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PROFESSOR: To begin today
I want to remind you,

00:00:24.120 --> 00:00:27.170
I need to write it down on
the board at least twice,

00:00:27.170 --> 00:00:33.230
of the fundamental
theorem of calculus.

00:00:33.230 --> 00:00:38.496
We called it FTC 1 because
it's the first version

00:00:38.496 --> 00:00:39.620
of the fundamental theorem.

00:00:39.620 --> 00:00:42.340
We'll be talking about
another version, called

00:00:42.340 --> 00:00:44.940
the second version, today.

00:00:44.940 --> 00:00:53.610
And what it says
is this: If F' = f,

00:00:53.610 --> 00:01:03.330
then the integral from a to b
of f(x) dx is equal to F(b) -

00:01:03.330 --> 00:01:03.830
F(a).

00:01:06.410 --> 00:01:10.750
So that's the fundamental
theorem of calculus.

00:01:10.750 --> 00:01:15.600
And the way we used
it last time was,

00:01:15.600 --> 00:01:22.090
this was used to
evaluate integrals.

00:01:22.090 --> 00:01:28.110
Not surprisingly,
that's how we used it.

00:01:28.110 --> 00:01:35.670
But today, I want to
reverse that point of view.

00:01:35.670 --> 00:01:38.900
We're going to read
the equation backwards,

00:01:38.900 --> 00:01:49.240
and we're going to
write it this way.

00:01:49.240 --> 00:02:01.010
And we're going to use f
to understand capital F.

00:02:01.010 --> 00:02:04.370
Or in other words,
the derivative

00:02:04.370 --> 00:02:07.700
to understand the function.

00:02:07.700 --> 00:02:13.380
So that's the reversal of point
of view that I'd like to make.

00:02:13.380 --> 00:02:16.310
And we'll make this
point in various ways.

00:02:16.310 --> 00:02:28.230
So information about
f, about F', gives us

00:02:28.230 --> 00:02:37.750
information about F. Now,
since there were questions

00:02:37.750 --> 00:02:39.740
about the mean
value theorem, I'm

00:02:39.740 --> 00:02:42.380
going to illustrate
this first by making

00:02:42.380 --> 00:02:46.710
a comparison between the
fundamental theorem of calculus

00:02:46.710 --> 00:02:50.600
and the mean value theorem.

00:02:50.600 --> 00:02:56.460
So we're going to compare this
fundamental theorem of calculus

00:02:56.460 --> 00:03:01.489
with what we call the
mean value theorem.

00:03:01.489 --> 00:03:02.905
And in order to
do that, I'm going

00:03:02.905 --> 00:03:05.070
to introduce a
couple of notations.

00:03:05.070 --> 00:03:08.870
I'll write delta
F as F(b) - F(a).

00:03:11.820 --> 00:03:17.770
And another highly imaginative
notation, delta x = b - a.

00:03:17.770 --> 00:03:21.760
So here's the change in F,
there's the change in x.

00:03:21.760 --> 00:03:25.690
And then, this
fundamental theorem

00:03:25.690 --> 00:03:30.800
can be written, of course, right
up above there is the formula.

00:03:30.800 --> 00:03:36.430
And it's the
formula for delta F.

00:03:36.430 --> 00:03:39.240
So this is what we call
the fundamental theorem

00:03:39.240 --> 00:03:44.090
of calculus.

00:03:44.090 --> 00:03:51.810
I'm going to divide
by delta x, now.

00:03:51.810 --> 00:03:56.510
And If I divide by delta x,
that's the same thing as 1 /

00:03:56.510 --> 00:04:02.880
(b-a) times the integral
from a to b of f(x) dx.

00:04:02.880 --> 00:04:05.110
So I've just rewritten
the formula here.

00:04:05.110 --> 00:04:11.750
And this expression here,
on the right-hand side,

00:04:11.750 --> 00:04:13.650
is a fairly important one.

00:04:13.650 --> 00:04:23.370
This is the average of f.

00:04:23.370 --> 00:04:29.490
That's the average value of f.

00:04:29.490 --> 00:04:32.560
Now, so this is
going to permit me

00:04:32.560 --> 00:04:35.980
to make the comparison between
the mean value theorem, which

00:04:35.980 --> 00:04:38.630
we don't have stated yet here.

00:04:38.630 --> 00:04:41.010
And the fundamental theorem.

00:04:41.010 --> 00:04:49.130
And I'll do it in the
form of inequalities.

00:04:49.130 --> 00:04:51.079
So right in the
middle here, I'm going

00:04:51.079 --> 00:04:52.370
to put the fundamental theorem.

00:04:52.370 --> 00:04:56.580
It says that delta F in
this notation is equal to,

00:04:56.580 --> 00:04:59.150
well if I multiply
by delta x again, I

00:04:59.150 --> 00:05:01.850
can write it as
the average of f--

00:05:01.850 --> 00:05:04.870
So I'm going to write it
as the average of F' here.

00:05:04.870 --> 00:05:07.100
Times delta x.

00:05:07.100 --> 00:05:09.120
So we have this
factor here, which

00:05:09.120 --> 00:05:11.870
is the average of F', or
the average of little f,

00:05:11.870 --> 00:05:13.810
it's the same thing.

00:05:13.810 --> 00:05:15.430
And then I multiplied
through again.

00:05:15.430 --> 00:05:20.810
So I put the thing
in the red box, here.

00:05:20.810 --> 00:05:29.970
STUDENT: [INAUDIBLE]

00:05:29.970 --> 00:05:37.190
PROFESSOR: Isn't what
the average of big F?

00:05:37.190 --> 00:05:39.990
So the question is,
why is this the average

00:05:39.990 --> 00:05:44.990
of little f rather than
the average of big F.

00:05:44.990 --> 00:05:50.130
So the average of a function
is the typical value.

00:05:50.130 --> 00:05:54.680
If, for example,
little f were constant,

00:05:54.680 --> 00:05:59.770
little f were constant, then
this integral would be--

00:05:59.770 --> 00:06:03.920
So the question is why
is this the average.

00:06:03.920 --> 00:06:08.970
And I'll take a little
second to explain that.

00:06:08.970 --> 00:06:13.050
But I think I'll
explain it over here.

00:06:13.050 --> 00:06:19.010
Because I'm going to erase it.

00:06:19.010 --> 00:06:27.270
So the idea of an
average is the following.

00:06:27.270 --> 00:06:32.281
For example, imagine
that a = 0 and b = n,

00:06:32.281 --> 00:06:33.600
let's say for example.

00:06:33.600 --> 00:06:46.850
And so we might sum the
function from 1 to n.

00:06:46.850 --> 00:06:49.520
Now, that would be the sum
of the values from 1 to n.

00:06:49.520 --> 00:06:54.960
But the average is,
we divide by n here.

00:06:54.960 --> 00:06:56.140
So this is the average.

00:06:56.140 --> 00:07:00.180
And this is a kind
of Riemann sum,

00:07:00.180 --> 00:07:06.290
representing the integral
from 0 to n, of f(x) dx.

00:07:06.290 --> 00:07:10.700
Where the increment,
delta x, is 1.

00:07:10.700 --> 00:07:13.470
So this is the notion of
an average value here,

00:07:13.470 --> 00:07:15.760
but in the continuum
setting, as opposed

00:07:15.760 --> 00:07:20.570
to the discrete setting.

00:07:20.570 --> 00:07:24.270
Whereas what's on
the left-hand side

00:07:24.270 --> 00:07:28.250
is the change in
F. The capital F.

00:07:28.250 --> 00:07:31.790
And this is the average
of the little f.

00:07:31.790 --> 00:07:33.860
So an average is a sum.

00:07:33.860 --> 00:07:39.520
And it's like an integral.

00:07:39.520 --> 00:07:42.720
So, in other words what I have
here is that the change in F

00:07:42.720 --> 00:07:45.620
is the average of its
infinitesimal change

00:07:45.620 --> 00:07:50.000
times the amount of time
elapsed, if you like.

00:07:50.000 --> 00:07:56.100
So this is the statement
of the fundamental theorem.

00:07:56.100 --> 00:07:56.960
Just rewritten.

00:07:56.960 --> 00:07:58.450
Exactly what I wrote there.

00:07:58.450 --> 00:08:01.780
But I multiplied
back by delta x.

00:08:01.780 --> 00:08:07.530
Now, let me compare this
with the mean value theorem.

00:08:07.530 --> 00:08:13.590
The mean value theorem
also is an equation.

00:08:13.590 --> 00:08:17.790
The mean value theorem says that
this is equal to F'(c) delta x.

00:08:21.560 --> 00:08:23.270
Now, I pulled a fast one on you.

00:08:23.270 --> 00:08:26.560
I used capital F's here
to make the analogy clear.

00:08:26.560 --> 00:08:30.710
But the role of the letter
is important to make

00:08:30.710 --> 00:08:32.200
the transition to
this comparison.

00:08:32.200 --> 00:08:35.010
We're talking about the
function capital F here.

00:08:35.010 --> 00:08:36.620
And its derivative.

00:08:36.620 --> 00:08:38.510
Now, this is true.

00:08:38.510 --> 00:08:42.510
So now I claim that this
thing is fairly specific.

00:08:42.510 --> 00:08:47.750
Whereas this, unfortunately,
is a little bit vague.

00:08:47.750 --> 00:08:51.240
And the reason why it's vague
is that c is just somewhere

00:08:51.240 --> 00:08:52.790
in the interval.

00:08:52.790 --> 00:09:01.010
So some c-- Sorry, this is
some c, in between a and b.

00:09:01.010 --> 00:09:04.475
So really, since we don't
know where this thing is,

00:09:04.475 --> 00:09:05.850
we don't know
which of the values

00:09:05.850 --> 00:09:07.690
it is, we can't say what it is.

00:09:07.690 --> 00:09:11.700
All we can do is
say, well for sure

00:09:11.700 --> 00:09:13.280
it's less than
the largest value,

00:09:13.280 --> 00:09:18.090
say, the maximum of
F', times delta x.

00:09:18.090 --> 00:09:20.720
And the only thing we can
say for sure on the other end

00:09:20.720 --> 00:09:23.720
is that it's less than
or equal to-- sorry,

00:09:23.720 --> 00:09:25.700
it's greater than or
equal to the minimum

00:09:25.700 --> 00:09:27.410
of F' times delta x.

00:09:27.410 --> 00:09:29.090
Over that same interval.

00:09:29.090 --> 00:09:39.650
This is over 0 less
than-- sorry, a < x < b.

00:09:39.650 --> 00:09:43.220
So that means that the
fundamental theorem of calculus

00:09:43.220 --> 00:09:45.950
is a much more specific thing.

00:09:45.950 --> 00:09:48.272
And indeed it gives
the same conclusion.

00:09:48.272 --> 00:09:50.230
It's much stronger than
the mean value theorem.

00:09:50.230 --> 00:09:52.330
It's way better than
the mean value theorem.

00:09:52.330 --> 00:09:55.100
In fact, as soon as
we have integrals,

00:09:55.100 --> 00:09:57.280
we can abandon the
mean value theorem.

00:09:57.280 --> 00:09:58.550
We don't want it.

00:09:58.550 --> 00:10:00.810
It's too simple-minded.

00:10:00.810 --> 00:10:03.700
And what we have is something
much more sophisticated,

00:10:03.700 --> 00:10:04.940
which we can use.

00:10:04.940 --> 00:10:05.760
Which is this.

00:10:05.760 --> 00:10:08.400
So it's obvious that
if this is the average,

00:10:08.400 --> 00:10:09.960
the average is less
than the maximum.

00:10:09.960 --> 00:10:14.610
So it's obvious that
it works just as well

00:10:14.610 --> 00:10:15.950
to draw this conclusion.

00:10:15.950 --> 00:10:20.010
And similarly over
here with the minimum.

00:10:20.010 --> 00:10:22.240
OK, the average is always
bigger than the minimum

00:10:22.240 --> 00:10:25.430
and smaller than the max.

00:10:25.430 --> 00:10:28.680
So this is the
connection, if you like.

00:10:28.680 --> 00:10:33.050
And I'm going to elaborate just
one step further by talking

00:10:33.050 --> 00:10:36.340
about the problem that
you had on the exam.

00:10:36.340 --> 00:10:39.640
So there was an Exam 2 problem.

00:10:39.640 --> 00:10:43.590
And I'll show you how it works
using the mean value theorem

00:10:43.590 --> 00:10:45.602
and how it works
using integrals.

00:10:45.602 --> 00:10:47.810
But I'm going to have to
use this notation capital F.

00:10:47.810 --> 00:10:51.850
So capital F', as
opposed to the little f,

00:10:51.850 --> 00:10:55.270
which was what was the
notation that was on your exam.

00:10:55.270 --> 00:10:57.620
So we had this situation here.

00:10:57.620 --> 00:11:01.380
These were the givens
of the problem.

00:11:01.380 --> 00:11:11.620
And then the question was,
the mean value theorem says,

00:11:11.620 --> 00:11:14.160
or implies, if you
like, it doesn't say it,

00:11:14.160 --> 00:11:25.370
but it implies it - implies
A is less than capital F of 4

00:11:25.370 --> 00:11:38.690
is less than B,
for which A and B?

00:11:38.690 --> 00:11:43.330
So let's take a look
at what it says.

00:11:43.330 --> 00:11:48.840
Well, the mean value
theorem says that F( F(4) -

00:11:48.840 --> 00:11:53.620
F(0) = F'(c) (4 - 0).

00:11:57.240 --> 00:12:03.770
This is this F' times delta
x, this is the change in x.

00:12:03.770 --> 00:12:08.100
And that's the same
thing as 1/(1+c) times 4.

00:12:12.750 --> 00:12:21.780
And so the range of values
of this number here is from /

00:12:21.780 --> 00:12:24.630
1/(1+0) times 4, that's 4.

00:12:24.630 --> 00:12:28.000
To, that's the largest value,
to the smallest that it gets,

00:12:28.000 --> 00:12:32.510
which is 1/(1+4) times 4.

00:12:32.510 --> 00:12:41.020
That's the range.

00:12:41.020 --> 00:12:54.410
And so the conclusion is that
F(4) - f(0) is between, well,

00:12:54.410 --> 00:12:55.090
let's see.

00:12:55.090 --> 00:12:59.040
It's between 4 and 4/5.

00:12:59.040 --> 00:13:01.850
Which are those two
numbers down there.

00:13:01.850 --> 00:13:03.920
And if you remember
that F(0) was 1,

00:13:03.920 --> 00:13:15.680
this is the same F(4)
is between 5 and 9/5.

00:13:15.680 --> 00:13:19.650
So that's the way that
you were supposed to solve

00:13:19.650 --> 00:13:22.570
the problem on the exam.

00:13:22.570 --> 00:13:25.760
On the other hand,
let's compare to what

00:13:25.760 --> 00:13:27.940
you would do with the
fundamental theorem

00:13:27.940 --> 00:13:31.140
of calculus.

00:13:31.140 --> 00:13:33.090
With the fundamentals
theorem of calculus,

00:13:33.090 --> 00:13:35.090
we have the following formula.

00:13:35.090 --> 00:13:41.710
F(4) - F(0) is equal to the
integral from 0 to 4 of dx /

00:13:41.710 --> 00:13:42.210
(1+x).

00:13:46.100 --> 00:13:52.700
That's what the
fundamental theorem says.

00:13:52.700 --> 00:13:58.550
And now I claim that we can
get these same types of results

00:13:58.550 --> 00:14:00.780
by a very elementary
observation.

00:14:00.780 --> 00:14:03.470
It's really the same
observation that I made up here,

00:14:03.470 --> 00:14:06.110
that the average is less
than or equal to the maximum.

00:14:06.110 --> 00:14:11.810
Which is that the biggest this
can ever be is, let's see.

00:14:11.810 --> 00:14:15.280
The biggest it is
when x is 0, that's 1.

00:14:15.280 --> 00:14:20.360
So the biggest it
ever gets is this.

00:14:20.360 --> 00:14:25.110
And that's equal to 4.

00:14:25.110 --> 00:14:25.660
Right?

00:14:25.660 --> 00:14:30.910
On the other hand, the
smallest it ever gets to be,

00:14:30.910 --> 00:14:36.770
it's equal to this.

00:14:36.770 --> 00:14:39.620
The smallest it ever gets
to be is the integral

00:14:39.620 --> 00:14:42.400
from 0 to 4 of 1/5 dx.

00:14:42.400 --> 00:14:46.110
Because that's the lowest
value that the integrand takes.

00:14:46.110 --> 00:14:48.390
When x = 4, it's 1/5.

00:14:48.390 --> 00:14:54.270
And that's equal to 4/5.

00:14:54.270 --> 00:14:56.440
Now, there's a
little tiny detail

00:14:56.440 --> 00:14:58.350
which is that really
we know that this

00:14:58.350 --> 00:15:01.560
is the area of some rectangle
and this is strictly smaller.

00:15:01.560 --> 00:15:03.990
And we know that these
inequalities are actually

00:15:03.990 --> 00:15:05.210
strict.

00:15:05.210 --> 00:15:07.620
But that's a minor point.

00:15:07.620 --> 00:15:12.270
And certainly not one that
we'll pay close attention to.

00:15:12.270 --> 00:15:17.910
But now, let me show you what
this looks like geometrically.

00:15:17.910 --> 00:15:22.270
So geometrically, we interpret
this as the area under a curve.

00:15:22.270 --> 00:15:31.850
Here's a piece of the
curve y = 1/(1+x).

00:15:31.850 --> 00:15:37.130
And it's going up to 4
and starting at 0 here.

00:15:37.130 --> 00:15:41.510
And the first
estimate that we made

00:15:41.510 --> 00:15:46.030
- that is, the upper bound
- was by trapping this

00:15:46.030 --> 00:15:53.200
in this big rectangle here.

00:15:53.200 --> 00:15:55.610
We compared it to the
constant function,

00:15:55.610 --> 00:15:57.990
which was 1 all the way across.

00:15:57.990 --> 00:16:00.300
This is y = 1.

00:16:00.300 --> 00:16:05.780
And then we also trapped
it from underneath

00:16:05.780 --> 00:16:08.600
by the function which
was at the bottom.

00:16:08.600 --> 00:16:14.430
And this was y = 1/5.

00:16:14.430 --> 00:16:18.410
And so what this really
is is, these things

00:16:18.410 --> 00:16:21.060
are the simplest
possible Riemann sum.

00:16:21.060 --> 00:16:22.860
Sort of a silly Riemann sum.

00:16:22.860 --> 00:16:34.390
This is a Riemann sum
with one rectangle.

00:16:34.390 --> 00:16:36.420
This is the simplest
possible one.

00:16:36.420 --> 00:16:38.580
And so this is a very,
very crude estimate.

00:16:38.580 --> 00:16:40.970
You can see it misses by a mile.

00:16:40.970 --> 00:16:42.840
The larger and
the smaller values

00:16:42.840 --> 00:16:46.370
are off by a factor of 5.

00:16:46.370 --> 00:16:50.660
But this one is called
the-- this one is the lower

00:16:50.660 --> 00:16:53.390
Riemann sum.

00:16:53.390 --> 00:16:59.560
And that one is less
than our actual integral.

00:16:59.560 --> 00:17:14.100
Which is less than
the upper Riemann sum.

00:17:14.100 --> 00:17:19.070
And you should, by now, have
looked at those upper and lower

00:17:19.070 --> 00:17:20.670
sums on your homework.

00:17:20.670 --> 00:17:22.460
So it's just the
rectangles underneath

00:17:22.460 --> 00:17:25.390
and the rectangles on top.

00:17:25.390 --> 00:17:27.370
So at this point,
we can literally

00:17:27.370 --> 00:17:28.662
abandon the mean value theorem.

00:17:28.662 --> 00:17:30.953
Because we have a much better
way of getting at things.

00:17:30.953 --> 00:17:32.770
If we chop things up
into more rectangles,

00:17:32.770 --> 00:17:35.550
we'll get much better
numerical approximations.

00:17:35.550 --> 00:17:38.587
And if we use simpleminded
expressions with integrals,

00:17:38.587 --> 00:17:40.670
we'll be able to figure
out any bound we could get

00:17:40.670 --> 00:17:42.610
using the mean value theorem.

00:17:42.610 --> 00:17:45.790
So that's not the relevance
of the mean value theorem.

00:17:45.790 --> 00:17:48.810
I'll explain to you why
we talked about it, even,

00:17:48.810 --> 00:17:51.310
in a few minutes.

00:17:51.310 --> 00:17:59.790
OK, are there any
questions before we go on?

00:17:59.790 --> 00:18:00.290
Yeah.

00:18:00.290 --> 00:18:07.000
STUDENT: [INAUDIBLE]

00:18:07.000 --> 00:18:09.860
PROFESSOR: I knew that the
range of c was from 0 to 4,

00:18:09.860 --> 00:18:11.980
I should have said
that right here.

00:18:11.980 --> 00:18:13.715
This is true for this theorem.

00:18:13.715 --> 00:18:16.720
The mean value theorem comes
with an extra statement,

00:18:16.720 --> 00:18:17.950
which I missed.

00:18:17.950 --> 00:18:21.960
Which is that this is for
some c between 0 and 4.

00:18:21.960 --> 00:18:23.690
So I know the range
is between 0 and 4.

00:18:23.690 --> 00:18:25.148
The reason why it's
between 0 and 4

00:18:25.148 --> 00:18:27.560
is that's part of the
mean value theorem.

00:18:27.560 --> 00:18:29.610
We started at 0, we ended at 4.

00:18:29.610 --> 00:18:32.420
So the c has to be
somewhere in between.

00:18:32.420 --> 00:18:42.057
That's part of the
mean value theorem.

00:18:42.057 --> 00:18:42.890
STUDENT: [INAUDIBLE]

00:18:42.890 --> 00:18:43.730
PROFESSOR: The
question is, do you

00:18:43.730 --> 00:18:45.820
exclude any values that
are above 4 and below 0.

00:18:45.820 --> 00:18:46.884
Yes, absolutely.

00:18:46.884 --> 00:18:48.550
The point is that in
order to figure out

00:18:48.550 --> 00:18:51.942
how F changes, capital F
changes, between 0 and 4,

00:18:51.942 --> 00:18:54.150
you need only pay attention
to the values in between.

00:18:54.150 --> 00:18:55.608
You don't have to
pay any attention

00:18:55.608 --> 00:18:59.740
to what the function is
doing below 0 or above 4.

00:18:59.740 --> 00:19:07.090
Those things are
strictly irrelevant.

00:19:07.090 --> 00:19:16.569
STUDENT: [INAUDIBLE]

00:19:16.569 --> 00:19:18.110
PROFESSOR: Yeah, I
mean it's strictly

00:19:18.110 --> 00:19:19.950
in between these two numbers.

00:19:19.950 --> 00:19:23.420
I have to understand what the
lowest and the highest one is.

00:19:23.420 --> 00:19:24.700
STUDENT: [INAUDIBLE]

00:19:24.700 --> 00:19:35.031
PROFESSOR: It's
approaching that, so.

00:19:35.031 --> 00:19:35.530
OK.

00:19:35.530 --> 00:19:39.650
So now, the next thing that
we're going to talk about

00:19:39.650 --> 00:19:41.820
is, since I've got
that 1 up there,

00:19:41.820 --> 00:19:44.180
that Fundamental
Theorem of Calculus 1, I

00:19:44.180 --> 00:20:05.070
need to talk about version 2.

00:20:05.070 --> 00:20:15.530
So here is the Fundamental
Theorem of Calculus version 2.

00:20:15.530 --> 00:20:20.660
I'm going to start out
with a function little f,

00:20:20.660 --> 00:20:28.002
and I'm going to assume
that it's continuous.

00:20:28.002 --> 00:20:30.210
And then I'm going to define
a new function, which is

00:20:30.210 --> 00:20:33.030
defined as a definite integral.

00:20:33.030 --> 00:20:40.310
G(x) is the integral
from a to x of f(t) dt.

00:20:40.310 --> 00:20:42.829
Now, I want to emphasize here
because it's the first time

00:20:42.829 --> 00:20:44.370
that I'm writing
something like this,

00:20:44.370 --> 00:20:47.300
that this is a fairly
complicated gadget.

00:20:47.300 --> 00:20:52.110
It plays a very basic and very
fundamental but simple role,

00:20:52.110 --> 00:20:54.490
but it nevertheless is
a little complicated.

00:20:54.490 --> 00:20:58.370
What's happening here is
that the upper limit I've now

00:20:58.370 --> 00:21:05.840
called x, and the variable t
is ranging between a and x,

00:21:05.840 --> 00:21:08.030
and that the a and
the x are fixed

00:21:08.030 --> 00:21:12.290
when I calculate the integral.

00:21:12.290 --> 00:21:14.730
And the t is what's
called the dummy variable.

00:21:14.730 --> 00:21:16.290
It's the variable
of integration.

00:21:16.290 --> 00:21:21.420
You'll see a lot of people who
will mix this x with this t.

00:21:21.420 --> 00:21:25.190
And if you do that,
you will get confused,

00:21:25.190 --> 00:21:28.150
potentially hopelessly
confused, in this class.

00:21:28.150 --> 00:21:32.310
In 18.02 you will be
completely lost if you do that.

00:21:32.310 --> 00:21:33.720
So don't do it.

00:21:33.720 --> 00:21:38.410
Don't mix these two guys up.

00:21:38.410 --> 00:21:42.089
It's actually done by
many people in textbooks,

00:21:42.089 --> 00:21:43.130
and it's fairly careless.

00:21:43.130 --> 00:21:45.230
Especially in
old-fashioned textbooks.

00:21:45.230 --> 00:21:48.610
But don't do it.

00:21:48.610 --> 00:21:50.570
So here we have this G(x).

00:21:50.570 --> 00:21:56.250
Now, remember, this G(x)
really does make sense.

00:21:56.250 --> 00:21:59.650
If you give me an a,
and you give me an x,

00:21:59.650 --> 00:22:01.960
I can figure out what this
is, because I can figure out

00:22:01.960 --> 00:22:02.830
the Riemann sum.

00:22:02.830 --> 00:22:05.090
So of course I need to know
what the function is, too.

00:22:05.090 --> 00:22:07.960
But anyway, we have a numerical
procedure for figuring out

00:22:07.960 --> 00:22:09.920
what the function G is.

00:22:09.920 --> 00:22:13.010
Now, as is suggested by this
mysterious letter x being

00:22:13.010 --> 00:22:16.350
in the place where it is,
I'm actually going to vary x.

00:22:16.350 --> 00:22:19.240
So the conclusion is
that if this is true,

00:22:19.240 --> 00:22:21.870
and this is just a parenthesis,
not part of the theorem.

00:22:21.870 --> 00:22:25.770
It's just an indication of
what the notation means.

00:22:25.770 --> 00:22:40.430
Then G' = f.

00:22:40.430 --> 00:22:43.030
Let me first explain what the
significance of this theorem

00:22:43.030 --> 00:22:48.280
is, from the point of view
of differential equations.

00:22:48.280 --> 00:23:04.420
G(x) solves the differential
equation y' = f(x).

00:23:04.420 --> 00:23:09.390
So y' = f, I shouldn't put
the x in if I got it here,

00:23:09.390 --> 00:23:13.240
with the condition y(a) = 0.

00:23:13.240 --> 00:23:19.000
So it solves this pair
of conditions here.

00:23:19.000 --> 00:23:21.860
The rate of change, and
the initial position

00:23:21.860 --> 00:23:23.280
is specified here.

00:23:23.280 --> 00:23:29.660
Because when you integrate
from a to a, you get 0 always.

00:23:29.660 --> 00:23:34.100
And what this theorem
says is you can always

00:23:34.100 --> 00:23:35.700
solve that equation.

00:23:35.700 --> 00:23:38.130
When we did differential
equations, I said that already.

00:23:38.130 --> 00:23:39.990
I said we'll treat
these as always solved.

00:23:39.990 --> 00:23:41.530
Well, here's the reason.

00:23:41.530 --> 00:23:45.080
We have a numerical procedure
for computing things like this.

00:23:45.080 --> 00:23:49.330
We could always
solve this equation.

00:23:49.330 --> 00:23:52.290
And the formula is a
fairly complicated gadget,

00:23:52.290 --> 00:23:58.400
but so far just associated
with Riemann sums.

00:23:58.400 --> 00:24:01.000
Alright, now.

00:24:01.000 --> 00:24:13.820
Let's just do one example.

00:24:13.820 --> 00:24:17.900
Unfortunately, not a complicated
example and maybe not

00:24:17.900 --> 00:24:21.230
persuasive as to why you would
care about this just yet.

00:24:21.230 --> 00:24:23.670
But nevertheless very important.

00:24:23.670 --> 00:24:26.470
Because this is the quiz
question which everybody gets

00:24:26.470 --> 00:24:29.530
wrong until they practice it.

00:24:29.530 --> 00:24:35.960
So the integral from,
say 1 to x, of dt / t^2.

00:24:38.710 --> 00:24:45.420
Let's try this one here.

00:24:45.420 --> 00:24:56.650
So here's an example of
this theorem, I claim.

00:24:56.650 --> 00:25:00.850
Now, this is a question
which challenges your ability

00:25:00.850 --> 00:25:04.740
to understand what
the question means.

00:25:04.740 --> 00:25:06.800
Because it's got
a lot of symbols.

00:25:06.800 --> 00:25:09.730
It's got the integration and
it's got the differentiation.

00:25:09.730 --> 00:25:15.700
However, what it really is
is an exercise in recopying.

00:25:15.700 --> 00:25:20.660
You look at it and you
write down the answer.

00:25:20.660 --> 00:25:24.570
And the reason is
that, by definition,

00:25:24.570 --> 00:25:29.190
this function in here is
a function of the form

00:25:29.190 --> 00:25:34.320
G(x) of the theorem over here.

00:25:34.320 --> 00:25:35.920
So this is the G(x).

00:25:35.920 --> 00:25:42.980
And by definition, we
said that G'(x) = f(x).

00:25:42.980 --> 00:25:46.170
Well, what's the f(x)?

00:25:46.170 --> 00:25:47.010
Look inside here.

00:25:47.010 --> 00:25:48.760
It's what's called
the integrand.

00:25:48.760 --> 00:25:53.640
This is the integral from
0 to x of f(t) dt, right?

00:25:53.640 --> 00:26:00.710
Where the f(t) is
equal to 1 / t^2.

00:26:00.710 --> 00:26:02.910
So your ability is challenged.

00:26:02.910 --> 00:26:06.060
You have to take that 1 /
t^2 and you have to plug

00:26:06.060 --> 00:26:09.960
in the letter x,
instead of t, for it.

00:26:09.960 --> 00:26:11.480
And then write it down.

00:26:11.480 --> 00:26:18.120
As I say, this is an exercise
in recopying what's there.

00:26:18.120 --> 00:26:19.917
So this is quite
easy to do, right?

00:26:19.917 --> 00:26:21.750
I mean, you just look
and you write it down.

00:26:21.750 --> 00:26:28.260
But nevertheless, it looks like
a long, elaborate object here.

00:26:28.260 --> 00:26:28.760
Pardon me?

00:26:28.760 --> 00:26:30.590
STUDENT: [INAUDIBLE]

00:26:30.590 --> 00:26:32.940
PROFESSOR: So the question
was, why did I integrate.

00:26:32.940 --> 00:26:34.000
STUDENT: [INAUDIBLE]

00:26:34.000 --> 00:26:36.990
PROFESSOR: Why did
I not integrate?

00:26:36.990 --> 00:26:37.720
Ah.

00:26:37.720 --> 00:26:38.870
Very good question.

00:26:38.870 --> 00:26:41.010
Why did I not integrate.

00:26:41.010 --> 00:26:45.300
The reason why I didn't
integrate is I didn't need to.

00:26:45.300 --> 00:26:47.665
Just as when you take the
antiderivative-- sorry,

00:26:47.665 --> 00:26:50.040
the derivative of something,
you take the antiderivative,

00:26:50.040 --> 00:26:51.750
you get back to the thing.

00:26:51.750 --> 00:26:54.947
So, in this case, we're taking
the antiderivative of something

00:26:54.947 --> 00:26:56.030
and we're differentiating.

00:26:56.030 --> 00:26:58.380
So we end back in the same
place where we started.

00:26:58.380 --> 00:27:01.780
We started with f(t),
we're ending with f.

00:27:01.780 --> 00:27:04.990
Little f.

00:27:04.990 --> 00:27:06.950
So you integrate, and
then differentiate.

00:27:06.950 --> 00:27:09.410
And you get back
to the same place.

00:27:09.410 --> 00:27:12.330
Now, the only difference between
this and the other version

00:27:12.330 --> 00:27:15.610
is, in this case when you
differentiate and integrate

00:27:15.610 --> 00:27:18.195
you could be off by a constant.

00:27:18.195 --> 00:27:19.570
That's what that
shift, why there

00:27:19.570 --> 00:27:21.420
are two pieces to this one.

00:27:21.420 --> 00:27:23.020
But there's never
an extra piece here.

00:27:23.020 --> 00:27:24.919
There's no plus c here.

00:27:24.919 --> 00:27:26.460
When you integrate
and differentiate,

00:27:26.460 --> 00:27:28.080
you kill whatever
the constant is.

00:27:28.080 --> 00:27:31.920
Because the derivative
of a constant is 0.

00:27:31.920 --> 00:27:36.220
So no matter what the constant
is, hiding inside of G,

00:27:36.220 --> 00:27:41.140
you're getting the same result.
So this is the basic idea.

00:27:41.140 --> 00:27:46.220
Now, I just want
to double-check it,

00:27:46.220 --> 00:27:52.124
using the Fundamental
Theorem of Calculus 1 here.

00:27:52.124 --> 00:27:53.790
So let's actually
evaluate the integral.

00:27:53.790 --> 00:27:54.880
So now I'm going
to do what you've

00:27:54.880 --> 00:27:56.185
suggested, which
is I'm just going

00:27:56.185 --> 00:27:57.370
to check whether it's true.

00:27:57.370 --> 00:27:59.560
No, no I am because I'm
going just double-check

00:27:59.560 --> 00:28:01.120
that it's consistent.

00:28:01.120 --> 00:28:03.447
It certainly is slower
this way, and we're not

00:28:03.447 --> 00:28:05.030
going to want to do
this all the time,

00:28:05.030 --> 00:28:06.810
but we might as well check one.

00:28:06.810 --> 00:28:09.370
So this is our integral.

00:28:09.370 --> 00:28:10.860
And we know how to do it.

00:28:10.860 --> 00:28:13.260
No, I need to do it.

00:28:13.260 --> 00:28:17.820
And this is -t^(-1),
evaluated at 1 and x.

00:28:17.820 --> 00:28:21.670
Again, there's something
subliminally here

00:28:21.670 --> 00:28:23.300
for you to think about.

00:28:23.300 --> 00:28:28.737
Which is that, remember, it's t
is ranging between 1 and t = x.

00:28:28.737 --> 00:28:30.820
And this is one of the big
reasons why this letter

00:28:30.820 --> 00:28:32.860
t has to be different from x.

00:28:32.860 --> 00:28:35.370
Because here it's
1 and there it's x.

00:28:35.370 --> 00:28:37.210
It's not x.

00:28:37.210 --> 00:28:38.890
So you can't put an x here.

00:28:38.890 --> 00:28:44.130
Again, this is t = 1 and
this is t = x over there.

00:28:44.130 --> 00:28:48.630
And now if I plug
that in, I get what?

00:28:48.630 --> 00:28:55.860
I get -1/x, and
then I get -(-1).

00:28:55.860 --> 00:28:59.090
So this is, let me get rid
of those little t's there.

00:28:59.090 --> 00:29:05.690
This is a little easier to read.

00:29:05.690 --> 00:29:07.030
And so now let's check it.

00:29:07.030 --> 00:29:07.730
It's d/dx.

00:29:07.730 --> 00:29:09.500
So here's what G(x) is.

00:29:09.500 --> 00:29:12.260
G(x) = 1 - 1/x.

00:29:12.260 --> 00:29:14.860
That's what G(x) is.

00:29:14.860 --> 00:29:20.500
And if I differentiate
that, I get +1 / x^2.

00:29:20.500 --> 00:29:26.170
That's it.

00:29:26.170 --> 00:29:40.810
You see the constant
washed away.

00:29:40.810 --> 00:29:42.070
So now, here's my job.

00:29:42.070 --> 00:29:44.440
My job is to prove
these theorems.

00:29:44.440 --> 00:29:45.809
I never did prove them for you.

00:29:45.809 --> 00:29:47.725
So, I'm going to prove
the Fundamental Theorem

00:29:47.725 --> 00:29:49.340
of Calculus.

00:29:49.340 --> 00:29:51.860
But I'm going to do 2 first.

00:29:51.860 --> 00:29:53.420
And then I'm going to do 1.

00:29:53.420 --> 00:29:56.490
And it's just going to take
me just one blackboard.

00:29:56.490 --> 00:30:00.310
It's not that hard.

00:30:00.310 --> 00:30:03.150
The proof is by picture.

00:30:03.150 --> 00:30:08.310
And, using the interpretation
as area under the curve.

00:30:08.310 --> 00:30:12.170
So if here's the
value of a, and this

00:30:12.170 --> 00:30:22.550
is the graph of the
function y equals f of x.

00:30:22.550 --> 00:30:26.380
Then I want to draw
three vertical lines.

00:30:26.380 --> 00:30:29.330
One of them is going to be at x.

00:30:29.330 --> 00:30:33.790
And one of them is going
to be at x + delta x.

00:30:33.790 --> 00:30:35.950
So here I have the
interval from 0

00:30:35.950 --> 00:30:39.510
to x, and next I have the
interval from x to delta

00:30:39.510 --> 00:30:42.900
x more than that.

00:30:42.900 --> 00:30:50.360
And now the pieces that I've
got are the area of this part.

00:30:50.360 --> 00:30:53.430
So this has area
which has a name.

00:30:53.430 --> 00:30:55.860
It's called G(x).

00:30:55.860 --> 00:31:00.790
By definition, G(x), which
is sitting right over here

00:31:00.790 --> 00:31:03.940
in the fundamental theorem,
is the integral from a

00:31:03.940 --> 00:31:06.100
to x of this function.

00:31:06.100 --> 00:31:07.890
So it's the area
under the curve.

00:31:07.890 --> 00:31:10.400
So that area is G(x).

00:31:10.400 --> 00:31:17.770
Now this other
chunk here, I claim

00:31:17.770 --> 00:31:23.430
that this is delta G. This
is the change in G. It's

00:31:23.430 --> 00:31:26.260
the value of G(x) that is the
area of the whole business all

00:31:26.260 --> 00:31:30.530
the way up to x + delta x
minus the first part, G(x).

00:31:30.530 --> 00:31:31.640
So it's what's left over.

00:31:31.640 --> 00:31:39.780
It's the incremental
amount of area there.

00:31:39.780 --> 00:31:45.830
And now I am going to carry out
a pretty standard estimation

00:31:45.830 --> 00:31:46.520
here.

00:31:46.520 --> 00:31:48.850
This is practically a rectangle.

00:31:48.850 --> 00:31:51.700
And it's got a base of delta
x, and so we need to figure out

00:31:51.700 --> 00:31:55.320
what its height is.

00:31:55.320 --> 00:32:02.120
This is delta G, and it's
approximately its base

00:32:02.120 --> 00:32:05.340
times its height.

00:32:05.340 --> 00:32:06.820
But what is the height?

00:32:06.820 --> 00:32:10.740
Well, the height is maybe either
this segment or this segment

00:32:10.740 --> 00:32:11.920
or something in between.

00:32:11.920 --> 00:32:13.720
But they're all about the same.

00:32:13.720 --> 00:32:17.070
So I'm just going to put in
the value at the first point.

00:32:17.070 --> 00:32:19.430
That's the left end there.

00:32:19.430 --> 00:32:25.550
So that's this
height here, is f(x).

00:32:25.550 --> 00:32:28.200
So this is f(x), and so
really I approximate it

00:32:28.200 --> 00:32:30.050
by that rectangle there.

00:32:30.050 --> 00:32:33.760
And now if I divide
and take the limit,

00:32:33.760 --> 00:32:38.510
as delta x goes to 0,
of delta G / delta x,

00:32:38.510 --> 00:32:40.270
it's going to equal f(x).

00:32:43.120 --> 00:32:48.920
And this is where I'm using
the fact that f is continuous.

00:32:48.920 --> 00:32:51.270
Because I need the
values nearby to be

00:32:51.270 --> 00:32:59.690
similar to the
value in the limit.

00:32:59.690 --> 00:33:00.650
OK, that's the end.

00:33:00.650 --> 00:33:05.020
This the end of the proof, so
I'll put a nice little Q.E.D.

00:33:05.020 --> 00:33:10.800
here.

00:33:10.800 --> 00:33:14.360
So we've done Fundamental
Theorem of Calculus 2,

00:33:14.360 --> 00:33:19.440
and now we're ready for
Fundamental Theorem of Calculus

00:33:19.440 --> 00:33:38.420
1.

00:33:38.420 --> 00:33:44.580
So now I still have it on
the blackboard to remind you.

00:33:44.580 --> 00:33:48.200
It says that the integral
of the derivative

00:33:48.200 --> 00:33:50.980
is the function, at least the
difference between the values

00:33:50.980 --> 00:33:54.830
of the function at two places.

00:33:54.830 --> 00:34:08.040
So the place where we start is
with this property that F' = f.

00:34:08.040 --> 00:34:10.600
That's the starting--
that's the hypothesis.

00:34:10.600 --> 00:34:12.710
Now, unfortunately,
I'm going to have

00:34:12.710 --> 00:34:14.420
to assume something
extra in order

00:34:14.420 --> 00:34:18.660
to use the Fundamental
Theorem of Calculus 2,

00:34:18.660 --> 00:34:27.980
which is I'm going to
assume that f is continuous.

00:34:27.980 --> 00:34:30.490
That's not really
necessary, but that's

00:34:30.490 --> 00:34:32.170
just a very minor
technical point,

00:34:32.170 --> 00:34:34.280
which I'm just going to ignore.

00:34:34.280 --> 00:34:40.610
So we're going to
start with F' = f.

00:34:40.610 --> 00:34:46.530
And then I'm going
to go somewhere else.

00:34:46.530 --> 00:34:52.620
I'm going to define
a new function, G(x),

00:34:52.620 --> 00:35:00.130
which is the integral
from a to x of f(t) dt.

00:35:00.130 --> 00:35:04.560
This is where we needed all
of the labor of Riemann sums.

00:35:04.560 --> 00:35:07.170
Because otherwise we don't
have a way of making sense out

00:35:07.170 --> 00:35:10.340
of what this even means.

00:35:10.340 --> 00:35:13.350
So hiding behind
this one sentence

00:35:13.350 --> 00:35:16.050
is the fact that we
actually have a number.

00:35:16.050 --> 00:35:18.430
We have a formula
for such functions.

00:35:18.430 --> 00:35:20.550
So there is a
function G(x) which,

00:35:20.550 --> 00:35:22.220
once you've produced
a little f for me,

00:35:22.220 --> 00:35:27.420
I can cook up a function
capital G for you.

00:35:27.420 --> 00:35:31.390
Now, we're going to apply this
Fundamental Theorem of Calculus

00:35:31.390 --> 00:35:34.650
2, the one that we've
already checked.

00:35:34.650 --> 00:35:36.030
So what does it say?

00:35:36.030 --> 00:35:46.510
It says that G' = f.

00:35:46.510 --> 00:35:49.820
And so now we're in the
following situation.

00:35:49.820 --> 00:35:54.030
We know that F'(x) = G'(x).

00:35:58.080 --> 00:36:00.360
That's what we've got so far.

00:36:00.360 --> 00:36:07.210
And now we have one last step to
get a good connection between F

00:36:07.210 --> 00:36:10.120
and G. Which is that we
can conclude that F(x)

00:36:10.120 --> 00:36:12.060
is G(x) plus a constant.

00:36:20.440 --> 00:36:26.520
Now, this little step
may seem innocuous

00:36:26.520 --> 00:36:32.610
but I remind you that this
is the spot that requires

00:36:32.610 --> 00:36:36.030
the mean value theorem.

00:36:36.030 --> 00:36:39.650
So in order not to lie
to you, we actually

00:36:39.650 --> 00:36:42.680
tell you what the underpinnings
of all of calculus are.

00:36:42.680 --> 00:36:45.152
And they're this: the
fact, if you like,

00:36:45.152 --> 00:36:47.110
that if two functions
have the same derivative,

00:36:47.110 --> 00:36:48.430
they differ by a constant.

00:36:48.430 --> 00:36:50.750
Or that if a function
has derivative 0,

00:36:50.750 --> 00:36:53.960
it's a constant itself.

00:36:53.960 --> 00:36:57.850
Now, that is the fundamental
step that's needed,

00:36:57.850 --> 00:36:59.740
the underlying
step that's needed.

00:36:59.740 --> 00:37:02.500
And, unfortunately, there
aren't any proofs of it

00:37:02.500 --> 00:37:06.050
that are less complicated than
using the mean value theorem.

00:37:06.050 --> 00:37:08.560
And so that's why we talk a
little bit about the mean value

00:37:08.560 --> 00:37:10.934
theorem, because we don't want
to lie to you about what's

00:37:10.934 --> 00:37:11.731
really going on.

00:37:11.731 --> 00:37:12.230
Yes.

00:37:12.230 --> 00:37:19.070
STUDENT: [INAUDIBLE]

00:37:19.070 --> 00:37:24.210
PROFESSOR: The question is how
did I get from here, to here.

00:37:24.210 --> 00:37:27.440
And the answer is that
if G' is little f,

00:37:27.440 --> 00:37:32.950
and we also know that F'
is little f, then F' is G'.

00:37:32.950 --> 00:37:37.090
OK.

00:37:37.090 --> 00:37:50.710
Other questions?

00:37:50.710 --> 00:37:52.750
Alright, so we're almost done.

00:37:52.750 --> 00:37:57.400
I just have to work out
the arithmetic here.

00:37:57.400 --> 00:38:02.030
So I start with F(b) - F(a).

00:38:04.810 --> 00:38:12.330
And that's equal to
(G(b) + c) - (G(a) + c).

00:38:18.130 --> 00:38:20.970
And then I cancel the c's.

00:38:20.970 --> 00:38:23.310
So I have here G(b) - G(a).

00:38:29.960 --> 00:38:32.665
And now I just have to
check what each of these is.

00:38:32.665 --> 00:38:35.470
So remember the
definition of G here.

00:38:35.470 --> 00:38:38.100
G(b) is just what we want.

00:38:38.100 --> 00:38:42.460
The integral from
a to b of f(x) dx.

00:38:42.460 --> 00:38:46.430
Well I called it f(t) dt,
that's the same as f(x) dx

00:38:46.430 --> 00:38:48.665
now, because I have
the limit being b

00:38:48.665 --> 00:38:52.900
and I'm allowed to use
x as the dummy variable.

00:38:52.900 --> 00:38:55.780
Now the other one,
I claim, is 0.

00:38:55.780 --> 00:38:59.680
Because it's the
integral from a to a.

00:38:59.680 --> 00:39:03.490
This one is the
integral from a to a.

00:39:03.490 --> 00:39:06.510
Which gives us 0.

00:39:06.510 --> 00:39:09.370
So this is just this minus
0, and that's the end.

00:39:09.370 --> 00:39:13.460
That's it.

00:39:13.460 --> 00:39:20.271
I started with F(b) - F(a),
I got to the integral.

00:39:20.271 --> 00:39:20.770
Question?

00:39:20.770 --> 00:39:27.370
STUDENT: [INAUDIBLE]

00:39:27.370 --> 00:39:32.610
PROFESSOR: How did I get from
F(b) - F(a), is (G(b) + c) -

00:39:32.610 --> 00:39:35.430
(G(a) + c), that's the question.

00:39:35.430 --> 00:39:40.190
STUDENT: [INAUDIBLE]

00:39:40.190 --> 00:39:44.160
PROFESSOR: Oh, sorry
this is an equals sign.

00:39:44.160 --> 00:39:47.090
Sorry, the second
line didn't draw.

00:39:47.090 --> 00:39:48.640
OK, equals.

00:39:48.640 --> 00:39:53.141
Because we're plugging in,
for F(x), the formula for it.

00:39:53.141 --> 00:39:53.640
Yes.

00:39:53.640 --> 00:39:57.310
STUDENT: [INAUDIBLE]

00:39:57.310 --> 00:39:59.630
PROFESSOR: This step here?

00:39:59.630 --> 00:40:04.730
Or this one?

00:40:04.730 --> 00:40:09.392
STUDENT: [INAUDIBLE]

00:40:09.392 --> 00:40:10.100
PROFESSOR: Right.

00:40:10.100 --> 00:40:12.090
So that was a good question.

00:40:12.090 --> 00:40:15.120
But the answer is that
that's the statement

00:40:15.120 --> 00:40:16.246
that we're aiming for.

00:40:16.246 --> 00:40:18.120
That's the Fundamental
Theorem of Calculus 1,

00:40:18.120 --> 00:40:19.614
which we don't know yet.

00:40:19.614 --> 00:40:22.030
So we're trying to prove it,
and that's why we haven't, we

00:40:22.030 --> 00:40:25.950
can't assume it.

00:40:25.950 --> 00:40:33.270
OK, so let me just notice that
in the example that we had,

00:40:33.270 --> 00:40:36.410
before we go on to
something else here.

00:40:36.410 --> 00:40:48.530
In the example above, what we
had was the following thing.

00:40:48.530 --> 00:40:58.070
We had, say, F(x) = -1/x.

00:40:58.070 --> 00:41:01.890
So F'(x) = 1 / x^2.

00:41:01.890 --> 00:41:08.850
And, say, G(x) = 1 - 1/x.

00:41:08.850 --> 00:41:11.970
And you can see that
either way you do that,

00:41:11.970 --> 00:41:14.310
if you integrate from
1 to 2, let's say,

00:41:14.310 --> 00:41:18.670
which is what we had
over there, dt / t^2,

00:41:18.670 --> 00:41:28.140
you're going to get either -1/t,
1 to 2, or, if you like, 1 -

00:41:28.140 --> 00:41:31.200
1/t, 1 to 2.

00:41:31.200 --> 00:41:33.930
So this is the F version,
this is the G version.

00:41:33.930 --> 00:41:45.410
And that's what plays itself
out here, in this general proof.

00:41:45.410 --> 00:41:49.640
Alright.

00:41:49.640 --> 00:41:54.910
So now I want to go back to
the theme for today, which

00:41:54.910 --> 00:42:01.290
is using little f to understand
capital F. In other words,

00:42:01.290 --> 00:42:05.890
using the derivative of F
to understand capital F.

00:42:05.890 --> 00:42:22.630
And I want to illustrate it by
some more complicated examples.

00:42:22.630 --> 00:42:27.510
So I guess I just erased it, but
we just took the antiderivative

00:42:27.510 --> 00:42:29.740
of 1 / t^2.

00:42:29.740 --> 00:42:34.340
And there's-- all of the
powers work easily, but one.

00:42:34.340 --> 00:42:39.190
And the tricky one
is the power 1 / x.

00:42:39.190 --> 00:42:43.280
So let's consider the
differential equation L' (x) =

00:42:43.280 --> 00:42:44.550
1 / x.

00:42:44.550 --> 00:42:54.490
And say, with the
initial value L(1) = 0.

00:42:54.490 --> 00:42:58.840
The solution, so the Fundamental
Theorem of Calculus 2

00:42:58.840 --> 00:43:07.330
tells us the solution
is this function here.

00:43:07.330 --> 00:43:14.070
L(x) equals the integral
from 1 to x, dt / t.

00:43:14.070 --> 00:43:16.490
That's how we solve
all such equations.

00:43:16.490 --> 00:43:19.290
We just integrate, take
the definite integral.

00:43:19.290 --> 00:43:27.400
And I'm starting at 1 because
I insisted that L(1) be 0.

00:43:27.400 --> 00:43:31.100
So that's the solution
to the problem.

00:43:31.100 --> 00:43:33.900
And now the thing
that's interesting here

00:43:33.900 --> 00:43:35.720
is that we started
from a polynomial.

00:43:35.720 --> 00:43:37.990
Or we started from a rational,
a ratio of polynomials;

00:43:37.990 --> 00:43:40.460
that is, 1 / t or 1 / x.

00:43:40.460 --> 00:43:42.490
And we get to a function
which is actually

00:43:42.490 --> 00:43:44.990
what's known as a
transcendental function.

00:43:44.990 --> 00:43:46.455
It's not an algebraic function.

00:43:46.455 --> 00:43:47.080
Yeah, question.

00:43:47.080 --> 00:43:57.000
STUDENT: [INAUDIBLE]

00:43:57.000 --> 00:44:04.970
PROFESSOR: The question is
why is this equal to that.

00:44:04.970 --> 00:44:08.210
And the answer is, it's
for the same reason

00:44:08.210 --> 00:44:10.200
that this is equal to that.

00:44:10.200 --> 00:44:14.030
It's the same reason as this.

00:44:14.030 --> 00:44:16.760
It's that the 1's cancel.

00:44:16.760 --> 00:44:19.800
We've taken the value of
something at 2 minus the value

00:44:19.800 --> 00:44:20.300
at 1.

00:44:20.300 --> 00:44:22.500
The value at 2 minus
the value at 1.

00:44:22.500 --> 00:44:24.720
And you'll get a 1 in the
one case, and you get a 1

00:44:24.720 --> 00:44:25.470
in the other case.

00:44:25.470 --> 00:44:27.580
And you subtract them
and they will cancel.

00:44:27.580 --> 00:44:28.620
They'll give you 0.

00:44:28.620 --> 00:44:31.380
These two things
really are equal.

00:44:31.380 --> 00:44:33.507
This is not a function
evaluated at one place,

00:44:33.507 --> 00:44:35.590
it's the difference between
the function evaluated

00:44:35.590 --> 00:44:37.025
at 2 and the value at 1.

00:44:37.025 --> 00:44:39.570
And whenever you subtract
two things like that,

00:44:39.570 --> 00:44:40.900
constants drop out.

00:44:40.900 --> 00:44:44.370
STUDENT: [INAUDIBLE]
PROFESSOR: That's right.

00:44:44.370 --> 00:44:46.030
If I put 2 here,
if I put c here,

00:44:46.030 --> 00:44:47.370
it would have been the same.

00:44:47.370 --> 00:44:49.010
It would just have dropped out.

00:44:49.010 --> 00:44:50.640
It's not there.

00:44:50.640 --> 00:44:53.260
And that's exactly this
arithmetic right here.

00:44:53.260 --> 00:44:55.520
It doesn't matter which
antiderivative you take.

00:44:55.520 --> 00:44:57.850
When you take the differences,
the c's will cancel.

00:44:57.850 --> 00:45:03.130
You always get the
same answer in the end.

00:45:03.130 --> 00:45:04.850
That's exactly why
I wrote this down,

00:45:04.850 --> 00:45:06.290
so that you would see that.

00:45:06.290 --> 00:45:12.890
It doesn't matter
which one you do.

00:45:12.890 --> 00:45:21.540
So, we still have a couple
of minutes left here.

00:45:21.540 --> 00:45:23.630
This is actually--
So let me go back.

00:45:23.630 --> 00:45:29.720
So here's the antiderivative
of 1 / x, with value 1 at 0.

00:45:29.720 --> 00:45:32.180
Now, in disguise, we know
what this function is.

00:45:32.180 --> 00:45:35.410
We know this function is
the logarithm function.

00:45:35.410 --> 00:45:37.580
But this is actually
a better way

00:45:37.580 --> 00:45:42.030
of deriving all of the
formulas for the logarithm.

00:45:42.030 --> 00:45:44.850
This is a much quicker and
more efficient way of doing it.

00:45:44.850 --> 00:45:47.280
We had to do it by very
laborious processes.

00:45:47.280 --> 00:45:51.690
This will allow us
to do it very easily.

00:45:51.690 --> 00:45:56.040
And so, I'm going to
do that next time.

00:45:56.040 --> 00:45:58.395
But rather than do
that now, I'm going

00:45:58.395 --> 00:46:10.280
to point out to you that we can
also get truly new functions.

00:46:10.280 --> 00:46:12.130
OK, so there are all
kinds of new functions.

00:46:12.130 --> 00:46:15.690
So this is the first example
of this kind would be,

00:46:15.690 --> 00:46:21.630
for example, to solve the
equation y' = e^(-x^2) with

00:46:21.630 --> 00:46:25.140
y(0) = 0, let's say.

00:46:25.140 --> 00:46:28.047
Now, the solution to
that is a function

00:46:28.047 --> 00:46:30.380
which again I can write down
by the fundamental theorem.

00:46:30.380 --> 00:46:48.160
It's the integral from
0 to x of e^(-t^2) dt.

00:46:48.160 --> 00:46:52.370
This is a very famous function.

00:46:52.370 --> 00:46:55.820
This shape here is
known as the bell curve.

00:46:55.820 --> 00:46:59.170
And it's the thing that comes
up in probability all the time.

00:46:59.170 --> 00:47:01.880
This shape e^(-x^2).

00:47:01.880 --> 00:47:04.590
And our function is
geometrically just the area

00:47:04.590 --> 00:47:06.310
under the curve here.

00:47:06.310 --> 00:47:09.080
This is F(x).

00:47:09.080 --> 00:47:12.350
If this place is x.

00:47:12.350 --> 00:47:13.860
So I have a
geometric definition,

00:47:13.860 --> 00:47:16.620
I have a way of constructing
what it is by Riemann sums.

00:47:16.620 --> 00:47:18.810
And I have a function here.

00:47:18.810 --> 00:47:26.850
But the curious thing about
F(x) is that F(x) cannot be

00:47:26.850 --> 00:47:34.470
expressed in terms of
any function you've seen

00:47:34.470 --> 00:47:35.360
previously.

00:47:35.360 --> 00:47:44.210
So logs, exponentials,
trig functions, cannot be.

00:47:44.210 --> 00:47:51.600
It's a totally new function.

00:47:51.600 --> 00:47:53.730
Nevertheless, we'll
be able to get

00:47:53.730 --> 00:47:56.030
any possible piece of
information we would want to,

00:47:56.030 --> 00:47:56.910
out of this function.

00:47:56.910 --> 00:47:58.770
It's perfectly
acceptable function,

00:47:58.770 --> 00:48:00.250
it will work just great for us.

00:48:00.250 --> 00:48:01.700
Just like any other function.

00:48:01.700 --> 00:48:03.540
Just like the log.

00:48:03.540 --> 00:48:08.890
And what this is analogous to
is the following kind of thing.

00:48:08.890 --> 00:48:12.650
If you take the circle, the
ancient Greeks, if you like,

00:48:12.650 --> 00:48:18.030
already understood that if you
have a circle of radius 1, then

00:48:18.030 --> 00:48:23.290
its area is pi.

00:48:23.290 --> 00:48:24.820
So that's a geometric
construction

00:48:24.820 --> 00:48:31.430
of what you could
call a new number.

00:48:31.430 --> 00:48:34.280
Which is outside of the realm
of what you might expect.

00:48:34.280 --> 00:48:37.840
And the weird thing
about this number pi

00:48:37.840 --> 00:48:49.780
is that it is not the root
of an algebraic equation

00:48:49.780 --> 00:48:57.260
with rational coefficients.

00:48:57.260 --> 00:49:00.540
It's what's called
transcendental.

00:49:00.540 --> 00:49:02.680
Meaning, it's just completely
outside of the realm

00:49:02.680 --> 00:49:04.050
of algebra.

00:49:04.050 --> 00:49:06.660
And, indeed, the
logarithm function

00:49:06.660 --> 00:49:08.790
is called a
transcendental function,

00:49:08.790 --> 00:49:11.350
because it's completely out
of the realm of algebra.

00:49:11.350 --> 00:49:14.710
It's only in calculus
that you come up

00:49:14.710 --> 00:49:16.190
with this kind of thing.

00:49:16.190 --> 00:49:20.030
So these kinds of
functions will have access

00:49:20.030 --> 00:49:23.130
to a huge class of new
functions here, all of which

00:49:23.130 --> 00:49:26.730
are important tools in
science and engineering.

00:49:26.730 --> 00:49:29.537
So, see you next time.