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PROFESSOR: OK, we're ready to
start the eleventh lecture.
9
00:00:25,380 --> 00:00:29,390
We're still in the
middle of sketching.
10
00:00:29,390 --> 00:00:32,910
And, indeed, one of the
reasons why we did not
11
00:00:32,910 --> 00:00:37,040
talk about hyperbolic
functions is
12
00:00:37,040 --> 00:00:39,540
that we're running just
a little bit behind.
13
00:00:39,540 --> 00:00:41,720
And we'll catch up
a tiny bit today.
14
00:00:41,720 --> 00:00:46,000
And I hope all the way
on Tuesday of next week.
15
00:00:46,000 --> 00:00:58,640
So let me pick up where we
left off, with sketching.
16
00:00:58,640 --> 00:01:05,220
So this is a continuation.
17
00:01:05,220 --> 00:01:07,020
I want to give you
one more example
18
00:01:07,020 --> 00:01:08,390
of how to sketch things.
19
00:01:08,390 --> 00:01:10,380
And then we'll go through
it systematically.
20
00:01:10,380 --> 00:01:14,830
So the second example that we
did as one example last time,
21
00:01:14,830 --> 00:01:18,060
is this.
22
00:01:18,060 --> 00:01:21,970
The function is (x+1)/(x+2).
23
00:01:21,970 --> 00:01:25,780
And I'm going to save
you the time right now.
24
00:01:25,780 --> 00:01:28,629
This is very typical
of me, especially
25
00:01:28,629 --> 00:01:30,170
if you're in a hurry
on an exam, I'll
26
00:01:30,170 --> 00:01:32,920
just tell you what
the derivative is.
27
00:01:32,920 --> 00:01:34,490
So in this case,
it's 1 / (x+2)^2.
28
00:01:37,760 --> 00:01:42,327
Now, the reason why I'm
bringing this example up,
29
00:01:42,327 --> 00:01:44,660
even though it'll turn out
to be a relatively simple one
30
00:01:44,660 --> 00:01:49,150
to sketch, is that
it's easy to fall
31
00:01:49,150 --> 00:01:53,890
into a black hole
with this problem.
32
00:01:53,890 --> 00:01:57,470
So let me just show you.
33
00:01:57,470 --> 00:01:59,600
This is not equal to 0.
34
00:01:59,600 --> 00:02:01,180
It's never equal to 0.
35
00:02:01,180 --> 00:02:08,940
So that means there
are no critical points.
36
00:02:08,940 --> 00:02:13,710
At this point,
students, many students
37
00:02:13,710 --> 00:02:16,250
who have been
trained like monkeys
38
00:02:16,250 --> 00:02:18,710
to do exactly what
they've been told,
39
00:02:18,710 --> 00:02:21,170
suddenly freeze and give up.
40
00:02:21,170 --> 00:02:23,970
Because there's nothing to do.
41
00:02:23,970 --> 00:02:28,000
So this is the one thing that
I have to train out of you.
42
00:02:28,000 --> 00:02:32,260
You can't just give
up at this point.
43
00:02:32,260 --> 00:02:35,060
So what would you suggest?
44
00:02:35,060 --> 00:02:38,100
Can anybody get us
out of this jam?
45
00:02:38,100 --> 00:02:38,780
Yeah.
46
00:02:38,780 --> 00:02:47,410
STUDENT: [INAUDIBLE]
47
00:02:47,410 --> 00:02:48,150
PROFESSOR: Right.
48
00:02:48,150 --> 00:02:52,810
So the suggestion was
to find the x-values
49
00:02:52,810 --> 00:02:55,800
where f(x) is undefined.
50
00:02:55,800 --> 00:03:00,680
In fact, so now that's a
fairly sophisticated way
51
00:03:00,680 --> 00:03:03,090
of putting the point
that I want to make,
52
00:03:03,090 --> 00:03:05,380
which is that what we
want to do is go back
53
00:03:05,380 --> 00:03:07,770
to our precalculus skills.
54
00:03:07,770 --> 00:03:11,150
And just plot points.
55
00:03:11,150 --> 00:03:13,500
So instead, you go
back to precalculus
56
00:03:13,500 --> 00:03:16,934
and you just plot some points.
57
00:03:16,934 --> 00:03:18,350
It's a perfectly
reasonable thing.
58
00:03:18,350 --> 00:03:21,970
Now, it turns out that the
most important point to plot
59
00:03:21,970 --> 00:03:24,730
is the one that's not there.
60
00:03:24,730 --> 00:03:29,300
Namely, the value of x = -2.
61
00:03:29,300 --> 00:03:31,780
Which is just what
was suggested.
62
00:03:31,780 --> 00:03:38,710
Namely, we plot the points where
the function is not defined.
63
00:03:38,710 --> 00:03:41,220
So how do we do that?
64
00:03:41,220 --> 00:03:43,670
Well, you have to think
about it for a second
65
00:03:43,670 --> 00:03:46,470
and I'll introduce some
new notation when I do it.
66
00:03:46,470 --> 00:03:49,290
If I evaluate 2 at this
place, actually I can't do it.
67
00:03:49,290 --> 00:03:51,250
I have to do it from
the left and the right.
68
00:03:51,250 --> 00:03:57,840
So if I plug in -2 on the
positive side, from the right,
69
00:03:57,840 --> 00:04:03,500
that's going to be equal
to -2 + 1 divided by -2,
70
00:04:03,500 --> 00:04:06,590
a little bit more
than -2, plus 2.
71
00:04:06,590 --> 00:04:11,110
Which is -1 divided by -
now, this denominator is -2,
72
00:04:11,110 --> 00:04:12,730
a little more than that, plus 2.
73
00:04:12,730 --> 00:04:20,200
So it's a little more than 0.
74
00:04:20,200 --> 00:04:24,720
And that is, well we'll
fill that in in a second.
75
00:04:24,720 --> 00:04:25,620
Everybody's puzzled.
76
00:04:25,620 --> 00:04:26,120
Yes.
77
00:04:26,120 --> 00:04:30,670
STUDENT: [INAUDIBLE]
78
00:04:30,670 --> 00:04:36,105
PROFESSOR: No,
that's the function.
79
00:04:36,105 --> 00:04:37,980
I'm plotting points,
I'm not differentiating.
80
00:04:37,980 --> 00:04:38,250
I've already differentiated it.
81
00:04:38,250 --> 00:04:39,630
I've already got something
that's a little puzzling.
82
00:04:39,630 --> 00:04:41,250
Now I'm focusing
on the weird spot.
83
00:04:41,250 --> 00:04:42,166
Yes, another question.
84
00:04:42,166 --> 00:04:47,652
STUDENT: Wouldn't it be
a little less than 0?
85
00:04:47,652 --> 00:04:49,610
PROFESSOR: Wouldn't it
be a little less than 0?
86
00:04:49,610 --> 00:04:52,800
OK, that's a very good point
and this is a matter of notation
87
00:04:52,800 --> 00:04:53,310
here.
88
00:04:53,310 --> 00:04:55,690
And a matter of parentheses.
89
00:04:55,690 --> 00:04:57,430
So wouldn't this be
a little less than 2.
90
00:04:57,430 --> 00:05:01,900
Well, if the parentheses
were this way; that is, 2+ ,
91
00:05:01,900 --> 00:05:06,790
with a minus after I did the
2+ , then it would be less.
92
00:05:06,790 --> 00:05:09,830
But it's this way.
93
00:05:09,830 --> 00:05:10,670
OK.
94
00:05:10,670 --> 00:05:14,420
So the notation is,
you have a number
95
00:05:14,420 --> 00:05:17,190
and you take the
plus part of it.
96
00:05:17,190 --> 00:05:21,550
That's the part which is a
little bit bigger than it.
97
00:05:21,550 --> 00:05:26,840
And so this is what I mean.
98
00:05:26,840 --> 00:05:29,870
And if you like, here I can
put in those parentheses too.
99
00:05:29,870 --> 00:05:31,270
Yeah, another question.
100
00:05:31,270 --> 00:05:34,730
STUDENT: [INAUDIBLE]
101
00:05:34,730 --> 00:05:37,253
PROFESSOR: Why doesn't
the top one have a plus?
102
00:05:37,253 --> 00:05:39,378
The only reason why the
top one doesn't have a plus
103
00:05:39,378 --> 00:05:42,860
is that I don't need
it to evaluate this.
104
00:05:42,860 --> 00:05:45,410
And when I take the limit, I
can just plug in the value.
105
00:05:45,410 --> 00:05:48,250
Whereas here, I'm
still uncertain.
106
00:05:48,250 --> 00:05:49,430
Because it's going to be 0.
107
00:05:49,430 --> 00:05:51,740
And I want to know
which side of 0 it's on.
108
00:05:51,740 --> 00:05:55,190
Whether it's on the positive
side or the negative side.
109
00:05:55,190 --> 00:05:58,310
So this one, I could have
written here a parentheses 2+,
110
00:05:58,310 --> 00:06:01,410
but then it would have
just simplified to -1.
111
00:06:01,410 --> 00:06:04,410
In the limit.
112
00:06:04,410 --> 00:06:07,190
So now, I've got a
negative number divided
113
00:06:07,190 --> 00:06:10,024
by a tiny positive number.
114
00:06:10,024 --> 00:06:11,940
And so, somebody want
to tell me what that is?
115
00:06:11,940 --> 00:06:16,750
Negative infinity.
116
00:06:16,750 --> 00:06:21,460
So, we just evaluated this
function from one side.
117
00:06:21,460 --> 00:06:24,830
And if you follow
through the other side,
118
00:06:24,830 --> 00:06:30,700
so this one here, you get
something very similar,
119
00:06:30,700 --> 00:06:34,120
except that this should be--
whoops, what did I do wrong?
120
00:06:34,120 --> 00:06:37,690
I meant this.
121
00:06:37,690 --> 00:06:40,500
I wanted -2, the
same base point,
122
00:06:40,500 --> 00:06:43,750
but I want to go from the left.
123
00:06:43,750 --> 00:06:47,780
So that's going to be
-2 + 1, same numerator.
124
00:06:47,780 --> 00:06:51,840
And then this -2 on
the left, plus 2,
125
00:06:51,840 --> 00:06:57,580
and that's going to come
out to be -1 / 0-, -,
126
00:06:57,580 --> 00:07:00,670
which is plus infinity.
127
00:07:00,670 --> 00:07:10,520
Or just plain infinity, we don't
have to put the plus sign in.
128
00:07:10,520 --> 00:07:13,090
So this is the first
part of the problem.
129
00:07:13,090 --> 00:07:16,190
And the second piece, to
get ourselves started,
130
00:07:16,190 --> 00:07:18,260
you could evaluate this
function at any point.
131
00:07:18,260 --> 00:07:21,256
This is just the most
interesting point, alright?
132
00:07:21,256 --> 00:07:22,880
This is just the most
interesting place
133
00:07:22,880 --> 00:07:24,936
to evaluate it.
134
00:07:24,936 --> 00:07:26,560
Now, the next thing
that I'd like to do
135
00:07:26,560 --> 00:07:32,130
is to pay attention to the ends.
136
00:07:32,130 --> 00:07:34,290
And I haven't really
said what the ends are.
137
00:07:34,290 --> 00:07:37,399
So the ends are just all the
way to the left and all the way
138
00:07:37,399 --> 00:07:37,940
to the right.
139
00:07:37,940 --> 00:07:42,180
So that means x going to
plus or minus infinity.
140
00:07:42,180 --> 00:07:44,430
So that's the second thing
I want to pay attention to.
141
00:07:44,430 --> 00:07:49,450
Again, this is a little bit
like a video screen here.
142
00:07:49,450 --> 00:07:52,400
And we're about to discover
something that's really
143
00:07:52,400 --> 00:07:55,654
off the screen, in both cases.
144
00:07:55,654 --> 00:07:58,070
We're taking care of what's
happening way to the left, way
145
00:07:58,070 --> 00:07:59,360
to the right, here.
146
00:07:59,360 --> 00:08:01,170
And up above, we
just took care what
147
00:08:01,170 --> 00:08:05,280
happens way up and way down.
148
00:08:05,280 --> 00:08:11,550
So on these ends, I need
to do some more analysis.
149
00:08:11,550 --> 00:08:15,480
Which is related to
a precalculus skill
150
00:08:15,480 --> 00:08:18,310
which is evaluating limits.
151
00:08:18,310 --> 00:08:21,070
And here, the way to
do it is to divide
152
00:08:21,070 --> 00:08:23,010
by x the numerator
and denominator.
153
00:08:23,010 --> 00:08:27,360
Write it as (1 +
1/x) / (1 + 2/x).
154
00:08:27,360 --> 00:08:29,650
And then you can see
what happens as x
155
00:08:29,650 --> 00:08:30,980
goes to plus or minus infinity.
156
00:08:30,980 --> 00:08:33,590
It just goes to 1.
157
00:08:33,590 --> 00:08:37,690
So, no matter whether x
is positive or negative.
158
00:08:37,690 --> 00:08:42,810
When it gets huge, these two
extra numbers here go to 0.
159
00:08:42,810 --> 00:08:44,550
And so, this tends to 1.
160
00:08:44,550 --> 00:08:47,560
So if you like, you
could abbreviate this
161
00:08:47,560 --> 00:08:52,830
as f plus or minus
infinity is equal to 1.
162
00:08:52,830 --> 00:08:54,610
So now, I get to draw this.
163
00:08:54,610 --> 00:08:56,900
And we draw this
using asymptotes.
164
00:08:56,900 --> 00:09:01,800
So there's a level
which is y = 1.
165
00:09:01,800 --> 00:09:06,790
And then there's
another line to draw.
166
00:09:06,790 --> 00:09:11,180
Which is x = -2.
167
00:09:15,190 --> 00:09:18,520
And now, what information
do I have so far?
168
00:09:18,520 --> 00:09:20,740
Well, the information
that I have so far
169
00:09:20,740 --> 00:09:26,530
is that when we're coming in
from the right, that's to -2,
170
00:09:26,530 --> 00:09:28,570
it plunges down
to minus infinity.
171
00:09:28,570 --> 00:09:33,170
So that's down like this.
172
00:09:33,170 --> 00:09:39,780
And I also know that it goes up
to infinity on the other side
173
00:09:39,780 --> 00:09:41,850
of the asymptote.
174
00:09:41,850 --> 00:09:48,130
And over here, I know it's
going out to the level 1.
175
00:09:48,130 --> 00:09:53,410
And here it's also
going to the level 1.
176
00:09:53,410 --> 00:09:57,160
Now, there's an issue.
177
00:09:57,160 --> 00:09:59,490
I can almost finish
this graph now.
178
00:09:59,490 --> 00:10:01,460
I almost have enough
information to finish it.
179
00:10:01,460 --> 00:10:03,530
But there's one
thing which is making
180
00:10:03,530 --> 00:10:06,780
me hesitate a little bit.
181
00:10:06,780 --> 00:10:10,160
And that is, I don't know,
for instance, over here,
182
00:10:10,160 --> 00:10:14,230
whether it's going to maybe
dip below and come back up.
183
00:10:14,230 --> 00:10:16,930
Or not.
184
00:10:16,930 --> 00:10:20,140
So what does it do here?
185
00:10:20,140 --> 00:10:24,750
Can anybody see?
186
00:10:24,750 --> 00:10:25,250
Yeah.
187
00:10:25,250 --> 00:10:29,900
STUDENT: [INAUDIBLE]
188
00:10:29,900 --> 00:10:32,220
PROFESSOR: It can't
dip below because there
189
00:10:32,220 --> 00:10:33,178
are no critical points.
190
00:10:33,178 --> 00:10:34,540
What a precisely correct answer.
191
00:10:34,540 --> 00:10:36,730
So that's exactly right.
192
00:10:36,730 --> 00:10:43,390
The point here is that
because f' is not 0,
193
00:10:43,390 --> 00:10:45,020
it can't double back on itself.
194
00:10:45,020 --> 00:10:49,790
Because there can't be any
of these horizontal tangents.
195
00:10:49,790 --> 00:11:00,560
It can't double back,
so it can't backtrack.
196
00:11:00,560 --> 00:11:07,190
So sorry, if f' is not
0, f can't backtrack.
197
00:11:07,190 --> 00:11:09,410
And so that means that it
doesn't look like this.
198
00:11:09,410 --> 00:11:14,320
It just goes like this.
199
00:11:14,320 --> 00:11:15,690
So that's basically it.
200
00:11:15,690 --> 00:11:17,630
And it's practically
the end of the problem.
201
00:11:17,630 --> 00:11:19,500
Goes like this.
202
00:11:19,500 --> 00:11:21,870
Now you can decorate
your thing, right?
203
00:11:21,870 --> 00:11:24,660
You may notice that maybe it
crosses here, the axes, you can
204
00:11:24,660 --> 00:11:26,790
actually evaluate these places.
205
00:11:26,790 --> 00:11:27,500
And so forth.
206
00:11:27,500 --> 00:11:31,130
We're looking right now
for qualitative behavior.
207
00:11:31,130 --> 00:11:34,280
In fact, you can see
where these places hit.
208
00:11:34,280 --> 00:11:36,660
And it's actually a little
higher up than I drew.
209
00:11:36,660 --> 00:11:40,140
Maybe I'll draw it accurately.
210
00:11:40,140 --> 00:11:44,970
As we'll see in a second.
211
00:11:44,970 --> 00:11:47,710
So that's what happens
to this function.
212
00:11:47,710 --> 00:11:51,860
Now, let's just take a look
in a little bit more detail,
213
00:11:51,860 --> 00:11:56,470
by double checking.
214
00:11:56,470 --> 00:11:58,470
So we're just going to
double check what happens
215
00:11:58,470 --> 00:12:01,280
to the sign of the derivative.
216
00:12:01,280 --> 00:12:03,460
And in the meantime, I'm
going to explain to you
217
00:12:03,460 --> 00:12:05,510
what the derivative
is and also talk
218
00:12:05,510 --> 00:12:07,045
about the second derivative.
219
00:12:07,045 --> 00:12:11,820
So first of all, the trick
for evaluating the derivative
220
00:12:11,820 --> 00:12:13,570
is an algebraic one.
221
00:12:13,570 --> 00:12:16,080
I mean, obviously you can do
this by the quotient rule.
222
00:12:16,080 --> 00:12:24,250
But I just point out that this
is the same thing as this.
223
00:12:24,250 --> 00:12:27,330
And now it has, whoops,
that should be a 2
224
00:12:27,330 --> 00:12:28,690
in the denominator.
225
00:12:28,690 --> 00:12:33,050
And so, now this has
the form 1 - 1/(x+2).
226
00:12:35,950 --> 00:12:39,600
So this makes it easy to
see what the derivative is.
227
00:12:39,600 --> 00:12:42,730
Because the derivative of
a constant is 0, right?
228
00:12:42,730 --> 00:12:49,300
So this is, derivative, is just
going to be, switch the sign.
229
00:12:49,300 --> 00:12:53,410
This is what I wrote before.
230
00:12:53,410 --> 00:12:55,230
And that explains it.
231
00:12:55,230 --> 00:12:57,610
But incidentally,
it also shows you
232
00:12:57,610 --> 00:13:04,890
that that this is a hyperbola.
233
00:13:04,890 --> 00:13:09,620
These are just two
curves of a hyperbola.
234
00:13:09,620 --> 00:13:12,240
So now, let's check the sign.
235
00:13:12,240 --> 00:13:14,566
It's already totally
obvious to us
236
00:13:14,566 --> 00:13:15,940
that this is just
a double check.
237
00:13:15,940 --> 00:13:18,710
We didn't actually even have
to pay any attention to this.
238
00:13:18,710 --> 00:13:19,790
It had better be true.
239
00:13:19,790 --> 00:13:22,220
This is just going to
check our arithmetic.
240
00:13:22,220 --> 00:13:24,570
Namely, it's increasing here.
241
00:13:24,570 --> 00:13:26,610
It's increasing there.
242
00:13:26,610 --> 00:13:27,830
That's got to be true.
243
00:13:27,830 --> 00:13:30,970
And, sure enough,
this is positive,
244
00:13:30,970 --> 00:13:32,930
as you can see it's
1 over a square.
245
00:13:32,930 --> 00:13:34,070
So it is increasing.
246
00:13:34,070 --> 00:13:35,630
So we checked it.
247
00:13:35,630 --> 00:13:39,100
But now, there's one more
thing that I want to just
248
00:13:39,100 --> 00:13:40,720
have you watch out about.
249
00:13:40,720 --> 00:13:46,980
So this means that
f is increasing.
250
00:13:46,980 --> 00:13:51,830
On the interval minus
infinity < x < -2.
251
00:13:51,830 --> 00:13:56,700
And also from -2 all
the way out to infinity.
252
00:13:56,700 --> 00:14:02,260
So I just want to warn
you, you cannot say,
253
00:14:02,260 --> 00:14:10,370
don't say f is increasing on
(minus infinity, infinity),
254
00:14:10,370 --> 00:14:12,020
or all x.
255
00:14:12,020 --> 00:14:14,826
OK, this is just not true.
256
00:14:14,826 --> 00:14:16,700
I've written it on the
board, but it's wrong.
257
00:14:16,700 --> 00:14:18,660
I'd better get rid of it.
258
00:14:18,660 --> 00:14:19,200
There it is.
259
00:14:19,200 --> 00:14:20,970
Get rid of it.
260
00:14:20,970 --> 00:14:24,555
And the reason is, so first
of all it's totally obvious.
261
00:14:24,555 --> 00:14:25,390
It's going up here.
262
00:14:25,390 --> 00:14:28,700
But then it went
zooming back down there.
263
00:14:28,700 --> 00:14:35,860
And here this was true,
but only if x is not -2.
264
00:14:35,860 --> 00:14:37,400
So there's a break.
265
00:14:37,400 --> 00:14:39,280
And you've got to pay
attention to the break.
266
00:14:39,280 --> 00:14:51,060
So basically, the moral here is
that if you ignore this place,
267
00:14:51,060 --> 00:14:54,390
it's like ignoring Mount
Everest, or the Grand Canyon.
268
00:14:54,390 --> 00:14:56,640
You're ignoring the
most important feature
269
00:14:56,640 --> 00:14:58,174
of this function here.
270
00:14:58,174 --> 00:14:59,590
If you're going
to be figuring out
271
00:14:59,590 --> 00:15:01,980
where things are going up
and down, which is basically
272
00:15:01,980 --> 00:15:04,460
all we're doing, you'd
better pay attention
273
00:15:04,460 --> 00:15:07,330
to these kinds of places.
274
00:15:07,330 --> 00:15:09,330
So don't ignore them.
275
00:15:09,330 --> 00:15:13,140
So that's the first remark.
276
00:15:13,140 --> 00:15:17,340
And now there's just a little
bit of decoration as well.
277
00:15:17,340 --> 00:15:19,990
Which is the role of
the second derivative.
278
00:15:19,990 --> 00:15:21,990
So we've written down the
first derivative here.
279
00:15:21,990 --> 00:15:33,490
The second derivative is
now -2 / (x + 2)^3, right?
280
00:15:33,490 --> 00:15:36,300
So I get that from
differentiating this formula up
281
00:15:36,300 --> 00:15:39,490
here for the first derivative.
282
00:15:39,490 --> 00:15:43,970
And now, of course, that's
also, only works for x not equal
283
00:15:43,970 --> 00:15:47,680
to -2.
284
00:15:47,680 --> 00:15:54,820
And now, we can see that this
is going to be negative, let's
285
00:15:54,820 --> 00:15:56,650
see, where is it negative?
286
00:15:56,650 --> 00:15:58,540
When this is a
positive quantity,
287
00:15:58,540 --> 00:16:04,590
so when -2 < x <
infinity, it's negative.
288
00:16:04,590 --> 00:16:07,770
And this is where
this thing is concave.
289
00:16:07,770 --> 00:16:08,670
Let's see.
290
00:16:08,670 --> 00:16:12,650
Did I say that right?
291
00:16:12,650 --> 00:16:13,330
Negative, right?
292
00:16:13,330 --> 00:16:19,180
This is concave down.
293
00:16:19,180 --> 00:16:19,930
Right.
294
00:16:19,930 --> 00:16:22,570
And similarly, if I
look at this expression,
295
00:16:22,570 --> 00:16:27,830
the numerator is always negative
but the denominator becomes
296
00:16:27,830 --> 00:16:31,130
negative as well when x < -2.
297
00:16:31,130 --> 00:16:33,710
So this becomes positive.
298
00:16:33,710 --> 00:16:36,690
So this case, it was
negative over positive.
299
00:16:36,690 --> 00:16:40,950
In this case it was negative
divided by negative.
300
00:16:40,950 --> 00:16:46,300
So here, this is in the range
minus infinity < x < -2 And
301
00:16:46,300 --> 00:16:52,240
here it's concave up.
302
00:16:52,240 --> 00:16:54,962
Now, again, this is just
consistent with what
303
00:16:54,962 --> 00:16:55,920
we're already guessing.
304
00:16:55,920 --> 00:16:57,628
Of course we already
know it in this case
305
00:16:57,628 --> 00:16:59,680
if we know that
this is a hyperbola.
306
00:16:59,680 --> 00:17:01,600
That it's going
to be concave down
307
00:17:01,600 --> 00:17:04,270
to the right of the vertical
line, dotted vertical line.
308
00:17:04,270 --> 00:17:07,770
And concave up to the left.
309
00:17:07,770 --> 00:17:10,640
So what extra piece
of information
310
00:17:10,640 --> 00:17:16,100
is it that this is giving us?
311
00:17:16,100 --> 00:17:17,800
Did I say this backwards?
312
00:17:17,800 --> 00:17:18,520
No.
313
00:17:18,520 --> 00:17:19,222
That's OK.
314
00:17:19,222 --> 00:17:21,430
So what extra piece of
information is this giving us?
315
00:17:21,430 --> 00:17:23,400
It looks like it's giving
us hardly anything.
316
00:17:23,400 --> 00:17:25,680
And it really is giving
us hardly anything.
317
00:17:25,680 --> 00:17:28,840
But it is giving us something
that's a little aesthetic.
318
00:17:28,840 --> 00:17:34,280
It's ruling out the
possibility of a wiggle.
319
00:17:34,280 --> 00:17:37,590
There isn't anything
like that in the curve.
320
00:17:37,590 --> 00:17:39,790
It can't shift from
curving this way
321
00:17:39,790 --> 00:17:41,590
to curving that way
to curving this way.
322
00:17:41,590 --> 00:17:42,860
That doesn't happen.
323
00:17:42,860 --> 00:17:59,070
So these properties say there's
no wiggle in the graph of that.
324
00:17:59,070 --> 00:17:59,790
Alright.
325
00:17:59,790 --> 00:18:01,390
So.
326
00:18:01,390 --> 00:18:01,890
Question.
327
00:18:01,890 --> 00:18:05,570
STUDENT: Do we define the
increasing and decreasing based
328
00:18:05,570 --> 00:18:09,250
purely on the
derivative, or the sort
329
00:18:09,250 --> 00:18:13,390
of more general definition
of picking any two points
330
00:18:13,390 --> 00:18:14,310
and seeing.
331
00:18:14,310 --> 00:18:16,610
Because sometimes there
can be an inconsistency
332
00:18:16,610 --> 00:18:20,300
between the two definitions.
333
00:18:20,300 --> 00:18:26,110
PROFESSOR: OK, so the
question is, in this course,
334
00:18:26,110 --> 00:18:29,290
are we going to define
positive derivative as being
335
00:18:29,290 --> 00:18:31,360
the same thing as increasing.
336
00:18:31,360 --> 00:18:33,030
And the answer is no.
337
00:18:33,030 --> 00:18:36,210
We'll try to use these
terms separately.
338
00:18:36,210 --> 00:18:40,330
What's always true is
that if f' is positive,
339
00:18:40,330 --> 00:18:42,530
then f is increasing.
340
00:18:42,530 --> 00:18:45,060
But the reverse is
not necessarily true.
341
00:18:45,060 --> 00:18:47,190
It could be very flat,
the derivative can be 0
342
00:18:47,190 --> 00:18:50,050
and still the function
can be increasing.
343
00:18:50,050 --> 00:18:53,730
OK, the derivative can
be 0 at a few places.
344
00:18:53,730 --> 00:18:59,300
For instance, like some cubics.
345
00:18:59,300 --> 00:19:05,370
Other questions?
346
00:19:05,370 --> 00:19:09,860
So that's as much as I
need to say in general.
347
00:19:09,860 --> 00:19:11,390
I mean, in a specific case.
348
00:19:11,390 --> 00:19:13,240
But I want to get
you a general scheme
349
00:19:13,240 --> 00:19:16,500
and I want to go through a
more complicated example that
350
00:19:16,500 --> 00:19:22,750
gets all the features
of this kind of thing.
351
00:19:22,750 --> 00:19:34,710
So let's talk about a general
strategy for sketching.
352
00:19:34,710 --> 00:19:40,300
So the first part of this
strategy, if you like,
353
00:19:40,300 --> 00:19:40,800
let's see.
354
00:19:40,800 --> 00:19:42,430
I have it all plotted out here.
355
00:19:42,430 --> 00:19:45,930
So I'm going to make sure
I get it exactly the way
356
00:19:45,930 --> 00:19:47,780
I wanted you to see.
357
00:19:47,780 --> 00:19:51,440
So I have, it's plotting.
358
00:19:51,440 --> 00:19:52,720
The plot thickens.
359
00:19:52,720 --> 00:19:54,040
Here we go.
360
00:19:54,040 --> 00:19:57,710
So plot, what is it that
you should plot first?
361
00:19:57,710 --> 00:20:01,070
Before you even think
about derivatives,
362
00:20:01,070 --> 00:20:08,250
you should plot discontinuities.
363
00:20:08,250 --> 00:20:18,490
Especially the infinite ones.
364
00:20:18,490 --> 00:20:20,210
That's the first
thing you should do.
365
00:20:20,210 --> 00:20:27,160
And then, you should plot
end points, for ends.
366
00:20:27,160 --> 00:20:30,580
For x going to plus
or minus infinity
367
00:20:30,580 --> 00:20:35,640
if there don't happen to be
any finite ends to the problem.
368
00:20:35,640 --> 00:20:44,600
And the third thing you can
do is plot any easy points.
369
00:20:44,600 --> 00:20:49,000
This is optional.
370
00:20:49,000 --> 00:20:50,710
At your discretion.
371
00:20:50,710 --> 00:20:53,810
You might, for instance,
on this example,
372
00:20:53,810 --> 00:20:59,550
plot the places where the
graph crosses the axis.
373
00:20:59,550 --> 00:21:04,540
If you want to.
374
00:21:04,540 --> 00:21:05,810
So that's the first part.
375
00:21:05,810 --> 00:21:08,190
And again, this is
all precalculus.
376
00:21:08,190 --> 00:21:18,050
So now, in the second part we're
going to solve this equation
377
00:21:18,050 --> 00:21:29,360
and we're going to plot the
critical points and values.
378
00:21:29,360 --> 00:21:32,810
In the problem which we just
discussed, there weren't any.
379
00:21:32,810 --> 00:21:38,640
So this part was empty.
380
00:21:38,640 --> 00:21:49,685
So the third step is to decide
whether f', sorry, whether,
381
00:21:49,685 --> 00:22:01,120
f' is positive or
negative on each interval.
382
00:22:01,120 --> 00:22:17,210
Between critical
points, discontinuities.
383
00:22:17,210 --> 00:22:22,560
The direction of the sign, in
this case it doesn't change.
384
00:22:22,560 --> 00:22:24,900
It goes up here and
it also goes up here.
385
00:22:24,900 --> 00:22:27,770
But it could go up here
and then come back down.
386
00:22:27,770 --> 00:22:31,460
So the direction can change
at every critical point.
387
00:22:31,460 --> 00:22:33,850
It can change at
every discontinuity.
388
00:22:33,850 --> 00:22:35,350
And you don't know.
389
00:22:35,350 --> 00:22:39,310
However, this
particular step has
390
00:22:39,310 --> 00:22:47,230
to be consistent with 1
and 2, with steps 1 and 2.
391
00:22:47,230 --> 00:22:51,800
In fact, it will
never, if you can
392
00:22:51,800 --> 00:22:56,970
succeed in doing steps 1 and
2, you'll never need step 3.
393
00:22:56,970 --> 00:23:02,190
All it's doing is
double-checking.
394
00:23:02,190 --> 00:23:05,770
So if you made an arithmetic
mistake somewhere,
395
00:23:05,770 --> 00:23:09,160
you'll be able to see it.
396
00:23:09,160 --> 00:23:10,870
So that's maybe the
most important thing.
397
00:23:10,870 --> 00:23:13,050
And it's actually the most
frustrating thing for me
398
00:23:13,050 --> 00:23:17,700
when I see people working on
problems, is they start step 3,
399
00:23:17,700 --> 00:23:21,140
they get it wrong, and then they
start trying to draw the graph
400
00:23:21,140 --> 00:23:22,480
and it doesn't work.
401
00:23:22,480 --> 00:23:23,620
Because it's inconsistent.
402
00:23:23,620 --> 00:23:25,880
And the reason is
some arithmetic error
403
00:23:25,880 --> 00:23:27,630
with the derivative
or something like that
404
00:23:27,630 --> 00:23:29,880
or some other misinterpretation.
405
00:23:29,880 --> 00:23:31,930
And then there's a total mess.
406
00:23:31,930 --> 00:23:34,170
If you start with
these two steps,
407
00:23:34,170 --> 00:23:36,454
then you're going to know
when you get to this step
408
00:23:36,454 --> 00:23:37,620
that you're making mistakes.
409
00:23:37,620 --> 00:23:41,860
People don't generally make as
many mistakes in the first two
410
00:23:41,860 --> 00:23:42,360
steps.
411
00:23:42,360 --> 00:23:45,220
Anyway, in fact you can
skip this step if you want.
412
00:23:45,220 --> 00:23:49,470
But that's at risk of not
double-checking your work.
413
00:23:49,470 --> 00:23:51,280
So what's the fourth step?
414
00:23:51,280 --> 00:23:59,400
Well, we take a look at whether
f'' is positive or negative.
415
00:23:59,400 --> 00:24:02,300
And so we're deciding on things
like whether it's concave
416
00:24:02,300 --> 00:24:07,640
up or down.
417
00:24:07,640 --> 00:24:15,120
And we have these
points, f''(x) = 0,
418
00:24:15,120 --> 00:24:24,570
which are called
inflection points.
419
00:24:24,570 --> 00:24:31,550
And the last step is just
to combine everything.
420
00:24:31,550 --> 00:24:35,710
So this is this the
scheme, the general scheme.
421
00:24:35,710 --> 00:24:58,850
And let's just carry it
out in a particular case.
422
00:24:58,850 --> 00:25:02,280
So here's the function that
I'm going to use as an example.
423
00:25:02,280 --> 00:25:08,250
I'll use f(x) = x / ln x.
424
00:25:08,250 --> 00:25:11,460
And because the
logarithm-- yeah, question.
425
00:25:11,460 --> 00:25:11,960
Yeah.
426
00:25:11,960 --> 00:25:18,660
STUDENT: [INAUDIBLE]
427
00:25:18,660 --> 00:25:21,350
PROFESSOR: The question
is, is this optional.
428
00:25:21,350 --> 00:25:25,570
So that's a good question.
429
00:25:25,570 --> 00:25:26,390
Is this optional.
430
00:25:26,390 --> 00:25:31,516
STUDENT: [INAUDIBLE]
431
00:25:31,516 --> 00:25:37,590
PROFESSOR: OK, the question
is is this optional,
432
00:25:37,590 --> 00:25:38,660
this kind of question.
433
00:25:38,660 --> 00:25:48,620
And the answer is,
it's more than just--
434
00:25:48,620 --> 00:25:51,700
so, in many instances, I'm
not going to ask you to.
435
00:25:51,700 --> 00:25:55,170
I strongly recommend that
if I don't ask you to do it,
436
00:25:55,170 --> 00:25:57,050
that you not try.
437
00:25:57,050 --> 00:26:01,050
Because it's usually awful to
find the second derivative.
438
00:26:01,050 --> 00:26:02,940
Any time you can get
away without computing
439
00:26:02,940 --> 00:26:06,330
a second derivative,
you're better off.
440
00:26:06,330 --> 00:26:07,834
So in many, many instances.
441
00:26:07,834 --> 00:26:09,500
On the other hand,
if I ask you to do it
442
00:26:09,500 --> 00:26:13,060
it's because I want you
to have the work to do it.
443
00:26:13,060 --> 00:26:16,470
But basically, if
nobody forces you to,
444
00:26:16,470 --> 00:26:22,130
I would say never
do step 4 here.
445
00:26:22,130 --> 00:26:26,750
Other questions.
446
00:26:26,750 --> 00:26:27,610
All right.
447
00:26:27,610 --> 00:26:29,150
So we're going to
force ourselves
448
00:26:29,150 --> 00:26:31,810
to do step 4, however,
in this instance.
449
00:26:31,810 --> 00:26:35,010
But maybe this will be
one of the few times.
450
00:26:35,010 --> 00:26:39,140
So here we go, just for
illustrative purposes.
451
00:26:39,140 --> 00:26:43,240
OK, now.
452
00:26:43,240 --> 00:26:46,140
So here's the function
that I want to discuss.
453
00:26:46,140 --> 00:26:49,120
And the range has
to be x positive,
454
00:26:49,120 --> 00:26:55,500
because the logarithm is not
defined for negative values.
455
00:26:55,500 --> 00:26:57,660
So the first thing
that I'm going to do
456
00:26:57,660 --> 00:27:02,560
is, I'd like to follow
the scheme here.
457
00:27:02,560 --> 00:27:04,770
Because if I don't
follow the scheme,
458
00:27:04,770 --> 00:27:06,490
I'm going to get
a little mixed up.
459
00:27:06,490 --> 00:27:13,980
So the first part is to
find the singularities.
460
00:27:13,980 --> 00:27:17,060
That is, the places
where f is infinite.
461
00:27:17,060 --> 00:27:20,720
And that's when the logarithm,
the denominator, vanishes.
462
00:27:20,720 --> 00:27:25,630
So that's f(1+), if you like.
463
00:27:25,630 --> 00:27:32,000
So that's 1 / ln(1+),
which is 1 / 0,
464
00:27:32,000 --> 00:27:34,280
with a little bit of
positiveness to it.
465
00:27:34,280 --> 00:27:37,270
Which is infinity.
466
00:27:37,270 --> 00:27:39,830
And second, we do
it the other way.
467
00:27:39,830 --> 00:27:46,850
And not surprisingly, this comes
out to be negative infinity.
468
00:27:46,850 --> 00:27:51,980
Now, the next thing I
want to do is the ends.
469
00:27:51,980 --> 00:27:56,980
So I call these the ends.
470
00:27:56,980 --> 00:28:01,380
And there are two of them.
471
00:28:01,380 --> 00:28:06,250
One of them is f(0)
from the right.
472
00:28:06,250 --> 00:28:08,300
f(0+).
473
00:28:08,300 --> 00:28:21,120
So that is 0+ / ln(0+),
which is 0+ divided by, well,
474
00:28:21,120 --> 00:28:25,360
ln(0+) is actually
minus infinity.
475
00:28:25,360 --> 00:28:27,180
That's what happens
to the logarithm, goes
476
00:28:27,180 --> 00:28:28,170
to minus infinity.
477
00:28:28,170 --> 00:28:31,160
So this is 0 over infinity,
which is definitely 0,
478
00:28:31,160 --> 00:28:37,100
there's no problem about
what happens to this.
479
00:28:37,100 --> 00:28:42,910
The other side, so this is the
end, this is the first end.
480
00:28:42,910 --> 00:28:44,920
The range is this.
481
00:28:44,920 --> 00:28:48,019
And I just did
the left endpoint.
482
00:28:48,019 --> 00:28:49,810
And so now I have to
do the right endpoint,
483
00:28:49,810 --> 00:28:51,870
I have to let x go to infinity.
484
00:28:51,870 --> 00:28:53,591
So if I let x go
to infinity, I'm
485
00:28:53,591 --> 00:28:55,090
just going to have
to think about it
486
00:28:55,090 --> 00:28:57,690
a little bit by plugging
in a very large number.
487
00:28:57,690 --> 00:29:01,960
I'll plug in 10^10,
to see what happens.
488
00:29:01,960 --> 00:29:07,890
So if I plug in 10^10 into x /
ln x, I get 10^10 / ln(10^10).
489
00:29:11,730 --> 00:29:17,590
Which is 10^10 / (10 ln(10)).
490
00:29:17,590 --> 00:29:20,110
So the denominator,
this number here,
491
00:29:20,110 --> 00:29:23,180
is about 2 point something.
492
00:29:23,180 --> 00:29:25,130
2.3 or so.
493
00:29:25,130 --> 00:29:27,470
So this is maybe 230
in the denominator,
494
00:29:27,470 --> 00:29:31,900
and this is a number
with ten 0's after it.
495
00:29:31,900 --> 00:29:33,530
So it's very, very large.
496
00:29:33,530 --> 00:29:35,300
I claim it's big.
497
00:29:35,300 --> 00:29:38,080
And that gives us the
clue that what's happening
498
00:29:38,080 --> 00:29:40,540
is that this thing is infinite.
499
00:29:40,540 --> 00:29:42,150
So, in other words,
our conclusion
500
00:29:42,150 --> 00:29:52,120
is that f of
infinity is infinity.
501
00:29:52,120 --> 00:29:59,650
So what do we have so
far for our function?
502
00:29:59,650 --> 00:30:03,300
We're just trying to build the
scaffolding of the function.
503
00:30:03,300 --> 00:30:07,270
And we're doing it by taking
the most important points.
504
00:30:07,270 --> 00:30:09,147
And from a mathematician's
point of view,
505
00:30:09,147 --> 00:30:10,980
the most important
points are the ones which
506
00:30:10,980 --> 00:30:13,490
are sort of infinitely obvious.
507
00:30:13,490 --> 00:30:15,190
For the ends of the problem.
508
00:30:15,190 --> 00:30:19,730
So that's where we're heading.
509
00:30:19,730 --> 00:30:22,600
We have a vertical
asymptote, which is at x = 1.
510
00:30:22,600 --> 00:30:29,150
So this gives us x = 1.
511
00:30:29,150 --> 00:30:34,270
And we have a value which
is that it's 0 here.
512
00:30:34,270 --> 00:30:38,640
And we also know that when
we come in from the-- sorry,
513
00:30:38,640 --> 00:30:42,330
so we come in from
the left, that's
514
00:30:42,330 --> 00:30:46,060
f, the one from the left,
we get negative infinity.
515
00:30:46,060 --> 00:30:47,550
So it's diving down.
516
00:30:47,550 --> 00:30:52,460
It's going down like this.
517
00:30:52,460 --> 00:30:55,925
And, furthermore, on the other
side we know it's climbing up.
518
00:30:55,925 --> 00:30:58,560
So it's going up like this.
519
00:30:58,560 --> 00:31:00,270
Just start a little higher.
520
00:31:00,270 --> 00:31:00,770
Right, so.
521
00:31:00,770 --> 00:31:02,520
So far, this is what we know.
522
00:31:02,520 --> 00:31:05,810
Oh, and there's one
other thing that we know.
523
00:31:05,810 --> 00:31:12,420
When we go to plus infinity,
it's going back up.
524
00:31:12,420 --> 00:31:15,150
So, so far we have this.
525
00:31:15,150 --> 00:31:17,750
Now, already it should
be pretty obvious what's
526
00:31:17,750 --> 00:31:19,619
going to happen
to this function.
527
00:31:19,619 --> 00:31:21,160
So there shouldn't
be many surprises.
528
00:31:21,160 --> 00:31:23,160
It's going to come
down like this.
529
00:31:23,160 --> 00:31:27,090
Go like this, it's going to
turn around and go back up.
530
00:31:27,090 --> 00:31:29,290
That's what we expect.
531
00:31:29,290 --> 00:31:33,600
So we don't know that yet,
but we're pretty sure.
532
00:31:33,600 --> 00:31:36,800
So at this point, we can start
looking at the critical points.
533
00:31:36,800 --> 00:31:41,994
We can do our step 2 here -
we need a little bit more room
534
00:31:41,994 --> 00:31:45,490
here - and see what's
happening with this function.
535
00:31:45,490 --> 00:31:49,220
So I have to differentiate it.
536
00:31:49,220 --> 00:31:52,070
And it's, this is
the quotient rule.
537
00:31:52,070 --> 00:31:54,880
So remember the function
is up here, x / ln x.
538
00:31:54,880 --> 00:31:59,400
So I have a (ln x)^2
in the denominator.
539
00:31:59,400 --> 00:32:02,430
And I get here the derivative
of x is 1, so we get 1 *
540
00:32:02,430 --> 00:32:07,174
ln x minus x times
the derivative of ln
541
00:32:07,174 --> 00:32:07,840
x, which is 1/x.
542
00:32:10,560 --> 00:32:16,850
So all told, that's
(ln x - 1) / (ln x)^2.
543
00:32:20,160 --> 00:32:27,770
So here's our derivative.
544
00:32:27,770 --> 00:32:35,080
And now, if I set this equal to
0, at least in the numerator,
545
00:32:35,080 --> 00:32:40,970
the numerator is 0 when x = e.
546
00:32:40,970 --> 00:32:43,490
The log of e is 1.
547
00:32:43,490 --> 00:32:46,290
So here's our critical point.
548
00:32:46,290 --> 00:32:51,320
And we have a critical
value, which is f(e).
549
00:32:51,320 --> 00:32:55,610
And that's going to be e / ln e.
550
00:32:55,610 --> 00:32:57,240
Which is e, again.
551
00:32:57,240 --> 00:32:59,090
Because ln e = 1.
552
00:32:59,090 --> 00:33:01,890
So now I can also plot
the critical point,
553
00:33:01,890 --> 00:33:03,110
which is down here.
554
00:33:03,110 --> 00:33:07,680
And there's only one of
them, and it's at (e, e).
555
00:33:07,680 --> 00:33:09,530
That's kind of
not to scale here,
556
00:33:09,530 --> 00:33:12,520
because my blackboard
isn't quite tall enough.
557
00:33:12,520 --> 00:33:15,360
It should be over here
and then, it's slope 1.
558
00:33:15,360 --> 00:33:17,140
But I dipped it down.
559
00:33:17,140 --> 00:33:18,730
So this is not to
scale, and indeed
560
00:33:18,730 --> 00:33:20,490
that's one of the
things that we're not
561
00:33:20,490 --> 00:33:22,680
going to attempt to do
with these pictures,
562
00:33:22,680 --> 00:33:24,680
is to make them to scale.
563
00:33:24,680 --> 00:33:29,560
So the scale's a
little squashed.
564
00:33:29,560 --> 00:33:32,710
So, so far I have
this critical point.
565
00:33:32,710 --> 00:33:36,180
And, in fact, I'm going
to label it with a C.
566
00:33:36,180 --> 00:33:38,030
Whenever I have a
critical point I'll just
567
00:33:38,030 --> 00:33:41,490
make sure that I remember
that that's what it is.
568
00:33:41,490 --> 00:33:45,120
And since there's only one,
the rest of this picture
569
00:33:45,120 --> 00:33:49,900
is now correct.
570
00:33:49,900 --> 00:33:54,880
That's the same mechanism that
we used for the hyperbola.
571
00:33:54,880 --> 00:33:56,977
Namely, we know there's
only one place where
572
00:33:56,977 --> 00:33:57,810
the derivative is 0.
573
00:33:57,810 --> 00:33:59,980
So that means there
no more horizontals,
574
00:33:59,980 --> 00:34:01,950
so there's no more backtracking.
575
00:34:01,950 --> 00:34:03,330
It has to come down to here.
576
00:34:03,330 --> 00:34:03,970
Get to there.
577
00:34:03,970 --> 00:34:06,060
This is the only place
it can turn around.
578
00:34:06,060 --> 00:34:06,834
Goes back up.
579
00:34:06,834 --> 00:34:09,000
It has to start here and
it has to go down to there.
580
00:34:09,000 --> 00:34:10,600
It can't go above 0.
581
00:34:10,600 --> 00:34:13,600
Do not pass go, do
not get positive.
582
00:34:13,600 --> 00:34:20,230
It has to head down here.
583
00:34:20,230 --> 00:34:21,690
So that's great.
584
00:34:21,690 --> 00:34:25,080
That means that this picture is
almost completely correct now.
585
00:34:25,080 --> 00:34:27,520
And the rest is more
or less decoration.
586
00:34:27,520 --> 00:34:30,250
We're pretty much done
with the way it looks,
587
00:34:30,250 --> 00:34:34,570
at least schematically.
588
00:34:34,570 --> 00:34:37,700
However, I am going to punish
you, because I warned you.
589
00:34:37,700 --> 00:34:40,120
We are going to go over
here and do this step 4
590
00:34:40,120 --> 00:34:44,216
and fix up the concavity.
591
00:34:44,216 --> 00:34:45,840
And we're also going
to do a little bit
592
00:34:45,840 --> 00:35:00,770
of that double-checking.
593
00:35:00,770 --> 00:35:04,960
So now, let's
again-- just, I want
594
00:35:04,960 --> 00:35:10,660
to emphasize-- We're going
to do a double-check.
595
00:35:10,660 --> 00:35:12,240
This is part 3.
596
00:35:12,240 --> 00:35:16,870
But in advance, I already
have, based on this picture
597
00:35:16,870 --> 00:35:19,000
I already know what
has to be true.
598
00:35:19,000 --> 00:35:35,450
That f is decreasing on 0 to 1.
f is also decreasing on 1 to e.
599
00:35:35,450 --> 00:35:45,490
And f is increasing
on e to infinity.
600
00:35:45,490 --> 00:35:49,144
So, already, because we
plotted a bunch of points
601
00:35:49,144 --> 00:35:51,060
and we know that there
aren't any places where
602
00:35:51,060 --> 00:35:52,518
the derivative
vanishes, we already
603
00:35:52,518 --> 00:35:55,500
know it goes down, down, up.
604
00:35:55,500 --> 00:35:56,970
That's what it's got to do.
605
00:35:56,970 --> 00:35:59,550
Now, we'll just make sure that
we didn't make any arithmetic
606
00:35:59,550 --> 00:36:00,720
mistakes, now.
607
00:36:00,720 --> 00:36:02,980
By actually computing
the derivative,
608
00:36:02,980 --> 00:36:04,650
or staring at it, anyway.
609
00:36:04,650 --> 00:36:10,350
And making sure
that it's correct.
610
00:36:10,350 --> 00:36:17,600
So first of all, we just
take a look at the numerator.
611
00:36:17,600 --> 00:36:26,570
So f', remember, was
(ln x - 1) / (ln x)^2.
612
00:36:26,570 --> 00:36:28,480
So the denominator is positive.
613
00:36:28,480 --> 00:36:32,650
So let's just take a
look at the three ranges.
614
00:36:32,650 --> 00:36:37,330
So we have 0 < x < 1.
615
00:36:37,330 --> 00:36:40,310
And on that range, the
logarithm is negative,
616
00:36:40,310 --> 00:36:45,230
so this is negative divided by
positive, which is negative.
617
00:36:45,230 --> 00:36:47,220
That's decreasing, that's good.
618
00:36:47,220 --> 00:36:50,160
And in fact, that also
works on the next range.
619
00:36:50,160 --> 00:36:55,357
1 < x < e, it's negative
divided by positive.
620
00:36:55,357 --> 00:36:57,190
And the only reason why
we skipped 1, again,
621
00:36:57,190 --> 00:36:58,432
is that it's undefined there.
622
00:36:58,432 --> 00:37:01,040
And there's something
dramatic happening there.
623
00:37:01,040 --> 00:37:05,240
And then, at the last range,
when x is bigger than e,
624
00:37:05,240 --> 00:37:07,770
that means the logarithm
is already bigger than 1.
625
00:37:07,770 --> 00:37:09,145
So the numerator
is now positive,
626
00:37:09,145 --> 00:37:13,650
and the denominator's still
positive, so it's increasing.
627
00:37:13,650 --> 00:37:22,910
So we've just double-checked
something that we already knew.
628
00:37:22,910 --> 00:37:26,540
Alright, so that's
pretty much all
629
00:37:26,540 --> 00:37:29,200
there is to say about step 3.
630
00:37:29,200 --> 00:37:33,980
So this is checking the
positivity and negativity.
631
00:37:33,980 --> 00:37:35,660
And now, step 4.
632
00:37:35,660 --> 00:37:38,480
There is one small point which
I want to make before we go on.
633
00:37:38,480 --> 00:37:42,110
Which is that
sometimes, you can't
634
00:37:42,110 --> 00:37:45,445
evaluate the function or its
derivative particularly well.
635
00:37:45,445 --> 00:37:48,359
So sometimes you can't
plot the points very well.
636
00:37:48,359 --> 00:37:50,150
And if you can't plot
the points very well,
637
00:37:50,150 --> 00:37:52,440
then you might
have to do 3 first,
638
00:37:52,440 --> 00:37:55,290
to figure out what's
going on a little bit.
639
00:37:55,290 --> 00:37:59,150
You might have to skip.
640
00:37:59,150 --> 00:38:02,400
So now we're going to go on
to the second derivative.
641
00:38:02,400 --> 00:38:07,180
But first, I want to
use an algebraic trick
642
00:38:07,180 --> 00:38:08,470
to rearrange the terms.
643
00:38:08,470 --> 00:38:10,820
And I want to notice
one more little point.
644
00:38:10,820 --> 00:38:16,320
Which I-- as I say, this is
decoration for the graph.
645
00:38:16,320 --> 00:38:18,290
So I want to
rewrite the formula.
646
00:38:18,290 --> 00:38:22,320
Maybe I'll do it
right over here.
647
00:38:22,320 --> 00:38:27,270
Another way of writing this
is 1/(ln x) - 1/(ln x)^2.
648
00:38:31,440 --> 00:38:35,240
So that's another way of
writing the derivative.
649
00:38:35,240 --> 00:38:38,520
And that allows me
to notice something
650
00:38:38,520 --> 00:38:40,830
that I missed, before.
651
00:38:40,830 --> 00:38:47,410
When I solved the equation ln x
- 1 - this is equal to 0 here,
652
00:38:47,410 --> 00:38:48,590
this equation here.
653
00:38:48,590 --> 00:38:51,460
I missed a possibility.
654
00:38:51,460 --> 00:38:54,020
I missed the possibility
that the denominator
655
00:38:54,020 --> 00:38:58,690
could be infinity.
656
00:38:58,690 --> 00:39:02,040
So actually, if the
denominator's infinity,
657
00:39:02,040 --> 00:39:05,460
as you can see from the
other expression there,
658
00:39:05,460 --> 00:39:09,000
it actually is true that
the derivative is 0.
659
00:39:09,000 --> 00:39:16,710
So also when x = 0+, the
slope is going to be 0.
660
00:39:16,710 --> 00:39:19,050
Let me just
emphasize that again.
661
00:39:19,050 --> 00:39:23,500
If you evaluate using this
other formula over here,
662
00:39:23,500 --> 00:39:28,410
this is 1/(ln(0+))
- 1/(ln(0+))^2.
663
00:39:31,540 --> 00:39:36,500
That's 1 over -infinity - minus
1 over infinity, if you like,
664
00:39:36,500 --> 00:39:37,310
squared.
665
00:39:37,310 --> 00:39:40,630
Anyway, it's 0.
666
00:39:40,630 --> 00:39:42,330
So this is 0.
667
00:39:42,330 --> 00:39:43,410
The slope is 0 there.
668
00:39:43,410 --> 00:39:46,640
That is a little piece of
decoration on our graph.
669
00:39:46,640 --> 00:39:50,330
It's telling us, going
back to our graph here,
670
00:39:50,330 --> 00:39:53,560
it's telling us this is coming
in with slope horizontal.
671
00:39:53,560 --> 00:39:57,530
So we're starting out this way.
672
00:39:57,530 --> 00:40:01,013
That's just a little
start here to the graph.
673
00:40:01,013 --> 00:40:02,620
It's a horizontal slope.
674
00:40:02,620 --> 00:40:07,940
So there really were two places
where the slope was horizontal.
675
00:40:07,940 --> 00:40:11,470
Now, with the help of
this second formula
676
00:40:11,470 --> 00:40:16,972
I can also differentiate
a second time.
677
00:40:16,972 --> 00:40:19,430
So it's a little bit easier to
do that if I differentiate 1
678
00:40:19,430 --> 00:40:28,050
over the log, that's -(ln
x)^(-2) 1/x + 2 (ln x)^(-3)
679
00:40:28,050 --> 00:40:28,550
1/x.
680
00:40:34,270 --> 00:40:39,090
And that, if I put it
over a common denominator,
681
00:40:39,090 --> 00:40:48,690
is x (ln x)^3 times,
let's see here,
682
00:40:48,690 --> 00:40:55,340
I guess I'll have to
take the 2 - ln x.
683
00:40:55,340 --> 00:40:57,260
So I've now
rewritten the formula
684
00:40:57,260 --> 00:41:03,450
for the second
derivative as a ratio.
685
00:41:03,450 --> 00:41:09,610
Now, to decide the sign, you
see there are two places where
686
00:41:09,610 --> 00:41:11,910
the sign flips.
687
00:41:11,910 --> 00:41:16,030
The numerator crosses
when the logarithm is 2,
688
00:41:16,030 --> 00:41:18,620
that's going to be when x = e^2.
689
00:41:18,620 --> 00:41:22,400
And the denominator
flips when x = 1,
690
00:41:22,400 --> 00:41:28,020
that's when the log flips
from positive to negative.
691
00:41:28,020 --> 00:41:32,280
So we have a couple
of ranges here.
692
00:41:32,280 --> 00:41:37,580
So, first of all, we have
the range from 0 to 1.
693
00:41:37,580 --> 00:41:42,270
And then we have the
range from 1 to e^2.
694
00:41:42,270 --> 00:41:46,380
And then we have the range
from e^2 all the way out
695
00:41:46,380 --> 00:41:49,780
to infinity.
696
00:41:49,780 --> 00:41:57,210
So between 0 and
1, the numerator
697
00:41:57,210 --> 00:41:59,630
is, well this is a
negative number and this,
698
00:41:59,630 --> 00:42:01,300
so minus a negative
number is positive,
699
00:42:01,300 --> 00:42:04,650
so the numerator is positive.
700
00:42:04,650 --> 00:42:08,030
And the denominator is negative,
because the log is negative
701
00:42:08,030 --> 00:42:09,660
and it's taken to
the third power.
702
00:42:09,660 --> 00:42:12,170
So this is a negative
number, so it's positive
703
00:42:12,170 --> 00:42:15,190
divided by negative,
which is less than 0.
704
00:42:15,190 --> 00:42:18,770
That means it's concave down.
705
00:42:18,770 --> 00:42:26,040
So this is a concave down part.
706
00:42:26,040 --> 00:42:28,230
And that's a good thing,
because over here this
707
00:42:28,230 --> 00:42:29,120
was concave down.
708
00:42:29,120 --> 00:42:30,560
So there are no wiggles.
709
00:42:30,560 --> 00:42:34,260
It goes straight
down, like this.
710
00:42:34,260 --> 00:42:41,590
And then the other two pieces
are f'' is equal to, well
711
00:42:41,590 --> 00:42:43,260
it's going to switch here.
712
00:42:43,260 --> 00:42:44,856
The denominator
becomes positive.
713
00:42:44,856 --> 00:42:48,190
So it's positive over positive.
714
00:42:48,190 --> 00:42:56,670
So this is concave up.
715
00:42:56,670 --> 00:42:58,410
And that's going over here.
716
00:42:58,410 --> 00:43:02,715
But notice that it's not the
bottom where it turns around,
717
00:43:02,715 --> 00:43:07,740
it's somewhere else.
718
00:43:07,740 --> 00:43:09,930
So there's another
transition here.
719
00:43:09,930 --> 00:43:12,090
This is e^2.
720
00:43:12,090 --> 00:43:15,090
This is e.
721
00:43:15,090 --> 00:43:20,440
So what happens at the end is,
again, the sign flips again.
722
00:43:20,440 --> 00:43:26,580
Because the numerator, now,
when x > e^2, becomes negative.
723
00:43:26,580 --> 00:43:29,965
And this is negative divided
by positive, which is negative.
724
00:43:29,965 --> 00:43:35,630
And this is concave down.
725
00:43:35,630 --> 00:43:39,100
And so we didn't quite
draw the graph right.
726
00:43:39,100 --> 00:43:40,990
There's an inflection
point right here,
727
00:43:40,990 --> 00:43:45,670
which I'll label with an
I. And it makes a turn
728
00:43:45,670 --> 00:43:47,340
the other way at that point.
729
00:43:47,340 --> 00:43:49,480
So there was a wiggle.
730
00:43:49,480 --> 00:43:51,210
There's the wiggle.
731
00:43:51,210 --> 00:43:53,710
Still going up, still
going to infinity.
732
00:43:53,710 --> 00:43:57,040
But kind of the slope
of a mountain, right?
733
00:43:57,040 --> 00:44:01,240
It's going the other way.
734
00:44:01,240 --> 00:44:03,510
This point happens
to be (e^2, e^2 / 2).
735
00:44:09,350 --> 00:44:11,980
So that's as detailed
as we'll ever get.
736
00:44:11,980 --> 00:44:14,760
And indeed, the
next game is going
737
00:44:14,760 --> 00:44:18,730
to be avoid being-- is to
avoid being this detailed.
738
00:44:18,730 --> 00:44:21,760
So let me introduce
the next subject.
739
00:44:21,760 --> 00:44:48,310
Which is maxima and minima.
740
00:44:48,310 --> 00:45:04,160
OK, now, maxima and minima,
maximum and minimum problems
741
00:45:04,160 --> 00:45:06,550
can be described graphically
in the following ways.
742
00:45:06,550 --> 00:45:13,150
Suppose you have a
function, right, here it is.
743
00:45:13,150 --> 00:45:14,420
OK?
744
00:45:14,420 --> 00:45:24,630
Now, find the maximum.
745
00:45:24,630 --> 00:45:30,740
And find the minimum.
746
00:45:30,740 --> 00:45:31,410
OK.
747
00:45:31,410 --> 00:45:38,880
So this problem is done.
748
00:45:38,880 --> 00:45:51,625
The point being, that it is
easy to find max and the min
749
00:45:51,625 --> 00:45:58,670
with the sketch.
750
00:45:58,670 --> 00:46:00,330
It's very easy.
751
00:46:00,330 --> 00:46:05,130
The goal, the problem, is that
the sketch is a lot of work.
752
00:46:05,130 --> 00:46:10,180
We just spent 20 minutes
sketching something.
753
00:46:10,180 --> 00:46:12,637
We would not like to
spend all that time
754
00:46:12,637 --> 00:46:14,970
every single time we want to
find a maximum and minimum.
755
00:46:14,970 --> 00:46:19,576
So the goal is to do it
with-- so our goal is
756
00:46:19,576 --> 00:46:25,170
to use shortcuts.
757
00:46:25,170 --> 00:46:30,650
And, indeed, as I said
earlier, we certainly
758
00:46:30,650 --> 00:46:33,840
never want to use the second
derivative if we can avoid it.
759
00:46:33,840 --> 00:46:36,450
And we don't want to
decorate the graph
760
00:46:36,450 --> 00:46:38,550
and do all of these
elaborate, subtle,
761
00:46:38,550 --> 00:46:40,900
things which make the graph
look nicer and really,
762
00:46:40,900 --> 00:46:42,640
or aesthetically appropriate.
763
00:46:42,640 --> 00:46:46,700
But are totally unnecessary
to see whether the graph is
764
00:46:46,700 --> 00:46:54,290
up or down.
765
00:46:54,290 --> 00:46:57,030
So essentially,
this whole business
766
00:46:57,030 --> 00:47:00,030
is out, which is a good thing.
767
00:47:00,030 --> 00:47:03,640
And, unfortunately,
those early parts
768
00:47:03,640 --> 00:47:06,170
are the parts that
people tend to ignore.
769
00:47:06,170 --> 00:47:10,150
Which are typically,
often, very important.
770
00:47:10,150 --> 00:47:22,940
So let me first tell
you the main point here.
771
00:47:22,940 --> 00:47:32,760
So the key idea.
772
00:47:32,760 --> 00:47:39,650
Key to finding maximum.
773
00:47:39,650 --> 00:47:41,850
So the key point
is, we only need
774
00:47:41,850 --> 00:48:00,130
to look at critical points.
775
00:48:00,130 --> 00:48:01,980
Well, that's actually
what it seems
776
00:48:01,980 --> 00:48:05,490
like in many calculus classes.
777
00:48:05,490 --> 00:48:06,810
But that's not true.
778
00:48:06,810 --> 00:48:15,590
This is not the end
of the sentence.
779
00:48:15,590 --> 00:48:35,320
And, end points, and
points of discontinuity.
780
00:48:35,320 --> 00:48:37,980
So you must watch out for those.
781
00:48:37,980 --> 00:48:41,580
If you look at the example
that I just drew here,
782
00:48:41,580 --> 00:48:43,680
which is the one
that I carried out,
783
00:48:43,680 --> 00:48:50,120
you can see that there are
actually five extreme points
784
00:48:50,120 --> 00:48:51,250
on this picture.
785
00:48:51,250 --> 00:48:52,750
So let's switch.
786
00:48:52,750 --> 00:48:58,050
So we'll take a look.
787
00:48:58,050 --> 00:49:04,840
There are five places where
the max or the min might be.
788
00:49:04,840 --> 00:49:08,050
There's this important point.
789
00:49:08,050 --> 00:49:10,710
This is, as I say, the
scaffolding of the function.
790
00:49:10,710 --> 00:49:13,650
There's this point, there
down at minus infinity.
791
00:49:13,650 --> 00:49:18,370
There's this, there's
this, and there's this.
792
00:49:18,370 --> 00:49:23,620
Only one out of five
is a critical point.
793
00:49:23,620 --> 00:49:25,784
So there's more that you
have to pay attention to
794
00:49:25,784 --> 00:49:26,450
on the function.
795
00:49:26,450 --> 00:49:28,960
And you always have
to keep the schema,
796
00:49:28,960 --> 00:49:31,310
the picture of the function,
in the back of your head.
797
00:49:31,310 --> 00:49:33,760
Even though this may be
the most interesting point,
798
00:49:33,760 --> 00:49:36,520
and the one that you're
going to be looking at.
799
00:49:36,520 --> 00:49:41,180
So we'll do a few examples
of that next time.