06:31,8 --> 00:06:34,12
whoops, what did I do wrong?
123
00:06:34,12 --> 00:06:37,69
I meant this.
124
00:06:37,69 --> 00:06:40,87
I want it -2 the same base
point, but I want to
125
00:06:40,87 --> 00:06:43,75
go from the left.
126
00:06:43,75 --> 00:06:47,78
So that's going to be -
2 + 1, same numerator.
127
00:06:47,78 --> 00:06:52,79
And then this - 2 on the left +
2, and that's going to come out
128
00:06:52,79 --> 00:07:00,67
to be - 1 / 0 -, which
is plus infinity.
129
00:07:00,67 --> 00:07:02,348
Or just plain infinity,
we don't have to
130
00:07:02,348 --> 00:07:10,52
put the plus sign.
131
00:07:10,52 --> 00:07:13,09
So this is the first
part of the problem.
132
00:07:13,09 --> 00:07:16,63
And the second piece, to get
ourselves started, you could
133
00:07:16,63 --> 00:07:18,26
evaluate this function
at any point.
134
00:07:18,26 --> 00:07:21,34
This is just the most
interesting point, alright?
135
00:07:21,34 --> 00:07:22,88
This is just the most
interesting place
136
00:07:22,88 --> 00:07:25,27
to evaluate it.
137
00:07:25,27 --> 00:07:27,7
Now, the next thing that
I'd like to do is to pay
138
00:07:27,7 --> 00:07:32,13
attention to the ends.
139
00:07:32,13 --> 00:07:34,29
And I haven't really
said what the ends are.
140
00:07:34,29 --> 00:07:37,22
So the ends are just all the
way to the left and all
141
00:07:37,22 --> 00:07:37,94
the way to the right.
142
00:07:37,94 --> 00:07:42,67
So that means x going to
plus or minus infinity.
143
00:07:42,67 --> 00:07:44,43
So that's the second thing I
want to pay attention to.
144
00:07:44,43 --> 00:07:49,45
Again, this is a little bit
like a video screen here.
145
00:07:49,45 --> 00:07:52,72
And we're about to discover
something that's really off
146
00:07:52,72 --> 00:07:55,75
the screen, in both cases.
147
00:07:55,75 --> 00:07:57,95
We're taking care of what's
happening way to the left,
148
00:07:57,95 --> 00:07:59,36
way to the right, here.
149
00:07:59,36 --> 00:08:01,82
And up above, we just
took care what happens
150
00:08:01,82 --> 00:08:05,28
way up and way down.
151
00:08:05,28 --> 00:08:11,55
So on these ends, I need
to do some more analysis.
152
00:08:11,55 --> 00:08:15,65
Which is related to a
precalculus skill which
153
00:08:15,65 --> 00:08:18,31
is evaluating limits.
154
00:08:18,31 --> 00:08:21,82
And here, the way to do
it is to divide by x the
155
00:08:21,82 --> 00:08:23,01
numerator and denominator.
156
00:08:23,01 --> 00:08:27,36
Write it as (1 + 1
/ x)/ (1 + 2 /x).
157
00:08:27,36 --> 00:08:29,95
And then you can see what
happens as x goes to
158
00:08:29,95 --> 00:08:30,98
plus or minus infinity.
159
00:08:30,98 --> 00:08:33,59
It just goes to 1.
160
00:08:33,59 --> 00:08:37,69
So, no matter whether x
is positive or negative.
161
00:08:37,69 --> 00:08:42,81
When it gets huge, these two
extra numbers here go to 0.
162
00:08:42,81 --> 00:08:44,55
And so, this tends to 1.
163
00:08:44,55 --> 00:08:47,79
So if you like, you could
abbreviate this as f
164
00:08:47,79 --> 00:08:52,83
(+ or - infinity) = 1.
165
00:08:52,83 --> 00:08:54,61
So now, I get to draw this.
166
00:08:54,61 --> 00:08:56,9
And we draw this
using asymptotes.
167
00:08:56,9 --> 00:09:01,8
So there's a level
which is y = 1.
168
00:09:01,8 --> 00:09:06,79
And then there's
another line to draw.
169
00:09:06,79 --> 00:09:15,19
Which is x = - 2.
170
00:09:15,19 --> 00:09:18,52
And now, what information
do I have so far?
171
00:09:18,52 --> 00:09:21,26
Well, the information that I
have so far is that when we're
172
00:09:21,26 --> 00:09:27,55
coming in from the right,
that's to - 2, it plunges
173
00:09:27,55 --> 00:09:28,57
down to minus infinity.
174
00:09:28,57 --> 00:09:33,17
So that's down like this.
175
00:09:33,17 --> 00:09:38,81
And I also know that it goes
up to infinity on the other
176
00:09:38,81 --> 00:09:41,85
side of the asymptote.
177
00:09:41,85 --> 00:09:48,13
And over here, I know it's
going out to the level 1.
178
00:09:48,13 --> 00:09:53,41
And here it's also
going to the level 1.
179
00:09:53,41 --> 00:09:57,16
Now, there's an issue.
180
00:09:57,16 --> 00:09:59,49
I can almost finish
this graph now.
181
00:09:59,49 --> 00:10:01,46
I almost have enough
information to finish it.
182
00:10:01,46 --> 00:10:04,45
But there's one thing which
is making me hesitate
183
00:10:04,45 --> 00:10:06,78
a little bit.
184
00:10:06,78 --> 00:10:10,6
And that is, I don't know, for
instance, over here, whether
185
00:10:10,6 --> 00:10:14,23
it's going to maybe dip
below and come back up.
186
00:10:14,23 --> 00:10:16,93
Or not.
187
00:10:16,93 --> 00:10:20,14
So what does it do here?
188
00:10:20,14 --> 00:10:25,04
Can anybody see?
189
00:10:25,04 --> 00:10:25,25
Yeah.
190
00:10:25,25 --> 00:10:29,9
STUDENT: [INAUDIBLE]
191
00:10:29,9 --> 00:10:32,26
PROFESSOR: It can't dip
below because there are
192
00:10:32,26 --> 00:10:32,38
no critical points.
193
00:10:32,38 --> 00:10:34,54
What a precisely
correct answer.
194
00:10:34,54 --> 00:10:36,73
So that's exactly right.
195
00:10:36,73 --> 00:10:44,02
The point here is that because
f' is not 0, it can't
196
00:10:44,02 --> 00:10:45,02
double back on itself.
197
00:10:45,02 --> 00:10:49,79
Because there can't be any of
these horizontal tangents.
198
00:10:49,79 --> 00:11:00,56
It can't double back,
so it can't backtrack.
199
00:11:00,56 --> 00:11:07,19
So sorry, if f' is not
0, f can't backtrack.
200
00:11:07,19 --> 00:11:09,41
And so that means that it
doesn't look like this.
201
00:11:09,41 --> 00:11:14,32
It just goes like this.
202
00:11:14,32 --> 00:11:15,69
So that's basically it.
203
00:11:15,69 --> 00:11:17,63
And it's practically the
end of the problem.
204
00:11:17,63 --> 00:11:19,5
Goes like this.
205
00:11:19,5 --> 00:11:21,87
Now you can decorate
your thing, right?
206
00:11:21,87 --> 00:11:24,66
You may notice that maybe it
crosses here, the axes, you can
207
00:11:24,66 --> 00:11:26,79
actually evaluate these places.
208
00:11:26,79 --> 00:11:27,5
And so forth.
209
00:11:27,5 --> 00:11:31,13
We're looking right now
for qualitative behavior.
210
00:11:31,13 --> 00:11:34,28
In fact, you can see
where these places hit.
211
00:11:34,28 --> 00:11:36,66
And it's actually a little
higher up than I drew.
212
00:11:36,66 --> 00:11:40,14
Maybe I'll draw it accurately.
213
00:11:40,14 --> 00:11:44,97
As we'll see in a second.
214
00:11:44,97 --> 00:11:47,71
So that's what happens
to this function.
215
00:11:47,71 --> 00:11:51,04
Now, let's just take a
look in a little bit more
216
00:11:51,04 --> 00:11:56,84
detail, by double checking.
217
00:11:56,84 --> 00:11:58,64
So we're just going to double
check what happens to the
218
00:11:58,64 --> 00:12:01,28
sign of the derivative.
219
00:12:01,28 --> 00:12:03,83
And in the meantime, I'm going
to explain to you what the
220
00:12:03,83 --> 00:12:07,045
derivative is and also talk
about the second derivative.
221
00:12:07,045 --> 00:12:11,82
So first of all, the trick for
evaluating the derivative
222
00:12:11,82 --> 00:12:13,57
is an algebraic one.
223
00:12:13,57 --> 00:12:16,08
I mean, obviously you can do
this by the quotient rule.
224
00:12:16,08 --> 00:12:24,25
But I just point out that this
is the same thing as this.
225
00:12:24,25 --> 00:12:27,33
And now it has, whoops,
that should be a 2
226
00:12:27,33 --> 00:12:28,69
in the denominator.
227
00:12:28,69 --> 00:12:35,95
And so, now this has the
form 1 - (1 / x + 2).
228
00:12:35,95 --> 00:12:39,6
So this makes it easy to see
what the derivative is.
229
00:12:39,6 --> 00:12:42,73
Because the derivative of
a constant is 0, right?
230
00:12:42,73 --> 00:12:49,3
So this is, derivative, is just
going to be, switch the sign.
231
00:12:49,3 --> 00:12:53,41
This is what I wrote before.
232
00:12:53,41 --> 00:12:55,23
And that explains it.
233
00:12:55,23 --> 00:12:58,14
But incidentally, it also
shows you that that
234
00:12:58,14 --> 00:13:04,89
this is a hyperbola.
235
00:13:04,89 --> 00:13:09,62
These are just two
curves of a hyperbola.
236
00:13:09,62 --> 00:13:12,24
So now, let's check the sign.
237
00:13:12,24 --> 00:13:14,9
It's already totally
obvious to us that this
238
00:13:14,9 --> 00:13:15,94
is just a double check.
239
00:13:15,94 --> 00:13:18,71
We didn't actually even have
to pay any attention to this.
240
00:13:18,71 --> 00:13:19,79
It had better be true.
241
00:13:19,79 --> 00:13:22,22
This is just going to
check our arithmetic.
242
00:13:22,22 --> 00:13:24,57
Namely, it's increasing here.
243
00:13:24,57 --> 00:13:26,61
It's increasing there.
244
00:13:26,61 --> 00:13:27,83
That's got to be true.
245
00:13:27,83 --> 00:13:31,47
And, sure enough, this is
positive, as you can see
246
00:13:31,47 --> 00:13:32,93
it's 1 over a square.
247
00:13:32,93 --> 00:13:34,07
So it is increasing.
248
00:13:34,07 --> 00:13:35,63
So we checked it.
249
00:13:35,63 --> 00:13:39,37
But now, there's one more thing
that I want to just have
250
00:13:39,37 --> 00:13:40,72
you watch out about.
251
00:13:40,72 --> 00:13:46,98
So this means that
f is increasing.
252
00:13:46,98 --> 00:13:51,83
On the interval minus
infinity < x < - 2.
253
00:13:51,83 --> 00:13:56,7
And also from - 2 all the
way out to infinity.
254
00:13:56,7 --> 00:14:03,36
So I just want to warn you, you
cannot say, don't say f is
255
00:14:03,36 --> 00:14:12,02
increasing on minus infinity
< infinity, for all x.
256
00:14:12,02 --> 00:14:15,09
OK, this is just not true.
257
00:14:15,09 --> 00:14:16,7
I've written it on the
board, but it's wrong.
258
00:14:16,7 --> 00:14:18,66
I'd better get rid of it.
259
00:14:18,66 --> 00:14:19,2
There it is.
260
00:14:19,2 --> 00:14:20,97
Get rid of it.
261
00:14:20,97 --> 00:14:24,555
And the reason is, so first
of all it's totally obvious.
262
00:14:24,555 --> 00:14:25,39
It's going up here.
263
00:14:25,39 --> 00:14:28,7
But then it went zooming
back down there.
264
00:14:28,7 --> 00:14:35,86
And here this was true,
but only if x is not - 2.
265
00:14:35,86 --> 00:14:37,4
So there's a break.
266
00:14:37,4 --> 00:14:39,28
And you've got to pay
attention to the break.
267
00:14:39,28 --> 00:14:51,06
So basically, the moral here is
that if you ignore this place,
268
00:14:51,06 --> 00:14:54,39
it's like ignoring Mount
Everest, or the Grand Canyon.
269
00:14:54,39 --> 00:14:56,77
You're ignoring the most
important feature of
270
00:14:56,77 --> 00:14:58,3
this function here.
271
00:14:58,3 --> 00:15:00,51
If you're going to be figuring
out where things are going up
272
00:15:00,51 --> 00:15:03,71
and down, which is basically
all we're doing, you'd better
273
00:15:03,71 --> 00:15:07,33
pay attention to these
kinds of places.
274
00:15:07,33 --> 00:15:09,33
So don't ignore them.
275
00:15:09,33 --> 00:15:13,14
So that's the first remark.
276
00:15:13,14 --> 00:15:17,34
And now there's just a little
bit of decoration as well.
277
00:15:17,34 --> 00:15:19,99
Which is the role of
the second derivative.
278
00:15:19,99 --> 00:15:21,99
So we've written down the
first derivative here.
279
00:15:21,99 --> 00:15:33,49
The second derivative is now
- 2 / (x + 2)^3, right?
280
00:15:33,49 --> 00:15:36,3
So I get that from
differentiating this formula up
281
00:15:36,3 --> 00:15:39,49
here for the first derivative.
282
00:15:39,49 --> 00:15:43,475
And now, of course, that's
also, only works for
283
00:15:43,475 --> 00:15:47,68
x not equal to - 2.
284
00:15:47,68 --> 00:15:54,82
And now, we can see that this
is going to be negative, let's
285
00:15:54,82 --> 00:15:56,65
see, where is it negative?
286
00:15:56,65 --> 00:16:01,66
When this is a positive
quantity, so when -2
287
00:16:01,66 --> 00:16:03,405
< x < infinity.
288
00:16:03,405 --> 00:16:04,59
It's negative.
289
00:16:04,59 --> 00:16:07,77
And this is where this
thing is concave.
290
00:16:07,77 --> 00:16:08,67
Let's see.
291
00:16:08,67 --> 00:16:12,65
Did I say that right?
292
00:16:12,65 --> 00:16:13,33
Negative, right?
293
00:16:13,33 --> 00:16:19,18
This is concave down.
294
00:16:19,18 --> 00:16:19,93
Right.
295
00:16:19,93 --> 00:16:23,4
And similarly, if I look at
this expression, the numerator
296
00:16:23,4 --> 00:16:28,18
is always negative but the
denominator becomes negative
297
00:16:28,18 --> 00:16:31,13
as well when x < -2.
298
00:16:31,13 --> 00:16:33,71
So this becomes positive.
299
00:16:33,71 --> 00:16:36,69
So this case, it was
negative over positive.
300
00:16:36,69 --> 00:16:40,95
In this case it was negative
divided by negative.
301
00:16:40,95 --> 00:16:46,14
So here, this is in the range
- infinity < x < -2.
302
00:16:46,14 --> 00:16:52,24
And here it's concave up.
303
00:16:52,24 --> 00:16:55,15
Now, again, this is just
consistent with what
304
00:16:55,15 --> 00:16:55,92
we're already guessing.
305
00:16:55,92 --> 00:16:57,77
Of course we already know
it in this case if we know
306
00:16:57,77 --> 00:16:59,68
that this is a hyperbola.
307
00:16:59,68 --> 00:17:02,31
That it's going to be concave
down to the right of the
308
00:17:02,31 --> 00:17:04,27
vertical line, dotted
vertical line.
309
00:17:04,27 --> 00:17:07,77
And concave up to the left.
310
00:17:07,77 --> 00:17:11,91
So what extra piece of
information is it that
311
00:17:11,91 --> 00:17:16,1
this is giving us?
312
00:17:16,1 --> 00:17:17,8
Did I say this backwards?
313
00:17:17,8 --> 00:17:18,52
No.
314
00:17:18,52 --> 00:17:19,57
That's OK.
315
00:17:19,57 --> 00:17:21,43
So what extra piece of
information is this giving us?
316
00:17:21,43 --> 00:17:23,4
It looks like it's giving
us hardly anything.
317
00:17:23,4 --> 00:17:25,68
And it really is giving
us hardly anything.
318
00:17:25,68 --> 00:17:28,84
But it is giving us something
that's a little aesthetic.
319
00:17:28,84 --> 00:17:34,28
It's ruling out the
possibility of a wiggle.
320
00:17:34,28 --> 00:17:37,59
There isn't anything
like that in the curve.
321
00:17:37,59 --> 00:17:40,286
It can't shift from curving
this way to curving that
322
00:17:40,286 --> 00:17:41,59
way to curving this way.
323
00:17:41,59 --> 00:17:42,86
That doesn't happen.
324
00:17:42,86 --> 00:17:59,07
So these properties say there's
no wiggle in the graph of that.
325
00:17:59,07 --> 00:17:59,79
Alright.
326
00:17:59,79 --> 00:18:01,41
So.
327
00:18:01,41 --> 00:18:01,89
Question.
328
00:18:01,89 --> 00:18:05,57
STUDENT: Do we define the
increasing and decreasing base
329
00:18:05,57 --> 00:18:10,63
purely on the derivative, or
the sort of more general
330
00:18:10,63 --> 00:18:14,31
definition of picking any
two points and seeing.
331
00:18:14,31 --> 00:18:17,07
Because sometimes there can
be an inconsistency between
332
00:18:17,07 --> 00:18:20,3
the two definitions.
333
00:18:20,3 --> 00:18:26,3
PROFESSOR: OK, so the question
is, in this course are we
334
00:18:26,3 --> 00:18:29,37
going to define positive
derivative as being the
335
00:18:29,37 --> 00:18:31,36
same thing as increasing.
336
00:18:31,36 --> 00:18:33,03
And the answer is no.
337
00:18:33,03 --> 00:18:36,21
We'll try to use these
terms separately.
338
00:18:36,21 --> 00:18:40,33
What's always true is
that if f' is positive,
339
00:18:40,33 --> 00:18:42,53
then f is increasing.
340
00:18:42,53 --> 00:18:45,06
But the reverse is not
necessarily true.
341
00:18:45,06 --> 00:18:47,79
It could be very flat, the
derivative can be 0 and still
342
00:18:47,79 --> 00:18:50,05
the function can be increasing.
343
00:18:50,05 --> 00:18:53,73
OK, the derivative can
be 0 at a few places.
344
00:18:53,73 --> 00:18:59,3
For instance, like some cubics.
345
00:18:59,3 --> 00:19:05,37
Other questions?
346
00:19:05,37 --> 00:19:09,86
So that's as much as I
need to say in general.
347
00:19:09,86 --> 00:19:11,39
I mean, in a specific case.
348
00:19:11,39 --> 00:19:14,6
But I want to get you a general
scheme and I want to go through
349
00:19:14,6 --> 00:19:20,32
a more complicated example
that gets all the features
350
00:19:20,32 --> 00:19:22,75
of this kind of thing.
351
00:19:22,75 --> 00:19:34,71
So let's talk about a general
strategy for sketching.
352
00:19:34,71 --> 00:19:38,6
So the first part of
this strategy, if
353
00:19:38,6 --> 00:19:40,8
you like, let's see.
354
00:19:40,8 --> 00:19:42,43
I have it all plotted out here.
355
00:19:42,43 --> 00:19:45,93
So I'm going to make sure
I get it exactly the way
356
00:19:45,93 --> 00:19:47,78
I wanted you to see.
357
00:19:47,78 --> 00:19:51,44
So I have, its plotting.
358
00:19:51,44 --> 00:19:52,72
The plot thickens.
359
00:19:52,72 --> 00:19:54,04
Here we go.
360
00:19:54,04 --> 00:19:57,71
So plot, what is it that
you should plot first?
361
00:19:57,71 --> 00:20:01,47
Before you even think about
derivatives, you should
362
00:20:01,47 --> 00:20:08,25
plot discontinuities.
363
00:20:08,25 --> 00:20:18,49
Especially the infinite ones.
364
00:20:18,49 --> 00:20:20,21
That's the first
thing you should do.
365
00:20:20,21 --> 00:20:27,16
And then, you should plot
end points, for ends.
366
00:20:27,16 --> 00:20:31,44
For x going to plus or minus
infinity if there don't
367
00:20:31,44 --> 00:20:35,64
happen to be any finite
ends to the problem.
368
00:20:35,64 --> 00:20:44,6
And the third thing you can
do is plot any easy points.
369
00:20:44,6 --> 00:20:49
This is optional.
370
00:20:49 --> 00:20:50,71
At your discretion.
371
00:20:50,71 --> 00:20:54,43
You might, for instance,
on this example, plot the
372
00:20:54,43 --> 00:20:59,55
places where the graph
crosses the axis.
373
00:20:59,55 --> 00:21:04,54
If you want to.
374
00:21:04,54 --> 00:21:05,81
So that's the first part.
375
00:21:05,81 --> 00:21:08,19
And again, this is
all precalculus.
376
00:21:08,19 --> 00:21:17,23
So now, in the second part
we're going to solve this
377
00:21:17,23 --> 00:21:22,82
equation and we're going
to plot the critical
378
00:21:22,82 --> 00:21:29,36
points and values.
379
00:21:29,36 --> 00:21:32,81
In the problem which we just
discussed, there weren't any.
380
00:21:32,81 --> 00:21:38,64
So this part was empty.
381
00:21:38,64 --> 00:21:50,2
So the third step is to decide
whether f', sorry, whether, f'
382
00:21:50,2 --> 00:22:01,12
is positive or negative
on each interval.
383
00:22:01,12 --> 00:22:17,21
Between critical points,
discontinuities.
384
00:22:17,21 --> 00:22:22,56
The direction of the sign, in
this case it doesn't change.
385
00:22:22,56 --> 00:22:24,9
It goes up here and it
also goes up here.
386
00:22:24,9 --> 00:22:27,77
But it could go up here
and then come back down.
387
00:22:27,77 --> 00:22:31,46
So the direction can change
at every critical point.
388
00:22:31,46 --> 00:22:33,85
It can change at
every discontinuity.
389
00:22:33,85 --> 00:22:35,35
And you don't know.
390
00:22:35,35 --> 00:22:44,52
However, this particular step
has to be consistent with 1
391
00:22:44,52 --> 00:22:47,23
and 2, with steps 1 and 2.
392
00:22:47,23 --> 00:22:53,205
In fact, it will never, if you
can succeed in doing steps 1
393
00:22:53,205 --> 00:22:56,97
and 2, you'll never
need step 3.
394
00:22:56,97 --> 00:23:02,19
All it's doing is
double-checking.
395
00:23:02,19 --> 00:23:05,99
So if you made an arithmetic
mistake somewhere, you'll
396
00:23:05,99 --> 00:23:09,16
be able to see it.
397
00:23:09,16 --> 00:23:10,87
So that's maybe the
most important thing.
398
00:23:10,87 --> 00:23:13,3
And it's actually the most
frustrating thing for me when I
399
00:23:13,3 --> 00:23:17,98
see people working on problems,
is they start step 3, they get
400
00:23:17,98 --> 00:23:21,14
it wrong, and then they start
trying to draw the graph
401
00:23:21,14 --> 00:23:22,48
and it doesn't work.
402
00:23:22,48 --> 00:23:23,62
Because it's inconsistent.
403
00:23:23,62 --> 00:23:26,28
And the reason is some
arithmetic error with the
404
00:23:26,28 --> 00:23:28,3
derivative or something
like that or some other
405
00:23:28,3 --> 00:23:29,88
misinterpretation.
406
00:23:29,88 --> 00:23:31,93
And then there's a total mess.
407
00:23:31,93 --> 00:23:35,04
If you start with these two
steps, then you're going to
408
00:23:35,04 --> 00:23:37,62
know when you get to this step
that you're making mistakes.
409
00:23:37,62 --> 00:23:40,31
People don't generally
make as many mistakes
410
00:23:40,31 --> 00:23:42,36
in the first two steps.
411
00:23:42,36 --> 00:23:45,22
Anyway, in fact you can skip
this step if you want.
412
00:23:45,22 --> 00:23:49,47
But that's at risk of not
double-checking your work.
413
00:23:49,47 --> 00:23:51,28
So what's the fourth step?
414
00:23:51,28 --> 00:23:59,4
Well, we take a look at whether
f'' is positive or negative.
415
00:23:59,4 --> 00:24:01,89
And so we're deciding on
things like whether it's
416
00:24:01,89 --> 00:24:07,64
concave up or down.
417
00:24:07,64 --> 00:24:15,49
And we have these points,
f'' ( x ) = 0, which are
418
00:24:15,49 --> 00:24:24,57
called inflection points.
419
00:24:24,57 --> 00:24:31,55
And the last step is just
to combine everything.
420
00:24:31,55 --> 00:24:35,71
So this is this the scheme,
the general scheme.
421
00:24:35,71 --> 00:24:58,85
And let's just carry it
out in a particular case.
422
00:24:58,85 --> 00:25:02,28
So here's the function that I'm
going to use as an example.
423
00:25:02,28 --> 00:25:08,25
I'll use f ( x ) = x / ln x.
424
00:25:08,25 --> 00:25:11,64
And because the logarithm
- yeah, question.
425
00:25:11,64 --> 00:25:11,96
Yeah.
426
00:25:11,96 --> 00:25:18,66
STUDENT: [INAUDIBLE]
427
00:25:18,66 --> 00:25:21,35
PROFESSOR: The question
is, is this optional.
428
00:25:21,35 --> 00:25:25,57
So that's a good question.
429
00:25:25,57 --> 00:25:26,39
Is this optional.
430
00:25:26,39 --> 00:25:31,516
STUDENT: [INAUDIBLE]
431
00:25:31,516 --> 00:25:37,59
PROFESSOR: OK, the question
is is this optional;
432
00:25:37,59 --> 00:25:38,66
this kind of question.
433
00:25:38,66 --> 00:25:48,94
And the answer is, it's
more than just -- so, in
434
00:25:48,94 --> 00:25:51,7
many instances, I'm not
going to ask you to.
435
00:25:51,7 --> 00:25:54,78
I strongly recommend that
if I don't ask you to do
436
00:25:54,78 --> 00:25:57,05
it, that you not try.
437
00:25:57,05 --> 00:26:01,05
Because it's usually awful to
find the second derivative.
438
00:26:01,05 --> 00:26:03,29
Any time you can get away
without computing a second
439
00:26:03,29 --> 00:26:06,33
derivative, you're better off.
440
00:26:06,33 --> 00:26:07,92
So in many, many instances.
441
00:26:07,92 --> 00:26:10,72
On the other hand, if I ask you
to do it it's because I want
442
00:26:10,72 --> 00:26:13,06
you to have the, work to do it.
443
00:26:13,06 --> 00:26:16,72
But basically, if nobody
forces you to, I would
444
00:26:16,72 --> 00:26:22,13
say never do step 4.
445
00:26:22,13 --> 00:26:26,75
Other questions.
446
00:26:26,75 --> 00:26:27,61
Alright.
447
00:26:27,61 --> 00:26:29,95
So we're going to force
ourselves to do step 4,
448
00:26:29,95 --> 00:26:31,81
however, in this instance.
449
00:26:31,81 --> 00:26:35,01
But maybe this will be
one of the few times.
450
00:26:35,01 --> 00:26:39,14
So here we go, just for
illustrative purposes.
451
00:26:39,14 --> 00:26:43,24
OK, now.
452
00:26:43,24 --> 00:26:46,14
So here's the function
that I want to discuss.
453
00:26:46,14 --> 00:26:50,01
And the range has to be x
positive, because the
454
00:26:50,01 --> 00:26:55,5
logarithm is not defined
for negative values.
455
00:26:55,5 --> 00:26:58,85
So the first thing that I'm
going to do is, I'd like to
456
00:26:58,85 --> 00:27:02,56
follow the scheme here.
457
00:27:02,56 --> 00:27:05,14
Because if I don't follow
the scheme, I'm going to
458
00:27:05,14 --> 00:27:06,49
get a little mixed up.
459
00:27:06,49 --> 00:27:13,98
So the first part is to
find the singularities.
460
00:27:13,98 --> 00:27:17,06
That is, the places
where f is infinite.
461
00:27:17,06 --> 00:27:20,72
And that's when the logarithm,
the denominator, vanishes.
462
00:27:20,72 --> 00:27:25,63
So that's f ( 1
+), if you like.
463
00:27:25,63 --> 00:27:32,37
So that's 1 / ln 1 +, which
is 1 / 0, with a little
464
00:27:32,37 --> 00:27:34,28
bit of positiveness to it.
465
00:27:34,28 --> 00:27:37,27
Which is infinity.
466
00:27:37,27 --> 00:27:39,83
And second, we do
it the other way.
467
00:27:39,83 --> 00:27:42,84
And not surprisingly,
this comes out to be
468
00:27:42,84 --> 00:27:46,85
negative infinity.
469
00:27:46,85 --> 00:27:51,98
Now, the next thing I
want to do is the ends.
470
00:27:51,98 --> 00:27:56,98
So I call these the ends.
471
00:27:56,98 --> 00:28:01,38
And there are two of them.
472
00:28:01,38 --> 00:28:08,3
One of them is f ( 0 )
from the right. f ( 0+).
473
00:28:08,3 --> 00:28:21,49
So that is 0 + / ln 0 +, which
is 0 plus divided by, well, ln
474
00:28:21,49 --> 00:28:25,36
0 + is actually minus infinity.
475
00:28:25,36 --> 00:28:27,18
That's what happens to
the logarithm, goes
476
00:28:27,18 --> 00:28:28,17
to minus infinity.
477
00:28:28,17 --> 00:28:31,25
So this is 0 over infinity,
which is definitely 0, there's
478
00:28:31,25 --> 00:28:37,1
no problem. about what happens.
479
00:28:37,1 --> 00:28:42,91
The other side, so this is the
end, this is the first end.
480
00:28:42,91 --> 00:28:44,92
The range is this.
481
00:28:44,92 --> 00:28:48,48
And I just did the
left endpoint.
482
00:28:48,48 --> 00:28:50,63
And so now I have to do the
right endpoint, I have
483
00:28:50,63 --> 00:28:51,87
to let x go to infinity.
484
00:28:51,87 --> 00:28:54,67
So if I let x go to infinity,
I'm just going to have to think
485
00:28:54,67 --> 00:28:56,52
about it a little bit by
plugging in a very
486
00:28:56,52 --> 00:28:57,69
large number.
487
00:28:57,69 --> 00:29:01,96
I'll plug in 10 ^ 10, just
to see what happens.
488
00:29:01,96 --> 00:29:11,73
So if I plug in 10 ^ 10 into x
ln x, I get 10 ^ 10 / ln 10^10.
489
00:29:11,73 --> 00:29:17,59
Which is 10 ^ 10
/ 10 ( log 10).
490
00:29:17,59 --> 00:29:23,18
So the denominator, this number
here, is about 2.something.
491
00:29:23,18 --> 00:29:25,13
2.3 or something.
492
00:29:25,13 --> 00:29:28,31
So this is maybe 230 in the
denominator, and this is a
493
00:29:28,31 --> 00:29:31,9
number with ten 0's after it.
494
00:29:31,9 --> 00:29:33,53
So it's very, very large.
495
00:29:33,53 --> 00:29:35,3
I claim it's big.
496
00:29:35,3 --> 00:29:38,73
And that gives us the clue
that what's happening is
497
00:29:38,73 --> 00:29:40,54
that this thing is infinite.
498
00:29:40,54 --> 00:29:43,766
So, in other words, our
conclusion is that f of
499
00:29:43,766 --> 00:29:52,12
infinity is infinity.
500
00:29:52,12 --> 00:29:59,65
So what do we have so
far for our function?
501
00:29:59,65 --> 00:30:03,3
We're just trying to build the
scaffolding of the function.
502
00:30:03,3 --> 00:30:07,27
And we're doing it by taking
the most important points.
503
00:30:07,27 --> 00:30:09,52
And from a mathematician's
point of view, the most
504
00:30:09,52 --> 00:30:11,38
important points are the
ones which are sort of
505
00:30:11,38 --> 00:30:13,49
infinitely obvious.
506
00:30:13,49 --> 00:30:15,19
For the ends of the problem.
507
00:30:15,19 --> 00:30:19,73
So that's where we're heading.
508
00:30:19,73 --> 00:30:22,6
We have a vertical asymptote,
which is at x = 1.
509
00:30:22,6 --> 00:30:29,15
So this gives us x = 1.
510
00:30:29,15 --> 00:30:34,27
And we have a value which
is that it's 0 here.
511
00:30:34,27 --> 00:30:38,97
And we also know that when we
come in from the - sorry, so we
512
00:30:38,97 --> 00:30:44,91
come in from the left, that's
f, the one from the left,
513
00:30:44,91 --> 00:30:46,06
we get negative infinity.
514
00:30:46,06 --> 00:30:47,55
So it's diving down.
515
00:30:47,55 --> 00:30:52,46
It's going down like this.
516
00:30:52,46 --> 00:30:55,925
And, furthermore, on the other
side we know it's climbing up.
517
00:30:55,925 --> 00:30:58,56
So it's going up like this.
518
00:30:58,56 --> 00:31:00,46
Just start a little higher.
519
00:31:00,46 --> 00:31:00,77
Right, so.
520
00:31:00,77 --> 00:31:02,52
So far, this is what we know.
521
00:31:02,52 --> 00:31:05,81
Oh, and there's one other
thing that we know.
522
00:31:05,81 --> 00:31:12,42
When we go to plus infinity,
it's going back up.
523
00:31:12,42 --> 00:31:15,15
So, so far we have this.
524
00:31:15,15 --> 00:31:17,85
Now, already it should be
pretty obvious what's going
525
00:31:17,85 --> 00:31:19,76
to happen to this function.
526
00:31:19,76 --> 00:31:21,16
So there shouldn't
be many surprises.
527
00:31:21,16 --> 00:31:23,16
It's going to come
down like this.
528
00:31:23,16 --> 00:31:27,09
Go like this, it's going to
turn around and go back up.
529
00:31:27,09 --> 00:31:29,29
That's what we expect.
530
00:31:29,29 --> 00:31:33,6
So we don't know that yet,
but we're pretty sure.
531
00:31:33,6 --> 00:31:36,8
So at this point, we can start
looking at the critical points.
532
00:31:36,8 --> 00:31:41,995
We can do our step 2 here -- we
need a little bit more room
533
00:31:41,995 --> 00:31:45,49
here -- and see what's
happening with this function.
534
00:31:45,49 --> 00:31:49,22
So I have to differentiate it.
535
00:31:49,22 --> 00:31:52,07
And it's, this is
the quotient rule.
536
00:31:52,07 --> 00:31:54,88
So remember the function
is up here, x / ln x.
537
00:31:54,88 --> 00:31:59,4
So I have a ln x^2
in the denominator.
538
00:31:59,4 --> 00:32:04,19
And I get here the derivative
of x is 1, so we get 1 ( ln x)
539
00:32:04,19 --> 00:32:10,56
- x ( the derivative of
ln x, which is 1 /x).
540
00:32:10,56 --> 00:32:20,16
So all told, that's
(ln x - 1) / ln x^2.
541
00:32:20,16 --> 00:32:27,77
So here's our derivative.
542
00:32:27,77 --> 00:32:35,08
And now, if I set this equal to
0, at least in the numerator,
543
00:32:35,08 --> 00:32:40,97
the numerator is 0 when x = e.
544
00:32:40,97 --> 00:32:43,49
The ln e = 1.
545
00:32:43,49 --> 00:32:46,29
So here's our critical point.
546
00:32:46,29 --> 00:32:51,32
And we have a critical
value, which is f(e).
547
00:32:51,32 --> 00:32:55,61
And that's going
to be e / ln e.
548
00:32:55,61 --> 00:32:57,24
Which is e, again.
549
00:32:57,24 --> 00:32:59,09
Because ln e = 1.
550
00:32:59,09 --> 00:33:01,89
So now I can also plot
the critical point,
551
00:33:01,89 --> 00:33:03,11
which is down here.
552
00:33:03,11 --> 00:33:07,68
And there's only one of
them, and it's at (e e).
553
00:33:07,68 --> 00:33:10,8
That's kind of not to scale
here, because my blackboard
554
00:33:10,8 --> 00:33:12,52
isn't quite tall enough.
555
00:33:12,52 --> 00:33:15,36
It should be over here
and then, it's slope 1.
556
00:33:15,36 --> 00:33:17,14
But I dipped it down.
557
00:33:17,14 --> 00:33:19,93
So this is not to scale, and
indeed that's one of the things
558
00:33:19,93 --> 00:33:22,68
that we're not going to attempt
to do with these pictures,
559
00:33:22,68 --> 00:33:24,68
is to make them to scale.
560
00:33:24,68 --> 00:33:29,56
So the scale's a
little squashed.
561
00:33:29,56 --> 00:33:32,71
So, so far I have
this critical point.
562
00:33:32,71 --> 00:33:36,18
And, in fact, I'm going
to label it with a c.
563
00:33:36,18 --> 00:33:38,46
Whenever I have a critical
point I'll just make sure
564
00:33:38,46 --> 00:33:41,49
that I remember that
that's what it is.
565
00:33:41,49 --> 00:33:44,79
And since there's only
one, the rest of this
566
00:33:44,79 --> 00:33:49,9
picture is now correct.
567
00:33:49,9 --> 00:33:54,88
That's the same mechanism that
we used for the hyperbola.
568
00:33:54,88 --> 00:33:57,14
Namely, we know there's
only one place where
569
00:33:57,14 --> 00:33:57,81
the derivative is 0.
570
00:33:57,81 --> 00:34:00,34
So that means there no more
horizontals, so there's
571
00:34:00,34 --> 00:34:01,95
no more backtracking.
572
00:34:01,95 --> 00:34:03,33
It has to come down to here.
573
00:34:03,33 --> 00:34:03,97
Get to there.
574
00:34:03,97 --> 00:34:06,06
This is the only place
it can turn around.
575
00:34:06,06 --> 00:34:07,17
Goes back up.
576
00:34:07,17 --> 00:34:09
It has to start here and it
has to go down to there.
577
00:34:09 --> 00:34:10,6
It can't go above 0.
578
00:34:10,6 --> 00:34:13,6
Do not pass go, do
not get positive.
579
00:34:13,6 --> 00:34:20,23
It has to head down here.
580
00:34:20,23 --> 00:34:21,69
So that's great.
581
00:34:21,69 --> 00:34:25,08
That means that this picture is
almost completely correct now.
582
00:34:25,08 --> 00:34:27,52
And the rest is more
or less decoration.
583
00:34:27,52 --> 00:34:30,25
We're pretty much done
with the way it looks,
584
00:34:30,25 --> 00:34:34,57
at least schematically.
585
00:34:34,57 --> 00:34:37,7
However, I am going to punish
you, because I warned you.
586
00:34:37,7 --> 00:34:40,72
We are going to go over here
and do this step 4 and
587
00:34:40,72 --> 00:34:44,28
fix up the concavity.
588
00:34:44,28 --> 00:34:45,93
And we're also going
to do a little bit of
589
00:34:45,93 --> 00:35:00,77
that double-checking.
590
00:35:00,77 --> 00:35:06,13
So now, let's again, just,
I want to emphasize.
591
00:35:06,13 --> 00:35:10,66
We're going to do
a double-check.
592
00:35:10,66 --> 00:35:12,24
This is part 3.
593
00:35:12,24 --> 00:35:17,4
But in advance, I already have,
based on this picture I already
594
00:35:17,4 --> 00:35:19
know what has to be true.
595
00:35:19 --> 00:35:35,45
That f is decreasing on 0 to 1.
f is also decreasing on 1 to e.
596
00:35:35,45 --> 00:35:45,49
And f is increasing
on e to infinity.
597
00:35:45,49 --> 00:35:49,88
So, already, because we've plot
a bunch of points and we know
598
00:35:49,88 --> 00:35:51,47
that there aren't any places
where the derivative
599
00:35:51,47 --> 00:35:55,5
vanishes, we already know it
goes down, down, up.
600
00:35:55,5 --> 00:35:56,97
That's what it's got to do.
601
00:35:56,97 --> 00:35:58,98
Now, we'll just make sure
that we didn't make any
602
00:35:58,98 --> 00:36:00,72
arithmetic mistakes, now.
603
00:36:00,72 --> 00:36:03,59
By actually computing the
derivative, or staring
604
00:36:03,59 --> 00:36:04,65
at it, anyway.
605
00:36:04,65 --> 00:36:10,35
And making sure
that it's correct.
606
00:36:10,35 --> 00:36:17,6
So first of all, we just take
a look at the numerator.
607
00:36:17,6 --> 00:36:26,655
So f,' remember, was (ln x - 1)
/ ln x^2, the quantity squared.
608
00:36:26,655 --> 00:36:28,48
So the denominator is positive.
609
00:36:28,48 --> 00:36:32,65
So let's just take a look
at the three ranges.
610
00:36:32,65 --> 00:36:37,33
So we have 0 < x < 1.
611
00:36:37,33 --> 00:36:40,57
And on that range, the
logarithm is negative, so
612
00:36:40,57 --> 00:36:45,23
this is negative divided by
positive, which is negative.
613
00:36:45,23 --> 00:36:47,22
That's decreasing, that's good.
614
00:36:47,22 --> 00:36:50,16
And in fact, that also
works on the next range.
615
00:36:50,16 --> 00:36:55,37
1 l< x < e, it's
negative divided by positive.
616
00:36:55,37 --> 00:36:57,72
And the only reason why we
skipped 1, again, is that
617
00:36:57,72 --> 00:36:58,433
it's undefined there.
618
00:36:58,433 --> 00:37:01,04
And there's something
dramatic happening there.
619
00:37:01,04 --> 00:37:05,42
And then, at the last range,
when x is bigger than e, that
620
00:37:05,42 --> 00:37:07,86
means the logarithm is
already bigger than 1.
621
00:37:07,86 --> 00:37:09,923
So the numerator is now
positive, and the denominator's
622
00:37:09,923 --> 00:37:13,65
still positive, so
it's increasing.
623
00:37:13,65 --> 00:37:22,91
So we've just double-checked
something that we already knew.
624
00:37:22,91 --> 00:37:26,94
Alright, so that's pretty
much all there is to
625
00:37:26,94 --> 00:37:29,2
say about step 3.
626
00:37:29,2 --> 00:37:33,98
So this is checking the
positivity and negativity.
627
00:37:33,98 --> 00:37:35,66
And now, step 4.
628
00:37:35,66 --> 00:37:38,48
There is one small point which
I want to make before we go on.
629
00:37:38,48 --> 00:37:42,9
Which is that sometimes,
you can't evaluate the
630
00:37:42,9 --> 00:37:45,445
function or its derivative
particularly well.
631
00:37:45,445 --> 00:37:48,62
So sometimes you can't plot
the points very well.
632
00:37:48,62 --> 00:37:50,41
And if you can't plot the
points very well, then you
633
00:37:50,41 --> 00:37:53,16
might have to do 3 first,
to figure out what's
634
00:37:53,16 --> 00:37:55,29
going on a little bit.
635
00:37:55,29 --> 00:37:59,15
You might have to skip.
636
00:37:59,15 --> 00:38:02,4
So now we're going to go
on the second derivative.
637
00:38:02,4 --> 00:38:07,34
But first, I want to use
an algebraic trick to
638
00:38:07,34 --> 00:38:08,47
rearrange the terms.
639
00:38:08,47 --> 00:38:10,82
And I want to notice
one more little point.
640
00:38:10,82 --> 00:38:16,32
Which I, as I say, this is
decoration for the graph.
641
00:38:16,32 --> 00:38:18,29
So I want to rewrite
the formula.
642
00:38:18,29 --> 00:38:22,32
Maybe I'll do it
right over here.
643
00:38:22,32 --> 00:38:31,44
Another way of writing this is
(1 / ln x) - (1 / (ln x)^2).
644
00:38:31,44 --> 00:38:35,24
So that's another way of
writing the derivative.
645
00:38:35,24 --> 00:38:38,64
And that allows me to
notice something that
646
00:38:38,64 --> 00:38:40,83
I missed, before.
647
00:38:40,83 --> 00:38:46,88
When I solved the equation ln
x - 1 - this is equal to 0
648
00:38:46,88 --> 00:38:48,59
here, this equation here.
649
00:38:48,59 --> 00:38:51,46
I missed a possibility.
650
00:38:51,46 --> 00:38:54,02
I missed the possibility
that the denominator
651
00:38:54,02 --> 00:38:58,69
could be infinity.
652
00:38:58,69 --> 00:39:02,11
So actually, if the
denominator's infinity, as
653
00:39:02,11 --> 00:39:05,6
you can see from the other
expression there, it
654
00:39:05,6 --> 00:39:09
actually is true that
the derivative is 0.
655
00:39:09 --> 00:39:16,71
So also when x = 0 +, the
slope is going to be 0.
656
00:39:16,71 --> 00:39:19,05
Let me just emphasize
that again.
657
00:39:19,05 --> 00:39:23,5
If you evaluate using this
other formula over here,
658
00:39:23,5 --> 00:39:31,54
this is (1 / ln 0+)
- (1 / (ln 0+)^2).
659
00:39:31,54 --> 00:39:37,31
That's 1 / - infinity - 1 /
infinity, if you like, squared.
660
00:39:37,31 --> 00:39:40,63
Anyway, it's 0.
661
00:39:40,63 --> 00:39:42,33
So this is 0.
662
00:39:42,33 --> 00:39:43,41
The slope is 0 there.
663
00:39:43,41 --> 00:39:46,64
That is a little piece of
decoration on our graph.
664
00:39:46,64 --> 00:39:50,92
It's telling us, going back to
our graph here, it's telling us
665
00:39:50,92 --> 00:39:53,56
this is coming in with
slope horizontal.
666
00:39:53,56 --> 00:39:57,53
So we're starting out this way.
667
00:39:57,53 --> 00:40:01,013
That's just a little
start here to the graph.
668
00:40:01,013 --> 00:40:02,62
It's a horizontal slope.
669
00:40:02,62 --> 00:40:07,94
So there really were two places
where the slope was horizontal.
670
00:40:07,94 --> 00:40:13,55
Now, with the help of this
second formula I can also
671
00:40:13,55 --> 00:40:17,05
differentiate a second time.
672
00:40:17,05 --> 00:40:19,69
So it's a little bit easier to
do that if I differentiate 1 /
673
00:40:19,69 --> 00:40:34,27
ln, that's -( ln x) ^ - 2 ( 1
/ x) + 2 (ln x) ^ -3 (1/x).
674
00:40:34,27 --> 00:40:42,15
And that, if I put it over a
common denominator, is x ln
675
00:40:42,15 --> 00:40:48,69
x^3 times, let's see here.
676
00:40:48,69 --> 00:40:55,34
I guess I'll have to
take the 2 - ln x.
677
00:40:55,34 --> 00:40:57,88
So I've now rewritten the
formula for the second
678
00:40:57,88 --> 00:41:03,45
derivative as a ratio.
679
00:41:03,45 --> 00:41:09,43
Now, to decide the sign, you
see there are two places
680
00:41:09,43 --> 00:41:11,91
where the sign flips.
681
00:41:11,91 --> 00:41:16,37
The numerator crosses when
the logarithm is 2, that's
682
00:41:16,37 --> 00:41:18,62
going to be when x = e ^2.
683
00:41:18,62 --> 00:41:24,27
And the denominator flips when
x = 1, that's when the log
684
00:41:24,27 --> 00:41:28,02
flips from positive
to negative.
685
00:41:28,02 --> 00:41:32,28
So we have a couple
of ranges here.
686
00:41:32,28 --> 00:41:37,58
So, first of all, we have
the range from 0 to 1.
687
00:41:37,58 --> 00:41:42,27
And then we have the
range from 1 to e^2.
688
00:41:42,27 --> 00:41:45,995
And then we have the
range from e ^2 all the
689
00:41:45,995 --> 00:41:49,78
way out to infinity.
690
00:41:49,78 --> 00:41:58,64
So between 0 and 1, the
numerator is, well this is a
691
00:41:58,64 --> 00:42:00,695
negative number in this, so
minus a negative number
692
00:42:00,695 --> 00:42:04,65
is positive, so the
numerator is positive.
693
00:42:04,65 --> 00:42:07,505
And the denominator is
negative, because the ln
694
00:42:07,505 --> 00:42:09,66
is negative it's taken
to the third power.
695
00:42:09,66 --> 00:42:12,372
So this is a negative numbers,
so it's positive divided by a
696
00:42:12,372 --> 00:42:15,19
negative number, which
is less than 0.
697
00:42:15,19 --> 00:42:18,77
That means it's concave down.
698
00:42:18,77 --> 00:42:26,04
So this is concave down plot.
699
00:42:26,04 --> 00:42:28,01
And that's a good thing,
because over here
700
00:42:28,01 --> 00:42:29,12
this is concave down.
701
00:42:29,12 --> 00:42:30,56
So there are no wiggles.
702
00:42:30,56 --> 00:42:34,26
It goes straight
down, like this.
703
00:42:34,26 --> 00:42:41,59
And then the other two pieces
are f'' is equal to, well
704
00:42:41,59 --> 00:42:43,26
it's going to switch here.
705
00:42:43,26 --> 00:42:44,856
The denominator
becomes positive.
706
00:42:44,856 --> 00:42:48,19
So it's positive over positive.
707
00:42:48,19 --> 00:42:56,67
So this is concave up.
708
00:42:56,67 --> 00:42:58,41
And that's going over here.
709
00:42:58,41 --> 00:43:02,715
But notice that it's not the
bottom where it turns around,
710
00:43:02,715 --> 00:43:07,74
it's somewhere else.
711
00:43:07,74 --> 00:43:09,93
So there's another
transition here.
712
00:43:09,93 --> 00:43:12,09
This is e ^2.
713
00:43:12,09 --> 00:43:15,09
This is e.
714
00:43:15,09 --> 00:43:20,44
So what happens at the end is,
again, the sign flips again.
715
00:43:20,44 --> 00:43:23,57
Because the numerator,
now, when x > e
716
00:43:23,57 --> 00:43:26,58
^2, becomes negative.
717
00:43:26,58 --> 00:43:29,965
And this is negative divided by
positive, which is negative.
718
00:43:29,965 --> 00:43:35,63
And part is concave down.
719
00:43:35,63 --> 00:43:39,1
And so we didn't quite
draw the graph right.
720
00:43:39,1 --> 00:43:41,18
There's an inflection
point right here, which
721
00:43:41,18 --> 00:43:45
I'll label with i.
722
00:43:45 --> 00:43:47,34
Makes a turn the other
way at that point.
723
00:43:47,34 --> 00:43:49,48
So there was a wiggle.
724
00:43:49,48 --> 00:43:51,21
There's the wiggle.
725
00:43:51,21 --> 00:43:53,71
Still going up, still
going to infinity.
726
00:43:53,71 --> 00:43:57,04
But kind of the slope of
the mountain, right?
727
00:43:57,04 --> 00:44:01,24
It's going the other way.
728
00:44:01,24 --> 00:44:09,35
This point happens to
be (e^2, e ^2 / 2).
729
00:44:09,35 --> 00:44:11,98
So that's as detailed
as we'll ever get.
730
00:44:11,98 --> 00:44:16,91
And indeed, the next game is
going to be avoid being, is to
731
00:44:16,91 --> 00:44:18,73
avoid being this detailed.
732
00:44:18,73 --> 00:44:21,76
So let me introduce
the next subject.
733
00:44:21,76 --> 00:44:48,31
Which is maxima and minima.
734
00:44:48,31 --> 00:45:04,16
OK, now, maxima and minima,
maximum and minimum problems
735
00:45:04,16 --> 00:45:06,55
can be described graphically
in the following ways.
736
00:45:06,55 --> 00:45:13,15
Suppose you have a function,
right, here it is.
737
00:45:13,15 --> 00:45:14,42
OK?
738
00:45:14,42 --> 00:45:24,63
Now, find the maximum.
739
00:45:24,63 --> 00:45:30,74
And find the minimum.
740
00:45:30,74 --> 00:45:31,41
OK.
741
00:45:31,41 --> 00:45:38,88
So this problem is done.
742
00:45:38,88 --> 00:45:50,54
The point being, that it is
easy to find max and the
743
00:45:50,54 --> 00:45:58,67
min with the sketch.
744
00:45:58,67 --> 00:46:00,33
It's very easy.
745
00:46:00,33 --> 00:46:05,13
The goal, the problem, is that
the sketch is a lot of work.
746
00:46:05,13 --> 00:46:10,18
We just spent 20 minutes
sketching something.
747
00:46:10,18 --> 00:46:13,24
We would not like to spend
all that time every single
748
00:46:13,24 --> 00:46:14,97
time we want to find a
maximum and minimum.
749
00:46:14,97 --> 00:46:19,576
So the goal is to do it
with, so our goal is
750
00:46:19,576 --> 00:46:25,17
to use shortcuts.
751
00:46:25,17 --> 00:46:31,52
And, indeed, as I said earlier,
we certainly never want to use
752
00:46:31,52 --> 00:46:33,84
the second derivative
if we can avoid it.
753
00:46:33,84 --> 00:46:37,27
And we don't want to decorate
the graph and do all of these
754
00:46:37,27 --> 00:46:40,62
elaborate, subtle, things which
make the graph look nicer and
755
00:46:40,62 --> 00:46:42,64
really, or aesthetically
appropriate.
756
00:46:42,64 --> 00:46:45,64
But are totally unnecessary
to see whether the
757
00:46:45,64 --> 00:46:54,29
graph is up or down.
758
00:46:54,29 --> 00:46:58,03
So essentially, this whole
business is out, which
759
00:46:58,03 --> 00:47:00,03
is a good thing.
760
00:47:00,03 --> 00:47:04,31
And, unfortunately, those early
parts are the parts that
761
00:47:04,31 --> 00:47:06,17
people tend to ignore.
762
00:47:06,17 --> 00:47:10,15
Which are typically,
often, very important.
763
00:47:10,15 --> 00:47:22,94
So let me first tell you
the main point here.
764
00:47:22,94 --> 00:47:32,76
So the key idea.
765
00:47:32,76 --> 00:47:39,65
Key to finding maximum.
766
00:47:39,65 --> 00:47:42,19
So the key point is,
we only need to look
767
00:47:42,19 --> 00:48:00,13
at critical points.
768
00:48:00,13 --> 00:48:04,54
Well, that's actually what
it seems like to, in
769
00:48:04,54 --> 00:48:05,49
many calculus classes.
770
00:48:05,49 --> 00:48:06,81
But that's not true.
771
00:48:06,81 --> 00:48:15,59
This is not the end
of the sentence.
772
00:48:15,59 --> 00:48:35,32
And, end points, and
points of discontinuity.
773
00:48:35,32 --> 00:48:37,98
So you must watch
out for those.
774
00:48:37,98 --> 00:48:42,37
If you look at the example that
I just drew here, which is the
775
00:48:42,37 --> 00:48:48,08
one that I carried out, you can
see that there are actually
776
00:48:48,08 --> 00:48:51,25
five extreme points
on this picture.
777
00:48:51,25 --> 00:48:52,75
So let's switch.
778
00:48:52,75 --> 00:48:58,05
So we'll take a look.
779
00:48:58,05 --> 00:49:04,84
There are five places where
the max or the min might be.
780
00:49:04,84 --> 00:49:08,05
There's this important point.
781
00:49:08,05 --> 00:49:10,71
This is, as I say, the
scaffolding of the function.
782
00:49:10,71 --> 00:49:13,65
There's this point, there
down at minus infinity.
783
00:49:13,65 --> 00:49:18,37
There's this, there's
this, and there's this.
784
00:49:18,37 --> 00:49:23,62
Only one out of five
is a critical point.
785
00:49:23,62 --> 00:49:25,51
So there's more that you
have to pay attention
786
00:49:25,51 --> 00:49:26,45
to on the function.
787
00:49:26,45 --> 00:49:29,71
And you always have to keep
the schema, the picture of
788
00:49:29,71 --> 00:49:31,31
the function, in the
back of your head.
789
00:49:31,31 --> 00:49:33,76
Even though this may be the
most interesting point,
790
00:49:33,76 --> 00:49:36,52
and the one that you're
going to be looking at.
791
00:49:36,52 --> 00:49:40,832
So we'll do a few examples
of that next time.
792
00:49:40,832 --> 00:49:41,715