1
00:00:00,370 --> 00:00:01,510
Approximation.
mmm.

2
00:00:08,110 --> 00:00:10,660
Way back in Calculus 1, we were finding

3
00:00:10,660 --> 00:00:14,620
linear approximations for a function near
a point a.

4
00:00:14,620 --> 00:00:15,470
Well how did that go?

5
00:00:15,470 --> 00:00:21,590
Well it look like this, f was x is
approximately f of a,

6
00:00:21,590 --> 00:00:27,600
right now linear approximation around the
point a plus the derivative at a

7
00:00:27,600 --> 00:00:33,310
times how much the input has to change to
go from a to x, so times x

8
00:00:33,310 --> 00:00:36,940
minus a.
And why does this make sense?

9
00:00:36,940 --> 00:00:40,510
Well, the derivative is recording
infinitesimally how the

10
00:00:40,510 --> 00:00:43,480
output changes as the input changes, so if

11
00:00:43,480 --> 00:00:48,340
I take the ratio of output change to input
change and multiply by the input change.

12
00:00:48,340 --> 00:00:50,690
This is approximately the output change.

13
00:00:50,690 --> 00:00:54,210
Then I'm adding it to the output at the
point a,

14
00:00:54,210 --> 00:00:57,690
and that'll give me about the value of the
function at x.

15
00:00:57,690 --> 00:00:58,480
But let me put a little different

16
00:00:58,480 --> 00:00:59,860
spin on this.

17
00:00:59,860 --> 00:01:02,150
Instead of thinking about, you know, what
the derivative

18
00:01:02,150 --> 00:01:06,750
means, the sub-ratio between output values
and input values.

19
00:01:06,750 --> 00:01:09,310
Let's just think a little more naively,
let's just think that

20
00:01:09,310 --> 00:01:12,220
I'm trying to write down a function which
has the same

21
00:01:12,220 --> 00:01:15,950
value as my function f at the point a and
which

22
00:01:15,950 --> 00:01:19,790
has the same derivative as my function f
at the point a.

23
00:01:19,790 --> 00:01:21,430
We'll talk about this a little more
precisely,

24
00:01:21,430 --> 00:01:24,550
let me give a name to this linear
approximation.

25
00:01:24,550 --> 00:01:27,050
Let's call that g of x.

26
00:01:27,050 --> 00:01:34,410
So g of x will be f of a plus f prime of a
times x minus a.

27
00:01:34,410 --> 00:01:39,270
Now, what do I know about this function g?
Well, what's the value of g at a?

28
00:01:39,270 --> 00:01:43,200
Alright, what happens if I plug in a for
x?

29
00:01:43,200 --> 00:01:44,650
Well then I've got a minus a.

30
00:01:44,650 --> 00:01:48,130
This is 0 in that case, so this whole term
is 0.

31
00:01:48,130 --> 00:01:49,570
So g of a is just

32
00:01:49,570 --> 00:01:55,670
this, f of a.
And what's the derivative of g at a?

33
00:01:56,680 --> 00:01:59,020
Well what's just the derivative of g?
Alright.

34
00:01:59,020 --> 00:02:03,910
Well it's the derivative of this constant
which is 0 plus the derivative of this.

35
00:02:03,910 --> 00:02:07,490
This is a constant so it's f prime of a
times the

36
00:02:07,490 --> 00:02:10,910
derivative of this, but what's the
derivative of this with respect to x?

37
00:02:10,910 --> 00:02:14,650
It's just 1.
So the derivative of g regardless of what

38
00:02:14,650 --> 00:02:19,440
x is, is just the derivative of f at a.
So look.

39
00:02:19,440 --> 00:02:24,920
The function g here is a function whose
value at a is equal to the

40
00:02:24,920 --> 00:02:31,180
value of f at a and whose derivative at a
is equal to the derivative of f at a.

41
00:02:31,180 --> 00:02:32,470
I can do better.

42
00:02:32,470 --> 00:02:35,620
So I'm going to define a new function.
I'm also going to call it g.

43
00:02:35,620 --> 00:02:40,200
But it's going to be defined this way.
So g of x

44
00:02:40,200 --> 00:02:46,330
will be defined as follows.
G of x will be f of a plus

45
00:02:46,330 --> 00:02:52,370
the derivative of f at a times x minus a.

46
00:02:52,370 --> 00:02:54,480
So this is just the linear approximation
to f at

47
00:02:54,480 --> 00:02:57,450
the point a, and I'm going to add an extra
term.

48
00:02:57,450 --> 00:03:06,740
Plus, the second derivative of f at a
divided by 2 times x minus a squared.

49
00:03:08,110 --> 00:03:09,910
Why is that a good idea?

50
00:03:09,910 --> 00:03:12,730
Well when I write it out like this,

51
00:03:12,730 --> 00:03:17,680
this new function g, has three really
great properties.

52
00:03:17,680 --> 00:03:23,080
First of all, g at the point a is just
equal to f of a.

53
00:03:23,080 --> 00:03:27,870
If I plug in an a for x, that kills this
term, and this term.

54
00:03:27,870 --> 00:03:33,260
All that I'm left with is just f of a.
Also, the derivative of

55
00:03:33,260 --> 00:03:38,920
g, at the point a, is equal to the
derivative of f, at the point a.

56
00:03:38,920 --> 00:03:42,230
Now that would have been true even if I
didn't have this extra term.

57
00:03:42,230 --> 00:03:47,740
But, by including this extra term, it also
turns out that the second derivative

58
00:03:47,740 --> 00:03:53,660
of g at the point a is equal to the second
derivative of f at the point a.

59
00:03:53,660 --> 00:03:56,000
Now, I mean actually, the second
derivative of

60
00:03:56,000 --> 00:03:58,340
g at any point is equal to the second

61
00:03:58,340 --> 00:04:02,330
derivative of f at the point a.
But in particular, this is true.

62
00:04:02,330 --> 00:04:06,720
So I've written down a function that not
only has the same value as

63
00:04:06,720 --> 00:04:10,480
f does at a, not only has the same
derivative as f does at a.

64
00:04:10,480 --> 00:04:14,210
But also has the same second derivative as
f does at a.

65
00:04:14,210 --> 00:04:16,920
And, I can just keep on going.

66
00:04:16,920 --> 00:04:17,770
So, here we go.

67
00:04:17,770 --> 00:04:21,125
I'm going to define yet another function
that I'll again call g of x.

68
00:04:21,125 --> 00:04:22,890
It's going to start out the exact same
way.

69
00:04:22,890 --> 00:04:23,490
It's going to

70
00:04:23,490 --> 00:04:29,240
start out with f of a plus the derivative
of f.

71
00:04:29,240 --> 00:04:32,260
At the point a times x minus a.

72
00:04:32,260 --> 00:04:36,110
So this is just the linear approximation
to f that we're used to.

73
00:04:36,110 --> 00:04:37,660
Then, last time I added just one more

74
00:04:37,660 --> 00:04:39,410
term, I'm going to add that term again,
the

75
00:04:39,410 --> 00:04:47,210
second derivative of f, at the point a
divided by 2 times x minus a squared.

76
00:04:47,210 --> 00:04:48,780
But I'll add another term.

77
00:04:48,780 --> 00:04:50,040
I'm going to add this term.

78
00:04:50,040 --> 00:04:52,710
The third derivative of the function f at
the

79
00:04:52,710 --> 00:04:57,570
point a divided by 6 times x minus a
cubed.

80
00:04:58,780 --> 00:05:01,550
Now the idea here is that this function g
is going to

81
00:05:01,550 --> 00:05:05,850
turn out to agree with the function f at
the point a.

82
00:05:05,850 --> 00:05:08,800
It's also going to have the same
derivative

83
00:05:08,800 --> 00:05:10,570
as the function f at the point a.

84
00:05:10,570 --> 00:05:14,000
It's also going to have the same second
derivative as the function

85
00:05:14,000 --> 00:05:16,810
f at the point a, and it's going to have
the

86
00:05:16,810 --> 00:05:21,040
same third derivative as the function f at
the point a.

87
00:05:21,040 --> 00:05:24,700
Now to see this all I have to do is
differentiate.

88
00:05:24,700 --> 00:05:30,390
Before I differentiate let me just point
out that g of a is equal to what?

89
00:05:30,390 --> 00:05:35,750
Well if I plug in a for x, that kills this
term, this term, and this term.

90
00:05:35,750 --> 00:05:36,920
These are all 0.

91
00:05:36,920 --> 00:05:39,400
So the only thing that's left is f

92
00:05:39,400 --> 00:05:40,770
of a.
Now.

93
00:05:40,770 --> 00:05:43,260
Let's try differentiating.
Let's differentiate this once.

94
00:05:43,260 --> 00:05:46,590
What's the derivative of g?

95
00:05:46,590 --> 00:05:49,135
Well the differentiate g is going to
differentiate this.

96
00:05:49,135 --> 00:05:54,190
So the derivative of this constant is
zero, the derivative of this constant

97
00:05:54,190 --> 00:05:59,580
times x minus a, well that's the constant
f prime of a times the derivative

98
00:05:59,580 --> 00:06:04,700
of x minus a which is one.
Plus what's the derivative of this term?

99
00:06:04,700 --> 00:06:09,960
Well that's this constant.
F double prime of f at a divided

100
00:06:09,960 --> 00:06:14,940
by 2 times the derivative of x minus a
squared, which is

101
00:06:14,940 --> 00:06:20,220
2 times x minus a, times the derivative of
the inside, which is just 1.

102
00:06:20,220 --> 00:06:23,730
Alright, and then I gotta add the
derivative of this term.

103
00:06:23,730 --> 00:06:25,770
Well, that's this constant.

104
00:06:25,770 --> 00:06:30,120
F triple prime at the point a divided by 6
times

105
00:06:30,120 --> 00:06:31,610
the derivative of this.

106
00:06:31,610 --> 00:06:34,800
Which is 3 times x minus a squared times

107
00:06:34,800 --> 00:06:36,890
the derivative of the inside, which is
just 1.

108
00:06:36,890 --> 00:06:40,320
And now, what happens if I plug in a.

109
00:06:41,570 --> 00:06:45,150
What is g prime at a?

110
00:06:45,150 --> 00:06:47,670
Well this term and this term both have an
x minus

111
00:06:47,670 --> 00:06:50,340
a so they die the only thing that survives
is this.

112
00:06:50,340 --> 00:06:55,230
So the derivative of the function g at the
point a is the same

113
00:06:55,230 --> 00:06:58,150
as the derivative of the function f at the
point a.

114
00:06:58,150 --> 00:06:59,490
Now what about the second derivative?

115
00:06:59,490 --> 00:07:03,848
I've just got to differentiate this again.
So what's g double prime of x?

116
00:07:03,848 --> 00:07:05,470
Well I don't need to differentiate the
zero, that's

117
00:07:05,470 --> 00:07:09,820
just zero, this is a constant so that's
also zero.

118
00:07:09,820 --> 00:07:11,480
What's the derivative of this?

119
00:07:11,480 --> 00:07:14,270
Well it's this which is just f double
prime of

120
00:07:14,270 --> 00:07:17,600
a, times the derivative of x minus a which
is 1.

121
00:07:17,600 --> 00:07:20,640
And then what's the derivative of this?

122
00:07:20,640 --> 00:07:26,050
Well it's this constant which is f triple
prime at a divided by 2,

123
00:07:26,050 --> 00:07:31,690
times the derivative of x minus a squared.
Which is 2

124
00:07:31,690 --> 00:07:37,710
times x minus a times the derivative of
the inside, which is just 1.

125
00:07:37,710 --> 00:07:42,270
Now, what is this when I plug in a for x?

126
00:07:42,270 --> 00:07:46,450
All right, what's g double prime at the
point a?

127
00:07:46,450 --> 00:07:49,700
Well, when I plug in a for x, that kills

128
00:07:49,700 --> 00:07:52,760
this term, and the only thing that
survives is this.

129
00:07:52,760 --> 00:07:56,490
So the 2nd derivative of the function g at
the point a

130
00:07:56,490 --> 00:07:59,750
is the same as the 2nd derivative of f at
the point a.

131
00:07:59,750 --> 00:08:01,200
Now, what about the 3rd derivative?

132
00:08:01,200 --> 00:08:03,310
Well, then, I just gotta differentiate
this.

133
00:08:03,310 --> 00:08:06,840
So, what's g triple prime at x?

134
00:08:06,840 --> 00:08:11,790
Well, it's 0 plus 0 plus the derivative of
this constant is 0.

135
00:08:11,790 --> 00:08:13,680
Plus the derivative of this term.

136
00:08:13,680 --> 00:08:16,870
Well, that's this constant times the
derivative of x minus a.

137
00:08:16,870 --> 00:08:20,980
Well, this constant is just f triple prime
at a

138
00:08:20,980 --> 00:08:24,180
times the derivative of x minus a which is
just one.

139
00:08:25,250 --> 00:08:31,820
So, the third derivative of g at the point
a is equal to

140
00:08:31,820 --> 00:08:37,040
the third derivative of F at the point A.
So this cubic polynomial not only

141
00:08:37,040 --> 00:08:40,840
matches the function's value, but it also
matches the first, second

142
00:08:40,840 --> 00:08:44,550
and third derivative of the function f at
the point a.

143
00:08:44,550 --> 00:08:46,710
Let me make this more concrete.

144
00:08:46,710 --> 00:08:51,200
Let me apply this to a specific function.
Here's a specific example.

145
00:08:51,200 --> 00:08:56,260
Let's apply this to the function f of x
equals sine x.

146
00:08:56,260 --> 00:08:59,030
I'm going to differentiate sine a bunch of
times.

147
00:08:59,030 --> 00:09:03,200
So the derivative of sine is cosine.

148
00:09:04,480 --> 00:09:07,760
The second derivative of sine, the
derivative of

149
00:09:07,760 --> 00:09:11,740
cosine, is minus sine, so that if we
choke.

150
00:09:11,740 --> 00:09:14,836
What's the third derivative of sine,
what's the

151
00:09:14,836 --> 00:09:18,730
derivative of minus sine, that's negative
cosine x.

152
00:09:18,730 --> 00:09:21,400
Now let's evaluate those at zero.

153
00:09:21,400 --> 00:09:27,440
Well the function's value at zero and
what's sine at zero, that's zero.

154
00:09:27,440 --> 00:09:29,730
What's the derivative at zero?

155
00:09:29,730 --> 00:09:34,420
What's co sin at zero, that's one.
What's the second derivative at zero?

156
00:09:34,420 --> 00:09:37,750
Well, that's negative sine of zero, that's
zero.

157
00:09:37,750 --> 00:09:39,970
What's the third derivative at zero?

158
00:09:39,970 --> 00:09:43,390
Well, that's negative cosine of zero,
that's negative one.

159
00:09:43,390 --> 00:09:45,020
Now I've got all the pieces ready.

160
00:09:45,020 --> 00:09:48,190
I'm ready to write down a polynomial that
has the same value,

161
00:09:48,190 --> 00:09:51,980
the same derivative, the same second
derivative, and the same third derivative.

162
00:09:51,980 --> 00:09:54,075
As sine does at the point zero.

163
00:09:54,075 --> 00:09:58,730
Well in this case, g of x is what?

164
00:09:58,730 --> 00:10:04,920
Well if the function's value is zero,
which is zero, plus the derivative of

165
00:10:04,920 --> 00:10:11,320
zero, which is one, times x minus a, which
is just x plus

166
00:10:11,320 --> 00:10:18,040
the second derivative at 0, which is 0
divided by 2 times x minus a squared,

167
00:10:19,670 --> 00:10:23,480
plus the third derivative of the function
at 0, which

168
00:10:23,480 --> 00:10:29,360
is negative 1, divided by 6 times x minus
a cubed.

169
00:10:29,360 --> 00:10:34,100
And the a is 0 in this case, so just x
cubed.

170
00:10:34,100 --> 00:10:39,745
I can simplify that a bit.
We'll now just get zero, I don't have to

171
00:10:39,745 --> 00:10:46,290
write that down, x which is just zero
minus x cubed over six.

172
00:10:46,290 --> 00:10:49,965
So there is my cubic polynomial.
So this is pretty great.

173
00:10:49,965 --> 00:10:54,510
I've got this polynomial and it's supposed
to be a pretty good approximation to sine.

174
00:10:54,510 --> 00:10:57,510
We'll let's take a look at a graph of sign
just to get an idea

175
00:10:57,510 --> 00:11:01,490
just to get an idea of just how good an
approximation this polynomial really is.

176
00:11:01,490 --> 00:11:04,320
Well, here's a graph of the function y
equals sin of x.

177
00:11:04,320 --> 00:11:06,810
It has that familiar sinasoidal shape.

178
00:11:06,810 --> 00:11:08,954
Let me super impose on this, the graph

179
00:11:08,954 --> 00:11:12,900
of the cubic polynomial that we've been
studying.

180
00:11:12,900 --> 00:11:15,010
Here's that that cubic polynomial.

181
00:11:15,010 --> 00:11:20,680
This thick, green curve is the graph of y
equals x minus x cubed over six.

182
00:11:20,680 --> 00:11:22,520
And remember how I got this polynomial.

183
00:11:22,520 --> 00:11:25,940
This polynomial is rigged so that the
value of this polynomial

184
00:11:25,940 --> 00:11:28,570
at zero is the same as the value of sine
at zero.

185
00:11:28,570 --> 00:11:31,040
The derivative of this polynomial at zero
is

186
00:11:31,040 --> 00:11:33,370
the same as the derivative of sine at
zero.

187
00:11:33,370 --> 00:11:35,840
Same goes for the second derivative and
the third derivative, right?

188
00:11:35,840 --> 00:11:37,910
If I differentiate this thing three times,
I get

189
00:11:37,910 --> 00:11:41,190
the same thing as this, differentiated
three times at zero.

190
00:11:43,160 --> 00:11:47,160
But it's interesting to note that even
though I've rigged this polynomial only to

191
00:11:47,160 --> 00:11:52,040
match up with sine at zero at least in
value for second or third derivative.

192
00:11:53,290 --> 00:11:56,120
This graph is sort of resembling sine.

193
00:11:56,120 --> 00:12:00,490
I mean, this green curve, especially
around here,

194
00:12:00,490 --> 00:12:03,260
is a pretty good approximation to the
graph

195
00:12:03,260 --> 00:12:04,080
of the sine function.

196
00:12:04,080 --> 00:12:06,010
I mean, it's obviously not great out here,

197
00:12:06,010 --> 00:12:08,760
but at least in here, it's looking pretty
good.

198
00:12:08,760 --> 00:12:11,210
We can also look at this numerically.

199
00:12:11,210 --> 00:12:17,830
For example.
Sine of a half a radion is approximately,

200
00:12:17,830 --> 00:12:24,550
to four decimal places, 0.4794, but what
is

201
00:12:24,550 --> 00:12:28,730
this approximating function at one half?
What's g

202
00:12:28,730 --> 00:12:30,340
of one half?

203
00:12:30,340 --> 00:12:35,700
Well, that's one half minus one half cubed
over 6.

204
00:12:35,700 --> 00:12:38,600
Well that's one half minus what is this?

205
00:12:38,600 --> 00:12:42,290
That's one half to the third, that's an
eighth

206
00:12:42,290 --> 00:12:46,530
or a sixth, that's one half minus a
fourty-eighth.

207
00:12:46,530 --> 00:12:52,196
So instead of writing a half, I could
write 24/48ths Minus a forty-eight.

208
00:12:52,196 --> 00:12:54,692
Well that's 23 forty-eights.

209
00:12:54,692 --> 00:13:00,448
Well what's 23 forty-eights?
That's approximately

210
00:13:00,448 --> 00:13:06,680
0.4792.
That is awfully close to sine of one half.

211
00:13:06,680 --> 00:13:08,430
And whether we're thinking graphically or

212
00:13:08,430 --> 00:13:11,840
numerically, we're seeing something
significant here.

213
00:13:11,840 --> 00:13:16,440
Well way back in calculus 1, we thought
about approximating a function like this.

214
00:13:16,440 --> 00:13:20,150
These linear approximations, but now we
getting the idea

215
00:13:20,150 --> 00:13:21,728
that it'd be even better.

216
00:13:21,728 --> 00:13:24,550
To approximate f not just by a degree

217
00:13:24,550 --> 00:13:27,860
1 polynomial, but by a higher degree
polynomial.

218
00:13:27,860 --> 00:13:31,540
And even better than polynomials are power
series.

219
00:13:31,540 --> 00:13:32,140
Well, exactly.

220
00:13:32,140 --> 00:13:36,368
A power series is a lot like a polynomial
that just keeps on going.

221
00:13:36,368 --> 00:13:40,208
So if a degree ten polynomial is doing
better job than just

222
00:13:40,208 --> 00:13:45,248
a linear approximation, and if a degree of
thousand polynomial would do an

223
00:13:45,248 --> 00:13:50,688
even better job, maybe the power series,
if we just kept on going forever, we do

224
00:13:50,688 --> 00:13:56,129
such a good job that the function would
actually be equal to some power series.

225
00:13:56,129 --> 00:13:58,343
That is a huge thing to hope for.

226
00:13:58,343 --> 00:14:00,979
But sometimes dreams come true.

227
00:14:00,979 --> 00:14:10,979
[SOUND]