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PROFESSOR: Hi.

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Our lesson, today, hopefully
will serve two purposes.

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00:00:33,330 --> 00:00:36,270
On the one hand, we will give
a nice application of the

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00:00:36,270 --> 00:00:36,940
chain rule.

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00:00:36,940 --> 00:00:40,970
Now that we've had two units
devoted to working with the

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chain rule, I thought you might
enjoy seeing how it's

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used in places which aren't
quite that obvious--

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00:00:48,360 --> 00:00:50,520
places that we wouldn't expect
to be used, at least.

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00:00:50,520 --> 00:00:54,670
And secondly, I would like to
pick as my application one

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00:00:54,670 --> 00:00:58,520
which comes up in many, many
different contexts.

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00:00:58,520 --> 00:01:01,750
And without further ado, the
topic I want to cover today is

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00:01:01,750 --> 00:01:04,950
called "Integrals Involving
Parameters".

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00:01:04,950 --> 00:01:06,430
Now, that sounds like
a big mouthful.

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00:01:06,430 --> 00:01:10,000
Let me motivate that for you,
first of all, physically, and

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00:01:10,000 --> 00:01:12,440
then in term of a couple
of geometric examples.

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00:01:12,440 --> 00:01:15,500
You all know from past
experience my great ability

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00:01:15,500 --> 00:01:18,700
with physical applications, so
I won't even try to find

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00:01:18,700 --> 00:01:20,050
anything profound here.

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00:01:20,050 --> 00:01:24,000
Let me just take a pseudo
example, pointing out what

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00:01:24,000 --> 00:01:27,570
type of situation we're trying
to deal with and leaving it to

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00:01:27,570 --> 00:01:30,500
your own backgrounds to see
places where the same

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00:01:30,500 --> 00:01:33,210
principle could have been
applied, but hopefully in a

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00:01:33,210 --> 00:01:35,400
more practical, meaningful
way for you.

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00:01:35,400 --> 00:01:39,390
Imagine for example, that we
have a platform that we're

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00:01:39,390 --> 00:01:41,830
looking along in the
x direction.

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00:01:41,830 --> 00:01:46,180
And we've punched holes in this
platform, say, and liquid

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00:01:46,180 --> 00:01:48,480
is trickling through these
various holes.

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00:01:48,480 --> 00:01:52,360
The holes are all on a
horizontal line this way.

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00:01:52,360 --> 00:01:54,870
What we're going to do is
we're going to focus our

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00:01:54,870 --> 00:01:58,020
attention on a particular
particle of the liquid, and

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00:01:58,020 --> 00:02:00,510
we're going to watch
it as it falls.

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00:02:00,510 --> 00:02:04,110
And what we would like to do is
find how far that particle

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00:02:04,110 --> 00:02:08,520
fell, say, during the first
second of its flight.

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00:02:08,520 --> 00:02:11,350
Obviously, from a calculus point
of view, the first thing

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00:02:11,350 --> 00:02:14,610
we have to do is know what the
velocity function is, because

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00:02:14,610 --> 00:02:17,880
ultimately, we would like to
integrate the velocity.

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00:02:17,880 --> 00:02:20,590
Now, notice that we would expect
the velocity to be

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00:02:20,590 --> 00:02:21,770
depending on time.

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00:02:21,770 --> 00:02:25,360
If this were a freely falling
situation, we'd expect the

49
00:02:25,360 --> 00:02:27,980
usual gravitational
type situation.

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00:02:27,980 --> 00:02:30,440
Or whatever the situation
happened to be, we would

51
00:02:30,440 --> 00:02:32,780
expect on the one hand that
the velocity does

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00:02:32,780 --> 00:02:33,970
depend on of time.

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00:02:33,970 --> 00:02:38,030
On the other hand, because of
how the streams are flowing--

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00:02:38,030 --> 00:02:40,770
in other words, we don't know
what's happening above here

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00:02:40,770 --> 00:02:43,760
that's causing the water to
shoot out-- we don't know what

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00:02:43,760 --> 00:02:46,670
the initial velocity is coming
out of each of these holes in

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00:02:46,670 --> 00:02:50,420
the sense that a different
opening may give rise to a

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00:02:50,420 --> 00:02:53,010
different velocity of
stream coming out.

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00:02:53,010 --> 00:02:55,790
All I'm trying to bring out
here is that as we try to

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focus our attention on a
particular opening, we find

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that the velocity of the
particle that will follow it

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during that first second is a
function both of its position

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x-- in other words, x sub 0 in
this case, because we're

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focusing at x equals x0
and the time t as t

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goes from 0 to 1.

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00:03:16,030 --> 00:03:18,570
And we then simply integrate
along the

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vertical direction here.

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00:03:20,040 --> 00:03:26,960
We find y as a function of x0
from 0 to 1-- v(x0, t) dt.

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00:03:26,960 --> 00:03:29,340
Once we're through integrating,
you see, notice

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that the integration is
with respect to t.

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00:03:32,610 --> 00:03:35,400
So when we're through
integrating, this being a

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definite integral, t
no longer appears.

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We have a function of x0 alone,
saying nothing more

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than the distance that the
particle falls during the

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first second is a function of
the position of the opening

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along the line here.

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Now, at any rate, the practical
application is not

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so much writing down this
equation as the

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inverse is the case.

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00:03:57,690 --> 00:04:02,650
Namely, in many practical
applications, we are given

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00:04:02,650 --> 00:04:04,280
this particular integral.

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00:04:04,280 --> 00:04:07,850
And for some reason or other,
want to determine

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00:04:07,850 --> 00:04:09,190
what v itself is.

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00:04:09,190 --> 00:04:14,250
In other words, we often want
to find the derivative given

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the integral.

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00:04:15,200 --> 00:04:16,190
All right.

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00:04:16,190 --> 00:04:18,649
Let's just let it go at that
for the time being.

88
00:04:18,649 --> 00:04:22,000
The important point is that I
want you to see an example of

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00:04:22,000 --> 00:04:22,530
an integral.

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00:04:22,530 --> 00:04:25,530
Let me just write this here
in more abstract form.

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00:04:25,530 --> 00:04:28,680
It appears to be a definite
integral, a to b.

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00:04:28,680 --> 00:04:32,290
The function inside the
integrand is apparently a

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00:04:32,290 --> 00:04:35,850
function of two variables,
x and y--

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00:04:35,850 --> 00:04:37,650
say, in this case x0 and t.

95
00:04:37,650 --> 00:04:40,730
One of the variables is a
variable of integration--

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00:04:40,730 --> 00:04:41,960
in this case, y--

97
00:04:41,960 --> 00:04:44,640
and the other variable is being
treated as a constant.

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00:04:44,640 --> 00:04:47,660
And that's where the word
parameter comes in. x is a

99
00:04:47,660 --> 00:04:51,600
parameter, meaning a variable
constant, in the sense that,

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00:04:51,600 --> 00:04:55,930
for this particular problem, x
is chosen to be in some domain

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00:04:55,930 --> 00:04:57,040
and remains fixed.

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00:04:57,040 --> 00:05:01,040
In other words, this is some
function of x when we're all

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00:05:01,040 --> 00:05:01,970
through here.

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00:05:01,970 --> 00:05:02,640
All right?

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00:05:02,640 --> 00:05:04,930
As a geometric example, imagine

106
00:05:04,930 --> 00:05:06,300
the following situation.

107
00:05:06,300 --> 00:05:09,030
We have a surface w = f(x,y).

108
00:05:09,030 --> 00:05:14,710
We take the plane x = x0 and
intersect this surface with

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00:05:14,710 --> 00:05:16,150
that particular plane.

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00:05:16,150 --> 00:05:17,770
We get a curve, you see.

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00:05:17,770 --> 00:05:22,450
Now, we look at that curve
corresponding to two points, p

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00:05:22,450 --> 00:05:25,990
and q, where p and q are
determined by the y

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00:05:25,990 --> 00:05:27,640
values a and b.

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00:05:27,640 --> 00:05:30,990
In other words, p corresponds
to y = a.

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00:05:30,990 --> 00:05:33,100
q corresponds to y = b.

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00:05:33,100 --> 00:05:35,800
And now, a very natural question
that might come up is

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00:05:35,800 --> 00:05:39,380
that we would like to find the
area of this particular plane

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00:05:39,380 --> 00:05:42,520
region, in other words,
the area of this slice

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00:05:42,520 --> 00:05:43,985
between p and q.

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00:05:43,985 --> 00:05:46,830
Now, you know, the first thing
I hope that you'll notice is

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00:05:46,830 --> 00:05:49,950
that because this shape of the
surface can be in many

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00:05:49,950 --> 00:05:53,490
different ways, the particular
cross section that we get does

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00:05:53,490 --> 00:05:55,690
depend on the choice of x0.

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00:05:55,690 --> 00:05:57,320
Different slices--

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00:05:57,320 --> 00:06:00,630
different planes x = x0 will
give us different curves of

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00:06:00,630 --> 00:06:01,520
intersection.

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00:06:01,520 --> 00:06:04,930
The point is that once we have
the curve intersection x is

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00:06:04,930 --> 00:06:06,730
being treated as
the parameter.

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00:06:06,730 --> 00:06:10,130
This particular curve is
given by what equation?

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00:06:10,130 --> 00:06:13,780
f is a function of x0 and y.

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00:06:13,780 --> 00:06:18,730
See, x = x0 for every
point on this curve.

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00:06:18,730 --> 00:06:22,090
And so the area of the region R
is the integral from a to b,

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00:06:22,090 --> 00:06:24,420
f(x0, y) dy.

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00:06:24,420 --> 00:06:26,950
And the question that very often
comes up is, how do you

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00:06:26,950 --> 00:06:31,180
find the derivative of A sub R
with respect to x0, noticing,

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00:06:31,180 --> 00:06:34,940
you see, that A sub R is a
function of x0 alone, the y

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00:06:34,940 --> 00:06:38,070
dropping out between the limits
a and b when we perform

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00:06:38,070 --> 00:06:40,270
the operation of integration.

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00:06:40,270 --> 00:06:43,890
A third place that this type
of situation occurs is in

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00:06:43,890 --> 00:06:46,100
solving certain differential
equations.

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00:06:46,100 --> 00:06:49,220
For example, suppose we're given
a particular curve and

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00:06:49,220 --> 00:06:54,460
that that curve determines a
region R between the lines x =

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00:06:54,460 --> 00:06:56,820
a and x = b.

144
00:06:56,820 --> 00:06:58,830
Suppose all we know about
the curve is its

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00:06:58,830 --> 00:07:00,170
slope at any point.

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00:07:00,170 --> 00:07:03,330
We know that its slope at any
point is given by dy/dx and

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00:07:03,330 --> 00:07:06,500
some function of x and y which
we don't necessarily have to

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00:07:06,500 --> 00:07:07,550
go into right now.

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00:07:07,550 --> 00:07:09,470
And let's suppose, for the
sake of argument, that we

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00:07:09,470 --> 00:07:12,060
solve this first order
differential equation.

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00:07:12,060 --> 00:07:14,840
What we'll find, if we're
lucky, is that y is some

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00:07:14,840 --> 00:07:17,900
function of x and an
arbitrary constant.

153
00:07:17,900 --> 00:07:20,090
Remember, once you have one
solution to a differential

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00:07:20,090 --> 00:07:23,760
equation, you have an infinite
family in terms of a one

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00:07:23,760 --> 00:07:27,370
parameter solution to a
differential equation.

156
00:07:27,370 --> 00:07:30,850
In other words, in finding the
area of the region R in this

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00:07:30,850 --> 00:07:34,690
case, there are many curves that
satisfy this particular

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00:07:34,690 --> 00:07:35,770
differential equation.

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00:07:35,770 --> 00:07:38,820
Until we know what specific
points are being referred to

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00:07:38,820 --> 00:07:41,510
over here, the best we know
for sure is what?

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00:07:41,510 --> 00:07:45,420
That these endpoints are a and
b, that the integrand is

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00:07:45,420 --> 00:07:49,650
f(x,c), and we're integrating
that with respect

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00:07:49,650 --> 00:07:52,130
to x from a to b.

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00:07:52,130 --> 00:07:55,630
I freudianly put a c in here,
because I think what I was

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00:07:55,630 --> 00:07:58,420
trying to emphasize for you is
that when you look at this

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00:07:58,420 --> 00:08:01,320
thing, observe that
this integral is a

167
00:08:01,320 --> 00:08:02,870
function of c alone.

168
00:08:02,870 --> 00:08:05,490
Namely, when you integrate this
thing, you integrate it

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00:08:05,490 --> 00:08:06,280
with respect to x.

170
00:08:06,280 --> 00:08:07,120
The x drops out.

171
00:08:07,120 --> 00:08:08,740
All you have left here is a c.

172
00:08:08,740 --> 00:08:11,570
A sub R, then, is
a function of c.

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00:08:11,570 --> 00:08:14,090
And in many cases, what we would
like to do is see how

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00:08:14,090 --> 00:08:17,320
fast the area changes
as a function of c.

175
00:08:17,320 --> 00:08:20,700
In other words, how do we change
the area as a function

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00:08:20,700 --> 00:08:23,740
of changing the arbitrary
constant c?

177
00:08:23,740 --> 00:08:27,410
And I have enough exercises in
the assignment to give you

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00:08:27,410 --> 00:08:28,950
concrete drill on this.

179
00:08:28,950 --> 00:08:31,910
All I'm trying to give you here
is an overview of the

180
00:08:31,910 --> 00:08:32,960
entire topic.

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00:08:32,960 --> 00:08:35,350
And the reason I want to give
you this overview is that it's

182
00:08:35,350 --> 00:08:37,929
hinted at in the textbook,
but this topic

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00:08:37,929 --> 00:08:38,940
is not covered there.

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00:08:38,940 --> 00:08:41,870
In fact, the reason that I had
you read that particular

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00:08:41,870 --> 00:08:44,920
section of the textbook before
the lecture this time-- you

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00:08:44,920 --> 00:08:46,700
notice that usually we start
with the lecture.

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00:08:46,700 --> 00:08:50,640
This time I had you read the
textbook first, because the

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00:08:50,640 --> 00:08:54,080
way the textbook covered this
topic is essentially nothing

189
00:08:54,080 --> 00:08:56,430
more than the way we
tackled a different

190
00:08:56,430 --> 00:08:58,040
problem last semester.

191
00:08:58,040 --> 00:09:00,910
And I want you to see that the
problem done in the Thomas

192
00:09:00,910 --> 00:09:03,620
text is not a new problem--

193
00:09:03,620 --> 00:09:05,380
it's one that we've
done before--

194
00:09:05,380 --> 00:09:08,500
but that with the tools that
we now have available, we

195
00:09:08,500 --> 00:09:10,740
could've tackled a more
significant problem.

196
00:09:10,740 --> 00:09:13,770
And that's the one I'm electing
to do here and what I

197
00:09:13,770 --> 00:09:15,890
want to show you the key
steps on, because

198
00:09:15,890 --> 00:09:17,140
they're not in the text.

199
00:09:17,140 --> 00:09:17,900
But I will leave the

200
00:09:17,900 --> 00:09:20,230
reinforcement for the exercise.

201
00:09:20,230 --> 00:09:23,790
At any rate, hopefully now, when
you see an integral of

202
00:09:23,790 --> 00:09:26,300
this type, it will not
bother you too much.

203
00:09:26,300 --> 00:09:28,320
In other words don't worry about
how do you integrate a

204
00:09:28,320 --> 00:09:30,060
function of two independent
variables?

205
00:09:30,060 --> 00:09:33,700
When you see something like
this, it means that there is

206
00:09:33,700 --> 00:09:37,930
some implicitly implied
domain for x.

207
00:09:37,930 --> 00:09:40,800
In other words, we have
some function of x.

208
00:09:40,800 --> 00:09:46,010
Let's say the domain of g might
very well be, say, all

209
00:09:46,010 --> 00:09:49,380
x's between two values,
say c and d.

210
00:09:49,380 --> 00:09:50,840
But who cares about
that right now?

211
00:09:50,840 --> 00:09:53,390
The important point is that
g is defined on a certain

212
00:09:53,390 --> 00:09:54,410
set of values x.

213
00:09:54,410 --> 00:09:58,870
And what it says is to compute
the output of the g machine.

214
00:09:58,870 --> 00:10:04,470
For the given x, you fix that
x and integrate f(x,y) dy

215
00:10:04,470 --> 00:10:05,880
between a and b.

216
00:10:05,880 --> 00:10:09,020
Notice, you see, during the
integration, x is being

217
00:10:09,020 --> 00:10:12,430
treated as a constant, so that
for all intents and purposes,

218
00:10:12,430 --> 00:10:14,680
this is an ordinary integral.

219
00:10:14,680 --> 00:10:17,780
But because x isn't a bona fide
constant, meaning what?

220
00:10:17,780 --> 00:10:21,080
It's a constant only in the
sense that once chosen, it

221
00:10:21,080 --> 00:10:24,180
remains fixed for this
particular integration.

222
00:10:24,180 --> 00:10:26,540
Different values of
x will give me

223
00:10:26,540 --> 00:10:28,030
different integrals here.

224
00:10:28,030 --> 00:10:30,910
And consequently a very natural
question that comes up

225
00:10:30,910 --> 00:10:34,710
is how does my function g--
which depends on x--

226
00:10:34,710 --> 00:10:38,130
how does that vary
as x varies?

227
00:10:38,130 --> 00:10:40,160
In other words, the key question
is simply this.

228
00:10:40,160 --> 00:10:44,060
First of all, given g defined
this way, one, does g prime

229
00:10:44,060 --> 00:10:45,120
even exist?

230
00:10:45,120 --> 00:10:47,260
Does dg/dx exist?

231
00:10:47,260 --> 00:10:52,620
And two, if it does
exist, what is it?

232
00:10:52,620 --> 00:10:54,840
In other words, the question
that we're raising is if we

233
00:10:54,840 --> 00:10:58,570
can find g prime of x, how do
we do it in terms of looking

234
00:10:58,570 --> 00:11:00,440
at the right hand side?

235
00:11:00,440 --> 00:11:02,900
And let me not try to guess
the answer here.

236
00:11:02,900 --> 00:11:05,940
The answer does turn out to be,
in this particular case,

237
00:11:05,940 --> 00:11:07,470
one that you might
have guessed.

238
00:11:07,470 --> 00:11:09,890
But I prefer to show you
that we don't have to

239
00:11:09,890 --> 00:11:11,500
guess, point one.

240
00:11:11,500 --> 00:11:13,620
Point two, if you do
guess, you won't

241
00:11:13,620 --> 00:11:14,610
always be that lucky.

242
00:11:14,610 --> 00:11:16,680
That's my finale for
today's lecture.

243
00:11:16,680 --> 00:11:18,760
But let me see if I
can survive to get

244
00:11:18,760 --> 00:11:20,140
to the finale first.

245
00:11:20,140 --> 00:11:22,510
Let me see how we'll tackle
a problem like this.

246
00:11:22,510 --> 00:11:27,220
First of all, to see if g prime
exists at some value x0,

247
00:11:27,220 --> 00:11:29,140
what we have to do is--

248
00:11:29,140 --> 00:11:31,520
way back to the very beginning
of Part 1--

249
00:11:31,520 --> 00:11:34,930
the same old definition for
an ordinary derivative.

250
00:11:34,930 --> 00:11:39,200
We have to compute the limit of
g of x sub 0 plus h minus

251
00:11:39,200 --> 00:11:43,880
g(x sub 0) over h, taking the
limit as h approaches 0.

252
00:11:43,880 --> 00:11:45,960
Notice what the g
machine does...

253
00:11:45,960 --> 00:11:50,230
What the g machine does is it
feeds x into the integrand

254
00:11:50,230 --> 00:11:54,960
here and integrates this with
respect to y from a to b.

255
00:11:54,960 --> 00:11:59,670
So if the input of my g machine
is x0 plus h, that

256
00:11:59,670 --> 00:12:04,360
means that the x is replaced
by x0 plus h here. g of x0

257
00:12:04,360 --> 00:12:07,210
plus h is simply integral
from a to b f of x0

258
00:12:07,210 --> 00:12:09,140
plus h comma y dy.

259
00:12:09,140 --> 00:12:13,700
Similarly, g(x0) is integral
from a to b f(x0, y) dy.

260
00:12:13,700 --> 00:12:16,080
I now want to form
this difference.

261
00:12:16,080 --> 00:12:17,190
And noticing--

262
00:12:17,190 --> 00:12:19,820
again, this is all calculus
of a single variable--

263
00:12:19,820 --> 00:12:23,470
that the difference of two
definite integrals is the

264
00:12:23,470 --> 00:12:26,670
definite integral of the
difference, I can conclude

265
00:12:26,670 --> 00:12:31,320
that g(x0 + h) minus g(x0)
is simply this single

266
00:12:31,320 --> 00:12:33,990
integral over here.

267
00:12:33,990 --> 00:12:37,060
Now, my next step in determining
g prime of x0 is I

268
00:12:37,060 --> 00:12:39,030
must divide this by h.

269
00:12:39,030 --> 00:12:43,990
Notice, by the way, that h is
an arbitrary increment, but

270
00:12:43,990 --> 00:12:46,240
once chosen, remains fixed.

271
00:12:46,240 --> 00:12:49,210
Notice that h is a constant
as far as this

272
00:12:49,210 --> 00:12:50,880
integration is concerned.

273
00:12:50,880 --> 00:12:56,050
Consequently, to divide by h, it
is permissible to bring the

274
00:12:56,050 --> 00:12:57,990
h inside the integrand.

275
00:12:57,990 --> 00:13:01,250
In other words, technically
speaking, the h should be

276
00:13:01,250 --> 00:13:04,750
here, but since h is a constant
with respect to y,

277
00:13:04,750 --> 00:13:08,000
the integral can have
the h brought in.

278
00:13:08,000 --> 00:13:10,080
Why do I want to bring
the h in here?

279
00:13:10,080 --> 00:13:12,910
Let me again telegraph what
I'm leading up to.

280
00:13:12,910 --> 00:13:16,310
Obviously, when I'm going to
compute g prime, my next step

281
00:13:16,310 --> 00:13:19,000
is to take the limit of this
as h approaches 0.

282
00:13:19,000 --> 00:13:23,230
With h in here, I look at this
and what I hope is that by

283
00:13:23,230 --> 00:13:26,960
this amount of time at least the
following minimum amount

284
00:13:26,960 --> 00:13:29,960
of material has rubbed off on
you in a second nature way--

285
00:13:29,960 --> 00:13:34,050
that if I look at this
expression in brackets as h

286
00:13:34,050 --> 00:13:38,250
approaches 0, this is precisely
the definition of

287
00:13:38,250 --> 00:13:45,110
what we mean by the partial of
f(x, y) with respect to x

288
00:13:45,110 --> 00:13:46,600
evaluated at (x0, y).

289
00:13:46,600 --> 00:13:47,460
See, this is what?

290
00:13:47,460 --> 00:13:50,500
The change in f-- see,
y is held constant.

291
00:13:50,500 --> 00:13:54,110
We're taking this over to change
from x0 to x0 plus h

292
00:13:54,110 --> 00:13:55,150
and dividing by h.

293
00:13:55,150 --> 00:13:57,760
This is a partial of f
with respect to x.

294
00:13:57,760 --> 00:13:59,900
That's why I want to
bring the h inside.

295
00:13:59,900 --> 00:14:02,780
So now, I say, OK,
g prime of x0, by

296
00:14:02,780 --> 00:14:04,540
definition, is this limit.

297
00:14:04,540 --> 00:14:07,255
I now want to take the limit
of this expression.

298
00:14:07,255 --> 00:14:10,770

299
00:14:10,770 --> 00:14:14,010
And by the way, notice what I'd
love to do now is to jump

300
00:14:14,010 --> 00:14:17,885
right in here and say, aha, this
is just the partial of f

301
00:14:17,885 --> 00:14:21,310
with respect to x evaluated
at (x0, y0).

302
00:14:21,310 --> 00:14:23,510
But the thing I would
like you to notice--

303
00:14:23,510 --> 00:14:26,700
and again, going back to Part
1 of course, one of the big

304
00:14:26,700 --> 00:14:29,420
things that we talked about
under the heading of uniform

305
00:14:29,420 --> 00:14:30,310
convergence.

306
00:14:30,310 --> 00:14:33,060
There is a very dangerous
thing in general to

307
00:14:33,060 --> 00:14:36,190
interchange the order of
limit and integration.

308
00:14:36,190 --> 00:14:37,110
This says what?

309
00:14:37,110 --> 00:14:40,580
First perform the integration,
and then take the limit.

310
00:14:40,580 --> 00:14:44,860
What we would like to be able to
do is first take the limit

311
00:14:44,860 --> 00:14:47,030
and then integrate the result.

312
00:14:47,030 --> 00:14:51,500
Now, we did see that, provided
the integrand was continuous,

313
00:14:51,500 --> 00:14:53,330
these operations were
permissible.

314
00:14:53,330 --> 00:14:55,150
But we'll talk about that
a little later.

315
00:14:55,150 --> 00:14:59,120
For the time being, let's simply
summarize by saying if

316
00:14:59,120 --> 00:15:02,660
the limit operation and the
integration operation can be

317
00:15:02,660 --> 00:15:06,720
interchanged, then
the derivative--

318
00:15:06,720 --> 00:15:07,630
see, this thing here is what?

319
00:15:07,630 --> 00:15:09,500
This is my g(x).

320
00:15:09,500 --> 00:15:12,640
The derivative of g(x) with
respect to x has the very

321
00:15:12,640 --> 00:15:16,080
delightful form that,
essentially, all I have to do

322
00:15:16,080 --> 00:15:19,340
is take the derivative
operation, come inside the

323
00:15:19,340 --> 00:15:23,500
integrand, and replace the
derivative with respect to x

324
00:15:23,500 --> 00:15:26,380
by the partial derivative
with respect to x.

325
00:15:26,380 --> 00:15:29,220
In other words, the derivative
of the integral from a to b,

326
00:15:29,220 --> 00:15:33,890
f(x, y) dy is the integral from
a to b, the partial of f

327
00:15:33,890 --> 00:15:36,250
with respect to x dy--

328
00:15:36,250 --> 00:15:39,480
provided, of course, that the
limit and the integration are

329
00:15:39,480 --> 00:15:40,510
interchangeable.

330
00:15:40,510 --> 00:15:44,820
In particular, this will be true
if f and f sub x exist

331
00:15:44,820 --> 00:15:48,590
and are continuous and see,
straighten out the range and

332
00:15:48,590 --> 00:15:50,830
the domain and what have
you once and for all.

333
00:15:50,830 --> 00:15:53,860
Notice that y is allowed to
exist between a and b.

334
00:15:53,860 --> 00:15:56,690
We've said that x is going
to exist on some domain

335
00:15:56,690 --> 00:15:58,040
between c and d.

336
00:15:58,040 --> 00:16:01,090
Notice that saying that y is
between a and b and x is

337
00:16:01,090 --> 00:16:04,410
between c and d geometrically
says that f is

338
00:16:04,410 --> 00:16:06,570
defined on a rectangle.

339
00:16:06,570 --> 00:16:07,320
See?

340
00:16:07,320 --> 00:16:07,970
OK.

341
00:16:07,970 --> 00:16:11,270
And then what we're saying is,
under these conditions, to

342
00:16:11,270 --> 00:16:15,620
integrate an integral with a
parameter, with respect to

343
00:16:15,620 --> 00:16:19,360
that parameter, all we have to
do is come inside the integral

344
00:16:19,360 --> 00:16:21,980
sign and differentiate--

345
00:16:21,980 --> 00:16:23,220
take the partial derivative--

346
00:16:23,220 --> 00:16:26,190
with respect to what is being
used as the parameter.

347
00:16:26,190 --> 00:16:28,540
In this case, it's x, which
is the parameter.

348
00:16:28,540 --> 00:16:30,880
Now, the only danger with this
particular thing-- and by the

349
00:16:30,880 --> 00:16:33,040
way, notice, not only is there
a danger here that I'm going

350
00:16:33,040 --> 00:16:33,380
to mention.

351
00:16:33,380 --> 00:16:36,260
The danger is this looks so
easy, you may be saying, why

352
00:16:36,260 --> 00:16:37,250
did he do it the hard way?

353
00:16:37,250 --> 00:16:38,840
Why didn't he just tell
us this was the right

354
00:16:38,840 --> 00:16:40,050
way of doing it?

355
00:16:40,050 --> 00:16:41,840
And the point is it just happens
to be one of those

356
00:16:41,840 --> 00:16:46,460
coincidences where the rigorous
way yields a logical

357
00:16:46,460 --> 00:16:49,240
answer which is consistent
with what is probably our

358
00:16:49,240 --> 00:16:51,230
intuitive guess.

359
00:16:51,230 --> 00:16:52,940
But it's not always going
to happen that way.

360
00:16:52,940 --> 00:16:55,545
And the example that I have in
mind now goes back to what we

361
00:16:55,545 --> 00:16:57,180
were talking about the beginning
of the lecture.

362
00:16:57,180 --> 00:16:59,300
Namely, I wanted to give
you an application

363
00:16:59,300 --> 00:17:00,360
of the chain rule.

364
00:17:00,360 --> 00:17:02,810
And here's where that
application comes in.

365
00:17:02,810 --> 00:17:04,319
I now call this--

366
00:17:04,319 --> 00:17:05,130
I don't know what to call it.

367
00:17:05,130 --> 00:17:07,540
So let's just call it variable
limits of integration.

368
00:17:07,540 --> 00:17:09,480
Same problem as before--

369
00:17:09,480 --> 00:17:12,540
it's going to cause the chain
rule to come in now.

370
00:17:12,540 --> 00:17:14,490
The only difference
is going to be--

371
00:17:14,490 --> 00:17:15,680
and that's just a forewarning.

372
00:17:15,680 --> 00:17:17,650
You don't have to know
that right now.

373
00:17:17,650 --> 00:17:19,240
I'm going to have the same
problem as before.

374
00:17:19,240 --> 00:17:21,130
What did I have before?

375
00:17:21,130 --> 00:17:25,280
I had that g(x) was integral
f(x, y) dy between the two

376
00:17:25,280 --> 00:17:27,050
constants a and b.

377
00:17:27,050 --> 00:17:30,760
Now, I'm going to let my
constants of integration also

378
00:17:30,760 --> 00:17:32,240
depend on the parameter.

379
00:17:32,240 --> 00:17:35,300
See, all the constants of
integration have to be our

380
00:17:35,300 --> 00:17:39,540
constants as far as
y is concerned.

381
00:17:39,540 --> 00:17:42,080
What I'm saying is what makes
this problem differ from the

382
00:17:42,080 --> 00:17:45,030
previous one is suppose that
it happens that instead of

383
00:17:45,030 --> 00:17:47,950
being given a nice rectangle to
play around with, I'm given

384
00:17:47,950 --> 00:17:50,520
a couple of curves like
this in the xy plane.

385
00:17:50,520 --> 00:17:53,360
See, this would be a of x.

386
00:17:53,360 --> 00:17:55,860
This would be y equals b(x).

387
00:17:55,860 --> 00:18:00,610
See, what I'm saying now is
that not only does the

388
00:18:00,610 --> 00:18:04,220
integrand depend on what value
of x I pick, but the limits of

389
00:18:04,220 --> 00:18:07,200
integration as I'm finding a
cross-sectional area of a

390
00:18:07,200 --> 00:18:08,180
surface, you see.

391
00:18:08,180 --> 00:18:10,990
The limits of the integral
themselves depend on the

392
00:18:10,990 --> 00:18:12,250
choice of x--

393
00:18:12,250 --> 00:18:15,820
constant, as far as y is
concerned, but depend on x.

394
00:18:15,820 --> 00:18:20,250
You see, now, what happens is
that my parameter appears in

395
00:18:20,250 --> 00:18:23,670
the limits as well as just
in the integrand.

396
00:18:23,670 --> 00:18:26,430
And now, you see, also what
this means is if I try the

397
00:18:26,430 --> 00:18:30,510
previous approach of computing
g(x plus delta x), et cetera,

398
00:18:30,510 --> 00:18:33,710
I'm in trouble, because the
only way I can combine two

399
00:18:33,710 --> 00:18:36,930
integrals and put them under the
same integral sign is if

400
00:18:36,930 --> 00:18:39,470
they're between the same
limits of integration.

401
00:18:39,470 --> 00:18:43,460
Notice here, for example, that
if I replace x by x plus delta

402
00:18:43,460 --> 00:18:47,080
x, I not only change the
integrand, but notice that the

403
00:18:47,080 --> 00:18:48,590
limits become what?

404
00:18:48,590 --> 00:18:54,330
a of x plus delta x, b(x plus
delta x)-- and those in

405
00:18:54,330 --> 00:18:56,680
general, unless a and b happen
to be constants,

406
00:18:56,680 --> 00:18:58,420
will vary with x.

407
00:18:58,420 --> 00:19:00,180
In fact, let's look
at it this way.

408
00:19:00,180 --> 00:19:04,000
This is the problem we should
have started with in the sense

409
00:19:04,000 --> 00:19:06,840
that constant limits of
integration are a

410
00:19:06,840 --> 00:19:08,640
special case of this.

411
00:19:08,640 --> 00:19:10,880
At any rate, what I wanted
to show you was that this

412
00:19:10,880 --> 00:19:14,100
particular problem can be
handled very nicely in terms

413
00:19:14,100 --> 00:19:15,390
of the chain rule.

414
00:19:15,390 --> 00:19:18,900
Namely, what we do here is we
observe that, first of all, y

415
00:19:18,900 --> 00:19:20,510
is not really a variable here.

416
00:19:20,510 --> 00:19:22,310
It's integrated out.

417
00:19:22,310 --> 00:19:26,240
So what we think of is let's
think of x as being some

418
00:19:26,240 --> 00:19:27,330
variable u.

419
00:19:27,330 --> 00:19:31,000
Let's think of b(x) as being
some variable v. Let's write

420
00:19:31,000 --> 00:19:33,400
down the function of
three independent

421
00:19:33,400 --> 00:19:35,590
variables u, v, and x.

422
00:19:35,590 --> 00:19:36,840
OK.

423
00:19:36,840 --> 00:19:39,240
What will that function be?

424
00:19:39,240 --> 00:19:43,100
Let u, v, and x be arbitrary,
independent variables.

425
00:19:43,100 --> 00:19:46,460
Look at the integral from
u to v f(x, y) dy.

426
00:19:46,460 --> 00:19:49,210
This is obviously dependent
upon u.

427
00:19:49,210 --> 00:19:52,780
It's dependent upon v. And
it's dependent upon x.

428
00:19:52,780 --> 00:19:55,340
The place that the chain rule
comes in is that in our

429
00:19:55,340 --> 00:19:58,810
particular problem, u and v
cannot be arbitrary, but

430
00:19:58,810 --> 00:20:01,900
rather u must be that particular
function a(x), and

431
00:20:01,900 --> 00:20:05,030
v must be the particular
function b(x).

432
00:20:05,030 --> 00:20:07,480
Consequently g(x)
is simply what?

433
00:20:07,480 --> 00:20:12,720
It's h(u, v, x), where u
is a(x) and v is b(x).

434
00:20:12,720 --> 00:20:16,450
Consequently, to find g prime of
x, what we want is h prime

435
00:20:16,450 --> 00:20:19,240
of x and to find h prime
as a function of x.

436
00:20:19,240 --> 00:20:20,410
See the chain rule here?

437
00:20:20,410 --> 00:20:22,540
u can be expressed
in terms of x.

438
00:20:22,540 --> 00:20:24,430
v can be expressed
in terms of x.

439
00:20:24,430 --> 00:20:27,180
Obviously, x is already
expressed in terms of x.

440
00:20:27,180 --> 00:20:31,380
So this is really implicitly
a function of x alone.

441
00:20:31,380 --> 00:20:34,300
So by the chain rule, what I'd
like to be able to do is to

442
00:20:34,300 --> 00:20:36,530
combine these three pieces
of information to

443
00:20:36,530 --> 00:20:38,410
find h prime of x.

444
00:20:38,410 --> 00:20:39,810
And remember how the
chain rule works.

445
00:20:39,810 --> 00:20:41,320
Now, I'm not going to
beat that to death.

446
00:20:41,320 --> 00:20:42,970
We've just had two
units on that.

447
00:20:42,970 --> 00:20:45,290
Let's just say it
rather quickly.

448
00:20:45,290 --> 00:20:49,780
g prime of x is the partial of
h with respect to u times u

449
00:20:49,780 --> 00:20:53,140
prime of x plus the partial of h
with respectively to v times

450
00:20:53,140 --> 00:20:57,030
v prime of x times the partial
of h with respect to x times x

451
00:20:57,030 --> 00:20:58,790
prime of x, which, of
course, is just 1.

452
00:20:58,790 --> 00:21:02,020
In other words, writing this
thing out, g prime of x is

453
00:21:02,020 --> 00:21:04,730
simply this.

454
00:21:04,730 --> 00:21:08,870
What is the partial of
h with respect to u?

455
00:21:08,870 --> 00:21:11,590
Let's come back here for a
second and remember what h is.

456
00:21:11,590 --> 00:21:12,790
h is this integral.

457
00:21:12,790 --> 00:21:15,610
I want to take the partial of
that with respect to u.

458
00:21:15,610 --> 00:21:18,950
That means I have to
investigate this.

459
00:21:18,950 --> 00:21:20,450
Now, here's the interesting
point.

460
00:21:20,450 --> 00:21:25,210
Whereas u, v, and x are
independent variables, what

461
00:21:25,210 --> 00:21:27,280
does it mean when you say you're
taking the partial with

462
00:21:27,280 --> 00:21:28,350
respect to u?

463
00:21:28,350 --> 00:21:33,650
It means that you're treating
v and x as constants.

464
00:21:33,650 --> 00:21:37,370
Now, if v and x are being
treated as constants, what I

465
00:21:37,370 --> 00:21:39,530
have is simply what?

466
00:21:39,530 --> 00:21:42,920
I'm taking a derivative with
respect to a variable where

467
00:21:42,920 --> 00:21:47,100
the only place the variable
appears is as the lower limit

468
00:21:47,100 --> 00:21:48,410
of the integrand.

469
00:21:48,410 --> 00:21:50,730
In other words, I claim that
that's nothing more

470
00:21:50,730 --> 00:21:53,460
than minus f(x, u).

471
00:21:53,460 --> 00:21:54,570
Then I go inside
the integrand.

472
00:21:54,570 --> 00:21:57,620
In other words, I differentiate
the integral.

473
00:21:57,620 --> 00:21:59,530
That leaves me just
the integrand--

474
00:21:59,530 --> 00:22:05,350
and replace the variable
by the variable of

475
00:22:05,350 --> 00:22:06,580
integration u here.

476
00:22:06,580 --> 00:22:08,250
And because it's the
lower limit, I put

477
00:22:08,250 --> 00:22:08,960
in the minus sign.

478
00:22:08,960 --> 00:22:11,280
Now, why did I go through
that very fast?

479
00:22:11,280 --> 00:22:13,840
That's why I had you read
this assignment first.

480
00:22:13,840 --> 00:22:17,300
Notice that the assignment in
the textbook does not touch

481
00:22:17,300 --> 00:22:21,390
what I'm talking about, but
rather seems to review that

482
00:22:21,390 --> 00:22:23,790
topic that we covered
under Part 1--

483
00:22:23,790 --> 00:22:26,050
that if you wanted to take an
ordinary derivative with

484
00:22:26,050 --> 00:22:31,480
respect to u, integral from u
to a, g(y) dy, the answer

485
00:22:31,480 --> 00:22:33,950
would just be minus g(u).

486
00:22:33,950 --> 00:22:35,770
And that's exactly what
I did in here.

487
00:22:35,770 --> 00:22:38,200
I treated v and x as
constants here.

488
00:22:38,200 --> 00:22:42,100
In other words, the only
variable in here was u.

489
00:22:42,100 --> 00:22:44,380
That will be emphasized again
in the exercises.

490
00:22:44,380 --> 00:22:47,700
In a similar way, the partial
of h with respect to v means

491
00:22:47,700 --> 00:22:48,920
this thing.

492
00:22:48,920 --> 00:22:52,020
Notice now that u is being
treated as a constant.

493
00:22:52,020 --> 00:22:54,070
x is being treated
as a constant.

494
00:22:54,070 --> 00:22:58,640
To differentiate this, my
variable appears only as an

495
00:22:58,640 --> 00:23:00,410
upper limit on the integrand.

496
00:23:00,410 --> 00:23:03,810
That means I come inside the
integral sign, replace the

497
00:23:03,810 --> 00:23:07,800
integral by just the function
itself, replacing what?

498
00:23:07,800 --> 00:23:13,050

499
00:23:13,050 --> 00:23:14,150
Replacing y--

500
00:23:14,150 --> 00:23:16,740
that's the only variables
of integration--

501
00:23:16,740 --> 00:23:21,450
by the upper limit v. In other
words, this is f(x, v).

502
00:23:21,450 --> 00:23:22,030
All right?

503
00:23:22,030 --> 00:23:25,770
And finally, the partial of h
with respect to x is this

504
00:23:25,770 --> 00:23:26,970
integral here.

505
00:23:26,970 --> 00:23:30,470
Notice now that u and v are
being treated as constants.

506
00:23:30,470 --> 00:23:33,990
With u and v being treated as
constants, that's the special

507
00:23:33,990 --> 00:23:37,070
case that we started out lecture
with, namely, to

508
00:23:37,070 --> 00:23:40,230
differentiate with respect to a
parameter when the parameter

509
00:23:40,230 --> 00:23:42,360
appears only as part
of the integrand.

510
00:23:42,360 --> 00:23:43,420
So how do we do that?

511
00:23:43,420 --> 00:23:44,900
We come inside.

512
00:23:44,900 --> 00:23:47,690
This is the interval from u to
v, the partial of f with

513
00:23:47,690 --> 00:23:50,050
respect to x dy.

514
00:23:50,050 --> 00:23:52,190
Putting the whole
thing together--

515
00:23:52,190 --> 00:23:56,760
recalling, among other things,
that du/dx, since u is a(x),

516
00:23:56,760 --> 00:24:01,090
is just a prime of x and that
dv/dx is b prime of x-- what

517
00:24:01,090 --> 00:24:02,440
this says--

518
00:24:02,440 --> 00:24:04,940
and again, I want to see the
beauty of the chain rule here,

519
00:24:04,940 --> 00:24:07,610
because at least to my way of
thinking, I don't see anything

520
00:24:07,610 --> 00:24:10,480
at all intuitive about the
result I'm going to show you.

521
00:24:10,480 --> 00:24:14,790
And that is as soon as you make
the limits of integration

522
00:24:14,790 --> 00:24:18,930
variable, to differentiate
an integral involving a

523
00:24:18,930 --> 00:24:19,450
parameter--

524
00:24:19,450 --> 00:24:22,970
you see again, what's the
parameter here? x.

525
00:24:22,970 --> 00:24:24,070
We integrate it with
respect to y.

526
00:24:24,070 --> 00:24:27,550
This is a constant as far
as y is concerned.

527
00:24:27,550 --> 00:24:30,500
See, intuitively, you might say,
gee, all you've got to do

528
00:24:30,500 --> 00:24:33,820
is take the derivative sign,
bring it inside, and this

529
00:24:33,820 --> 00:24:35,450
should be the answer.

530
00:24:35,450 --> 00:24:36,920
See, it's the same
as we did before.

531
00:24:36,920 --> 00:24:39,510
The point is-- and this is where
many serious mistakes

532
00:24:39,510 --> 00:24:43,010
are made in problems involving
integrals of this type--

533
00:24:43,010 --> 00:24:45,750
is that the reason that our
intuitive way happened to be

534
00:24:45,750 --> 00:24:50,790
right in the simpler case was
that these were constants with

535
00:24:50,790 --> 00:24:51,890
respect to x.

536
00:24:51,890 --> 00:24:52,930
Now, they're variables.

537
00:24:52,930 --> 00:24:55,810
Well, it turns out, if you just
wrote this thing down,

538
00:24:55,810 --> 00:24:56,900
you would be wrong.

539
00:24:56,900 --> 00:24:58,510
What is the correction factor?

540
00:24:58,510 --> 00:25:00,380
Again, come back to here.

541
00:25:00,380 --> 00:25:04,760
The correction factor is this
here, which we've just started

542
00:25:04,760 --> 00:25:07,030
to compute.

543
00:25:07,030 --> 00:25:08,600
Again, just saying it--

544
00:25:08,600 --> 00:25:11,640
if you wrote this term down to
get the correct answer, you

545
00:25:11,640 --> 00:25:13,180
would have to tack on what?

546
00:25:13,180 --> 00:25:16,810
b prime of x times f(x, b(x)).

547
00:25:16,810 --> 00:25:18,270
That means what?

548
00:25:18,270 --> 00:25:20,790
You think of a y as
being over here.

549
00:25:20,790 --> 00:25:24,685
For the particular value of
x, you replace y by b(x).

550
00:25:24,685 --> 00:25:27,140
In other words, you look at f(x,
y), and every place you

551
00:25:27,140 --> 00:25:29,130
see a y, replace it by b(x).

552
00:25:29,130 --> 00:25:31,490
And then subtract that
from that a prime of

553
00:25:31,490 --> 00:25:34,910
x times f(x, a(x)).

554
00:25:34,910 --> 00:25:38,280
Now again, I suspect that, for
many of you, it's the first

555
00:25:38,280 --> 00:25:40,800
time that you've seen something
like this, because I

556
00:25:40,800 --> 00:25:44,850
say it's a topic which I believe
was a natural one to

557
00:25:44,850 --> 00:25:47,560
occur in the textbook, but
for some reason it

558
00:25:47,560 --> 00:25:48,830
doesn't appear there.

559
00:25:48,830 --> 00:25:52,470
Because of the importance of
the concept, the number of

560
00:25:52,470 --> 00:25:55,190
times it appears in physical
applications, the number of

561
00:25:55,190 --> 00:25:58,480
times that one has to
differentiate with respect to

562
00:25:58,480 --> 00:25:59,270
an integral--

563
00:25:59,270 --> 00:26:02,190
I don't know the physical
applications well enough to

564
00:26:02,190 --> 00:26:05,410
lecture on them, but it does
occur in probability theory,

565
00:26:05,410 --> 00:26:08,660
among other places, it appears
in any subject involving

566
00:26:08,660 --> 00:26:11,040
integral equations
and the like--

567
00:26:11,040 --> 00:26:14,610
that I wanted to give you the
experience of seeing what the

568
00:26:14,610 --> 00:26:16,960
concept means, to have
you hear me say it.

569
00:26:16,960 --> 00:26:22,320
And then I will spend the
exercises trying to drive home

570
00:26:22,320 --> 00:26:25,180
the computational know how so
that you will be able to do

571
00:26:25,180 --> 00:26:29,280
these things at least in a
mechanical way, independently

572
00:26:29,280 --> 00:26:32,800
of whether the theory made that
much sense because of the

573
00:26:32,800 --> 00:26:35,380
lack of physical example
motivation other than what I

574
00:26:35,380 --> 00:26:36,410
did at the beginning.

575
00:26:36,410 --> 00:26:38,880
At any rate, keep in mind,
though, that in terms of our

576
00:26:38,880 --> 00:26:41,740
present topic, where we're
talking about the chain rule,

577
00:26:41,740 --> 00:26:45,500
this is a certainly noble
application to show an

578
00:26:45,500 --> 00:26:48,530
important place in the physical
world where knowledge

579
00:26:48,530 --> 00:26:51,340
of the chain rule plays
a very important role.

580
00:26:51,340 --> 00:26:54,220
And with that, I might just as
well conclude today's lecture.

581
00:26:54,220 --> 00:26:55,990
And until next time, good bye.

582
00:26:55,990 --> 00:26:58,740

583
00:26:58,740 --> 00:27:01,940
Funding for the publication of
this video was provided by the

584
00:27:01,940 --> 00:27:05,990
Gabriella and Paul Rosenbaum
Foundation.

585
00:27:05,990 --> 00:27:10,170
Help OCW continue to provide
free and open access to MIT

586
00:27:10,170 --> 00:27:17,870
courses by making a donation
at ocw.mit.edu/donate.

587
00:27:17,870 --> 00:27:19,147