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Professor: In today's lecture I
want to develop several more

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00:00:26,45 --> 00:00:29,77
formulas that will allow
us to reach our goal of

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00:00:29,77 --> 00:00:32,41
differentiating everything.

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00:00:32,41 --> 00:00:42,39
So these are derivative
formulas, and they

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00:00:42,39 --> 00:00:45,56
come in two flavors.

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00:00:45,56 --> 00:00:53,88
The first kind is specific, so
some specific function we're

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00:00:53,88 --> 00:00:55,46
giving the derivative of.

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00:00:55,46 --> 00:01:00,74
And that would be, for
example, x^n or (1/x) .

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00:01:00,74 --> 00:01:05,22
Those are the ones that we did
a couple of lectures ago.

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00:01:05,22 --> 00:01:11,72
And then there are general
formulas, and the general ones

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00:01:11,72 --> 00:01:14,82
don't actually give you a
formula for a specific function

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00:01:14,82 --> 00:01:19,08
but tell you something like, if
you take two functions and add

20
00:01:19,08 --> 00:01:23,47
them together, their derivative
is the sum of the derivatives.

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00:01:23,47 --> 00:01:30,95
Or if you multiply by a
constant, for example, so (cu),

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00:01:30,95 --> 00:01:38,99
the derivative of that is
(cu)' where c is constant.

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00:01:38,99 --> 00:01:43,42
All right, so these kinds of
formulas are very useful,

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00:01:43,42 --> 00:01:46,07
both the specific and
the general kind.

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00:01:46,07 --> 00:02:03,63
For example, we need both
kinds for polynomials.

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00:02:03,63 --> 00:02:06,17
And more generally, pretty much
any set of forumulas that we

27
00:02:06,17 --> 00:02:08,97
give you, will give you a few
functions to start out with and

28
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then you'll be able to generate
lots more by these

29
00:02:11,48 --> 00:02:16,48
general formulas.

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00:02:16,48 --> 00:02:21,22
So today, we wanna concentrate
on the trig functions, and

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00:02:21,22 --> 00:02:27,57
so we'll start out with
some specific formulas.

32
00:02:27,57 --> 00:02:30,24
And they're going to be the
formulas for the derivative

33
00:02:30,24 --> 00:02:37,91
of the sine function and
the cosine function.

34
00:02:37,91 --> 00:02:41,24
So that's what we'll spend the
first part of the lecture on,

35
00:02:41,24 --> 00:02:46,51
and at the same time I hope to
get you very used to dealing

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00:02:46,51 --> 00:02:49,41
with trig functions, although
that's something that you

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00:02:49,41 --> 00:02:55,63
should think of as
a gradual process.

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00:02:55,63 --> 00:02:59,06
Alright, so in order to
calculate these, I'm gonna

39
00:02:59,06 --> 00:03:03,72
start over here and just
start the calculation.

40
00:03:03,72 --> 00:03:05,27
So here we go.

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00:03:05,27 --> 00:03:08,11
Let's check what happens
with the sine function.

42
00:03:08,11 --> 00:03:11,07
So, I take sin (x

43
00:03:11,07 --> 00:03:22,09
delta x), I subtract sin x
and I divide by delta x.

44
00:03:22,09 --> 00:03:24,47
Right, so this is the
difference quotient and

45
00:03:24,47 --> 00:03:26,635
eventually I'm gonna have
to take the limit as

46
00:03:26,635 --> 00:03:29,32
delta x goes to 0.

47
00:03:29,32 --> 00:03:35,07
And there's really only one
thing we can do with this to

48
00:03:35,07 --> 00:03:40,22
simplify or change it, and that
is to use the sum formula

49
00:03:40,22 --> 00:03:42,44
for the sine function.

50
00:03:42,44 --> 00:03:43,56
So, that's this.

51
00:03:43,56 --> 00:03:54,49
That's sin x co delta x plus

52
00:03:54,49 --> 00:03:56,07
Oh, that's not what it is?

53
00:03:56,07 --> 00:03:57,94
OK, so what is it?

54
00:03:57,94 --> 00:04:01,04
Sin x sin delta x.

55
00:04:01,04 --> 00:04:02,94
OK, good.

56
00:04:02,94 --> 00:04:07,5
Plus cosine.

57
00:04:07,5 --> 00:04:09,29
No?

58
00:04:09,29 --> 00:04:10,56
Oh, OK.

59
00:04:10,56 --> 00:04:14,47
So which is it?

60
00:04:14,47 --> 00:04:16,22
OK.

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00:04:16,22 --> 00:04:17,3
Alright, let's take a vote.

62
00:04:17,3 --> 00:04:20,02
Is it sine, sine, or
is it sine, cosine?

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Audience: [INAUDIBLE]

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00:04:22,58 --> 00:04:31,1
Professor: OK, so is this
going to be... cosine.

65
00:04:31,1 --> 00:04:34,45
All right, you better remember
these formulas, alright?

66
00:04:34,45 --> 00:04:37,52
OK, turns out that
it's sine, cosine.

67
00:04:37,52 --> 00:04:37,79
All right.

68
00:04:37,79 --> 00:04:39,64
Cosine, sine.

69
00:04:39,64 --> 00:04:47,31
So here we go, no gotta do
x here, sin (delta x).

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00:04:47,31 --> 00:04:51,1
Alright, so now there's lots of
places to get confused here,

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00:04:51,1 --> 00:04:55,16
and you're gonna need to
make sure you get it right.

72
00:04:55,16 --> 00:04:59,23
Alright, so we're gonna put
those in parentheses here.

73
00:04:59,23 --> 00:05:06,95
Sin (a + b) is sin a (cos b)

74
00:05:06,95 --> 00:05:10,31
cos a (sin b).

75
00:05:10,31 --> 00:05:13,95
All right, now that's what I
did over here, except the

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00:05:13,95 --> 00:05:21,43
letter x was a, and the
letter b was delta x.

77
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Now that's just the first part.

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That's just this part
of the expression.

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I still have to
remember the - sin x.

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That comes at the end.

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00:05:30,05 --> 00:05:32,12
Minus sin x.

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00:05:32,12 --> 00:05:34,7
And then, I have to
remember the denominator,

83
00:05:34,7 --> 00:05:37,9
which is delta x.

84
00:05:37,9 --> 00:05:43,04
OK?

85
00:05:43,04 --> 00:05:47,25
Alright, so now...

86
00:05:47,25 --> 00:05:50,11
The next thing we're gonna
do is we're gonna try

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to group the terms.

88
00:05:52,28 --> 00:05:57,95
And the difficulty with all
such arguments is the following

89
00:05:57,95 --> 00:06:02,15
one: any tricky limit is
basically 0 / 0 when you

90
00:06:02,15 --> 00:06:03,2
set delta x equal to 0.

91
00:06:03,2 --> 00:06:06,48
If I set delta x equal to
0, this is sin x - sin x.

92
00:06:06,48 --> 00:06:08,52
So it's a 0 / 0 term.

93
00:06:08,52 --> 00:06:10,85
Here we have various things
which are 0 and various

94
00:06:10,85 --> 00:06:12,17
things which are non-zero.

95
00:06:12,17 --> 00:06:17,88
We must group the terms so
that a 0 stays over a 0.

96
00:06:17,88 --> 00:06:19,22
Otherwise, we're
gonna have no hope.

97
00:06:19,22 --> 00:06:21,99
If we get some 1 / 0 term,
we'll get something

98
00:06:21,99 --> 00:06:24,16
meaningless in the limit.

99
00:06:24,16 --> 00:06:27,86
So I claim that the right thing
to do here is to notice, and

100
00:06:27,86 --> 00:06:31,63
I'll just point out
this one thing.

101
00:06:31,63 --> 00:06:35,58
When delta x goes to 0,
this cosine of 0 is 1.

102
00:06:35,58 --> 00:06:38,86
So it doesn't cancel
unless we throw in this

103
00:06:38,86 --> 00:06:40,13
extra sine term here.

104
00:06:40,13 --> 00:06:43,77
So I'm going to use this
common factor, and

105
00:06:43,77 --> 00:06:44,67
combine those terms.

106
00:06:44,67 --> 00:06:46,9
So this is really the only
thing you're gonna have to

107
00:06:46,9 --> 00:06:48,8
check in this particular
calculation.

108
00:06:48,8 --> 00:06:52,92
So we have the common factor of
sin x, and that multiplies

109
00:06:52,92 --> 00:06:55,995
something that will cancel,
which is (cos delta

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00:06:55,995 --> 00:06:59,17
x - 1) / delta x.

111
00:06:59,17 --> 00:07:04,61
That's the first term, and now
what's left, well there's a cos

112
00:07:04,61 --> 00:07:08,36
x that factors out, and then
the other factor is (sin

113
00:07:08,36 --> 00:07:14,04
delta x) / (delta x).

114
00:07:14,04 --> 00:07:20,85
OK, now does anyone remember
from last time what

115
00:07:20,85 --> 00:07:25
this thing goes to?

116
00:07:25 --> 00:07:27,58
How many people say 1?

117
00:07:27,58 --> 00:07:29,34
How many people say 0?

118
00:07:29,34 --> 00:07:31,18
All right, it's 0.

119
00:07:31,18 --> 00:07:33,6
That's my favorite
number, alright?

120
00:07:33,6 --> 00:07:34,04
0.

121
00:07:34,04 --> 00:07:36,18
It's the easiest
number to deal with.

122
00:07:36,18 --> 00:07:45,98
So this goes 0, and that's what
happens as delta x tends to 0.

123
00:07:45,98 --> 00:07:47,15
How about this one?

124
00:07:47,15 --> 00:07:51,75
This one goes to 1, my second
favorite number, almost as

125
00:07:51,75 --> 00:07:54,35
easy to deal with as 0.

126
00:07:54,35 --> 00:07:56,75
And these things are
picked for a reason.

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00:07:56,75 --> 00:07:58,03
They're the simplest
numbers to deal with.

128
00:07:58,03 --> 00:08:06,82
So altogether, this thing as
delta x goes to 0 goes to what?

129
00:08:06,82 --> 00:08:09,44
I want a single person to
answer, a brave volunteer.

130
00:08:09,44 --> 00:08:10,33
Alright, back there.

131
00:08:10,33 --> 00:08:12,07
Student: Cosine

132
00:08:12,07 --> 00:08:14,65
Professor: Cosine, because
this factor is 0.

133
00:08:14,65 --> 00:08:17,56
It cancels and this factor
has a 1, so it's cosine.

134
00:08:17,56 --> 00:08:20,23
So it's cos x.

135
00:08:20,23 --> 00:08:25,84
So our conclusion over here -
and I'll put it in orange -

136
00:08:25,84 --> 00:08:34,92
is that the derivative of
the sine is the cosine.

137
00:08:34,92 --> 00:08:37,84
OK, now I still wanna label
these very important

138
00:08:37,84 --> 00:08:39,22
limit facts here.

139
00:08:39,22 --> 00:08:43,03
This one we'll call A, and this
one we're going to call B,

140
00:08:43,03 --> 00:08:44,34
because we haven't
checked them yet.

141
00:08:44,34 --> 00:08:46,52
I promised you I would
do that, and I'll have

142
00:08:46,52 --> 00:08:48,46
to do that this time.

143
00:08:48,46 --> 00:08:52,49
So we're relying on those
things being true.

144
00:08:52,49 --> 00:08:56,14
Now I'm gonna do the same thing
with the cosine function,

145
00:08:56,14 --> 00:08:58,24
except in order to do it
I'm gonna have to remember

146
00:08:58,24 --> 00:09:00,93
the sum rule for cosine.

147
00:09:00,93 --> 00:09:03,62
So we're gonna do almost
the same calculation here.

148
00:09:03,62 --> 00:09:05,99
We're gonna see that that will
work out, but now you have to

149
00:09:05,99 --> 00:09:14,23
remember that cos (a + b) = cos
cos, no it's not cosine^2,

150
00:09:14,23 --> 00:09:16,76
because there are two
different quantities here.

151
00:09:16,76 --> 00:09:23,65
It's (cos a cos b)
- (sin a sin b).

152
00:09:23,65 --> 00:09:31,28
All right, so you'll have to
be willing to call those

153
00:09:31,28 --> 00:09:34,8
forth at will right now.

154
00:09:34,8 --> 00:09:36,46
So let's do the cosine now.

155
00:09:36,46 --> 00:09:45,5
So that's cosine ((x + delta
x) - cos x) / delta x.

156
00:09:45,5 --> 00:09:48,11
OK, there's the difference
quotient for the

157
00:09:48,11 --> 00:09:49,55
cosine function.

158
00:09:49,55 --> 00:09:51,776
And now I'm gonna do the same
thing I did before except I'm

159
00:09:51,776 --> 00:09:54,05
going to apply the second
rule, that is the

160
00:09:54,05 --> 00:09:55,77
sum rule for cosine.

161
00:09:55,77 --> 00:10:00,38
And that's gonna give me
(cos x cos delta x) -

162
00:10:00,38 --> 00:10:03,9
(sin x sin delta x).

163
00:10:03,9 --> 00:10:08,94
And I have to remember again
to subtract the cosine

164
00:10:08,94 --> 00:10:11,59
divided by this delta x.

165
00:10:11,59 --> 00:10:16,93
And now I'm going to regroup
just the way I did before, and

166
00:10:16,93 --> 00:10:20,86
I get the common factor of
cosine multiplying ((cosine

167
00:10:20,86 --> 00:10:25,18
delta x - 1) / delta x).

168
00:10:25,18 --> 00:10:30,91
And here I get the sin x
but actually it's - sin x.

169
00:10:30,91 --> 00:10:36,32
And then I have (sin
delta x) / delta x.

170
00:10:36,32 --> 00:10:36,9
All right?

171
00:10:36,9 --> 00:10:38,79
The only difference is
this minus sign which

172
00:10:38,79 --> 00:10:43,09
I stuck inside there.

173
00:10:43,09 --> 00:10:45,546
Well that's not the only
difference, but it's

174
00:10:45,546 --> 00:10:48,44
a crucial difference.

175
00:10:48,44 --> 00:10:54,7
OK, again by A we get that this
is 0 as delta x tends to 0.

176
00:10:54,7 --> 00:10:56,37
And this is 1.

177
00:10:56,37 --> 00:10:59,59
Those are the properties
I called A and B.

178
00:10:59,59 --> 00:11:07,33
And so the result here as
delta x tends to 0 is that

179
00:11:07,33 --> 00:11:08,47
we get negative sine x.

180
00:11:08,47 --> 00:11:11,8
That's the factor.

181
00:11:11,8 --> 00:11:18,88
So this guy is negative sine x.

182
00:11:18,88 --> 00:11:24,56
I'll put a little box
around that too.

183
00:11:24,56 --> 00:11:29,44
Alright, now these formulas
take a little bit of getting

184
00:11:29,44 --> 00:11:36,04
used to, but before I do that
I'm gonna explain to you

185
00:11:36,04 --> 00:11:38,48
the proofs of A and B.

186
00:11:38,48 --> 00:11:44,47
So we'll get ourselves
started by mentioning that.

187
00:11:44,47 --> 00:11:49,23
Maybe before I do that though,
I want to show you how A and B

188
00:11:49,23 --> 00:11:51,71
fit into the proofs
of these theorems.

189
00:11:51,71 --> 00:12:06,03
So, let me just make
some remarks here.

190
00:12:06,03 --> 00:12:12,68
So this is just a remark but
it's meant to help you to frame

191
00:12:12,68 --> 00:12:15,6
how these proofs worked.

192
00:12:15,6 --> 00:12:18,48
So, first of all, I want to
point out that if you take the

193
00:12:18,48 --> 00:12:28,83
rate of change of sin x, no
let's start with cosine because

194
00:12:28,83 --> 00:12:30,45
a little bit less obvious.

195
00:12:30,45 --> 00:12:34
If I take the rate of change of
cos x, so in other words this

196
00:12:34 --> 00:12:41,92
derivative at x = 0, then by
definition this is a certain

197
00:12:41,92 --> 00:12:45,3
limit as delta x goes to 0.

198
00:12:45,3 --> 00:12:46,86
So which one is it?

199
00:12:46,86 --> 00:12:51,08
Well I have to
evaluate cosine at 0

200
00:12:51,08 --> 00:12:53,39
delta x, but that's
just delta x.

201
00:12:53,39 --> 00:12:56,24
And I have to subtract
cosine at 0.

202
00:12:56,24 --> 00:13:00,04
That's the base point,
but that's just 1.

203
00:13:00,04 --> 00:13:03,05
And then I have to
divide by delta x.

204
00:13:03,05 --> 00:13:06,85
And lo and behold you can see
that this is exactly the limit

205
00:13:06,85 --> 00:13:08,22
that we had over there.

206
00:13:08,22 --> 00:13:15,88
This is the one that we know is
0 by what we call property A.

207
00:13:15,88 --> 00:13:23,15
And similarly, if I take the
derivative of (sin x) at x= 0,

208
00:13:23,15 --> 00:13:27
then that's going to be the
limit as delta x goes to 0

209
00:13:27 --> 00:13:30,7
of sine delta x / delta x.

210
00:13:30,7 --> 00:13:34,74
And that's because I should
be subtracting sine of

211
00:13:34,74 --> 00:13:38,03
0 but sine of 0 is 0.

212
00:13:38,03 --> 00:13:38,37
Right?

213
00:13:38,37 --> 00:13:46,37
So this is going to be
1 by our property B.

214
00:13:46,37 --> 00:13:51,03
And so the remark that I want
to make, in addition to this,

215
00:13:51,03 --> 00:13:55,2
is something about the
structure of these two proofs.

216
00:13:55,2 --> 00:14:12,99
Which is the derivatives of
sine and cosine at x = 0

217
00:14:12,99 --> 00:14:25,5
give all values of d/dx
sin x, d/dx cos x.

218
00:14:25,5 --> 00:14:27,92
So that's really what this
argument is showing us, is that

219
00:14:27,92 --> 00:14:31,93
we just need one rate of change
at one place and then we work

220
00:14:31,93 --> 00:14:38,77
out all the rest of them.

221
00:14:38,77 --> 00:14:40,99
So that's really the
substance of this proof.

222
00:14:40,99 --> 00:14:43,67
That of course really then
shows that it boils down to

223
00:14:43,67 --> 00:14:48,02
showing what this rate of
change is in these two cases.

224
00:14:48,02 --> 00:14:51,39
So now there's enough suspense
that we want to make sure

225
00:14:51,39 --> 00:15:08,01
that we know that those
answers are correct.

226
00:15:08,01 --> 00:15:12,18
OK, so let's demonstrate
both of them.

227
00:15:12,18 --> 00:15:16,12
I'll start with B.

228
00:15:16,12 --> 00:15:18,34
I need to figure
out property B.

229
00:15:18,34 --> 00:15:22,04
Now, we only have one
alternative as to a type of

230
00:15:22,04 --> 00:15:25,23
proof that we can give of this
kind of result, and that's

231
00:15:25,23 --> 00:15:30,44
because we only have one way of
describing sine and cosine

232
00:15:30,44 --> 00:15:32,54
functions, that is
geometrically.

233
00:15:32,54 --> 00:15:42,62
So we have to give
a geometric proof.

234
00:15:42,62 --> 00:15:45,533
And to write down a geometric
proof we are going to

235
00:15:45,533 --> 00:15:47,15
have to draw a picture.

236
00:15:47,15 --> 00:15:50,43
And the first step in the
proof, really, is to replace

237
00:15:50,43 --> 00:15:55,63
this variable delta x which is
going to 0 with another name

238
00:15:55,63 --> 00:15:57,7
which is suggestive of what
we're gonna do which is the

239
00:15:57,7 --> 00:16:00,95
letter theta for an angle.

240
00:16:00,95 --> 00:16:03,93
OK, so let's draw a picture
of what it is that

241
00:16:03,93 --> 00:16:05,95
we're going to do.

242
00:16:05,95 --> 00:16:07,98
Here is the circle.

243
00:16:07,98 --> 00:16:10,74
And here is the origin.

244
00:16:10,74 --> 00:16:14,38
And here's some little angle,
well I'll draw it a little

245
00:16:14,38 --> 00:16:16,11
larger so it's visible.

246
00:16:16,11 --> 00:16:19,43
Here's theta, alright?

247
00:16:19,43 --> 00:16:21,01
And this is the unit circle.

248
00:16:21,01 --> 00:16:25,91
I won't write that down on here
but that's the unit circle.

249
00:16:25,91 --> 00:16:29,78
And now sin theta is this
vertical distance here.

250
00:16:29,78 --> 00:16:32,945
Maybe, I'll draw it in a
different color so that

251
00:16:32,945 --> 00:16:34,75
we can see it all.

252
00:16:34,75 --> 00:16:38,4
OK so here's this distance.

253
00:16:38,4 --> 00:16:45,82
This distance is sin theta.

254
00:16:45,82 --> 00:16:48,36
OK?

255
00:16:48,36 --> 00:16:52,71
Now almost the only other thing
we have to write down in this

256
00:16:52,71 --> 00:16:56,55
picture to have it work out is
that we have to recognize that

257
00:16:56,55 --> 00:17:02,21
when theta is the angle, that's
also the arc length of this

258
00:17:02,21 --> 00:17:04,32
piece of the circle when
measured in radians.

259
00:17:04,32 --> 00:17:13,73
So this length here is
also arc length theta.

260
00:17:13,73 --> 00:17:14,82
That little piece in there.

261
00:17:14,82 --> 00:17:18,58
So maybe I'll use a different
color for that to indicate it.

262
00:17:18,58 --> 00:17:25,56
So that's orange and that's
this little chunk there.

263
00:17:25,56 --> 00:17:26,87
So those are the two pieces.

264
00:17:26,87 --> 00:17:36,52
Now in order to persuade you
now that the limit is what it's

265
00:17:36,52 --> 00:17:38,96
supposed to be, I'm going to
extend the picture

266
00:17:38,96 --> 00:17:39,63
just a little bit.

267
00:17:39,63 --> 00:17:42,69
I'm going to double it, just
for my own linguistic sake and

268
00:17:42,69 --> 00:17:44,24
so that I can tell you a story.

269
00:17:44,24 --> 00:17:46,69
Alright, so that
you'll remember this.

270
00:17:46,69 --> 00:17:50
So I'm going to take a theta
angle below and I'll have

271
00:17:50 --> 00:17:53,67
another copy of sin
theta down here.

272
00:17:53,67 --> 00:18:03,1
And now the total picture
is really like a bow and

273
00:18:03,1 --> 00:18:04,88
its bow string there.

274
00:18:04,88 --> 00:18:05,22
Alright?

275
00:18:05,22 --> 00:18:11,05
So what we have here is a
length of 2 sin theta.

276
00:18:11,05 --> 00:18:13,63
So maybe I'll write it
this way, 2 sin theta.

277
00:18:13,63 --> 00:18:15,12
I just doubled it.

278
00:18:15,12 --> 00:18:25,64
And here I have underneath,
whoops, I got it backwards.

279
00:18:25,64 --> 00:18:27,04
Sorry about that.

280
00:18:27,04 --> 00:18:29,39
Trying to be fancy with
the colored chalk and I

281
00:18:29,39 --> 00:18:30,31
have it reversed here.

282
00:18:30,31 --> 00:18:32,18
So this is not 2 sin theta.

283
00:18:32,18 --> 00:18:33,54
2 sin theta is the vertical.

284
00:18:33,54 --> 00:18:34,91
That's the green.

285
00:18:34,91 --> 00:18:37,17
So let's try that again.

286
00:18:37,17 --> 00:18:41,19
This is 2 sin theta, alright?

287
00:18:41,19 --> 00:18:44,93
And then in the denominator I
have the arc length which is

288
00:18:44,93 --> 00:18:50,68
theta is the first half and
so double it is 2 theta.

289
00:18:50,68 --> 00:18:51,31
Alright?

290
00:18:51,31 --> 00:18:56,63
So if you like, this is
the bow and up here we

291
00:18:56,63 --> 00:19:04,29
have the bow string.

292
00:19:04,29 --> 00:19:07,74
And of course we can
cancel the 2's.

293
00:19:07,74 --> 00:19:11,25
That's equal to sin
theta / theta.

294
00:19:11,25 --> 00:19:17,9
And so now why does this tend
to 1 as theta goes to 0?

295
00:19:17,9 --> 00:19:24,1
Well, it's because as the angle
theta gets very small, this

296
00:19:24,1 --> 00:19:28,88
curved piece looks more and
more like a straight one.

297
00:19:28,88 --> 00:19:29,64
Alright?

298
00:19:29,64 --> 00:19:32,81
And if you get very, very close
here the green segment and

299
00:19:32,81 --> 00:19:34,61
the orange segment
would just merge.

300
00:19:34,61 --> 00:19:36,85
They would be practically
on top of each other.

301
00:19:36,85 --> 00:19:42,36
And they have closer and closer
and closer to the same length.

302
00:19:42,36 --> 00:19:51,79
So that's why this is true.

303
00:19:51,79 --> 00:20:02,32
I guess I'll articulate that
by saying that short curves

304
00:20:02,32 --> 00:20:06,55
are nearly straight.

305
00:20:06,55 --> 00:20:10
Alright, so that's the
principle that we're using.

306
00:20:10 --> 00:20:18,97
Or short pieces of curves, if
you like, are nearly straight.

307
00:20:18,97 --> 00:20:23,64
So if you like, this
is the principle.

308
00:20:23,64 --> 00:20:30,85
So short pieces of curves.

309
00:20:30,85 --> 00:20:31,54
Alright?

310
00:20:31,54 --> 00:20:39,39
So now I also need to
give you a proof of A.

311
00:20:39,39 --> 00:20:43,99
And that has to do with
this cosine function here.

312
00:20:43,99 --> 00:20:49,84
This is the property A.

313
00:20:49,84 --> 00:20:53,98
So I'm going to do this by
flipping it around, because it

314
00:20:53,98 --> 00:20:56,07
turns out that this numerator
is a negative number.

315
00:20:56,07 --> 00:20:58,655
If I want to interpret it as
a length, I'm gonna want

316
00:20:58,655 --> 00:21:00,34
a positive quantity.

317
00:21:00,34 --> 00:21:04,61
So I'm gonna write down (1 -
cos theta) here and then I'm

318
00:21:04,61 --> 00:21:08,32
gonna divide by theta there.

319
00:21:08,32 --> 00:21:10,72
Again I'm gonna make some
kind of interpretation.

320
00:21:10,72 --> 00:21:15,13
Now this time I'm going to draw
the same sort of bow and arrow

321
00:21:15,13 --> 00:21:18,97
arrangement, but maybe I'll
exaggerate it a little bit.

322
00:21:18,97 --> 00:21:23,8
So here's the vertex of the
sector, but we'll maybe

323
00:21:23,8 --> 00:21:31,78
make it a little longer.

324
00:21:31,78 --> 00:21:35,59
Alright, so here it is, and
here was that middle line

325
00:21:35,59 --> 00:21:36,82
which was the unit...

326
00:21:36,82 --> 00:21:38,31
Whoops.

327
00:21:38,31 --> 00:21:40,88
OK, I think I'm going
to have to tilt it up.

328
00:21:40,88 --> 00:21:47
OK, let's try from here.

329
00:21:47 --> 00:21:51,85
Alright, well you know on your
pencil and paper it will look

330
00:21:51,85 --> 00:21:53,81
better than it does
on my blackboard.

331
00:21:53,81 --> 00:21:55,19
OK, so here we are.

332
00:21:55,19 --> 00:21:56,77
Here's this shape.

333
00:21:56,77 --> 00:22:01,66
Now this angle is supposed
to be theta and this

334
00:22:01,66 --> 00:22:03,35
angle is another theta.

335
00:22:03,35 --> 00:22:06,3
So here we have a length
which is again theta and

336
00:22:06,3 --> 00:22:07,91
another length which
is theta over here.

337
00:22:07,91 --> 00:22:11,4
That's the same as in the
other picture, except we've

338
00:22:11,4 --> 00:22:13,29
exaggerated a bit here.

339
00:22:13,29 --> 00:22:15,66
And now we have this vertical
line, which again I'm gonna

340
00:22:15,66 --> 00:22:18,24
draw in green, the bow string.

341
00:22:18,24 --> 00:22:24,82
But notice that as the vertex
gets farther and farther away,

342
00:22:24,82 --> 00:22:27,3
the curved line gets closer
and closer to being

343
00:22:27,3 --> 00:22:28,06
a vertical line.

344
00:22:28,06 --> 00:22:31,55
That's sort of the flip
side, by expansion, of

345
00:22:31,55 --> 00:22:33,11
the zoom in principle.

346
00:22:33,11 --> 00:22:35,49
The principle that curves
are nearly straight

347
00:22:35,49 --> 00:22:36,95
when you zoom in.

348
00:22:36,95 --> 00:22:39,2
If you zoom out that would
mean sending this vertex

349
00:22:39,2 --> 00:22:42,2
way, way out somewhere.

350
00:22:42,2 --> 00:22:45,51
The curved line, the piece
of the circle, gets

351
00:22:45,51 --> 00:22:48,57
more and more straight.

352
00:22:48,57 --> 00:22:53,63
And now let me show you where
this numerator (1 - cos

353
00:22:53,63 --> 00:22:57,62
theta) is on this picture.

354
00:22:57,62 --> 00:23:01,04
So where is it?

355
00:23:01,04 --> 00:23:03,68
Well, this whole distance is 1.

356
00:23:03,68 --> 00:23:07,61
But the distance from
the vertex to the green

357
00:23:07,61 --> 00:23:09,87
is cosine of theta.

358
00:23:09,87 --> 00:23:12,65
Right, because this is theta,
so dropping down the

359
00:23:12,65 --> 00:23:16,83
perpendicular this distance
back to the origin

360
00:23:16,83 --> 00:23:17,63
is cos theta.

361
00:23:17,63 --> 00:23:23,23
So this little tiny, bitty
segment here is basically the

362
00:23:23,23 --> 00:23:28,82
gap between the curve and
the vertical segment.

363
00:23:28,82 --> 00:23:35,82
So the gap = 1 - cos theta.

364
00:23:35,82 --> 00:23:41,47
So now you can see that as this
point gets farther away, if

365
00:23:41,47 --> 00:23:44,44
this got sent off to the Stata
Center, you would hardly be

366
00:23:44,44 --> 00:23:45,84
able to tell the difference.

367
00:23:45,84 --> 00:23:48,77
The bow string would coincide
with the bow and this little

368
00:23:48,77 --> 00:23:52,43
gap between the bow string and
the bow would be tending to 0.

369
00:23:52,43 --> 00:23:55,878
And that's the statement
that this tends to 0

370
00:23:55,878 --> 00:23:58,54
as theta tends to 0.

371
00:23:58,54 --> 00:24:00,13
The scaled version of that.

372
00:24:00,13 --> 00:24:01,25
Yeah, question down here.

373
00:24:01,25 --> 00:24:01,421
Student: Doesn't the
denominator also

374
00:24:01,421 --> 00:24:04,8
tend to 0 though?

375
00:24:04,8 --> 00:24:10,19
Professor: Ah, the question is
"doesn't the denominator also

376
00:24:10,19 --> 00:24:14,51
tend to 0?" And the
answer is yes.

377
00:24:14,51 --> 00:24:19,2
In my strange analogy with
zooming in, what I did was

378
00:24:19,2 --> 00:24:20,39
I zoomed out the picture.

379
00:24:20,39 --> 00:24:25,55
So in other words, if you
imagine you're taking this and

380
00:24:25,55 --> 00:24:28,78
you're putting it under a
microscope over here and you're

381
00:24:28,78 --> 00:24:31
looking at something where
theta is getting smaller and

382
00:24:31 --> 00:24:33,34
smaller and smaller
and smaller.

383
00:24:33,34 --> 00:24:34,37
Alright?

384
00:24:34,37 --> 00:24:39,5
But now because I want my
picture, I expanded my picture.

385
00:24:39,5 --> 00:24:42,96
So the ratio is the
thing that's preserved.

386
00:24:42,96 --> 00:24:50,47
So if I make it so that
this gap is tiny...

387
00:24:50,47 --> 00:24:52,41
Let me say this one more time.

388
00:24:52,41 --> 00:24:57,03
I'm afraid I've made life
complicated for myself.

389
00:24:57,03 --> 00:25:02,68
If I simply let this theta tend
in to 0, that would be the same

390
00:25:02,68 --> 00:25:05,2
effect as making this closer
and closer in and then the

391
00:25:05,2 --> 00:25:06,46
vertical would approach.

392
00:25:06,46 --> 00:25:09,58
But I want to keep on blowing
up the picture so that I can

393
00:25:09,58 --> 00:25:13,26
see the difference between
the vertical and the curve.

394
00:25:13,26 --> 00:25:16,79
So that's very much like if you
are on a video screen and you

395
00:25:16,79 --> 00:25:18,74
zoom in, zoom in, zoom
in, and zoom in.

396
00:25:18,74 --> 00:25:20,24
So the question is what
would that look like?

397
00:25:20,24 --> 00:25:23,35
That has the same effect
as sending this point out

398
00:25:23,35 --> 00:25:27,84
farther and farther in that
direction, to the left.

399
00:25:27,84 --> 00:25:31,22
And so I'm just trying to
visualize it for you by leaving

400
00:25:31,22 --> 00:25:33,59
the theta at this scale, but
actually the scale of the

401
00:25:33,59 --> 00:25:36,29
picture is then changing
when I do that.

402
00:25:36,29 --> 00:25:40,8
So theta is going to 0, but I
I'm rescaling so that it's of a

403
00:25:40,8 --> 00:25:44,25
size that we can look at it,
And then imagine what's

404
00:25:44,25 --> 00:25:46,37
happening to it.

405
00:25:46,37 --> 00:25:47,75
OK, does that answer
your question?

406
00:25:47,75 --> 00:25:47,947
Student: My question then is
that seems to prove that

407
00:25:47,947 --> 00:25:54,25
that limit is equal to 0/0.

408
00:25:54,25 --> 00:26:01,51
Professor: It proves more
than it is equal to 0 / 0.

409
00:26:01,51 --> 00:26:03,53
It's the ratio of this
little short thing to

410
00:26:03,53 --> 00:26:06,09
this longer thing.

411
00:26:06,09 --> 00:26:08,92
And this is getting much, much
shorter than this total length.

412
00:26:08,92 --> 00:26:11,05
You're absolutely right that
we're comparing two quantities

413
00:26:11,05 --> 00:26:13,325
which are going to 0, but one
of them is much smaller

414
00:26:13,325 --> 00:26:14,45
than the other.

415
00:26:14,45 --> 00:26:16,73
In the other case we compared
two quantities which were both

416
00:26:16,73 --> 00:26:19,84
going to 0 and they both end up
being about equal in length.

417
00:26:19,84 --> 00:26:23,06
Here the previous one
was this green one.

418
00:26:23,06 --> 00:26:26,93
Here it's this little tiny bit
here and it's way shorter

419
00:26:26,93 --> 00:26:32,84
than the 2 theta distance.

420
00:26:32,84 --> 00:26:33,9
Yeah, another question.

421
00:26:33,9 --> 00:26:34,117
Student: (Cos theta -1) /
(cos theta) is the same as

422
00:26:34,117 --> 00:26:35,02
(1- cos theta) / theta?

423
00:26:35,02 --> 00:26:45,74
Professor: Cos
theta - 1 over...

424
00:26:45,74 --> 00:26:49,78
Student: [INAUDIBLE]

425
00:26:49,78 --> 00:26:56,79
Professor: So here, what I
wrote is (cos delta x - 1) /

426
00:26:56,79 --> 00:27:01,62
delta x, OK, and I claimed
that it goes to 0.

427
00:27:01,62 --> 00:27:12,37
Here, I wrote minus that, that
is I replaced delta x by theta.

428
00:27:12,37 --> 00:27:22,72
But then I wrote this thing.

429
00:27:22,72 --> 00:27:26,55
So (cos theta - 1) - 1 is
the negative of this.

430
00:27:26,55 --> 00:27:28,07
Alright?

431
00:27:28,07 --> 00:27:30,32
And if I show that this goes to
0, it's the same as showing

432
00:27:30,32 --> 00:27:33,2
the other one goes to 0.

433
00:27:33,2 --> 00:27:33,87
Another question?

434
00:27:33,87 --> 00:27:39,05
Student: [INAUDIBLE]

435
00:27:39,05 --> 00:27:42,32
Professor: So the question
is, what about this

436
00:27:42,32 --> 00:27:44,22
business about arc length.

437
00:27:44,22 --> 00:27:48,61
So the word arc length, that
orange shape is an arc.

438
00:27:48,61 --> 00:27:51,59
And we're just talking about
the length of that arc, and so

439
00:27:51,59 --> 00:27:53,22
we're calling it arc length.

440
00:27:53,22 --> 00:27:54,89
That's what the word are
length means, it just means

441
00:27:54,89 --> 00:27:55,17
the length of the arc.

442
00:27:55,17 --> 00:28:03,62
Student: [INAUDIBLE]

443
00:28:03,62 --> 00:28:06,03
Professor: Why is
this length theta?

444
00:28:06,03 --> 00:28:08,75
Ah, ok so this is a very
important point, and in fact

445
00:28:08,75 --> 00:28:11,48
it's the very next point
that I wanted to make.

446
00:28:11,48 --> 00:28:15,71
Namely, notice that in this
calculation it was very

447
00:28:15,71 --> 00:28:19,7
important that we used length.

448
00:28:19,7 --> 00:28:24,48
And that means that the way
that we're measuring theta, is

449
00:28:24,48 --> 00:28:32,33
in what are known as radians.

450
00:28:32,33 --> 00:28:36,7
Right, so that applies to both
B and A, it's a scale change in

451
00:28:36,7 --> 00:28:39,47
A and doesn't really matter
but in B it's very important.

452
00:28:39,47 --> 00:28:45,36
The only way that this orange
length is comparable to this

453
00:28:45,36 --> 00:28:49,99
green length, the vertical is
comparable to the arc, is if

454
00:28:49,99 --> 00:28:53,52
we measure them in terms of
the same notion of length.

455
00:28:53,52 --> 00:28:56,58
If we measure them in degrees,
for example, it would

456
00:28:56,58 --> 00:28:58,89
be completely wrong.

457
00:28:58,89 --> 00:29:03,72
We divide up the angles into
360 , and that's wrong

458
00:29:03,72 --> 00:29:04,32
unit of measure.

459
00:29:04,32 --> 00:29:07,99
The correct measures is the
length along the unit circle,

460
00:29:07,99 --> 00:29:09,59
which is what radians are.

461
00:29:09,59 --> 00:29:21,49
And so this is only true
if we use radians.

462
00:29:21,49 --> 00:29:33,65
So again, a little warning
here, that this is in radians.

463
00:29:33,65 --> 00:29:41,06
Now here x is in radians.

464
00:29:41,06 --> 00:29:45,69
The formulas are just wrong
if you use other units.

465
00:29:45,69 --> 00:29:46,25
Ah yeah?

466
00:29:46,25 --> 00:29:55,48
Student: [INAUDIBLE].

467
00:29:55,48 --> 00:29:57,6
Professor: OK so the second
question is why is this

468
00:29:57,6 --> 00:30:00,63
crazy length here 1.

469
00:30:00,63 --> 00:30:08,69
And the reason is that the
relationship between this

470
00:30:08,69 --> 00:30:12,71
picture up here and this
picture down here, is that I'm

471
00:30:12,71 --> 00:30:16,15
drawing a different shape.

472
00:30:16,15 --> 00:30:19,3
Namely, what I'm really
imagining here is a much,

473
00:30:19,3 --> 00:30:21,46
much smaller theta.

474
00:30:21,46 --> 00:30:22,34
OK?

475
00:30:22,34 --> 00:30:25,43
And then I'm blowing
that up in scale.

476
00:30:25,43 --> 00:30:28,87
So this scale of this picture
down here is very different

477
00:30:28,87 --> 00:30:31,43
from the scale of the
picture up there.

478
00:30:31,43 --> 00:30:36,41
And if the angle is very, very,
very small then one has to be

479
00:30:36,41 --> 00:30:39,43
very, very long in order for
me to finish the circle.

480
00:30:39,43 --> 00:30:42,79
So, in other words, this
length is 1 because that's

481
00:30:42,79 --> 00:30:44,69
what I'm insisting on.

482
00:30:44,69 --> 00:30:47,91
So, I'm claiming that that's
how I define this circle,

483
00:30:47,91 --> 00:30:52,49
to be of unit radius.

484
00:30:52,49 --> 00:30:53,2
Another question?

485
00:30:53,2 --> 00:31:04,24
Student: [INAUDIBLE]

486
00:31:04,24 --> 00:31:06,258
the ratio between 1 - theta
and theta and theta will

487
00:31:06,258 --> 00:31:07,36
get closer and closer to 1.

488
00:31:07,36 --> 00:31:08
I don't understand [INAUDIBLE].

489
00:31:08 --> 00:31:22,47
Professor: OK, so the question
is it's hard to visualize

490
00:31:22,47 --> 00:31:25,81
this fact here.

491
00:31:25,81 --> 00:31:30,9
So let me let me take you
through a couple of steps,

492
00:31:30,9 --> 00:31:33,37
because I think probably other
people are also having trouble

493
00:31:33,37 --> 00:31:34,93
with this visualization.

494
00:31:34,93 --> 00:31:36,96
The first part of the
visualization I'm gonna

495
00:31:36,96 --> 00:31:39,57
try to demonstrate on
this picture up here.

496
00:31:39,57 --> 00:31:42,05
The first part of the
visualization is that I should

497
00:31:42,05 --> 00:31:46,89
think of a beak of a bird
closing down, getting

498
00:31:46,89 --> 00:31:47,88
narrower and narrower.

499
00:31:47,88 --> 00:31:51,79
So in other words, the angle
theta has to be getting smaller

500
00:31:51,79 --> 00:31:54,05
and smaller and smaller.

501
00:31:54,05 --> 00:31:55,78
OK, that's the first step.

502
00:31:55,78 --> 00:31:58,65
So that's the process that
we're talking about.

503
00:31:58,65 --> 00:32:02,66
Now, in order to draw that,
once theta gets incredibly

504
00:32:02,66 --> 00:32:06,03
narrow, in order to depict that
I have to blow the whole

505
00:32:06,03 --> 00:32:07,7
picture back up in order
be able to see it.

506
00:32:07,7 --> 00:32:09,43
Otherwise it just
disappears on me.

507
00:32:09,43 --> 00:32:12,09
In fact in the limit theta
= 0, it's meaningless.

508
00:32:12,09 --> 00:32:13,08
It's just a flat line.

509
00:32:13,08 --> 00:32:15,5
That's the whole problem
with these tricky limits.

510
00:32:15,5 --> 00:32:18,1
They're meaningless right
at the (0, 0) level.

511
00:32:18,1 --> 00:32:22,01
It's only just a little away
that they're actually useful,

512
00:32:22,01 --> 00:32:25,89
that you get useful geometric
information out of them.

513
00:32:25,89 --> 00:32:27,3
So we're just a little away.

514
00:32:27,3 --> 00:32:30,34
So that's what this picture
down below in part

515
00:32:30,34 --> 00:32:31,16
A is meant to be.

516
00:32:31,16 --> 00:32:33,4
It's supposed to be that
theta is open a tiny

517
00:32:33,4 --> 00:32:35,19
crack, just a little bit.

518
00:32:35,19 --> 00:32:37,44
And the smallest I can draw it
on the board for you to

519
00:32:37,44 --> 00:32:40,16
visualize it is using the whole
length of the blackboard

520
00:32:40,16 --> 00:32:41,39
here for that.

521
00:32:41,39 --> 00:32:44,17
So I've opened a little tiny
bit and by the time we get to

522
00:32:44,17 --> 00:32:45,71
the other end of the
blackboard, of course

523
00:32:45,71 --> 00:32:46,51
it's fairly wide.

524
00:32:46,51 --> 00:32:50,52
But this angle theta is
a very small angle.

525
00:32:50,52 --> 00:32:50,79
Alright?

526
00:32:50,79 --> 00:32:56,67
So I'm trying to imagine what
happens as this collapses.

527
00:32:56,67 --> 00:33:00,32
Now, when I imagine that I
have to imagine a geometric

528
00:33:00,32 --> 00:33:03,39
interpretation of both the
numerator and the denominator

529
00:33:03,39 --> 00:33:06,02
of this quantity here.

530
00:33:06,02 --> 00:33:08,3
And just see what happens.

531
00:33:08,3 --> 00:33:14,02
Now I claimed the numerator is
this little tiny bit over here

532
00:33:14,02 --> 00:33:19,03
and the denominator is actually
half of this whole length here.

533
00:33:19,03 --> 00:33:21,15
But the factor of 2 doesn't
matter when you're

534
00:33:21,15 --> 00:33:24,25
seeing whether something
tends to 0 or not.

535
00:33:24,25 --> 00:33:24,99
Alright?

536
00:33:24,99 --> 00:33:27,16
And I claimed that if you stare
at this, it's clear that this

537
00:33:27,16 --> 00:33:32,75
is much shorter than that
vertical curve there.

538
00:33:32,75 --> 00:33:35,24
And I'm claiming, so this is
what you have to imagine, is

539
00:33:35,24 --> 00:33:39,01
this as it gets smaller and
smaller and smaller still that

540
00:33:39,01 --> 00:33:41,59
has the same effect of this
thing going way, way way,

541
00:33:41,59 --> 00:33:45,51
farther away and this vertical
curve getting closer and closer

542
00:33:45,51 --> 00:33:47,16
and closer to the green.

543
00:33:47,16 --> 00:33:52,53
And so that the gap between
them gets tiny and goes to 0.

544
00:33:52,53 --> 00:33:53,44
Alright?

545
00:33:53,44 --> 00:33:56,73
So not only does it go to 0,
that's not enough for us, but

546
00:33:56,73 --> 00:34:01,54
it also goes to 0 faster
than this theta goes to 0.

547
00:34:01,54 --> 00:34:05,92
And I hope the evidence is
pretty strong here because it's

548
00:34:05,92 --> 00:34:10,22
so tiny already at this stage.

549
00:34:10,22 --> 00:34:12,35
Alright.

550
00:34:12,35 --> 00:34:16,81
We are going to move forward
and you'll have to ponder

551
00:34:16,81 --> 00:34:18,42
these things some other time.

552
00:34:18,42 --> 00:34:21,35
So I'm gonna give you an even
harder thing to visualize

553
00:34:21,35 --> 00:34:26,6
now so be prepared.

554
00:34:26,6 --> 00:34:36,6
OK, so now, the next thing
that i'd like to do is to

555
00:34:36,6 --> 00:34:37,7
give you a second proof.

556
00:34:37,7 --> 00:34:43,43
Because it really is important
I think to understand this

557
00:34:43,43 --> 00:34:48,71
particular fact more thoroughly
and also to get a lot of

558
00:34:48,71 --> 00:34:51,45
practice with sines
and cosines.

559
00:34:51,45 --> 00:34:57,4
So I'm gonna give you a
geometric proof of the formula

560
00:34:57,4 --> 00:35:11,01
for sine here, for the
derivative of sine.

561
00:35:11,01 --> 00:35:13,42
So here we go.

562
00:35:13,42 --> 00:35:26,28
This is a geometric
proof of this fact.

563
00:35:26,28 --> 00:35:29,4
This is for all theta.

564
00:35:29,4 --> 00:35:33,84
So far we only did it for
theta = 0 and now we're going

565
00:35:33,84 --> 00:35:36,36
to do it for all theta.

566
00:35:36,36 --> 00:35:39,55
So this is a different
proof, but it uses exactly

567
00:35:39,55 --> 00:35:42,42
the same principles.

568
00:35:42,42 --> 00:35:45,39
Right?

569
00:35:45,39 --> 00:35:52,41
So, I want do this by drawing
another picture, and the

570
00:35:52,41 --> 00:35:59,305
picture is going to describe Y,
which is sin theta, which is if

571
00:35:59,305 --> 00:36:22,16
you like the vertical position
of some circular motion.

572
00:36:22,16 --> 00:36:27,17
So I'm imagining that something
is going around in a circle.

573
00:36:27,17 --> 00:36:30,62
Some particle is going
around in a circle.

574
00:36:30,62 --> 00:36:36,46
And so here's the circle,
here the origin.

575
00:36:36,46 --> 00:36:37,46
This is the unit distance.

576
00:36:37,46 --> 00:36:43,2
And right now it happens
to be at this location P.

577
00:36:43,2 --> 00:36:46,16
Maybe we'll put P a
little over here.

578
00:36:46,16 --> 00:36:50,26
And here's the angle theta.

579
00:36:50,26 --> 00:36:51,56
And now we're going to move it.

580
00:36:51,56 --> 00:36:55,04
We're going to vary theta
and we're interested in

581
00:36:55,04 --> 00:36:56,92
the rate of change of Y.

582
00:36:56,92 --> 00:37:00,76
So Y is the height P he
but we're gonna move it

583
00:37:00,76 --> 00:37:01,87
to another location.

584
00:37:01,87 --> 00:37:07,16
We'll move it along
the circle to Q.

585
00:37:07,16 --> 00:37:07,4
Right?

586
00:37:07,4 --> 00:37:09,2
So here it is.

587
00:37:09,2 --> 00:37:12,36
Here's the thing.

588
00:37:12,36 --> 00:37:14,45
So how far did we move it?

589
00:37:14,45 --> 00:37:18,57
Well we moved it by an
angle delta theta.

590
00:37:18,57 --> 00:37:21,37
So we started theta, theta is
going to be fixed in this

591
00:37:21,37 --> 00:37:23,99
argument, and we're going to
move a little bit delta theta.

592
00:37:23,99 --> 00:37:26,37
And now we're just gonna
try to figure out how

593
00:37:26,37 --> 00:37:28,51
far the thing moved.

594
00:37:28,51 --> 00:37:31,89
Well, in order to do that we've
got to keep track of the the

595
00:37:31,89 --> 00:37:34,59
height, the vertical
displacement here.

596
00:37:34,59 --> 00:37:38,335
So we're going to draw this
right angle here, this

597
00:37:38,335 --> 00:37:40,13
is the position R.

598
00:37:40,13 --> 00:37:45,71
And then this distance
here is the change in Y.

599
00:37:45,71 --> 00:37:46
Alright?

600
00:37:46 --> 00:37:50,46
So the picture is we
have something moving

601
00:37:50,46 --> 00:37:52,11
around a unit circle.

602
00:37:52,11 --> 00:37:53,68
A point moving around
a unit circle.

603
00:37:53,68 --> 00:37:56,37
It starts at P it moves to Q.

604
00:37:56,37 --> 00:37:59,095
It moves from angle
theta to angle theta

605
00:37:59,095 --> 00:37:59,59
delta theta.

606
00:37:59,59 --> 00:38:05,77
And the issue is how
much does Y move?

607
00:38:05,77 --> 00:38:07,34
And the formula for
Y is sin theta.

608
00:38:07,34 --> 00:38:29,71
So that's telling us the rate
of change of sin theta.

609
00:38:29,71 --> 00:38:34,37
Alright, well so let's just
try to think a little

610
00:38:34,37 --> 00:38:35,98
bit about what this is.

611
00:38:35,98 --> 00:38:37,99
So, first of all, I've already
said this and I'm going

612
00:38:37,99 --> 00:38:39,3
to repeat it here.

613
00:38:39,3 --> 00:38:41,65
Delta Y is PR.

614
00:38:41,65 --> 00:38:44,49
It's going from P and
going straight up to R.

615
00:38:44,49 --> 00:38:47,08
That's how far Y moves.

616
00:38:47,08 --> 00:38:48,25
That's the change in Y.

617
00:38:48,25 --> 00:38:52,91
That's what I said up in the
right hand corner there.

618
00:38:52,91 --> 00:38:53,47
Oops.

619
00:38:53,47 --> 00:38:56,43
I said PR but I wrote PQ.

620
00:38:56,43 --> 00:38:59,41
Alright, that's
not a good idea.

621
00:38:59,41 --> 00:38:59,63
Alright.

622
00:38:59,63 --> 00:39:03,09
So delta Y is PR.

623
00:39:03,09 --> 00:39:07,16
And now I want to draw the
diagram again one time.

624
00:39:07,16 --> 00:39:15,665
So here's Q, here's
R, and here's P, and

625
00:39:15,665 --> 00:39:17,3
here's my triangle.

626
00:39:17,3 --> 00:39:24,71
And now what i'd like to do is
draw this curve here which is a

627
00:39:24,71 --> 00:39:26,97
piece of the arc of the circle.

628
00:39:26,97 --> 00:39:30,6
But really what I want to keep
in mind is something that I did

629
00:39:30,6 --> 00:39:33,3
also in all these
other arguments.

630
00:39:33,3 --> 00:39:35,98
Which is, maybe I should have
called this orange, that

631
00:39:35,98 --> 00:39:38,51
I'm gonna think of the
straight line between.

632
00:39:38,51 --> 00:39:41,66
So it's the straight line
approximation to the curve that

633
00:39:41,66 --> 00:39:45,08
we're always interested in.

634
00:39:45,08 --> 00:39:47,5
So the straight line is much
simpler, because then we

635
00:39:47,5 --> 00:39:48,61
just have a triangle here.

636
00:39:48,61 --> 00:39:52,2
And in fact it's a
right triangle.

637
00:39:52,2 --> 00:39:54,35
Right, so we have the geometry
of a right triangle which

638
00:39:54,35 --> 00:39:59,21
is going to now let us do
all of our calculations.

639
00:39:59,21 --> 00:40:04,31
OK, so now the key step is this
same principle that we already

640
00:40:04,31 --> 00:40:09,04
used which is that short pieces
of curves are nearly straight.

641
00:40:09,04 --> 00:40:12
So that means that this piece
of the circular arc here from P

642
00:40:12 --> 00:40:16,12
to Q is practically the same as
the straight segment

643
00:40:16,12 --> 00:40:19,23
from P to Q.

644
00:40:19,23 --> 00:40:24,04
So, that's this principal.

645
00:40:24,04 --> 00:40:25,75
Well, let's put it over here.

646
00:40:25,75 --> 00:40:29,84
Is that PQ is practically
the same as the straight

647
00:40:29,84 --> 00:40:33,19
segment from P to Q.

648
00:40:33,19 --> 00:40:35,82
So how are we going
to use that?

649
00:40:35,82 --> 00:40:37,88
We want to use that
quantitatively in

650
00:40:37,88 --> 00:40:39,08
the following way.

651
00:40:39,08 --> 00:40:42,49
What we want to notice is that
the distance from P to Q is

652
00:40:42,49 --> 00:40:46,37
approximately delta theta.

653
00:40:46,37 --> 00:40:46,62
Right?

654
00:40:46,62 --> 00:40:49,53
Because the arc length along
that curve, the length of

655
00:40:49,53 --> 00:40:50,68
the curve is delta theta.

656
00:40:50,68 --> 00:40:53,74
So the length of the
green which is PQ is

657
00:40:53,74 --> 00:40:55,05
almost delta theta.

658
00:40:55,05 --> 00:41:01,69
So this is essentially delta
theta, this distance here.

659
00:41:01,69 --> 00:41:05,81
Now the second step, which is
a little trickier, is that

660
00:41:05,81 --> 00:41:08,98
we have to work out
what this angle is.

661
00:41:08,98 --> 00:41:11,64
So our goal, and I'm gonna put
it one step below because I'm

662
00:41:11,64 --> 00:41:14,94
gonna put the geometric
reasoning in between, is I need

663
00:41:14,94 --> 00:41:20,98
to figure out what
the angle QPR is.

664
00:41:20,98 --> 00:41:24,71
If I can figure out what this
angle is, then I'll be able to

665
00:41:24,71 --> 00:41:27,23
figure out what this vertical
distance is because I'll know

666
00:41:27,23 --> 00:41:30,12
the hypotenuse and I'll know
the angle so I'll be able to

667
00:41:30,12 --> 00:41:36,61
figure out what the side
of the triangle is.

668
00:41:36,61 --> 00:41:40,22
So now let me show you why
that's possible to do.

669
00:41:40,22 --> 00:41:43,4
So in order to do that first of
all I'm gonna trade the boards

670
00:41:43,4 --> 00:41:50,6
and show you where
the line PQ is.

671
00:41:50,6 --> 00:41:54,37
So the line PQ is here.

672
00:41:54,37 --> 00:41:56,47
That's the whole thing.

673
00:41:56,47 --> 00:42:00,19
And the key point about this
line that I need you to realize

674
00:42:00,19 --> 00:42:05,57
is that it's practically
perpendicular, it's almost

675
00:42:05,57 --> 00:42:08,91
perpendicular, to
this ray here.

676
00:42:08,91 --> 00:42:09,55
Alright?

677
00:42:09,55 --> 00:42:12,42
It's not quite because
the distance between

678
00:42:12,42 --> 00:42:13,42
P to Q is non-zero.

679
00:42:13,42 --> 00:42:15,09
So it isn't quite, but
in the limit it's going

680
00:42:15,09 --> 00:42:17,07
to be perpendicular.

681
00:42:17,07 --> 00:42:18,1
Exactly perpendicular.

682
00:42:18,1 --> 00:42:20,98
The tangent line to the circle.

683
00:42:20,98 --> 00:42:31,04
So the key thing that I'm going
to use is that PQ is almost

684
00:42:31,04 --> 00:42:35,29
perpendicular to OP.

685
00:42:35,29 --> 00:42:35,63
Alright?

686
00:42:35,63 --> 00:42:38,28
The ray from the origin is
basically perpendicular

687
00:42:38,28 --> 00:42:39,9
to that green line.

688
00:42:39,9 --> 00:42:43,51
And then the second thing I'm
going to use is something

689
00:42:43,51 --> 00:42:53,23
that's obvious which is
that PR is vertical.

690
00:42:53,23 --> 00:42:53,62
OK?

691
00:42:53,62 --> 00:42:58,08
So those are the two pieces of
geometry that I need to see.

692
00:42:58,08 --> 00:43:02,47
And now notice what's happening
upstairs on the picture

693
00:43:02,47 --> 00:43:05,05
here in the upper right.

694
00:43:05,05 --> 00:43:09,88
What I have is the angle
theta is the angle between

695
00:43:09,88 --> 00:43:12,91
the horizontal and OP.

696
00:43:12,91 --> 00:43:14,35
That's angle theta.

697
00:43:14,35 --> 00:43:17,99
If I rotate it by ninety
degree, the horizontal

698
00:43:17,99 --> 00:43:18,88
becomes vertical.

699
00:43:18,88 --> 00:43:22,97
It becomes PR and the other
thing rotated by 90 degrees

700
00:43:22,97 --> 00:43:24,81
becomes the green line.

701
00:43:24,81 --> 00:43:30,08
So the angle that I'm talking
about I get by taking this guy

702
00:43:30,08 --> 00:43:32,47
and rotating it by 90 degrees.

703
00:43:32,47 --> 00:43:33,8
It's the same angle.

704
00:43:33,8 --> 00:43:38,23
So that means that this angle
here is essentially theta.

705
00:43:38,23 --> 00:43:39,88
That's what this angle is.

706
00:43:39,88 --> 00:43:41,84
Let me repeat that
one more time.

707
00:43:41,84 --> 00:43:45,1
We started out with an angle
that looks like this, which

708
00:43:45,1 --> 00:43:47,97
is the horizontal that's
the origin straight

709
00:43:47,97 --> 00:43:48,6
out horizontally.

710
00:43:48,6 --> 00:43:50,56
That's the thing labeled 1.

711
00:43:50,56 --> 00:43:54,94
That distance there.

712
00:43:54,94 --> 00:43:56,43
That's my right arm
which is down here.

713
00:43:56,43 --> 00:43:59,97
My left arm is pointing up
and it's going from the

714
00:43:59,97 --> 00:44:03,35
origin to the point P.

715
00:44:03,35 --> 00:44:07,62
So here's the horizontal
and the angle between

716
00:44:07,62 --> 00:44:09,37
them is theta.

717
00:44:09,37 --> 00:44:13,18
And now, what I claim is is
that if I rotate by 90 degrees

718
00:44:13,18 --> 00:44:17,38
up, like this, without changing
anything - so that was what

719
00:44:17,38 --> 00:44:21,16
I did - the horizontal
will become a vertical.

720
00:44:21,16 --> 00:44:22,99
That's PR.

721
00:44:22,99 --> 00:44:25,03
That's going up, PR.

722
00:44:25,03 --> 00:44:32,08
And if I rotate OP 90
degrees, that's exactly PQ.

723
00:44:32,08 --> 00:44:33,54
OK?

724
00:44:33,54 --> 00:44:42,56
So let me draw it
on there one time.

725
00:44:42,56 --> 00:44:45,22
Let's do it with
some arrows here.

726
00:44:45,22 --> 00:44:52,46
So I started out with this and
then, we'll label this as

727
00:44:52,46 --> 00:45:00,5
orange, OK so red to orange,
and then I rotate by 90 degrees

728
00:45:00,5 --> 00:45:06,08
and the red becomes this
starting from P and the orange

729
00:45:06,08 --> 00:45:11,37
rotates around 90 degrees and
becomes this thing here.

730
00:45:11,37 --> 00:45:12,19
Alright?

731
00:45:12,19 --> 00:45:16,252
So this angle here is the
same as the other one

732
00:45:16,252 --> 00:45:18,46
which I've just drawn.

733
00:45:18,46 --> 00:45:27,03
Different vertices
for the angles.

734
00:45:27,03 --> 00:45:28,21
OK?

735
00:45:28,21 --> 00:45:31,23
Well I didn't say that
all arguments were

736
00:45:31,23 --> 00:45:36,45
supposed to be easy.

737
00:45:36,45 --> 00:45:39,55
Alright, so I claim that the
conclusion is that this angle

738
00:45:39,55 --> 00:45:43,2
is approximately theta.

739
00:45:43,2 --> 00:45:46,36
And now we can finish our
calculation, because we have

740
00:45:46,36 --> 00:45:49,54
something with the hypotenuse
being delta theta and the angle

741
00:45:49,54 --> 00:45:54,35
being theta and so this segment
here PR is approximately the

742
00:45:54,35 --> 00:46:02,43
hypotenuse length times
the cosine of the angle.

743
00:46:02,43 --> 00:46:05,74
And that is exactly
what we wanted.

744
00:46:05,74 --> 00:46:09,84
If we divide, we divide by
delta theta, we get (delta

745
00:46:09,84 --> 00:46:17,03
Y) / (delta theta) is
approximately cos theta.

746
00:46:17,03 --> 00:46:20,7
And that's the same thing as...

747
00:46:20,7 --> 00:46:23,235
So what we want in the limit is
exactly the delta theta going

748
00:46:23,235 --> 00:46:28,02
to 0 of (delta y) / (delta
theta) = cos theta.

749
00:46:28,02 --> 00:46:32,27
So we get an approximation on
a scale that we can visualize

750
00:46:32,27 --> 00:46:39,59
and in the limit the
formula is exact.

751
00:46:39,59 --> 00:46:44,06
OK, so that is a geometric
argument for the same result.

752
00:46:44,06 --> 00:46:48,22
Namely that the derivative
of sine is cosine.

753
00:46:48,22 --> 00:46:48,44
Yeah?

754
00:46:48,44 --> 00:46:51,59
Student: [INAUDIBLE].

755
00:46:51,59 --> 00:46:54,09
Professor: You will have to
do some kind of geometric

756
00:46:54,09 --> 00:46:55,84
proofs sometimes.

757
00:46:55,84 --> 00:46:59,73
When you'll really need
this is probably in 18.02.

758
00:46:59,73 --> 00:47:03,02
So you'll need to make
reasoning like this.

759
00:47:03,02 --> 00:47:05,73
This is, for example, the way
that you actually develop

760
00:47:05,73 --> 00:47:08,2
the theory of arc length.

761
00:47:08,2 --> 00:47:13,25
Dealing with delta x's and
delta y's is a common tool.

762
00:47:13,25 --> 00:47:18,73
Alright, I have one more thing
that I want to talk about

763
00:47:18,73 --> 00:47:25,07
today, which is some
general rules.

764
00:47:25,07 --> 00:47:28,23
We took a little bit more time
than I expected with this.

765
00:47:28,23 --> 00:47:32,41
So what I'm gonna do is just
tell you the rules and we'll

766
00:47:32,41 --> 00:47:36,33
discuss them in a few days.

767
00:47:36,33 --> 00:47:50,18
So let me tell you
the general rules.

768
00:47:50,18 --> 00:48:00,17
So these were the specific ones
and here are some general ones.

769
00:48:00,17 --> 00:48:08,49
So the first one is
called the product rule.

770
00:48:08,49 --> 00:48:11,01
And what it says is that if you
take the product of two

771
00:48:11,01 --> 00:48:15,533
functions and differentiate
them, you get the derivative of

772
00:48:15,533 --> 00:48:18,92
one times the other plus
the other times the

773
00:48:18,92 --> 00:48:22,06
derivative of the one.

774
00:48:22,06 --> 00:48:24,87
Now the way that you should
remember this, and the way that

775
00:48:24,87 --> 00:48:30
I'll carry out the proof, is
that you should think of it

776
00:48:30 --> 00:48:40,01
is you change one at a time.

777
00:48:40,01 --> 00:48:42,69
And this is a very useful way
of thinking about

778
00:48:42,69 --> 00:48:46,91
differentiation when you have
things which depend on more

779
00:48:46,91 --> 00:48:49,66
than one function.

780
00:48:49,66 --> 00:48:53,75
So this is a general procedure.

781
00:48:53,75 --> 00:48:59,35
The second formula that I
wanted to mention is called

782
00:48:59,35 --> 00:49:07,47
the quotient rule and
that says the following.

783
00:49:07,47 --> 00:49:13,22
That u / v' has a
formula as well.

784
00:49:13,22 --> 00:49:21,28
And the formula is
((u'v - uv' ) / v^2).

785
00:49:21,28 --> 00:49:23,24
So this is our second formula.

786
00:49:23,24 --> 00:49:31,5
Let me just mention, both of
them are extremely valuable and

787
00:49:31,5 --> 00:49:33,17
you'll use them all the time.

788
00:49:33,17 --> 00:49:43,6
This one of course only
works when v is not 0.

789
00:49:43,6 --> 00:49:47,55
Alright, so because we're of
time we're not gonna prove

790
00:49:47,55 --> 00:49:49,78
these today but we'll prove
these next time and you're

791
00:49:49,78 --> 00:49:52,17
definitely going to be
responsible for these

792
00:49:52,17 --> 00:49:53,81
kinds of proofs.

793
00:49:53,81 --> 00:49:54,445