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PROF.
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JERISON: We're starting
a new unit today.
00:00:26.950 --> 00:00:39.050
And, so this is Unit 2, and
it's called Applications
00:00:39.050 --> 00:00:48.810
of Differentiation.
00:00:48.810 --> 00:00:51.200
OK.
00:00:51.200 --> 00:00:57.400
So, the first application, and
we're going to do two today,
00:00:57.400 --> 00:01:04.030
is what are known as
linear approximations.
00:01:04.030 --> 00:01:06.310
Whoops, that should
have two p's in it.
00:01:06.310 --> 00:01:12.460
Approximations.
00:01:12.460 --> 00:01:16.360
So, that can be summarized
with one formula,
00:01:16.360 --> 00:01:19.040
but it's going to take
us at least half an hour
00:01:19.040 --> 00:01:21.960
to explain how this
formula is used.
00:01:21.960 --> 00:01:24.100
So here's the formula.
00:01:24.100 --> 00:01:34.390
It's f(x) is approximately equal
to its value at a base point
00:01:34.390 --> 00:01:38.260
plus the derivative
times x - x_0.
00:01:38.260 --> 00:01:38.760
Right?
00:01:38.760 --> 00:01:42.720
So this is the main formula.
00:01:42.720 --> 00:01:44.310
For right now.
00:01:44.310 --> 00:01:52.140
Put it in a box.
00:01:52.140 --> 00:01:57.430
And let me just describe
what it means, first.
00:01:57.430 --> 00:01:59.830
And then I'll describe
what it means again,
00:01:59.830 --> 00:02:01.780
and several other times.
00:02:01.780 --> 00:02:04.690
So, first of all,
what it means is
00:02:04.690 --> 00:02:11.140
that if you have a
curve, which is y = f(x),
00:02:11.140 --> 00:02:18.860
it's approximately the
same as its tangent line.
00:02:18.860 --> 00:02:37.020
So this other side is the
equation of the tangent line.
00:02:37.020 --> 00:02:43.090
So let's give an example.
00:02:43.090 --> 00:02:50.190
I'm going to take the
function f(x), which is ln x,
00:02:50.190 --> 00:02:53.980
and then its derivative is 1/x.
00:02:58.990 --> 00:03:03.800
And, so let's take the
base point x_0 = 1.
00:03:03.800 --> 00:03:05.470
That's pretty much
the only place where
00:03:05.470 --> 00:03:08.900
we know the logarithm for sure.
00:03:08.900 --> 00:03:13.360
And so, what we plug in
here now, are the values.
00:03:13.360 --> 00:03:17.890
So f(1) is the log of 0.
00:03:17.890 --> 00:03:20.750
Or, sorry, the log
of 1, which is 0.
00:03:20.750 --> 00:03:28.130
And f'(1), well,
that's 1/1, which is 1.
00:03:28.130 --> 00:03:31.100
So now we have an
approximation formula which,
00:03:31.100 --> 00:03:34.020
if I copy down
what's right up here,
00:03:34.020 --> 00:03:40.560
it's going to be ln x is
approximately, so f(0)
00:03:40.560 --> 00:03:44.480
is 0, right?
00:03:44.480 --> 00:03:49.700
Plus 1 times (x - 1).
00:03:49.700 --> 00:03:52.910
So I plugged in here,
for x_0, three places.
00:03:52.910 --> 00:04:00.670
I evaluated the coefficients and
this is the dependent variable.
00:04:00.670 --> 00:04:03.720
So, all told, if you
like, what I have here
00:04:03.720 --> 00:04:11.100
is that the logarithm of
x is approximately x - 1.
00:04:11.100 --> 00:04:16.660
And let me draw a
picture of this.
00:04:16.660 --> 00:04:22.310
So here's the graph of ln x.
00:04:22.310 --> 00:04:26.920
And then, I'll draw in the
tangent line at the place
00:04:26.920 --> 00:04:30.350
that we're considering,
which is x = 1.
00:04:30.350 --> 00:04:33.030
So here's the tangent line.
00:04:33.030 --> 00:04:35.195
And I've separated a
little bit, but really
00:04:35.195 --> 00:04:37.551
I probably should have drawn
it a little closer there,
00:04:37.551 --> 00:04:38.050
to show you.
00:04:38.050 --> 00:04:42.870
The whole point is that
these two are nearby.
00:04:42.870 --> 00:04:44.400
But they're not
nearby everywhere.
00:04:44.400 --> 00:04:50.130
So this is the line y = x - 1.
00:04:50.130 --> 00:04:51.960
Right, that's the tangent line.
00:04:51.960 --> 00:04:55.240
They're nearby only
when x is near 1.
00:04:55.240 --> 00:04:58.010
So say in this
little realm here.
00:04:58.010 --> 00:05:05.050
So when x is approximately
1, this is true.
00:05:05.050 --> 00:05:06.580
Once you get a
little farther away,
00:05:06.580 --> 00:05:08.413
this straight line,
this straight green line
00:05:08.413 --> 00:05:10.540
will separate from the graph.
00:05:10.540 --> 00:05:14.610
But near this place
they're close together.
00:05:14.610 --> 00:05:17.920
So the idea, again, is that
the curve, the curved line,
00:05:17.920 --> 00:05:19.770
is approximately
the tangent line.
00:05:19.770 --> 00:05:25.350
And this is one example of it.
00:05:25.350 --> 00:05:29.850
All right, so I want to
explain this in one more way.
00:05:29.850 --> 00:05:32.690
And then we want to
discuss it systematically.
00:05:32.690 --> 00:05:37.090
So the second way that
I want to describe this
00:05:37.090 --> 00:05:39.310
requires me to remind
you what the definition
00:05:39.310 --> 00:05:41.290
of the derivative is.
00:05:41.290 --> 00:05:46.370
So, the definition
of a derivative
00:05:46.370 --> 00:05:53.360
is that it's the limit, as delta
x goes to 0, of delta f / delta
00:05:53.360 --> 00:05:56.410
x, that's one way of
writing it, all right?
00:05:56.410 --> 00:06:01.260
And this is the
way we defined it.
00:06:01.260 --> 00:06:03.860
And one of the things that
we did in the first unit
00:06:03.860 --> 00:06:09.070
was we looked at this backwards.
00:06:09.070 --> 00:06:12.890
We used the derivative knowing
the derivatives of functions
00:06:12.890 --> 00:06:14.250
to evaluate some limits.
00:06:14.250 --> 00:06:17.860
So you were supposed
to do that on your.
00:06:17.860 --> 00:06:21.200
In our test, there were
some examples there,
00:06:21.200 --> 00:06:23.200
at least one example,
where that was the easiest
00:06:23.200 --> 00:06:26.140
way to do the problem.
00:06:26.140 --> 00:06:28.850
So in other words, you can
read this equation both ways.
00:06:28.850 --> 00:06:31.770
This is really, of course, the
same equation written twice.
00:06:31.770 --> 00:06:34.970
Now, what's new about
what we're going to do now
00:06:34.970 --> 00:06:40.150
is that we're going to take
this expression here, delta f
00:06:40.150 --> 00:06:42.910
/ delta x, and
we're going to say
00:06:42.910 --> 00:06:45.810
well, when delta x
is fairly near 0,
00:06:45.810 --> 00:06:47.360
this expression is
going to be fairly
00:06:47.360 --> 00:06:49.450
close to the limiting value.
00:06:49.450 --> 00:06:53.760
So this is
approximately f'(x_0).
00:06:53.760 --> 00:07:00.730
So that, I claim, is the same
as what's in the box in pink
00:07:00.730 --> 00:07:02.810
that I have over here.
00:07:02.810 --> 00:07:10.840
So this approximation formula
here is the same as this one.
00:07:10.840 --> 00:07:13.760
This is an average
rate of change,
00:07:13.760 --> 00:07:16.420
and this is an infinitesimal
rate of change.
00:07:16.420 --> 00:07:17.980
And they're nearly the same.
00:07:17.980 --> 00:07:19.230
That's the claim.
00:07:19.230 --> 00:07:22.950
So you'll have various exercises
in which this approximation is
00:07:22.950 --> 00:07:25.220
the useful one to use.
00:07:25.220 --> 00:07:28.860
And I will, as I said, I'll be
illustrating this a little bit
00:07:28.860 --> 00:07:29.610
today.
00:07:29.610 --> 00:07:34.060
Now, let me just explain why
those two formulas in the boxes
00:07:34.060 --> 00:07:36.560
are the same.
00:07:36.560 --> 00:07:41.110
So let's just start over
here and explain that.
00:07:41.110 --> 00:07:47.150
So the smaller box is the same
thing if I multiply through
00:07:47.150 --> 00:07:55.490
by delta x, as delta f is
approximately f'(x_0) delta x.
00:07:55.490 --> 00:07:57.290
And now if I just
write out what this
00:07:57.290 --> 00:08:09.820
is, it's f(x),
right, minus f(x_0),
00:08:09.820 --> 00:08:11.630
I'm going to write it this way.
00:08:11.630 --> 00:08:16.450
Which is approximately
f'(x_0), and this is x - x_0.
00:08:16.450 --> 00:08:25.670
So here I'm using the
notations delta x is x - x0.
00:08:25.670 --> 00:08:28.160
And so this is the
change in f, this
00:08:28.160 --> 00:08:32.270
is just rewriting
what delta x is.
00:08:32.270 --> 00:08:36.480
And now the last step is
just to put the constant
00:08:36.480 --> 00:08:37.430
on the other side.
00:08:37.430 --> 00:08:47.010
So f(x) is approximately
f(x_0) + f'(x_0)(x - x_0).
00:08:47.010 --> 00:08:51.920
So this is exactly what I had
just to begin with, right?
00:08:51.920 --> 00:08:53.570
So these two are
just algebraically
00:08:53.570 --> 00:08:56.690
the same statement.
00:08:56.690 --> 00:09:00.660
That's one another
way of looking at it.
00:09:00.660 --> 00:09:05.100
All right, so now,
I want to go through
00:09:05.100 --> 00:09:08.590
some systematic
discussion here of
00:09:08.590 --> 00:09:12.250
several linear approximations,
which you're going
00:09:12.250 --> 00:09:14.850
to be wanting to memorize.
00:09:14.850 --> 00:09:18.020
And rather than it's being
hard to memorize these,
00:09:18.020 --> 00:09:19.760
it's supposed to remind you.
00:09:19.760 --> 00:09:22.440
So that you'll have a lot
of extra reinforcement
00:09:22.440 --> 00:09:25.240
in remembering
derivatives of all kinds.
00:09:25.240 --> 00:09:31.240
So, when we carry out these
systematic discussions,
00:09:31.240 --> 00:09:33.060
we want to make
things absolutely as
00:09:33.060 --> 00:09:34.550
simple as possible.
00:09:34.550 --> 00:09:36.640
And so one of the
things that we do
00:09:36.640 --> 00:09:40.380
is we always use the
base point to be x_0.
00:09:40.380 --> 00:09:44.180
So I'm always going
to have x_0 = 0
00:09:44.180 --> 00:09:48.920
in this standard list of
formulas that I'm going to use.
00:09:48.920 --> 00:09:52.030
And if I put x_0 =
0, then this formula
00:09:52.030 --> 00:09:56.130
becomes f(x), a little
bit simpler to read.
00:09:56.130 --> 00:09:59.980
It becomes f(x)
is f(0) + f'(0) x.
00:10:03.520 --> 00:10:05.950
So this is probably
the form that you'll
00:10:05.950 --> 00:10:10.680
want to remember most.
00:10:10.680 --> 00:10:12.580
That's again, just the
linear approximation.
00:10:12.580 --> 00:10:16.010
But one always has
to remember, and this
00:10:16.010 --> 00:10:22.650
is a very important thing, this
one only worked near x is 1.
00:10:22.650 --> 00:10:29.170
This approximation here really
only works when x is near x_0.
00:10:29.170 --> 00:10:31.600
So that's a little addition
that you need to throw in.
00:10:31.600 --> 00:10:38.810
So this one works
when x is near 0.
00:10:38.810 --> 00:10:40.770
You can't expect it
to be true far away.
00:10:40.770 --> 00:10:42.790
The curve can go
anywhere it wants,
00:10:42.790 --> 00:10:46.500
when it's far away from
the point of tangency.
00:10:46.500 --> 00:10:49.060
So, OK, so let's work this out.
00:10:49.060 --> 00:10:51.560
Let's do it for
the sine function,
00:10:51.560 --> 00:10:56.401
for the cosine function,
and for e^x, to begin with.
00:10:56.401 --> 00:10:56.900
Yeah.
00:10:56.900 --> 00:10:57.400
Question.
00:10:57.400 --> 00:11:02.522
STUDENT: [INAUDIBLE]
00:11:02.522 --> 00:11:03.022
PROF.
00:11:03.022 --> 00:11:03.126
JERISON: Yeah.
00:11:03.126 --> 00:11:04.125
When does this one work.
00:11:04.125 --> 00:11:07.410
Well, so the question was,
when does this one work.
00:11:07.410 --> 00:11:12.100
Again, this is when x
is approximately x_0.
00:11:12.100 --> 00:11:18.130
Because it's actually the
same as this one over here.
00:11:18.130 --> 00:11:20.050
OK.
00:11:20.050 --> 00:11:23.010
And indeed, that's
what's going on when
00:11:23.010 --> 00:11:24.870
we take this limiting value.
00:11:24.870 --> 00:11:26.760
Delta x going to 0 is the same.
00:11:26.760 --> 00:11:27.980
Delta x small.
00:11:27.980 --> 00:11:37.050
So another way of saying it
is, the delta x is small.
00:11:37.050 --> 00:11:41.030
Now, exactly what we mean by
small will also be explained.
00:11:41.030 --> 00:11:45.240
But it is a matter to
some extent of intuition
00:11:45.240 --> 00:11:47.620
as to how much, how good it is.
00:11:47.620 --> 00:11:49.650
In practical cases,
people will really
00:11:49.650 --> 00:11:52.950
care about how small it is
before the approximation is
00:11:52.950 --> 00:11:53.970
useful.
00:11:53.970 --> 00:11:56.710
And that's a serious issue.
00:11:56.710 --> 00:12:00.200
All right, so let me carry out
these approximations for x.
00:12:00.200 --> 00:12:06.710
Again, this is
always for x near 0.
00:12:06.710 --> 00:12:08.930
So all of these are
going to be for x near 0.
00:12:08.930 --> 00:12:10.790
So in order to make
this computation,
00:12:10.790 --> 00:12:15.170
I have to evaluate the function.
00:12:15.170 --> 00:12:17.817
I need to plug in
two numbers here.
00:12:17.817 --> 00:12:19.150
In order to get this expression.
00:12:19.150 --> 00:12:23.070
I need to know what f(0) is and
I need to know what f'(0) is.
00:12:23.070 --> 00:12:26.110
If this is the function f(x),
then I'm going to make a little
00:12:26.110 --> 00:12:30.615
table over to the right here
with f' and then I'm going
00:12:30.615 --> 00:12:33.160
to evaluate f(0), and
then I'm going to evaluate
00:12:33.160 --> 00:12:38.150
f'(0), and then read off
what the answers are.
00:12:38.150 --> 00:12:41.450
Right, so first of all if
the function is sine x,
00:12:41.450 --> 00:12:44.170
the derivative is cosine x.
00:12:44.170 --> 00:12:49.690
The value of f(0),
that's sine of 0, is 0.
00:12:49.690 --> 00:12:51.550
The derivative is cosine.
00:12:51.550 --> 00:12:54.060
Cosine of 0 is 1.
00:12:54.060 --> 00:12:55.350
So there we go.
00:12:55.350 --> 00:12:58.850
So now we have the
coefficients 0 and 1.
00:12:58.850 --> 00:13:01.070
So this number is 0.
00:13:01.070 --> 00:13:04.480
And this number is 1.
00:13:04.480 --> 00:13:11.310
So what we get here is 0 + 1x,
so this is approximately x.
00:13:11.310 --> 00:13:18.080
There's the linear
approximation to sin x.
00:13:18.080 --> 00:13:20.570
Similarly, so now this
is a routine matter
00:13:20.570 --> 00:13:22.440
to just read this
off for this table.
00:13:22.440 --> 00:13:23.940
We'll do it for the
cosine function.
00:13:23.940 --> 00:13:30.230
If you differentiate the
cosine, what you get is -sin x.
00:13:30.230 --> 00:13:34.940
The value at 0 is 1, so
that's cosine of 0 at 1.
00:13:34.940 --> 00:13:39.540
The value of this
minus sine at 0 is 0.
00:13:39.540 --> 00:13:43.890
So this is going back
over here, 1 + 0x,
00:13:43.890 --> 00:13:48.170
so this is approximately 1.
00:13:48.170 --> 00:13:52.300
This linear function
happens to be constant.
00:13:52.300 --> 00:13:58.730
And finally, if I do need e^x,
its derivative is again e^x,
00:13:58.730 --> 00:14:02.840
and its value at 0 is 1, the
value of the derivative at 0 is
00:14:02.840 --> 00:14:04.180
also 1.
00:14:04.180 --> 00:14:09.500
So both of the terms here,
f(0) and f'(0), they're both 1
00:14:09.500 --> 00:14:15.400
and we get 1 + x.
00:14:15.400 --> 00:14:18.360
So these are the
linear approximations.
00:14:18.360 --> 00:14:19.610
You can memorize these.
00:14:19.610 --> 00:14:23.940
You'll probably remember them
either this way or that way.
00:14:23.940 --> 00:14:26.130
This collection of
information here
00:14:26.130 --> 00:14:28.556
encodes the same
collection of information
00:14:28.556 --> 00:14:29.430
as we have over here.
00:14:29.430 --> 00:14:31.460
For the values of the
function and the values
00:14:31.460 --> 00:14:36.310
of their derivatives at 0.
00:14:36.310 --> 00:14:39.370
So let me just emphasize again
the geometric point of view
00:14:39.370 --> 00:14:48.840
by drawing pictures
of these results.
00:14:48.840 --> 00:14:56.605
So first of all, for the sine
function, here's the sine
00:14:56.605 --> 00:15:03.500
- well, close enough.
00:15:03.500 --> 00:15:07.170
So that's - boy, now that is
quite some sine, isn't it?
00:15:07.170 --> 00:15:10.570
I should try to make the two
bumps be the same height,
00:15:10.570 --> 00:15:11.870
roughly speaking.
00:15:11.870 --> 00:15:15.450
Anyway the tangent line
we're talking about is here.
00:15:15.450 --> 00:15:17.730
And this is y = x.
00:15:17.730 --> 00:15:22.870
And this is the function sine x.
00:15:22.870 --> 00:15:28.870
And near 0, those things
coincide pretty closely.
00:15:28.870 --> 00:15:34.040
The cosine function, I'll
put that underneath, I guess.
00:15:34.040 --> 00:15:35.110
I think I can fit it.
00:15:35.110 --> 00:15:39.390
Make it a little smaller here.
00:15:39.390 --> 00:15:44.850
So for the cosine
function, we're up here.
00:15:44.850 --> 00:15:48.500
It's y = 1.
00:15:48.500 --> 00:15:51.990
Well, no wonder the
tangent line is constant.
00:15:51.990 --> 00:15:54.630
It's horizontal.
00:15:54.630 --> 00:15:57.920
The tangent line is horizontal,
so the function corresponding
00:15:57.920 --> 00:15:59.560
is constant.
00:15:59.560 --> 00:16:04.890
So this is y = cos x.
00:16:04.890 --> 00:16:14.790
And finally, if I draw y = e^x,
that's coming down like this.
00:16:14.790 --> 00:16:17.870
And the tangent line is here.
00:16:17.870 --> 00:16:19.410
And it's y = 1 + x.
00:16:19.410 --> 00:16:24.700
The value is 1 and
the slope is 1.
00:16:24.700 --> 00:16:28.030
So this is how to remember
it graphically if you like.
00:16:28.030 --> 00:16:34.810
This analytic picture
is extremely important
00:16:34.810 --> 00:16:37.950
and will help you to
deal with sines, cosines
00:16:37.950 --> 00:16:41.090
and exponentials.
00:16:41.090 --> 00:16:41.730
Yes, question.
00:16:41.730 --> 00:16:45.806
STUDENT: [INAUDIBLE]
00:16:45.806 --> 00:16:46.306
PROF.
00:16:46.306 --> 00:16:48.181
JERISON: The question
is what do you normally
00:16:48.181 --> 00:16:50.350
use linear approximations for.
00:16:50.350 --> 00:16:51.140
Good question.
00:16:51.140 --> 00:16:52.260
We're getting there.
00:16:52.260 --> 00:16:54.210
First, we're getting a
little library of them
00:16:54.210 --> 00:16:56.220
and I'll give you
a few examples.
00:16:56.220 --> 00:17:02.540
OK, so now, I need
to finish the catalog
00:17:02.540 --> 00:17:05.700
with two more examples which
are just a little bit, slightly
00:17:05.700 --> 00:17:07.620
more challenging.
00:17:07.620 --> 00:17:09.860
And a little bit less obvious.
00:17:09.860 --> 00:17:22.356
So, the next couple that we're
going to do are ln(1+x) and (1
00:17:22.356 --> 00:17:25.530
+ x)^r.
00:17:25.530 --> 00:17:28.130
OK, these are the last two
that we're going to write down.
00:17:28.130 --> 00:17:30.850
And that you need
to think about.
00:17:30.850 --> 00:17:34.930
Now, the procedure is
the same as over here.
00:17:34.930 --> 00:17:39.230
Namely, I have to write down
f' and I have to write down
00:17:39.230 --> 00:17:41.992
f'(0) and I have to
write down f'(0).
00:17:41.992 --> 00:17:43.450
And then I'll have
the coefficients
00:17:43.450 --> 00:17:46.970
to be able to fill in
what the approximation is.
00:17:46.970 --> 00:17:51.840
So f' = 1 / (1+x), in the
case of the logarithm.
00:17:51.840 --> 00:17:57.010
And f(0), if I plug in,
that's log of 1, which is 0.
00:17:57.010 --> 00:18:01.190
And f' if I plug
in 0 here, I get 1.
00:18:01.190 --> 00:18:04.850
And similarly if I do it for
this one, I get r(1+x)^(r-1).
00:18:07.850 --> 00:18:12.320
And when I plug in f(0),
I get 1^r, which is 1.
00:18:12.320 --> 00:18:18.850
And here I get r
(1)^(r-1), which is r.
00:18:18.850 --> 00:18:22.830
So the corresponding statement
here is that ln(1+x) is
00:18:22.830 --> 00:18:24.790
approximately x.
00:18:24.790 --> 00:18:31.140
And (1+x)^r is
approximately 1 + rx.
00:18:31.140 --> 00:18:35.660
That's 0 + 1x and
here we have 1 + rx.
00:18:41.370 --> 00:18:44.320
And now, I do want
to make a connection,
00:18:44.320 --> 00:18:47.120
explain to you what's going
on here and the connection
00:18:47.120 --> 00:18:48.750
with the first example.
00:18:48.750 --> 00:18:50.800
We already did the
logarithm once.
00:18:50.800 --> 00:18:53.520
And let's just point out
that these two computations
00:18:53.520 --> 00:18:57.470
are the same, or
practically the same.
00:18:57.470 --> 00:19:02.580
Here I use the base point
1, but because of my,
00:19:02.580 --> 00:19:05.420
sort of, convenient
form, which will end up,
00:19:05.420 --> 00:19:07.180
I claim, being much
more convenient
00:19:07.180 --> 00:19:09.310
for pretty much
every purpose, we
00:19:09.310 --> 00:19:14.510
want to do these things
near x is approximately 0.
00:19:14.510 --> 00:19:19.040
You cannot expand the logarithm
and understand a tangent line
00:19:19.040 --> 00:19:22.730
for it at x equals 0, because
it goes down to minus infinity.
00:19:22.730 --> 00:19:27.260
Similarly, if you
try to graph (1+x)^r,
00:19:27.260 --> 00:19:30.690
x^r without the 1 here,
you'll discover that sometimes
00:19:30.690 --> 00:19:33.260
the slope is infinite,
and so forth.
00:19:33.260 --> 00:19:35.500
So this is a bad
choice of point.
00:19:35.500 --> 00:19:39.030
1 is a much better choice
of a place to expand around.
00:19:39.030 --> 00:19:42.150
And then we shift things so
that it looks like it's x = 0,
00:19:42.150 --> 00:19:43.600
by shifting by the 1.
00:19:43.600 --> 00:19:50.950
So the connection with the
previous example is that
00:19:50.950 --> 00:19:57.020
the-- what we wrote before I
could write as ln u = u - 1.
00:19:57.020 --> 00:20:00.890
Right, that's just recopying
what I have over here.
00:20:00.890 --> 00:20:04.930
Except with the letter u
rather than the letter x.
00:20:04.930 --> 00:20:12.790
And then I plug in, u = 1 + x.
00:20:12.790 --> 00:20:14.920
And then that, if
I copy it down,
00:20:14.920 --> 00:20:16.880
you see that I have
a u in place of 1+x,
00:20:16.880 --> 00:20:19.020
that's the same as this.
00:20:19.020 --> 00:20:22.880
And if I write out u-1,
if I subtract 1 from u,
00:20:22.880 --> 00:20:23.924
that means that it's x.
00:20:23.924 --> 00:20:25.840
So that's what's on the
right-hand side there.
00:20:25.840 --> 00:20:27.720
So these are the
same computation,
00:20:27.720 --> 00:20:38.860
I've just changed the variable.
00:20:38.860 --> 00:20:44.220
So now I want to try to
address the question that was
00:20:44.220 --> 00:20:47.380
asked about how this is used.
00:20:47.380 --> 00:20:49.370
And what the importance is.
00:20:49.370 --> 00:20:58.010
And what I'm going to do is
just give you one example here.
00:20:58.010 --> 00:21:02.690
And then try to emphasize.
00:21:02.690 --> 00:21:05.930
The first way in which
this is a useful idea.
00:21:05.930 --> 00:21:10.460
So, or maybe this is
the second example.
00:21:10.460 --> 00:21:13.330
If you like.
00:21:13.330 --> 00:21:16.460
So we'll call this
Example 2, maybe.
00:21:16.460 --> 00:21:19.070
So let's just take
the logarithm of 1.1.
00:21:19.070 --> 00:21:22.220
Just a second.
00:21:22.220 --> 00:21:25.710
Let's take the logarithm of 1.1.
00:21:25.710 --> 00:21:30.150
So I claim that, according to
our rules, I can glance at this
00:21:30.150 --> 00:21:33.680
and I can immediately see
that it's approximately 1/10.
00:21:33.680 --> 00:21:35.710
So what did I use here?
00:21:35.710 --> 00:21:42.630
I used that ln(1+x)
is approximately x,
00:21:42.630 --> 00:21:46.281
and the value of x
that I used was 1/10.
00:21:46.281 --> 00:21:46.780
Right?
00:21:46.780 --> 00:21:48.850
So that is the
formula, so I should
00:21:48.850 --> 00:21:54.560
put a box around these
two formulas too.
00:21:54.560 --> 00:21:57.730
That's this formula here,
applied with x = 1/10.
00:21:57.730 --> 00:22:01.680
And I'm claiming that 1/10 is
a sufficiently small number,
00:22:01.680 --> 00:22:08.845
sufficiently close to 0,
that this is an OK statement.
00:22:08.845 --> 00:22:10.220
So the first
question that I want
00:22:10.220 --> 00:22:12.000
to ask you is,
which do you think
00:22:12.000 --> 00:22:14.470
is a more complicated thing.
00:22:14.470 --> 00:22:19.244
The left-hand side or
the right-hand side.
00:22:19.244 --> 00:22:21.160
I claim that this is a
more complicated thing,
00:22:21.160 --> 00:22:24.070
you'd have to go to a calculator
to punch out and figure out
00:22:24.070 --> 00:22:25.170
what this thing is.
00:22:25.170 --> 00:22:26.230
This is easy.
00:22:26.230 --> 00:22:28.730
You know what a tenth is.
00:22:28.730 --> 00:22:31.530
So the distinction
that I want to make
00:22:31.530 --> 00:22:37.190
is that this half, this
part, this is hard.
00:22:37.190 --> 00:22:40.700
And this is easy.
00:22:40.700 --> 00:22:43.090
Now, that may look
contradictory,
00:22:43.090 --> 00:22:45.610
but I want to just do
it right above as well.
00:22:45.610 --> 00:22:48.940
This is hard.
00:22:48.940 --> 00:22:52.160
And this is easy.
00:22:52.160 --> 00:22:52.870
OK.
00:22:52.870 --> 00:22:56.850
This looks uglier, but
actually this is the hard one.
00:22:56.850 --> 00:22:58.650
And this is giving us
information about it.
00:22:58.650 --> 00:23:00.930
Now, let me show
you why that's true.
00:23:00.930 --> 00:23:02.530
Look down this column here.
00:23:02.530 --> 00:23:05.230
These are the hard
ones, hard functions.
00:23:05.230 --> 00:23:07.360
These are the easy functions.
00:23:07.360 --> 00:23:09.970
What's easier than this?
00:23:09.970 --> 00:23:11.260
Nothing.
00:23:11.260 --> 00:23:11.760
OK.
00:23:11.760 --> 00:23:12.590
Well, yeah, 0.
00:23:12.590 --> 00:23:14.480
That's easier.
00:23:14.480 --> 00:23:16.060
Over here it gets even worse.
00:23:16.060 --> 00:23:21.090
These are the hard functions
and these are the easy ones.
00:23:21.090 --> 00:23:24.910
So that's the main advantage
of linear approximation
00:23:24.910 --> 00:23:27.730
is you get something much
simpler to deal with.
00:23:27.730 --> 00:23:30.380
And if you've made a
valid approximation
00:23:30.380 --> 00:23:33.510
you can make much
progress on problems.
00:23:33.510 --> 00:23:35.780
OK, we'll be doing
some more examples,
00:23:35.780 --> 00:23:38.990
but I saw some more questions
before I made that point.
00:23:38.990 --> 00:23:39.490
Yeah.
00:23:39.490 --> 00:23:42.373
STUDENT: [INAUDIBLE]
00:23:42.373 --> 00:23:42.873
PROF.
00:23:42.873 --> 00:23:46.080
JERISON: Is this
ln of 1.1 or what?
00:23:46.080 --> 00:23:48.410
STUDENT: [INAUDIBLE]
00:23:48.410 --> 00:23:49.115
PROF.
00:23:49.115 --> 00:23:52.580
JERISON: This is a parens there.
00:23:52.580 --> 00:23:56.430
It's ln of 1.1, it's the
digital number, right.
00:23:56.430 --> 00:23:59.820
I guess I've never used that
before a decimal point, have I?
00:23:59.820 --> 00:24:04.834
I don't know.
00:24:04.834 --> 00:24:05.500
Other questions.
00:24:05.500 --> 00:24:11.910
STUDENT: [INAUDIBLE]
00:24:11.910 --> 00:24:12.410
PROF.
00:24:12.410 --> 00:24:12.990
JERISON: OK.
00:24:12.990 --> 00:24:14.900
So let's continue here.
00:24:14.900 --> 00:24:18.570
Let me give you some more
examples, where it becomes
00:24:18.570 --> 00:24:21.460
even more vivid if you like.
00:24:21.460 --> 00:24:24.340
That this approximation
is giving us something
00:24:24.340 --> 00:24:30.190
a little simpler to deal with.
00:24:30.190 --> 00:24:34.960
So here's Example 3.
00:24:34.960 --> 00:24:48.400
I want to, I'll find the linear
approximation near x = 0.
00:24:48.400 --> 00:24:52.340
I also - when I write this
expression near x = 0,
00:24:52.340 --> 00:24:55.020
that's the same thing as this.
00:24:55.020 --> 00:24:58.940
That's the same thing as
saying x is approximately 0 -
00:24:58.940 --> 00:25:06.360
of the function e^(-3x)
divided by square root 1+x.
00:25:09.170 --> 00:25:17.390
So here's a function.
00:25:17.390 --> 00:25:17.890
OK.
00:25:17.890 --> 00:25:21.990
Now, what I claim I want
to use for the purposes
00:25:21.990 --> 00:25:26.830
of this approximation,
are just the sum
00:25:26.830 --> 00:25:32.110
of the approximation formulas
that we've already derived.
00:25:32.110 --> 00:25:33.850
And just to combine
them algebraically.
00:25:33.850 --> 00:25:35.350
So I'm not going
to do any calculus,
00:25:35.350 --> 00:25:37.080
I'm just going to remember.
00:25:37.080 --> 00:25:41.580
So with e^(-3x), it's pretty
clear that I should be using
00:25:41.580 --> 00:25:44.570
this formula for e^x.
00:25:44.570 --> 00:25:47.820
For the other one, it may be
slightly less obvious but we
00:25:47.820 --> 00:25:53.470
have powers of 1+x over here.
00:25:53.470 --> 00:25:55.510
So let's plug those in.
00:25:55.510 --> 00:26:04.640
I'll put this up so that
you can remember it.
00:26:04.640 --> 00:26:10.810
And we're going to carry
out this approximation.
00:26:10.810 --> 00:26:16.380
So, first of all, I'm going
to write this so that it's
00:26:16.380 --> 00:26:17.910
slightly more suggestive.
00:26:17.910 --> 00:26:23.630
Namely, I'm going to
write it as a product.
00:26:23.630 --> 00:26:27.220
And there you can
now see the exponent.
00:26:27.220 --> 00:26:31.870
In this case, r = 1/2, eh
-1/2, that we're going to use.
00:26:31.870 --> 00:26:32.900
OK.
00:26:32.900 --> 00:26:39.880
So now I have
e^(-3x) (1+x)^(-1/2),
00:26:39.880 --> 00:26:41.940
and that's going to
be approximately--
00:26:41.940 --> 00:26:44.220
well I'm going to
use this formula.
00:26:44.220 --> 00:26:48.760
I have to use it correctly. x is
replaced by -3x, so this is 1 -
00:26:48.760 --> 00:26:50.160
3x.
00:26:50.160 --> 00:26:52.270
And then over here,
I can just copy
00:26:52.270 --> 00:26:57.900
verbatim the other approximation
formula with r = -1/2.
00:26:57.900 --> 00:27:05.670
So this is times 1 - 1/2 x.
00:27:05.670 --> 00:27:11.080
And now I'm going to carry
out the multiplication.
00:27:11.080 --> 00:27:17.100
So this is 1 - 3x
- 1/2 x + 3/2 x^2.
00:27:27.310 --> 00:27:32.090
So now, here's our formula.
00:27:32.090 --> 00:27:34.340
So now this isn't
where things stop.
00:27:34.340 --> 00:27:36.960
And indeed, in this
kind of arithmetic
00:27:36.960 --> 00:27:39.420
that I'm describing
now, things are
00:27:39.420 --> 00:27:43.780
easier than they are in
ordinary algebra, in arithmetic.
00:27:43.780 --> 00:27:47.770
The reason is that there's
another step, which
00:27:47.770 --> 00:27:49.200
I'm now going to perform.
00:27:49.200 --> 00:27:54.602
Which is that I'm going to
throw away this term here.
00:27:54.602 --> 00:27:55.560
I'm going to ignore it.
00:27:55.560 --> 00:27:57.480
In fact, I didn't even
have to work it out.
00:27:57.480 --> 00:27:59.070
Because I'm going
to throw it away.
00:27:59.070 --> 00:28:01.520
So the reason is
that already, when
00:28:01.520 --> 00:28:03.424
I passed from this
expression to this one,
00:28:03.424 --> 00:28:05.340
that is from this type
of thing to this thing,
00:28:05.340 --> 00:28:07.940
I was already throwing away
quadratic and higher-ordered
00:28:07.940 --> 00:28:09.460
terms.
00:28:09.460 --> 00:28:12.650
So this isn't the
only quadratic term.
00:28:12.650 --> 00:28:13.640
There are tons of them.
00:28:13.640 --> 00:28:14.920
I have to ignore
all of them if I'm
00:28:14.920 --> 00:28:16.128
going to ignore some of them.
00:28:16.128 --> 00:28:20.240
And in fact, I only want to
be left with the linear stuff.
00:28:20.240 --> 00:28:23.220
Because that's all I'm really
getting a valid computation
00:28:23.220 --> 00:28:24.080
for.
00:28:24.080 --> 00:28:28.060
So, this is approximately
1 minus, so let's see.
00:28:28.060 --> 00:28:32.450
It's a total of 7/2 x.
00:28:32.450 --> 00:28:36.410
And this is the answer.
00:28:36.410 --> 00:28:38.290
This is the linear part.
00:28:38.290 --> 00:28:42.800
So the x^2 term is negligible.
00:28:42.800 --> 00:28:46.680
So we drop x^2 term.
00:28:46.680 --> 00:28:55.712
Terms, and higher.
00:28:55.712 --> 00:28:57.420
All of those terms
should be lower-order.
00:28:57.420 --> 00:29:00.180
If you imagine x is
1/10, or maybe 1/100,
00:29:00.180 --> 00:29:04.470
then these terms will end
up being much smaller.
00:29:04.470 --> 00:29:08.970
So we have a rather
crude approach.
00:29:08.970 --> 00:29:10.530
And that's really
the simplicity,
00:29:10.530 --> 00:29:15.360
and that's the savings.
00:29:15.360 --> 00:29:21.020
So now, since this unit
is called Applications,
00:29:21.020 --> 00:29:24.380
and these are indeed
applications to math,
00:29:24.380 --> 00:29:30.360
I also wanted to give you
a real-life application.
00:29:30.360 --> 00:29:34.290
Or a place where linear
approximations come up
00:29:34.290 --> 00:29:46.590
in real life.
00:29:46.590 --> 00:29:50.560
So maybe we'll call
this Example 4.
00:29:50.560 --> 00:29:57.270
This is supposedly
a real-life example.
00:29:57.270 --> 00:30:06.580
I'll try to persuade
you that it is.
00:30:06.580 --> 00:30:09.840
So I like this example because
it's got a lot of math,
00:30:09.840 --> 00:30:11.170
as well as physics in it.
00:30:11.170 --> 00:30:17.000
So here I am, on the
surface of the earth.
00:30:17.000 --> 00:30:24.610
And here is a satellite
going this way.
00:30:24.610 --> 00:30:30.790
At some velocity, v.
And this satellite
00:30:30.790 --> 00:30:33.630
has a clock on it because
this is a GPS satellite.
00:30:33.630 --> 00:30:37.720
And it has a time, T, OK?
00:30:37.720 --> 00:30:41.160
But I have a watch, in
fact it's right here.
00:30:41.160 --> 00:30:44.030
And I have a time which I keep.
00:30:44.030 --> 00:30:48.170
Which is T', And there's
an interesting relationship
00:30:48.170 --> 00:30:56.650
between T and T', which
is called time dilation.
00:30:56.650 --> 00:31:04.860
And this is from
special relativity.
00:31:04.860 --> 00:31:06.470
And it's the following formula.
00:31:06.470 --> 00:31:13.720
T' = T divided by the
square root of 1 - v^2/c^2,
00:31:13.720 --> 00:31:17.230
where v is the velocity
of the satellite,
00:31:17.230 --> 00:31:22.960
and c is the speed of light.
00:31:22.960 --> 00:31:28.080
So now I'd like to get a
rough idea of how different
00:31:28.080 --> 00:31:34.980
my watch is from the
clock on the satellite.
00:31:34.980 --> 00:31:38.540
So I'm going to use
this same approximation,
00:31:38.540 --> 00:31:40.990
we've already used it once.
00:31:40.990 --> 00:31:42.010
I'm going to write t.
00:31:42.010 --> 00:31:43.990
But now let me just remind you.
00:31:43.990 --> 00:31:46.809
The situation here is, we
have something of the form
00:31:46.809 --> 00:31:47.350
(1-u)^(-1/2).
00:31:52.410 --> 00:31:55.760
That's what's happening when
I multiply through here.
00:31:55.760 --> 00:31:59.500
So with u = v^2 / c^2.
00:32:02.080 --> 00:32:05.240
So in real life, of
course, the expression
00:32:05.240 --> 00:32:07.780
that you're going to use
the linear approximation on
00:32:07.780 --> 00:32:10.280
isn't necessarily itself linear.
00:32:10.280 --> 00:32:11.990
It can be any physical quantity.
00:32:11.990 --> 00:32:15.940
So in this case it's v
squared over c squared.
00:32:15.940 --> 00:32:18.189
And now the
approximation formula
00:32:18.189 --> 00:32:20.230
says that if this is
approximately equal to, well
00:32:20.230 --> 00:32:21.540
again it's the same rule.
00:32:21.540 --> 00:32:25.870
There's an r and then x
is -u, so this is - - 1/2,
00:32:25.870 --> 00:32:34.610
so it's 1 + 1/2 u.
00:32:34.610 --> 00:32:40.350
So this is approximately,
by the same rule, this is T,
00:32:40.350 --> 00:32:55.800
T' is approximately t
T(1 + 1/2 v^2/c^2) Now,
00:32:55.800 --> 00:32:58.150
I promised you that this
would be a real-life problem.
00:32:58.150 --> 00:33:02.520
So the question is when people
were designing these GPS
00:33:02.520 --> 00:33:06.666
systems, they run clocks
in the satellites.
00:33:06.666 --> 00:33:08.790
You're down there, you're
making your measurements,
00:33:08.790 --> 00:33:12.270
you're talking to
the satellite by--
00:33:12.270 --> 00:33:15.310
or you're receiving its
signals from its radio.
00:33:15.310 --> 00:33:19.010
The question is, is this
going to cause problems
00:33:19.010 --> 00:33:23.670
in the transmission.
00:33:23.670 --> 00:33:25.580
And there are dozens
of such problems
00:33:25.580 --> 00:33:27.180
that you have to check for.
00:33:27.180 --> 00:33:29.950
So in this case, what
actually happened
00:33:29.950 --> 00:33:35.010
is that v is about 4
kilometers per second.
00:33:35.010 --> 00:33:38.740
That's how fast the GPS
satellites actually go.
00:33:38.740 --> 00:33:41.430
In fact, they had to decide to
put them at a certain altitude
00:33:41.430 --> 00:33:43.950
and they could've tweaked
this if they had put them
00:33:43.950 --> 00:33:46.040
at different places.
00:33:46.040 --> 00:33:55.330
Anyway, the speed of light is
3 * 10^5 kilometers per second.
00:33:55.330 --> 00:34:01.100
So this number, v^2 / c^2
is approximately 10^(-10).
00:34:05.710 --> 00:34:11.160
Now, if you actually keep
track of how much of an error
00:34:11.160 --> 00:34:15.530
that would make in a GPS
location, what you would find
00:34:15.530 --> 00:34:17.820
is maybe it's a millimeter
or something like that.
00:34:17.820 --> 00:34:20.080
So in fact it doesn't matter.
00:34:20.080 --> 00:34:21.380
So that's nice.
00:34:21.380 --> 00:34:23.180
But in fact the
engineers who were
00:34:23.180 --> 00:34:26.870
designing these systems actually
did use this very computation.
00:34:26.870 --> 00:34:29.270
Exactly this.
00:34:29.270 --> 00:34:31.640
And the way that
they used it was,
00:34:31.640 --> 00:34:35.190
they decided that because
the clocks were different,
00:34:35.190 --> 00:34:38.740
when the satellite broadcasts
its radio frequency,
00:34:38.740 --> 00:34:40.350
that frequency would be shifted.
00:34:40.350 --> 00:34:41.500
Would be offset.
00:34:41.500 --> 00:34:44.426
And they decided that the
fidelity was so important
00:34:44.426 --> 00:34:46.050
that they would send
the satellites off
00:34:46.050 --> 00:34:49.120
with this kind of,
exactly this, offset.
00:34:49.120 --> 00:34:51.460
To compensate for the
way the signal is.
00:34:51.460 --> 00:34:53.360
So from the point of
view of good reception
00:34:53.360 --> 00:34:56.950
on your little GPS device, they
changed the frequency at which
00:34:56.950 --> 00:35:00.160
the transmitter
in the satellites,
00:35:00.160 --> 00:35:04.990
according to exactly this rule.
00:35:04.990 --> 00:35:08.120
And incidentally, the reason
why they didn't-- they ignored
00:35:08.120 --> 00:35:11.010
higher-order terms, the
sort of quadratic terms,
00:35:11.010 --> 00:35:17.460
is that if you take u^2
that's a size 10^(-20).
00:35:17.460 --> 00:35:20.104
And that really is
totally negligible.
00:35:20.104 --> 00:35:22.020
That doesn't matter to
any measurement at all.
00:35:22.020 --> 00:35:25.210
That's on the order
of nanometers,
00:35:25.210 --> 00:35:30.200
and it's not important for
any of the uses to which GPS
00:35:30.200 --> 00:35:32.510
is put.
00:35:32.510 --> 00:35:40.470
OK, so that's a real example of
a use of linear approximations.
00:35:40.470 --> 00:35:42.720
So. let's take a
little pause here.
00:35:42.720 --> 00:35:44.850
I'm going to switch
gears and talk
00:35:44.850 --> 00:35:46.610
about quadratic approximations.
00:35:46.610 --> 00:35:48.900
But before I do that, let's
have some more questions.
00:35:48.900 --> 00:35:49.400
Yeah.
00:35:49.400 --> 00:36:03.780
STUDENT: [INAUDIBLE]
00:36:03.780 --> 00:36:04.566
PROF.
00:36:04.566 --> 00:36:08.040
JERISON: OK, so the
question was asked,
00:36:08.040 --> 00:36:11.580
suppose I did this
by different method.
00:36:11.580 --> 00:36:15.840
Suppose I applied the
original formula here.
00:36:15.840 --> 00:36:18.050
Namely, I define
the function f(x),
00:36:18.050 --> 00:36:22.140
which was this function here.
00:36:22.140 --> 00:36:25.050
And then I plugged
in its value at x = 0
00:36:25.050 --> 00:36:28.000
and the value of its
derivative at x = 0.
00:36:28.000 --> 00:36:32.510
So the answer is, yes, it's
also true that if I call this
00:36:32.510 --> 00:36:37.940
function f f(x), then it
must be true that the linear
00:36:37.940 --> 00:36:45.910
approximation is f(x_0) plus
f' of - I'm sorry, it's at 0,
00:36:45.910 --> 00:36:49.340
so it's f(0), f'(0) times x.
00:36:49.340 --> 00:36:50.550
So that should be true.
00:36:50.550 --> 00:36:52.810
That's the formula
that we're using.
00:36:52.810 --> 00:36:57.170
It's up there in the pink also.
00:36:57.170 --> 00:36:58.590
So this is the formula.
00:36:58.590 --> 00:37:00.650
So now, what about f(0)?
00:37:00.650 --> 00:37:04.350
Well, if I plug in
0 here, I get 1 * 1.
00:37:04.350 --> 00:37:05.940
So this thing is 1.
00:37:05.940 --> 00:37:07.550
So that's no surprise.
00:37:07.550 --> 00:37:11.260
And that's what I got.
00:37:11.260 --> 00:37:15.600
If I computed f',
by the product rule
00:37:15.600 --> 00:37:19.150
it would be an annoying,
somewhat long, computation.
00:37:19.150 --> 00:37:21.510
And because of
what we just done,
00:37:21.510 --> 00:37:23.130
we know what it has to be.
00:37:23.130 --> 00:37:25.990
It has to be negative 7/2.
00:37:25.990 --> 00:37:28.280
Because this is a
shortcut for doing it.
00:37:28.280 --> 00:37:29.900
This is faster than doing that.
00:37:29.900 --> 00:37:32.190
But of course, that's a
legal way of doing it.
00:37:32.190 --> 00:37:33.780
When you get to
second derivatives,
00:37:33.780 --> 00:37:36.210
you'll quickly discover that
this method that I've just
00:37:36.210 --> 00:37:38.950
described is
complicated, but far
00:37:38.950 --> 00:37:41.330
superior to differentiating
this expression twice.
00:37:41.330 --> 00:37:46.087
STUDENT: [INAUDIBLE] PROF.
00:37:46.087 --> 00:37:48.420
JERISON: Would you have to
throw away an x^2 term if you
00:37:48.420 --> 00:37:49.560
differentiated?
00:37:49.560 --> 00:37:50.470
No.
00:37:50.470 --> 00:37:53.220
And in fact, we didn't
really have to do that here.
00:37:53.220 --> 00:37:55.385
If you differentiate
and then plug in x = 0.
00:37:55.385 --> 00:37:57.510
So if you differentiate
this and you plug in x = 0,
00:37:57.510 --> 00:37:58.970
you get -7/2.
00:37:58.970 --> 00:38:01.349
You differentiate this
and you plug in x = 0,
00:38:01.349 --> 00:38:03.140
this term still drops
out because it's just
00:38:03.140 --> 00:38:05.370
a 3x when you differentiate.
00:38:05.370 --> 00:38:08.270
And then you plug in
x = 0, it's gone too.
00:38:08.270 --> 00:38:10.650
And similarly, if you're
up here, it goes away
00:38:10.650 --> 00:38:12.410
and similarly over
here it goes away.
00:38:12.410 --> 00:38:18.555
So the higher-order terms never
influence this computation
00:38:18.555 --> 00:38:19.055
here.
00:38:19.055 --> 00:38:27.430
This just captures the linear
features of the function.
00:38:27.430 --> 00:38:30.980
So now I want to go on to
quadratic approximation.
00:38:30.980 --> 00:38:44.500
And now we're going to
elaborate on this formula.
00:38:44.500 --> 00:38:46.040
So, linear approximation.
00:38:46.040 --> 00:38:49.840
Well, that should have
been linear approximation.
00:38:49.840 --> 00:38:50.530
Liner.
00:38:50.530 --> 00:38:51.680
That's interesting.
00:38:51.680 --> 00:38:54.070
OK, so that was wrong.
00:38:54.070 --> 00:38:59.700
But now we're going to
change it to quadratic.
00:38:59.700 --> 00:39:04.280
So, suppose we talk about a
quadratic approximation here.
00:39:04.280 --> 00:39:07.450
Now, the quadratic
approximation is
00:39:07.450 --> 00:39:15.430
going to be just an elaboration,
one more step of detail.
00:39:15.430 --> 00:39:16.270
From the linear.
00:39:16.270 --> 00:39:18.060
In other words,
it's an extension
00:39:18.060 --> 00:39:20.230
of the linear approximation.
00:39:20.230 --> 00:39:24.320
And so we're adding
one more term here.
00:39:24.320 --> 00:39:26.650
And the extra term
turns out to be related
00:39:26.650 --> 00:39:28.990
to the second derivative.
00:39:28.990 --> 00:39:34.340
But there's a factor of 2.
00:39:34.340 --> 00:39:39.090
So this is the formula for
the quadratic approximation.
00:39:39.090 --> 00:39:46.450
And this chunk of it, of
course, is the linear part.
00:39:46.450 --> 00:39:54.190
This time I'll spell
'linear' correctly.
00:39:54.190 --> 00:39:56.030
So the linear part
is the first piece.
00:39:56.030 --> 00:40:05.050
And the quadratic part
is the second piece.
00:40:05.050 --> 00:40:09.630
I want to develop this
same catalog of functions
00:40:09.630 --> 00:40:11.140
as I had before.
00:40:11.140 --> 00:40:14.640
In other words, I want
to extend our formulas
00:40:14.640 --> 00:40:19.660
to the higher-order terms.
00:40:19.660 --> 00:40:26.070
And if you do that
for this example here,
00:40:26.070 --> 00:40:28.180
maybe I'll even illustrate
with this example
00:40:28.180 --> 00:40:31.050
before I go on, if you
do it with this example
00:40:31.050 --> 00:40:39.320
here, just to give you a
flavor for what goes on,
00:40:39.320 --> 00:40:41.140
what turns out to be the case.
00:40:41.140 --> 00:40:45.390
So this is the linear version.
00:40:45.390 --> 00:40:48.220
And now I'm going to compare
it to the quadratic version.
00:40:48.220 --> 00:40:55.540
So the quadratic version
turns out to be this.
00:40:55.540 --> 00:40:58.760
That's what turns out to be
the quadratic approximation.
00:40:58.760 --> 00:41:03.100
And when I use
this example here,
00:41:03.100 --> 00:41:09.400
so this is 1.1, which is the
same as ln of 1 + 1/10, right?
00:41:09.400 --> 00:41:17.430
So that's approximately
1/10 - 1/2 (1/10)^2.
00:41:17.430 --> 00:41:19.170
So 1/200.
00:41:19.170 --> 00:41:21.960
So that turns out,
instead of being
00:41:21.960 --> 00:41:29.160
1/10, that's point, what is it,
.095 or something like that.
00:41:29.160 --> 00:41:31.370
It's a little bit less.
00:41:31.370 --> 00:41:36.240
It's not .1, but
it's pretty close.
00:41:36.240 --> 00:41:39.350
So if you like,
the correction is
00:41:39.350 --> 00:41:48.900
lower in the decimal expansion.
00:41:48.900 --> 00:41:53.650
Now let me actually
check a few of these.
00:41:53.650 --> 00:41:54.940
I'll carry them out.
00:41:54.940 --> 00:41:58.670
And what I'm going to
probably save for next time
00:41:58.670 --> 00:42:08.020
is explaining to you, so this
is why this factor of 1/2,
00:42:08.020 --> 00:42:10.610
and we're going
to do this later.
00:42:10.610 --> 00:42:11.530
Do this next time.
00:42:11.530 --> 00:42:17.230
You can certainly do well to
stick with this presentation
00:42:17.230 --> 00:42:18.470
for one more lecture.
00:42:18.470 --> 00:42:22.210
So we can see this reinforced.
00:42:22.210 --> 00:42:32.580
So now I'm going to work
out these derivatives
00:42:32.580 --> 00:42:34.630
of the higher-order terms.
00:42:34.630 --> 00:42:39.450
And let me do it for the
x approximately 0 case.
00:42:39.450 --> 00:42:47.990
So first of all, I want to
add in the extra term here.
00:42:47.990 --> 00:42:50.830
Here's the extra term.
00:42:50.830 --> 00:42:53.780
For the quadratic part.
00:42:53.780 --> 00:42:57.050
And now in order to figure
out what's going on,
00:42:57.050 --> 00:43:03.350
I'm going to need to compute,
also, second derivatives.
00:43:03.350 --> 00:43:05.150
So here I need a
second derivative.
00:43:05.150 --> 00:43:11.465
And I need to throw in the value
of that second derivative at 0.
00:43:11.465 --> 00:43:13.340
So this is what I'm
going to need to compute.
00:43:13.340 --> 00:43:17.449
So if I do it, for example,
for the sine function,
00:43:17.449 --> 00:43:18.740
I already have the linear part.
00:43:18.740 --> 00:43:20.290
I need this last bit.
00:43:20.290 --> 00:43:22.570
So I differentiate the
sine function twice
00:43:22.570 --> 00:43:25.180
and I get, I claim
minus the sine function.
00:43:25.180 --> 00:43:26.900
The first derivative
is the cosine
00:43:26.900 --> 00:43:29.250
and the cosine derivative
is minus the sine.
00:43:29.250 --> 00:43:34.180
And when I evaluate it at
0, I get, lo and behold, 0.
00:43:34.180 --> 00:43:35.540
Sine of 0 is 0.
00:43:35.540 --> 00:43:40.361
So actually the quadratic
approximation is the same.
00:43:40.361 --> 00:43:40.860
0x^2.
00:43:40.860 --> 00:43:43.070
There's no x^2 term here.
00:43:43.070 --> 00:43:46.510
So that's why this is such
a terrific approximation.
00:43:46.510 --> 00:43:48.890
It's also the quadratic
approximation.
00:43:48.890 --> 00:43:53.460
For the cosine function,
if you differentiate twice,
00:43:53.460 --> 00:43:56.300
you get the derivative is
minus the sign and derivative
00:43:56.300 --> 00:44:00.170
of that is minus the cosine.
00:44:00.170 --> 00:44:03.060
So that's f''.
00:44:03.060 --> 00:44:09.600
And now, if I evaluate
that at 0, I get -1.
00:44:09.600 --> 00:44:11.530
And so the term that I
have to plug in here,
00:44:11.530 --> 00:44:15.240
this -1 is the coefficient
that appears right here.
00:44:15.240 --> 00:44:23.350
So I need a -1/2 x^2 extra.
00:44:23.350 --> 00:44:26.100
And if you do it for
the e^x, you get an e^x,
00:44:26.100 --> 00:44:39.450
and you got a 1 and so
you get 1/2 x^2 here.
00:44:39.450 --> 00:44:42.329
I'm going to finish these
two in just a second,
00:44:42.329 --> 00:44:43.745
but I first want
to tell you about
00:44:43.745 --> 00:44:56.480
the geometric significance
of this quadratic term.
00:44:56.480 --> 00:44:58.790
So here we go.
00:44:58.790 --> 00:45:18.430
Geometric significance
(of the quadratic term).
00:45:18.430 --> 00:45:21.100
So the geometric
significance is best
00:45:21.100 --> 00:45:25.670
to describe just by
drawing a picture here.
00:45:25.670 --> 00:45:29.300
And I'm going to draw the
picture of the cosine function.
00:45:29.300 --> 00:45:34.270
And remember we already
had the tangent line.
00:45:34.270 --> 00:45:38.620
So the tangent line was
this horizontal here.
00:45:38.620 --> 00:45:40.350
And that was y = 1.
00:45:40.350 --> 00:45:42.880
But you can see intuitively,
that doesn't even
00:45:42.880 --> 00:45:46.130
tell you whether this function
is above or below 1 there.
00:45:46.130 --> 00:45:47.437
Doesn't tell you much.
00:45:47.437 --> 00:45:50.020
It's sort of begging for there
to be a little more information
00:45:50.020 --> 00:45:52.470
to tell us what the
function is doing nearby.
00:45:52.470 --> 00:45:57.470
And indeed, that's what this
second expression does for us.
00:45:57.470 --> 00:46:00.850
It's some kind of
parabola underneath here.
00:46:00.850 --> 00:46:05.420
So this is y = 1 - 1/2 x^2.
00:46:05.420 --> 00:46:08.890
Which is a much better
fit to the curve
00:46:08.890 --> 00:46:12.740
than the horizontal line.
00:46:12.740 --> 00:46:23.750
And this is, if you like,
this is the best fit parabola.
00:46:23.750 --> 00:46:28.510
So it's going to be the
closest parabola to the curve.
00:46:28.510 --> 00:46:31.370
And that's more or
less the significance.
00:46:31.370 --> 00:46:34.600
It's much, much closer.
00:46:34.600 --> 00:46:40.220
All right, I want
to give you, well,
00:46:40.220 --> 00:46:43.040
I think we'll save these other
derivations for next time
00:46:43.040 --> 00:46:44.880
because I think we're
out of time now.
00:46:44.880 --> 00:46:47.110
So we'll do these next time.