1
00:00:00 --> 00:00:00,37
2
00:00:00,37 --> 00:00:03,61
The following is provided under
a Creative Commons License.
3
00:00:03,61 --> 00:00:06,74
Your support will help MIT
OpenCourseWare continue to
4
00:00:06,74 --> 00:00:09,95
offer high quality educational
resources for free.
5
00:00:09,95 --> 00:00:12,84
To make a donation or to view
additional materials from
6
00:00:12,84 --> 00:00:15,91
hundreds of MIT courses, visit
MIT OpenCourseWare
7
00:00:15,91 --> 00:00:19,35
at ocw.mit.edu.
8
00:00:19,35 --> 00:00:26,45
Professor: In today's lecture I
want to develop several more
9
00:00:26,45 --> 00:00:29,77
formulas that will allow
us to reach our goal of
10
00:00:29,77 --> 00:00:32,41
differentiating everything.
11
00:00:32,41 --> 00:00:42,39
So these are derivative
formulas, and they
12
00:00:42,39 --> 00:00:45,56
come in two flavors.
13
00:00:45,56 --> 00:00:53,88
The first kind is specific, so
some specific function we're
14
00:00:53,88 --> 00:00:55,46
giving the derivative of.
15
00:00:55,46 --> 00:01:00,74
And that would be, for
example, x^n or (1/x) .
16
00:01:00,74 --> 00:01:05,22
Those are the ones that we did
a couple of lectures ago.
17
00:01:05,22 --> 00:01:11,72
And then there are general
formulas, and the general ones
18
00:01:11,72 --> 00:01:14,82
don't actually give you a
formula for a specific function
19
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.
21
00:01:23,47 --> 00:01:30,95
Or if you multiply by a
constant, for example, so (cu),
22
00:01:30,95 --> 00:01:38,99
the derivative of that is
(cu)' where c is constant.
23
00:01:38,99 --> 00:01:43,42
All right, so these kinds of
formulas are very useful,
24
00:01:43,42 --> 00:01:46,07
both the specific and
the general kind.
25
00:01:46,07 --> 00:02:03,63
For example, we need both
kinds for polynomials.
26
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
00:02:08,97 --> 00:02:11,48
then you'll be able to generate
lots more by these
29
00:02:11,48 --> 00:02:16,48
general formulas.
30
00:02:16,48 --> 00:02:21,22
So today, we wanna concentrate
on the trig functions, and
31
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
36
00:02:46,51 --> 00:02:49,41
with trig functions, although
that's something that you
37
00:02:49,41 --> 00:02:55,63
should think of as
a gradual process.
38
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.
41
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.
61
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?
63
00:04:20,02 --> 00:04:22,58
Audience: [INAUDIBLE]
64
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).
70
00:04:47,31 --> 00:04:51,1
Alright, so now there's lots of
places to get confused here,
71
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
76
00:05:13,95 --> 00:05:21,43
letter x was a, and the
letter b was delta x.
77
00:05:21,43 --> 00:05:23,56
Now that's just the first part.
78
00:05:23,56 --> 00:05:26,98
That's just this part
of the expression.
79
00:05:26,98 --> 00:05:29,21
I still have to
remember the - sin x.
80
00:05:29,21 --> 00:05:30,05
That comes at the end.
81
00:05:30,05 --> 00:05:32,12
Minus sin x.
82
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
87
00:05:50,11 --> 00:05:52,28
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
110
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.
127
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