1
00:00:00 --> 00:00:00
2
00:00:00 --> 00:00:02
The following content is
provided under a Creative
3
00:00:02 --> 00:00:03
Commons license.
4
00:00:03 --> 00:00:06
Your support will help MIT
OpenCourseWare continue to
5
00:00:06 --> 00:00:10
offer high quality educational
resources for free.
6
00:00:10 --> 00:00:12
To make a donation or to view
additional materials from
7
00:00:12 --> 00:00:16
hundreds of MIT courses, visit
MIT OpenCourseWare
8
00:00:16 --> 00:00:22
at ocw.mit.edu.
9
00:00:22 --> 00:00:22
PROF.
10
00:00:22 --> 00:00:25
JERISON: So, we're ready to
begin Lecture 10, and what
11
00:00:25 --> 00:00:29
I'm going to begin with
is by finishing up some
12
00:00:29 --> 00:00:33
things from last time.
13
00:00:33 --> 00:00:42
We'll talk about
approximations, and I want to
14
00:00:42 --> 00:00:50
fill in a number of comments
and get you a little bit more
15
00:00:50 --> 00:00:54
oriented in the point of view
that I'm trying to express
16
00:00:54 --> 00:00:55
about approximations.
17
00:00:55 --> 00:00:59
So, first of all, I want to
remind you of the actual
18
00:00:59 --> 00:01:03
applied example that I
wrote down last time.
19
00:01:03 --> 00:01:08
So that was this business here.
20
00:01:08 --> 00:01:11
There was something from
special relativity.
21
00:01:11 --> 00:01:15
And the approximation that
we used was the linear
22
00:01:15 --> 00:01:19
approximation, with a -
1/2 power that comes
23
00:01:19 --> 00:01:21
out to be t( 1
24
00:01:21 --> 00:01:23
1/2 v^2 / C^2).
25
00:01:23 --> 00:01:27
26
00:01:27 --> 00:01:30
I want to reiterate why
this is a useful way
27
00:01:30 --> 00:01:31
of thinking of things.
28
00:01:31 --> 00:01:34
And why this is that this
comes up in real life.
29
00:01:34 --> 00:01:37
Why this is maybe more
important than everything
30
00:01:37 --> 00:01:39
that I've taught you
about technically so far.
31
00:01:39 --> 00:01:47
So, first of all, what this is
telling us is the change in t /
32
00:01:47 --> 00:01:51
t, if you do the arithmetic
here and subtract t that's
33
00:01:51 --> 00:01:55
using the change in
t is t' - t here.
34
00:01:55 --> 00:01:59
If you work that out, this
is approximately the
35
00:01:59 --> 00:02:02
same as 1/2 (v^2 / C^2).
36
00:02:04 --> 00:02:06
So what is this saying?
37
00:02:06 --> 00:02:08
This is saying that if you have
this satellite, which is going
38
00:02:08 --> 00:02:14
at speed v, and little c is the
speed of light, then the change
39
00:02:14 --> 00:02:19
in the watch down here on
earth, relative to the time on
40
00:02:19 --> 00:02:22
the satellite, is going to
be proportional to
41
00:02:22 --> 00:02:24
this ratio here.
42
00:02:24 --> 00:02:25
So, physically,
this makes sense.
43
00:02:25 --> 00:02:28
This is time divided by time.
44
00:02:28 --> 00:02:30
And this is velocity squared
divided by velocity squared.
45
00:02:30 --> 00:02:32
So, in each case, the
units divide out.
46
00:02:32 --> 00:02:34
So this is a
dimensionless quantity.
47
00:02:34 --> 00:02:36
And this is a
dimensionless quantity.
48
00:02:36 --> 00:02:41
And the only point here that
we're trying to make is just
49
00:02:41 --> 00:02:43
this notion of proportionality.
50
00:02:43 --> 00:02:45
So I want to write this down.
51
00:02:45 --> 00:02:48
Just, in summary.
52
00:02:48 --> 00:02:51
So the error fraction, if you
like, which is sort of the
53
00:02:51 --> 00:02:55
number of significant digits
that we have in our
54
00:02:55 --> 00:03:04
measurement, is proportional,
in this case, to this quantity.
55
00:03:04 --> 00:03:09
It happens to be proportional
to this quantity here.
56
00:03:09 --> 00:03:16
And the factor is,
happens to be, 1/2.
57
00:03:16 --> 00:03:20
So these proportionality
factors are what
58
00:03:20 --> 00:03:21
we're looking for.
59
00:03:21 --> 00:03:22
Their rates of change.
60
00:03:22 --> 00:03:24
Their rates of change of
something with respect
61
00:03:24 --> 00:03:25
to something else.
62
00:03:25 --> 00:03:29
Now, on your homework, you
have something rather
63
00:03:29 --> 00:03:30
similar to this.
64
00:03:30 --> 00:03:39
So in Problem, on Part 2b, Part
II, Problem 1, there's the
65
00:03:39 --> 00:03:41
speed of a pitch, right?
66
00:03:41 --> 00:03:44
And the speed of the pitch
is changing depending on
67
00:03:44 --> 00:03:45
how high the mound is.
68
00:03:45 --> 00:03:48
And the point here is that
that's approximately
69
00:03:48 --> 00:03:52
proportional to the change
in the height of the mound.
70
00:03:52 --> 00:03:55
In that problem, we had this
delta h, that was the x
71
00:03:55 --> 00:03:56
variable in that problem.
72
00:03:56 --> 00:03:59
And what you're trying to
figure out is what the constant
73
00:03:59 --> 00:04:03
of proportionality is.
74
00:04:03 --> 00:04:04
That's what you're aiming
for in this problem.
75
00:04:04 --> 00:04:08
So there's a linear
relationship, approximately,
76
00:04:08 --> 00:04:11
to all intents and purposes
this is an equality.
77
00:04:11 --> 00:04:14
Because the lower order
terms are unimportant
78
00:04:14 --> 00:04:14
for the problem.
79
00:04:14 --> 00:04:16
Just as over here, this
function is a little
80
00:04:16 --> 00:04:18
bit complicated.
81
00:04:18 --> 00:04:19
This function is a
little more simple.
82
00:04:19 --> 00:04:22
For the purposes of this
problem, they are the same.
83
00:04:22 --> 00:04:28
Because the errors are
negligible for the particular
84
00:04:28 --> 00:04:29
problem that we're working on.
85
00:04:29 --> 00:04:34
So we might as well work with
the simpler relationship.
86
00:04:34 --> 00:04:37
And similarly, over here, so
you could do this with, in
87
00:04:37 --> 00:04:40
this case with square roots,
it's not so hard here with
88
00:04:40 --> 00:04:41
reciprocals of square roots.
89
00:04:41 --> 00:04:45
It's also not terribly hard
to do it numerically.
90
00:04:45 --> 00:04:48
And the reason why we're not
doing it numerically is
91
00:04:48 --> 00:04:51
that, as I say, this is
something that happens
92
00:04:51 --> 00:04:53
all across engineering.
93
00:04:53 --> 00:04:56
People are looking for these
linear relationships between
94
00:04:56 --> 00:05:00
the change in some input and
the change in the output.
95
00:05:00 --> 00:05:03
And if you don't make these
simplifications, then when you
96
00:05:03 --> 00:05:07
get, say, a dozen of them
together, you can't figure
97
00:05:07 --> 00:05:09
out what's going on.
98
00:05:09 --> 00:05:12
In this case the design of the
satellite, it's very important.
99
00:05:12 --> 00:05:15
The speed actually
isn't just one speed.
100
00:05:15 --> 00:05:19
Because it's the relative
speed of u to the satellite.
101
00:05:19 --> 00:05:21
And you might be, it depends on
your angle of sight with the
102
00:05:21 --> 00:05:22
satellite what the speed is.
103
00:05:22 --> 00:05:24
So it varies quite a bit.
104
00:05:24 --> 00:05:26
So you really need
this rule of thumb.
105
00:05:26 --> 00:05:28
Then there are all kinds
of other considerations
106
00:05:28 --> 00:05:29
in this question.
107
00:05:29 --> 00:05:31
Like, for example, there's the
fact that we're sitting on
108
00:05:31 --> 00:05:34
Earth and so we're rotating
around on what's called
109
00:05:34 --> 00:05:36
a non-inertial frame.
110
00:05:36 --> 00:05:38
So there's the question
of that acceleration.
111
00:05:38 --> 00:05:41
There's the question that the
gravity that I experience here
112
00:05:41 --> 00:05:44
on Earth is not the same
as up at the satellite.
113
00:05:44 --> 00:05:48
And that also creates a
difference in time, as a
114
00:05:48 --> 00:05:49
result of general relativity.
115
00:05:49 --> 00:05:54
So all of these considerations
come down to formulas which
116
00:05:54 --> 00:05:56
are this complicated or
maybe a tiny bit more.
117
00:05:56 --> 00:05:58
Not really that much.
118
00:05:58 --> 00:06:00
And then people simplify them
enormously to these very
119
00:06:00 --> 00:06:02
simple-minded rules.
120
00:06:02 --> 00:06:05
And they don't keep track
of what's going on.
121
00:06:05 --> 00:06:08
So in order to design the
system, you must make these
122
00:06:08 --> 00:06:10
simplifications, otherwise
you can't even think
123
00:06:10 --> 00:06:12
about what's going on.
124
00:06:12 --> 00:06:13
This comes up in everything.
125
00:06:13 --> 00:06:17
In weather forecasting,
economic forecasting.
126
00:06:17 --> 00:06:19
Figuring out whether there's
going to be an asteroid that's
127
00:06:19 --> 00:06:22
going to hit the Earth.
128
00:06:22 --> 00:06:24
Every single one of these
things involves dozens of
129
00:06:24 --> 00:06:27
these considerations.
130
00:06:27 --> 00:06:30
OK, there was a question
that I saw, here.
131
00:06:30 --> 00:06:30
Yes.
132
00:06:30 --> 00:06:38
STUDENT: [INAUDIBLE]
133
00:06:38 --> 00:06:39
PROF.
134
00:06:39 --> 00:06:40
JERISON: Yeah.
135
00:06:40 --> 00:06:42
Basically, any problem where
you have a derivative, the rate
136
00:06:42 --> 00:06:46
of change also depends upon
what the base point is.
137
00:06:46 --> 00:06:46
That's the question.
138
00:06:46 --> 00:06:50
You're saying, doesn't this
delta v also depend, I had a
139
00:06:50 --> 00:06:51
base point in that problem.
140
00:06:51 --> 00:06:53
I happened to decide that
pitchers pitch on average
141
00:06:53 --> 00:06:55
about 90 miles an hour.
142
00:06:55 --> 00:06:58
Whereas, in fact, some pitchers
pitch at 100 miles an hour,
143
00:06:58 --> 00:07:00
some pitch at 80 miles an hour,
and of course they vary
144
00:07:00 --> 00:07:02
the speed of the pitch.
145
00:07:02 --> 00:07:03
And so, this varies
a little bit.
146
00:07:03 --> 00:07:05
In fact, that's sort of
a second order effect.
147
00:07:05 --> 00:07:08
It does change the constant
of proportionality.
148
00:07:08 --> 00:07:11
It's a rate of change at
a different base point.
149
00:07:11 --> 00:07:14
Which we're considering fixed.
150
00:07:14 --> 00:07:17
In fact, that's sort of
a second order effect.
151
00:07:17 --> 00:07:19
When you actually do the
computations, what you discover
152
00:07:19 --> 00:07:21
is that it doesn't make
that much difference.
153
00:07:21 --> 00:07:22
To the a.
154
00:07:22 --> 00:07:25
And that's something that
you get from experience.
155
00:07:25 --> 00:07:28
That it turns out, which things
matter and which things don't.
156
00:07:28 --> 00:07:31
And yet again, that's exactly
the same sort of consideration
157
00:07:31 --> 00:07:34
but at the next order of what
I'm talking about here is.
158
00:07:34 --> 00:07:36
You have to have enough
experience with numbers to know
159
00:07:36 --> 00:07:39
that if you take, if you vary
something a little bit it's not
160
00:07:39 --> 00:07:42
going to change the answer that
you're looking for very much.
161
00:07:42 --> 00:07:45
And that's exactly the
point that I'm making.
162
00:07:45 --> 00:07:51
So I can't make them all
at once, all such points.
163
00:07:51 --> 00:07:55
So that's my pitch for
understanding things from
164
00:07:55 --> 00:07:56
this point of view.
165
00:07:56 --> 00:08:02
Now, we're going to go on, now,
to quadratic approximations,
166
00:08:02 --> 00:08:13
which are a little
more complicated.
167
00:08:13 --> 00:08:15
So, we talked a little
bit about this last time
168
00:08:15 --> 00:08:16
but I didn't finish.
169
00:08:16 --> 00:08:19
So I want to finish this up.
170
00:08:19 --> 00:08:23
And the first thing that
I should say is that you
171
00:08:23 --> 00:08:35
use the when the linear
approximation is not enough.
172
00:08:35 --> 00:08:38
OK, so, that's something that
you really need to get a
173
00:08:38 --> 00:08:40
little experience with.
174
00:08:40 --> 00:08:43
In economics, I told you
they use logarithms.
175
00:08:43 --> 00:08:46
So sometimes they use
log linear functions.
176
00:08:46 --> 00:08:48
Sometimes they use log
quadratic functions when the
177
00:08:48 --> 00:08:49
log linear ones don't work.
178
00:08:49 --> 00:08:52
So most modeling in
economics is with log
179
00:08:52 --> 00:08:53
quadratic functions.
180
00:08:53 --> 00:08:55
And if you've made it any
more complicated than
181
00:08:55 --> 00:08:56
that, it's useless.
182
00:08:56 --> 00:08:57
And it's a mess.
183
00:08:57 --> 00:08:58
And people don't do it.
184
00:08:58 --> 00:09:02
So they stick with the
quadratic ones, typically.
185
00:09:02 --> 00:09:07
So the basic formula here, and
I'm going to take the base
186
00:09:07 --> 00:09:11
point to be 0, is that f ( x
) is approximately f ( 0 )
187
00:09:12 --> 00:09:14
f' ( 0 )x.
188
00:09:14 --> 00:09:16
That's the linear part.
189
00:09:16 --> 00:09:17
Plus this extra term.
190
00:09:17 --> 00:09:20
Which is f'' ( 0 ) / 2x^2.
191
00:09:21 --> 00:09:27
And this is supposed
to work for x near 0.
192
00:09:27 --> 00:09:34
So it shows in the base point
as simply as possible.
193
00:09:34 --> 00:09:38
So here's more or less where
we left off last time.
194
00:09:38 --> 00:09:42
And one thing that I said I was
going to explain, which I will
195
00:09:42 --> 00:09:48
now, is why it's
(1/2) f'' ( 0 ).
196
00:09:48 --> 00:09:51
So we need to know that.
197
00:09:51 --> 00:09:54
So let's work that out
here first of all.
198
00:09:54 --> 00:09:57
So I'm just going to
do it by example.
199
00:09:57 --> 00:10:00
So if you like, the answer
is just, well, what happens
200
00:10:00 --> 00:10:03
when you have a parabola?
201
00:10:03 --> 00:10:06
A parabola's a quadratic.
202
00:10:06 --> 00:10:09
It had better, its quadratic
approximation had
203
00:10:09 --> 00:10:10
better be itself.
204
00:10:10 --> 00:10:11
It's got to be the best one.
205
00:10:11 --> 00:10:13
So it's got to be itself.
206
00:10:13 --> 00:10:15
So this formula, if it's
going to work, has
207
00:10:15 --> 00:10:19
to work on the nose.
208
00:10:19 --> 00:10:21
For quadratic functions.
209
00:10:21 --> 00:10:23
So, let's take a look.
210
00:10:23 --> 00:10:26
If I differentiate, I get b
211
00:10:26 --> 00:10:28
2cx.
212
00:10:28 --> 00:10:32
If I differentiate a
second time, I get 2c.
213
00:10:32 --> 00:10:34
And now let's plug it in.
214
00:10:34 --> 00:10:39
Well, we can recover, what is
it that we want to recover?
215
00:10:39 --> 00:10:42
We want to recover these
numbers a, b and c using the
216
00:10:42 --> 00:10:45
derivatives evaluated at 0.
217
00:10:45 --> 00:10:49
So let's see.
218
00:10:49 --> 00:10:52
It's pretty easy,
actually. f ( 0 ) = a.
219
00:10:52 --> 00:10:53
That's on the nose.
220
00:10:53 --> 00:10:56
If you plug in x = 0
here, these terms drop
221
00:10:56 --> 00:10:58
out and you get a.
222
00:10:58 --> 00:11:02
And now, f' ( 0 ),
whoops that was wrong.
223
00:11:02 --> 00:11:05
So I wrote f' but
what I meant was f.
224
00:11:05 --> 00:11:07
So f ( 0 ) is a.
225
00:11:07 --> 00:11:13
Let's back up. f ( 0 ) is a, so
if I plug in x = 0 I get a.
226
00:11:13 --> 00:11:18
Now, f' ( 0 ), that's this next
formula here, f' ( 0 ), I
227
00:11:18 --> 00:11:21
plug in 0 here, and I get b.
228
00:11:21 --> 00:11:22
That's also good.
229
00:11:22 --> 00:11:24
And that's exactly what
the linear approximation
230
00:11:24 --> 00:11:25
is supposed to be.
231
00:11:25 --> 00:11:28
But now you notice, f'' is 2c.
232
00:11:28 --> 00:11:34
So to recover c, I
better take half of it.
233
00:11:34 --> 00:11:35
And that's it.
234
00:11:35 --> 00:11:38
That's the reason.
235
00:11:38 --> 00:11:40
There's no chance that any
other formula could work.
236
00:11:40 --> 00:11:45
And this one does.
237
00:11:45 --> 00:11:50
So that's the explanation
for the formula.
238
00:11:50 --> 00:11:54
And now I remind you that I had
a collection of basic formulas
239
00:11:54 --> 00:11:55
written on the board.
240
00:11:55 --> 00:12:00
And I want to just make sure
we know all of them again.
241
00:12:00 --> 00:12:07
So, first of all, there was
sine x is approximately x.
242
00:12:07 --> 00:12:10
Cosine x is approximately
1 - 1/2 x^2.
243
00:12:12 --> 00:12:15
And e ^ x is approximately 1
244
00:12:15 --> 00:12:17
x
245
00:12:17 --> 00:12:17
1/2 x^2.
246
00:12:19 --> 00:12:22
So those were three that
I mentioned last time.
247
00:12:22 --> 00:12:28
And, again, this is
all for x near 0.
248
00:12:28 --> 00:12:31
All for x near 0 only.
249
00:12:31 --> 00:12:35
These are wildly wrong, Far
away, but near 0 they're nice,
250
00:12:35 --> 00:12:37
good, quadratic approximations.
251
00:12:37 --> 00:12:39
Now, the other two
approximations that I want to
252
00:12:39 --> 00:12:46
mention are the logarithm and
we use the base point shifted.
253
00:12:46 --> 00:12:50
So we can put it at x near 0.
254
00:12:50 --> 00:12:52
And this one - sorry,
this is an approximately
255
00:12:52 --> 00:12:55
equals sign there.
256
00:12:55 --> 00:12:57
Turns out to be x - 1/2 x^2.
257
00:12:59 --> 00:13:02
And the last one is one (1
258
00:13:02 --> 00:13:07
x) ^ r, which turns
out to be 1
259
00:13:07 --> 00:13:08
rx
260
00:13:09 --> 00:13:12
r ( r - 1) / 2x^2.
261
00:13:14 --> 00:13:20
Now, eventually, your mind will
converge on all of these and
262
00:13:20 --> 00:13:23
you'll find them relatively
easy to memorize.
263
00:13:23 --> 00:13:25
But it'll take some
getting used to.
264
00:13:25 --> 00:13:30
And I'm not claiming that you
should recognize them and
265
00:13:30 --> 00:13:32
understand them all now.
266
00:13:32 --> 00:13:35
But I'm going to put a
giant box around this.
267
00:13:35 --> 00:13:42
STUDENT: [INAUDIBLE]
268
00:13:42 --> 00:13:42
PROF.
269
00:13:42 --> 00:13:42
JERISON: Yes.
270
00:13:42 --> 00:13:45
So the question was, you get
all of these if you use
271
00:13:45 --> 00:13:45
that equation there.
272
00:13:45 --> 00:13:48
That's exactly what are you
going to do. so I already
273
00:13:48 --> 00:13:52
did it actually for
these three, last time.
274
00:13:52 --> 00:13:55
But I didn't do it
yet for these two.
275
00:13:55 --> 00:13:58
But I will do it in
about two minutes.
276
00:13:58 --> 00:14:00
Well, maybe five minutes.
277
00:14:00 --> 00:14:10
But first I want to explain
just a few things about these.
278
00:14:10 --> 00:14:12
They all follow from
the basic formula.
279
00:14:12 --> 00:14:16
In fact, that one deserves a
pink box too, doesn't it.
280
00:14:16 --> 00:14:18
That one's pretty important.
281
00:14:18 --> 00:14:19
Alright.
282
00:14:19 --> 00:14:23
Yeah.
283
00:14:23 --> 00:14:25
Maybe even some
little sparkles.
284
00:14:25 --> 00:14:32
Alright.
285
00:14:32 --> 00:14:33
OK.
286
00:14:33 --> 00:14:36
So that's pretty important.
287
00:14:36 --> 00:14:39
Almost as important as the
more basic one without
288
00:14:39 --> 00:14:41
this term here.
289
00:14:41 --> 00:14:47
So now, let me just tell you
a little bit more about
290
00:14:47 --> 00:14:54
the significance.
291
00:14:54 --> 00:14:56
Again, this is just to
reinforce something that
292
00:14:56 --> 00:14:57
we've already done.
293
00:14:57 --> 00:14:59
But it's closely related
to what you're doing
294
00:14:59 --> 00:15:00
on your problem set.
295
00:15:00 --> 00:15:05
So it's worth your
while to recall this.
296
00:15:05 --> 00:15:10
So, there's this expression
that we were dealing with.
297
00:15:10 --> 00:15:13
And we talked about
it in lecture.
298
00:15:13 --> 00:15:19
And we showed that this tends
to e as k goes to infinity.
299
00:15:19 --> 00:15:21
So that's what we
showed in lecture.
300
00:15:21 --> 00:15:25
And the way that we did that
was, we took the logarithm
301
00:15:25 --> 00:15:29
and we wrote it as k
times, sorry, the ln of 1
302
00:15:29 --> 00:15:32
(1 / k).
303
00:15:32 --> 00:15:35
And then we evaluated
the limit of this.
304
00:15:35 --> 00:15:38
And I want to do this
limit again, using
305
00:15:38 --> 00:15:40
linear approximation.
306
00:15:40 --> 00:15:41
To show you how easy it is.
307
00:15:41 --> 00:15:44
If you just remember the
linear approximation.
308
00:15:44 --> 00:15:46
And then we'll explain
where the quadratic
309
00:15:46 --> 00:15:48
approximation comes in.
310
00:15:48 --> 00:15:52
So I claim that this is
approximately equal
311
00:15:52 --> 00:15:59
to k ( 1 / k).
312
00:15:59 --> 00:16:00
Now, why is that?
313
00:16:00 --> 00:16:04
Well, that's just this
linear approximation.
314
00:16:04 --> 00:16:05
So what did I use here?
315
00:16:05 --> 00:16:07
I used ln of 1
316
00:16:07 --> 00:16:10
x is approximately x.
317
00:16:10 --> 00:16:14
For x = 1 / k.
318
00:16:14 --> 00:16:18
That's what I used in
this approximation here.
319
00:16:18 --> 00:16:20
And that's the linear
approximation to the
320
00:16:20 --> 00:16:24
natural logarithm.
321
00:16:24 --> 00:16:27
And this number is relatively
easy to evaluate.
322
00:16:27 --> 00:16:28
I know how to do it.
323
00:16:28 --> 00:16:31
It's equal to 1.
324
00:16:31 --> 00:16:34
That's the same, well, so
where does this work?
325
00:16:34 --> 00:16:37
This works where this
thing is near 0.
326
00:16:37 --> 00:16:41
Which is when k is
going to infinity.
327
00:16:41 --> 00:16:44
This thing is working only
when k is going to infinity.
328
00:16:44 --> 00:16:46
So what it's really saying,
this approximation formula,
329
00:16:46 --> 00:16:50
it's really saying that as
we go to infinity, in k,
330
00:16:50 --> 00:16:54
this thing is going to 1.
331
00:16:54 --> 00:16:58
As k goes to infinity.
332
00:16:58 --> 00:17:00
So that's what it's saying.
333
00:17:00 --> 00:17:01
That's the substance there.
334
00:17:01 --> 00:17:05
And that's how we want to
use it, in many instances.
335
00:17:05 --> 00:17:06
Just to evaluate limits.
336
00:17:06 --> 00:17:09
We also want to realize
that it's nearby when k
337
00:17:09 --> 00:17:12
is pretty large, like 100
or something like that.
338
00:17:12 --> 00:17:16
Now, so that's the idea of
the linear approximation.
339
00:17:16 --> 00:17:21
Now, if you want to get the
rate of convergence here,
340
00:17:21 --> 00:17:27
so the rate of what's
called convergence.
341
00:17:27 --> 00:17:34
So convergence means how fast
this is going towards that.
342
00:17:34 --> 00:17:36
What I have to do is
take the difference.
343
00:17:36 --> 00:17:40
I have to take ln ak, and I
have to subtract 1 from it.
344
00:17:40 --> 00:17:43
And I know that this is going
to 0, and the question
345
00:17:43 --> 00:17:47
is how big is this.
346
00:17:47 --> 00:17:51
We want it to be very small.
347
00:17:51 --> 00:17:56
And the answer we're going to
get, so the answer just uses
348
00:17:56 --> 00:18:03
the quadratic approximation.
349
00:18:03 --> 00:18:06
So if I just have a little
bit more detail, then this
350
00:18:06 --> 00:18:10
expression here, in other
words, I have the next
351
00:18:10 --> 00:18:11
higher order term.
352
00:18:11 --> 00:18:13
This is like 1 / k,
this is like 1 / k^2.
353
00:18:15 --> 00:18:21
Then I can understand how big
the difference is between the
354
00:18:21 --> 00:18:24
expression that I've
got and its limit.
355
00:18:24 --> 00:18:26
And so that's what's
on your homework.
356
00:18:26 --> 00:18:31
This is on your problem set.
357
00:18:31 --> 00:18:35
OK, so that is more or less an
explanation for one of the
358
00:18:35 --> 00:18:38
things that quadratic
approximations are good for.
359
00:18:38 --> 00:18:42
And I'm going to give you
one more illustration.
360
00:18:42 --> 00:18:45
One more illustration.
361
00:18:45 --> 00:18:47
And then we'll actually
check these formulas.
362
00:18:47 --> 00:18:48
Yeah, another question.
363
00:18:48 --> 00:18:55
STUDENT: [INAUDIBLE]
364
00:18:55 --> 00:18:56
PROF.
365
00:18:56 --> 00:18:58
JERISON: That's a very
good question here.
366
00:18:58 --> 00:19:02
When they, which in this case
means maybe, me, when I give
367
00:19:02 --> 00:19:10
you a question, does one
specify whether you want
368
00:19:10 --> 00:19:14
to use a linear or a
quadratic approximation.
369
00:19:14 --> 00:19:18
The answer is, in real life
when you're faced with a
370
00:19:18 --> 00:19:23
problem like this, where some
satellite is orbiting and you
371
00:19:23 --> 00:19:25
want to know the effects of
gravity or something like
372
00:19:25 --> 00:19:28
that, nobody is going
to tell you anything.
373
00:19:28 --> 00:19:30
They're not even going to
tell you whether a linear
374
00:19:30 --> 00:19:33
approximation is relevant,
or a quadratic or anything.
375
00:19:33 --> 00:19:36
So you're on your own.
376
00:19:36 --> 00:19:40
When I give you a question,
at least for right now, I'm
377
00:19:40 --> 00:19:42
always going to tell you.
378
00:19:42 --> 00:19:47
But as time goes on I'd like
you to get used to when it's
379
00:19:47 --> 00:19:49
enough to get away with
a linear approximation.
380
00:19:49 --> 00:19:53
And you should only use a
quadratic approximation if
381
00:19:53 --> 00:19:55
somebody forces you to.
382
00:19:55 --> 00:19:57
You should always start
trying with a linear one.
383
00:19:57 --> 00:20:00
Because the quadratic ones are
much more complicated as you'll
384
00:20:00 --> 00:20:03
see in this next example.
385
00:20:03 --> 00:20:06
OK, so the example that I want
to use is, you're going to be
386
00:20:06 --> 00:20:09
stuck with it because I'm
asking for the quadratic.
387
00:20:09 --> 00:20:16
So we're going to find the
quadratic approximation
388
00:20:16 --> 00:20:23
near, for x near 0.
389
00:20:23 --> 00:20:24
To what?
390
00:20:24 --> 00:20:29
Well, this is the same
function that we used
391
00:20:29 --> 00:20:30
in the last lecture.
392
00:20:30 --> 00:20:34
I think this was
it. e ^ - 3x (1
393
00:20:34 --> 00:20:37
x) ^ - 1/2.
394
00:20:37 --> 00:20:40
OK.
395
00:20:40 --> 00:20:45
So, unfortunately, I stuck
it in the wrong place
396
00:20:45 --> 00:20:47
to be able to fit this
very long formula here.
397
00:20:47 --> 00:20:51
So I'm going to switch it.
398
00:20:51 --> 00:20:57
I'm just going to
write it here.
399
00:20:57 --> 00:21:00
And we're going to just
do the approximation.
400
00:21:00 --> 00:21:03
So we're going to say
quadratic, in parentheses.
401
00:21:03 --> 00:21:08
And we'll say x near 0.
402
00:21:08 --> 00:21:12
So that's what I want.
403
00:21:12 --> 00:21:15
So now, here's what
I have to do.
404
00:21:15 --> 00:21:18
Well, I have to write in the
quadratic approximation for e
405
00:21:18 --> 00:21:25
^ - 3x, and I'm going to use
this formula right here.
406
00:21:25 --> 00:21:27
And so that's (1
407
00:21:27 --> 00:21:29
(- 3x)
408
00:21:30 --> 00:21:33
(- 3x)^2 / 2).
409
00:21:33 --> 00:21:39
And the other factor, I'm going
to have to use this formula
410
00:21:39 --> 00:21:43
down here. because r is - 1/2.
411
00:21:43 --> 00:21:48
And so that's (1 - 1/2 x
412
00:21:48 --> 00:21:53
1/2 ( - 1/2)( - 3/2)x^2).
413
00:21:59 --> 00:22:09
So this is the r term, and
this is the r - 1 term.
414
00:22:09 --> 00:22:12
And now I'm going to do
something which is the
415
00:22:12 --> 00:22:15
only good thing about
quadratic approximations.
416
00:22:15 --> 00:22:18
They're messy, they're long,
there's nothing particularly
417
00:22:18 --> 00:22:19
good about them.
418
00:22:19 --> 00:22:21
But there is one good
thing about them.
419
00:22:21 --> 00:22:25
Which is that you always get to
ignore the higher order terms.
420
00:22:25 --> 00:22:29
So even though this looks like
a very ugly multiplication,
421
00:22:29 --> 00:22:31
I can do it in my head.
422
00:22:31 --> 00:22:33
Just watching it.
423
00:22:33 --> 00:22:38
Because I get a 1 * 1, I'm
forced with that term here.
424
00:22:38 --> 00:22:41
And then I get the cross
terms which are linear,
425
00:22:41 --> 00:22:44
which is - 3x - 1/2 x.
426
00:22:44 --> 00:22:46
We already did that when
we calculated the linear
427
00:22:46 --> 00:22:50
approximation, so that's
this times the 1 and
428
00:22:50 --> 00:22:51
this times that 1.
429
00:22:51 --> 00:22:54
And now I have three
cross-terms which
430
00:22:54 --> 00:22:55
are quadratic.
431
00:22:55 --> 00:22:58
So one of them is these two
linear terms are multiplying.
432
00:22:58 --> 00:23:00
So that's plus 3/2 x^2.
433
00:23:02 --> 00:23:05
That's (- 3)( - 1/2).
434
00:23:05 --> 00:23:07
And then there's this
term, multiplying the
435
00:23:07 --> 00:23:10
1, that's plus 9/2 x^2.
436
00:23:11 --> 00:23:15
And then there's one last term,
which is this monster here.
437
00:23:15 --> 00:23:23
Multiplying 1, and
that is - 3/8.
438
00:23:23 --> 00:23:31
So the great thing is, we drop
x^3, x ^ 4, etc., terms.
439
00:23:31 --> 00:23:37
Yeah?
440
00:23:37 --> 00:23:39
STUDENT: [INAUDIBLE]
441
00:23:39 --> 00:23:40
PROF.
442
00:23:40 --> 00:23:42
JERISON: OK, well
so copy it down.
443
00:23:42 --> 00:23:45
And you work it out
as I'm doing it now.
444
00:23:45 --> 00:23:47
So what I did is, I
multiplied 1 by 1.
445
00:23:47 --> 00:23:50
I'm using the
distributive law here.
446
00:23:50 --> 00:23:51
That was this one.
447
00:23:51 --> 00:23:55
I multiplied this 3x by this
one, that was that term.
448
00:23:55 --> 00:23:58
I multiplied this by
this, that's that term.
449
00:23:58 --> 00:24:01
And then I multiplied
this by this.
450
00:24:01 --> 00:24:03
In other words, 2 x terms
that gave me an x^2
451
00:24:03 --> 00:24:06
and a (- 3)( - 1/2).
452
00:24:06 --> 00:24:08
And I'm going to
stop at that point.
453
00:24:08 --> 00:24:10
Because the point is it's
just all the rest of
454
00:24:10 --> 00:24:12
the terms that come up.
455
00:24:12 --> 00:24:14
Now, the reason, the only
reason why it's easy is
456
00:24:14 --> 00:24:16
that I only have to go
up to x squared term.
457
00:24:16 --> 00:24:21
I don't have to do
the higher ones.
458
00:24:21 --> 00:24:22
Another question,
way back here.
459
00:24:22 --> 00:24:23
Yeah, right there.
460
00:24:23 --> 00:24:29
STUDENT: [INAUDIBLE]
461
00:24:29 --> 00:24:29
PROF.
462
00:24:29 --> 00:24:29
JERISON: OK.
463
00:24:29 --> 00:24:38
So somebody can check
my arithmetic, too.
464
00:24:38 --> 00:24:39
Good.
465
00:24:39 --> 00:24:40
STUDENT: [INAUDIBLE]
466
00:24:40 --> 00:24:40
PROF.
467
00:24:40 --> 00:24:43
JERISON: Why do I get to drop
all the higher-order terms.
468
00:24:43 --> 00:24:46
So, that's because the
situation where I'm going to
469
00:24:46 --> 00:24:52
apply this is the situation
in which x is, say, 1/100.
470
00:24:52 --> 00:24:55
So if here's about 1/100.
471
00:24:55 --> 00:24:57
Here's something which
is on the order of 100.
472
00:24:57 --> 00:24:58
This is on the
order of 1/100^2.
473
00:25:00 --> 00:25:02
1/100^2, all of these terms.
474
00:25:02 --> 00:25:06
Now, these cubic and
quartic terms are of
475
00:25:06 --> 00:25:09
the order of 1/100^3.
476
00:25:10 --> 00:25:12
That's 10 ^ - 6.
477
00:25:12 --> 00:25:14
And the point is that
I'm not claiming that I
478
00:25:14 --> 00:25:16
have an exact answer.
479
00:25:16 --> 00:25:19
And I'm going to drop things
of that order of magnitude.
480
00:25:19 --> 00:25:22
So I'm saving everything
up to 4 decimal places.
481
00:25:22 --> 00:25:29
I'm throwing away things which
are 6 decimal places out.
482
00:25:29 --> 00:25:30
Does that answer your question?
483
00:25:30 --> 00:25:35
STUDENT: [INAUDIBLE]
484
00:25:35 --> 00:25:35
PROF.
485
00:25:35 --> 00:25:36
JERISON: So.
486
00:25:36 --> 00:25:39
That's the situation, and now
you can combine the terms.
487
00:25:39 --> 00:25:43
I mean, it's not very
impressive here.
488
00:25:43 --> 00:25:48
This is equal to 1
- 7/2 x, maybe
489
00:25:50 --> 00:25:53
51/8 x^2.
490
00:25:54 --> 00:25:57
If I've made that, if those
minus signs hadn't canceled,
491
00:25:57 --> 00:25:59
I would have gotten
the wrong answer here.
492
00:25:59 --> 00:26:00
Anyway.
493
00:26:00 --> 00:26:03
So, this is a 2 here, sorry.
494
00:26:03 --> 00:26:04
7/2.
495
00:26:04 --> 00:26:07
This is the linear
approximation we got last
496
00:26:07 --> 00:26:09
time and here's the extra
information that we got
497
00:26:09 --> 00:26:11
from this calculation.
498
00:26:11 --> 00:26:17
Which is this 51/8 term.
499
00:26:17 --> 00:26:20
Right, you have to accept that
there's a certain degree of
500
00:26:20 --> 00:26:23
complexity to this problem and
the answer is sufficiently
501
00:26:23 --> 00:26:26
complicated so it can't be less
arithmetic because we get this
502
00:26:26 --> 00:26:29
peculiar 51/8 there, right.
503
00:26:29 --> 00:26:33
So one of the things to realize
is that these kinds of
504
00:26:33 --> 00:26:37
problems, because they involve
many, many terms are always
505
00:26:37 --> 00:26:43
going to involve a little bit
of complicated arithmetic.
506
00:26:43 --> 00:26:48
Last little bit, I did promise
you that I was going to derive
507
00:26:48 --> 00:26:51
these two relations, as I said.
508
00:26:51 --> 00:26:53
Did the ones in
the left column.
509
00:26:53 --> 00:26:56
So let's carry that out.
510
00:26:56 --> 00:27:00
And as someone just pointed
out, it all comes from
511
00:27:00 --> 00:27:01
this formula here.
512
00:27:01 --> 00:27:07
So let's just check it.
513
00:27:07 --> 00:27:12
So we'll start with
the ln function.
514
00:27:12 --> 00:27:16
This is the function,
f, and then f' is 1/1
515
00:27:17 --> 00:27:18
x.
516
00:27:18 --> 00:27:24
And f'', so this is f',
this is f'', is - 1/1
517
00:27:24 --> 00:27:25
x^2.
518
00:27:25 --> 00:27:28
519
00:27:28 --> 00:27:31
And now I have to
plug in x = 0.
520
00:27:31 --> 00:27:35
So at x = 0 this is
ln 1, which is 0.
521
00:27:35 --> 00:27:37
So this is at x = 0.
522
00:27:37 --> 00:27:41
I'm getting 0 here, I
plug in 0 and I get 1.
523
00:27:41 --> 00:27:45
And here, I plug in
0 and I get - 1.
524
00:27:45 --> 00:27:49
So now I go and I look up at
that formula, which is way
525
00:27:49 --> 00:27:50
in that upper corner there.
526
00:27:50 --> 00:27:53
And I see that the coefficient
on the constant is 0.
527
00:27:53 --> 00:27:55
The coefficient on x is 1.
528
00:27:55 --> 00:27:59
And then the other coefficient,
the very last one, is - 1/2.
529
00:27:59 --> 00:28:01
So this is the - 1 here.
530
00:28:01 --> 00:28:05
And then in the formula,
there's a 2 in the denominator.
531
00:28:05 --> 00:28:07
So it's half of whatever
I get for this second
532
00:28:07 --> 00:28:10
derivative, at 0.
533
00:28:10 --> 00:28:13
So this is the approximation
formula, which is way up
534
00:28:13 --> 00:28:17
in that corner there.
535
00:28:17 --> 00:28:19
Similarly, if I do it for (1
536
00:28:19 --> 00:28:23
x) ^ r, I have to differentiate
that I get r( 1
537
00:28:23 --> 00:28:28
x) ^ r - 1, and
then r ( r - 1( x
538
00:28:29 --> 00:28:31
1) ^ r - 2.
539
00:28:31 --> 00:28:33
So here are the derivatives.
540
00:28:33 --> 00:28:41
And so if I evaluate
them at x = 0, I get 1.
541
00:28:41 --> 00:28:43
That's 1 ^ r = 1.
542
00:28:43 --> 00:28:47
And here I get r. (1 ^ r - 1)r.
543
00:28:49 --> 00:28:59
And here, I plug in x =
0 and I get r ( r - 1).
544
00:28:59 --> 00:29:02
So again, the pattern is
right above it here.
545
00:29:02 --> 00:29:05
The 1 is there, the r is there.
546
00:29:05 --> 00:29:08
And then instead of r ( r
- 1), I have half that.
547
00:29:08 --> 00:29:21
For the coefficient.
548
00:29:21 --> 00:29:22
So these are just examples.
549
00:29:22 --> 00:29:24
Obviously if we had a more
complicated functional,
550
00:29:24 --> 00:29:26
we might carry this out.
551
00:29:26 --> 00:29:29
But as a practical matter, we
try to stick with the ones in
552
00:29:29 --> 00:29:42
the pink box and just use
algebra to get other formulas.
553
00:29:42 --> 00:29:46
So I want to shift gears now
and treat the subject that was
554
00:29:46 --> 00:29:48
supposed to be this lecture.
555
00:29:48 --> 00:29:53
And we're not quite caught up,
but we will try to do our best
556
00:29:53 --> 00:29:54
to do as much as we can today.
557
00:29:54 --> 00:29:59
So the next topic is
curve sketching.
558
00:29:59 --> 00:30:18
And so let's get
started with that.
559
00:30:18 --> 00:30:23
So now, happily in this
subject, there are more
560
00:30:23 --> 00:30:26
pictures and it's a little
bit more geometric.
561
00:30:26 --> 00:30:30
And there's relatively
little computation.
562
00:30:30 --> 00:30:33
So let's hope we can do this.
563
00:30:33 --> 00:30:37
So I want to, so here we
go, we'll start with
564
00:30:37 --> 00:30:44
curve sketching.
565
00:30:44 --> 00:30:47
And the goal here
566
00:30:47 --> 00:30:56
STUDENT: [INAUDIBLE]
567
00:30:56 --> 00:30:56
PROF.
568
00:30:56 --> 00:31:00
JERISON: So that's like,
liner, the last time.
569
00:31:00 --> 00:31:09
That's kind of sketchy
spelling, isn't it?
570
00:31:09 --> 00:31:12
Yeah, there are certain kinds
of things which I can't spell.
571
00:31:12 --> 00:31:16
But, alright.
572
00:31:16 --> 00:31:18
Sketching.
573
00:31:18 --> 00:31:19
Alright.
574
00:31:19 --> 00:31:21
So here's our goal.
575
00:31:21 --> 00:31:38
Our goal is to draw the graph
of f, using f' and f''.
576
00:31:38 --> 00:31:42
577
00:31:42 --> 00:31:47
Whether they're
positive or negative.
578
00:31:47 --> 00:31:48
So that's it.
579
00:31:48 --> 00:31:52
This is the goal here.
580
00:31:52 --> 00:31:57
However, there is a big warning
that I want to give you.
581
00:31:57 --> 00:32:06
And this is one that
unfortunately I now have to
582
00:32:06 --> 00:32:08
make you unlearn, especially
those that you that have
583
00:32:08 --> 00:32:11
actually had a little bit of
calculus before, I want to make
584
00:32:11 --> 00:32:13
you unlearn some of your
instincts that you developed.
585
00:32:13 --> 00:32:15
So this will be harder for
those of you who have
586
00:32:15 --> 00:32:19
actually done this before.
587
00:32:19 --> 00:32:22
But for the rest of you, it
will be relatively easy.
588
00:32:22 --> 00:32:35
Which is, don't abandon
your precalculus skills.
589
00:32:35 --> 00:32:42
And common sense.
590
00:32:42 --> 00:32:46
So there's a great deal
of common sense in this.
591
00:32:46 --> 00:32:50
And it actually trumps
some of the calculus.
592
00:32:50 --> 00:32:56
The calculus just fills in what
you didn't quite know yet.
593
00:32:56 --> 00:32:59
So I will try to
illustrate this.
594
00:32:59 --> 00:33:02
And because we're running a bit
late, I won't get to the some
595
00:33:02 --> 00:33:05
of the main punchlines
until next lecture.
596
00:33:05 --> 00:33:07
But I want you to do it.
597
00:33:07 --> 00:33:09
So for now, I'm just going
to tell you about the
598
00:33:09 --> 00:33:10
general principles.
599
00:33:10 --> 00:33:14
And in the process I'm going
to introduce the terminology.
600
00:33:14 --> 00:33:17
Just, the words that we need
to use to describe what
601
00:33:17 --> 00:33:18
is that we're doing.
602
00:33:18 --> 00:33:19
And there's also a certain
amount of carelessness
603
00:33:19 --> 00:33:23
with that in many of the
treatments that you'll see.
604
00:33:23 --> 00:33:24
And a lot of hastiness.
605
00:33:24 --> 00:33:29
So just be a little patient
and we will do this.
606
00:33:29 --> 00:33:33
So, the first principle
is the following.
607
00:33:33 --> 00:33:40
If f' is positive,
then f is increasing.
608
00:33:40 --> 00:33:44
That's a straightforward idea,
and it's closely related to
609
00:33:44 --> 00:33:48
this tangent line approximation
or the linear approximation
610
00:33:48 --> 00:33:49
that I just did.
611
00:33:49 --> 00:33:50
You can just imagine.
612
00:33:50 --> 00:33:53
Here's the tangent line,
here's the function.
613
00:33:53 --> 00:33:56
And if the tangent line is
pointing up, then the function
614
00:33:56 --> 00:33:58
is also going up, too.
615
00:33:58 --> 00:34:00
So that's all that's
going on here.
616
00:34:00 --> 00:34:10
Similarly, if f' is negative,
then f is decreasing.
617
00:34:10 --> 00:34:12
And that's the basic idea.
618
00:34:12 --> 00:34:17
Now, the second step is also
fairly straightforward.
619
00:34:17 --> 00:34:21
It's just a second order
effect of the same type.
620
00:34:21 --> 00:34:27
If you have f'' as positive,
then that means that
621
00:34:27 --> 00:34:33
f' is increasing.
622
00:34:33 --> 00:34:36
That's the same principle
applied one step up.
623
00:34:36 --> 00:34:37
Right?
624
00:34:37 --> 00:34:41
Because if f'' is positive,
that means it's the
625
00:34:41 --> 00:34:42
derivative of f'.
626
00:34:42 --> 00:34:45
So it's the same
principle just repeated.
627
00:34:45 --> 00:34:49
And now I just want to
draw a picture of this.
628
00:34:49 --> 00:34:52
Here's a picture
of it, I claim.
629
00:34:52 --> 00:34:54
And it looks like
something's going down.
630
00:34:54 --> 00:34:56
And I did that on purpose.
631
00:34:56 --> 00:34:58
But there is something
that's increasing here.
632
00:34:58 --> 00:35:02
Which is, the slope is
very steep negative here.
633
00:35:02 --> 00:35:06
And it's less steep
negative over here.
634
00:35:06 --> 00:35:10
So we have the slope which is
some negative number, say, - 4.
635
00:35:10 --> 00:35:13
And here it's - 3.
636
00:35:13 --> 00:35:14
So it's increasing.
637
00:35:14 --> 00:35:18
It's getting less negative,
and maybe eventually it'll
638
00:35:18 --> 00:35:19
curve up the other way.
639
00:35:19 --> 00:35:23
And this is a picture of what
I'm talking about here.
640
00:35:23 --> 00:35:25
That's what it means to say
that f' is increasing.
641
00:35:25 --> 00:35:28
The slope is getting larger.
642
00:35:28 --> 00:35:34
And the way to describe a curve
like this is that it's concave.
643
00:35:34 --> 00:35:41
So f is concave up.
644
00:35:41 --> 00:35:49
And similarly, f'' negative is
going to be the same thing as f
645
00:35:49 --> 00:35:59
concave, or implies
f concave down.
646
00:35:59 --> 00:36:04
So those are the ways in which
derivatives will help us
647
00:36:04 --> 00:36:08
qualitatively to draw graphs.
648
00:36:08 --> 00:36:10
But as I said before, we still
have to use a little bit of
649
00:36:10 --> 00:36:13
common sense when we
draw the graphs.
650
00:36:13 --> 00:36:15
These are just the additional
bits of help that we
651
00:36:15 --> 00:36:17
have from calculus.
652
00:36:17 --> 00:36:25
In drawing pictures.
653
00:36:25 --> 00:36:34
So I'm going to go through
one example to introduce
654
00:36:34 --> 00:36:36
all the notations.
655
00:36:36 --> 00:36:40
And then eventually, so
probably at the beginning of
656
00:36:40 --> 00:36:45
next time, I'll give you a
systematic strategy that's
657
00:36:45 --> 00:36:49
going to work when what I'm
describing now goes wrong,
658
00:36:49 --> 00:36:52
or a little bit wrong.
659
00:36:52 --> 00:36:58
So let's begin with a
straightforward example.
660
00:36:58 --> 00:37:00
So, the first example that
I'll give you is the
661
00:37:00 --> 00:37:04
function f (x) = 3x - x^3.
662
00:37:04 --> 00:37:07
663
00:37:07 --> 00:37:11
Just, as I said, to be able to
introduce all the notations.
664
00:37:11 --> 00:37:16
Now, if you differentiate
it, you get 3 - 3x^2.
665
00:37:17 --> 00:37:20
And I can factor that.
666
00:37:20 --> 00:37:23
This is 3 ( 1 - x)( 1
667
00:37:24 --> 00:37:26
x).
668
00:37:26 --> 00:37:27
OK?
669
00:37:27 --> 00:37:33
And so, I can decide
whether the derivative
670
00:37:33 --> 00:37:37
is positive or negative.
671
00:37:37 --> 00:37:38
Easily enough.
672
00:37:38 --> 00:37:50
Namely, just staring at this, I
can see that when -1 < x < 1,
673
00:37:50 --> 00:37:53
in that range there, both
these numbers, both these
674
00:37:53 --> 00:37:55
factors, are positive.
675
00:37:55 --> 00:37:57
1 - x is a positive
number and 1
676
00:37:58 --> 00:37:59
x is a positive number.
677
00:37:59 --> 00:38:04
So, in this range, f'
( x ) is positive.
678
00:38:04 --> 00:38:09
So this thing is, so
f is increasing.
679
00:38:09 --> 00:38:14
And similarly, in the other
ranges, if x is very, very
680
00:38:14 --> 00:38:17
large, this becomes, if it
crosses 1, in fact, this
681
00:38:17 --> 00:38:19
becomes, this factor
becomes negative and this
682
00:38:19 --> 00:38:21
one stays positive.
683
00:38:21 --> 00:38:29
So when x > 1, we have
that f ' (x) is negative.
684
00:38:29 --> 00:38:35
And so f is decreasing.
685
00:38:35 --> 00:38:39
And the same thing goes
for the other side.
686
00:38:39 --> 00:38:42
When it's less than - 1,
that also works this way.
687
00:38:42 --> 00:38:45
Because when it's less
than - 1, this number
688
00:38:45 --> 00:38:46
factors positive.
689
00:38:46 --> 00:38:50
But the other one is negative.
690
00:38:50 --> 00:38:56
So in both of these cases, we
get that it's decreasing.
691
00:38:56 --> 00:39:07
So now, here's the schematic
picture of this function.
692
00:39:07 --> 00:39:12
So here's - 1, here's 1.
693
00:39:12 --> 00:39:19
It's going to go
down, up, down.
694
00:39:19 --> 00:39:21
That's what it's doing.
695
00:39:21 --> 00:39:23
Maybe I'll just leave
it alone like this.
696
00:39:23 --> 00:39:28
That's what it looks like.
697
00:39:28 --> 00:39:31
So, this is the kind of
information we can get
698
00:39:31 --> 00:39:32
right off the bat.
699
00:39:32 --> 00:39:38
And you notice immediately that
it's very important, from the
700
00:39:38 --> 00:39:40
features of the function, the
sort of key features of the
701
00:39:40 --> 00:39:44
function that we see here,
are these two places.
702
00:39:44 --> 00:39:49
Maybe I'll even mark
them in a, like this.
703
00:39:49 --> 00:39:59
And these things are
turning points.
704
00:39:59 --> 00:40:00
So what are they?
705
00:40:00 --> 00:40:03
Well, they're just
the points where the
706
00:40:03 --> 00:40:05
derivative changes sign.
707
00:40:05 --> 00:40:07
Where it's negative here
and it's positive there,
708
00:40:07 --> 00:40:09
so there it must be 0.
709
00:40:09 --> 00:40:12
So we have a definition, and
this is the most important
710
00:40:12 --> 00:40:21
definition in this subject,
which is that is if f' (x0) =
711
00:40:21 --> 00:40:33
0, we call x0 a critical point.
712
00:40:33 --> 00:40:36
The word 'turning point' is not
used just because, in fact,
713
00:40:36 --> 00:40:38
it doesn't have to turn
around at those points.
714
00:40:38 --> 00:40:42
But certainly, if it turns
around then this will happen.
715
00:40:42 --> 00:40:46
And we also have another
notation, which is the number
716
00:40:46 --> 00:40:59
y0 which is f ( x0 ) is
called a critical value.
717
00:40:59 --> 00:41:02
So these are the key numbers
that we're going to have to
718
00:41:02 --> 00:41:18
work out in order to understand
what the function looks like.
719
00:41:18 --> 00:41:27
So what I'm going to
do is just plot them.
720
00:41:27 --> 00:41:30
We're going to plot the
critical points and the values.
721
00:41:30 --> 00:41:34
Well, we found the critical
points relatively easily.
722
00:41:34 --> 00:41:37
I didn't write it down here
but it's pretty obvious.
723
00:41:37 --> 00:41:43
If you set f(x) = 0, that
implies that (1 - x)( 1
724
00:41:44 --> 00:41:47
x) = 0, which
implies that x is
725
00:41:47 --> 00:41:50
or - 1.
726
00:41:50 --> 00:41:52
So those are known as
the critical points.
727
00:41:52 --> 00:41:56
And now, in order to get the
critical values here, I have to
728
00:41:56 --> 00:42:02
plug in f (1), for instance,
the function is 3x - x^2, so
729
00:42:02 --> 00:42:07
there's this 3 * 1
- 1^3, which is 2.
730
00:42:07 --> 00:42:17
And f ( - 1), which is 3 ( -
1) - (- 1)^3, which is - 2.
731
00:42:17 --> 00:42:21
And so I can plot
the function here.
732
00:42:21 --> 00:42:26
So here's the point - 1 and
here's, up here, is 2.
733
00:42:26 --> 00:42:28
So this is - whoops,
which one is it?
734
00:42:28 --> 00:42:29
Yeah.
735
00:42:29 --> 00:42:32
This is - 1, so it's down here.
736
00:42:32 --> 00:42:35
So it's (- 1, - 2).
737
00:42:35 --> 00:42:41
And then over here, I
have the point (1, 2).
738
00:42:41 --> 00:42:47
Alright, now, what information
do I get from - so I've
739
00:42:47 --> 00:42:50
now plotted two, I claim,
very interesting points.
740
00:42:50 --> 00:42:55
What information do
I get from this?
741
00:42:55 --> 00:42:59
The answer is, I know
something very nearby.
742
00:42:59 --> 00:43:02
Because I've already checked
that the thing is coming down
743
00:43:02 --> 00:43:04
from the left, and
coming back up.
744
00:43:04 --> 00:43:07
And so it must be
shaped like this.
745
00:43:07 --> 00:43:08
Over here.
746
00:43:08 --> 00:43:11
The tangent line is 0, it's
going to be level there.
747
00:43:11 --> 00:43:14
And similarly over here,
it's going to do that.
748
00:43:14 --> 00:43:22
So this is what we know so far,
based on what we've computed.
749
00:43:22 --> 00:43:22
Question.
750
00:43:22 --> 00:43:37
STUDENT: [INAUDIBLE]
751
00:43:37 --> 00:43:37
PROF.
752
00:43:37 --> 00:43:38
JERISON: The question is,
what happens if there's
753
00:43:38 --> 00:43:38
a sharp corner.
754
00:43:38 --> 00:43:45
The answer is, calculus, it's
not called a critical point.
755
00:43:45 --> 00:43:47
It's a something else.
756
00:43:47 --> 00:43:50
And it's a very
important point, too.
757
00:43:50 --> 00:43:52
And we will be discussing
those kinds of points.
758
00:43:52 --> 00:43:54
There are much more dramatic
instances of that.
759
00:43:54 --> 00:43:56
That's part of what
we're going to say.
760
00:43:56 --> 00:43:58
But I just want to
save that, alright.
761
00:43:58 --> 00:44:02
We will be discussing.
762
00:44:02 --> 00:44:03
Yeah.
763
00:44:03 --> 00:44:03
Question.
764
00:44:03 --> 00:44:08
STUDENT: [INAUDIBLE]
765
00:44:08 --> 00:44:08
PROF.
766
00:44:08 --> 00:44:11
JERISON: The question that was
asked was, how did I know at
767
00:44:11 --> 00:44:14
the critical point that it's
concave down over here and
768
00:44:14 --> 00:44:17
concave up over here.
769
00:44:17 --> 00:44:22
The answer is that I actually
did not use the second
770
00:44:22 --> 00:44:24
derivative yet.
771
00:44:24 --> 00:44:26
What I used is another
piece of information.
772
00:44:26 --> 00:44:28
I used the information
that I derived over here.
773
00:44:28 --> 00:44:32
That f' is positive,
where f' is positive and
774
00:44:32 --> 00:44:33
where it's negative.
775
00:44:33 --> 00:44:36
So what I know is that
the graph is going down
776
00:44:36 --> 00:44:39
to the left of - 1.
777
00:44:39 --> 00:44:41
It's going up to
the right, here.
778
00:44:41 --> 00:44:44
It's going up here and
it's going down there.
779
00:44:44 --> 00:44:47
I did not use the
second derivative.
780
00:44:47 --> 00:44:49
I used the first derivative.
781
00:44:49 --> 00:44:52
OK, but I didn't just use
the fact that there was
782
00:44:52 --> 00:44:56
a turning point here.
783
00:44:56 --> 00:44:57
So, actually, I was using
the fact that it was
784
00:44:57 --> 00:44:58
a turning point.
785
00:44:58 --> 00:45:00
I wasn't using the fact
that it had the second
786
00:45:00 --> 00:45:01
derivative, though.
787
00:45:01 --> 00:45:01
OK.
788
00:45:01 --> 00:45:03
For now.
789
00:45:03 --> 00:45:09
You can also see it by the
second derivative as well.
790
00:45:09 --> 00:45:14
So now, the next thing that
I'd like to do, I need to
791
00:45:14 --> 00:45:16
finish off this graph.
792
00:45:16 --> 00:45:19
And I just want to do it a
little bit carefully here.
793
00:45:19 --> 00:45:22
In the order that
is reasonable.
794
00:45:22 --> 00:45:27
Now, you might happen to
notice, and there's nothing
795
00:45:27 --> 00:45:35
wrong with this, so let's
even fill in a guess.
796
00:45:35 --> 00:45:37
In order to fill in a guess,
though, and have it be even
797
00:45:37 --> 00:45:39
vaguely right, I do have to
notice that this thing
798
00:45:39 --> 00:45:41
crosses, this function
crosses the origin.
799
00:45:41 --> 00:45:48
The function f(x) = 3x -
x^3 happens also have the
800
00:45:48 --> 00:45:50
property that f ( 0 ) = 0.
801
00:45:50 --> 00:45:52
Again, common sense.
802
00:45:52 --> 00:45:54
You're allowed to use
your common sense.
803
00:45:54 --> 00:45:56
You're allowed to notice
a value of the function
804
00:45:56 --> 00:45:58
and put it in.
805
00:45:58 --> 00:46:00
So there's nothing
wrong with that.
806
00:46:00 --> 00:46:03
If you happen to
have such a value.
807
00:46:03 --> 00:46:06
So, now we can guess what our
function is going to look like.
808
00:46:06 --> 00:46:10
It's going to maybe
come down like this.
809
00:46:10 --> 00:46:11
Come up like this.
810
00:46:11 --> 00:46:13
And come down like this.
811
00:46:13 --> 00:46:15
That could be what
it looks like.
812
00:46:15 --> 00:46:17
But, you know, another
possibility is it sort
813
00:46:17 --> 00:46:19
of comes along here
and goes out that way.
814
00:46:19 --> 00:46:22
Comes along here and goes
out that way, who knows?
815
00:46:22 --> 00:46:25
It happens, by the way,
that it's an odd function.
816
00:46:25 --> 00:46:25
Right?
817
00:46:25 --> 00:46:26
Those are all odd powers.
818
00:46:26 --> 00:46:28
So, actually, it's symmetric
on the right half
819
00:46:28 --> 00:46:29
and the left half.
820
00:46:29 --> 00:46:31
And crosses at 0.
821
00:46:31 --> 00:46:33
So everything that we do on the
right is going to be the same
822
00:46:33 --> 00:46:34
as what happens on the left.
823
00:46:34 --> 00:46:36
That's another piece
of common sense.
824
00:46:36 --> 00:46:39
You want to make use of that
as much as possible, whenever
825
00:46:39 --> 00:46:40
you're drawing anything.
826
00:46:40 --> 00:46:42
Don't want to throw
out information.
827
00:46:42 --> 00:46:46
So this function
happens to be odd.
828
00:46:46 --> 00:46:48
Odd, and f ( 0 ) = 0.
829
00:46:48 --> 00:46:52
I'm considering those to be
kinds of precalculus skills
830
00:46:52 --> 00:47:00
that I want you to use
as much as you can.
831
00:47:00 --> 00:47:03
So now, here's the first
feature which is unfortunately
832
00:47:03 --> 00:47:08
ignored in most
discussions of functions.
833
00:47:08 --> 00:47:11
And it's strange, because
nowadays we have
834
00:47:11 --> 00:47:13
graphing things.
835
00:47:13 --> 00:47:19
And it's really the only part
of the exercise that you
836
00:47:19 --> 00:47:26
couldn't do, at least on this
relatively simpleminded level,
837
00:47:26 --> 00:47:28
with a graphing calculator.
838
00:47:28 --> 00:47:33
And that is what I would call
the ends of the problem.
839
00:47:33 --> 00:47:37
So what happens off the
screen, is the question.
840
00:47:37 --> 00:47:39
And that basically is the
theoretical part of the problem
841
00:47:39 --> 00:47:41
that you have to address.
842
00:47:41 --> 00:47:42
You can program this.
843
00:47:42 --> 00:47:45
You can draw all the
pictures that you want.
844
00:47:45 --> 00:47:48
But what you won't see is
what's off the screen.
845
00:47:48 --> 00:47:50
You need to know something
to figure out what's
846
00:47:50 --> 00:47:51
off the screen.
847
00:47:51 --> 00:47:54
So, in this case, I'm talking
about what's off the screen
848
00:47:54 --> 00:48:01
going to the right, or
going to the left.
849
00:48:01 --> 00:48:06
So let's check the ends.
850
00:48:06 --> 00:48:07
So here, let's
just take a look.
851
00:48:07 --> 00:48:12
We have the function f(x),
which is, sorry, 3x - x^3.
852
00:48:12 --> 00:48:14
Again this is a precalculus
sort of thing.
853
00:48:14 --> 00:48:16
And we're imagining now,
let's just do x goes
854
00:48:16 --> 00:48:18
to plus infinity.
855
00:48:18 --> 00:48:19
So what happens here.
856
00:48:19 --> 00:48:25
When x is gigantic, this term
is completely negligible.
857
00:48:25 --> 00:48:30
And it just behaves like - x^3,
which goes to minus infinity
858
00:48:30 --> 00:48:32
as x goes to plus infinity.
859
00:48:32 --> 00:48:41
And similarly, f (x) goes
to plus infinity if x
860
00:48:41 --> 00:48:46
goes to minus infinity.
861
00:48:46 --> 00:48:49
Now let me pull down this
picture again, and show
862
00:48:49 --> 00:48:53
you what piece of the
information we've got.
863
00:48:53 --> 00:48:55
We now know that it is
heading up this way.
864
00:48:55 --> 00:48:58
It doesn't go like this,
it goes up like that.
865
00:48:58 --> 00:49:00
And I'm going to put an
arrow for it, And it's
866
00:49:00 --> 00:49:02
going down like this.
867
00:49:02 --> 00:49:06
Heading down to minus infinity
as x goes out farther
868
00:49:06 --> 00:49:07
to the right.
869
00:49:07 --> 00:49:15
And going out to plus infinity
as x goes farther to the left.
870
00:49:15 --> 00:49:21
So now there's hardly anything
left of this function
871
00:49:21 --> 00:49:21
to describe.
872
00:49:21 --> 00:49:27
There's really nothing left
except maybe decoration.
873
00:49:27 --> 00:49:31
And we kind of like that
decoration, so we will
874
00:49:31 --> 00:49:32
pay attention to it.
875
00:49:32 --> 00:49:35
And to do that, we'll have to
check the second derivative.
876
00:49:35 --> 00:49:39
So if we differentiate a second
time, the first derivative
877
00:49:39 --> 00:49:41
was, remember, 3 - 3x^2.
878
00:49:42 --> 00:49:53
So the second
derivative is - 6x.
879
00:49:53 --> 00:50:02
So now we notice that f'' (x)
is negative if x is positive.
880
00:50:02 --> 00:50:04
And f'' ( x)
881
00:50:04 --> 00:50:08
is positive if x is negative.
882
00:50:08 --> 00:50:13
And so in this part
it's concave down.
883
00:50:13 --> 00:50:18
And in this part
it's concave up.
884
00:50:18 --> 00:50:21
And now I'm going to
switch the boards so that
885
00:50:21 --> 00:50:24
you'll, and draw it.
886
00:50:24 --> 00:50:30
And you see that it was
begging to be this way.
887
00:50:30 --> 00:50:32
So we'll fill in the
rest of it here.
888
00:50:32 --> 00:50:35
Maybe in a nice color here.
889
00:50:35 --> 00:50:38
So this is the whole graph and
this is the correct graph.
890
00:50:38 --> 00:50:41
It comes down in one swoop
down here, and comes up here.
891
00:50:41 --> 00:50:46
And then it changes to concave
down right at the origin.
892
00:50:46 --> 00:50:49
So this point is of interest,
not only because it's the place
893
00:50:49 --> 00:50:52
where it crosses the axis, but
it's also what's called
894
00:50:52 --> 00:51:00
an inflection point.
895
00:51:00 --> 00:51:04
Inflection point, that's a
point where because f'' at
896
00:51:04 --> 00:51:07
that place is equal to 0.
897
00:51:07 --> 00:51:10
So it's a place where the
second derivative is 0.
898
00:51:10 --> 00:51:15
We also consider those to
be interesting points.
899
00:51:15 --> 00:51:21
Now, so let me just making
one closing remark here.
900
00:51:21 --> 00:51:26
Which is that all of this
information fits together.
901
00:51:26 --> 00:51:29
And we're going to have much,
much harder examples of this
902
00:51:29 --> 00:51:32
where you'll actually have to
think about what's going on.
903
00:51:32 --> 00:51:35
But there's a lot of
stuff protecting you.
904
00:51:35 --> 00:51:39
And functions will behave
themselves and turn
905
00:51:39 --> 00:51:40
around appropriately.
906
00:51:40 --> 00:51:43
Anyway, we'll talk
about it next time.
907
00:51:43 --> 00:51:43