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PROF.
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00:00:22 --> 00:00:26
JERISON: We're starting
a new unit today.
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00:00:26 --> 00:00:39
And, so this is Unit 2, and
it's called Applications
12
00:00:39 --> 00:00:48
of Differentiation.
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00:00:48 --> 00:00:51
OK.
14
00:00:51 --> 00:00:56
So, the first application,
and we're going to do two
15
00:00:56 --> 00:01:04
today, is what are known
as linear approximations.
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00:01:04 --> 00:01:06
Whoops, that should
have two p's in it.
17
00:01:06 --> 00:01:12
Approximations.
18
00:01:12 --> 00:01:16
So, that can be summarized with
one formula, but it's going to
19
00:01:16 --> 00:01:19
take us at least half an hour
to explain how this
20
00:01:19 --> 00:01:21
formula is used.
21
00:01:21 --> 00:01:24
So here's the formula.
22
00:01:24 --> 00:01:33
It's f(x) is approximately
equal to f(x0)
23
00:01:34 --> 00:01:38
f'(x)( x - x0).
24
00:01:38 --> 00:01:38
Right?
25
00:01:38 --> 00:01:42
So this is the main formula.
26
00:01:42 --> 00:01:44
For right now.
27
00:01:44 --> 00:01:52
Put it in a box.
28
00:01:52 --> 00:01:57
And let me just describe
what it means, first.
29
00:01:57 --> 00:02:00
And then I'll describe
what it means again, and
30
00:02:00 --> 00:02:01
several other times.
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00:02:01 --> 00:02:07
So, first of all, what it means
is that if you have a curve,
32
00:02:07 --> 00:02:14
which is y = f(x), it's
approximately the same
33
00:02:14 --> 00:02:18
as its tangent line.
34
00:02:18 --> 00:02:37
So this other side is the
equation of the tangent line.
35
00:02:37 --> 00:02:43
So let's give an example.
36
00:02:43 --> 00:02:51
I'm going to take the function
f(x), which is ln x, and then
37
00:02:51 --> 00:02:58
its derivative is 1 / x.
38
00:02:58 --> 00:03:03
And, so let's take the
base point x0 = 1.
39
00:03:03 --> 00:03:07
That's pretty much the only
place where we know the
40
00:03:07 --> 00:03:08
logarithm for sure.
41
00:03:08 --> 00:03:13
And so, what we plug in
here now, are the values.
42
00:03:13 --> 00:03:17
So f (1) is the ln of 0.
43
00:03:17 --> 00:03:20
Or, sorry, the ln
of 1, which is 0.
44
00:03:20 --> 00:03:28
And f'(1), well, that's
1/1, which is 1.
45
00:03:28 --> 00:03:32
So now we have an approximation
formula which, if I copy down
46
00:03:32 --> 00:03:36
what's right up here, it's
going to be ln x is
47
00:03:36 --> 00:03:43
approximately, so
f(0) is 0, right?
48
00:03:44 --> 00:03:49
1 (x - 1).
49
00:03:49 --> 00:03:52
So I plugged in here,
for x0, three places.
50
00:03:52 --> 00:03:59
I evaluated the coefficients
and this is the
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dependent variable.
52
00:04:00 --> 00:04:04
So, all told, if you like,
what I have here is that
53
00:04:04 --> 00:04:11
the logarithm of x is
approximately x - 1.
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00:04:11 --> 00:04:16
And let me draw a
picture of this.
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00:04:16 --> 00:04:22
So here's the graph of ln x.
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00:04:22 --> 00:04:26
And then, I'll draw in the
tangent line at the place
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00:04:26 --> 00:04:30
that we're considering,
which is x = 1.
58
00:04:30 --> 00:04:33
So here's the tangent line.
59
00:04:33 --> 00:04:35
And I've separated a little
bit, but really I probably
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00:04:35 --> 00:04:38
should have drawn it a little
closer there, to show you
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00:04:38 --> 00:04:42
the whole point is that
these two are nearby.
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00:04:42 --> 00:04:44
But they're not
nearby everywhere.
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00:04:44 --> 00:04:50
So this is the line y = x - 1.
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00:04:50 --> 00:04:51
Right, that's the tangent line.
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00:04:51 --> 00:04:55
They're nearby only
when x is near 1.
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00:04:55 --> 00:04:58
So say in this
little realm here.
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00:04:58 --> 00:05:05
So when x is approximately
1, this is true.
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00:05:05 --> 00:05:07
Once you get a little farther
away, this straight line, this
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straight green line will
separate from the graph.
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00:05:10 --> 00:05:14
But near this place
they're close together.
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00:05:14 --> 00:05:18
So the idea, again, is that the
curve, the curved line, is
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00:05:18 --> 00:05:19
approximately the tangent line.
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00:05:19 --> 00:05:25
And this is one example of it.
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00:05:25 --> 00:05:29
All right, so I want to
explain this in one more way.
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00:05:29 --> 00:05:32
And then we want to discuss
it systematically.
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00:05:32 --> 00:05:37
So the second way that I want
to describe this requires me to
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00:05:37 --> 00:05:41
remind you what the definition
of the derivative is.
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00:05:41 --> 00:05:46
So, the definition of a
derivative is that it's the
79
00:05:46 --> 00:05:54
limit, as delta x goes to 0, of
delta f / delta x, that's one
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00:05:54 --> 00:05:56
way of writing it, all right?
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00:05:56 --> 00:06:01
And this is the way
we defined it.
82
00:06:01 --> 00:06:04
And one of the things that we
did in the first unit was we
83
00:06:04 --> 00:06:09
looked at this backwards.
84
00:06:09 --> 00:06:12
We used the derivative knowing
the derivatives of functions
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00:06:12 --> 00:06:14
to evaluate some limits.
86
00:06:14 --> 00:06:17
So you were supposed
to do that on your.
87
00:06:17 --> 00:06:21
In our test, there were some
examples there, at least one
88
00:06:21 --> 00:06:26
example, where that was the
easiest way to do the problem.
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00:06:26 --> 00:06:28
So in other words, you can
read this equation both ways.
90
00:06:28 --> 00:06:31
This is really, of course, the
same equation written twice.
91
00:06:31 --> 00:06:36
Now, what's new about what
we're going to do now is that
92
00:06:36 --> 00:06:40
we're going to take this
expression here, delta f /
93
00:06:40 --> 00:06:44
delta x, and we're going to say
well, when delta x is fairly
94
00:06:44 --> 00:06:47
near 0, this expression is
going to be fairly close
95
00:06:47 --> 00:06:49
to the limiting value.
96
00:06:49 --> 00:06:51
So this is
approximately f'(x0).
97
00:06:53 --> 00:06:59
So that, I claim, is the same
as what's in the box in
98
00:06:59 --> 00:07:02
pink that I have over here.
99
00:07:02 --> 00:07:10
So this approximation formula
here is the same as this one.
100
00:07:10 --> 00:07:14
This is an average rate of
change, and this is an
101
00:07:14 --> 00:07:16
infinitesimal rate of change.
102
00:07:16 --> 00:07:17
And they're nearly the same.
103
00:07:17 --> 00:07:19
That's the claim.
104
00:07:19 --> 00:07:21
So you'll have various
exercises in which this
105
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approximation is the
useful one to use.
106
00:07:25 --> 00:07:27
And I will, as I said,
I'll be illustrating
107
00:07:27 --> 00:07:29
this a little bit today.
108
00:07:29 --> 00:07:33
Now, let me just explain
why those two formulas in
109
00:07:33 --> 00:07:36
the boxes are the same.
110
00:07:36 --> 00:07:41
So let's just start over
here and explain that.
111
00:07:41 --> 00:07:47
So the smaller box is the same
thing if I multiply through by
112
00:07:47 --> 00:07:50
delta x, as delta f is
approximately f'(
113
00:07:50 --> 00:07:55
x0 ) (delta x).
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00:07:55 --> 00:08:04
And now if I just write out
what this is, it's f ( x )
115
00:08:04 --> 00:08:11
right, - f ( x0), I'm going
to write it this way.
116
00:08:11 --> 00:08:16
Which is approximately f' ( x0
), and this is x minus x0.
117
00:08:16 --> 00:08:25
So here I'm using the
notations delta x is x - x0.
118
00:08:25 --> 00:08:29
And so this is the change in
f, this is just rewriting
119
00:08:29 --> 00:08:32
what delta x is.
120
00:08:32 --> 00:08:36
And now the last step is
just to put the constant
121
00:08:36 --> 00:08:37
on the other side.
122
00:08:37 --> 00:08:42
So f ( x ) is
approximately f ( x0 )
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00:08:44 --> 00:08:47
f'(x0)( x - x0).
124
00:08:47 --> 00:08:51
So this is exactly what I had
just to begin with, right?
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00:08:51 --> 00:08:53
So these two are just
algebraically the
126
00:08:53 --> 00:08:56
same statement.
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00:08:56 --> 00:09:00
That's one another way
of looking at it.
128
00:09:00 --> 00:09:06
All right, so now, I want to
go through some systematic
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00:09:06 --> 00:09:11
discussion here of several
linear approximations,
130
00:09:11 --> 00:09:14
which you're going to
be wanting to memorize.
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00:09:14 --> 00:09:18
And rather than it's being
hard to memorize these, it's
132
00:09:18 --> 00:09:19
supposed to remind you.
133
00:09:19 --> 00:09:22
So that you'll have a lot
of extra reinforcement in
134
00:09:22 --> 00:09:25
remembering derivatives
of all kinds.
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00:09:25 --> 00:09:31
So, when we carry out these
systematic discussions, we want
136
00:09:31 --> 00:09:34
to make things absolutely
as simple as possible.
137
00:09:34 --> 00:09:38
And so one of the things
that we do is we always use
138
00:09:38 --> 00:09:40
the base point to be x0.
139
00:09:40 --> 00:09:46
So I'm always going to have x0
= 0 in this standard list of
140
00:09:46 --> 00:09:48
formulas that I'm going to use.
141
00:09:48 --> 00:09:54
And if I put x0 = 0, then
this formula becomes f(x), a
142
00:09:54 --> 00:09:56
little bit simpler to read.
143
00:09:56 --> 00:09:59
It becomes f ( x ) = f ( 0 )
144
00:09:59 --> 00:10:01
f' ( 0 )x.
145
00:10:03 --> 00:10:06
So this is probably the
form that you'll want
146
00:10:06 --> 00:10:10
to remember most.
147
00:10:10 --> 00:10:12
That's again, just the
linear approximation.
148
00:10:12 --> 00:10:17
But one always has to remember,
and this is a very important
149
00:10:17 --> 00:10:22
thing, this one only
worked near x is 1.
150
00:10:22 --> 00:10:29
This approximation here really
only works when x is near x0.
151
00:10:29 --> 00:10:31
So that's a little addition
that you need to throw in.
152
00:10:31 --> 00:10:38
So this one works
when x is near 0.
153
00:10:38 --> 00:10:40
You can't expect it
to be true far away.
154
00:10:40 --> 00:10:44
The curve can go anywhere it
wants, when it's far away
155
00:10:44 --> 00:10:46
from the point of tangency.
156
00:10:46 --> 00:10:49
So, OK, so let's work this out.
157
00:10:49 --> 00:10:52
Let's do it for the sine
function, for the cosine
158
00:10:52 --> 00:10:56
function, and for e
^ x, to begin with.
159
00:10:56 --> 00:10:56
Yeah.
160
00:10:56 --> 00:10:57
Question.
161
00:10:57 --> 00:11:02
STUDENT: [INAUDIBLE]
162
00:11:02 --> 00:11:03
PROF.
163
00:11:03 --> 00:11:03
JERISON: Yeah.
164
00:11:03 --> 00:11:04
When does this one work.
165
00:11:04 --> 00:11:07
Well, so the question was,
when does this one work.
166
00:11:07 --> 00:11:12
Again, this is when x
is approximately x0.
167
00:11:12 --> 00:11:18
Because it's actually the
same as this one over here.
168
00:11:18 --> 00:11:20
OK.
169
00:11:20 --> 00:11:23
And indeed, that's what's
going on when we take
170
00:11:23 --> 00:11:24
this limiting value.
171
00:11:24 --> 00:11:26
Delta x going to 0 is the same.
172
00:11:26 --> 00:11:27
Delta x small.
173
00:11:27 --> 00:11:37
So another way of saying it
is, the delta x is small.
174
00:11:37 --> 00:11:41
Now, exactly what we mean by
small will also be explained.
175
00:11:41 --> 00:11:45
But it is a matter to some
extent of intuition as to
176
00:11:45 --> 00:11:47
how much, how good it is.
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00:11:47 --> 00:11:50
In practical cases, people
will really care about how
178
00:11:50 --> 00:11:53
small it is before the
approximation is useful.
179
00:11:53 --> 00:11:56
And that's a serious issue.
180
00:11:56 --> 00:12:00
All right, so let me carry out
these approximations for x.
181
00:12:00 --> 00:12:06
Again, this is always
for x near 0.
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00:12:06 --> 00:12:08
So all of these are going
to be for x near 0.
183
00:12:08 --> 00:12:11
So in order to make this
computation, I have to
184
00:12:11 --> 00:12:15
evaluate the function.
185
00:12:15 --> 00:12:17
I need to plug in
two numbers here.
186
00:12:17 --> 00:12:19
In order to get
this expression.
187
00:12:19 --> 00:12:20
I need to know what f(
0 ) and I need to know
188
00:12:20 --> 00:12:23
what f' ( 0 ) is.
189
00:12:23 --> 00:12:25
If this is the function f ( x
), then I'm going to make a
190
00:12:25 --> 00:12:30
little table over to the right
here with f' and then I'm going
191
00:12:30 --> 00:12:34
to evaluate f ( 0 ), and then
I'm going to evaluate f' ( 0 ),
192
00:12:34 --> 00:12:38
and then read off what
the answers are.
193
00:12:38 --> 00:12:41
Right, so first of all if
the function is sine x, the
194
00:12:41 --> 00:12:44
derivative is cosine x.
195
00:12:44 --> 00:12:49
The value of f ( 0 ),
that's sine of 0, is 0.
196
00:12:49 --> 00:12:51
The derivative is cosine.
197
00:12:51 --> 00:12:54
Cosine of 0 is 1.
198
00:12:54 --> 00:12:55
So there we go.
199
00:12:55 --> 00:12:58
So now we have the
coefficients 0 and 1.
200
00:12:58 --> 00:13:01
So this number is 0.
201
00:13:01 --> 00:13:04
And this number is 1.
202
00:13:04 --> 00:13:06
So what we get here is 0
203
00:13:07 --> 00:13:11
1x, so this is approximately x.
204
00:13:11 --> 00:13:18
There's the linear
approximation to sine x.
205
00:13:18 --> 00:13:21
Similarly, so now this is a
routine matter to just read
206
00:13:21 --> 00:13:22
this off for this table.
207
00:13:22 --> 00:13:23
We'll do it for the
cosine function.
208
00:13:23 --> 00:13:26
If you differentiate
the cosine, what you
209
00:13:26 --> 00:13:30
get is - sine x.
210
00:13:30 --> 00:13:34
The value at 0 is 1, so
that's cosine of 0 at 1.
211
00:13:34 --> 00:13:39
The value of this
= sine at 0 is 0.
212
00:13:39 --> 00:13:42
So this is going
back over here, 1
213
00:13:42 --> 00:13:48
0x, so this is approximately 1.
214
00:13:48 --> 00:13:52
This linear function
happens to be constant.
215
00:13:52 --> 00:13:57
And finally, if I do need e ^
x, its derivative is again e ^
216
00:13:57 --> 00:14:02
x, and its value at 0 is 1, the
value of the derivative
217
00:14:02 --> 00:14:04
at 0 is also 1.
218
00:14:04 --> 00:14:08
So both of the terms here, f
( 0 ) and f' ( 0 ), they're
219
00:14:08 --> 00:14:11
both 1 and we get 1
220
00:14:13 --> 00:14:15
x.
221
00:14:15 --> 00:14:18
So these are the linear
approximations.
222
00:14:18 --> 00:14:19
You can memorize these.
223
00:14:19 --> 00:14:23
You'll probably remember them
either this way or that way.
224
00:14:23 --> 00:14:27
This collection of information
here encodes the same
225
00:14:27 --> 00:14:29
collection of information
as we have over here.
226
00:14:29 --> 00:14:31
For the values of the function
and the values of their
227
00:14:31 --> 00:14:36
derivatives at 0.
228
00:14:36 --> 00:14:39
So let me just emphasize again
the geometric point of view
229
00:14:39 --> 00:14:48
by drawing pictures
of these results.
230
00:14:48 --> 00:14:56
So first of all, for the sine
function, here's the sine
231
00:14:56 --> 00:15:03
- well, close enough.
232
00:15:03 --> 00:15:07
So that's - boy, now that is
quite some sine, isn't it?
233
00:15:07 --> 00:15:10
I should try to make the two
bumps be the same height,
234
00:15:10 --> 00:15:11
roughly speaking.
235
00:15:11 --> 00:15:15
Anyway the tangent line we're
talking about is here.
236
00:15:15 --> 00:15:17
And this is y = x.
237
00:15:17 --> 00:15:22
And this is the
function sine x.
238
00:15:22 --> 00:15:28
And near 0, those things
coincide pretty closely.
239
00:15:28 --> 00:15:34
The cosine function, I'll put
that underneath, I guess.
240
00:15:34 --> 00:15:35
I think I can fit it.
241
00:15:35 --> 00:15:39
Make it a little smaller here.
242
00:15:39 --> 00:15:44
So for the cosine
function, we're up here.
243
00:15:44 --> 00:15:48
It's y = 1.
244
00:15:48 --> 00:15:51
Well, no wonder the
tangent line is constant.
245
00:15:51 --> 00:15:54
It's horizontal.
246
00:15:54 --> 00:15:56
The tangent line is
horizontal, so the function
247
00:15:56 --> 00:15:59
corresponding is constant.
248
00:15:59 --> 00:16:04
So this is y = cosine x.
249
00:16:04 --> 00:16:14
And finally, if I draw y = e^x,
that's coming down like this.
250
00:16:14 --> 00:16:17
And the tangent line is here.
251
00:16:17 --> 00:16:18
And it's y = 1
252
00:16:18 --> 00:16:19
x.
253
00:16:19 --> 00:16:24
The value is 1 and
the slope is 1.
254
00:16:24 --> 00:16:28
So this is how to remember
it graphically if you like.
255
00:16:28 --> 00:16:35
This analytic picture is
extremely important and will
256
00:16:35 --> 00:16:41
help you to deal with sines,
cosines and exponentials.
257
00:16:41 --> 00:16:41
Yes, question.
258
00:16:41 --> 00:16:46
STUDENT: [INAUDIBLE]
259
00:16:46 --> 00:16:46
PROF.
260
00:16:46 --> 00:16:48
JERISON: The question is what
do you normally use linear
261
00:16:48 --> 00:16:50
approximations for.
262
00:16:50 --> 00:16:51
Good question.
263
00:16:51 --> 00:16:52
We're getting there.
264
00:16:52 --> 00:16:54
First, we're getting a little
library of them and I'll
265
00:16:54 --> 00:16:56
give you a few examples.
266
00:16:56 --> 00:17:03
OK, so now, I need to finish
the catalog with two more
267
00:17:03 --> 00:17:05
examples which are just a
little bit, slightly
268
00:17:05 --> 00:17:07
more challenging.
269
00:17:07 --> 00:17:09
And a little bit less obvious.
270
00:17:09 --> 00:17:17
So, the next couple that
we're going to do are ln (1
271
00:17:17 --> 00:17:22
x) and (1
272
00:17:22 --> 00:17:23
x)^r.
273
00:17:25 --> 00:17:28
OK, these are the last two that
we're going to write down.
274
00:17:28 --> 00:17:30
And that you need
to think about.
275
00:17:30 --> 00:17:34
Now, the procedure is
the same as over here.
276
00:17:34 --> 00:17:39
Namely, I have to write down f'
and I have to write down f ( 0
277
00:17:39 --> 00:17:42
) and I have to right
down f' ( 0 ).
278
00:17:42 --> 00:17:44
And then I'll have the
coefficients to be able to fill
279
00:17:44 --> 00:17:46
in what the approximation is.
280
00:17:46 --> 00:17:49
So f' = 1 / 1
281
00:17:49 --> 00:17:51
x, in the case of
the logarithm.
282
00:17:51 --> 00:17:57
And f ( 0 ), if I plug in,
that's ln of 1, which is 0.
283
00:17:57 --> 00:18:01
And f' if I plug in
0 here, I get 1.
284
00:18:01 --> 00:18:05
And similarly if I do it
for this one, I get r (1
285
00:18:05 --> 00:18:07
x) ^ r - 1.
286
00:18:07 --> 00:18:12
And when I plug in f ( 0 ),
I get 1 ^ r, which is 1.
287
00:18:12 --> 00:18:18
And here I get r ( 1 )
^ r - 1, which is r.
288
00:18:18 --> 00:18:22
So the corresponding
statement here is that ln 1
289
00:18:22 --> 00:18:24
x is approximately x.
290
00:18:24 --> 00:18:26
And (1
291
00:18:26 --> 00:18:28
x) ^ r is approximately 1
292
00:18:28 --> 00:18:29
rx.
293
00:18:31 --> 00:18:31
That's 0
294
00:18:32 --> 00:18:35
1x and here we have 1
295
00:18:35 --> 00:18:41
r x.
296
00:18:41 --> 00:18:45
And now, I do want to make a
connection, explain to you
297
00:18:45 --> 00:18:47
what's going on here and the
connection with the
298
00:18:47 --> 00:18:48
first example.
299
00:18:48 --> 00:18:50
We already did the
logarithm once.
300
00:18:50 --> 00:18:53
And let's just point out that
these two computations are the
301
00:18:53 --> 00:18:57
same, or practically the same.
302
00:18:57 --> 00:19:02
Here I use the base point 1,
but because of my, sort of,
303
00:19:02 --> 00:19:06
convenient form, which will end
up, I claim, being much more
304
00:19:06 --> 00:19:09
convenient for pretty much
every purpose, we want to do
305
00:19:09 --> 00:19:14
these things near x
is approximately 0.
306
00:19:14 --> 00:19:19
You cannot expand the logarithm
and understand a tangent line
307
00:19:19 --> 00:19:22
for it at x equals 0, because
it goes down to minus infinity.
308
00:19:22 --> 00:19:25
Similarly, if you
try to graph (1
309
00:19:25 --> 00:19:30
x) ^ r, or x ^ r without the 1
here, you'll discover that
310
00:19:30 --> 00:19:33
sometimes the slope is
infinite, and so forth.
311
00:19:33 --> 00:19:35
So this is a bad
choice of point.
312
00:19:35 --> 00:19:39
1 is a much better choice of
a place to expand around.
313
00:19:39 --> 00:19:41
And then we shift things so
that it looks like it's x =
314
00:19:41 --> 00:19:43
0, by shifting by the 1.
315
00:19:43 --> 00:19:51
So the connection with the
previous example is that the,
316
00:19:51 --> 00:19:57
what we wrote before I could
write as the ln u = u - 1.
317
00:19:57 --> 00:20:00
Right, that's just recopying
what I have over here.
318
00:20:00 --> 00:20:04
Except with the letter u
rather than the letter x.
319
00:20:04 --> 00:20:11
And then I plug in, u = 1
320
00:20:11 --> 00:20:12
x.
321
00:20:12 --> 00:20:15
And then that, if I copy
it down, you see that I
322
00:20:15 --> 00:20:16
have a u in place of 1
323
00:20:16 --> 00:20:19
x, that's the same as this.
324
00:20:19 --> 00:20:22
And if I write out u - 1,
if I subtract 1 from u,
325
00:20:22 --> 00:20:23
that means that it's x.
326
00:20:23 --> 00:20:25
So that's what's on the
right-hand side there.
327
00:20:25 --> 00:20:31
So these are the same
computation, I've just
328
00:20:31 --> 00:20:38
changed the variable.
329
00:20:38 --> 00:20:45
So now I want to try to address
the question that was asked
330
00:20:45 --> 00:20:47
about how this is used.
331
00:20:47 --> 00:20:49
And what the importance is.
332
00:20:49 --> 00:20:58
And what I'm going to do is
just give you one example here.
333
00:20:58 --> 00:21:02
And then try to emphasize.
334
00:21:02 --> 00:21:05
The first way in which
this is a useful idea.
335
00:21:05 --> 00:21:10
So, or maybe this is
the second example.
336
00:21:10 --> 00:21:13
If you like.
337
00:21:13 --> 00:21:16
So we'll call this
Example 2, maybe.
338
00:21:16 --> 00:21:19
So let's just take the
logarithm of 1.1.
339
00:21:19 --> 00:21:22
Just a second.
340
00:21:22 --> 00:21:25
Let's take the
logarithm of 1.1.
341
00:21:25 --> 00:21:30
So I claim that, according to
our rules, I can glance at this
342
00:21:30 --> 00:21:33
and I can immediately see
that it's approximately 1/10.
343
00:21:33 --> 00:21:35
So what did I use here?
344
00:21:35 --> 00:21:38
I used that the ln (1
345
00:21:39 --> 00:21:44
x) is approximately x,
and the value of x
346
00:21:44 --> 00:21:46
that I used was 1/10.
347
00:21:46 --> 00:21:46
Right?
348
00:21:46 --> 00:21:49
So that is the formula, so
I should put a box around
349
00:21:49 --> 00:21:54
these two formulas too.
350
00:21:54 --> 00:21:57
That's this formula here,
applied with x = 1/10.
351
00:21:57 --> 00:22:01
And I'm claiming that 1/10 is
a sufficiently small number,
352
00:22:01 --> 00:22:09
sufficiently close to 0
this is an OK statement.
353
00:22:09 --> 00:22:11
So the first question that I
want to ask you is, which do
354
00:22:11 --> 00:22:14
you think is a more
complicated thing.
355
00:22:14 --> 00:22:19
The left-hand side or
the right-hand side.
356
00:22:19 --> 00:22:21
I claim that this is a more
complicated thing, you'd
357
00:22:21 --> 00:22:23
have to go to a calculator
to punch out and figure
358
00:22:23 --> 00:22:25
out what this thing is.
359
00:22:25 --> 00:22:26
This is easy.
360
00:22:26 --> 00:22:28
You know what a tenth is.
361
00:22:28 --> 00:22:33
So the distinction that I want
to make is that this half,
362
00:22:33 --> 00:22:37
this part, this is hard.
363
00:22:37 --> 00:22:40
And this is easy.
364
00:22:40 --> 00:22:43
Now, that may look
contradictory, but I want to
365
00:22:43 --> 00:22:45
just do it right above as well.
366
00:22:45 --> 00:22:48
This is hard.
367
00:22:48 --> 00:22:52
And this is easy.
368
00:22:52 --> 00:22:52
OK.
369
00:22:52 --> 00:22:56
This looks uglier, but actually
this is the hard one.
370
00:22:56 --> 00:22:58
And this is giving us
information about it.
371
00:22:58 --> 00:23:00
Now, let me show you
why that's true.
372
00:23:00 --> 00:23:02
Look down this column here.
373
00:23:02 --> 00:23:05
These are the hard
ones, hard functions.
374
00:23:05 --> 00:23:07
These are the easy functions.
375
00:23:07 --> 00:23:09
What's easier than this?
376
00:23:09 --> 00:23:11
Nothing.
377
00:23:11 --> 00:23:11
OK.
378
00:23:11 --> 00:23:12
Well, yeah, 0.
379
00:23:12 --> 00:23:14
That's easier.
380
00:23:14 --> 00:23:16
Over here it gets even worse.
381
00:23:16 --> 00:23:21
These are the hard functions
and these are the easy ones.
382
00:23:21 --> 00:23:25
So that's the main advantage of
linear approximation is you get
383
00:23:25 --> 00:23:27
something much simpler
to deal with.
384
00:23:27 --> 00:23:31
And if you've made a valid
approximation you can make
385
00:23:31 --> 00:23:33
much progress on problems.
386
00:23:33 --> 00:23:36
OK, we'll be doing some more
examples, but I saw some more
387
00:23:36 --> 00:23:38
questions before I
made that point.
388
00:23:38 --> 00:23:39
Yeah.
389
00:23:39 --> 00:23:42
STUDENT: [INAUDIBLE]
390
00:23:42 --> 00:23:42
PROF.
391
00:23:42 --> 00:23:46
JERISON: Is this is
ln of 1.1 or what?
392
00:23:46 --> 00:23:48
STUDENT: [INAUDIBLE]
393
00:23:48 --> 00:23:49
PROF.
394
00:23:49 --> 00:23:52
JERISON: This is
a parens there.
395
00:23:52 --> 00:23:56
It's ln of 1.1, it's the
digital number, right.
396
00:23:56 --> 00:23:59
I guess I've never used that
before a decimal point, have I?
397
00:23:59 --> 00:24:04
I don't know.
398
00:24:04 --> 00:24:05
Other questions.
399
00:24:05 --> 00:24:12
STUDENT: [INAUDIBLE]
400
00:24:12 --> 00:24:12
PROF.
401
00:24:12 --> 00:24:12
JERISON: OK.
402
00:24:12 --> 00:24:14
So let's continue here.
403
00:24:14 --> 00:24:18
Let me give you some more
examples, where it becomes
404
00:24:18 --> 00:24:21
even more vivid if you like.
405
00:24:21 --> 00:24:24
That this approximation is
giving us something a little
406
00:24:24 --> 00:24:30
simpler to deal with.
407
00:24:30 --> 00:24:34
So here's Example 3.
408
00:24:34 --> 00:24:48
I want to, I'll find the linear
approximation near x = 0.
409
00:24:48 --> 00:24:52
I also, when I write this
expression near x = 0, that's
410
00:24:52 --> 00:24:55
the same thing as this.
411
00:24:55 --> 00:24:58
That's the same thing as
saying x is approximately 0.
412
00:24:58 --> 00:25:07
Of the function (e ^ - 3x)
/ the square root of 1
413
00:25:08 --> 00:25:09
x.
414
00:25:09 --> 00:25:17
So here's a function.
415
00:25:17 --> 00:25:17
OK.
416
00:25:17 --> 00:25:22
Now, what I claim I want to
use for the purposes of this
417
00:25:22 --> 00:25:29
approximation, are just the sum
of the approximation formulas
418
00:25:29 --> 00:25:32
that we've already derived.
419
00:25:32 --> 00:25:33
And just to combine
them algebraically.
420
00:25:33 --> 00:25:35
So I'm not going to do
any calculus, I'm just
421
00:25:35 --> 00:25:37
going to remember.
422
00:25:37 --> 00:25:41
So with either the - 3x, it's
pretty clear that I should be
423
00:25:41 --> 00:25:44
using this formula for e ^ x.
424
00:25:44 --> 00:25:47
For the other one, it may
be slightly less obvious
425
00:25:47 --> 00:25:50
but we have powers of 1
426
00:25:50 --> 00:25:53
x over here.
427
00:25:53 --> 00:25:55
So let's plug those again.
428
00:25:55 --> 00:26:04
I'll put this up so that
you can remember it.
429
00:26:04 --> 00:26:10
And we're going to carry
out this approximation.
430
00:26:10 --> 00:26:16
So, first of all, I'm going
to write this so that it's
431
00:26:16 --> 00:26:17
slightly more suggestive.
432
00:26:17 --> 00:26:23
Namely, I'm going to
write it as a product.
433
00:26:23 --> 00:26:27
And there you can now
see the exponent.
434
00:26:27 --> 00:26:31
In this case, r = 1/2. - 1/2
that we're going to use.
435
00:26:31 --> 00:26:32
OK.
436
00:26:32 --> 00:26:37
So now I have e ^ - 3x (1
437
00:26:38 --> 00:26:42
x) ^ -1/2, and that's going to
be approximately, well I'm
438
00:26:42 --> 00:26:44
going to use this formula.
439
00:26:44 --> 00:26:48
I have to use it correctly. x
is replaced by - 3x, so this is
440
00:26:48 --> 00:26:53
1 - 3x And then over here, I
can just copy verbatim the
441
00:26:53 --> 00:26:54
other approximation formula.
442
00:26:54 --> 00:26:57
With r = - 1/2.
443
00:26:57 --> 00:27:05
So this is times 1 - 1/2x.
444
00:27:05 --> 00:27:11
And now I'm going to carry
out the multiplication.
445
00:27:11 --> 00:27:15
So this is 1 - 3x - 1/2x
446
00:27:16 --> 00:27:17
3/2x^2.
447
00:27:27 --> 00:27:32
So now, here's our formula.
448
00:27:32 --> 00:27:34
So now this isn't
where things stop.
449
00:27:34 --> 00:27:38
And indeed, in this kind of
arithmetic that I'm describing
450
00:27:38 --> 00:27:41
now, things are easier
than they are in ordinary
451
00:27:41 --> 00:27:43
algebra, in arithmetic.
452
00:27:43 --> 00:27:47
The reason is that there's
another step, which I'm
453
00:27:47 --> 00:27:49
now going to perform.
454
00:27:49 --> 00:27:54
Which is that I'm going to
throw away this term here.
455
00:27:54 --> 00:27:55
I'm going to ignore it.
456
00:27:55 --> 00:27:57
In fact, I didn't even
have to work it out.
457
00:27:57 --> 00:27:59
Because I'm going
to throw it away.
458
00:27:59 --> 00:28:02
So the reason is that already,
when I passed from this
459
00:28:02 --> 00:28:05
expression to this one, that is
from this type of thing to this
460
00:28:05 --> 00:28:07
thing, I was already throwing
away quadratic and
461
00:28:07 --> 00:28:09
higher-ordered terms.
462
00:28:09 --> 00:28:12
So this isn't the
only quadratic term.
463
00:28:12 --> 00:28:13
There are tons of them.
464
00:28:13 --> 00:28:15
I have to ignore all of
them if I'm going to
465
00:28:15 --> 00:28:15
ignore some of them.
466
00:28:15 --> 00:28:20
And in fact, I only want to be
left with the linear stuff.
467
00:28:20 --> 00:28:22
Because that's all I'm
really getting a valid
468
00:28:22 --> 00:28:24
computation for.
469
00:28:24 --> 00:28:28
So, this is approximately
1 minus, so let's see.
470
00:28:28 --> 00:28:32
It's a total of 7/2x.
471
00:28:32 --> 00:28:36
And this is the answer.
472
00:28:36 --> 00:28:38
This is the linear part.
473
00:28:38 --> 00:28:42
So the x^2 term is negligible.
474
00:28:42 --> 00:28:46
So we drop x^2 term.
475
00:28:46 --> 00:28:56
Terms, and higher.
476
00:28:56 --> 00:28:57
All of those terms
should be lower-order.
477
00:28:57 --> 00:29:00
If you imagine x is 1/10, or
maybe 1/100, then these terms
478
00:29:00 --> 00:29:04
will end up being much smaller.
479
00:29:04 --> 00:29:08
So we have a rather
crude approach.
480
00:29:08 --> 00:29:10
And that's really
the simplicity, and
481
00:29:10 --> 00:29:15
that's the savings.
482
00:29:15 --> 00:29:21
So now, since this unit is
called Applications, and these
483
00:29:21 --> 00:29:26
are indeed applications to
math, I also wanted to give
484
00:29:26 --> 00:29:30
you a real-life application.
485
00:29:30 --> 00:29:33
Or a place where linear
approximations come
486
00:29:33 --> 00:29:46
up in real life.
487
00:29:46 --> 00:29:50
So maybe we'll call
this example 4.
488
00:29:50 --> 00:29:57
This is supposedly a
real-life example.
489
00:29:57 --> 00:30:06
I'll try to persuade
you that it is.
490
00:30:06 --> 00:30:09
So I like this example because
it's got a lot of math,
491
00:30:09 --> 00:30:11
as well as physics in it.
492
00:30:11 --> 00:30:17
So here I am, on the
surface of the earth.
493
00:30:17 --> 00:30:24
And here is a satellite
going this way.
494
00:30:24 --> 00:30:28
At some velocity, v.
495
00:30:28 --> 00:30:32
And this satellite has a
clock on it because this
496
00:30:32 --> 00:30:33
is a GPS satellite.
497
00:30:33 --> 00:30:37
And it has a time, t, OK?
498
00:30:37 --> 00:30:41
But I have a watch, in
fact it's right here.
499
00:30:41 --> 00:30:44
And I have a time which I keep.
500
00:30:44 --> 00:30:44
Which is t'.
501
00:30:45 --> 00:30:51
And there's an interesting
relationship between t and t',
502
00:30:51 --> 00:30:56
which is called time dilation.
503
00:30:56 --> 00:31:04
And this is from
special relativity.
504
00:31:04 --> 00:31:11
And it's the following formula.
t' = t / the square root of 1 -
505
00:31:11 --> 00:31:17
(v^2 / C^2), where v is the
velocity of the satellite,
506
00:31:17 --> 00:31:22
and C is the speed of light.
507
00:31:22 --> 00:31:29
So now I'd like to get a rough
idea of how different my watch
508
00:31:29 --> 00:31:34
is from the clock
on the satellite.
509
00:31:34 --> 00:31:38
So I'm going to use this
same approximation, we've
510
00:31:38 --> 00:31:40
already used it once.
511
00:31:40 --> 00:31:42
I'm going to write t.
512
00:31:42 --> 00:31:43
But now let me just remind you.
513
00:31:43 --> 00:31:46
The situation here is, we
have something of the
514
00:31:46 --> 00:31:52
form (1 - u) ^ - 1/2.
515
00:31:52 --> 00:31:55
That's what's happening when
I multiply through here.
516
00:31:55 --> 00:31:59
So with u = v^2 / C^2.
517
00:31:59 --> 00:32:02
518
00:32:02 --> 00:32:05
So in real life, of course, the
expression that you're going to
519
00:32:05 --> 00:32:08
use the linear approximation on
isn't necessarily
520
00:32:08 --> 00:32:10
itself linear.
521
00:32:10 --> 00:32:11
It can be any
physical quantity.
522
00:32:11 --> 00:32:15
So in this case it's v
squared over C squared.
523
00:32:15 --> 00:32:19
And now the approximation
formula says that if this is
524
00:32:19 --> 00:32:21
approximately equal to, well
again it's the same rule.
525
00:32:21 --> 00:32:27
There's an r and then x is - u,
so this is - - 1/2, so it's 1
526
00:32:28 --> 00:32:34
1/2 u.
527
00:32:34 --> 00:32:40
So this is approximately, by
the same rule, this is t,
528
00:32:40 --> 00:32:43
t' is approximately t ( 1
529
00:32:44 --> 00:32:46
1/2 v^2 / C^2).
530
00:32:46 --> 00:32:53
531
00:32:53 --> 00:32:58
Now, I promised you that this
would be a real life problem.
532
00:32:58 --> 00:33:02
So the question is when people
were designing these GPS
533
00:33:02 --> 00:33:06
systems, they run clocks
in the satellites.
534
00:33:06 --> 00:33:08
You're down there, you're
making your measurements,
535
00:33:08 --> 00:33:13
you're talking to the satellite
by, or you're receiving its
536
00:33:13 --> 00:33:15
signals from its radio.
537
00:33:15 --> 00:33:19
The question is, is this
going to cause problems
538
00:33:19 --> 00:33:23
in the transmission.
539
00:33:23 --> 00:33:25
And there are dozens of
such problems that you
540
00:33:25 --> 00:33:27
have to check for.
541
00:33:27 --> 00:33:32
So in this case, what actually
happened is that v is about
542
00:33:32 --> 00:33:35
4 kilometers per second.
543
00:33:35 --> 00:33:38
That's how fast the GPS
satellites actually go.
544
00:33:38 --> 00:33:41
In fact, they had to decide to
put them at a certain altitude
545
00:33:41 --> 00:33:43
and they could've tweaked this
if they had put them
546
00:33:43 --> 00:33:46
at different places.
547
00:33:46 --> 00:33:55
Anyway, the speed of light is 3
( 10^5) kilometers per second.
548
00:33:55 --> 00:34:05
So this number, v^2 / C^2 is
approximately 10 ^ - 10.
549
00:34:05 --> 00:34:11
Now, if you actually keep track
of how much of an error that
550
00:34:11 --> 00:34:16
would make in a GPS location,
what you would find is maybe
551
00:34:16 --> 00:34:17
it's a millimeter or
something like that.
552
00:34:17 --> 00:34:20
So in fact it doesn't matter.
553
00:34:20 --> 00:34:21
So that's nice.
554
00:34:21 --> 00:34:24
But in fact the engineers who
were designing these systems
555
00:34:24 --> 00:34:26
actually did use this
very computation.
556
00:34:26 --> 00:34:29
Exactly this.
557
00:34:29 --> 00:34:33
And the way that they used it
was, they decided that because
558
00:34:33 --> 00:34:37
the clocks were different,
when the satellite broadcasts
559
00:34:37 --> 00:34:40
its radio frequency, that
frequency would be shifted.
560
00:34:40 --> 00:34:41
Would be offset.
561
00:34:41 --> 00:34:44
And they decided that the
fidelity was so important that
562
00:34:44 --> 00:34:47
they would send the satellites
off with this kind of,
563
00:34:47 --> 00:34:49
exactly this, offset.
564
00:34:49 --> 00:34:51
To compensate for the
way the signal is.
565
00:34:51 --> 00:34:53
So from the point of view of
good reception on your little
566
00:34:53 --> 00:34:58
GPS device, they changed the
frequency at which the
567
00:34:58 --> 00:35:04
transmitter in the satellites,
according to exactly this rule.
568
00:35:04 --> 00:35:08
And incidentally, the reason
why they didn't, they ignored
569
00:35:08 --> 00:35:12
higher-order terms, the sort of
quadratic terms, is that if you
570
00:35:12 --> 00:35:17
take u^2 that's a
size 10 ^ - 20.
571
00:35:17 --> 00:35:20
And that really is
totally negligible.
572
00:35:20 --> 00:35:22
That doesn't matter to
any measurement at all.
573
00:35:22 --> 00:35:26
That's on the order of
nanometers, and it's not
574
00:35:26 --> 00:35:32
important for any of the
uses to which GPS is put.
575
00:35:32 --> 00:35:40
OK, so that's a real example of
a use of linear approximations.
576
00:35:40 --> 00:35:42
So. let's take a
little pause here.
577
00:35:42 --> 00:35:45
I'm going to switch gears
and talk about quadratic
578
00:35:45 --> 00:35:46
approximations.
579
00:35:46 --> 00:35:48
But before I do that, let's
have some more questions.
580
00:35:48 --> 00:35:49
Yeah.
581
00:35:49 --> 00:36:03
STUDENT: [INAUDIBLE]
582
00:36:03 --> 00:36:04
PROF.
583
00:36:04 --> 00:36:08
JERISON: OK, so the question
was asked, suppose I did
584
00:36:08 --> 00:36:11
this by different method.
585
00:36:11 --> 00:36:15
Suppose I applied the
original formula here.
586
00:36:15 --> 00:36:18
Namely, I define the
function f (x), which
587
00:36:18 --> 00:36:22
was this function here.
588
00:36:22 --> 00:36:25
And then I plugged in its value
at x = 0 and the value of
589
00:36:25 --> 00:36:28
its derivative at x = 0.
590
00:36:28 --> 00:36:32
So the answer is, yes, it's
also true that if I call this
591
00:36:32 --> 00:36:37
function f ( x ), then it must
be true that the linear
592
00:36:37 --> 00:36:43
approximation is f (x0 )
593
00:36:43 --> 00:36:46
f' of - I'm sorry, it's
at 0, so it's f ( 0 )
594
00:36:46 --> 00:36:47
f' ( 0 )x.
595
00:36:49 --> 00:36:50
So that should be true.
596
00:36:50 --> 00:36:52
That's the formula
that we're using.
597
00:36:52 --> 00:36:57
It's up there in the pink also.
598
00:36:57 --> 00:36:58
So this is the formula.
599
00:36:58 --> 00:37:00
So now, what about f ( 0 )?
600
00:37:00 --> 00:37:04
Well, if I plug in 0
here, I get 1 * 1.
601
00:37:04 --> 00:37:05
So this thing is 1.
602
00:37:05 --> 00:37:07
So that's no surprise.
603
00:37:07 --> 00:37:11
And that's what I got.
604
00:37:11 --> 00:37:16
If I computed f', by the
product rule it would be
605
00:37:16 --> 00:37:19
an annoying, somewhat
long, computation.
606
00:37:19 --> 00:37:21
And because of what we
just done, we know
607
00:37:21 --> 00:37:23
what it has to be.
608
00:37:23 --> 00:37:25
It has to be negative 7/2.
609
00:37:25 --> 00:37:28
Because this is a
shortcut for doing it.
610
00:37:28 --> 00:37:29
This is faster than doing that.
611
00:37:29 --> 00:37:32
But of course, that's a
legal way of doing it.
612
00:37:32 --> 00:37:34
When you get to second
derivatives, you'll quickly
613
00:37:34 --> 00:37:37
discover that this method that
I've just described is
614
00:37:37 --> 00:37:40
complicated, but far superior
to differentiating this
615
00:37:40 --> 00:37:41
expression twice.
616
00:37:41 --> 00:37:46
STUDENT: [INAUDIBLE]
617
00:37:46 --> 00:37:46
PROF.
618
00:37:46 --> 00:37:48
JERISON: Would you have to
throw away an x^2 term
619
00:37:48 --> 00:37:49
if you differentiated?
620
00:37:49 --> 00:37:50
No.
621
00:37:50 --> 00:37:53
And in fact, we didn't really
have to do that here.
622
00:37:53 --> 00:37:55
If you differentiate and
then plug in x = 0.
623
00:37:55 --> 00:37:57
So if you differentiate
this and you plug in
624
00:37:57 --> 00:37:58
x = 0, you get - 7/2.
625
00:37:58 --> 00:38:01
You differentiate this and you
plug in x = 0, this term still
626
00:38:01 --> 00:38:05
drops out because it's just a
3x when you differentiate.
627
00:38:05 --> 00:38:08
And then you plug in
x = 0, it's gone to.
628
00:38:08 --> 00:38:10
And similarly, if you're up
here, it goes away and
629
00:38:10 --> 00:38:12
similarly over here
it goes away.
630
00:38:12 --> 00:38:17
So the higher-order terms
never influence this
631
00:38:17 --> 00:38:19
computation here.
632
00:38:19 --> 00:38:27
This just captures the linear
features of the function.
633
00:38:27 --> 00:38:30
So now I want to go on to
quadratic approximation.
634
00:38:30 --> 00:38:44
And now we're going to
elaborate on this formula.
635
00:38:44 --> 00:38:46
So, linear approximation.
636
00:38:46 --> 00:38:49
Well, that should have been
linear approximation.
637
00:38:49 --> 00:38:50
Liner.
638
00:38:50 --> 00:38:51
That's interesting.
639
00:38:51 --> 00:38:54
OK, so that was wrong.
640
00:38:54 --> 00:38:59
But now we're going to
change it to quadratic.
641
00:38:59 --> 00:39:04
So, suppose we talk about a
quadratic approximation here.
642
00:39:04 --> 00:39:08
Now, the quadratic
approximation is going to be
643
00:39:08 --> 00:39:15
just an elaboration, one
more step of detail.
644
00:39:15 --> 00:39:16
From the linear.
645
00:39:16 --> 00:39:18
In other words, it's
an extension of the
646
00:39:18 --> 00:39:20
linear approximation.
647
00:39:20 --> 00:39:24
And so we're adding
one more term here.
648
00:39:24 --> 00:39:26
And the extra term turns
out to be related to
649
00:39:26 --> 00:39:28
the second derivative.
650
00:39:28 --> 00:39:34
But there's a factor of 2.
651
00:39:34 --> 00:39:39
So this is the formula for
the quadratic approximation.
652
00:39:39 --> 00:39:46
And this chunk of it, of
course, is the linear part.
653
00:39:46 --> 00:39:54
This time I'll spell
'linear' correctly.
654
00:39:54 --> 00:39:56
So the linear part
is the first piece.
655
00:39:56 --> 00:40:05
And the quadratic part
is the second piece.
656
00:40:05 --> 00:40:09
I want to develop this
same catalog of functions
657
00:40:09 --> 00:40:11
as I had before.
658
00:40:11 --> 00:40:16
In other words, I want to
extend our formulas to
659
00:40:16 --> 00:40:19
the higher-order terms.
660
00:40:19 --> 00:40:26
And if you do that for this
example here, maybe I'll even
661
00:40:26 --> 00:40:30
illustrate with this example
before I go on, if you do it
662
00:40:30 --> 00:40:34
with this example here, just to
give you a flavor for what goes
663
00:40:34 --> 00:40:41
on, what turns out
to be the case.
664
00:40:41 --> 00:40:45
So this is the linear version.
665
00:40:45 --> 00:40:48
And now I'm going to compare
it to the quadratic version.
666
00:40:48 --> 00:40:55
So the quadratic version
turns out to be this.
667
00:40:55 --> 00:40:58
That's what turns out to be
the quadratic approximation.
668
00:40:58 --> 00:41:05
And when I use this example
here, so this is 1.1, which
669
00:41:05 --> 00:41:07
is the same as ln of 1
670
00:41:07 --> 00:41:09
1/10, right?
671
00:41:09 --> 00:41:15
So that's approximately
1/10 - 1/2 (1/10)^2.
672
00:41:17 --> 00:41:19
So 1/200.
673
00:41:19 --> 00:41:24
So that turns out, instead of
being 1/10, that's point,
674
00:41:24 --> 00:41:29
what is it, .095 or
something like that.
675
00:41:29 --> 00:41:31
It's a little bit less.
676
00:41:31 --> 00:41:36
It's not 0.1, but
it's pretty close.
677
00:41:36 --> 00:41:40
So if you like, the
correction is lower in
678
00:41:40 --> 00:41:48
the decimal expansion.
679
00:41:48 --> 00:41:53
Now let me actually
check a few of these.
680
00:41:53 --> 00:41:54
I'll carry them out.
681
00:41:54 --> 00:41:59
And what I'm going to probably
save for next time is
682
00:41:59 --> 00:42:08
explaining to you, so this is
y, this factor of 1/2, and
683
00:42:08 --> 00:42:10
we're going to do this later.
684
00:42:10 --> 00:42:11
Do this next time.
685
00:42:11 --> 00:42:17
You can certainly do well to
stick with this presentation
686
00:42:17 --> 00:42:18
for one more lecture.
687
00:42:18 --> 00:42:22
So we can see this reinforced.
688
00:42:22 --> 00:42:32
So now I'm going to work
out these derivatives of
689
00:42:32 --> 00:42:34
the higher-order terms.
690
00:42:34 --> 00:42:39
And let me do it for the
x approximately 0 case.
691
00:42:39 --> 00:42:47
So first of all, I want to
add in the extra term here.
692
00:42:47 --> 00:42:50
Here's the extra term.
693
00:42:50 --> 00:42:53
For the quadratic part.
694
00:42:53 --> 00:42:57
And now in order to figure out
what's going on, I'm going to
695
00:42:57 --> 00:43:03
need to compute, also,
second derivatives.
696
00:43:03 --> 00:43:05
So here I need a
second derivative.
697
00:43:05 --> 00:43:07
And I need to throw
in the value of that
698
00:43:07 --> 00:43:11
second derivative at 0.
699
00:43:11 --> 00:43:13
So this is what I'm going
to need to compute.
700
00:43:13 --> 00:43:17
So if I do it, for example, for
the sine function, I already
701
00:43:17 --> 00:43:18
have the linear part.
702
00:43:18 --> 00:43:20
I need this last bit.
703
00:43:20 --> 00:43:23
So I differentiate the sine
function twice and I get, I
704
00:43:23 --> 00:43:25
claim minus the sine function.
705
00:43:25 --> 00:43:27
The first derivative is the
cosine and the cosine
706
00:43:27 --> 00:43:29
derivative is minus the sine.
707
00:43:29 --> 00:43:34
And when I evaluate it at 0,
I get, lo and behold, 0.
708
00:43:34 --> 00:43:35
Sine of 0 is 0.
709
00:43:35 --> 00:43:39
So actually the quadratic
approximation is the same.
710
00:43:40 --> 00:43:40
0x^2.
711
00:43:40 --> 00:43:43
There's no x^2 term here.
712
00:43:43 --> 00:43:46
So that's why this is such
a terrific approximation.
713
00:43:46 --> 00:43:48
It's also the quadratic
approximation.
714
00:43:48 --> 00:43:54
For the cosine function, if you
differentiate twice, you get
715
00:43:54 --> 00:44:00
the derivative is -sin and
derivative of that is - cos.
716
00:44:00 --> 00:44:09
So that's f'' And now if I
evaluate that at 0, I get - 1.
717
00:44:09 --> 00:44:12
And so the term that I have
to plug in here, this - 1
718
00:44:12 --> 00:44:15
is the coefficient that
appears right here.
719
00:44:15 --> 00:44:23
So I need a - 1/2 x^2 extra.
720
00:44:23 --> 00:44:26
And if you do it for the e
^ x, you get an e ^ x, and
721
00:44:26 --> 00:44:29
you got a 1 and so you get
722
00:44:29 --> 00:44:39
1/2 x^2 here.
723
00:44:39 --> 00:44:43
I'm going to finish these two
in just a second, but I first
724
00:44:43 --> 00:44:46
want to tell you about the
geometric significance
725
00:44:46 --> 00:44:56
of this quadratic term.
726
00:44:56 --> 00:44:58
So here we go.
727
00:44:58 --> 00:45:18
Geometric significance
(of the quadratic term).
728
00:45:18 --> 00:45:22
So the geometric significance
is best to describe just
729
00:45:22 --> 00:45:25
by drawing a picture here.
730
00:45:25 --> 00:45:29
And I'm going to draw the
picture of the cosine function.
731
00:45:29 --> 00:45:34
And remember we already
had the tangent line.
732
00:45:34 --> 00:45:38
So the tangent line was
this horizontal here.
733
00:45:38 --> 00:45:40
And that was y = 1.
734
00:45:40 --> 00:45:43
But you can see intuitively,
that doesn't even tell you
735
00:45:43 --> 00:45:46
whether this function is
above or below 1 there.
736
00:45:46 --> 00:45:47
Doesn't tell you much.
737
00:45:47 --> 00:45:50
It's sort of begging for there
to be a little more information
738
00:45:50 --> 00:45:52
to tell us what the
function is doing nearby.
739
00:45:52 --> 00:45:57
And indeed, that's what this
second expression does for us.
740
00:45:57 --> 00:46:00
It's some kind of parabola
underneath here.
741
00:46:00 --> 00:46:03
So this is y = 1 - 1/2 x^2.
742
00:46:05 --> 00:46:09
Which is a much better
fit to the curve than
743
00:46:09 --> 00:46:12
the horizontal line.
744
00:46:12 --> 00:46:23
And this is, if you like, this
is the best fit parabola.
745
00:46:23 --> 00:46:28
So it's going to be the closest
parabola to the curve.
746
00:46:28 --> 00:46:31
And that's more or less
the significance.
747
00:46:31 --> 00:46:34
It's much, much closer.
748
00:46:34 --> 00:46:41
All right, I want to give you,
well, I think we'll save these
749
00:46:41 --> 00:46:43
other derivations for next time
because I think we're
750
00:46:43 --> 00:46:44
out of time now.
751
00:46:44 --> 00:46:47
So we'll do these next time.
752
00:46:47 --> 00:46:47