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PROFESSOR: What we're
going to talk about today
00:00:23.380 --> 00:00:26.750
is a continuation of last time.
00:00:26.750 --> 00:00:29.600
I want to review Newton's
method because I want to talk
00:00:29.600 --> 00:00:41.220
to you about its accuracy.
00:00:41.220 --> 00:00:46.090
So if you remember, the way
Newton's method works is this.
00:00:46.090 --> 00:00:49.280
If you have a curve
and you want to know
00:00:49.280 --> 00:00:52.642
whether it crosses the axis.
00:00:52.642 --> 00:00:54.100
And you don't know
where this point
00:00:54.100 --> 00:01:00.100
is, this point which I'll
call x here, what you do
00:01:00.100 --> 00:01:01.250
is you take a guess.
00:01:01.250 --> 00:01:03.590
Maybe you take a point x_0 here.
00:01:03.590 --> 00:01:06.640
And then you go down to
this point on the graph,
00:01:06.640 --> 00:01:08.602
and you draw the tangent line.
00:01:08.602 --> 00:01:10.560
I'll draw these in a
couple of different colors
00:01:10.560 --> 00:01:13.180
so that you can see the
difference between them.
00:01:13.180 --> 00:01:14.410
So here's a tangent line.
00:01:14.410 --> 00:01:18.280
It's coming out like that.
00:01:18.280 --> 00:01:20.750
And that one is going
to get a little closer
00:01:20.750 --> 00:01:23.090
to our target point.
00:01:23.090 --> 00:01:26.550
But now the trick is,
and this is rather hard
00:01:26.550 --> 00:01:30.060
to see because the scale
gets small incredibly fast,
00:01:30.060 --> 00:01:32.500
is that if you go
right up from that,
00:01:32.500 --> 00:01:34.420
and you do this same
trick over again.
00:01:34.420 --> 00:01:38.400
That is, this is your
second guess, x_1, and now
00:01:38.400 --> 00:01:41.360
you draw the second
tangent line.
00:01:41.360 --> 00:01:44.680
Which is going to
come down this way.
00:01:44.680 --> 00:01:46.290
That's really close.
00:01:46.290 --> 00:01:48.570
You can see here
on the chalkboard,
00:01:48.570 --> 00:01:51.740
it's practically the
same as the dot of x.
00:01:51.740 --> 00:01:54.610
So that's the next guess.
00:01:54.610 --> 00:01:56.860
Which is x_2.
00:01:56.860 --> 00:02:03.620
And I want to analyze,
now, how close it gets.
00:02:03.620 --> 00:02:05.990
And just describe
to you how it works.
00:02:05.990 --> 00:02:09.060
So let me just remind
you of the formulas, too.
00:02:09.060 --> 00:02:11.150
It's worth having
them in your head.
00:02:11.150 --> 00:02:19.240
So the formula for
the next one is this.
00:02:19.240 --> 00:02:23.570
And then the idea is just
to repeat this process.
00:02:23.570 --> 00:02:28.720
Which has a fancy name,
in algorithms, which
00:02:28.720 --> 00:02:31.210
is to iterate, if you like.
00:02:31.210 --> 00:02:32.530
So we repeat the process.
00:02:32.530 --> 00:02:35.720
And that means, for example,
we generate x_2 from x_1
00:02:35.720 --> 00:02:41.730
by the same formula.
00:02:41.730 --> 00:02:43.170
And we did this last time.
00:02:43.170 --> 00:02:47.930
And, more generally, the (n+1)st
is generated from the nth
00:02:47.930 --> 00:02:55.150
guess, by this formula here.
00:02:55.150 --> 00:03:02.820
So what I'd like to do is just
draw the picture of one step
00:03:02.820 --> 00:03:03.920
a little bit more closely.
00:03:03.920 --> 00:03:05.580
So I want to blow up
the picture, which
00:03:05.580 --> 00:03:10.880
is above me there.
00:03:10.880 --> 00:03:11.950
That's a little too high.
00:03:11.950 --> 00:03:16.750
Where are my erasers?
00:03:16.750 --> 00:03:18.390
Got to get it a little
lower than that,
00:03:18.390 --> 00:03:21.890
since I'm going to depict
everything above the line here.
00:03:21.890 --> 00:03:24.180
So here's my curve coming down.
00:03:24.180 --> 00:03:30.620
And suppose that x_1 is
here, so this is directly
00:03:30.620 --> 00:03:32.170
above it is this point here.
00:03:32.170 --> 00:03:35.040
And then as I drew
it, this green
00:03:35.040 --> 00:03:38.530
tangent coming down like that.
00:03:38.530 --> 00:03:42.970
It's a little bit closer,
and this was the place, x_2,
00:03:42.970 --> 00:03:46.440
and then here was
x, our target, which
00:03:46.440 --> 00:03:49.390
is where the curve crosses as
opposed to the straight tangent
00:03:49.390 --> 00:03:52.600
line crossing.
00:03:52.600 --> 00:03:56.310
So that's the picture that
I want you to keep in mind.
00:03:56.310 --> 00:04:00.440
And now, we're just going to
do a very qualitative kind
00:04:00.440 --> 00:04:03.450
of error analysis.
00:04:03.450 --> 00:04:12.960
So here's our error analysis.
00:04:12.960 --> 00:04:19.355
And we're starting out, the
distance between x_1 and x
00:04:19.355 --> 00:04:20.480
is what we want to measure.
00:04:20.480 --> 00:04:23.530
In other words, how close we
are to where we're heading.
00:04:23.530 --> 00:04:26.600
And so I've called that, I'm
going to call that Error 1.
00:04:26.600 --> 00:04:31.440
That's x - x1, in
absolute value.
00:04:31.440 --> 00:04:37.010
And then, E_2 would be x
- x_2, in absolute value.
00:04:37.010 --> 00:04:39.100
And so forth.
00:04:39.100 --> 00:04:44.190
And, last time, when I was
estimating the size of this--
00:04:44.190 --> 00:04:45.870
So E_n would be whatever it was.
00:04:45.870 --> 00:04:51.900
Last time, remember, we
did it for a specific case.
00:04:51.900 --> 00:04:56.020
So last time, I actually
wrote down the numbers.
00:04:56.020 --> 00:04:57.700
And they were these
numbers, maybe
00:04:57.700 --> 00:05:00.365
you could call them E_n,
which was the absolute value
00:05:00.365 --> 00:05:03.270
of square root of 5 minus x_n.
00:05:03.270 --> 00:05:06.610
These are the sizes that I
was writing down last time.
00:05:06.610 --> 00:05:11.620
And I just want to talk about
in general what to expect.
00:05:11.620 --> 00:05:13.480
That worked
amazingly well, and I
00:05:13.480 --> 00:05:16.990
want to show you that that's
true pretty much in general.
00:05:16.990 --> 00:05:23.660
So the first distance, again,
is E_1, is this distance here.
00:05:23.660 --> 00:05:25.190
That's the E_1.
00:05:25.190 --> 00:05:27.570
And then this shorter
distance, here,
00:05:27.570 --> 00:05:37.410
this little bit, which I'll
mark maybe in green, is E_2.
00:05:37.410 --> 00:05:40.970
So how much shorter
is E_1 than E_2?
00:05:40.970 --> 00:05:44.080
Well, the idea is pretty simple.
00:05:44.080 --> 00:05:46.895
It's that if this distance
and this vertical distance,
00:05:46.895 --> 00:05:49.520
they are probably about the same
as the perpendicular distance.
00:05:49.520 --> 00:05:51.550
And this is basically
the situation
00:05:51.550 --> 00:05:54.140
of a curve touching
a tangent line.
00:05:54.140 --> 00:05:58.010
Then the separation is
going to be quadratic.
00:05:58.010 --> 00:06:00.150
And that's basically
what's going to happen.
00:06:00.150 --> 00:06:03.780
So, in other words
the distance E_2
00:06:03.780 --> 00:06:06.530
is going to be about the
square of the distance E_1.
00:06:06.530 --> 00:06:10.579
And that's really the
only feature of this
00:06:10.579 --> 00:06:11.620
that I want to point out.
00:06:11.620 --> 00:06:14.175
So, approximately,
this is the situation
00:06:14.175 --> 00:06:17.880
that we're going to get.
00:06:17.880 --> 00:06:21.690
And so what that
means is, and maybe
00:06:21.690 --> 00:06:24.380
thinking from last
time, what we had was
00:06:24.380 --> 00:06:25.760
something roughly like this.
00:06:25.760 --> 00:06:29.060
You have an E_0, you have
an E_1, you have an E_2,
00:06:29.060 --> 00:06:31.590
you have an E_3, and so forth.
00:06:31.590 --> 00:06:34.370
Maybe I'll write down E_4 here.
00:06:34.370 --> 00:06:37.330
And last time, this
was about 10^(-1).
00:06:37.330 --> 00:06:39.650
So the expectation
based on this rule
00:06:39.650 --> 00:06:42.710
is that the next error's the
square of the previous one.
00:06:42.710 --> 00:06:44.500
So that's 10^(-2).
00:06:44.500 --> 00:06:47.140
The next one is the square
of the previous one.
00:06:47.140 --> 00:06:49.090
So that's 10^(-4).
00:06:49.090 --> 00:06:52.069
And the next one is the square
of that, that's 10^(-8).
00:06:52.069 --> 00:06:53.110
And this one is 10^(-16).
00:06:55.980 --> 00:06:59.550
So the thing that's impressive
about this list of numbers
00:06:59.550 --> 00:07:02.780
is you can see that the
number of digits of accuracy
00:07:02.780 --> 00:07:09.590
doubles at each stage.
00:07:09.590 --> 00:07:20.760
Accuracy doubles at each step.
00:07:20.760 --> 00:07:23.710
The number of digits of
accuracy doubles at each step.
00:07:23.710 --> 00:07:28.150
So, very, very quickly
you get past the accuracy
00:07:28.150 --> 00:07:31.150
of your calculator, as you
saw on your problem set.
00:07:31.150 --> 00:07:34.220
And this thing
works beautifully.
00:07:34.220 --> 00:07:42.700
So, let me just summarize by
saying that Newton's method
00:07:42.700 --> 00:07:49.680
works very well.
00:07:49.680 --> 00:07:52.250
By which I mean
this kind of rate.
00:07:52.250 --> 00:07:55.450
And I want to be just
slightly specific.
00:07:55.450 --> 00:08:00.490
If-- there are really two
conditions disguised in this,
00:08:00.490 --> 00:08:01.480
that are going on.
00:08:01.480 --> 00:08:09.890
One is that f' has to be,
not to big-- to be not small.
00:08:09.890 --> 00:08:19.624
And f'' has to be not too big.
00:08:19.624 --> 00:08:21.290
That's roughly speaking
what's going on.
00:08:21.290 --> 00:08:23.320
I'll explain these
in just a second.
00:08:23.320 --> 00:08:34.050
And x_0 starts nearby.
00:08:34.050 --> 00:08:37.490
Nearby the target value x.
00:08:37.490 --> 00:08:42.050
So that's really
what's going on here.
00:08:42.050 --> 00:08:44.610
So let me just
illustrate to you.
00:08:44.610 --> 00:08:48.920
So I'm not going to explain
this, except to say the reason
00:08:48.920 --> 00:08:51.120
why this second
derivative gets involved
00:08:51.120 --> 00:08:53.360
is that it's how
curved the curve is,
00:08:53.360 --> 00:08:55.330
that how far away you get.
00:08:55.330 --> 00:08:57.507
If the second
derivative were 0, that
00:08:57.507 --> 00:08:58.840
would be the best possible case.
00:08:58.840 --> 00:09:00.660
Then we would get
it on the nose.
00:09:00.660 --> 00:09:02.890
If the second derivative
is not too big,
00:09:02.890 --> 00:09:05.360
that means the quadratic
part is not too big.
00:09:05.360 --> 00:09:07.820
So we don't get away very
far from the green line
00:09:07.820 --> 00:09:14.110
to the curve itself.
00:09:14.110 --> 00:09:17.460
The other thing to
say is, as I said,
00:09:17.460 --> 00:09:19.050
that x_0 needs to start nearby.
00:09:19.050 --> 00:09:21.520
So I'll explain
that by explaining
00:09:21.520 --> 00:09:23.310
what maybe could go wrong.
00:09:23.310 --> 00:09:40.080
So the ways the method can fail,
and one example which actually
00:09:40.080 --> 00:09:45.080
would have happened in our
example from last time,
00:09:45.080 --> 00:09:51.750
which was y = x^2 - 5, is
suppose we'd started x_0 over
00:09:51.750 --> 00:09:54.840
here.
00:09:54.840 --> 00:09:57.430
Then this thing would
have gone off to the left,
00:09:57.430 --> 00:10:00.713
and we would have landed
on not the square root
00:10:00.713 --> 00:10:03.030
of 5 but the other root.
00:10:03.030 --> 00:10:11.750
So if it's too far away,
then we get the wrong root.
00:10:11.750 --> 00:10:15.590
So that's an example,
explaining that the x_0 needs
00:10:15.590 --> 00:10:18.710
to start near the root
that we're talking about.
00:10:18.710 --> 00:10:21.460
Otherwise, the
method doesn't know
00:10:21.460 --> 00:10:22.940
which root you're asking for.
00:10:22.940 --> 00:10:24.580
It only knows where you started.
00:10:24.580 --> 00:10:27.310
So it may go off
to the wrong place.
00:10:27.310 --> 00:10:34.430
OK, it can't read your mind.
00:10:34.430 --> 00:10:35.350
Yes, question.
00:10:35.350 --> 00:10:42.610
STUDENT: [INAUDIBLE]
00:10:42.610 --> 00:10:44.460
PROFESSOR: Oh, good question.
00:10:44.460 --> 00:10:49.550
So the question was, what if the
first error is larger than 1.
00:10:49.550 --> 00:10:51.220
Are you in trouble?
00:10:51.220 --> 00:10:56.000
And the answer is,
absolutely, yes.
00:10:56.000 --> 00:10:58.110
If you have quadratic
behavior, you can see.
00:10:58.110 --> 00:11:00.800
If you have a
quadratic nearby, it's
00:11:00.800 --> 00:11:02.820
pretty close to
the straight line.
00:11:02.820 --> 00:11:05.900
But far away, a parabola is
miles from a straight line.
00:11:05.900 --> 00:11:08.180
It's way, way, way far away.
00:11:08.180 --> 00:11:14.010
So if you're foolish
enough to start over here,
00:11:14.010 --> 00:11:17.150
you may have some
trouble making progress.
00:11:17.150 --> 00:11:20.430
Actually, it isn't, when
I-- that little wiggle
00:11:20.430 --> 00:11:22.010
there just meant
proportional to.
00:11:22.010 --> 00:11:24.570
In fact, in the particular
case of a parabola,
00:11:24.570 --> 00:11:25.810
it manages to get back.
00:11:25.810 --> 00:11:27.150
It saves itself.
00:11:27.150 --> 00:11:30.010
But there's no guarantee
of that sort of thing.
00:11:30.010 --> 00:11:32.790
You really do want to
start reasonably close.
00:11:32.790 --> 00:11:33.350
Yep.
00:11:33.350 --> 00:11:40.140
STUDENT: [INAUDIBLE]
00:11:40.140 --> 00:11:42.580
PROFESSOR: What you have to
do is you have to watch out.
00:11:42.580 --> 00:11:46.790
That is, it's hard to know what
assumptions to make about x_0.
00:11:46.790 --> 00:11:50.250
You plug it into the machine
and you see what you get.
00:11:50.250 --> 00:11:53.227
And either it works
or it doesn't.
00:11:53.227 --> 00:11:55.560
You can tell that it's marching
toward a specific place,
00:11:55.560 --> 00:11:58.400
and you can tell that that place
probably is a zero, usually.
00:11:58.400 --> 00:12:00.510
But maybe it's not the
one you were looking for.
00:12:00.510 --> 00:12:02.384
So in other words, you
have to use your head.
00:12:02.384 --> 00:12:05.230
You run the program and
then you see what it does.
00:12:05.230 --> 00:12:08.000
And if you're lucky-- the
problem is, if you have no idea
00:12:08.000 --> 00:12:12.270
where the zero is, you may
just wander around forever.
00:12:12.270 --> 00:12:14.720
As we'll see in a second.
00:12:14.720 --> 00:12:20.870
So the next example
here is the following.
00:12:20.870 --> 00:12:24.590
I said that f' has
to be not too small.
00:12:24.590 --> 00:12:26.970
There's a real
catastrophe hiding just
00:12:26.970 --> 00:12:28.250
inside this picture.
00:12:28.250 --> 00:12:30.935
Which is the transition between
when you find the positive root
00:12:30.935 --> 00:12:32.890
and when you find the
negative root here.
00:12:32.890 --> 00:12:35.480
Which is, if you're
right down here.
00:12:35.480 --> 00:12:37.860
If you were foolish
enough to get 0,
00:12:37.860 --> 00:12:41.060
then what's going to
happen is your tangent line
00:12:41.060 --> 00:12:41.940
is horizontal.
00:12:41.940 --> 00:12:44.610
It doesn't even meet the axis.
00:12:44.610 --> 00:12:47.840
So in the formula, you can
see that's a catastrophe.
00:12:47.840 --> 00:12:53.030
Because there's an f'
in the denominator.
00:12:53.030 --> 00:12:53.530
So that's 0.
00:12:53.530 --> 00:12:54.791
That's undefined.
00:12:54.791 --> 00:12:56.290
It's not surprising,
it's consistent
00:12:56.290 --> 00:12:59.480
that the parallel line
doesn't meet the axis.
00:12:59.480 --> 00:13:01.940
And you have no x_1.
00:13:01.940 --> 00:13:06.370
So you had-- So if you
like, another point here
00:13:06.370 --> 00:13:12.870
is that f' = 0 is a disaster.
00:13:12.870 --> 00:13:24.310
A disaster for the method.
00:13:24.310 --> 00:13:28.620
Because the next-- So
say, if f(x_0) = 0,
00:13:28.620 --> 00:13:34.700
then x_1 is undefined.
00:13:34.700 --> 00:13:39.170
And finally, there's one other
weird thing that can happen.
00:13:39.170 --> 00:13:43.460
Which is, which I'll just draw
a picture of schematically.
00:13:43.460 --> 00:13:47.150
Which you can get from a wiggle.
00:13:47.150 --> 00:13:49.260
So this wiggle has three roots.
00:13:49.260 --> 00:13:52.060
The way I've drawn it.
00:13:52.060 --> 00:13:56.720
And it can be that you can
start over here with your x_0.
00:13:56.720 --> 00:14:01.200
And draw your tangent line
and go over here to an x_1.
00:14:01.200 --> 00:14:05.510
And then that tangent line will
take you right back to the x_0.
00:14:05.510 --> 00:14:09.560
I didn't draw it quite right,
but that's about right.
00:14:09.560 --> 00:14:11.330
So it goes over like this.
00:14:11.330 --> 00:14:13.520
So let me draw the
two tangent lines, so
00:14:13.520 --> 00:14:14.760
that you can see it properly.
00:14:14.760 --> 00:14:16.614
Sorry, I messed it up.
00:14:16.614 --> 00:14:18.030
So here are the
two tangent lines.
00:14:18.030 --> 00:14:20.870
This guy and this guy.
00:14:20.870 --> 00:14:25.950
And it just goes back and
forth. x_0 cycles to x_1,
00:14:25.950 --> 00:14:27.520
and x_1 goes back to x_0.
00:14:27.520 --> 00:14:30.200
We have a cycle.
00:14:30.200 --> 00:14:32.090
And it never goes anywhere.
00:14:32.090 --> 00:14:34.915
This is, the grass
is always greener.
00:14:34.915 --> 00:14:36.540
It's over here, it
thinks, oh, I really
00:14:36.540 --> 00:14:39.000
would prefer to go to
this zero and then it
00:14:39.000 --> 00:14:40.840
thinks oh, I want to go back.
00:14:40.840 --> 00:14:43.650
And it goes back and
forth, and back and forth.
00:14:43.650 --> 00:14:47.060
Grass is always greener on
the other side of the fence.
00:14:47.060 --> 00:14:50.090
Never, never gets anywhere.
00:14:50.090 --> 00:14:52.510
So those are the sorts of
things that can go wrong
00:14:52.510 --> 00:14:53.490
with Newton's method.
00:14:53.490 --> 00:14:55.210
Nevertheless, it's fantastic.
00:14:55.210 --> 00:14:59.740
It works very well, in
a lot of situations.
00:14:59.740 --> 00:15:03.510
And solves basically any
equation that you can imagine,
00:15:03.510 --> 00:15:10.894
numerically.
00:15:10.894 --> 00:15:12.060
Next we're going to move on.
00:15:12.060 --> 00:15:13.518
We're going to move
on to something
00:15:13.518 --> 00:15:15.730
which is a little theoretical.
00:15:15.730 --> 00:15:18.170
Which is the mean value theorem.
00:15:18.170 --> 00:15:22.410
And that will allow
us in just a day or so
00:15:22.410 --> 00:15:26.150
to launch into the
ideas of integration,
00:15:26.150 --> 00:15:31.140
which is the whole second
half of the course.
00:15:31.140 --> 00:15:50.200
So let's get started with that.
00:15:50.200 --> 00:15:57.030
The mean value theorem will
henceforth be abbreviated MVT.
00:15:57.030 --> 00:15:58.630
So I don't have
to write quite as
00:15:58.630 --> 00:16:03.750
much every time I refer to it.
00:16:03.750 --> 00:16:07.290
The mean value
theorem, colloquially,
00:16:07.290 --> 00:16:09.530
says the following.
00:16:09.530 --> 00:16:23.040
If you go from Boston to LA,
which I think a lot of Red Sox
00:16:23.040 --> 00:16:31.110
fans are going to want to do
soon, so that's 3,000 miles.
00:16:31.110 --> 00:16:47.800
In 6 hours, then
at some time you
00:16:47.800 --> 00:16:55.640
are going at a certain speed.
00:16:55.640 --> 00:17:01.360
The average of this speed.
00:17:01.360 --> 00:17:08.850
Average, so, speed, which
in this case is what?
00:17:08.850 --> 00:17:10.770
So we're going at
the average speed.
00:17:10.770 --> 00:17:16.890
That's 3,000 miles
times 6 hours,
00:17:16.890 --> 00:17:21.420
so that's 500 miles per hour.
00:17:21.420 --> 00:17:23.340
Exactly.
00:17:23.340 --> 00:17:26.442
So some time on your journey--
of course, some of the time
00:17:26.442 --> 00:17:28.150
you're going more than
500 miles an hour,
00:17:28.150 --> 00:17:29.610
sometimes you are going less.
00:17:29.610 --> 00:17:33.220
And some time you must've
been going 500 miles an hour
00:17:33.220 --> 00:17:35.110
exactly.
00:17:35.110 --> 00:17:37.520
That's the mean value theorem.
00:17:37.520 --> 00:17:39.890
The reason why it's
called mean value theorem
00:17:39.890 --> 00:17:55.440
is that word mean is the
same as the word average.
00:17:55.440 --> 00:18:08.160
So now I'm going to state it in
math symbols, the same theorem.
00:18:08.160 --> 00:18:10.470
And it's a formula.
00:18:10.470 --> 00:18:16.275
It says that the
difference quotient
00:18:16.275 --> 00:18:20.410
- so this is the
distance traveled
00:18:20.410 --> 00:18:25.570
divided by the time elapsed,
that's the average speed -
00:18:25.570 --> 00:18:35.320
is equal to the infinitesimal
speed for some time in between.
00:18:35.320 --> 00:18:48.590
So some c, which is in between
a and b-- I'm not quite done.
00:18:48.590 --> 00:18:53.130
It's a real theorem,
it has hypotheses.
00:18:53.130 --> 00:18:56.230
I've told you the
conclusion first,
00:18:56.230 --> 00:18:58.030
but there are some
hypotheses, they're
00:18:58.030 --> 00:18:59.650
straightforward hypotheses.
00:18:59.650 --> 00:19:03.320
Provided f is
differentiable, that is,
00:19:03.320 --> 00:19:12.820
it has a derivative in
the interval a < x < b.
00:19:12.820 --> 00:19:21.560
And continuous in a <= x <= b.
00:19:29.110 --> 00:19:32.080
There has to be a sense that
you can make out of the speed,
00:19:32.080 --> 00:19:36.140
or the rate of change of f
at each intermediate point.
00:19:36.140 --> 00:19:40.200
And in order for the values
at the ends to make sense,
00:19:40.200 --> 00:19:41.200
it has to be continuous.
00:19:41.200 --> 00:19:44.470
There has to be a link
between the values at the ends
00:19:44.470 --> 00:19:47.409
and what's going on in between.
00:19:47.409 --> 00:19:48.950
If it were discontinuous,
there would
00:19:48.950 --> 00:19:52.330
be no relation between
the left and right values
00:19:52.330 --> 00:19:55.970
and the rest of the function.
00:19:55.970 --> 00:20:00.160
So here's the theorem,
conclusion and its hypothesis.
00:20:00.160 --> 00:20:11.780
And it means what I said
more colloquially up above.
00:20:11.780 --> 00:20:14.540
Now, I'm going to prove
this theorem immediately.
00:20:14.540 --> 00:20:18.110
At least, give a geometric
intuitive argument,
00:20:18.110 --> 00:20:21.220
which is not very different
from the one that's given
00:20:21.220 --> 00:20:26.410
in a very systematic treatment.
00:20:26.410 --> 00:20:34.280
So here's the proof of
the mean value theorem.
00:20:34.280 --> 00:20:36.860
It's really just a picture.
00:20:36.860 --> 00:20:42.880
So here's a place, and here's
another place on the graph.
00:20:42.880 --> 00:20:47.180
And the graph is going
along like this, let's say.
00:20:47.180 --> 00:20:50.700
And this line here
is the secant line.
00:20:50.700 --> 00:20:55.310
So this is (a, f(a)) down here.
00:20:55.310 --> 00:20:59.320
And this is (b, f(b)) up there.
00:20:59.320 --> 00:21:02.720
And this segment is
the secant, its slope
00:21:02.720 --> 00:21:04.480
is the slope that
we're aiming for.
00:21:04.480 --> 00:21:09.400
The slope of that line is the
left-hand side of this formula
00:21:09.400 --> 00:21:11.190
here.
00:21:11.190 --> 00:21:14.062
So we need to find
something with that slope.
00:21:14.062 --> 00:21:16.520
And what we need to find is a
tangent line with that slope,
00:21:16.520 --> 00:21:18.340
because what's on
the right-hand side
00:21:18.340 --> 00:21:20.220
is the slope of a tangent line.
00:21:20.220 --> 00:21:22.340
So here's how we construct it.
00:21:22.340 --> 00:21:27.240
We take a parallel
line, down here.
00:21:27.240 --> 00:21:31.125
And then we just translate
it up, leaving it parallel,
00:21:31.125 --> 00:21:32.750
we move it up.
00:21:32.750 --> 00:21:34.080
Towards this one.
00:21:34.080 --> 00:21:38.410
Until it touches.
00:21:38.410 --> 00:21:43.040
And where it touches, at this
point of tangency, down there,
00:21:43.040 --> 00:21:47.490
I've just found my value of c.
00:21:47.490 --> 00:21:49.830
And you can see that if the
tangent line is parallel
00:21:49.830 --> 00:21:53.180
to this line, that's exactly
the equation we want.
00:21:53.180 --> 00:21:59.910
So this thing has slope f'(c).
00:21:59.910 --> 00:22:07.150
And this other one has slope
equal to this complicated
00:22:07.150 --> 00:22:15.970
expression, (f(b)
- f(a)) / (b - a).
00:22:15.970 --> 00:22:19.840
That is almost the
end of the proof.
00:22:19.840 --> 00:22:25.330
There's one problem.
00:22:25.330 --> 00:22:29.880
So, again, we move
a parallel line up.
00:22:29.880 --> 00:22:43.310
Move up the parallel
line until it touches.
00:22:43.310 --> 00:22:46.300
There's a little subtlety here,
which I just want to emphasize.
00:22:46.300 --> 00:22:49.210
Which is that that dotted
line keeps on going here.
00:22:49.210 --> 00:22:51.220
But when we bring
it up, we're going
00:22:51.220 --> 00:22:54.500
to ignore what's
happening outside of a.
00:22:54.500 --> 00:22:56.870
And beyond b, alright?
00:22:56.870 --> 00:23:01.590
So we're just going to
ignore the rest of the graph.
00:23:01.590 --> 00:23:06.090
But there is one thing
that can go wrong.
00:23:06.090 --> 00:23:16.800
So if it does not touch, then
the picture looks likes this.
00:23:16.800 --> 00:23:18.230
Here are the same two points.
00:23:18.230 --> 00:23:20.240
And the graph is all above.
00:23:20.240 --> 00:23:22.060
And we brought up our thing.
00:23:22.060 --> 00:23:23.960
And it went like that.
00:23:23.960 --> 00:23:27.480
So we didn't construct
a tangent line.
00:23:27.480 --> 00:23:29.520
If this happens.
00:23:29.520 --> 00:23:31.980
So we're in trouble,
in that point.
00:23:31.980 --> 00:23:37.010
In this situation, sorry.
00:23:37.010 --> 00:23:40.860
But there's a trick, which
is a straightforward trick.
00:23:40.860 --> 00:23:55.810
Then bring the tangent
lines down from the top.
00:23:55.810 --> 00:23:58.040
So parallel lines,
sorry, not tangent lines.
00:23:58.040 --> 00:24:06.580
Parallel lines.
00:24:06.580 --> 00:24:11.600
From above.
00:24:11.600 --> 00:24:16.380
So, that's the whole story.
00:24:16.380 --> 00:24:43.460
That's how we cook up this point
c, with the right properties.
00:24:43.460 --> 00:24:46.640
I want to point out just
one more theoretical thing.
00:24:46.640 --> 00:24:50.087
And then the rest, we're going
to be drawing conclusions.
00:24:50.087 --> 00:24:51.670
So there's one more
theoretical remark
00:24:51.670 --> 00:24:55.490
about the proof, which
is something that is
00:24:55.490 --> 00:24:59.130
fairly important to understand.
00:24:59.130 --> 00:25:01.640
When you understand a
proof, you should always
00:25:01.640 --> 00:25:06.170
be thinking about why the
hypotheses are necessary.
00:25:06.170 --> 00:25:08.280
Where do I use the hypothesis.
00:25:08.280 --> 00:25:10.890
And I want to give you an
example the proof doesn't
00:25:10.890 --> 00:25:15.350
work to show you that the
hypothesis is an important one.
00:25:15.350 --> 00:25:17.230
So the example is the following.
00:25:17.230 --> 00:25:18.710
I'll just take a
function which is
00:25:18.710 --> 00:25:22.400
two straight lines like this.
00:25:22.400 --> 00:25:27.920
And if you try to perform
this trick with these things,
00:25:27.920 --> 00:25:32.380
then it's going to come up
and it's going to touch here.
00:25:32.380 --> 00:25:36.070
But the problem is that
the tangent line is not
00:25:36.070 --> 00:25:36.740
defined here.
00:25:36.740 --> 00:25:38.660
There are lots of
tangents, and there's
00:25:38.660 --> 00:25:40.610
no derivative at this point.
00:25:40.610 --> 00:25:44.860
So the derivative
doesn't exist here.
00:25:44.860 --> 00:25:57.230
So this is the claim that one
bad point ruins the proof.
00:25:57.230 --> 00:26:08.730
We need f' to exist at all--
so, f'(x) to exist at all x
00:26:08.730 --> 00:26:14.340
in between.
00:26:14.340 --> 00:26:30.980
Can't get away even with
one defective point.
00:26:30.980 --> 00:26:40.620
Now it's time to draw
some consequences.
00:26:40.620 --> 00:26:45.670
And the main
consequence is going
00:26:45.670 --> 00:26:57.880
to have to do with
applications to graphing.
00:26:57.880 --> 00:27:01.090
But we'll see tomorrow and
for the rest of the course
00:27:01.090 --> 00:27:03.920
that this is even
more significant.
00:27:03.920 --> 00:27:09.790
It's significant to all
the rest of calculus.
00:27:09.790 --> 00:27:12.440
I'm going to list three
consequences which you're
00:27:12.440 --> 00:27:14.890
quite familiar with already.
00:27:14.890 --> 00:27:27.550
So, the first one is if f' is
positive, then f is increasing.
00:27:27.550 --> 00:27:40.430
And the second one is if f' is
negative, then f is decreasing.
00:27:40.430 --> 00:27:44.130
And the last one seems
like the simplest.
00:27:44.130 --> 00:27:48.460
But even this one alone
is the key to everything.
00:27:48.460 --> 00:28:03.340
If f' = 0, then f is constant.
00:28:03.340 --> 00:28:13.490
These are three consequences,
now, of the mean value theorem.
00:28:13.490 --> 00:28:17.130
And let me show you
how they're proved.
00:28:17.130 --> 00:28:22.870
I just told you that they
were true, maybe, a while ago.
00:28:22.870 --> 00:28:27.030
And certainly I
mentioned the first two.
00:28:27.030 --> 00:28:30.200
The last one was so simple
that we maybe just swept it
00:28:30.200 --> 00:28:30.810
under the rug.
00:28:30.810 --> 00:28:36.570
You did use it on a
problem set, once or twice.
00:28:36.570 --> 00:28:39.140
But it turns out that this
actually requires proof,
00:28:39.140 --> 00:28:48.010
and we're going to give
the proof right now.
00:28:48.010 --> 00:28:52.190
The way that the proof goes
is simply to write down,
00:28:52.190 --> 00:28:54.500
to rewrite star.
00:28:54.500 --> 00:28:59.440
Rewrite our formula.
00:28:59.440 --> 00:29:05.440
Which says that (f(b) -
f(a)) / (b - a) = f'(c).
00:29:10.050 --> 00:29:13.730
And you see I've written
it from left to right here
00:29:13.730 --> 00:29:16.110
to say that the right-hand
side information
00:29:16.110 --> 00:29:17.590
about the derivative
is going to be
00:29:17.590 --> 00:29:19.340
giving the information
about the function.
00:29:19.340 --> 00:29:22.230
That's the way I'm
going to read it.
00:29:22.230 --> 00:29:26.360
In order to express
this, though, I'm
00:29:26.360 --> 00:29:30.760
going to just rewrite it
a couple of times here.
00:29:30.760 --> 00:29:37.360
So here's f(a), multiplying
through by the denominator.
00:29:37.360 --> 00:29:40.470
And now I'm going to write
it in another customary form
00:29:40.470 --> 00:29:42.300
for the mean value theorem.
00:29:42.300 --> 00:29:46.120
Which is f(b) =
f(a) + f'(c) (b-a).
00:29:51.300 --> 00:29:53.524
So here's another version.
00:29:53.524 --> 00:29:55.440
I should probably have
put this one in the box
00:29:55.440 --> 00:29:59.870
to begin with anyway.
00:29:59.870 --> 00:30:03.800
And, just changing it
around algebraically,
00:30:03.800 --> 00:30:07.160
it's this fact here.
00:30:07.160 --> 00:30:13.310
They're the same thing.
00:30:13.310 --> 00:30:17.760
And now with the formula
written in this form,
00:30:17.760 --> 00:30:24.060
I claim that I can
check these three facts.
00:30:24.060 --> 00:30:26.680
Let's start with the first one.
00:30:26.680 --> 00:30:33.580
I'm going to set things
up always so that a < b.
00:30:33.580 --> 00:30:36.840
And that's the setup
of the theorem.
00:30:36.840 --> 00:30:42.010
And so that means that
b - a is positive.
00:30:42.010 --> 00:30:48.190
Which means that this factor
over here is a positive number.
00:30:48.190 --> 00:30:56.000
If f' is positive,
which is what happens
00:30:56.000 --> 00:30:58.830
in the first case, that's the
assumption that we're making,
00:30:58.830 --> 00:31:01.210
then this is a positive number.
00:31:01.210 --> 00:31:02.750
And so f(b) > f(a).
00:31:08.190 --> 00:31:09.565
Which means that
it's increasing.
00:31:09.565 --> 00:31:13.900
It goes up as the value goes up.
00:31:13.900 --> 00:31:20.780
Similarly, if f'(c) is negative,
then this is a positive times
00:31:20.780 --> 00:31:22.830
a negative number,
this is negative.
00:31:22.830 --> 00:31:25.610
f(b) < f(a).
00:31:25.610 --> 00:31:37.850
So it goes the other way.
00:31:37.850 --> 00:31:39.520
Maybe I'll write this way.
00:31:39.520 --> 00:31:49.910
And finally, if f'(c)
= 0, then f(b) = f(a).
00:31:49.910 --> 00:31:52.110
Which if you apply it
to all possible ends,
00:31:52.110 --> 00:31:56.110
means if you can do it for
every interval, which you can,
00:31:56.110 --> 00:31:57.940
then that means
that f is constant.
00:31:57.940 --> 00:32:12.640
It never gets to change values.
00:32:12.640 --> 00:32:17.260
Well you might have believed
these facts already.
00:32:17.260 --> 00:32:20.670
But I just want to emphasize
to you that this turns out
00:32:20.670 --> 00:32:23.670
to be the one key link
between infinitesimals,
00:32:23.670 --> 00:32:27.610
between limits and these
actual differences.
00:32:27.610 --> 00:32:30.240
Before, we were saying that
the difference quotient
00:32:30.240 --> 00:32:32.530
was approximately equal
to the derivative.
00:32:32.530 --> 00:32:35.520
Now we're saying that it's
exactly equal to a derivative.
00:32:35.520 --> 00:32:38.930
Although we don't know
exactly which point to use.
00:32:38.930 --> 00:32:47.150
It's some point in between.
00:32:47.150 --> 00:32:49.900
I'm going to be deducing some
other consequences in a second,
00:32:49.900 --> 00:32:52.187
but let me stop for
second to make sure
00:32:52.187 --> 00:32:53.270
that everybody's on board.
00:32:53.270 --> 00:32:56.880
Especially since I've
finished the blackboards here.
00:32:56.880 --> 00:32:59.510
Before we-- everybody happy?
00:32:59.510 --> 00:33:00.100
One question.
00:33:00.100 --> 00:33:08.617
STUDENT: [INAUDIBLE]
00:33:08.617 --> 00:33:10.950
PROFESSOR: I'm just going to
repeat your question first.
00:33:10.950 --> 00:33:12.950
I'm a little bit
confused, you said,
00:33:12.950 --> 00:33:16.230
about what guarantees that
there's a point of tangency.
00:33:16.230 --> 00:33:19.030
That's what you said.
00:33:19.030 --> 00:33:20.780
So do you want to
elaborate, or do you
00:33:20.780 --> 00:33:23.152
want to want to stop
with what you just said?
00:33:23.152 --> 00:33:24.360
What is it that confuses you?
00:33:24.360 --> 00:33:28.764
STUDENT: [INAUDIBLE]
00:33:28.764 --> 00:33:29.430
PROFESSOR: Yeah.
00:33:29.430 --> 00:33:43.500
STUDENT: [INAUDIBLE]
00:33:43.500 --> 00:33:46.305
PROFESSOR: So I'm not claiming
that there's only one point.
00:33:46.305 --> 00:33:48.180
This could wiggle a lot
of times and it maybe
00:33:48.180 --> 00:33:49.990
touches at ten places.
00:33:49.990 --> 00:33:54.640
In other words, it's OK with me
if it touches more than once.
00:33:54.640 --> 00:33:56.830
Then I just have more,
the more the merrier.
00:33:56.830 --> 00:33:59.550
In other words, I don't
want there necessarily only
00:33:59.550 --> 00:34:00.050
to be one.
00:34:00.050 --> 00:34:02.720
It could come down like this.
00:34:02.720 --> 00:34:05.140
And touch a second time.
00:34:05.140 --> 00:34:09.290
Is that what was concerning you?
00:34:09.290 --> 00:34:11.580
So in mathematics,
when we claim that this
00:34:11.580 --> 00:34:14.410
is true for some point,
we don't necessarily
00:34:14.410 --> 00:34:16.210
mean that it doesn't
work for others.
00:34:16.210 --> 00:34:18.310
In fact, if the
function is constant,
00:34:18.310 --> 00:34:25.960
this is 0 and in fact this
equation is true for every c.
00:34:25.960 --> 00:34:30.840
That satisfies your question?
00:34:30.840 --> 00:34:33.620
The fact that this point exists
actually is a touchy point.
00:34:33.620 --> 00:34:35.320
I just convinced
you of it visually.
00:34:35.320 --> 00:34:39.030
It's a geometric issue, whether
you're allowed to do this.
00:34:39.030 --> 00:34:41.840
Indeed, it has to do
with the existence
00:34:41.840 --> 00:34:44.150
of tangent lines
and more analysis
00:34:44.150 --> 00:34:46.120
than we can do in this class.
00:34:46.120 --> 00:34:46.620
Yeah.
00:34:46.620 --> 00:34:47.328
Another question.
00:34:47.328 --> 00:34:48.880
STUDENT: [INAUDIBLE]
00:34:48.880 --> 00:34:50.437
PROFESSOR: Pardon me.
00:34:50.437 --> 00:34:51.270
STUDENT: [INAUDIBLE]
00:34:51.270 --> 00:34:52.686
PROFESSOR: The
question is, what's
00:34:52.686 --> 00:34:56.780
the difference between this
and the linear approximation.
00:34:56.780 --> 00:35:11.010
And I think, let me see
if I can describe that.
00:35:11.010 --> 00:35:12.520
I'll leave the
theorem on the board.
00:35:12.520 --> 00:35:14.490
I'm going to get rid of
the colloquial version
00:35:14.490 --> 00:35:19.380
of the theorem.
00:35:19.380 --> 00:35:26.200
And I'll try to describe to
you the difference between this
00:35:26.200 --> 00:35:32.130
and the linear approximation.
00:35:32.130 --> 00:35:34.220
I was planning to
do that in a while,
00:35:34.220 --> 00:35:36.890
but we'll do it right now since
that's what you're asking.
00:35:36.890 --> 00:35:37.660
That's fine.
00:35:37.660 --> 00:35:45.570
So here's the situation.
00:35:45.570 --> 00:35:51.660
The linear approximation,
so let's say comparison
00:35:51.660 --> 00:35:57.390
with linear approximation.
00:35:57.390 --> 00:35:59.710
They're very closely related.
00:35:59.710 --> 00:36:02.710
The linear approximation says
the change in f over the change
00:36:02.710 --> 00:36:06.290
in x, that's the left-hand
side of this thing,
00:36:06.290 --> 00:36:09.980
is approximately f'(a).
00:36:09.980 --> 00:36:21.200
For b near a, and
b - a = delta x.
00:36:21.200 --> 00:36:23.110
This statement, which
is in the box, which
00:36:23.110 --> 00:36:25.980
is sitting right up
there, is the statement
00:36:25.980 --> 00:36:31.330
that this change in f is
actually equal to something.
00:36:31.330 --> 00:36:33.180
Not approximately equal to it.
00:36:33.180 --> 00:36:37.670
It's equal to f' of some c.
00:36:37.670 --> 00:36:41.330
And the problem here is that
we don't know exactly which c.
00:36:41.330 --> 00:36:43.930
This is for some c.
00:36:43.930 --> 00:36:53.190
Between a and b.
00:36:53.190 --> 00:36:54.810
Right, so.
00:36:54.810 --> 00:36:59.190
That's the difference
between the two.
00:36:59.190 --> 00:37:19.700
And let me elaborate
a little bit.
00:37:19.700 --> 00:37:27.050
If you're trying to understand
what (f(b) - f(a)) / (b -
00:37:27.050 --> 00:37:32.840
a) is, the mean value theorem
is telling you for sure that
00:37:32.840 --> 00:37:35.530
it's equal to this f'(c).
00:37:35.530 --> 00:37:38.230
So that means it's less
than or equal to the largest
00:37:38.230 --> 00:37:44.280
possible value on the-- largest
value you can get, for sure.
00:37:44.280 --> 00:37:48.110
And this is on the
whole interval.
00:37:48.110 --> 00:37:49.980
And I'm going to
include the ends,
00:37:49.980 --> 00:37:54.050
because when you take a max it's
sometimes achieved at the ends.
00:37:54.050 --> 00:37:58.770
And similarly, because it's
f'(c), it's definitely bigger
00:37:58.770 --> 00:38:07.030
than the min on this
same interval here.
00:38:07.030 --> 00:38:13.420
This is all you can say based
on the mean value theorem.
00:38:13.420 --> 00:38:14.620
All you know is this.
00:38:14.620 --> 00:38:17.710
And colloquially,
what that means
00:38:17.710 --> 00:38:25.690
is that the average speed
is between the maximum
00:38:25.690 --> 00:38:29.000
and the minimum.
00:38:29.000 --> 00:38:31.450
Not very surprising.
00:38:31.450 --> 00:38:33.120
The mean value
theorem is supposed
00:38:33.120 --> 00:38:36.480
to be very intuitively obvious.
00:38:36.480 --> 00:38:39.370
It's saying the average
speed is trapped
00:38:39.370 --> 00:38:42.850
between the maximum speed
and the minimum speed.
00:38:42.850 --> 00:38:44.830
For sure, that's
something, that's
00:38:44.830 --> 00:38:48.340
why-- incidentally this
wasn't really proved when
00:38:48.340 --> 00:38:50.350
Newton and Leibniz were around.
00:38:50.350 --> 00:38:52.860
But, let's write this
so that you can read it.
00:38:52.860 --> 00:39:01.350
Average speed is between
the max and the min.
00:39:01.350 --> 00:39:04.960
But nobody had any trouble,
they didn't disbelieve it
00:39:04.960 --> 00:39:09.750
because it's a
very natural thing.
00:39:09.750 --> 00:39:16.980
Now if, for example, I take any
kind of linear approximation,
00:39:16.980 --> 00:39:25.670
say, for instance, e^x
is approximately 1 + x.
00:39:25.670 --> 00:39:30.620
Then I'm making the guess-- no,
I don't want to say this yet.
00:39:30.620 --> 00:39:35.830
That's not going to explain
it to you well enough.
00:39:35.830 --> 00:39:38.690
What we're saying, so this
is the mean value here.
00:39:38.690 --> 00:39:40.840
This is what the mean
value theorem says.
00:39:40.840 --> 00:39:47.270
And here's the
linear approximation.
00:39:47.270 --> 00:39:52.140
The linear approximation is
saying that the average speed
00:39:52.140 --> 00:39:59.550
is approximately the
initial speed, or possibly
00:39:59.550 --> 00:40:01.900
the final speed.
00:40:01.900 --> 00:40:07.140
So if a is the left endpoint,
then it's the initial speed.
00:40:07.140 --> 00:40:09.630
If it happens to be the right
endpoint, if the value of x
00:40:09.630 --> 00:40:13.460
is to the left then
it's the final speed.
00:40:13.460 --> 00:40:16.640
So those are the-- so you can
see it's approximately right.
00:40:16.640 --> 00:40:19.180
Because the speed, when
you're on a short interval,
00:40:19.180 --> 00:40:20.530
shouldn't be varying very much.
00:40:20.530 --> 00:40:22.696
The max and the min should
be pretty close together.
00:40:22.696 --> 00:40:25.630
So that's why the linear
approximation is reasonable.
00:40:25.630 --> 00:40:27.800
And this is telling
you absolutely,
00:40:27.800 --> 00:40:34.140
it's no less than the min
and no more than the max.
00:40:34.140 --> 00:40:34.640
Yeah.
00:40:34.640 --> 00:40:40.965
STUDENT: [INAUDIBLE]
00:40:40.965 --> 00:40:42.090
PROFESSOR: The little kink?
00:40:42.090 --> 00:40:46.074
STUDENT: [INAUDIBLE]
00:40:46.074 --> 00:40:47.740
PROFESSOR: If you
approach from the top.
00:40:47.740 --> 00:40:50.507
So if it's still under here
I can show you it again.
00:40:50.507 --> 00:40:51.590
Oh yeah, it's still there.
00:40:51.590 --> 00:40:52.090
Good.
00:40:52.090 --> 00:40:54.630
STUDENT: [INAUDIBLE]
00:40:54.630 --> 00:40:56.830
PROFESSOR: Oh, the one
with the wiggle on top?
00:40:56.830 --> 00:40:57.872
Yeah, this one you can't.
00:40:57.872 --> 00:40:59.704
Because there's nothing
to touch and it also
00:40:59.704 --> 00:41:02.160
fails from the bottom because
there's this bad point.
00:41:02.160 --> 00:41:04.217
From the top, it could work.
00:41:04.217 --> 00:41:05.550
It can certainly work both ways.
00:41:05.550 --> 00:41:07.760
So, for example.
00:41:07.760 --> 00:41:09.940
See if you're a
machine, you maybe
00:41:09.940 --> 00:41:11.310
don't have a way of doing this.
00:41:11.310 --> 00:41:14.330
But if you're a human being
you can spot all the places.
00:41:14.330 --> 00:41:17.990
There are a bunch of spots
where the slope is right.
00:41:17.990 --> 00:41:20.770
And it's perfectly OK.
00:41:20.770 --> 00:41:21.780
All of them work.
00:41:21.780 --> 00:41:25.139
STUDENT: [INAUDIBLE]
00:41:25.139 --> 00:41:26.930
PROFESSOR: It's not
that the c is the same.
00:41:26.930 --> 00:41:29.470
It's just we've now found
one, two, three, four,
00:41:29.470 --> 00:41:31.160
five c's for which it works.
00:41:31.160 --> 00:41:40.390
STUDENT: [INAUDIBLE] PROFESSOR:
If you're asked to find a c,
00:41:40.390 --> 00:41:44.897
so first of all that's
kind of a phony question.
00:41:44.897 --> 00:41:46.730
There are some questions
on your problem set
00:41:46.730 --> 00:41:48.340
which ask you to find a c.
00:41:48.340 --> 00:41:51.850
That actually is struggling
to get you to understand what
00:41:51.850 --> 00:41:54.310
the statement of the
mean value theorem is,
00:41:54.310 --> 00:41:58.890
but you should not pay a lot of
attention to those questions.
00:41:58.890 --> 00:42:01.870
They're not very impressive.
00:42:01.870 --> 00:42:04.860
But, of course, you would have
to find all the-- if it asked
00:42:04.860 --> 00:42:06.110
you to find one, you find one.
00:42:06.110 --> 00:42:10.950
If you can find some more, fine.
00:42:10.950 --> 00:42:13.470
You can pick whichever
one you want.
00:42:13.470 --> 00:42:16.560
Mean value theorem
just doesn't care.
00:42:16.560 --> 00:42:18.190
The mean value
theorem doesn't care
00:42:18.190 --> 00:42:21.060
because actually, the mean
value theorem is never
00:42:21.060 --> 00:42:28.020
used except to-- in real life,
except in this context here.
00:42:28.020 --> 00:42:31.225
You can never nail
down which c it
00:42:31.225 --> 00:42:33.140
is, so the only
thing you can say
00:42:33.140 --> 00:42:35.980
is that you're going slower
than the maximum speed
00:42:35.980 --> 00:42:40.414
and faster than
the minimum speed.
00:42:40.414 --> 00:42:41.330
Sorry, say that again?
00:42:41.330 --> 00:42:47.057
STUDENT: [INAUDIBLE]
00:42:47.057 --> 00:42:48.890
PROFESSOR: If you're
asked for a specific c,
00:42:48.890 --> 00:42:51.080
you have to find a specific c.
00:42:51.080 --> 00:42:53.070
And it has to be in the range.
00:42:53.070 --> 00:43:04.420
In between, it
has to be in here.
00:43:04.420 --> 00:43:07.880
So now I want to tell you about
another kind of application,
00:43:07.880 --> 00:43:11.100
which is really just
a consequence of what
00:43:11.100 --> 00:43:22.070
I've described here.
00:43:22.070 --> 00:43:26.050
I should emphasize, by
the way, this, probably,
00:43:26.050 --> 00:43:27.420
should be doing this.
00:43:27.420 --> 00:43:32.120
I guess we've never
used this color here.
00:43:32.120 --> 00:43:32.950
It's popular.
00:43:32.950 --> 00:43:33.640
This is pink.
00:43:33.640 --> 00:43:35.480
So this one is so good.
00:43:35.480 --> 00:43:47.360
So since we're going to do this.
00:43:47.360 --> 00:43:50.960
So the reason why the
exclamation points
00:43:50.960 --> 00:43:54.150
are temporary, this is
such an obvious fact.
00:43:54.150 --> 00:43:57.857
But this is the way
that you're going
00:43:57.857 --> 00:43:59.440
to want to use the
mean value theorem,
00:43:59.440 --> 00:44:01.650
and this is the only way
you need to understand
00:44:01.650 --> 00:44:02.670
the mean value theorem.
00:44:02.670 --> 00:44:06.520
On your test, or ever
in your whole life.
00:44:06.520 --> 00:44:10.630
So this is the way
it will be used.
00:44:10.630 --> 00:44:17.140
As I will make very clear
when we review for the exam.
00:44:17.140 --> 00:44:18.714
In practice what
happens is you even
00:44:18.714 --> 00:44:21.130
forget about the mean value
theorem, and what you remember
00:44:21.130 --> 00:44:24.360
is these three properties here.
00:44:24.360 --> 00:44:26.000
Which are themselves
consequences
00:44:26.000 --> 00:44:27.720
of the mean value theorem.
00:44:27.720 --> 00:44:31.950
So these are the ones that
I want to illustrate now.
00:44:31.950 --> 00:44:35.710
In my next discussion here.
00:44:35.710 --> 00:44:42.420
I'm just going to talk
about inequalities.
00:44:42.420 --> 00:44:46.930
Inequalities are relationships
between functions.
00:44:46.930 --> 00:44:50.170
And I'm going to prove a couple
of them using the properties
00:44:50.170 --> 00:44:52.300
over there, the
properties that functions
00:44:52.300 --> 00:44:56.900
with positive derivatives
are increasing.
00:44:56.900 --> 00:44:58.900
Here's an example.
00:44:58.900 --> 00:45:08.390
e^x > 1 + x, where x > 0.
00:45:08.390 --> 00:45:10.940
The proof is the following.
00:45:10.940 --> 00:45:16.070
I consider-- So here's a proof.
00:45:16.070 --> 00:45:21.550
I consider the function f(x),
which is the difference.
00:45:21.550 --> 00:45:22.490
e^x - (1+x).
00:45:27.360 --> 00:45:37.140
I observe that it starts at f(0)
equal to, well, that's e^0 -
00:45:37.140 --> 00:45:42.110
(1+0), which is 0.
00:45:42.110 --> 00:45:45.510
And, it keeps on going.
00:45:45.510 --> 00:45:50.310
f'(x) is e^x, if I differentiate
here, the 1 goes away.
00:45:50.310 --> 00:45:54.117
I get minus 1.
00:45:54.117 --> 00:45:55.700
That's the derivative
of the function.
00:45:55.700 --> 00:46:03.750
And this function, because e^x
> 1 for x positive, is positive.
00:46:03.750 --> 00:46:07.095
As x gets bigger and bigger,
this rate of increase
00:46:07.095 --> 00:46:08.290
is positive.
00:46:08.290 --> 00:46:13.940
And therefore, three
dots, that's therefore,
00:46:13.940 --> 00:46:23.350
f(x) is bigger than its
starting place, for x > 0.
00:46:23.350 --> 00:46:27.100
If it's increasing, then
that's-- in particular,
00:46:27.100 --> 00:46:28.730
it's increasing starting from 0.
00:46:28.730 --> 00:46:30.340
So this is true.
00:46:30.340 --> 00:46:36.000
Now, all I have to do is read
what this inequality says.
00:46:36.000 --> 00:46:39.860
And what it says is that
e^x, just plug in for f(x),
00:46:39.860 --> 00:46:45.130
which is right here, minus (1+x)
is greater than the starting
00:46:45.130 --> 00:46:48.590
value, which was 0.
00:46:48.590 --> 00:46:52.050
Now, I put the thing that's
negative on the other side.
00:46:52.050 --> 00:47:01.550
So that's the same
thing as e^x > 1 + x.
00:47:01.550 --> 00:47:04.330
That's a typical inequality.
00:47:04.330 --> 00:47:11.150
And now, we'll use
this principle again.
00:47:11.150 --> 00:47:12.900
Oh gee, I erased
the wrong thing.
00:47:12.900 --> 00:47:15.490
I erased the statement
and not the proof.
00:47:15.490 --> 00:47:23.340
Well, hide the proof.
00:47:23.340 --> 00:47:27.500
The next thing I want to prove
to you is that e^x > 1 + x +
00:47:27.500 --> 00:47:28.000
x^2 / 2.
00:47:33.090 --> 00:47:34.300
So, how do I do that?
00:47:34.300 --> 00:47:42.174
I introduce a function g(x),
which is e^x minus this.
00:47:42.174 --> 00:47:44.340
And now, I'm just going to
do exactly the same thing
00:47:44.340 --> 00:47:45.780
I did before.
00:47:45.780 --> 00:47:51.870
Which is, I get started
with g(0), which is 1 - 1.
00:47:51.870 --> 00:47:53.680
Which is 0.
00:47:53.680 --> 00:48:00.690
And g'(x) is e^x minus - now,
look at what happens when I
00:48:00.690 --> 00:48:03.590
differentiate this.
00:48:03.590 --> 00:48:04.870
The 1 goes away.
00:48:04.870 --> 00:48:10.600
The x gives me a 1, and the
x^2 / 2 gives me a plus x.
00:48:10.600 --> 00:48:18.190
And this one is positive for
x > 0, because of step 1.
00:48:18.190 --> 00:48:21.840
Because of the previous
one that I did.
00:48:21.840 --> 00:48:28.340
So this one is increasing.
g is increasing.
00:48:28.340 --> 00:48:33.350
Which says that g(x) > g(0).
00:48:33.350 --> 00:48:36.620
And if you just read that
off, it's exactly the same
00:48:36.620 --> 00:48:41.690
as our inequality here.
e^x > 1 + x + x^2 / 2.
00:48:48.270 --> 00:48:54.010
Now, you can keep on going
with this essentially forever.
00:48:54.010 --> 00:48:58.510
And let me just write
down what you get.
00:48:58.510 --> 00:49:04.772
You get e^x is greater
than 1 plus x plus x^2 / 2,
00:49:04.772 --> 00:49:06.480
the next one turns
out to be x^3 / (3*2).
00:49:10.460 --> 00:49:11.730
x^4 / (4*3*2).
00:49:14.850 --> 00:49:16.820
And you can do
whatever you want.
00:49:16.820 --> 00:49:19.490
You can do others.
00:49:19.490 --> 00:49:22.860
And this is like the
tortoise and the hare.
00:49:22.860 --> 00:49:27.740
This is the tortoise, and this
is the hare, it's always ahead.
00:49:27.740 --> 00:49:31.910
But eventually, if you go
infinitely far, it catches up.
00:49:31.910 --> 00:49:38.190
So this turns out to be exactly
equal to e^x in the limit.
00:49:38.190 --> 00:49:41.280
And we'll talk about that
maybe at the end of the course.