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Professor: In today's
lecture I want
00:00:25.150 --> 00:00:27.715
to develop several
more formulas that
00:00:27.715 --> 00:00:32.410
will allow us to reach our goal
of differentiating everything.
00:00:32.410 --> 00:00:41.800
So these are
derivative formulas,
00:00:41.800 --> 00:00:45.560
and they come in two flavors.
00:00:45.560 --> 00:00:52.070
The first kind is
specific, so some specific
00:00:52.070 --> 00:00:55.460
function we're giving
the derivative of.
00:00:55.460 --> 00:01:00.740
And that would be, for
example, x^n or (1/x) .
00:01:00.740 --> 00:01:05.220
Those are the ones that we
did a couple of lectures ago.
00:01:05.220 --> 00:01:10.510
And then there are
general formulas,
00:01:10.510 --> 00:01:12.400
and the general
ones don't actually
00:01:12.400 --> 00:01:14.820
give you a formula for
a specific function
00:01:14.820 --> 00:01:18.750
but tell you something like,
if you take two functions
00:01:18.750 --> 00:01:20.840
and add them together,
their derivative
00:01:20.840 --> 00:01:23.470
is the sum of the derivatives.
00:01:23.470 --> 00:01:27.220
Or if you multiply by a
constant, for example,
00:01:27.220 --> 00:01:31.490
so c times u, the
derivative of that
00:01:31.490 --> 00:01:38.990
is c times u' where
c is constant.
00:01:38.990 --> 00:01:41.630
All right, so these
kinds of formulas
00:01:41.630 --> 00:01:46.070
are very useful, both the
specific and the general kind.
00:01:46.070 --> 00:02:03.630
For example, we need both
kinds for polynomials.
00:02:03.630 --> 00:02:05.755
And more generally, pretty
much any set of formulas
00:02:05.755 --> 00:02:07.713
that we give you, will
give you a few functions
00:02:07.713 --> 00:02:09.180
to start out with
and then you'll
00:02:09.180 --> 00:02:16.480
be able to generate lots more
by these general formulas.
00:02:16.480 --> 00:02:21.000
So today, we wanna concentrate
on the trig functions,
00:02:21.000 --> 00:02:27.570
and so we'll start out with
some specific formulas.
00:02:27.570 --> 00:02:29.580
And they're going
to be the formulas
00:02:29.580 --> 00:02:33.000
for the derivative
of the sine function
00:02:33.000 --> 00:02:37.910
and the cosine function.
00:02:37.910 --> 00:02:41.240
So that's what we'll spend the
first part of the lecture on,
00:02:41.240 --> 00:02:46.510
and at the same time I hope to
get you very used to dealing
00:02:46.510 --> 00:02:49.130
with trig functions,
although that's something
00:02:49.130 --> 00:02:55.630
that you should think
of as a gradual process.
00:02:55.630 --> 00:02:58.310
Alright, so in order
to calculate these,
00:02:58.310 --> 00:03:03.720
I'm gonna start over here and
just start the calculation.
00:03:03.720 --> 00:03:05.270
So here we go.
00:03:05.270 --> 00:03:08.110
Let's check what happens
with the sine function.
00:03:08.110 --> 00:03:15.590
So, I take sin (x + delta
x), I subtract sin x
00:03:15.590 --> 00:03:22.090
and I divide by delta x.
00:03:22.090 --> 00:03:24.270
Right, so this is the
difference quotient
00:03:24.270 --> 00:03:26.850
and eventually I'm gonna have
to take the limit as delta
00:03:26.850 --> 00:03:29.320
x goes to 0.
00:03:29.320 --> 00:03:31.850
And there's really
only one thing
00:03:31.850 --> 00:03:36.830
we can do with this to
simplify or change it,
00:03:36.830 --> 00:03:42.440
and that is to use the sum
formula for the sine function.
00:03:42.440 --> 00:03:43.560
So, that's this.
00:03:43.560 --> 00:03:48.680
That's sin x cos delta x plus--
00:03:54.490 --> 00:03:56.070
Oh, that's not what it is?
00:03:56.070 --> 00:04:01.040
OK, so what is it?
sin x sin delta x.
00:04:01.040 --> 00:04:02.940
OK, good.
00:04:02.940 --> 00:04:07.500
Plus cosine.
00:04:07.500 --> 00:04:09.290
No?
00:04:09.290 --> 00:04:10.560
Oh, OK.
00:04:10.560 --> 00:04:14.470
So which is it?
00:04:14.470 --> 00:04:16.176
OK.
00:04:16.176 --> 00:04:17.300
Alright, let's take a vote.
00:04:17.300 --> 00:04:20.020
Is it sine, sine, or
is it sine, cosine?
00:04:20.020 --> 00:04:22.580
Audience: [INAUDIBLE]
00:04:22.580 --> 00:04:31.100
Professor: OK, so is this
going to be... cosine.
00:04:31.100 --> 00:04:34.450
All right, you better remember
these formulas, alright?
00:04:34.450 --> 00:04:37.291
OK, turns out that
it's sine, cosine.
00:04:37.291 --> 00:04:37.790
All right.
00:04:37.790 --> 00:04:39.640
Cosine, sine.
00:04:39.640 --> 00:04:47.310
So here we go, no gotta
do x here, sin (delta x).
00:04:47.310 --> 00:04:51.100
Alright, so now there's lots
of places to get confused here,
00:04:51.100 --> 00:04:55.160
and you're gonna need to
make sure you get it right.
00:04:55.160 --> 00:04:59.230
Alright, so we're gonna put
those in parentheses here.
00:04:59.230 --> 00:05:10.310
sin (a + b) is sin a
cos b plus cos a sin b.
00:05:10.310 --> 00:05:12.740
All right, now that's
what I did over here,
00:05:12.740 --> 00:05:21.430
except the letter x was a,
and the letter b was delta x.
00:05:21.430 --> 00:05:23.560
Now that's just the first part.
00:05:23.560 --> 00:05:26.980
That's just this part
of the expression.
00:05:26.980 --> 00:05:29.134
I still have to remember
the minus sin x.
00:05:29.134 --> 00:05:30.050
That comes at the end.
00:05:30.050 --> 00:05:32.120
Minus sin x.
00:05:32.120 --> 00:05:37.900
And then, I have to remember the
denominator, which is delta x.
00:05:37.900 --> 00:05:43.040
OK?
00:05:43.040 --> 00:05:47.250
Alright, so now...
00:05:47.250 --> 00:05:49.710
The next thing we're
gonna do is we're
00:05:49.710 --> 00:05:52.280
gonna try to group the terms.
00:05:52.280 --> 00:05:58.150
And the difficulty with all such
arguments is the following one:
00:05:58.150 --> 00:06:02.060
any tricky limit is
basically 0 / 0 when
00:06:02.060 --> 00:06:03.200
you set delta x equal to 0.
00:06:03.200 --> 00:06:06.480
If I set delta x equal to
0, this is sin x - sin x.
00:06:06.480 --> 00:06:08.520
So it's a 0 / 0 term.
00:06:08.520 --> 00:06:10.040
Here we have
various things which
00:06:10.040 --> 00:06:12.170
are 0 and various things
which are non-zero.
00:06:12.170 --> 00:06:17.721
We must group the terms so
that a 0 stays over a 0.
00:06:17.721 --> 00:06:19.220
Otherwise, we're
gonna have no hope.
00:06:19.220 --> 00:06:21.990
If we get some 1 / 0
term, we'll get something
00:06:21.990 --> 00:06:24.160
meaningless in the limit.
00:06:24.160 --> 00:06:27.790
So I claim that the right
thing to do here is to notice,
00:06:27.790 --> 00:06:31.630
and I'll just point
out this one thing.
00:06:31.630 --> 00:06:35.580
When delta x goes to 0,
this cosine of 0 is 1.
00:06:35.580 --> 00:06:39.630
So it doesn't cancel unless we
throw in this extra sine term
00:06:39.630 --> 00:06:40.130
here.
00:06:40.130 --> 00:06:43.620
So I'm going to use
this common factor,
00:06:43.620 --> 00:06:44.670
and combine those terms.
00:06:44.670 --> 00:06:46.480
So this is really
the only thing you're
00:06:46.480 --> 00:06:48.800
gonna have to check in this
particular calculation.
00:06:48.800 --> 00:06:50.760
So we have the
common factor of sin
00:06:50.760 --> 00:06:54.540
x, and that multiplies
something that will cancel,
00:06:54.540 --> 00:06:59.170
which is (cos delta
x - 1) / delta x.
00:06:59.170 --> 00:07:03.300
That's the first term,
and now what's left,
00:07:03.300 --> 00:07:06.470
well there's a cos
x that factors out,
00:07:06.470 --> 00:07:14.040
and then the other factor is
(sin delta x) / (delta x).
00:07:14.040 --> 00:07:20.850
OK, now does anyone
remember from last time what
00:07:20.850 --> 00:07:25.000
this thing goes to?
00:07:25.000 --> 00:07:27.580
How many people say 1?
00:07:27.580 --> 00:07:29.340
How many people say 0?
00:07:29.340 --> 00:07:31.180
All right, it's 0.
00:07:31.180 --> 00:07:33.540
That's my favorite
number, alright?
00:07:33.540 --> 00:07:34.040
0.
00:07:34.040 --> 00:07:36.180
It's the easiest
number to deal with.
00:07:36.180 --> 00:07:39.200
So this goes to
0, and that's what
00:07:39.200 --> 00:07:45.980
happens as delta x tends to 0.
00:07:45.980 --> 00:07:47.150
How about this one?
00:07:47.150 --> 00:07:51.750
This one goes to 1, my second
favorite number, almost as
00:07:51.750 --> 00:07:54.350
easy to deal with as 0.
00:07:54.350 --> 00:07:56.280
And these things are
picked for a reason.
00:07:56.280 --> 00:07:58.030
They're the simplest
numbers to deal with.
00:07:58.030 --> 00:08:06.820
So altogether, this thing as
delta x goes to 0 goes to what?
00:08:06.820 --> 00:08:09.440
I want a single person to
answer, a brave volunteer.
00:08:09.440 --> 00:08:10.330
Alright, back there.
00:08:10.330 --> 00:08:12.070
Student: Cosine
00:08:12.070 --> 00:08:14.650
Professor: Cosine,
because this factor is 0.
00:08:14.650 --> 00:08:17.560
It cancels and this factor
has a 1, so it's cosine.
00:08:17.560 --> 00:08:20.230
So it's cos x.
00:08:20.230 --> 00:08:25.840
So our conclusion over here
- and I'll put it in orange -
00:08:25.840 --> 00:08:34.920
is that the derivative of
the sine is the cosine.
00:08:34.920 --> 00:08:38.170
OK, now I still wanna label
these very important limit
00:08:38.170 --> 00:08:39.220
facts here.
00:08:39.220 --> 00:08:41.650
This one we'll call
A, and this one we're
00:08:41.650 --> 00:08:44.340
going to call B, because we
haven't checked them yet.
00:08:44.340 --> 00:08:46.340
I promised you I would
do that, and I'll
00:08:46.340 --> 00:08:48.460
have to do that this time.
00:08:48.460 --> 00:08:52.490
So we're relying on
those things being true.
00:08:52.490 --> 00:08:56.140
Now I'm gonna do the same
thing with the cosine function,
00:08:56.140 --> 00:08:58.730
except in order to do it I'm
gonna have to remember the sum
00:08:58.730 --> 00:09:00.930
rule for cosine.
00:09:00.930 --> 00:09:03.474
So we're gonna do almost
the same calculation here.
00:09:03.474 --> 00:09:05.140
We're gonna see that
that will work out,
00:09:05.140 --> 00:09:12.390
but now you have to remember
that cos (a + b) = cos cos,
00:09:12.390 --> 00:09:15.350
no it's not cosine^2, because
there are two different
00:09:15.350 --> 00:09:16.760
quantities here.
00:09:16.760 --> 00:09:23.650
It's cos a cos b - sin a sin b.
00:09:23.650 --> 00:09:31.280
All right, so you'll have
to be willing to call those
00:09:31.280 --> 00:09:34.800
forth at will right now.
00:09:34.800 --> 00:09:36.460
So let's do the cosine now.
00:09:36.460 --> 00:09:45.500
So that's cos (x + delta x)
- cos x divided by delta x.
00:09:45.500 --> 00:09:47.830
OK, there's the
difference quotient
00:09:47.830 --> 00:09:49.318
for the cosine function.
00:09:49.318 --> 00:09:51.776
And now I'm gonna do the same
thing I did before except I'm
00:09:51.776 --> 00:09:53.700
going to apply the
second rule, that
00:09:53.700 --> 00:09:55.770
is the sum rule for cosine.
00:09:55.770 --> 00:10:03.401
And that's gonna give me cos x
cos delta x - sin x sin delta
00:10:03.401 --> 00:10:03.900
x.
00:10:03.900 --> 00:10:09.300
And I have to remember again
to subtract the cosine divided
00:10:09.300 --> 00:10:11.590
by this delta x.
00:10:11.590 --> 00:10:16.370
And now I'm going to regroup
just the way I did before,
00:10:16.370 --> 00:10:22.050
and I get the common factor of
cosine multiplying (cos delta x
00:10:22.050 --> 00:10:25.180
- 1) / delta x.
00:10:25.180 --> 00:10:30.910
And here I get the sin x
but actually it's -sin x.
00:10:30.910 --> 00:10:36.320
And then I have (sin
delta x) / delta x.
00:10:36.320 --> 00:10:36.900
All right?
00:10:36.900 --> 00:10:38.570
The only difference
is this minus sign
00:10:38.570 --> 00:10:42.740
which I stuck inside there.
00:10:42.740 --> 00:10:44.240
Well that's not the
only difference,
00:10:44.240 --> 00:10:48.440
but it's a crucial difference.
00:10:48.440 --> 00:10:54.700
OK, again by A we get that this
is 0 as delta x tends to 0.
00:10:54.700 --> 00:10:56.370
And this is 1.
00:10:56.370 --> 00:10:59.590
Those are the properties
I called A and B.
00:10:59.590 --> 00:11:04.840
And so the result here
as delta x tends to 0
00:11:04.840 --> 00:11:08.470
is that we get negative sin x.
00:11:08.470 --> 00:11:11.800
That's the factor.
00:11:11.800 --> 00:11:18.880
So this guy is negative sin x.
00:11:18.880 --> 00:11:24.560
I'll put a little
box around that too.
00:11:24.560 --> 00:11:29.040
Alright, now these
formulas take a little bit
00:11:29.040 --> 00:11:34.220
of getting used
to, but before I do
00:11:34.220 --> 00:11:38.480
that I'm gonna explain to
you the proofs of A and B.
00:11:38.480 --> 00:11:44.470
So we'll get ourselves
started by mentioning that.
00:11:44.470 --> 00:11:47.570
Maybe before I do
that though, I want
00:11:47.570 --> 00:11:51.710
to show you how A and B fit into
the proofs of these theorems.
00:11:51.710 --> 00:12:06.030
So, let me just make
some remarks here.
00:12:06.030 --> 00:12:09.020
So this is just
a remark but it's
00:12:09.020 --> 00:12:15.600
meant to help you to frame
how these proofs worked.
00:12:15.600 --> 00:12:17.650
So, first of all,
I want to point out
00:12:17.650 --> 00:12:19.750
that if you take
the rate of change
00:12:19.750 --> 00:12:28.450
of sin x, no let's
start with cosine
00:12:28.450 --> 00:12:30.450
because a little
bit less obvious.
00:12:30.450 --> 00:12:33.820
If I take the rate of change
of cos x, so in other words
00:12:33.820 --> 00:12:41.180
this derivative at x =
0, then by definition
00:12:41.180 --> 00:12:45.300
this is a certain limit
as delta x goes to 0.
00:12:45.300 --> 00:12:46.860
So which one is it?
00:12:46.860 --> 00:12:51.170
Well I have to evaluate
cosine at 0 + delta
00:12:51.170 --> 00:12:53.390
x, but that's just delta x.
00:12:53.390 --> 00:12:56.240
And I have to
subtract cosine at 0.
00:12:56.240 --> 00:13:00.040
That's the base point,
but that's just 1.
00:13:00.040 --> 00:13:03.050
And then I have to
divide by delta x.
00:13:03.050 --> 00:13:06.850
And lo and behold you can see
that this is exactly the limit
00:13:06.850 --> 00:13:08.220
that we had over there.
00:13:08.220 --> 00:13:15.880
This is the one that we know is
0 by what we call property A.
00:13:15.880 --> 00:13:23.150
And similarly, if I take the
derivative of sin x at x=0,
00:13:23.150 --> 00:13:27.000
then that's going to be the
limit as delta x goes to 0
00:13:27.000 --> 00:13:30.700
of sin delta x / delta x.
00:13:30.700 --> 00:13:35.130
And that's because I should
be subtracting sine of 0
00:13:35.130 --> 00:13:37.871
but sine of 0 is 0.
00:13:37.871 --> 00:13:38.370
Right?
00:13:38.370 --> 00:13:48.510
So this is going to be 1 by our
property B. And so the remark
00:13:48.510 --> 00:13:51.030
that I want to make,
in addition to this,
00:13:51.030 --> 00:13:55.200
is something about the
structure of these two proofs.
00:13:55.200 --> 00:14:12.990
Which is the derivatives
of sine and cosine at x = 0
00:14:12.990 --> 00:14:25.500
give all values of
d/dx sin x, d/dx cos x.
00:14:25.500 --> 00:14:27.630
So that's really what this
argument is showing us,
00:14:27.630 --> 00:14:31.420
is that we just need one
rate of change at one place
00:14:31.420 --> 00:14:38.770
and then we work out
all the rest of them.
00:14:38.770 --> 00:14:40.990
So that's really the
substance of this proof.
00:14:40.990 --> 00:14:43.570
That of course really then
shows that it boils down
00:14:43.570 --> 00:14:48.020
to showing what this rate of
change is in these two cases.
00:14:48.020 --> 00:14:50.730
So now there's enough
suspense that we
00:14:50.730 --> 00:15:08.010
want to make sure that we know
that those answers are correct.
00:15:08.010 --> 00:15:12.180
OK, so let's demonstrate
both of them.
00:15:12.180 --> 00:15:18.540
I'll start with B. I need to
figure out property B. Now,
00:15:18.540 --> 00:15:22.470
we only have one alternative
as to a type of proof
00:15:22.470 --> 00:15:24.710
that we can give of
this kind of result,
00:15:24.710 --> 00:15:29.070
and that's because we only
have one way of describing
00:15:29.070 --> 00:15:32.540
sine and cosine functions,
that is geometrically.
00:15:32.540 --> 00:15:42.620
So we have to give
a geometric proof.
00:15:42.620 --> 00:15:45.190
And to write down
a geometric proof
00:15:45.190 --> 00:15:47.150
we are going to have
to draw a picture.
00:15:47.150 --> 00:15:49.550
And the first step
in the proof, really,
00:15:49.550 --> 00:15:51.750
is to replace this
variable delta
00:15:51.750 --> 00:15:55.800
x which is going to 0
with another name which
00:15:55.800 --> 00:15:58.370
is suggestive of what we're
gonna do which is the letter
00:15:58.370 --> 00:16:00.950
theta for an angle.
00:16:00.950 --> 00:16:03.560
OK, so let's draw
a picture of what
00:16:03.560 --> 00:16:05.950
it is that we're going to do.
00:16:05.950 --> 00:16:07.980
Here is the circle.
00:16:07.980 --> 00:16:10.740
And here is the origin.
00:16:10.740 --> 00:16:13.900
And here's some little
angle, well I'll
00:16:13.900 --> 00:16:16.110
draw it a little
larger so it's visible.
00:16:16.110 --> 00:16:19.430
Here's theta, alright?
00:16:19.430 --> 00:16:21.010
And this is the unit circle.
00:16:21.010 --> 00:16:25.910
I won't write that down on here
but that's the unit circle.
00:16:25.910 --> 00:16:29.780
And now sin theta is this
vertical distance here.
00:16:29.780 --> 00:16:32.740
Maybe, I'll draw it
in a different color
00:16:32.740 --> 00:16:34.750
so that we can see it all.
00:16:34.750 --> 00:16:38.400
OK so here's this distance.
00:16:38.400 --> 00:16:45.820
This distance is sin theta.
00:16:45.820 --> 00:16:48.360
OK?
00:16:48.360 --> 00:16:51.440
Now almost the
only other thing we
00:16:51.440 --> 00:16:54.770
have to write down in this
picture to have it work out
00:16:54.770 --> 00:16:58.500
is that we have to recognize
that when theta is the angle,
00:16:58.500 --> 00:17:03.279
that's also the arc length
of this piece of the circle
00:17:03.279 --> 00:17:04.320
when measured in radians.
00:17:04.320 --> 00:17:13.696
So this length here is
also arc length theta.
00:17:13.696 --> 00:17:14.820
That little piece in there.
00:17:14.820 --> 00:17:18.580
So maybe I'll use a different
color for that to indicate it.
00:17:18.580 --> 00:17:25.560
So that's orange and that's
this little chunk there.
00:17:25.560 --> 00:17:26.870
So those are the two pieces.
00:17:26.870 --> 00:17:36.250
Now in order to persuade
you now that the limit is
00:17:36.250 --> 00:17:37.964
what it's supposed
to be, I'm going
00:17:37.964 --> 00:17:39.630
to extend the picture
just a little bit.
00:17:39.630 --> 00:17:42.430
I'm going to double it, just
for my own linguistic sake
00:17:42.430 --> 00:17:44.240
and so that I can
tell you a story.
00:17:44.240 --> 00:17:46.690
Alright, so that
you'll remember this.
00:17:46.690 --> 00:17:49.640
So I'm going to take
a theta angle below
00:17:49.640 --> 00:17:53.670
and I'll have another copy
of sin theta down here.
00:17:53.670 --> 00:18:00.790
And now the total
picture is really
00:18:00.790 --> 00:18:04.721
like a bow and its
bow string there.
00:18:04.721 --> 00:18:05.220
Alright?
00:18:05.220 --> 00:18:11.050
So what we have here is
a length of 2 sin theta.
00:18:11.050 --> 00:18:13.630
So maybe I'll write it
this way, 2 sin theta.
00:18:13.630 --> 00:18:15.120
I just doubled it.
00:18:15.120 --> 00:18:25.640
And here I have underneath,
whoops, I got it backwards.
00:18:25.640 --> 00:18:27.040
Sorry about that.
00:18:27.040 --> 00:18:29.144
Trying to be fancy
with the colored chalk
00:18:29.144 --> 00:18:30.310
and I have it reversed here.
00:18:30.310 --> 00:18:32.180
So this is not 2 sin theta.
00:18:32.180 --> 00:18:33.540
2 sin theta is the vertical.
00:18:33.540 --> 00:18:34.910
That's the green.
00:18:34.910 --> 00:18:37.170
So let's try that again.
00:18:37.170 --> 00:18:41.190
This is 2 sin theta, alright?
00:18:41.190 --> 00:18:44.450
And then in the denominator
I have the arc length
00:18:44.450 --> 00:18:50.680
which is theta is the first half
and so double it is 2 theta.
00:18:50.680 --> 00:18:51.310
Alright?
00:18:51.310 --> 00:18:56.630
So if you like, this is
the bow and up here we
00:18:56.630 --> 00:19:04.290
have the bow string.
00:19:04.290 --> 00:19:07.740
And of course we
can cancel the 2's.
00:19:07.740 --> 00:19:11.250
That's equal to
sin theta / theta.
00:19:11.250 --> 00:19:17.900
And so now why does this
tend to 1 as theta goes to 0?
00:19:17.900 --> 00:19:23.670
Well, it's because as the
angle theta gets very small,
00:19:23.670 --> 00:19:28.880
this curved piece looks more
and more like a straight one.
00:19:28.880 --> 00:19:29.640
Alright?
00:19:29.640 --> 00:19:32.630
And if you get very, very
close here the green segment
00:19:32.630 --> 00:19:34.610
and the orange segment
would just merge.
00:19:34.610 --> 00:19:36.850
They would be practically
on top of each other.
00:19:36.850 --> 00:19:42.360
And they have closer and closer
and closer to the same length.
00:19:42.360 --> 00:19:51.790
So that's why this is true.
00:19:51.790 --> 00:20:03.200
I guess I'll articulate that
by saying that short curves are
00:20:03.200 --> 00:20:06.550
nearly straight.
00:20:06.550 --> 00:20:10.000
Alright, so that's the
principle that we're using.
00:20:10.000 --> 00:20:18.970
Or short pieces of curves, if
you like, are nearly straight.
00:20:18.970 --> 00:20:23.640
So if you like, this
is the principle.
00:20:23.640 --> 00:20:30.850
So short pieces of curves.
00:20:30.850 --> 00:20:31.540
Alright?
00:20:31.540 --> 00:20:39.390
So now I also need to
give you a proof of A.
00:20:39.390 --> 00:20:43.990
And that has to do with
this cosine function here.
00:20:43.990 --> 00:20:51.905
This is the property
A. So I'm going
00:20:51.905 --> 00:20:53.487
to do this by
flipping it around,
00:20:53.487 --> 00:20:56.070
because it turns out that this
numerator is a negative number.
00:20:56.070 --> 00:20:58.060
If I want to interpret
it as a length,
00:20:58.060 --> 00:21:00.340
I'm gonna want a
positive quantity.
00:21:00.340 --> 00:21:04.180
So I'm gonna write
down 1 - cos theta here
00:21:04.180 --> 00:21:08.320
and then I'm gonna
divide by theta there.
00:21:08.320 --> 00:21:10.720
Again I'm gonna make some
kind of interpretation.
00:21:10.720 --> 00:21:15.130
Now this time I'm going to draw
the same sort of bow and arrow
00:21:15.130 --> 00:21:18.970
arrangement, but maybe I'll
exaggerate it a little bit.
00:21:18.970 --> 00:21:23.190
So here's the vertex
of the sector,
00:21:23.190 --> 00:21:31.780
but we'll maybe make
it a little longer.
00:21:31.780 --> 00:21:35.590
Alright, so here it is, and
here was that middle line
00:21:35.590 --> 00:21:38.310
which was the unit-- Whoops.
00:21:38.310 --> 00:21:40.880
OK, I think I'm going
to have to tilt it up.
00:21:40.880 --> 00:21:47.000
OK, let's try from here.
00:21:47.000 --> 00:21:51.570
Alright, well you know
on your pencil and paper
00:21:51.570 --> 00:21:53.810
it will look better than
it does on my blackboard.
00:21:53.810 --> 00:21:55.190
OK, so here we are.
00:21:55.190 --> 00:21:56.770
Here's this shape.
00:21:56.770 --> 00:22:01.430
Now this angle is
supposed to be theta
00:22:01.430 --> 00:22:03.350
and this angle is another theta.
00:22:03.350 --> 00:22:05.895
So here we have a length
which is again theta
00:22:05.895 --> 00:22:07.910
and another length which
is theta over here.
00:22:07.910 --> 00:22:09.670
That's the same as
in the other picture,
00:22:09.670 --> 00:22:13.290
except we've
exaggerated a bit here.
00:22:13.290 --> 00:22:15.010
And now we have
this vertical line,
00:22:15.010 --> 00:22:18.240
which again I'm gonna draw
in green, the bow string.
00:22:18.240 --> 00:22:24.820
But notice that as the vertex
gets farther and farther away,
00:22:24.820 --> 00:22:27.019
the curved line gets
closer and closer
00:22:27.019 --> 00:22:28.060
to being a vertical line.
00:22:28.060 --> 00:22:30.670
That's sort of the flip
side, by expansion,
00:22:30.670 --> 00:22:33.110
of the zoom in principle.
00:22:33.110 --> 00:22:35.130
The principle that
curves are nearly
00:22:35.130 --> 00:22:36.950
straight when you zoom in.
00:22:36.950 --> 00:22:39.370
If you zoom out that would
mean sending this vertex way,
00:22:39.370 --> 00:22:42.200
way out somewhere.
00:22:42.200 --> 00:22:44.370
The curved line, the
piece of the circle,
00:22:44.370 --> 00:22:48.570
gets more and more straight.
00:22:48.570 --> 00:22:53.630
And now let me show you
where this numerator 1 - cos
00:22:53.630 --> 00:22:57.620
theta is on this picture.
00:22:57.620 --> 00:23:01.040
So where is it?
00:23:01.040 --> 00:23:03.680
Well, this whole distance is 1.
00:23:03.680 --> 00:23:07.610
But the distance from
the vertex to the green
00:23:07.610 --> 00:23:09.870
is cosine of theta.
00:23:09.870 --> 00:23:12.430
Right, because this is
theta, so dropping down
00:23:12.430 --> 00:23:16.830
the perpendicular this
distance back to the origin
00:23:16.830 --> 00:23:17.630
is cos theta.
00:23:17.630 --> 00:23:20.890
So this little
tiny, bitty segment
00:23:20.890 --> 00:23:26.540
here is basically the
gap between the curve
00:23:26.540 --> 00:23:28.820
and the vertical segment.
00:23:28.820 --> 00:23:35.820
So the gap is equal
to 1 - cos theta.
00:23:35.820 --> 00:23:40.640
So now you can see that as
this point gets farther away,
00:23:40.640 --> 00:23:43.670
if this got sent off
to the Stata Center,
00:23:43.670 --> 00:23:45.840
you would hardly be able
to tell the difference.
00:23:45.840 --> 00:23:48.120
The bow string would
coincide with the bow
00:23:48.120 --> 00:23:50.480
and this little gap
between the bow string
00:23:50.480 --> 00:23:52.430
and the bow would
be tending to 0.
00:23:52.430 --> 00:23:54.282
And that's the
statement that this
00:23:54.282 --> 00:23:58.540
tends to 0 as theta tends to 0.
00:23:58.540 --> 00:24:00.130
The scaled version of that.
00:24:00.130 --> 00:24:01.250
Yeah, question down here.
00:24:01.250 --> 00:24:04.800
Student: Doesn't the denominator
also tend to 0 though?
00:24:04.800 --> 00:24:10.190
Professor: Ah, the question is
"doesn't the denominator also
00:24:10.190 --> 00:24:11.670
tend to 0?"
00:24:11.670 --> 00:24:14.510
And the answer is yes.
00:24:14.510 --> 00:24:18.660
In my strange analogy with
zooming in, what I did
00:24:18.660 --> 00:24:20.390
was I zoomed out the picture.
00:24:20.390 --> 00:24:25.450
So in other words, if you
imagine you're taking this
00:24:25.450 --> 00:24:28.209
and you're putting it under
a microscope over here
00:24:28.209 --> 00:24:29.750
and you're looking
at something where
00:24:29.750 --> 00:24:31.910
theta is getting smaller
and smaller and smaller
00:24:31.910 --> 00:24:33.340
and smaller.
00:24:33.340 --> 00:24:34.370
Alright?
00:24:34.370 --> 00:24:39.500
But now because I want my
picture, I expanded my picture.
00:24:39.500 --> 00:24:42.960
So the ratio is the
thing that's preserved.
00:24:42.960 --> 00:24:50.470
So if I make it so that
this gap is tiny...
00:24:50.470 --> 00:24:52.410
Let me say this one more time.
00:24:52.410 --> 00:24:57.030
I'm afraid I've made life
complicated for myself.
00:24:57.030 --> 00:25:01.040
If I simply let this
theta tend in to 0,
00:25:01.040 --> 00:25:04.044
that would be the same
effect as making this
00:25:04.044 --> 00:25:06.460
closer and closer in and then
the vertical would approach.
00:25:06.460 --> 00:25:09.040
But I want to keep on
blowing up the picture
00:25:09.040 --> 00:25:11.940
so that I can see the
difference between the vertical
00:25:11.940 --> 00:25:13.260
and the curve.
00:25:13.260 --> 00:25:16.480
So that's very much like if
you are on a video screen
00:25:16.480 --> 00:25:18.740
and you zoom in, zoom
in, zoom in, and zoom in.
00:25:18.740 --> 00:25:20.615
So the question is what
would that look like?
00:25:20.615 --> 00:25:23.350
That has the same effect
as sending this point out
00:25:23.350 --> 00:25:27.840
farther and farther in that
direction, to the left.
00:25:27.840 --> 00:25:29.430
And so I'm just
trying to visualize it
00:25:29.430 --> 00:25:32.410
for you by leaving the
theta at this scale,
00:25:32.410 --> 00:25:34.090
but actually the
scale of the picture
00:25:34.090 --> 00:25:36.290
is then changing when I do that.
00:25:36.290 --> 00:25:39.380
So theta is going
to 0, but I I'm
00:25:39.380 --> 00:25:42.790
rescaling so that it's of a
size that we can look at it,
00:25:42.790 --> 00:25:46.292
And then imagine
what's happening to it.
00:25:46.292 --> 00:25:47.750
OK, does that answer
your question?
00:25:47.750 --> 00:25:50.592
Student: My question
then is that seems
00:25:50.592 --> 00:25:54.250
to prove that that
limit is equal to 0/0.
00:25:54.250 --> 00:26:01.510
Professor: It proves more
than it is equal to 0/0.
00:26:01.510 --> 00:26:04.925
It's the ratio of this little
short thing to this longer
00:26:04.925 --> 00:26:06.090
thing.
00:26:06.090 --> 00:26:08.920
And this is getting much, much
shorter than this total length.
00:26:08.920 --> 00:26:09.990
You're absolutely
right that we're
00:26:09.990 --> 00:26:12.573
comparing two quantities which
are going to 0, but one of them
00:26:12.573 --> 00:26:14.357
is much smaller than the other.
00:26:14.357 --> 00:26:16.190
In the other case we
compared two quantities
00:26:16.190 --> 00:26:18.148
which were both going to
0 and they both end up
00:26:18.148 --> 00:26:19.840
being about equal in length.
00:26:19.840 --> 00:26:23.060
Here the previous one
was this green one.
00:26:23.060 --> 00:26:25.000
Here it's this
little tiny bit here
00:26:25.000 --> 00:26:32.840
and it's way shorter than
the 2 theta distance.
00:26:32.840 --> 00:26:33.900
Yeah, another question.
00:26:33.900 --> 00:26:34.745
Student: cos theta - 1 over
cos theta is the same as 1-
00:26:34.745 --> 00:26:35.620
cos theta over theta?
00:26:35.620 --> 00:26:45.740
Professor: cos theta - 1 over...
00:26:45.740 --> 00:26:49.780
Student: [INAUDIBLE]
00:26:49.780 --> 00:26:56.690
Professor: So here, what I
wrote is (cos delta x - 1)
00:26:56.690 --> 00:27:01.620
/ delta x, OK, and I
claimed that it goes to 0.
00:27:01.620 --> 00:27:12.370
Here, I wrote minus that, that
is I replaced delta x by theta.
00:27:12.370 --> 00:27:22.720
But then I wrote this thing.
00:27:22.720 --> 00:27:26.550
So (cos theta - 1) minus
1 is the negative of this.
00:27:26.550 --> 00:27:28.070
Alright?
00:27:28.070 --> 00:27:29.510
And if I show that
this goes to 0,
00:27:29.510 --> 00:27:33.162
it's the same as showing
the other one goes to 0.
00:27:33.162 --> 00:27:33.870
Another question?
00:27:33.870 --> 00:27:39.050
Student: [INAUDIBLE]
00:27:39.050 --> 00:27:42.640
Professor: So the question
is, what about this business
00:27:42.640 --> 00:27:44.220
about arc length.
00:27:44.220 --> 00:27:48.610
So the word arc length,
that orange shape is an arc.
00:27:48.610 --> 00:27:51.350
And we're just talking about
the length of that arc,
00:27:51.350 --> 00:27:52.837
and so we're calling
it arc length.
00:27:52.837 --> 00:27:54.420
That's what the word
arc length means,
00:27:54.420 --> 00:27:55.920
it just means the
length of the arc.
00:27:55.920 --> 00:28:03.620
Student: [INAUDIBLE]
00:28:03.620 --> 00:28:06.030
Professor: Why is
this length theta?
00:28:06.030 --> 00:28:08.380
Ah, OK so this is a
very important point,
00:28:08.380 --> 00:28:11.480
and in fact it's the very next
point that I wanted to make.
00:28:11.480 --> 00:28:15.130
Namely, notice that
in this calculation
00:28:15.130 --> 00:28:19.700
it was very important
that we used length.
00:28:19.700 --> 00:28:24.140
And that means that the way
that we're measuring theta,
00:28:24.140 --> 00:28:32.330
is in what are known as radians.
00:28:32.330 --> 00:28:36.870
Right, so that applies to both B
and A, it's a scale change in A
00:28:36.870 --> 00:28:39.470
and doesn't really matter
but in B it's very important.
00:28:39.470 --> 00:28:42.980
The only way that
this orange length
00:28:42.980 --> 00:28:46.140
is comparable to
this green length,
00:28:46.140 --> 00:28:48.600
the vertical is
comparable to the arc,
00:28:48.600 --> 00:28:53.520
is if we measure them in terms
of the same notion of length.
00:28:53.520 --> 00:28:56.360
If we measure them in
degrees, for example,
00:28:56.360 --> 00:28:58.890
it would be completely wrong.
00:28:58.890 --> 00:29:02.720
We divide up the angles
into 360 degrees,
00:29:02.720 --> 00:29:04.320
and that's the wrong
unit of measure.
00:29:04.320 --> 00:29:07.990
The correct measure is the
length along the unit circle,
00:29:07.990 --> 00:29:09.590
which is what radians are.
00:29:09.590 --> 00:29:21.490
And so this is only
true if we use radians.
00:29:21.490 --> 00:29:33.650
So again, a little warning
here, that this is in radians.
00:29:33.650 --> 00:29:41.060
Now here x is in radians.
00:29:41.060 --> 00:29:45.690
The formulas are just wrong
if you use other units.
00:29:45.690 --> 00:29:46.250
Ah yeah?
00:29:46.250 --> 00:29:55.480
Student: [INAUDIBLE].
00:29:55.480 --> 00:29:57.290
Professor: OK so the
second question is why
00:29:57.290 --> 00:30:00.630
is this crazy length here 1.
00:30:00.630 --> 00:30:08.000
And the reason is
that the relationship
00:30:08.000 --> 00:30:11.780
between this picture up here
and this picture down here,
00:30:11.780 --> 00:30:16.150
is that I'm drawing
a different shape.
00:30:16.150 --> 00:30:18.800
Namely, what I'm
really imagining here
00:30:18.800 --> 00:30:21.460
is a much, much smaller theta.
00:30:21.460 --> 00:30:22.340
OK?
00:30:22.340 --> 00:30:25.430
And then I'm blowing
that up in scale.
00:30:25.430 --> 00:30:27.880
So this scale of this
picture down here
00:30:27.880 --> 00:30:31.430
is very different from the
scale of the picture up there.
00:30:31.430 --> 00:30:34.850
And if the angle is
very, very, very small
00:30:34.850 --> 00:30:37.835
then one has to be
very, very long in order
00:30:37.835 --> 00:30:39.430
for me to finish the circle.
00:30:39.430 --> 00:30:41.820
So, in other words,
this length is 1
00:30:41.820 --> 00:30:44.690
because that's what
I'm insisting on.
00:30:44.690 --> 00:30:47.910
So, I'm claiming that that's
how I define this circle,
00:30:47.910 --> 00:30:52.490
to be of unit radius.
00:30:52.490 --> 00:30:53.200
Another question?
00:30:53.200 --> 00:31:05.194
Student: [INAUDIBLE]
the ratio between 1 -
00:31:05.194 --> 00:31:07.360
cos theta and theta will
get closer and closer to 1.
00:31:07.360 --> 00:31:08.651
I don't understand [INAUDIBLE].
00:31:08.651 --> 00:31:21.550
Professor: OK, so
the question is it's
00:31:21.550 --> 00:31:25.810
hard to visualize
this fact here.
00:31:25.810 --> 00:31:30.829
So let me, let me take you
through a couple of steps,
00:31:30.829 --> 00:31:33.370
because I think probably other
people are also having trouble
00:31:33.370 --> 00:31:34.930
with this visualization.
00:31:34.930 --> 00:31:36.830
The first part of
the visualization I'm
00:31:36.830 --> 00:31:39.570
gonna try to demonstrate
on this picture up here.
00:31:39.570 --> 00:31:41.440
The first part of
the visualization
00:31:41.440 --> 00:31:45.350
is that I should think
of a beak of a bird
00:31:45.350 --> 00:31:47.880
closing down, getting
narrower and narrower.
00:31:47.880 --> 00:31:50.430
So in other words,
the angle theta
00:31:50.430 --> 00:31:54.050
has to be getting smaller
and smaller and smaller.
00:31:54.050 --> 00:31:55.780
OK, that's the first step.
00:31:55.780 --> 00:31:58.650
So that's the process
that we're talking about.
00:31:58.650 --> 00:32:03.880
Now, in order to draw that, once
theta gets incredibly narrow,
00:32:03.880 --> 00:32:06.450
in order to depict that I have
to blow the whole picture back
00:32:06.450 --> 00:32:07.700
up in order be able to see it.
00:32:07.700 --> 00:32:09.430
Otherwise it just
disappears on me.
00:32:09.430 --> 00:32:12.090
In fact in the limit theta
= 0, it's meaningless.
00:32:12.090 --> 00:32:13.080
It's just a flat line.
00:32:13.080 --> 00:32:15.500
That's the whole problem
with these tricky limits.
00:32:15.500 --> 00:32:18.100
They're meaningless right
at the zero-zero level.
00:32:18.100 --> 00:32:22.010
It's only just a little away
that they're actually useful,
00:32:22.010 --> 00:32:25.890
that you get useful geometric
information out of them.
00:32:25.890 --> 00:32:27.300
So we're just a little away.
00:32:27.300 --> 00:32:31.160
So that's what this picture down
below in part A is meant to be.
00:32:31.160 --> 00:32:34.135
It's supposed to be that theta
is open a tiny crack, just
00:32:34.135 --> 00:32:35.161
a little bit.
00:32:35.161 --> 00:32:37.285
And the smallest I can draw
it on the board for you
00:32:37.285 --> 00:32:40.160
to visualize it is using the
whole length of the blackboard
00:32:40.160 --> 00:32:41.390
here for that.
00:32:41.390 --> 00:32:43.390
So I've opened a little
tiny bit and by the time
00:32:43.390 --> 00:32:45.139
we get to the other
end of the blackboard,
00:32:45.139 --> 00:32:46.510
of course it's fairly wide.
00:32:46.510 --> 00:32:50.291
But this angle theta
is a very small angle.
00:32:50.291 --> 00:32:50.790
Alright?
00:32:50.790 --> 00:32:56.670
So I'm trying to imagine what
happens as this collapses.
00:32:56.670 --> 00:32:59.670
Now, when I imagine
that I have to imagine
00:32:59.670 --> 00:33:02.190
a geometric interpretation
of both the numerator
00:33:02.190 --> 00:33:06.020
and the denominator
of this quantity here.
00:33:06.020 --> 00:33:08.300
And just see what happens.
00:33:08.300 --> 00:33:14.020
Now I claimed the numerator is
this little tiny bit over here
00:33:14.020 --> 00:33:19.004
and the denominator is actually
half of this whole length here.
00:33:19.004 --> 00:33:20.420
But the factor of
2 doesn't matter
00:33:20.420 --> 00:33:24.250
when you're seeing whether
something tends to 0 or not.
00:33:24.250 --> 00:33:24.990
Alright?
00:33:24.990 --> 00:33:26.656
And I claimed that
if you stare at this,
00:33:26.656 --> 00:33:28.460
it's clear that
this is much shorter
00:33:28.460 --> 00:33:32.750
than that vertical curve there.
00:33:32.750 --> 00:33:35.140
And I'm claiming, so this
is what you have to imagine,
00:33:35.140 --> 00:33:38.820
is this as it gets smaller
and smaller and smaller still
00:33:38.820 --> 00:33:41.590
that has the same effect of
this thing going way, way way,
00:33:41.590 --> 00:33:45.510
farther away and this vertical
curve getting closer and closer
00:33:45.510 --> 00:33:47.160
and closer to the green.
00:33:47.160 --> 00:33:52.530
And so that the gap between
them gets tiny and goes to 0.
00:33:52.530 --> 00:33:53.440
Alright?
00:33:53.440 --> 00:33:56.580
So not only does it go to
0, that's not enough for us,
00:33:56.580 --> 00:34:01.540
but it also goes to 0 faster
than this theta goes to 0.
00:34:01.540 --> 00:34:05.280
And I hope the evidence
is pretty strong here
00:34:05.280 --> 00:34:10.220
because it's so tiny
already at this stage.
00:34:10.220 --> 00:34:12.350
Alright.
00:34:12.350 --> 00:34:16.140
We are going to move
forward and you'll
00:34:16.140 --> 00:34:18.420
have to ponder these
things some other time.
00:34:18.420 --> 00:34:20.640
So I'm gonna give you
an even harder thing
00:34:20.640 --> 00:34:26.600
to visualize now so be prepared.
00:34:26.600 --> 00:34:36.310
OK, so now, the next
thing that I'd like to do
00:34:36.310 --> 00:34:37.700
is to give you a second proof.
00:34:37.700 --> 00:34:39.430
Because it really
is important, I
00:34:39.430 --> 00:34:47.650
think, to understand this
particular fact more thoroughly
00:34:47.650 --> 00:34:49.310
and also to get
a lot of practice
00:34:49.310 --> 00:34:51.450
with sines and cosines.
00:34:51.450 --> 00:34:55.930
So I'm gonna give
you a geometric proof
00:34:55.930 --> 00:35:11.010
of the formula for sine here,
for the derivative of sine.
00:35:11.010 --> 00:35:13.420
So here we go.
00:35:13.420 --> 00:35:26.280
This is a geometric
proof of this fact.
00:35:26.280 --> 00:35:29.400
This is for all theta.
00:35:29.400 --> 00:35:33.100
So far we only did
it for theta = 0
00:35:33.100 --> 00:35:36.360
and now we're going to
do it for all theta.
00:35:36.360 --> 00:35:38.930
So this is a different
proof, but it uses
00:35:38.930 --> 00:35:42.420
exactly the same principles.
00:35:42.420 --> 00:35:45.390
Right?
00:35:45.390 --> 00:35:51.350
So, I want do this by
drawing another picture,
00:35:51.350 --> 00:35:54.790
and the picture is
going to describe
00:35:54.790 --> 00:35:59.370
y, which is sin
theta, which is if you
00:35:59.370 --> 00:36:22.160
like the vertical position
of some circular motion.
00:36:22.160 --> 00:36:27.170
So I'm imagining that something
is going around in a circle.
00:36:27.170 --> 00:36:30.620
Some particle is going
around in a circle.
00:36:30.620 --> 00:36:36.377
And so here's the
circle, here the origin.
00:36:36.377 --> 00:36:37.460
This is the unit distance.
00:36:37.460 --> 00:36:43.640
And right now it happens to
be at this location P. Maybe
00:36:43.640 --> 00:36:46.160
we'll put P a little over here.
00:36:46.160 --> 00:36:50.260
And here's the angle theta.
00:36:50.260 --> 00:36:51.560
And now we're going to move it.
00:36:51.560 --> 00:36:54.490
We're going to vary
theta and we're
00:36:54.490 --> 00:36:56.920
interested in the
rate of change of y.
00:36:56.920 --> 00:37:00.290
So y is the height
of P but we're
00:37:00.290 --> 00:37:01.870
gonna move it to
another location.
00:37:01.870 --> 00:37:07.400
We'll move it along
the circle to Q. Right?
00:37:07.400 --> 00:37:09.200
So here it is.
00:37:09.200 --> 00:37:12.360
Here's the thing.
00:37:12.360 --> 00:37:14.450
So how far did we move it?
00:37:14.450 --> 00:37:18.570
Well we moved it by
an angle delta theta.
00:37:18.570 --> 00:37:20.540
So we started theta,
theta is going
00:37:20.540 --> 00:37:22.313
to be fixed in this
argument, and we're
00:37:22.313 --> 00:37:23.990
going to move a little
bit delta theta.
00:37:23.990 --> 00:37:26.180
And now we're just
gonna try to figure out
00:37:26.180 --> 00:37:28.510
how far the thing moved.
00:37:28.510 --> 00:37:30.440
Well, in order to
do that we've got
00:37:30.440 --> 00:37:34.590
to keep track of the height,
the vertical displacement here.
00:37:34.590 --> 00:37:38.334
So we're going to draw
this right angle here, this
00:37:38.334 --> 00:37:42.440
is the position R. And
then this distance here
00:37:42.440 --> 00:37:45.500
is the change in y.
00:37:45.500 --> 00:37:46.000
Alright?
00:37:46.000 --> 00:37:50.190
So the picture is
we have something
00:37:50.190 --> 00:37:52.110
moving around a unit circle.
00:37:52.110 --> 00:37:53.680
A point moving
around a unit circle.
00:37:53.680 --> 00:37:57.635
It starts at P, it moves to
Q. It moves from angle theta
00:37:57.635 --> 00:37:59.590
to angle theta plus delta theta.
00:37:59.590 --> 00:38:05.770
And the issue is how
much does y move?
00:38:05.770 --> 00:38:07.340
And the formula
for y is sin theta.
00:38:07.340 --> 00:38:29.710
So that's telling us the
rate of change of sin theta.
00:38:29.710 --> 00:38:34.500
Alright, well so let's just
try to think a little bit
00:38:34.500 --> 00:38:35.980
about what this is.
00:38:35.980 --> 00:38:37.736
So, first of all,
I've already said this
00:38:37.736 --> 00:38:39.300
and I'm going to repeat it here.
00:38:39.300 --> 00:38:41.650
Delta y is PR.
00:38:41.650 --> 00:38:44.490
It's going from P and
going straight up to R.
00:38:44.490 --> 00:38:47.080
That's how far y moves.
00:38:47.080 --> 00:38:49.640
That's the change in y
That's what I said up
00:38:49.640 --> 00:38:52.910
in the right hand corner there.
00:38:52.910 --> 00:38:53.470
Oops.
00:38:53.470 --> 00:38:56.430
I said PR but I wrote PQ.
00:38:56.430 --> 00:38:59.130
Alright, that's not a good idea.
00:38:59.130 --> 00:38:59.630
Alright.
00:38:59.630 --> 00:39:03.090
So delta Y is PR.
00:39:03.090 --> 00:39:07.160
And now I want to draw the
diagram again one time.
00:39:07.160 --> 00:39:16.030
So here's Q, here's R,
and here's P, and here's
00:39:16.030 --> 00:39:17.300
my triangle.
00:39:17.300 --> 00:39:24.300
And now what I'd like to
do is draw this curve here
00:39:24.300 --> 00:39:26.970
which is a piece of
the arc of the circle.
00:39:26.970 --> 00:39:29.560
But really what I
want to keep in mind
00:39:29.560 --> 00:39:33.300
is something that I did also
in all these other arguments.
00:39:33.300 --> 00:39:35.330
Which is, maybe I
should have called
00:39:35.330 --> 00:39:38.011
this orange, that I'm gonna
think of the straight line
00:39:38.011 --> 00:39:38.510
between.
00:39:38.510 --> 00:39:41.165
So it's the straight line
approximation to the curve
00:39:41.165 --> 00:39:45.009
that we're always interested in.
00:39:45.009 --> 00:39:46.550
So the straight line
is much simpler,
00:39:46.550 --> 00:39:48.610
because then we just
have a triangle here.
00:39:48.610 --> 00:39:52.017
And in fact it's
a right triangle.
00:39:52.017 --> 00:39:54.350
Right, so we have the geometry
of a right triangle which
00:39:54.350 --> 00:39:59.210
is going to now let us do
all of our calculations.
00:39:59.210 --> 00:40:03.640
OK, so now the key step
is this same principle
00:40:03.640 --> 00:40:07.560
that we already used which is
that short pieces of curves
00:40:07.560 --> 00:40:09.040
are nearly straight.
00:40:09.040 --> 00:40:12.000
So that means that this piece
of the circular arc here from P
00:40:12.000 --> 00:40:17.323
to Q is practically the same
as the straight segment from P
00:40:17.323 --> 00:40:24.290
to Q. So, that's this
principle that - well,
00:40:24.290 --> 00:40:27.580
let's put it over
here - Is that PQ
00:40:27.580 --> 00:40:33.190
is practically the same as the
straight segment from P to Q.
00:40:33.190 --> 00:40:35.820
So how are we going to use that?
00:40:35.820 --> 00:40:37.750
We want to use
that quantitatively
00:40:37.750 --> 00:40:39.080
in the following way.
00:40:39.080 --> 00:40:42.350
What we want to notice is
that the distance from P to Q
00:40:42.350 --> 00:40:46.120
is approximately delta theta.
00:40:46.120 --> 00:40:46.620
Right?
00:40:46.620 --> 00:40:49.056
Because the arc length
along that curve,
00:40:49.056 --> 00:40:50.680
the length of the
curve is delta theta.
00:40:50.680 --> 00:40:55.050
So the length of the green which
is PQ is almost delta theta.
00:40:55.050 --> 00:41:01.690
So this is essentially delta
theta, this distance here.
00:41:01.690 --> 00:41:05.360
Now the second step, which
is a little trickier,
00:41:05.360 --> 00:41:08.980
is that we have to work
out what this angle is.
00:41:08.980 --> 00:41:11.640
So our goal, and I'm gonna put
it one step below because I'm
00:41:11.640 --> 00:41:14.280
gonna put the geometric
reasoning in between,
00:41:14.280 --> 00:41:20.980
is I need to figure out
what the angle QPR is.
00:41:20.980 --> 00:41:23.340
If I can figure out
what this angle is,
00:41:23.340 --> 00:41:26.630
then I'll be able to figure out
what this vertical distance is
00:41:26.630 --> 00:41:28.840
because I'll know the
hypotenuse and I'll
00:41:28.840 --> 00:41:30.900
know the angle so I'll be
able to figure out what
00:41:30.900 --> 00:41:36.610
the side of the triangle is.
00:41:36.610 --> 00:41:40.220
So now let me show you
why that's possible to do.
00:41:40.220 --> 00:41:43.400
So in order to do that first of
all I'm gonna trade the boards
00:41:43.400 --> 00:41:50.600
and show you where
the line PQ is.
00:41:50.600 --> 00:41:54.370
So the line PQ is here.
00:41:54.370 --> 00:41:56.470
That's the whole thing.
00:41:56.470 --> 00:42:00.190
And the key point about this
line that I need you to realize
00:42:00.190 --> 00:42:04.230
is that it's practically
perpendicular,
00:42:04.230 --> 00:42:08.910
it's almost perpendicular,
to this ray here.
00:42:08.910 --> 00:42:09.550
Alright?
00:42:09.550 --> 00:42:12.540
It's not quite because the
distance between P to Q
00:42:12.540 --> 00:42:13.262
is non-zero.
00:42:13.262 --> 00:42:14.720
So it isn't quite,
but in the limit
00:42:14.720 --> 00:42:17.070
it's going to be perpendicular.
00:42:17.070 --> 00:42:18.100
Exactly perpendicular.
00:42:18.100 --> 00:42:20.980
The tangent line to the circle.
00:42:20.980 --> 00:42:26.820
So the key thing
that I'm going to use
00:42:26.820 --> 00:42:35.131
is that PQ is almost
perpendicular to OP.
00:42:35.131 --> 00:42:35.630
Alright?
00:42:35.630 --> 00:42:37.710
The ray from the
origin is basically
00:42:37.710 --> 00:42:39.900
perpendicular to
that green line.
00:42:39.900 --> 00:42:42.540
And then the second
thing I'm going to use
00:42:42.540 --> 00:42:53.121
is something that's obvious
which is that PR is vertical.
00:42:53.121 --> 00:42:53.620
OK?
00:42:53.620 --> 00:42:58.080
So those are the two pieces of
geometry that I need to see.
00:42:58.080 --> 00:43:02.680
And now notice what's happening
upstairs on the picture here
00:43:02.680 --> 00:43:05.050
in the upper right.
00:43:05.050 --> 00:43:09.550
What I have is the
angle theta is the angle
00:43:09.550 --> 00:43:12.910
between the horizontal and OP.
00:43:12.910 --> 00:43:14.350
That's angle theta.
00:43:14.350 --> 00:43:17.990
If I rotate it by ninety
degree, the horizontal
00:43:17.990 --> 00:43:18.880
becomes vertical.
00:43:18.880 --> 00:43:21.730
It becomes PR and
the other thing
00:43:21.730 --> 00:43:24.810
rotated by 90 degrees
becomes the green line.
00:43:24.810 --> 00:43:30.080
So the angle that I'm talking
about I get by taking this guy
00:43:30.080 --> 00:43:32.470
and rotating it by 90 degrees.
00:43:32.470 --> 00:43:33.800
It's the same angle.
00:43:33.800 --> 00:43:38.230
So that means that this angle
here is essentially theta.
00:43:38.230 --> 00:43:39.880
That's what this angle is.
00:43:39.880 --> 00:43:41.840
Let me repeat that
one more time.
00:43:41.840 --> 00:43:43.290
We started out
with an angle that
00:43:43.290 --> 00:43:46.540
looks like this, which is
the horizontal-- that's
00:43:46.540 --> 00:43:48.600
the origin straight
out horizontally.
00:43:48.600 --> 00:43:50.560
That's the thing labeled 1.
00:43:50.560 --> 00:43:54.805
That distance there.
00:43:54.805 --> 00:43:56.430
That's my right arm
which is down here.
00:43:56.430 --> 00:44:01.180
My left arm is pointing up
and it's going from the origin
00:44:01.180 --> 00:44:06.430
to the point P. So
here's the horizontal
00:44:06.430 --> 00:44:09.370
and the angle between
them is theta.
00:44:09.370 --> 00:44:13.930
And now, what I claim is that
if I rotate by 90 degrees up,
00:44:13.930 --> 00:44:16.840
like this, without
changing anything -
00:44:16.840 --> 00:44:18.860
so that was what I
did - the horizontal
00:44:18.860 --> 00:44:21.160
will become a vertical.
00:44:21.160 --> 00:44:22.990
That's PR.
00:44:22.990 --> 00:44:25.030
That's going up, PR.
00:44:25.030 --> 00:44:32.080
And if I rotate OP 90
degrees, that's exactly PQ.
00:44:32.080 --> 00:44:33.540
OK?
00:44:33.540 --> 00:44:42.560
So let me draw it
on there one time.
00:44:42.560 --> 00:44:45.220
Let's do it with
some arrows here.
00:44:45.220 --> 00:44:49.620
So I started out with
this and then, we'll
00:44:49.620 --> 00:44:56.470
label this as orange,
OK so red to orange,
00:44:56.470 --> 00:45:01.840
and then I rotate by
90 degrees and the red
00:45:01.840 --> 00:45:07.060
becomes this starting from P
and the orange rotates around 90
00:45:07.060 --> 00:45:11.370
degrees and becomes
this thing here.
00:45:11.370 --> 00:45:12.190
Alright?
00:45:12.190 --> 00:45:16.252
So this angle here is
the same as the other one
00:45:16.252 --> 00:45:18.460
which I've just drawn.
00:45:18.460 --> 00:45:27.030
Different vertices
for the angles.
00:45:27.030 --> 00:45:28.210
OK?
00:45:28.210 --> 00:45:31.100
Well I didn't say
that all arguments
00:45:31.100 --> 00:45:36.450
were supposed to be easy.
00:45:36.450 --> 00:45:38.180
Alright, so I claim
that the conclusion
00:45:38.180 --> 00:45:43.200
is that this angle is
approximately theta.
00:45:43.200 --> 00:45:45.670
And now we can finish
our calculation,
00:45:45.670 --> 00:45:48.300
because we have something with
the hypotenuse being delta
00:45:48.300 --> 00:45:53.340
theta and the angle being theta
and so this segment here PR is
00:45:53.340 --> 00:45:56.830
approximately the
hypotenuse length
00:45:56.830 --> 00:46:02.430
times the cosine of the angle.
00:46:02.430 --> 00:46:05.740
And that is exactly
what we wanted.
00:46:05.740 --> 00:46:10.830
If we divide, we divide by
delta theta, we get (delta y)
00:46:10.830 --> 00:46:17.030
/ (delta theta) is
approximately cos theta.
00:46:17.030 --> 00:46:20.700
And that's the same thing as...
00:46:20.700 --> 00:46:22.370
So what we want in
the limit is exactly
00:46:22.370 --> 00:46:24.900
the delta theta going to 0
of (delta y) / (delta theta)
00:46:24.900 --> 00:46:28.020
is equal to cos theta.
00:46:28.020 --> 00:46:32.270
So we get an approximation on
a scale that we can visualize
00:46:32.270 --> 00:46:39.590
and in the limit the
formula is exact.
00:46:39.590 --> 00:46:44.060
OK, so that is a geometric
argument for the same result.
00:46:44.060 --> 00:46:47.940
Namely that the derivative
of sine is cosine.
00:46:47.940 --> 00:46:48.440
Yeah?
00:46:48.440 --> 00:46:51.590
Student: [INAUDIBLE].
00:46:51.590 --> 00:46:54.420
Professor: You will have to do
some kind of geometric proofs
00:46:54.420 --> 00:46:55.840
sometimes.
00:46:55.840 --> 00:46:59.730
When you'll really need
this is probably in 18.02.
00:46:59.730 --> 00:47:03.020
So you'll need to make
reasoning like this.
00:47:03.020 --> 00:47:05.730
This is, for example, the
way that you actually develop
00:47:05.730 --> 00:47:08.200
the theory of arc length.
00:47:08.200 --> 00:47:13.250
Dealing with delta x's and
delta y's is a common tool.
00:47:13.250 --> 00:47:17.810
Alright, I have one
more thing that I
00:47:17.810 --> 00:47:25.070
want to talk about today,
which is some general rules.
00:47:25.070 --> 00:47:28.230
We took a little bit more time
than I expected with this.
00:47:28.230 --> 00:47:31.780
So what I'm gonna do is
just tell you the rules
00:47:31.780 --> 00:47:36.330
and we'll discuss
them in a few days.
00:47:36.330 --> 00:47:50.180
So let me tell you
the general rules.
00:47:50.180 --> 00:48:00.170
So these were the specific ones
and here are some general ones.
00:48:00.170 --> 00:48:08.490
So the first one is
called the product rule.
00:48:08.490 --> 00:48:11.010
And what it says is that if
you take the product of two
00:48:11.010 --> 00:48:13.630
functions and
differentiate them,
00:48:13.630 --> 00:48:18.180
you get the derivative of
one times the other plus
00:48:18.180 --> 00:48:22.060
the other times the
derivative of the one.
00:48:22.060 --> 00:48:24.350
Now the way that you
should remember this,
00:48:24.350 --> 00:48:27.970
and the way that I'll
carry out the proof,
00:48:27.970 --> 00:48:40.010
is that you should think of it
is you change one at a time.
00:48:40.010 --> 00:48:44.080
And this is a very useful way of
thinking about differentiation
00:48:44.080 --> 00:48:46.530
when you have
things which depend
00:48:46.530 --> 00:48:49.660
on more than one function.
00:48:49.660 --> 00:48:53.750
So this is a general procedure.
00:48:53.750 --> 00:48:59.050
The second formula that
I wanted to mention
00:48:59.050 --> 00:49:07.470
is called the quotient rule
and that says the following.
00:49:07.470 --> 00:49:13.220
That (u / v) prime
has a formula as well.
00:49:13.220 --> 00:49:21.280
And the formula is
(u'v - uv' ) / v^2.
00:49:21.280 --> 00:49:23.240
So this is our second formula.
00:49:23.240 --> 00:49:31.360
Let me just mention, both of
them are extremely valuable
00:49:31.360 --> 00:49:33.170
and you'll use
them all the time.
00:49:33.170 --> 00:49:43.600
This one of course only
works when v is not 0.
00:49:43.600 --> 00:49:46.926
Alright, so because
we're out of time
00:49:46.926 --> 00:49:48.300
we're not gonna
prove these today
00:49:48.300 --> 00:49:50.508
but we'll prove these next
time and you're definitely
00:49:50.508 --> 00:49:53.760
going to be responsible
for these kinds of proofs.