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Professor: In today's
lecture I want
8
00:00:25,150 --> 00:00:27,715
to develop several
more formulas that
9
00:00:27,715 --> 00:00:32,410
will allow us to reach our goal
of differentiating everything.
10
00:00:32,410 --> 00:00:41,800
So these are
derivative formulas,
11
00:00:41,800 --> 00:00:45,560
and they come in two flavors.
12
00:00:45,560 --> 00:00:52,070
The first kind is
specific, so some specific
13
00:00:52,070 --> 00:00:55,460
function we're giving
the derivative of.
14
00:00:55,460 --> 00:01:00,740
And that would be, for
example, x^n or (1/x) .
15
00:01:00,740 --> 00:01:05,220
Those are the ones that we
did a couple of lectures ago.
16
00:01:05,220 --> 00:01:10,510
And then there are
general formulas,
17
00:01:10,510 --> 00:01:12,400
and the general
ones don't actually
18
00:01:12,400 --> 00:01:14,820
give you a formula for
a specific function
19
00:01:14,820 --> 00:01:18,750
but tell you something like,
if you take two functions
20
00:01:18,750 --> 00:01:20,840
and add them together,
their derivative
21
00:01:20,840 --> 00:01:23,470
is the sum of the derivatives.
22
00:01:23,470 --> 00:01:27,220
Or if you multiply by a
constant, for example,
23
00:01:27,220 --> 00:01:31,490
so c times u, the
derivative of that
24
00:01:31,490 --> 00:01:38,990
is c times u' where
c is constant.
25
00:01:38,990 --> 00:01:41,630
All right, so these
kinds of formulas
26
00:01:41,630 --> 00:01:46,070
are very useful, both the
specific and the general kind.
27
00:01:46,070 --> 00:02:03,630
For example, we need both
kinds for polynomials.
28
00:02:03,630 --> 00:02:05,755
And more generally, pretty
much any set of formulas
29
00:02:05,755 --> 00:02:07,713
that we give you, will
give you a few functions
30
00:02:07,713 --> 00:02:09,180
to start out with
and then you'll
31
00:02:09,180 --> 00:02:16,480
be able to generate lots more
by these general formulas.
32
00:02:16,480 --> 00:02:21,000
So today, we wanna concentrate
on the trig functions,
33
00:02:21,000 --> 00:02:27,570
and so we'll start out with
some specific formulas.
34
00:02:27,570 --> 00:02:29,580
And they're going
to be the formulas
35
00:02:29,580 --> 00:02:33,000
for the derivative
of the sine function
36
00:02:33,000 --> 00:02:37,910
and the cosine function.
37
00:02:37,910 --> 00:02:41,240
So that's what we'll spend the
first part of the lecture on,
38
00:02:41,240 --> 00:02:46,510
and at the same time I hope to
get you very used to dealing
39
00:02:46,510 --> 00:02:49,130
with trig functions,
although that's something
40
00:02:49,130 --> 00:02:55,630
that you should think
of as a gradual process.
41
00:02:55,630 --> 00:02:58,310
Alright, so in order
to calculate these,
42
00:02:58,310 --> 00:03:03,720
I'm gonna start over here and
just start the calculation.
43
00:03:03,720 --> 00:03:05,270
So here we go.
44
00:03:05,270 --> 00:03:08,110
Let's check what happens
with the sine function.
45
00:03:08,110 --> 00:03:15,590
So, I take sin (x + delta
x), I subtract sin x
46
00:03:15,590 --> 00:03:22,090
and I divide by delta x.
47
00:03:22,090 --> 00:03:24,270
Right, so this is the
difference quotient
48
00:03:24,270 --> 00:03:26,850
and eventually I'm gonna have
to take the limit as delta
49
00:03:26,850 --> 00:03:29,320
x goes to 0.
50
00:03:29,320 --> 00:03:31,850
And there's really
only one thing
51
00:03:31,850 --> 00:03:36,830
we can do with this to
simplify or change it,
52
00:03:36,830 --> 00:03:42,440
and that is to use the sum
formula for the sine function.
53
00:03:42,440 --> 00:03:43,560
So, that's this.
54
00:03:43,560 --> 00:03:48,680
That's sin x cos delta x plus--
55
00:03:54,490 --> 00:03:56,070
Oh, that's not what it is?
56
00:03:56,070 --> 00:04:01,040
OK, so what is it?
sin x sin delta x.
57
00:04:01,040 --> 00:04:02,940
OK, good.
58
00:04:02,940 --> 00:04:07,500
Plus cosine.
59
00:04:07,500 --> 00:04:09,290
No?
60
00:04:09,290 --> 00:04:10,560
Oh, OK.
61
00:04:10,560 --> 00:04:14,470
So which is it?
62
00:04:14,470 --> 00:04:16,176
OK.
63
00:04:16,176 --> 00:04:17,300
Alright, let's take a vote.
64
00:04:17,300 --> 00:04:20,020
Is it sine, sine, or
is it sine, cosine?
65
00:04:20,020 --> 00:04:22,580
Audience: [INAUDIBLE]
66
00:04:22,580 --> 00:04:31,100
Professor: OK, so is this
going to be... cosine.
67
00:04:31,100 --> 00:04:34,450
All right, you better remember
these formulas, alright?
68
00:04:34,450 --> 00:04:37,291
OK, turns out that
it's sine, cosine.
69
00:04:37,291 --> 00:04:37,790
All right.
70
00:04:37,790 --> 00:04:39,640
Cosine, sine.
71
00:04:39,640 --> 00:04:47,310
So here we go, no gotta
do x here, sin (delta x).
72
00:04:47,310 --> 00:04:51,100
Alright, so now there's lots
of places to get confused here,
73
00:04:51,100 --> 00:04:55,160
and you're gonna need to
make sure you get it right.
74
00:04:55,160 --> 00:04:59,230
Alright, so we're gonna put
those in parentheses here.
75
00:04:59,230 --> 00:05:10,310
sin (a + b) is sin a
cos b plus cos a sin b.
76
00:05:10,310 --> 00:05:12,740
All right, now that's
what I did over here,
77
00:05:12,740 --> 00:05:21,430
except the letter x was a,
and the letter b was delta x.
78
00:05:21,430 --> 00:05:23,560
Now that's just the first part.
79
00:05:23,560 --> 00:05:26,980
That's just this part
of the expression.
80
00:05:26,980 --> 00:05:29,134
I still have to remember
the minus sin x.
81
00:05:29,134 --> 00:05:30,050
That comes at the end.
82
00:05:30,050 --> 00:05:32,120
Minus sin x.
83
00:05:32,120 --> 00:05:37,900
And then, I have to remember the
denominator, which is delta x.
84
00:05:37,900 --> 00:05:43,040
OK?
85
00:05:43,040 --> 00:05:47,250
Alright, so now...
86
00:05:47,250 --> 00:05:49,710
The next thing we're
gonna do is we're
87
00:05:49,710 --> 00:05:52,280
gonna try to group the terms.
88
00:05:52,280 --> 00:05:58,150
And the difficulty with all such
arguments is the following one:
89
00:05:58,150 --> 00:06:02,060
any tricky limit is
basically 0 / 0 when
90
00:06:02,060 --> 00:06:03,200
you set delta x equal to 0.
91
00:06:03,200 --> 00:06:06,480
If I set delta x equal to
0, this is sin x - sin x.
92
00:06:06,480 --> 00:06:08,520
So it's a 0 / 0 term.
93
00:06:08,520 --> 00:06:10,040
Here we have
various things which
94
00:06:10,040 --> 00:06:12,170
are 0 and various things
which are non-zero.
95
00:06:12,170 --> 00:06:17,721
We must group the terms so
that a 0 stays over a 0.
96
00:06:17,721 --> 00:06:19,220
Otherwise, we're
gonna have no hope.
97
00:06:19,220 --> 00:06:21,990
If we get some 1 / 0
term, we'll get something
98
00:06:21,990 --> 00:06:24,160
meaningless in the limit.
99
00:06:24,160 --> 00:06:27,790
So I claim that the right
thing to do here is to notice,
100
00:06:27,790 --> 00:06:31,630
and I'll just point
out this one thing.
101
00:06:31,630 --> 00:06:35,580
When delta x goes to 0,
this cosine of 0 is 1.
102
00:06:35,580 --> 00:06:39,630
So it doesn't cancel unless we
throw in this extra sine term
103
00:06:39,630 --> 00:06:40,130
here.
104
00:06:40,130 --> 00:06:43,620
So I'm going to use
this common factor,
105
00:06:43,620 --> 00:06:44,670
and combine those terms.
106
00:06:44,670 --> 00:06:46,480
So this is really
the only thing you're
107
00:06:46,480 --> 00:06:48,800
gonna have to check in this
particular calculation.
108
00:06:48,800 --> 00:06:50,760
So we have the
common factor of sin
109
00:06:50,760 --> 00:06:54,540
x, and that multiplies
something that will cancel,
110
00:06:54,540 --> 00:06:59,170
which is (cos delta
x - 1) / delta x.
111
00:06:59,170 --> 00:07:03,300
That's the first term,
and now what's left,
112
00:07:03,300 --> 00:07:06,470
well there's a cos
x that factors out,
113
00:07:06,470 --> 00:07:14,040
and then the other factor is
(sin delta x) / (delta x).
114
00:07:14,040 --> 00:07:20,850
OK, now does anyone
remember from last time what
115
00:07:20,850 --> 00:07:25,000
this thing goes to?
116
00:07:25,000 --> 00:07:27,580
How many people say 1?
117
00:07:27,580 --> 00:07:29,340
How many people say 0?
118
00:07:29,340 --> 00:07:31,180
All right, it's 0.
119
00:07:31,180 --> 00:07:33,540
That's my favorite
number, alright?
120
00:07:33,540 --> 00:07:34,040
0.
121
00:07:34,040 --> 00:07:36,180
It's the easiest
number to deal with.
122
00:07:36,180 --> 00:07:39,200
So this goes to
0, and that's what
123
00:07:39,200 --> 00:07:45,980
happens as delta x tends to 0.
124
00:07:45,980 --> 00:07:47,150
How about this one?
125
00:07:47,150 --> 00:07:51,750
This one goes to 1, my second
favorite number, almost as
126
00:07:51,750 --> 00:07:54,350
easy to deal with as 0.
127
00:07:54,350 --> 00:07:56,280
And these things are
picked for a reason.
128
00:07:56,280 --> 00:07:58,030
They're the simplest
numbers to deal with.
129
00:07:58,030 --> 00:08:06,820
So altogether, this thing as
delta x goes to 0 goes to what?
130
00:08:06,820 --> 00:08:09,440
I want a single person to
answer, a brave volunteer.
131
00:08:09,440 --> 00:08:10,330
Alright, back there.
132
00:08:10,330 --> 00:08:12,070
Student: Cosine
133
00:08:12,070 --> 00:08:14,650
Professor: Cosine,
because this factor is 0.
134
00:08:14,650 --> 00:08:17,560
It cancels and this factor
has a 1, so it's cosine.
135
00:08:17,560 --> 00:08:20,230
So it's cos x.
136
00:08:20,230 --> 00:08:25,840
So our conclusion over here
- and I'll put it in orange -
137
00:08:25,840 --> 00:08:34,920
is that the derivative of
the sine is the cosine.
138
00:08:34,920 --> 00:08:38,170
OK, now I still wanna label
these very important limit
139
00:08:38,170 --> 00:08:39,220
facts here.
140
00:08:39,220 --> 00:08:41,650
This one we'll call
A, and this one we're
141
00:08:41,650 --> 00:08:44,340
going to call B, because we
haven't checked them yet.
142
00:08:44,340 --> 00:08:46,340
I promised you I would
do that, and I'll
143
00:08:46,340 --> 00:08:48,460
have to do that this time.
144
00:08:48,460 --> 00:08:52,490
So we're relying on
those things being true.
145
00:08:52,490 --> 00:08:56,140
Now I'm gonna do the same
thing with the cosine function,
146
00:08:56,140 --> 00:08:58,730
except in order to do it I'm
gonna have to remember the sum
147
00:08:58,730 --> 00:09:00,930
rule for cosine.
148
00:09:00,930 --> 00:09:03,474
So we're gonna do almost
the same calculation here.
149
00:09:03,474 --> 00:09:05,140
We're gonna see that
that will work out,
150
00:09:05,140 --> 00:09:12,390
but now you have to remember
that cos (a + b) = cos cos,
151
00:09:12,390 --> 00:09:15,350
no it's not cosine^2, because
there are two different
152
00:09:15,350 --> 00:09:16,760
quantities here.
153
00:09:16,760 --> 00:09:23,650
It's cos a cos b - sin a sin b.
154
00:09:23,650 --> 00:09:31,280
All right, so you'll have
to be willing to call those
155
00:09:31,280 --> 00:09:34,800
forth at will right now.
156
00:09:34,800 --> 00:09:36,460
So let's do the cosine now.
157
00:09:36,460 --> 00:09:45,500
So that's cos (x + delta x)
- cos x divided by delta x.
158
00:09:45,500 --> 00:09:47,830
OK, there's the
difference quotient
159
00:09:47,830 --> 00:09:49,318
for the cosine function.
160
00:09:49,318 --> 00:09:51,776
And now I'm gonna do the same
thing I did before except I'm
161
00:09:51,776 --> 00:09:53,700
going to apply the
second rule, that
162
00:09:53,700 --> 00:09:55,770
is the sum rule for cosine.
163
00:09:55,770 --> 00:10:03,401
And that's gonna give me cos x
cos delta x - sin x sin delta
164
00:10:03,401 --> 00:10:03,900
x.
165
00:10:03,900 --> 00:10:09,300
And I have to remember again
to subtract the cosine divided
166
00:10:09,300 --> 00:10:11,590
by this delta x.
167
00:10:11,590 --> 00:10:16,370
And now I'm going to regroup
just the way I did before,
168
00:10:16,370 --> 00:10:22,050
and I get the common factor of
cosine multiplying (cos delta x
169
00:10:22,050 --> 00:10:25,180
- 1) / delta x.
170
00:10:25,180 --> 00:10:30,910
And here I get the sin x
but actually it's -sin x.
171
00:10:30,910 --> 00:10:36,320
And then I have (sin
delta x) / delta x.
172
00:10:36,320 --> 00:10:36,900
All right?
173
00:10:36,900 --> 00:10:38,570
The only difference
is this minus sign
174
00:10:38,570 --> 00:10:42,740
which I stuck inside there.
175
00:10:42,740 --> 00:10:44,240
Well that's not the
only difference,
176
00:10:44,240 --> 00:10:48,440
but it's a crucial difference.
177
00:10:48,440 --> 00:10:54,700
OK, again by A we get that this
is 0 as delta x tends to 0.
178
00:10:54,700 --> 00:10:56,370
And this is 1.
179
00:10:56,370 --> 00:10:59,590
Those are the properties
I called A and B.
180
00:10:59,590 --> 00:11:04,840
And so the result here
as delta x tends to 0
181
00:11:04,840 --> 00:11:08,470
is that we get negative sin x.
182
00:11:08,470 --> 00:11:11,800
That's the factor.
183
00:11:11,800 --> 00:11:18,880
So this guy is negative sin x.
184
00:11:18,880 --> 00:11:24,560
I'll put a little
box around that too.
185
00:11:24,560 --> 00:11:29,040
Alright, now these
formulas take a little bit
186
00:11:29,040 --> 00:11:34,220
of getting used
to, but before I do
187
00:11:34,220 --> 00:11:38,480
that I'm gonna explain to
you the proofs of A and B.
188
00:11:38,480 --> 00:11:44,470
So we'll get ourselves
started by mentioning that.
189
00:11:44,470 --> 00:11:47,570
Maybe before I do
that though, I want
190
00:11:47,570 --> 00:11:51,710
to show you how A and B fit into
the proofs of these theorems.
191
00:11:51,710 --> 00:12:06,030
So, let me just make
some remarks here.
192
00:12:06,030 --> 00:12:09,020
So this is just
a remark but it's
193
00:12:09,020 --> 00:12:15,600
meant to help you to frame
how these proofs worked.
194
00:12:15,600 --> 00:12:17,650
So, first of all,
I want to point out
195
00:12:17,650 --> 00:12:19,750
that if you take
the rate of change
196
00:12:19,750 --> 00:12:28,450
of sin x, no let's
start with cosine
197
00:12:28,450 --> 00:12:30,450
because a little
bit less obvious.
198
00:12:30,450 --> 00:12:33,820
If I take the rate of change
of cos x, so in other words
199
00:12:33,820 --> 00:12:41,180
this derivative at x =
0, then by definition
200
00:12:41,180 --> 00:12:45,300
this is a certain limit
as delta x goes to 0.
201
00:12:45,300 --> 00:12:46,860
So which one is it?
202
00:12:46,860 --> 00:12:51,170
Well I have to evaluate
cosine at 0 + delta
203
00:12:51,170 --> 00:12:53,390
x, but that's just delta x.
204
00:12:53,390 --> 00:12:56,240
And I have to
subtract cosine at 0.
205
00:12:56,240 --> 00:13:00,040
That's the base point,
but that's just 1.
206
00:13:00,040 --> 00:13:03,050
And then I have to
divide by delta x.
207
00:13:03,050 --> 00:13:06,850
And lo and behold you can see
that this is exactly the limit
208
00:13:06,850 --> 00:13:08,220
that we had over there.
209
00:13:08,220 --> 00:13:15,880
This is the one that we know is
0 by what we call property A.
210
00:13:15,880 --> 00:13:23,150
And similarly, if I take the
derivative of sin x at x=0,
211
00:13:23,150 --> 00:13:27,000
then that's going to be the
limit as delta x goes to 0
212
00:13:27,000 --> 00:13:30,700
of sin delta x / delta x.
213
00:13:30,700 --> 00:13:35,130
And that's because I should
be subtracting sine of 0
214
00:13:35,130 --> 00:13:37,871
but sine of 0 is 0.
215
00:13:37,871 --> 00:13:38,370
Right?
216
00:13:38,370 --> 00:13:48,510
So this is going to be 1 by our
property B. And so the remark
217
00:13:48,510 --> 00:13:51,030
that I want to make,
in addition to this,
218
00:13:51,030 --> 00:13:55,200
is something about the
structure of these two proofs.
219
00:13:55,200 --> 00:14:12,990
Which is the derivatives
of sine and cosine at x = 0
220
00:14:12,990 --> 00:14:25,500
give all values of
d/dx sin x, d/dx cos x.
221
00:14:25,500 --> 00:14:27,630
So that's really what this
argument is showing us,
222
00:14:27,630 --> 00:14:31,420
is that we just need one
rate of change at one place
223
00:14:31,420 --> 00:14:38,770
and then we work out
all the rest of them.
224
00:14:38,770 --> 00:14:40,990
So that's really the
substance of this proof.
225
00:14:40,990 --> 00:14:43,570
That of course really then
shows that it boils down
226
00:14:43,570 --> 00:14:48,020
to showing what this rate of
change is in these two cases.
227
00:14:48,020 --> 00:14:50,730
So now there's enough
suspense that we
228
00:14:50,730 --> 00:15:08,010
want to make sure that we know
that those answers are correct.
229
00:15:08,010 --> 00:15:12,180
OK, so let's demonstrate
both of them.
230
00:15:12,180 --> 00:15:18,540
I'll start with B. I need to
figure out property B. Now,
231
00:15:18,540 --> 00:15:22,470
we only have one alternative
as to a type of proof
232
00:15:22,470 --> 00:15:24,710
that we can give of
this kind of result,
233
00:15:24,710 --> 00:15:29,070
and that's because we only
have one way of describing
234
00:15:29,070 --> 00:15:32,540
sine and cosine functions,
that is geometrically.
235
00:15:32,540 --> 00:15:42,620
So we have to give
a geometric proof.
236
00:15:42,620 --> 00:15:45,190
And to write down
a geometric proof
237
00:15:45,190 --> 00:15:47,150
we are going to have
to draw a picture.
238
00:15:47,150 --> 00:15:49,550
And the first step
in the proof, really,
239
00:15:49,550 --> 00:15:51,750
is to replace this
variable delta
240
00:15:51,750 --> 00:15:55,800
x which is going to 0
with another name which
241
00:15:55,800 --> 00:15:58,370
is suggestive of what we're
gonna do which is the letter
242
00:15:58,370 --> 00:16:00,950
theta for an angle.
243
00:16:00,950 --> 00:16:03,560
OK, so let's draw
a picture of what
244
00:16:03,560 --> 00:16:05,950
it is that we're going to do.
245
00:16:05,950 --> 00:16:07,980
Here is the circle.
246
00:16:07,980 --> 00:16:10,740
And here is the origin.
247
00:16:10,740 --> 00:16:13,900
And here's some little
angle, well I'll
248
00:16:13,900 --> 00:16:16,110
draw it a little
larger so it's visible.
249
00:16:16,110 --> 00:16:19,430
Here's theta, alright?
250
00:16:19,430 --> 00:16:21,010
And this is the unit circle.
251
00:16:21,010 --> 00:16:25,910
I won't write that down on here
but that's the unit circle.
252
00:16:25,910 --> 00:16:29,780
And now sin theta is this
vertical distance here.
253
00:16:29,780 --> 00:16:32,740
Maybe, I'll draw it
in a different color
254
00:16:32,740 --> 00:16:34,750
so that we can see it all.
255
00:16:34,750 --> 00:16:38,400
OK so here's this distance.
256
00:16:38,400 --> 00:16:45,820
This distance is sin theta.
257
00:16:45,820 --> 00:16:48,360
OK?
258
00:16:48,360 --> 00:16:51,440
Now almost the
only other thing we
259
00:16:51,440 --> 00:16:54,770
have to write down in this
picture to have it work out
260
00:16:54,770 --> 00:16:58,500
is that we have to recognize
that when theta is the angle,
261
00:16:58,500 --> 00:17:03,279
that's also the arc length
of this piece of the circle
262
00:17:03,279 --> 00:17:04,320
when measured in radians.
263
00:17:04,320 --> 00:17:13,696
So this length here is
also arc length theta.
264
00:17:13,696 --> 00:17:14,820
That little piece in there.
265
00:17:14,820 --> 00:17:18,580
So maybe I'll use a different
color for that to indicate it.
266
00:17:18,580 --> 00:17:25,560
So that's orange and that's
this little chunk there.
267
00:17:25,560 --> 00:17:26,870
So those are the two pieces.
268
00:17:26,870 --> 00:17:36,250
Now in order to persuade
you now that the limit is
269
00:17:36,250 --> 00:17:37,964
what it's supposed
to be, I'm going
270
00:17:37,964 --> 00:17:39,630
to extend the picture
just a little bit.
271
00:17:39,630 --> 00:17:42,430
I'm going to double it, just
for my own linguistic sake
272
00:17:42,430 --> 00:17:44,240
and so that I can
tell you a story.
273
00:17:44,240 --> 00:17:46,690
Alright, so that
you'll remember this.
274
00:17:46,690 --> 00:17:49,640
So I'm going to take
a theta angle below
275
00:17:49,640 --> 00:17:53,670
and I'll have another copy
of sin theta down here.
276
00:17:53,670 --> 00:18:00,790
And now the total
picture is really
277
00:18:00,790 --> 00:18:04,721
like a bow and its
bow string there.
278
00:18:04,721 --> 00:18:05,220
Alright?
279
00:18:05,220 --> 00:18:11,050
So what we have here is
a length of 2 sin theta.
280
00:18:11,050 --> 00:18:13,630
So maybe I'll write it
this way, 2 sin theta.
281
00:18:13,630 --> 00:18:15,120
I just doubled it.
282
00:18:15,120 --> 00:18:25,640
And here I have underneath,
whoops, I got it backwards.
283
00:18:25,640 --> 00:18:27,040
Sorry about that.
284
00:18:27,040 --> 00:18:29,144
Trying to be fancy
with the colored chalk
285
00:18:29,144 --> 00:18:30,310
and I have it reversed here.
286
00:18:30,310 --> 00:18:32,180
So this is not 2 sin theta.
287
00:18:32,180 --> 00:18:33,540
2 sin theta is the vertical.
288
00:18:33,540 --> 00:18:34,910
That's the green.
289
00:18:34,910 --> 00:18:37,170
So let's try that again.
290
00:18:37,170 --> 00:18:41,190
This is 2 sin theta, alright?
291
00:18:41,190 --> 00:18:44,450
And then in the denominator
I have the arc length
292
00:18:44,450 --> 00:18:50,680
which is theta is the first half
and so double it is 2 theta.
293
00:18:50,680 --> 00:18:51,310
Alright?
294
00:18:51,310 --> 00:18:56,630
So if you like, this is
the bow and up here we
295
00:18:56,630 --> 00:19:04,290
have the bow string.
296
00:19:04,290 --> 00:19:07,740
And of course we
can cancel the 2's.
297
00:19:07,740 --> 00:19:11,250
That's equal to
sin theta / theta.
298
00:19:11,250 --> 00:19:17,900
And so now why does this
tend to 1 as theta goes to 0?
299
00:19:17,900 --> 00:19:23,670
Well, it's because as the
angle theta gets very small,
300
00:19:23,670 --> 00:19:28,880
this curved piece looks more
and more like a straight one.
301
00:19:28,880 --> 00:19:29,640
Alright?
302
00:19:29,640 --> 00:19:32,630
And if you get very, very
close here the green segment
303
00:19:32,630 --> 00:19:34,610
and the orange segment
would just merge.
304
00:19:34,610 --> 00:19:36,850
They would be practically
on top of each other.
305
00:19:36,850 --> 00:19:42,360
And they have closer and closer
and closer to the same length.
306
00:19:42,360 --> 00:19:51,790
So that's why this is true.
307
00:19:51,790 --> 00:20:03,200
I guess I'll articulate that
by saying that short curves are
308
00:20:03,200 --> 00:20:06,550
nearly straight.
309
00:20:06,550 --> 00:20:10,000
Alright, so that's the
principle that we're using.
310
00:20:10,000 --> 00:20:18,970
Or short pieces of curves, if
you like, are nearly straight.
311
00:20:18,970 --> 00:20:23,640
So if you like, this
is the principle.
312
00:20:23,640 --> 00:20:30,850
So short pieces of curves.
313
00:20:30,850 --> 00:20:31,540
Alright?
314
00:20:31,540 --> 00:20:39,390
So now I also need to
give you a proof of A.
315
00:20:39,390 --> 00:20:43,990
And that has to do with
this cosine function here.
316
00:20:43,990 --> 00:20:51,905
This is the property
A. So I'm going
317
00:20:51,905 --> 00:20:53,487
to do this by
flipping it around,
318
00:20:53,487 --> 00:20:56,070
because it turns out that this
numerator is a negative number.
319
00:20:56,070 --> 00:20:58,060
If I want to interpret
it as a length,
320
00:20:58,060 --> 00:21:00,340
I'm gonna want a
positive quantity.
321
00:21:00,340 --> 00:21:04,180
So I'm gonna write
down 1 - cos theta here
322
00:21:04,180 --> 00:21:08,320
and then I'm gonna
divide by theta there.
323
00:21:08,320 --> 00:21:10,720
Again I'm gonna make some
kind of interpretation.
324
00:21:10,720 --> 00:21:15,130
Now this time I'm going to draw
the same sort of bow and arrow
325
00:21:15,130 --> 00:21:18,970
arrangement, but maybe I'll
exaggerate it a little bit.
326
00:21:18,970 --> 00:21:23,190
So here's the vertex
of the sector,
327
00:21:23,190 --> 00:21:31,780
but we'll maybe make
it a little longer.
328
00:21:31,780 --> 00:21:35,590
Alright, so here it is, and
here was that middle line
329
00:21:35,590 --> 00:21:38,310
which was the unit-- Whoops.
330
00:21:38,310 --> 00:21:40,880
OK, I think I'm going
to have to tilt it up.
331
00:21:40,880 --> 00:21:47,000
OK, let's try from here.
332
00:21:47,000 --> 00:21:51,570
Alright, well you know
on your pencil and paper
333
00:21:51,570 --> 00:21:53,810
it will look better than
it does on my blackboard.
334
00:21:53,810 --> 00:21:55,190
OK, so here we are.
335
00:21:55,190 --> 00:21:56,770
Here's this shape.
336
00:21:56,770 --> 00:22:01,430
Now this angle is
supposed to be theta
337
00:22:01,430 --> 00:22:03,350
and this angle is another theta.
338
00:22:03,350 --> 00:22:05,895
So here we have a length
which is again theta
339
00:22:05,895 --> 00:22:07,910
and another length which
is theta over here.
340
00:22:07,910 --> 00:22:09,670
That's the same as
in the other picture,
341
00:22:09,670 --> 00:22:13,290
except we've
exaggerated a bit here.
342
00:22:13,290 --> 00:22:15,010
And now we have
this vertical line,
343
00:22:15,010 --> 00:22:18,240
which again I'm gonna draw
in green, the bow string.
344
00:22:18,240 --> 00:22:24,820
But notice that as the vertex
gets farther and farther away,
345
00:22:24,820 --> 00:22:27,019
the curved line gets
closer and closer
346
00:22:27,019 --> 00:22:28,060
to being a vertical line.
347
00:22:28,060 --> 00:22:30,670
That's sort of the flip
side, by expansion,
348
00:22:30,670 --> 00:22:33,110
of the zoom in principle.
349
00:22:33,110 --> 00:22:35,130
The principle that
curves are nearly
350
00:22:35,130 --> 00:22:36,950
straight when you zoom in.
351
00:22:36,950 --> 00:22:39,370
If you zoom out that would
mean sending this vertex way,
352
00:22:39,370 --> 00:22:42,200
way out somewhere.
353
00:22:42,200 --> 00:22:44,370
The curved line, the
piece of the circle,
354
00:22:44,370 --> 00:22:48,570
gets more and more straight.
355
00:22:48,570 --> 00:22:53,630
And now let me show you
where this numerator 1 - cos
356
00:22:53,630 --> 00:22:57,620
theta is on this picture.
357
00:22:57,620 --> 00:23:01,040
So where is it?
358
00:23:01,040 --> 00:23:03,680
Well, this whole distance is 1.
359
00:23:03,680 --> 00:23:07,610
But the distance from
the vertex to the green
360
00:23:07,610 --> 00:23:09,870
is cosine of theta.
361
00:23:09,870 --> 00:23:12,430
Right, because this is
theta, so dropping down
362
00:23:12,430 --> 00:23:16,830
the perpendicular this
distance back to the origin
363
00:23:16,830 --> 00:23:17,630
is cos theta.
364
00:23:17,630 --> 00:23:20,890
So this little
tiny, bitty segment
365
00:23:20,890 --> 00:23:26,540
here is basically the
gap between the curve
366
00:23:26,540 --> 00:23:28,820
and the vertical segment.
367
00:23:28,820 --> 00:23:35,820
So the gap is equal
to 1 - cos theta.
368
00:23:35,820 --> 00:23:40,640
So now you can see that as
this point gets farther away,
369
00:23:40,640 --> 00:23:43,670
if this got sent off
to the Stata Center,
370
00:23:43,670 --> 00:23:45,840
you would hardly be able
to tell the difference.
371
00:23:45,840 --> 00:23:48,120
The bow string would
coincide with the bow
372
00:23:48,120 --> 00:23:50,480
and this little gap
between the bow string
373
00:23:50,480 --> 00:23:52,430
and the bow would
be tending to 0.
374
00:23:52,430 --> 00:23:54,282
And that's the
statement that this
375
00:23:54,282 --> 00:23:58,540
tends to 0 as theta tends to 0.
376
00:23:58,540 --> 00:24:00,130
The scaled version of that.
377
00:24:00,130 --> 00:24:01,250
Yeah, question down here.
378
00:24:01,250 --> 00:24:04,800
Student: Doesn't the denominator
also tend to 0 though?
379
00:24:04,800 --> 00:24:10,190
Professor: Ah, the question is
"doesn't the denominator also
380
00:24:10,190 --> 00:24:11,670
tend to 0?"
381
00:24:11,670 --> 00:24:14,510
And the answer is yes.
382
00:24:14,510 --> 00:24:18,660
In my strange analogy with
zooming in, what I did
383
00:24:18,660 --> 00:24:20,390
was I zoomed out the picture.
384
00:24:20,390 --> 00:24:25,450
So in other words, if you
imagine you're taking this
385
00:24:25,450 --> 00:24:28,209
and you're putting it under
a microscope over here
386
00:24:28,209 --> 00:24:29,750
and you're looking
at something where
387
00:24:29,750 --> 00:24:31,910
theta is getting smaller
and smaller and smaller
388
00:24:31,910 --> 00:24:33,340
and smaller.
389
00:24:33,340 --> 00:24:34,370
Alright?
390
00:24:34,370 --> 00:24:39,500
But now because I want my
picture, I expanded my picture.
391
00:24:39,500 --> 00:24:42,960
So the ratio is the
thing that's preserved.
392
00:24:42,960 --> 00:24:50,470
So if I make it so that
this gap is tiny...
393
00:24:50,470 --> 00:24:52,410
Let me say this one more time.
394
00:24:52,410 --> 00:24:57,030
I'm afraid I've made life
complicated for myself.
395
00:24:57,030 --> 00:25:01,040
If I simply let this
theta tend in to 0,
396
00:25:01,040 --> 00:25:04,044
that would be the same
effect as making this
397
00:25:04,044 --> 00:25:06,460
closer and closer in and then
the vertical would approach.
398
00:25:06,460 --> 00:25:09,040
But I want to keep on
blowing up the picture
399
00:25:09,040 --> 00:25:11,940
so that I can see the
difference between the vertical
400
00:25:11,940 --> 00:25:13,260
and the curve.
401
00:25:13,260 --> 00:25:16,480
So that's very much like if
you are on a video screen
402
00:25:16,480 --> 00:25:18,740
and you zoom in, zoom
in, zoom in, and zoom in.
403
00:25:18,740 --> 00:25:20,615
So the question is what
would that look like?
404
00:25:20,615 --> 00:25:23,350
That has the same effect
as sending this point out
405
00:25:23,350 --> 00:25:27,840
farther and farther in that
direction, to the left.
406
00:25:27,840 --> 00:25:29,430
And so I'm just
trying to visualize it
407
00:25:29,430 --> 00:25:32,410
for you by leaving the
theta at this scale,
408
00:25:32,410 --> 00:25:34,090
but actually the
scale of the picture
409
00:25:34,090 --> 00:25:36,290
is then changing when I do that.
410
00:25:36,290 --> 00:25:39,380
So theta is going
to 0, but I I'm
411
00:25:39,380 --> 00:25:42,790
rescaling so that it's of a
size that we can look at it,
412
00:25:42,790 --> 00:25:46,292
And then imagine
what's happening to it.
413
00:25:46,292 --> 00:25:47,750
OK, does that answer
your question?
414
00:25:47,750 --> 00:25:50,592
Student: My question
then is that seems
415
00:25:50,592 --> 00:25:54,250
to prove that that
limit is equal to 0/0.
416
00:25:54,250 --> 00:26:01,510
Professor: It proves more
than it is equal to 0/0.
417
00:26:01,510 --> 00:26:04,925
It's the ratio of this little
short thing to this longer
418
00:26:04,925 --> 00:26:06,090
thing.
419
00:26:06,090 --> 00:26:08,920
And this is getting much, much
shorter than this total length.
420
00:26:08,920 --> 00:26:09,990
You're absolutely
right that we're
421
00:26:09,990 --> 00:26:12,573
comparing two quantities which
are going to 0, but one of them
422
00:26:12,573 --> 00:26:14,357
is much smaller than the other.
423
00:26:14,357 --> 00:26:16,190
In the other case we
compared two quantities
424
00:26:16,190 --> 00:26:18,148
which were both going to
0 and they both end up
425
00:26:18,148 --> 00:26:19,840
being about equal in length.
426
00:26:19,840 --> 00:26:23,060
Here the previous one
was this green one.
427
00:26:23,060 --> 00:26:25,000
Here it's this
little tiny bit here
428
00:26:25,000 --> 00:26:32,840
and it's way shorter than
the 2 theta distance.
429
00:26:32,840 --> 00:26:33,900
Yeah, another question.
430
00:26:33,900 --> 00:26:34,745
Student: cos theta - 1 over
cos theta is the same as 1-
431
00:26:34,745 --> 00:26:35,620
cos theta over theta?
432
00:26:35,620 --> 00:26:45,740
Professor: cos theta - 1 over...
433
00:26:45,740 --> 00:26:49,780
Student: [INAUDIBLE]
434
00:26:49,780 --> 00:26:56,690
Professor: So here, what I
wrote is (cos delta x - 1)
435
00:26:56,690 --> 00:27:01,620
/ delta x, OK, and I
claimed that it goes to 0.
436
00:27:01,620 --> 00:27:12,370
Here, I wrote minus that, that
is I replaced delta x by theta.
437
00:27:12,370 --> 00:27:22,720
But then I wrote this thing.
438
00:27:22,720 --> 00:27:26,550
So (cos theta - 1) minus
1 is the negative of this.
439
00:27:26,550 --> 00:27:28,070
Alright?
440
00:27:28,070 --> 00:27:29,510
And if I show that
this goes to 0,
441
00:27:29,510 --> 00:27:33,162
it's the same as showing
the other one goes to 0.
442
00:27:33,162 --> 00:27:33,870
Another question?
443
00:27:33,870 --> 00:27:39,050
Student: [INAUDIBLE]
444
00:27:39,050 --> 00:27:42,640
Professor: So the question
is, what about this business
445
00:27:42,640 --> 00:27:44,220
about arc length.
446
00:27:44,220 --> 00:27:48,610
So the word arc length,
that orange shape is an arc.
447
00:27:48,610 --> 00:27:51,350
And we're just talking about
the length of that arc,
448
00:27:51,350 --> 00:27:52,837
and so we're calling
it arc length.
449
00:27:52,837 --> 00:27:54,420
That's what the word
arc length means,
450
00:27:54,420 --> 00:27:55,920
it just means the
length of the arc.
451
00:27:55,920 --> 00:28:03,620
Student: [INAUDIBLE]
452
00:28:03,620 --> 00:28:06,030
Professor: Why is
this length theta?
453
00:28:06,030 --> 00:28:08,380
Ah, OK so this is a
very important point,
454
00:28:08,380 --> 00:28:11,480
and in fact it's the very next
point that I wanted to make.
455
00:28:11,480 --> 00:28:15,130
Namely, notice that
in this calculation
456
00:28:15,130 --> 00:28:19,700
it was very important
that we used length.
457
00:28:19,700 --> 00:28:24,140
And that means that the way
that we're measuring theta,
458
00:28:24,140 --> 00:28:32,330
is in what are known as radians.
459
00:28:32,330 --> 00:28:36,870
Right, so that applies to both B
and A, it's a scale change in A
460
00:28:36,870 --> 00:28:39,470
and doesn't really matter
but in B it's very important.
461
00:28:39,470 --> 00:28:42,980
The only way that
this orange length
462
00:28:42,980 --> 00:28:46,140
is comparable to
this green length,
463
00:28:46,140 --> 00:28:48,600
the vertical is
comparable to the arc,
464
00:28:48,600 --> 00:28:53,520
is if we measure them in terms
of the same notion of length.
465
00:28:53,520 --> 00:28:56,360
If we measure them in
degrees, for example,
466
00:28:56,360 --> 00:28:58,890
it would be completely wrong.
467
00:28:58,890 --> 00:29:02,720
We divide up the angles
into 360 degrees,
468
00:29:02,720 --> 00:29:04,320
and that's the wrong
unit of measure.
469
00:29:04,320 --> 00:29:07,990
The correct measure is the
length along the unit circle,
470
00:29:07,990 --> 00:29:09,590
which is what radians are.
471
00:29:09,590 --> 00:29:21,490
And so this is only
true if we use radians.
472
00:29:21,490 --> 00:29:33,650
So again, a little warning
here, that this is in radians.
473
00:29:33,650 --> 00:29:41,060
Now here x is in radians.
474
00:29:41,060 --> 00:29:45,690
The formulas are just wrong
if you use other units.
475
00:29:45,690 --> 00:29:46,250
Ah yeah?
476
00:29:46,250 --> 00:29:55,480
Student: [INAUDIBLE].
477
00:29:55,480 --> 00:29:57,290
Professor: OK so the
second question is why
478
00:29:57,290 --> 00:30:00,630
is this crazy length here 1.
479
00:30:00,630 --> 00:30:08,000
And the reason is
that the relationship
480
00:30:08,000 --> 00:30:11,780
between this picture up here
and this picture down here,
481
00:30:11,780 --> 00:30:16,150
is that I'm drawing
a different shape.
482
00:30:16,150 --> 00:30:18,800
Namely, what I'm
really imagining here
483
00:30:18,800 --> 00:30:21,460
is a much, much smaller theta.
484
00:30:21,460 --> 00:30:22,340
OK?
485
00:30:22,340 --> 00:30:25,430
And then I'm blowing
that up in scale.
486
00:30:25,430 --> 00:30:27,880
So this scale of this
picture down here
487
00:30:27,880 --> 00:30:31,430
is very different from the
scale of the picture up there.
488
00:30:31,430 --> 00:30:34,850
And if the angle is
very, very, very small
489
00:30:34,850 --> 00:30:37,835
then one has to be
very, very long in order
490
00:30:37,835 --> 00:30:39,430
for me to finish the circle.
491
00:30:39,430 --> 00:30:41,820
So, in other words,
this length is 1
492
00:30:41,820 --> 00:30:44,690
because that's what
I'm insisting on.
493
00:30:44,690 --> 00:30:47,910
So, I'm claiming that that's
how I define this circle,
494
00:30:47,910 --> 00:30:52,490
to be of unit radius.
495
00:30:52,490 --> 00:30:53,200
Another question?
496
00:30:53,200 --> 00:31:05,194
Student: [INAUDIBLE]
the ratio between 1 -
497
00:31:05,194 --> 00:31:07,360
cos theta and theta will
get closer and closer to 1.
498
00:31:07,360 --> 00:31:08,651
I don't understand [INAUDIBLE].
499
00:31:08,651 --> 00:31:21,550
Professor: OK, so
the question is it's
500
00:31:21,550 --> 00:31:25,810
hard to visualize
this fact here.
501
00:31:25,810 --> 00:31:30,829
So let me, let me take you
through a couple of steps,
502
00:31:30,829 --> 00:31:33,370
because I think probably other
people are also having trouble
503
00:31:33,370 --> 00:31:34,930
with this visualization.
504
00:31:34,930 --> 00:31:36,830
The first part of
the visualization I'm
505
00:31:36,830 --> 00:31:39,570
gonna try to demonstrate
on this picture up here.
506
00:31:39,570 --> 00:31:41,440
The first part of
the visualization
507
00:31:41,440 --> 00:31:45,350
is that I should think
of a beak of a bird
508
00:31:45,350 --> 00:31:47,880
closing down, getting
narrower and narrower.
509
00:31:47,880 --> 00:31:50,430
So in other words,
the angle theta
510
00:31:50,430 --> 00:31:54,050
has to be getting smaller
and smaller and smaller.
511
00:31:54,050 --> 00:31:55,780
OK, that's the first step.
512
00:31:55,780 --> 00:31:58,650
So that's the process
that we're talking about.
513
00:31:58,650 --> 00:32:03,880
Now, in order to draw that, once
theta gets incredibly narrow,
514
00:32:03,880 --> 00:32:06,450
in order to depict that I have
to blow the whole picture back
515
00:32:06,450 --> 00:32:07,700
up in order be able to see it.
516
00:32:07,700 --> 00:32:09,430
Otherwise it just
disappears on me.
517
00:32:09,430 --> 00:32:12,090
In fact in the limit theta
= 0, it's meaningless.
518
00:32:12,090 --> 00:32:13,080
It's just a flat line.
519
00:32:13,080 --> 00:32:15,500
That's the whole problem
with these tricky limits.
520
00:32:15,500 --> 00:32:18,100
They're meaningless right
at the zero-zero level.
521
00:32:18,100 --> 00:32:22,010
It's only just a little away
that they're actually useful,
522
00:32:22,010 --> 00:32:25,890
that you get useful geometric
information out of them.
523
00:32:25,890 --> 00:32:27,300
So we're just a little away.
524
00:32:27,300 --> 00:32:31,160
So that's what this picture down
below in part A is meant to be.
525
00:32:31,160 --> 00:32:34,135
It's supposed to be that theta
is open a tiny crack, just
526
00:32:34,135 --> 00:32:35,161
a little bit.
527
00:32:35,161 --> 00:32:37,285
And the smallest I can draw
it on the board for you
528
00:32:37,285 --> 00:32:40,160
to visualize it is using the
whole length of the blackboard
529
00:32:40,160 --> 00:32:41,390
here for that.
530
00:32:41,390 --> 00:32:43,390
So I've opened a little
tiny bit and by the time
531
00:32:43,390 --> 00:32:45,139
we get to the other
end of the blackboard,
532
00:32:45,139 --> 00:32:46,510
of course it's fairly wide.
533
00:32:46,510 --> 00:32:50,291
But this angle theta
is a very small angle.
534
00:32:50,291 --> 00:32:50,790
Alright?
535
00:32:50,790 --> 00:32:56,670
So I'm trying to imagine what
happens as this collapses.
536
00:32:56,670 --> 00:32:59,670
Now, when I imagine
that I have to imagine
537
00:32:59,670 --> 00:33:02,190
a geometric interpretation
of both the numerator
538
00:33:02,190 --> 00:33:06,020
and the denominator
of this quantity here.
539
00:33:06,020 --> 00:33:08,300
And just see what happens.
540
00:33:08,300 --> 00:33:14,020
Now I claimed the numerator is
this little tiny bit over here
541
00:33:14,020 --> 00:33:19,004
and the denominator is actually
half of this whole length here.
542
00:33:19,004 --> 00:33:20,420
But the factor of
2 doesn't matter
543
00:33:20,420 --> 00:33:24,250
when you're seeing whether
something tends to 0 or not.
544
00:33:24,250 --> 00:33:24,990
Alright?
545
00:33:24,990 --> 00:33:26,656
And I claimed that
if you stare at this,
546
00:33:26,656 --> 00:33:28,460
it's clear that
this is much shorter
547
00:33:28,460 --> 00:33:32,750
than that vertical curve there.
548
00:33:32,750 --> 00:33:35,140
And I'm claiming, so this
is what you have to imagine,
549
00:33:35,140 --> 00:33:38,820
is this as it gets smaller
and smaller and smaller still
550
00:33:38,820 --> 00:33:41,590
that has the same effect of
this thing going way, way way,
551
00:33:41,590 --> 00:33:45,510
farther away and this vertical
curve getting closer and closer
552
00:33:45,510 --> 00:33:47,160
and closer to the green.
553
00:33:47,160 --> 00:33:52,530
And so that the gap between
them gets tiny and goes to 0.
554
00:33:52,530 --> 00:33:53,440
Alright?
555
00:33:53,440 --> 00:33:56,580
So not only does it go to
0, that's not enough for us,
556
00:33:56,580 --> 00:34:01,540
but it also goes to 0 faster
than this theta goes to 0.
557
00:34:01,540 --> 00:34:05,280
And I hope the evidence
is pretty strong here
558
00:34:05,280 --> 00:34:10,220
because it's so tiny
already at this stage.
559
00:34:10,220 --> 00:34:12,350
Alright.
560
00:34:12,350 --> 00:34:16,140
We are going to move
forward and you'll
561
00:34:16,140 --> 00:34:18,420
have to ponder these
things some other time.
562
00:34:18,420 --> 00:34:20,640
So I'm gonna give you
an even harder thing
563
00:34:20,640 --> 00:34:26,600
to visualize now so be prepared.
564
00:34:26,600 --> 00:34:36,310
OK, so now, the next
thing that I'd like to do
565
00:34:36,310 --> 00:34:37,700
is to give you a second proof.
566
00:34:37,700 --> 00:34:39,430
Because it really
is important, I
567
00:34:39,430 --> 00:34:47,650
think, to understand this
particular fact more thoroughly
568
00:34:47,650 --> 00:34:49,310
and also to get
a lot of practice
569
00:34:49,310 --> 00:34:51,450
with sines and cosines.
570
00:34:51,450 --> 00:34:55,930
So I'm gonna give
you a geometric proof
571
00:34:55,930 --> 00:35:11,010
of the formula for sine here,
for the derivative of sine.
572
00:35:11,010 --> 00:35:13,420
So here we go.
573
00:35:13,420 --> 00:35:26,280
This is a geometric
proof of this fact.
574
00:35:26,280 --> 00:35:29,400
This is for all theta.
575
00:35:29,400 --> 00:35:33,100
So far we only did
it for theta = 0
576
00:35:33,100 --> 00:35:36,360
and now we're going to
do it for all theta.
577
00:35:36,360 --> 00:35:38,930
So this is a different
proof, but it uses
578
00:35:38,930 --> 00:35:42,420
exactly the same principles.
579
00:35:42,420 --> 00:35:45,390
Right?
580
00:35:45,390 --> 00:35:51,350
So, I want do this by
drawing another picture,
581
00:35:51,350 --> 00:35:54,790
and the picture is
going to describe
582
00:35:54,790 --> 00:35:59,370
y, which is sin
theta, which is if you
583
00:35:59,370 --> 00:36:22,160
like the vertical position
of some circular motion.
584
00:36:22,160 --> 00:36:27,170
So I'm imagining that something
is going around in a circle.
585
00:36:27,170 --> 00:36:30,620
Some particle is going
around in a circle.
586
00:36:30,620 --> 00:36:36,377
And so here's the
circle, here the origin.
587
00:36:36,377 --> 00:36:37,460
This is the unit distance.
588
00:36:37,460 --> 00:36:43,640
And right now it happens to
be at this location P. Maybe
589
00:36:43,640 --> 00:36:46,160
we'll put P a little over here.
590
00:36:46,160 --> 00:36:50,260
And here's the angle theta.
591
00:36:50,260 --> 00:36:51,560
And now we're going to move it.
592
00:36:51,560 --> 00:36:54,490
We're going to vary
theta and we're
593
00:36:54,490 --> 00:36:56,920
interested in the
rate of change of y.
594
00:36:56,920 --> 00:37:00,290
So y is the height
of P but we're
595
00:37:00,290 --> 00:37:01,870
gonna move it to
another location.
596
00:37:01,870 --> 00:37:07,400
We'll move it along
the circle to Q. Right?
597
00:37:07,400 --> 00:37:09,200
So here it is.
598
00:37:09,200 --> 00:37:12,360
Here's the thing.
599
00:37:12,360 --> 00:37:14,450
So how far did we move it?
600
00:37:14,450 --> 00:37:18,570
Well we moved it by
an angle delta theta.
601
00:37:18,570 --> 00:37:20,540
So we started theta,
theta is going
602
00:37:20,540 --> 00:37:22,313
to be fixed in this
argument, and we're
603
00:37:22,313 --> 00:37:23,990
going to move a little
bit delta theta.
604
00:37:23,990 --> 00:37:26,180
And now we're just
gonna try to figure out
605
00:37:26,180 --> 00:37:28,510
how far the thing moved.
606
00:37:28,510 --> 00:37:30,440
Well, in order to
do that we've got
607
00:37:30,440 --> 00:37:34,590
to keep track of the height,
the vertical displacement here.
608
00:37:34,590 --> 00:37:38,334
So we're going to draw
this right angle here, this
609
00:37:38,334 --> 00:37:42,440
is the position R. And
then this distance here
610
00:37:42,440 --> 00:37:45,500
is the change in y.
611
00:37:45,500 --> 00:37:46,000
Alright?
612
00:37:46,000 --> 00:37:50,190
So the picture is
we have something
613
00:37:50,190 --> 00:37:52,110
moving around a unit circle.
614
00:37:52,110 --> 00:37:53,680
A point moving
around a unit circle.
615
00:37:53,680 --> 00:37:57,635
It starts at P, it moves to
Q. It moves from angle theta
616
00:37:57,635 --> 00:37:59,590
to angle theta plus delta theta.
617
00:37:59,590 --> 00:38:05,770
And the issue is how
much does y move?
618
00:38:05,770 --> 00:38:07,340
And the formula
for y is sin theta.
619
00:38:07,340 --> 00:38:29,710
So that's telling us the
rate of change of sin theta.
620
00:38:29,710 --> 00:38:34,500
Alright, well so let's just
try to think a little bit
621
00:38:34,500 --> 00:38:35,980
about what this is.
622
00:38:35,980 --> 00:38:37,736
So, first of all,
I've already said this
623
00:38:37,736 --> 00:38:39,300
and I'm going to repeat it here.
624
00:38:39,300 --> 00:38:41,650
Delta y is PR.
625
00:38:41,650 --> 00:38:44,490
It's going from P and
going straight up to R.
626
00:38:44,490 --> 00:38:47,080
That's how far y moves.
627
00:38:47,080 --> 00:38:49,640
That's the change in y
That's what I said up
628
00:38:49,640 --> 00:38:52,910
in the right hand corner there.
629
00:38:52,910 --> 00:38:53,470
Oops.
630
00:38:53,470 --> 00:38:56,430
I said PR but I wrote PQ.
631
00:38:56,430 --> 00:38:59,130
Alright, that's not a good idea.
632
00:38:59,130 --> 00:38:59,630
Alright.
633
00:38:59,630 --> 00:39:03,090
So delta Y is PR.
634
00:39:03,090 --> 00:39:07,160
And now I want to draw the
diagram again one time.
635
00:39:07,160 --> 00:39:16,030
So here's Q, here's R,
and here's P, and here's
636
00:39:16,030 --> 00:39:17,300
my triangle.
637
00:39:17,300 --> 00:39:24,300
And now what I'd like to
do is draw this curve here
638
00:39:24,300 --> 00:39:26,970
which is a piece of
the arc of the circle.
639
00:39:26,970 --> 00:39:29,560
But really what I
want to keep in mind
640
00:39:29,560 --> 00:39:33,300
is something that I did also
in all these other arguments.
641
00:39:33,300 --> 00:39:35,330
Which is, maybe I
should have called
642
00:39:35,330 --> 00:39:38,011
this orange, that I'm gonna
think of the straight line
643
00:39:38,011 --> 00:39:38,510
between.
644
00:39:38,510 --> 00:39:41,165
So it's the straight line
approximation to the curve
645
00:39:41,165 --> 00:39:45,009
that we're always interested in.
646
00:39:45,009 --> 00:39:46,550
So the straight line
is much simpler,
647
00:39:46,550 --> 00:39:48,610
because then we just
have a triangle here.
648
00:39:48,610 --> 00:39:52,017
And in fact it's
a right triangle.
649
00:39:52,017 --> 00:39:54,350
Right, so we have the geometry
of a right triangle which
650
00:39:54,350 --> 00:39:59,210
is going to now let us do
all of our calculations.
651
00:39:59,210 --> 00:40:03,640
OK, so now the key step
is this same principle
652
00:40:03,640 --> 00:40:07,560
that we already used which is
that short pieces of curves
653
00:40:07,560 --> 00:40:09,040
are nearly straight.
654
00:40:09,040 --> 00:40:12,000
So that means that this piece
of the circular arc here from P
655
00:40:12,000 --> 00:40:17,323
to Q is practically the same
as the straight segment from P
656
00:40:17,323 --> 00:40:24,290
to Q. So, that's this
principle that - well,
657
00:40:24,290 --> 00:40:27,580
let's put it over
here - Is that PQ
658
00:40:27,580 --> 00:40:33,190
is practically the same as the
straight segment from P to Q.
659
00:40:33,190 --> 00:40:35,820
So how are we going to use that?
660
00:40:35,820 --> 00:40:37,750
We want to use
that quantitatively
661
00:40:37,750 --> 00:40:39,080
in the following way.
662
00:40:39,080 --> 00:40:42,350
What we want to notice is
that the distance from P to Q
663
00:40:42,350 --> 00:40:46,120
is approximately delta theta.
664
00:40:46,120 --> 00:40:46,620
Right?
665
00:40:46,620 --> 00:40:49,056
Because the arc length
along that curve,
666
00:40:49,056 --> 00:40:50,680
the length of the
curve is delta theta.
667
00:40:50,680 --> 00:40:55,050
So the length of the green which
is PQ is almost delta theta.
668
00:40:55,050 --> 00:41:01,690
So this is essentially delta
theta, this distance here.
669
00:41:01,690 --> 00:41:05,360
Now the second step, which
is a little trickier,
670
00:41:05,360 --> 00:41:08,980
is that we have to work
out what this angle is.
671
00:41:08,980 --> 00:41:11,640
So our goal, and I'm gonna put
it one step below because I'm
672
00:41:11,640 --> 00:41:14,280
gonna put the geometric
reasoning in between,
673
00:41:14,280 --> 00:41:20,980
is I need to figure out
what the angle QPR is.
674
00:41:20,980 --> 00:41:23,340
If I can figure out
what this angle is,
675
00:41:23,340 --> 00:41:26,630
then I'll be able to figure out
what this vertical distance is
676
00:41:26,630 --> 00:41:28,840
because I'll know the
hypotenuse and I'll
677
00:41:28,840 --> 00:41:30,900
know the angle so I'll be
able to figure out what
678
00:41:30,900 --> 00:41:36,610
the side of the triangle is.
679
00:41:36,610 --> 00:41:40,220
So now let me show you
why that's possible to do.
680
00:41:40,220 --> 00:41:43,400
So in order to do that first of
all I'm gonna trade the boards
681
00:41:43,400 --> 00:41:50,600
and show you where
the line PQ is.
682
00:41:50,600 --> 00:41:54,370
So the line PQ is here.
683
00:41:54,370 --> 00:41:56,470
That's the whole thing.
684
00:41:56,470 --> 00:42:00,190
And the key point about this
line that I need you to realize
685
00:42:00,190 --> 00:42:04,230
is that it's practically
perpendicular,
686
00:42:04,230 --> 00:42:08,910
it's almost perpendicular,
to this ray here.
687
00:42:08,910 --> 00:42:09,550
Alright?
688
00:42:09,550 --> 00:42:12,540
It's not quite because the
distance between P to Q
689
00:42:12,540 --> 00:42:13,262
is non-zero.
690
00:42:13,262 --> 00:42:14,720
So it isn't quite,
but in the limit
691
00:42:14,720 --> 00:42:17,070
it's going to be perpendicular.
692
00:42:17,070 --> 00:42:18,100
Exactly perpendicular.
693
00:42:18,100 --> 00:42:20,980
The tangent line to the circle.
694
00:42:20,980 --> 00:42:26,820
So the key thing
that I'm going to use
695
00:42:26,820 --> 00:42:35,131
is that PQ is almost
perpendicular to OP.
696
00:42:35,131 --> 00:42:35,630
Alright?
697
00:42:35,630 --> 00:42:37,710
The ray from the
origin is basically
698
00:42:37,710 --> 00:42:39,900
perpendicular to
that green line.
699
00:42:39,900 --> 00:42:42,540
And then the second
thing I'm going to use
700
00:42:42,540 --> 00:42:53,121
is something that's obvious
which is that PR is vertical.
701
00:42:53,121 --> 00:42:53,620
OK?
702
00:42:53,620 --> 00:42:58,080
So those are the two pieces of
geometry that I need to see.
703
00:42:58,080 --> 00:43:02,680
And now notice what's happening
upstairs on the picture here
704
00:43:02,680 --> 00:43:05,050
in the upper right.
705
00:43:05,050 --> 00:43:09,550
What I have is the
angle theta is the angle
706
00:43:09,550 --> 00:43:12,910
between the horizontal and OP.
707
00:43:12,910 --> 00:43:14,350
That's angle theta.
708
00:43:14,350 --> 00:43:17,990
If I rotate it by ninety
degree, the horizontal
709
00:43:17,990 --> 00:43:18,880
becomes vertical.
710
00:43:18,880 --> 00:43:21,730
It becomes PR and
the other thing
711
00:43:21,730 --> 00:43:24,810
rotated by 90 degrees
becomes the green line.
712
00:43:24,810 --> 00:43:30,080
So the angle that I'm talking
about I get by taking this guy
713
00:43:30,080 --> 00:43:32,470
and rotating it by 90 degrees.
714
00:43:32,470 --> 00:43:33,800
It's the same angle.
715
00:43:33,800 --> 00:43:38,230
So that means that this angle
here is essentially theta.
716
00:43:38,230 --> 00:43:39,880
That's what this angle is.
717
00:43:39,880 --> 00:43:41,840
Let me repeat that
one more time.
718
00:43:41,840 --> 00:43:43,290
We started out
with an angle that
719
00:43:43,290 --> 00:43:46,540
looks like this, which is
the horizontal-- that's
720
00:43:46,540 --> 00:43:48,600
the origin straight
out horizontally.
721
00:43:48,600 --> 00:43:50,560
That's the thing labeled 1.
722
00:43:50,560 --> 00:43:54,805
That distance there.
723
00:43:54,805 --> 00:43:56,430
That's my right arm
which is down here.
724
00:43:56,430 --> 00:44:01,180
My left arm is pointing up
and it's going from the origin
725
00:44:01,180 --> 00:44:06,430
to the point P. So
here's the horizontal
726
00:44:06,430 --> 00:44:09,370
and the angle between
them is theta.
727
00:44:09,370 --> 00:44:13,930
And now, what I claim is that
if I rotate by 90 degrees up,
728
00:44:13,930 --> 00:44:16,840
like this, without
changing anything -
729
00:44:16,840 --> 00:44:18,860
so that was what I
did - the horizontal
730
00:44:18,860 --> 00:44:21,160
will become a vertical.
731
00:44:21,160 --> 00:44:22,990
That's PR.
732
00:44:22,990 --> 00:44:25,030
That's going up, PR.
733
00:44:25,030 --> 00:44:32,080
And if I rotate OP 90
degrees, that's exactly PQ.
734
00:44:32,080 --> 00:44:33,540
OK?
735
00:44:33,540 --> 00:44:42,560
So let me draw it
on there one time.
736
00:44:42,560 --> 00:44:45,220
Let's do it with
some arrows here.
737
00:44:45,220 --> 00:44:49,620
So I started out with
this and then, we'll
738
00:44:49,620 --> 00:44:56,470
label this as orange,
OK so red to orange,
739
00:44:56,470 --> 00:45:01,840
and then I rotate by
90 degrees and the red
740
00:45:01,840 --> 00:45:07,060
becomes this starting from P
and the orange rotates around 90
741
00:45:07,060 --> 00:45:11,370
degrees and becomes
this thing here.
742
00:45:11,370 --> 00:45:12,190
Alright?
743
00:45:12,190 --> 00:45:16,252
So this angle here is
the same as the other one
744
00:45:16,252 --> 00:45:18,460
which I've just drawn.
745
00:45:18,460 --> 00:45:27,030
Different vertices
for the angles.
746
00:45:27,030 --> 00:45:28,210
OK?
747
00:45:28,210 --> 00:45:31,100
Well I didn't say
that all arguments
748
00:45:31,100 --> 00:45:36,450
were supposed to be easy.
749
00:45:36,450 --> 00:45:38,180
Alright, so I claim
that the conclusion
750
00:45:38,180 --> 00:45:43,200
is that this angle is
approximately theta.
751
00:45:43,200 --> 00:45:45,670
And now we can finish
our calculation,
752
00:45:45,670 --> 00:45:48,300
because we have something with
the hypotenuse being delta
753
00:45:48,300 --> 00:45:53,340
theta and the angle being theta
and so this segment here PR is
754
00:45:53,340 --> 00:45:56,830
approximately the
hypotenuse length
755
00:45:56,830 --> 00:46:02,430
times the cosine of the angle.
756
00:46:02,430 --> 00:46:05,740
And that is exactly
what we wanted.
757
00:46:05,740 --> 00:46:10,830
If we divide, we divide by
delta theta, we get (delta y)
758
00:46:10,830 --> 00:46:17,030
/ (delta theta) is
approximately cos theta.
759
00:46:17,030 --> 00:46:20,700
And that's the same thing as...
760
00:46:20,700 --> 00:46:22,370
So what we want in
the limit is exactly
761
00:46:22,370 --> 00:46:24,900
the delta theta going to 0
of (delta y) / (delta theta)
762
00:46:24,900 --> 00:46:28,020
is equal to cos theta.
763
00:46:28,020 --> 00:46:32,270
So we get an approximation on
a scale that we can visualize
764
00:46:32,270 --> 00:46:39,590
and in the limit the
formula is exact.
765
00:46:39,590 --> 00:46:44,060
OK, so that is a geometric
argument for the same result.
766
00:46:44,060 --> 00:46:47,940
Namely that the derivative
of sine is cosine.
767
00:46:47,940 --> 00:46:48,440
Yeah?
768
00:46:48,440 --> 00:46:51,590
Student: [INAUDIBLE].
769
00:46:51,590 --> 00:46:54,420
Professor: You will have to do
some kind of geometric proofs
770
00:46:54,420 --> 00:46:55,840
sometimes.
771
00:46:55,840 --> 00:46:59,730
When you'll really need
this is probably in 18.02.
772
00:46:59,730 --> 00:47:03,020
So you'll need to make
reasoning like this.
773
00:47:03,020 --> 00:47:05,730
This is, for example, the
way that you actually develop
774
00:47:05,730 --> 00:47:08,200
the theory of arc length.
775
00:47:08,200 --> 00:47:13,250
Dealing with delta x's and
delta y's is a common tool.
776
00:47:13,250 --> 00:47:17,810
Alright, I have one
more thing that I
777
00:47:17,810 --> 00:47:25,070
want to talk about today,
which is some general rules.
778
00:47:25,070 --> 00:47:28,230
We took a little bit more time
than I expected with this.
779
00:47:28,230 --> 00:47:31,780
So what I'm gonna do is
just tell you the rules
780
00:47:31,780 --> 00:47:36,330
and we'll discuss
them in a few days.
781
00:47:36,330 --> 00:47:50,180
So let me tell you
the general rules.
782
00:47:50,180 --> 00:48:00,170
So these were the specific ones
and here are some general ones.
783
00:48:00,170 --> 00:48:08,490
So the first one is
called the product rule.
784
00:48:08,490 --> 00:48:11,010
And what it says is that if
you take the product of two
785
00:48:11,010 --> 00:48:13,630
functions and
differentiate them,
786
00:48:13,630 --> 00:48:18,180
you get the derivative of
one times the other plus
787
00:48:18,180 --> 00:48:22,060
the other times the
derivative of the one.
788
00:48:22,060 --> 00:48:24,350
Now the way that you
should remember this,
789
00:48:24,350 --> 00:48:27,970
and the way that I'll
carry out the proof,
790
00:48:27,970 --> 00:48:40,010
is that you should think of it
is you change one at a time.
791
00:48:40,010 --> 00:48:44,080
And this is a very useful way of
thinking about differentiation
792
00:48:44,080 --> 00:48:46,530
when you have
things which depend
793
00:48:46,530 --> 00:48:49,660
on more than one function.
794
00:48:49,660 --> 00:48:53,750
So this is a general procedure.
795
00:48:53,750 --> 00:48:59,050
The second formula that
I wanted to mention
796
00:48:59,050 --> 00:49:07,470
is called the quotient rule
and that says the following.
797
00:49:07,470 --> 00:49:13,220
That (u / v) prime
has a formula as well.
798
00:49:13,220 --> 00:49:21,280
And the formula is
(u'v - uv' ) / v^2.
799
00:49:21,280 --> 00:49:23,240
So this is our second formula.
800
00:49:23,240 --> 00:49:31,360
Let me just mention, both of
them are extremely valuable
801
00:49:31,360 --> 00:49:33,170
and you'll use
them all the time.
802
00:49:33,170 --> 00:49:43,600
This one of course only
works when v is not 0.
803
00:49:43,600 --> 00:49:46,926
Alright, so because
we're out of time
804
00:49:46,926 --> 00:49:48,300
we're not gonna
prove these today
805
00:49:48,300 --> 00:49:50,508
but we'll prove these next
time and you're definitely
806
00:49:50,508 --> 00:49:53,760
going to be responsible
for these kinds of proofs.