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PROF.
9
00:00:22,076 --> 00:00:26,950
JERISON: We're starting
a new unit today.
10
00:00:26,950 --> 00:00:39,050
And, so this is Unit 2, and
it's called Applications
11
00:00:39,050 --> 00:00:48,810
of Differentiation.
12
00:00:48,810 --> 00:00:51,200
OK.
13
00:00:51,200 --> 00:00:57,400
So, the first application, and
we're going to do two today,
14
00:00:57,400 --> 00:01:04,030
is what are known as
linear approximations.
15
00:01:04,030 --> 00:01:06,310
Whoops, that should
have two p's in it.
16
00:01:06,310 --> 00:01:12,460
Approximations.
17
00:01:12,460 --> 00:01:16,360
So, that can be summarized
with one formula,
18
00:01:16,360 --> 00:01:19,040
but it's going to take
us at least half an hour
19
00:01:19,040 --> 00:01:21,960
to explain how this
formula is used.
20
00:01:21,960 --> 00:01:24,100
So here's the formula.
21
00:01:24,100 --> 00:01:34,390
It's f(x) is approximately equal
to its value at a base point
22
00:01:34,390 --> 00:01:38,260
plus the derivative
times x - x_0.
23
00:01:38,260 --> 00:01:38,760
Right?
24
00:01:38,760 --> 00:01:42,720
So this is the main formula.
25
00:01:42,720 --> 00:01:44,310
For right now.
26
00:01:44,310 --> 00:01:52,140
Put it in a box.
27
00:01:52,140 --> 00:01:57,430
And let me just describe
what it means, first.
28
00:01:57,430 --> 00:01:59,830
And then I'll describe
what it means again,
29
00:01:59,830 --> 00:02:01,780
and several other times.
30
00:02:01,780 --> 00:02:04,690
So, first of all,
what it means is
31
00:02:04,690 --> 00:02:11,140
that if you have a
curve, which is y = f(x),
32
00:02:11,140 --> 00:02:18,860
it's approximately the
same as its tangent line.
33
00:02:18,860 --> 00:02:37,020
So this other side is the
equation of the tangent line.
34
00:02:37,020 --> 00:02:43,090
So let's give an example.
35
00:02:43,090 --> 00:02:50,190
I'm going to take the
function f(x), which is ln x,
36
00:02:50,190 --> 00:02:53,980
and then its derivative is 1/x.
37
00:02:58,990 --> 00:03:03,800
And, so let's take the
base point x_0 = 1.
38
00:03:03,800 --> 00:03:05,470
That's pretty much
the only place where
39
00:03:05,470 --> 00:03:08,900
we know the logarithm for sure.
40
00:03:08,900 --> 00:03:13,360
And so, what we plug in
here now, are the values.
41
00:03:13,360 --> 00:03:17,890
So f(1) is the log of 0.
42
00:03:17,890 --> 00:03:20,750
Or, sorry, the log
of 1, which is 0.
43
00:03:20,750 --> 00:03:28,130
And f'(1), well,
that's 1/1, which is 1.
44
00:03:28,130 --> 00:03:31,100
So now we have an
approximation formula which,
45
00:03:31,100 --> 00:03:34,020
if I copy down
what's right up here,
46
00:03:34,020 --> 00:03:40,560
it's going to be ln x is
approximately, so f(0)
47
00:03:40,560 --> 00:03:44,480
is 0, right?
48
00:03:44,480 --> 00:03:49,700
Plus 1 times (x - 1).
49
00:03:49,700 --> 00:03:52,910
So I plugged in here,
for x_0, three places.
50
00:03:52,910 --> 00:04:00,670
I evaluated the coefficients and
this is the dependent variable.
51
00:04:00,670 --> 00:04:03,720
So, all told, if you
like, what I have here
52
00:04:03,720 --> 00:04:11,100
is that the logarithm of
x is approximately x - 1.
53
00:04:11,100 --> 00:04:16,660
And let me draw a
picture of this.
54
00:04:16,660 --> 00:04:22,310
So here's the graph of ln x.
55
00:04:22,310 --> 00:04:26,920
And then, I'll draw in the
tangent line at the place
56
00:04:26,920 --> 00:04:30,350
that we're considering,
which is x = 1.
57
00:04:30,350 --> 00:04:33,030
So here's the tangent line.
58
00:04:33,030 --> 00:04:35,195
And I've separated a
little bit, but really
59
00:04:35,195 --> 00:04:37,551
I probably should have drawn
it a little closer there,
60
00:04:37,551 --> 00:04:38,050
to show you.
61
00:04:38,050 --> 00:04:42,870
The whole point is that
these two are nearby.
62
00:04:42,870 --> 00:04:44,400
But they're not
nearby everywhere.
63
00:04:44,400 --> 00:04:50,130
So this is the line y = x - 1.
64
00:04:50,130 --> 00:04:51,960
Right, that's the tangent line.
65
00:04:51,960 --> 00:04:55,240
They're nearby only
when x is near 1.
66
00:04:55,240 --> 00:04:58,010
So say in this
little realm here.
67
00:04:58,010 --> 00:05:05,050
So when x is approximately
1, this is true.
68
00:05:05,050 --> 00:05:06,580
Once you get a
little farther away,
69
00:05:06,580 --> 00:05:08,413
this straight line,
this straight green line
70
00:05:08,413 --> 00:05:10,540
will separate from the graph.
71
00:05:10,540 --> 00:05:14,610
But near this place
they're close together.
72
00:05:14,610 --> 00:05:17,920
So the idea, again, is that
the curve, the curved line,
73
00:05:17,920 --> 00:05:19,770
is approximately
the tangent line.
74
00:05:19,770 --> 00:05:25,350
And this is one example of it.
75
00:05:25,350 --> 00:05:29,850
All right, so I want to
explain this in one more way.
76
00:05:29,850 --> 00:05:32,690
And then we want to
discuss it systematically.
77
00:05:32,690 --> 00:05:37,090
So the second way that
I want to describe this
78
00:05:37,090 --> 00:05:39,310
requires me to remind
you what the definition
79
00:05:39,310 --> 00:05:41,290
of the derivative is.
80
00:05:41,290 --> 00:05:46,370
So, the definition
of a derivative
81
00:05:46,370 --> 00:05:53,360
is that it's the limit, as delta
x goes to 0, of delta f / delta
82
00:05:53,360 --> 00:05:56,410
x, that's one way of
writing it, all right?
83
00:05:56,410 --> 00:06:01,260
And this is the
way we defined it.
84
00:06:01,260 --> 00:06:03,860
And one of the things that
we did in the first unit
85
00:06:03,860 --> 00:06:09,070
was we looked at this backwards.
86
00:06:09,070 --> 00:06:12,890
We used the derivative knowing
the derivatives of functions
87
00:06:12,890 --> 00:06:14,250
to evaluate some limits.
88
00:06:14,250 --> 00:06:17,860
So you were supposed
to do that on your.
89
00:06:17,860 --> 00:06:21,200
In our test, there were
some examples there,
90
00:06:21,200 --> 00:06:23,200
at least one example,
where that was the easiest
91
00:06:23,200 --> 00:06:26,140
way to do the problem.
92
00:06:26,140 --> 00:06:28,850
So in other words, you can
read this equation both ways.
93
00:06:28,850 --> 00:06:31,770
This is really, of course, the
same equation written twice.
94
00:06:31,770 --> 00:06:34,970
Now, what's new about
what we're going to do now
95
00:06:34,970 --> 00:06:40,150
is that we're going to take
this expression here, delta f
96
00:06:40,150 --> 00:06:42,910
/ delta x, and
we're going to say
97
00:06:42,910 --> 00:06:45,810
well, when delta x
is fairly near 0,
98
00:06:45,810 --> 00:06:47,360
this expression is
going to be fairly
99
00:06:47,360 --> 00:06:49,450
close to the limiting value.
100
00:06:49,450 --> 00:06:53,760
So this is
approximately f'(x_0).
101
00:06:53,760 --> 00:07:00,730
So that, I claim, is the same
as what's in the box in pink
102
00:07:00,730 --> 00:07:02,810
that I have over here.
103
00:07:02,810 --> 00:07:10,840
So this approximation formula
here is the same as this one.
104
00:07:10,840 --> 00:07:13,760
This is an average
rate of change,
105
00:07:13,760 --> 00:07:16,420
and this is an infinitesimal
rate of change.
106
00:07:16,420 --> 00:07:17,980
And they're nearly the same.
107
00:07:17,980 --> 00:07:19,230
That's the claim.
108
00:07:19,230 --> 00:07:22,950
So you'll have various exercises
in which this approximation is
109
00:07:22,950 --> 00:07:25,220
the useful one to use.
110
00:07:25,220 --> 00:07:28,860
And I will, as I said, I'll be
illustrating this a little bit
111
00:07:28,860 --> 00:07:29,610
today.
112
00:07:29,610 --> 00:07:34,060
Now, let me just explain why
those two formulas in the boxes
113
00:07:34,060 --> 00:07:36,560
are the same.
114
00:07:36,560 --> 00:07:41,110
So let's just start over
here and explain that.
115
00:07:41,110 --> 00:07:47,150
So the smaller box is the same
thing if I multiply through
116
00:07:47,150 --> 00:07:55,490
by delta x, as delta f is
approximately f'(x_0) delta x.
117
00:07:55,490 --> 00:07:57,290
And now if I just
write out what this
118
00:07:57,290 --> 00:08:09,820
is, it's f(x),
right, minus f(x_0),
119
00:08:09,820 --> 00:08:11,630
I'm going to write it this way.
120
00:08:11,630 --> 00:08:16,450
Which is approximately
f'(x_0), and this is x - x_0.
121
00:08:16,450 --> 00:08:25,670
So here I'm using the
notations delta x is x - x0.
122
00:08:25,670 --> 00:08:28,160
And so this is the
change in f, this
123
00:08:28,160 --> 00:08:32,270
is just rewriting
what delta x is.
124
00:08:32,270 --> 00:08:36,480
And now the last step is
just to put the constant
125
00:08:36,480 --> 00:08:37,430
on the other side.
126
00:08:37,430 --> 00:08:47,010
So f(x) is approximately
f(x_0) + f'(x_0)(x - x_0).
127
00:08:47,010 --> 00:08:51,920
So this is exactly what I had
just to begin with, right?
128
00:08:51,920 --> 00:08:53,570
So these two are
just algebraically
129
00:08:53,570 --> 00:08:56,690
the same statement.
130
00:08:56,690 --> 00:09:00,660
That's one another
way of looking at it.
131
00:09:00,660 --> 00:09:05,100
All right, so now,
I want to go through
132
00:09:05,100 --> 00:09:08,590
some systematic
discussion here of
133
00:09:08,590 --> 00:09:12,250
several linear approximations,
which you're going
134
00:09:12,250 --> 00:09:14,850
to be wanting to memorize.
135
00:09:14,850 --> 00:09:18,020
And rather than it's being
hard to memorize these,
136
00:09:18,020 --> 00:09:19,760
it's supposed to remind you.
137
00:09:19,760 --> 00:09:22,440
So that you'll have a lot
of extra reinforcement
138
00:09:22,440 --> 00:09:25,240
in remembering
derivatives of all kinds.
139
00:09:25,240 --> 00:09:31,240
So, when we carry out these
systematic discussions,
140
00:09:31,240 --> 00:09:33,060
we want to make
things absolutely as
141
00:09:33,060 --> 00:09:34,550
simple as possible.
142
00:09:34,550 --> 00:09:36,640
And so one of the
things that we do
143
00:09:36,640 --> 00:09:40,380
is we always use the
base point to be x_0.
144
00:09:40,380 --> 00:09:44,180
So I'm always going
to have x_0 = 0
145
00:09:44,180 --> 00:09:48,920
in this standard list of
formulas that I'm going to use.
146
00:09:48,920 --> 00:09:52,030
And if I put x_0 =
0, then this formula
147
00:09:52,030 --> 00:09:56,130
becomes f(x), a little
bit simpler to read.
148
00:09:56,130 --> 00:09:59,980
It becomes f(x)
is f(0) + f'(0) x.
149
00:10:03,520 --> 00:10:05,950
So this is probably
the form that you'll
150
00:10:05,950 --> 00:10:10,680
want to remember most.
151
00:10:10,680 --> 00:10:12,580
That's again, just the
linear approximation.
152
00:10:12,580 --> 00:10:16,010
But one always has
to remember, and this
153
00:10:16,010 --> 00:10:22,650
is a very important thing, this
one only worked near x is 1.
154
00:10:22,650 --> 00:10:29,170
This approximation here really
only works when x is near x_0.
155
00:10:29,170 --> 00:10:31,600
So that's a little addition
that you need to throw in.
156
00:10:31,600 --> 00:10:38,810
So this one works
when x is near 0.
157
00:10:38,810 --> 00:10:40,770
You can't expect it
to be true far away.
158
00:10:40,770 --> 00:10:42,790
The curve can go
anywhere it wants,
159
00:10:42,790 --> 00:10:46,500
when it's far away from
the point of tangency.
160
00:10:46,500 --> 00:10:49,060
So, OK, so let's work this out.
161
00:10:49,060 --> 00:10:51,560
Let's do it for
the sine function,
162
00:10:51,560 --> 00:10:56,401
for the cosine function,
and for e^x, to begin with.
163
00:10:56,401 --> 00:10:56,900
Yeah.
164
00:10:56,900 --> 00:10:57,400
Question.
165
00:10:57,400 --> 00:11:02,522
STUDENT: [INAUDIBLE]
166
00:11:02,522 --> 00:11:03,022
PROF.
167
00:11:03,022 --> 00:11:03,126
JERISON: Yeah.
168
00:11:03,126 --> 00:11:04,125
When does this one work.
169
00:11:04,125 --> 00:11:07,410
Well, so the question was,
when does this one work.
170
00:11:07,410 --> 00:11:12,100
Again, this is when x
is approximately x_0.
171
00:11:12,100 --> 00:11:18,130
Because it's actually the
same as this one over here.
172
00:11:18,130 --> 00:11:20,050
OK.
173
00:11:20,050 --> 00:11:23,010
And indeed, that's
what's going on when
174
00:11:23,010 --> 00:11:24,870
we take this limiting value.
175
00:11:24,870 --> 00:11:26,760
Delta x going to 0 is the same.
176
00:11:26,760 --> 00:11:27,980
Delta x small.
177
00:11:27,980 --> 00:11:37,050
So another way of saying it
is, the delta x is small.
178
00:11:37,050 --> 00:11:41,030
Now, exactly what we mean by
small will also be explained.
179
00:11:41,030 --> 00:11:45,240
But it is a matter to
some extent of intuition
180
00:11:45,240 --> 00:11:47,620
as to how much, how good it is.
181
00:11:47,620 --> 00:11:49,650
In practical cases,
people will really
182
00:11:49,650 --> 00:11:52,950
care about how small it is
before the approximation is
183
00:11:52,950 --> 00:11:53,970
useful.
184
00:11:53,970 --> 00:11:56,710
And that's a serious issue.
185
00:11:56,710 --> 00:12:00,200
All right, so let me carry out
these approximations for x.
186
00:12:00,200 --> 00:12:06,710
Again, this is
always for x near 0.
187
00:12:06,710 --> 00:12:08,930
So all of these are
going to be for x near 0.
188
00:12:08,930 --> 00:12:10,790
So in order to make
this computation,
189
00:12:10,790 --> 00:12:15,170
I have to evaluate the function.
190
00:12:15,170 --> 00:12:17,817
I need to plug in
two numbers here.
191
00:12:17,817 --> 00:12:19,150
In order to get this expression.
192
00:12:19,150 --> 00:12:23,070
I need to know what f(0) is and
I need to know what f'(0) is.
193
00:12:23,070 --> 00:12:26,110
If this is the function f(x),
then I'm going to make a little
194
00:12:26,110 --> 00:12:30,615
table over to the right here
with f' and then I'm going
195
00:12:30,615 --> 00:12:33,160
to evaluate f(0), and
then I'm going to evaluate
196
00:12:33,160 --> 00:12:38,150
f'(0), and then read off
what the answers are.
197
00:12:38,150 --> 00:12:41,450
Right, so first of all if
the function is sine x,
198
00:12:41,450 --> 00:12:44,170
the derivative is cosine x.
199
00:12:44,170 --> 00:12:49,690
The value of f(0),
that's sine of 0, is 0.
200
00:12:49,690 --> 00:12:51,550
The derivative is cosine.
201
00:12:51,550 --> 00:12:54,060
Cosine of 0 is 1.
202
00:12:54,060 --> 00:12:55,350
So there we go.
203
00:12:55,350 --> 00:12:58,850
So now we have the
coefficients 0 and 1.
204
00:12:58,850 --> 00:13:01,070
So this number is 0.
205
00:13:01,070 --> 00:13:04,480
And this number is 1.
206
00:13:04,480 --> 00:13:11,310
So what we get here is 0 + 1x,
so this is approximately x.
207
00:13:11,310 --> 00:13:18,080
There's the linear
approximation to sin x.
208
00:13:18,080 --> 00:13:20,570
Similarly, so now this
is a routine matter
209
00:13:20,570 --> 00:13:22,440
to just read this
off for this table.
210
00:13:22,440 --> 00:13:23,940
We'll do it for the
cosine function.
211
00:13:23,940 --> 00:13:30,230
If you differentiate the
cosine, what you get is -sin x.
212
00:13:30,230 --> 00:13:34,940
The value at 0 is 1, so
that's cosine of 0 at 1.
213
00:13:34,940 --> 00:13:39,540
The value of this
minus sine at 0 is 0.
214
00:13:39,540 --> 00:13:43,890
So this is going back
over here, 1 + 0x,
215
00:13:43,890 --> 00:13:48,170
so this is approximately 1.
216
00:13:48,170 --> 00:13:52,300
This linear function
happens to be constant.
217
00:13:52,300 --> 00:13:58,730
And finally, if I do need e^x,
its derivative is again e^x,
218
00:13:58,730 --> 00:14:02,840
and its value at 0 is 1, the
value of the derivative at 0 is
219
00:14:02,840 --> 00:14:04,180
also 1.
220
00:14:04,180 --> 00:14:09,500
So both of the terms here,
f(0) and f'(0), they're both 1
221
00:14:09,500 --> 00:14:15,400
and we get 1 + x.
222
00:14:15,400 --> 00:14:18,360
So these are the
linear approximations.
223
00:14:18,360 --> 00:14:19,610
You can memorize these.
224
00:14:19,610 --> 00:14:23,940
You'll probably remember them
either this way or that way.
225
00:14:23,940 --> 00:14:26,130
This collection of
information here
226
00:14:26,130 --> 00:14:28,556
encodes the same
collection of information
227
00:14:28,556 --> 00:14:29,430
as we have over here.
228
00:14:29,430 --> 00:14:31,460
For the values of the
function and the values
229
00:14:31,460 --> 00:14:36,310
of their derivatives at 0.
230
00:14:36,310 --> 00:14:39,370
So let me just emphasize again
the geometric point of view
231
00:14:39,370 --> 00:14:48,840
by drawing pictures
of these results.
232
00:14:48,840 --> 00:14:56,605
So first of all, for the sine
function, here's the sine
233
00:14:56,605 --> 00:15:03,500
- well, close enough.
234
00:15:03,500 --> 00:15:07,170
So that's - boy, now that is
quite some sine, isn't it?
235
00:15:07,170 --> 00:15:10,570
I should try to make the two
bumps be the same height,
236
00:15:10,570 --> 00:15:11,870
roughly speaking.
237
00:15:11,870 --> 00:15:15,450
Anyway the tangent line
we're talking about is here.
238
00:15:15,450 --> 00:15:17,730
And this is y = x.
239
00:15:17,730 --> 00:15:22,870
And this is the function sine x.
240
00:15:22,870 --> 00:15:28,870
And near 0, those things
coincide pretty closely.
241
00:15:28,870 --> 00:15:34,040
The cosine function, I'll
put that underneath, I guess.
242
00:15:34,040 --> 00:15:35,110
I think I can fit it.
243
00:15:35,110 --> 00:15:39,390
Make it a little smaller here.
244
00:15:39,390 --> 00:15:44,850
So for the cosine
function, we're up here.
245
00:15:44,850 --> 00:15:48,500
It's y = 1.
246
00:15:48,500 --> 00:15:51,990
Well, no wonder the
tangent line is constant.
247
00:15:51,990 --> 00:15:54,630
It's horizontal.
248
00:15:54,630 --> 00:15:57,920
The tangent line is horizontal,
so the function corresponding
249
00:15:57,920 --> 00:15:59,560
is constant.
250
00:15:59,560 --> 00:16:04,890
So this is y = cos x.
251
00:16:04,890 --> 00:16:14,790
And finally, if I draw y = e^x,
that's coming down like this.
252
00:16:14,790 --> 00:16:17,870
And the tangent line is here.
253
00:16:17,870 --> 00:16:19,410
And it's y = 1 + x.
254
00:16:19,410 --> 00:16:24,700
The value is 1 and
the slope is 1.
255
00:16:24,700 --> 00:16:28,030
So this is how to remember
it graphically if you like.
256
00:16:28,030 --> 00:16:34,810
This analytic picture
is extremely important
257
00:16:34,810 --> 00:16:37,950
and will help you to
deal with sines, cosines
258
00:16:37,950 --> 00:16:41,090
and exponentials.
259
00:16:41,090 --> 00:16:41,730
Yes, question.
260
00:16:41,730 --> 00:16:45,806
STUDENT: [INAUDIBLE]
261
00:16:45,806 --> 00:16:46,306
PROF.
262
00:16:46,306 --> 00:16:48,181
JERISON: The question
is what do you normally
263
00:16:48,181 --> 00:16:50,350
use linear approximations for.
264
00:16:50,350 --> 00:16:51,140
Good question.
265
00:16:51,140 --> 00:16:52,260
We're getting there.
266
00:16:52,260 --> 00:16:54,210
First, we're getting a
little library of them
267
00:16:54,210 --> 00:16:56,220
and I'll give you
a few examples.
268
00:16:56,220 --> 00:17:02,540
OK, so now, I need
to finish the catalog
269
00:17:02,540 --> 00:17:05,700
with two more examples which
are just a little bit, slightly
270
00:17:05,700 --> 00:17:07,620
more challenging.
271
00:17:07,620 --> 00:17:09,860
And a little bit less obvious.
272
00:17:09,860 --> 00:17:22,356
So, the next couple that we're
going to do are ln(1+x) and (1
273
00:17:22,356 --> 00:17:25,530
+ x)^r.
274
00:17:25,530 --> 00:17:28,130
OK, these are the last two
that we're going to write down.
275
00:17:28,130 --> 00:17:30,850
And that you need
to think about.
276
00:17:30,850 --> 00:17:34,930
Now, the procedure is
the same as over here.
277
00:17:34,930 --> 00:17:39,230
Namely, I have to write down
f' and I have to write down
278
00:17:39,230 --> 00:17:41,992
f'(0) and I have to
write down f'(0).
279
00:17:41,992 --> 00:17:43,450
And then I'll have
the coefficients
280
00:17:43,450 --> 00:17:46,970
to be able to fill in
what the approximation is.
281
00:17:46,970 --> 00:17:51,840
So f' = 1 / (1+x), in the
case of the logarithm.
282
00:17:51,840 --> 00:17:57,010
And f(0), if I plug in,
that's log of 1, which is 0.
283
00:17:57,010 --> 00:18:01,190
And f' if I plug
in 0 here, I get 1.
284
00:18:01,190 --> 00:18:04,850
And similarly if I do it for
this one, I get r(1+x)^(r-1).
285
00:18:07,850 --> 00:18:12,320
And when I plug in f(0),
I get 1^r, which is 1.
286
00:18:12,320 --> 00:18:18,850
And here I get r
(1)^(r-1), which is r.
287
00:18:18,850 --> 00:18:22,830
So the corresponding statement
here is that ln(1+x) is
288
00:18:22,830 --> 00:18:24,790
approximately x.
289
00:18:24,790 --> 00:18:31,140
And (1+x)^r is
approximately 1 + rx.
290
00:18:31,140 --> 00:18:35,660
That's 0 + 1x and
here we have 1 + rx.
291
00:18:41,370 --> 00:18:44,320
And now, I do want
to make a connection,
292
00:18:44,320 --> 00:18:47,120
explain to you what's going
on here and the connection
293
00:18:47,120 --> 00:18:48,750
with the first example.
294
00:18:48,750 --> 00:18:50,800
We already did the
logarithm once.
295
00:18:50,800 --> 00:18:53,520
And let's just point out
that these two computations
296
00:18:53,520 --> 00:18:57,470
are the same, or
practically the same.
297
00:18:57,470 --> 00:19:02,580
Here I use the base point
1, but because of my,
298
00:19:02,580 --> 00:19:05,420
sort of, convenient
form, which will end up,
299
00:19:05,420 --> 00:19:07,180
I claim, being much
more convenient
300
00:19:07,180 --> 00:19:09,310
for pretty much
every purpose, we
301
00:19:09,310 --> 00:19:14,510
want to do these things
near x is approximately 0.
302
00:19:14,510 --> 00:19:19,040
You cannot expand the logarithm
and understand a tangent line
303
00:19:19,040 --> 00:19:22,730
for it at x equals 0, because
it goes down to minus infinity.
304
00:19:22,730 --> 00:19:27,260
Similarly, if you
try to graph (1+x)^r,
305
00:19:27,260 --> 00:19:30,690
x^r without the 1 here,
you'll discover that sometimes
306
00:19:30,690 --> 00:19:33,260
the slope is infinite,
and so forth.
307
00:19:33,260 --> 00:19:35,500
So this is a bad
choice of point.
308
00:19:35,500 --> 00:19:39,030
1 is a much better choice
of a place to expand around.
309
00:19:39,030 --> 00:19:42,150
And then we shift things so
that it looks like it's x = 0,
310
00:19:42,150 --> 00:19:43,600
by shifting by the 1.
311
00:19:43,600 --> 00:19:50,950
So the connection with the
previous example is that
312
00:19:50,950 --> 00:19:57,020
the-- what we wrote before I
could write as ln u = u - 1.
313
00:19:57,020 --> 00:20:00,890
Right, that's just recopying
what I have over here.
314
00:20:00,890 --> 00:20:04,930
Except with the letter u
rather than the letter x.
315
00:20:04,930 --> 00:20:12,790
And then I plug in, u = 1 + x.
316
00:20:12,790 --> 00:20:14,920
And then that, if
I copy it down,
317
00:20:14,920 --> 00:20:16,880
you see that I have
a u in place of 1+x,
318
00:20:16,880 --> 00:20:19,020
that's the same as this.
319
00:20:19,020 --> 00:20:22,880
And if I write out u-1,
if I subtract 1 from u,
320
00:20:22,880 --> 00:20:23,924
that means that it's x.
321
00:20:23,924 --> 00:20:25,840
So that's what's on the
right-hand side there.
322
00:20:25,840 --> 00:20:27,720
So these are the
same computation,
323
00:20:27,720 --> 00:20:38,860
I've just changed the variable.
324
00:20:38,860 --> 00:20:44,220
So now I want to try to
address the question that was
325
00:20:44,220 --> 00:20:47,380
asked about how this is used.
326
00:20:47,380 --> 00:20:49,370
And what the importance is.
327
00:20:49,370 --> 00:20:58,010
And what I'm going to do is
just give you one example here.
328
00:20:58,010 --> 00:21:02,690
And then try to emphasize.
329
00:21:02,690 --> 00:21:05,930
The first way in which
this is a useful idea.
330
00:21:05,930 --> 00:21:10,460
So, or maybe this is
the second example.
331
00:21:10,460 --> 00:21:13,330
If you like.
332
00:21:13,330 --> 00:21:16,460
So we'll call this
Example 2, maybe.
333
00:21:16,460 --> 00:21:19,070
So let's just take
the logarithm of 1.1.
334
00:21:19,070 --> 00:21:22,220
Just a second.
335
00:21:22,220 --> 00:21:25,710
Let's take the logarithm of 1.1.
336
00:21:25,710 --> 00:21:30,150
So I claim that, according to
our rules, I can glance at this
337
00:21:30,150 --> 00:21:33,680
and I can immediately see
that it's approximately 1/10.
338
00:21:33,680 --> 00:21:35,710
So what did I use here?
339
00:21:35,710 --> 00:21:42,630
I used that ln(1+x)
is approximately x,
340
00:21:42,630 --> 00:21:46,281
and the value of x
that I used was 1/10.
341
00:21:46,281 --> 00:21:46,780
Right?
342
00:21:46,780 --> 00:21:48,850
So that is the
formula, so I should
343
00:21:48,850 --> 00:21:54,560
put a box around these
two formulas too.
344
00:21:54,560 --> 00:21:57,730
That's this formula here,
applied with x = 1/10.
345
00:21:57,730 --> 00:22:01,680
And I'm claiming that 1/10 is
a sufficiently small number,
346
00:22:01,680 --> 00:22:08,845
sufficiently close to 0,
that this is an OK statement.
347
00:22:08,845 --> 00:22:10,220
So the first
question that I want
348
00:22:10,220 --> 00:22:12,000
to ask you is,
which do you think
349
00:22:12,000 --> 00:22:14,470
is a more complicated thing.
350
00:22:14,470 --> 00:22:19,244
The left-hand side or
the right-hand side.
351
00:22:19,244 --> 00:22:21,160
I claim that this is a
more complicated thing,
352
00:22:21,160 --> 00:22:24,070
you'd have to go to a calculator
to punch out and figure out
353
00:22:24,070 --> 00:22:25,170
what this thing is.
354
00:22:25,170 --> 00:22:26,230
This is easy.
355
00:22:26,230 --> 00:22:28,730
You know what a tenth is.
356
00:22:28,730 --> 00:22:31,530
So the distinction
that I want to make
357
00:22:31,530 --> 00:22:37,190
is that this half, this
part, this is hard.
358
00:22:37,190 --> 00:22:40,700
And this is easy.
359
00:22:40,700 --> 00:22:43,090
Now, that may look
contradictory,
360
00:22:43,090 --> 00:22:45,610
but I want to just do
it right above as well.
361
00:22:45,610 --> 00:22:48,940
This is hard.
362
00:22:48,940 --> 00:22:52,160
And this is easy.
363
00:22:52,160 --> 00:22:52,870
OK.
364
00:22:52,870 --> 00:22:56,850
This looks uglier, but
actually this is the hard one.
365
00:22:56,850 --> 00:22:58,650
And this is giving us
information about it.
366
00:22:58,650 --> 00:23:00,930
Now, let me show
you why that's true.
367
00:23:00,930 --> 00:23:02,530
Look down this column here.
368
00:23:02,530 --> 00:23:05,230
These are the hard
ones, hard functions.
369
00:23:05,230 --> 00:23:07,360
These are the easy functions.
370
00:23:07,360 --> 00:23:09,970
What's easier than this?
371
00:23:09,970 --> 00:23:11,260
Nothing.
372
00:23:11,260 --> 00:23:11,760
OK.
373
00:23:11,760 --> 00:23:12,590
Well, yeah, 0.
374
00:23:12,590 --> 00:23:14,480
That's easier.
375
00:23:14,480 --> 00:23:16,060
Over here it gets even worse.
376
00:23:16,060 --> 00:23:21,090
These are the hard functions
and these are the easy ones.
377
00:23:21,090 --> 00:23:24,910
So that's the main advantage
of linear approximation
378
00:23:24,910 --> 00:23:27,730
is you get something much
simpler to deal with.
379
00:23:27,730 --> 00:23:30,380
And if you've made a
valid approximation
380
00:23:30,380 --> 00:23:33,510
you can make much
progress on problems.
381
00:23:33,510 --> 00:23:35,780
OK, we'll be doing
some more examples,
382
00:23:35,780 --> 00:23:38,990
but I saw some more questions
before I made that point.
383
00:23:38,990 --> 00:23:39,490
Yeah.
384
00:23:39,490 --> 00:23:42,373
STUDENT: [INAUDIBLE]
385
00:23:42,373 --> 00:23:42,873
PROF.
386
00:23:42,873 --> 00:23:46,080
JERISON: Is this
ln of 1.1 or what?
387
00:23:46,080 --> 00:23:48,410
STUDENT: [INAUDIBLE]
388
00:23:48,410 --> 00:23:49,115
PROF.
389
00:23:49,115 --> 00:23:52,580
JERISON: This is a parens there.
390
00:23:52,580 --> 00:23:56,430
It's ln of 1.1, it's the
digital number, right.
391
00:23:56,430 --> 00:23:59,820
I guess I've never used that
before a decimal point, have I?
392
00:23:59,820 --> 00:24:04,834
I don't know.
393
00:24:04,834 --> 00:24:05,500
Other questions.
394
00:24:05,500 --> 00:24:11,910
STUDENT: [INAUDIBLE]
395
00:24:11,910 --> 00:24:12,410
PROF.
396
00:24:12,410 --> 00:24:12,990
JERISON: OK.
397
00:24:12,990 --> 00:24:14,900
So let's continue here.
398
00:24:14,900 --> 00:24:18,570
Let me give you some more
examples, where it becomes
399
00:24:18,570 --> 00:24:21,460
even more vivid if you like.
400
00:24:21,460 --> 00:24:24,340
That this approximation
is giving us something
401
00:24:24,340 --> 00:24:30,190
a little simpler to deal with.
402
00:24:30,190 --> 00:24:34,960
So here's Example 3.
403
00:24:34,960 --> 00:24:48,400
I want to, I'll find the linear
approximation near x = 0.
404
00:24:48,400 --> 00:24:52,340
I also - when I write this
expression near x = 0,
405
00:24:52,340 --> 00:24:55,020
that's the same thing as this.
406
00:24:55,020 --> 00:24:58,940
That's the same thing as
saying x is approximately 0 -
407
00:24:58,940 --> 00:25:06,360
of the function e^(-3x)
divided by square root 1+x.
408
00:25:09,170 --> 00:25:17,390
So here's a function.
409
00:25:17,390 --> 00:25:17,890
OK.
410
00:25:17,890 --> 00:25:21,990
Now, what I claim I want
to use for the purposes
411
00:25:21,990 --> 00:25:26,830
of this approximation,
are just the sum
412
00:25:26,830 --> 00:25:32,110
of the approximation formulas
that we've already derived.
413
00:25:32,110 --> 00:25:33,850
And just to combine
them algebraically.
414
00:25:33,850 --> 00:25:35,350
So I'm not going
to do any calculus,
415
00:25:35,350 --> 00:25:37,080
I'm just going to remember.
416
00:25:37,080 --> 00:25:41,580
So with e^(-3x), it's pretty
clear that I should be using
417
00:25:41,580 --> 00:25:44,570
this formula for e^x.
418
00:25:44,570 --> 00:25:47,820
For the other one, it may be
slightly less obvious but we
419
00:25:47,820 --> 00:25:53,470
have powers of 1+x over here.
420
00:25:53,470 --> 00:25:55,510
So let's plug those in.
421
00:25:55,510 --> 00:26:04,640
I'll put this up so that
you can remember it.
422
00:26:04,640 --> 00:26:10,810
And we're going to carry
out this approximation.
423
00:26:10,810 --> 00:26:16,380
So, first of all, I'm going
to write this so that it's
424
00:26:16,380 --> 00:26:17,910
slightly more suggestive.
425
00:26:17,910 --> 00:26:23,630
Namely, I'm going to
write it as a product.
426
00:26:23,630 --> 00:26:27,220
And there you can
now see the exponent.
427
00:26:27,220 --> 00:26:31,870
In this case, r = 1/2, eh
-1/2, that we're going to use.
428
00:26:31,870 --> 00:26:32,900
OK.
429
00:26:32,900 --> 00:26:39,880
So now I have
e^(-3x) (1+x)^(-1/2),
430
00:26:39,880 --> 00:26:41,940
and that's going to
be approximately--
431
00:26:41,940 --> 00:26:44,220
well I'm going to
use this formula.
432
00:26:44,220 --> 00:26:48,760
I have to use it correctly. x is
replaced by -3x, so this is 1 -
433
00:26:48,760 --> 00:26:50,160
3x.
434
00:26:50,160 --> 00:26:52,270
And then over here,
I can just copy
435
00:26:52,270 --> 00:26:57,900
verbatim the other approximation
formula with r = -1/2.
436
00:26:57,900 --> 00:27:05,670
So this is times 1 - 1/2 x.
437
00:27:05,670 --> 00:27:11,080
And now I'm going to carry
out the multiplication.
438
00:27:11,080 --> 00:27:17,100
So this is 1 - 3x
- 1/2 x + 3/2 x^2.
439
00:27:27,310 --> 00:27:32,090
So now, here's our formula.
440
00:27:32,090 --> 00:27:34,340
So now this isn't
where things stop.
441
00:27:34,340 --> 00:27:36,960
And indeed, in this
kind of arithmetic
442
00:27:36,960 --> 00:27:39,420
that I'm describing
now, things are
443
00:27:39,420 --> 00:27:43,780
easier than they are in
ordinary algebra, in arithmetic.
444
00:27:43,780 --> 00:27:47,770
The reason is that there's
another step, which
445
00:27:47,770 --> 00:27:49,200
I'm now going to perform.
446
00:27:49,200 --> 00:27:54,602
Which is that I'm going to
throw away this term here.
447
00:27:54,602 --> 00:27:55,560
I'm going to ignore it.
448
00:27:55,560 --> 00:27:57,480
In fact, I didn't even
have to work it out.
449
00:27:57,480 --> 00:27:59,070
Because I'm going
to throw it away.
450
00:27:59,070 --> 00:28:01,520
So the reason is
that already, when
451
00:28:01,520 --> 00:28:03,424
I passed from this
expression to this one,
452
00:28:03,424 --> 00:28:05,340
that is from this type
of thing to this thing,
453
00:28:05,340 --> 00:28:07,940
I was already throwing away
quadratic and higher-ordered
454
00:28:07,940 --> 00:28:09,460
terms.
455
00:28:09,460 --> 00:28:12,650
So this isn't the
only quadratic term.
456
00:28:12,650 --> 00:28:13,640
There are tons of them.
457
00:28:13,640 --> 00:28:14,920
I have to ignore
all of them if I'm
458
00:28:14,920 --> 00:28:16,128
going to ignore some of them.
459
00:28:16,128 --> 00:28:20,240
And in fact, I only want to
be left with the linear stuff.
460
00:28:20,240 --> 00:28:23,220
Because that's all I'm really
getting a valid computation
461
00:28:23,220 --> 00:28:24,080
for.
462
00:28:24,080 --> 00:28:28,060
So, this is approximately
1 minus, so let's see.
463
00:28:28,060 --> 00:28:32,450
It's a total of 7/2 x.
464
00:28:32,450 --> 00:28:36,410
And this is the answer.
465
00:28:36,410 --> 00:28:38,290
This is the linear part.
466
00:28:38,290 --> 00:28:42,800
So the x^2 term is negligible.
467
00:28:42,800 --> 00:28:46,680
So we drop x^2 term.
468
00:28:46,680 --> 00:28:55,712
Terms, and higher.
469
00:28:55,712 --> 00:28:57,420
All of those terms
should be lower-order.
470
00:28:57,420 --> 00:29:00,180
If you imagine x is
1/10, or maybe 1/100,
471
00:29:00,180 --> 00:29:04,470
then these terms will end
up being much smaller.
472
00:29:04,470 --> 00:29:08,970
So we have a rather
crude approach.
473
00:29:08,970 --> 00:29:10,530
And that's really
the simplicity,
474
00:29:10,530 --> 00:29:15,360
and that's the savings.
475
00:29:15,360 --> 00:29:21,020
So now, since this unit
is called Applications,
476
00:29:21,020 --> 00:29:24,380
and these are indeed
applications to math,
477
00:29:24,380 --> 00:29:30,360
I also wanted to give you
a real-life application.
478
00:29:30,360 --> 00:29:34,290
Or a place where linear
approximations come up
479
00:29:34,290 --> 00:29:46,590
in real life.
480
00:29:46,590 --> 00:29:50,560
So maybe we'll call
this Example 4.
481
00:29:50,560 --> 00:29:57,270
This is supposedly
a real-life example.
482
00:29:57,270 --> 00:30:06,580
I'll try to persuade
you that it is.
483
00:30:06,580 --> 00:30:09,840
So I like this example because
it's got a lot of math,
484
00:30:09,840 --> 00:30:11,170
as well as physics in it.
485
00:30:11,170 --> 00:30:17,000
So here I am, on the
surface of the earth.
486
00:30:17,000 --> 00:30:24,610
And here is a satellite
going this way.
487
00:30:24,610 --> 00:30:30,790
At some velocity, v.
And this satellite
488
00:30:30,790 --> 00:30:33,630
has a clock on it because
this is a GPS satellite.
489
00:30:33,630 --> 00:30:37,720
And it has a time, T, OK?
490
00:30:37,720 --> 00:30:41,160
But I have a watch, in
fact it's right here.
491
00:30:41,160 --> 00:30:44,030
And I have a time which I keep.
492
00:30:44,030 --> 00:30:48,170
Which is T', And there's
an interesting relationship
493
00:30:48,170 --> 00:30:56,650
between T and T', which
is called time dilation.
494
00:30:56,650 --> 00:31:04,860
And this is from
special relativity.
495
00:31:04,860 --> 00:31:06,470
And it's the following formula.
496
00:31:06,470 --> 00:31:13,720
T' = T divided by the
square root of 1 - v^2/c^2,
497
00:31:13,720 --> 00:31:17,230
where v is the velocity
of the satellite,
498
00:31:17,230 --> 00:31:22,960
and c is the speed of light.
499
00:31:22,960 --> 00:31:28,080
So now I'd like to get a
rough idea of how different
500
00:31:28,080 --> 00:31:34,980
my watch is from the
clock on the satellite.
501
00:31:34,980 --> 00:31:38,540
So I'm going to use
this same approximation,
502
00:31:38,540 --> 00:31:40,990
we've already used it once.
503
00:31:40,990 --> 00:31:42,010
I'm going to write t.
504
00:31:42,010 --> 00:31:43,990
But now let me just remind you.
505
00:31:43,990 --> 00:31:46,809
The situation here is, we
have something of the form
506
00:31:46,809 --> 00:31:47,350
(1-u)^(-1/2).
507
00:31:52,410 --> 00:31:55,760
That's what's happening when
I multiply through here.
508
00:31:55,760 --> 00:31:59,500
So with u = v^2 / c^2.
509
00:32:02,080 --> 00:32:05,240
So in real life, of
course, the expression
510
00:32:05,240 --> 00:32:07,780
that you're going to use
the linear approximation on
511
00:32:07,780 --> 00:32:10,280
isn't necessarily itself linear.
512
00:32:10,280 --> 00:32:11,990
It can be any physical quantity.
513
00:32:11,990 --> 00:32:15,940
So in this case it's v
squared over c squared.
514
00:32:15,940 --> 00:32:18,189
And now the
approximation formula
515
00:32:18,189 --> 00:32:20,230
says that if this is
approximately equal to, well
516
00:32:20,230 --> 00:32:21,540
again it's the same rule.
517
00:32:21,540 --> 00:32:25,870
There's an r and then x
is -u, so this is - - 1/2,
518
00:32:25,870 --> 00:32:34,610
so it's 1 + 1/2 u.
519
00:32:34,610 --> 00:32:40,350
So this is approximately,
by the same rule, this is T,
520
00:32:40,350 --> 00:32:55,800
T' is approximately t
T(1 + 1/2 v^2/c^2) Now,
521
00:32:55,800 --> 00:32:58,150
I promised you that this
would be a real-life problem.
522
00:32:58,150 --> 00:33:02,520
So the question is when people
were designing these GPS
523
00:33:02,520 --> 00:33:06,666
systems, they run clocks
in the satellites.
524
00:33:06,666 --> 00:33:08,790
You're down there, you're
making your measurements,
525
00:33:08,790 --> 00:33:12,270
you're talking to
the satellite by--
526
00:33:12,270 --> 00:33:15,310
or you're receiving its
signals from its radio.
527
00:33:15,310 --> 00:33:19,010
The question is, is this
going to cause problems
528
00:33:19,010 --> 00:33:23,670
in the transmission.
529
00:33:23,670 --> 00:33:25,580
And there are dozens
of such problems
530
00:33:25,580 --> 00:33:27,180
that you have to check for.
531
00:33:27,180 --> 00:33:29,950
So in this case, what
actually happened
532
00:33:29,950 --> 00:33:35,010
is that v is about 4
kilometers per second.
533
00:33:35,010 --> 00:33:38,740
That's how fast the GPS
satellites actually go.
534
00:33:38,740 --> 00:33:41,430
In fact, they had to decide to
put them at a certain altitude
535
00:33:41,430 --> 00:33:43,950
and they could've tweaked
this if they had put them
536
00:33:43,950 --> 00:33:46,040
at different places.
537
00:33:46,040 --> 00:33:55,330
Anyway, the speed of light is
3 * 10^5 kilometers per second.
538
00:33:55,330 --> 00:34:01,100
So this number, v^2 / c^2
is approximately 10^(-10).
539
00:34:05,710 --> 00:34:11,160
Now, if you actually keep
track of how much of an error
540
00:34:11,160 --> 00:34:15,530
that would make in a GPS
location, what you would find
541
00:34:15,530 --> 00:34:17,820
is maybe it's a millimeter
or something like that.
542
00:34:17,820 --> 00:34:20,080
So in fact it doesn't matter.
543
00:34:20,080 --> 00:34:21,380
So that's nice.
544
00:34:21,380 --> 00:34:23,180
But in fact the
engineers who were
545
00:34:23,180 --> 00:34:26,870
designing these systems actually
did use this very computation.
546
00:34:26,870 --> 00:34:29,270
Exactly this.
547
00:34:29,270 --> 00:34:31,640
And the way that
they used it was,
548
00:34:31,640 --> 00:34:35,190
they decided that because
the clocks were different,
549
00:34:35,190 --> 00:34:38,740
when the satellite broadcasts
its radio frequency,
550
00:34:38,740 --> 00:34:40,350
that frequency would be shifted.
551
00:34:40,350 --> 00:34:41,500
Would be offset.
552
00:34:41,500 --> 00:34:44,426
And they decided that the
fidelity was so important
553
00:34:44,426 --> 00:34:46,050
that they would send
the satellites off
554
00:34:46,050 --> 00:34:49,120
with this kind of,
exactly this, offset.
555
00:34:49,120 --> 00:34:51,460
To compensate for the
way the signal is.
556
00:34:51,460 --> 00:34:53,360
So from the point of
view of good reception
557
00:34:53,360 --> 00:34:56,950
on your little GPS device, they
changed the frequency at which
558
00:34:56,950 --> 00:35:00,160
the transmitter
in the satellites,
559
00:35:00,160 --> 00:35:04,990
according to exactly this rule.
560
00:35:04,990 --> 00:35:08,120
And incidentally, the reason
why they didn't-- they ignored
561
00:35:08,120 --> 00:35:11,010
higher-order terms, the
sort of quadratic terms,
562
00:35:11,010 --> 00:35:17,460
is that if you take u^2
that's a size 10^(-20).
563
00:35:17,460 --> 00:35:20,104
And that really is
totally negligible.
564
00:35:20,104 --> 00:35:22,020
That doesn't matter to
any measurement at all.
565
00:35:22,020 --> 00:35:25,210
That's on the order
of nanometers,
566
00:35:25,210 --> 00:35:30,200
and it's not important for
any of the uses to which GPS
567
00:35:30,200 --> 00:35:32,510
is put.
568
00:35:32,510 --> 00:35:40,470
OK, so that's a real example of
a use of linear approximations.
569
00:35:40,470 --> 00:35:42,720
So. let's take a
little pause here.
570
00:35:42,720 --> 00:35:44,850
I'm going to switch
gears and talk
571
00:35:44,850 --> 00:35:46,610
about quadratic approximations.
572
00:35:46,610 --> 00:35:48,900
But before I do that, let's
have some more questions.
573
00:35:48,900 --> 00:35:49,400
Yeah.
574
00:35:49,400 --> 00:36:03,780
STUDENT: [INAUDIBLE]
575
00:36:03,780 --> 00:36:04,566
PROF.
576
00:36:04,566 --> 00:36:08,040
JERISON: OK, so the
question was asked,
577
00:36:08,040 --> 00:36:11,580
suppose I did this
by different method.
578
00:36:11,580 --> 00:36:15,840
Suppose I applied the
original formula here.
579
00:36:15,840 --> 00:36:18,050
Namely, I define
the function f(x),
580
00:36:18,050 --> 00:36:22,140
which was this function here.
581
00:36:22,140 --> 00:36:25,050
And then I plugged
in its value at x = 0
582
00:36:25,050 --> 00:36:28,000
and the value of its
derivative at x = 0.
583
00:36:28,000 --> 00:36:32,510
So the answer is, yes, it's
also true that if I call this
584
00:36:32,510 --> 00:36:37,940
function f f(x), then it
must be true that the linear
585
00:36:37,940 --> 00:36:45,910
approximation is f(x_0) plus
f' of - I'm sorry, it's at 0,
586
00:36:45,910 --> 00:36:49,340
so it's f(0), f'(0) times x.
587
00:36:49,340 --> 00:36:50,550
So that should be true.
588
00:36:50,550 --> 00:36:52,810
That's the formula
that we're using.
589
00:36:52,810 --> 00:36:57,170
It's up there in the pink also.
590
00:36:57,170 --> 00:36:58,590
So this is the formula.
591
00:36:58,590 --> 00:37:00,650
So now, what about f(0)?
592
00:37:00,650 --> 00:37:04,350
Well, if I plug in
0 here, I get 1 * 1.
593
00:37:04,350 --> 00:37:05,940
So this thing is 1.
594
00:37:05,940 --> 00:37:07,550
So that's no surprise.
595
00:37:07,550 --> 00:37:11,260
And that's what I got.
596
00:37:11,260 --> 00:37:15,600
If I computed f',
by the product rule
597
00:37:15,600 --> 00:37:19,150
it would be an annoying,
somewhat long, computation.
598
00:37:19,150 --> 00:37:21,510
And because of
what we just done,
599
00:37:21,510 --> 00:37:23,130
we know what it has to be.
600
00:37:23,130 --> 00:37:25,990
It has to be negative 7/2.
601
00:37:25,990 --> 00:37:28,280
Because this is a
shortcut for doing it.
602
00:37:28,280 --> 00:37:29,900
This is faster than doing that.
603
00:37:29,900 --> 00:37:32,190
But of course, that's a
legal way of doing it.
604
00:37:32,190 --> 00:37:33,780
When you get to
second derivatives,
605
00:37:33,780 --> 00:37:36,210
you'll quickly discover that
this method that I've just
606
00:37:36,210 --> 00:37:38,950
described is
complicated, but far
607
00:37:38,950 --> 00:37:41,330
superior to differentiating
this expression twice.
608
00:37:41,330 --> 00:37:46,087
STUDENT: [INAUDIBLE] PROF.
609
00:37:46,087 --> 00:37:48,420
JERISON: Would you have to
throw away an x^2 term if you
610
00:37:48,420 --> 00:37:49,560
differentiated?
611
00:37:49,560 --> 00:37:50,470
No.
612
00:37:50,470 --> 00:37:53,220
And in fact, we didn't
really have to do that here.
613
00:37:53,220 --> 00:37:55,385
If you differentiate
and then plug in x = 0.
614
00:37:55,385 --> 00:37:57,510
So if you differentiate
this and you plug in x = 0,
615
00:37:57,510 --> 00:37:58,970
you get -7/2.
616
00:37:58,970 --> 00:38:01,349
You differentiate this
and you plug in x = 0,
617
00:38:01,349 --> 00:38:03,140
this term still drops
out because it's just
618
00:38:03,140 --> 00:38:05,370
a 3x when you differentiate.
619
00:38:05,370 --> 00:38:08,270
And then you plug in
x = 0, it's gone too.
620
00:38:08,270 --> 00:38:10,650
And similarly, if you're
up here, it goes away
621
00:38:10,650 --> 00:38:12,410
and similarly over
here it goes away.
622
00:38:12,410 --> 00:38:18,555
So the higher-order terms never
influence this computation
623
00:38:18,555 --> 00:38:19,055
here.
624
00:38:19,055 --> 00:38:27,430
This just captures the linear
features of the function.
625
00:38:27,430 --> 00:38:30,980
So now I want to go on to
quadratic approximation.
626
00:38:30,980 --> 00:38:44,500
And now we're going to
elaborate on this formula.
627
00:38:44,500 --> 00:38:46,040
So, linear approximation.
628
00:38:46,040 --> 00:38:49,840
Well, that should have
been linear approximation.
629
00:38:49,840 --> 00:38:50,530
Liner.
630
00:38:50,530 --> 00:38:51,680
That's interesting.
631
00:38:51,680 --> 00:38:54,070
OK, so that was wrong.
632
00:38:54,070 --> 00:38:59,700
But now we're going to
change it to quadratic.
633
00:38:59,700 --> 00:39:04,280
So, suppose we talk about a
quadratic approximation here.
634
00:39:04,280 --> 00:39:07,450
Now, the quadratic
approximation is
635
00:39:07,450 --> 00:39:15,430
going to be just an elaboration,
one more step of detail.
636
00:39:15,430 --> 00:39:16,270
From the linear.
637
00:39:16,270 --> 00:39:18,060
In other words,
it's an extension
638
00:39:18,060 --> 00:39:20,230
of the linear approximation.
639
00:39:20,230 --> 00:39:24,320
And so we're adding
one more term here.
640
00:39:24,320 --> 00:39:26,650
And the extra term
turns out to be related
641
00:39:26,650 --> 00:39:28,990
to the second derivative.
642
00:39:28,990 --> 00:39:34,340
But there's a factor of 2.
643
00:39:34,340 --> 00:39:39,090
So this is the formula for
the quadratic approximation.
644
00:39:39,090 --> 00:39:46,450
And this chunk of it, of
course, is the linear part.
645
00:39:46,450 --> 00:39:54,190
This time I'll spell
'linear' correctly.
646
00:39:54,190 --> 00:39:56,030
So the linear part
is the first piece.
647
00:39:56,030 --> 00:40:05,050
And the quadratic part
is the second piece.
648
00:40:05,050 --> 00:40:09,630
I want to develop this
same catalog of functions
649
00:40:09,630 --> 00:40:11,140
as I had before.
650
00:40:11,140 --> 00:40:14,640
In other words, I want
to extend our formulas
651
00:40:14,640 --> 00:40:19,660
to the higher-order terms.
652
00:40:19,660 --> 00:40:26,070
And if you do that
for this example here,
653
00:40:26,070 --> 00:40:28,180
maybe I'll even illustrate
with this example
654
00:40:28,180 --> 00:40:31,050
before I go on, if you
do it with this example
655
00:40:31,050 --> 00:40:39,320
here, just to give you a
flavor for what goes on,
656
00:40:39,320 --> 00:40:41,140
what turns out to be the case.
657
00:40:41,140 --> 00:40:45,390
So this is the linear version.
658
00:40:45,390 --> 00:40:48,220
And now I'm going to compare
it to the quadratic version.
659
00:40:48,220 --> 00:40:55,540
So the quadratic version
turns out to be this.
660
00:40:55,540 --> 00:40:58,760
That's what turns out to be
the quadratic approximation.
661
00:40:58,760 --> 00:41:03,100
And when I use
this example here,
662
00:41:03,100 --> 00:41:09,400
so this is 1.1, which is the
same as ln of 1 + 1/10, right?
663
00:41:09,400 --> 00:41:17,430
So that's approximately
1/10 - 1/2 (1/10)^2.
664
00:41:17,430 --> 00:41:19,170
So 1/200.
665
00:41:19,170 --> 00:41:21,960
So that turns out,
instead of being
666
00:41:21,960 --> 00:41:29,160
1/10, that's point, what is it,
.095 or something like that.
667
00:41:29,160 --> 00:41:31,370
It's a little bit less.
668
00:41:31,370 --> 00:41:36,240
It's not .1, but
it's pretty close.
669
00:41:36,240 --> 00:41:39,350
So if you like,
the correction is
670
00:41:39,350 --> 00:41:48,900
lower in the decimal expansion.
671
00:41:48,900 --> 00:41:53,650
Now let me actually
check a few of these.
672
00:41:53,650 --> 00:41:54,940
I'll carry them out.
673
00:41:54,940 --> 00:41:58,670
And what I'm going to
probably save for next time
674
00:41:58,670 --> 00:42:08,020
is explaining to you, so this
is why this factor of 1/2,
675
00:42:08,020 --> 00:42:10,610
and we're going
to do this later.
676
00:42:10,610 --> 00:42:11,530
Do this next time.
677
00:42:11,530 --> 00:42:17,230
You can certainly do well to
stick with this presentation
678
00:42:17,230 --> 00:42:18,470
for one more lecture.
679
00:42:18,470 --> 00:42:22,210
So we can see this reinforced.
680
00:42:22,210 --> 00:42:32,580
So now I'm going to work
out these derivatives
681
00:42:32,580 --> 00:42:34,630
of the higher-order terms.
682
00:42:34,630 --> 00:42:39,450
And let me do it for the
x approximately 0 case.
683
00:42:39,450 --> 00:42:47,990
So first of all, I want to
add in the extra term here.
684
00:42:47,990 --> 00:42:50,830
Here's the extra term.
685
00:42:50,830 --> 00:42:53,780
For the quadratic part.
686
00:42:53,780 --> 00:42:57,050
And now in order to figure
out what's going on,
687
00:42:57,050 --> 00:43:03,350
I'm going to need to compute,
also, second derivatives.
688
00:43:03,350 --> 00:43:05,150
So here I need a
second derivative.
689
00:43:05,150 --> 00:43:11,465
And I need to throw in the value
of that second derivative at 0.
690
00:43:11,465 --> 00:43:13,340
So this is what I'm
going to need to compute.
691
00:43:13,340 --> 00:43:17,449
So if I do it, for example,
for the sine function,
692
00:43:17,449 --> 00:43:18,740
I already have the linear part.
693
00:43:18,740 --> 00:43:20,290
I need this last bit.
694
00:43:20,290 --> 00:43:22,570
So I differentiate the
sine function twice
695
00:43:22,570 --> 00:43:25,180
and I get, I claim
minus the sine function.
696
00:43:25,180 --> 00:43:26,900
The first derivative
is the cosine
697
00:43:26,900 --> 00:43:29,250
and the cosine derivative
is minus the sine.
698
00:43:29,250 --> 00:43:34,180
And when I evaluate it at
0, I get, lo and behold, 0.
699
00:43:34,180 --> 00:43:35,540
Sine of 0 is 0.
700
00:43:35,540 --> 00:43:40,361
So actually the quadratic
approximation is the same.
701
00:43:40,361 --> 00:43:40,860
0x^2.
702
00:43:40,860 --> 00:43:43,070
There's no x^2 term here.
703
00:43:43,070 --> 00:43:46,510
So that's why this is such
a terrific approximation.
704
00:43:46,510 --> 00:43:48,890
It's also the quadratic
approximation.
705
00:43:48,890 --> 00:43:53,460
For the cosine function,
if you differentiate twice,
706
00:43:53,460 --> 00:43:56,300
you get the derivative is
minus the sign and derivative
707
00:43:56,300 --> 00:44:00,170
of that is minus the cosine.
708
00:44:00,170 --> 00:44:03,060
So that's f''.
709
00:44:03,060 --> 00:44:09,600
And now, if I evaluate
that at 0, I get -1.
710
00:44:09,600 --> 00:44:11,530
And so the term that I
have to plug in here,
711
00:44:11,530 --> 00:44:15,240
this -1 is the coefficient
that appears right here.
712
00:44:15,240 --> 00:44:23,350
So I need a -1/2 x^2 extra.
713
00:44:23,350 --> 00:44:26,100
And if you do it for
the e^x, you get an e^x,
714
00:44:26,100 --> 00:44:39,450
and you got a 1 and so
you get 1/2 x^2 here.
715
00:44:39,450 --> 00:44:42,329
I'm going to finish these
two in just a second,
716
00:44:42,329 --> 00:44:43,745
but I first want
to tell you about
717
00:44:43,745 --> 00:44:56,480
the geometric significance
of this quadratic term.
718
00:44:56,480 --> 00:44:58,790
So here we go.
719
00:44:58,790 --> 00:45:18,430
Geometric significance
(of the quadratic term).
720
00:45:18,430 --> 00:45:21,100
So the geometric
significance is best
721
00:45:21,100 --> 00:45:25,670
to describe just by
drawing a picture here.
722
00:45:25,670 --> 00:45:29,300
And I'm going to draw the
picture of the cosine function.
723
00:45:29,300 --> 00:45:34,270
And remember we already
had the tangent line.
724
00:45:34,270 --> 00:45:38,620
So the tangent line was
this horizontal here.
725
00:45:38,620 --> 00:45:40,350
And that was y = 1.
726
00:45:40,350 --> 00:45:42,880
But you can see intuitively,
that doesn't even
727
00:45:42,880 --> 00:45:46,130
tell you whether this function
is above or below 1 there.
728
00:45:46,130 --> 00:45:47,437
Doesn't tell you much.
729
00:45:47,437 --> 00:45:50,020
It's sort of begging for there
to be a little more information
730
00:45:50,020 --> 00:45:52,470
to tell us what the
function is doing nearby.
731
00:45:52,470 --> 00:45:57,470
And indeed, that's what this
second expression does for us.
732
00:45:57,470 --> 00:46:00,850
It's some kind of
parabola underneath here.
733
00:46:00,850 --> 00:46:05,420
So this is y = 1 - 1/2 x^2.
734
00:46:05,420 --> 00:46:08,890
Which is a much better
fit to the curve
735
00:46:08,890 --> 00:46:12,740
than the horizontal line.
736
00:46:12,740 --> 00:46:23,750
And this is, if you like,
this is the best fit parabola.
737
00:46:23,750 --> 00:46:28,510
So it's going to be the
closest parabola to the curve.
738
00:46:28,510 --> 00:46:31,370
And that's more or
less the significance.
739
00:46:31,370 --> 00:46:34,600
It's much, much closer.
740
00:46:34,600 --> 00:46:40,220
All right, I want
to give you, well,
741
00:46:40,220 --> 00:46:43,040
I think we'll save these other
derivations for next time
742
00:46:43,040 --> 00:46:44,880
because I think we're
out of time now.
743
00:46:44,880 --> 00:46:47,110
So we'll do these next time.