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PROFESSOR: What we're going
to talk about today is a
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continuation of last time.
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I want to review Newton's
method because I want to talk
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00:00:29 --> 00:00:41
to you about its accuracy.
13
00:00:41 --> 00:00:46
So if you remember, the way
Newton's method works is this.
14
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If you have a curve and
you want to know whether
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it crosses the axis.
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And you don't know where this
point is, this point which
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I'll call x here, what you
do is you take a guess.
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00:01:01 --> 00:01:03
Maybe you take a point x0 here.
19
00:01:03 --> 00:01:06
And then you go down to this
point on the graph. and
20
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you draw the tangent line.
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I'll draw these in a couple of
different colors so that you
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can see the difference
between them.
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So here's a tangent line.
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It's coming out like that.
25
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And that one is going
to get a little closer
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to our target point.
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But now the trick is, and this
is rather hard to see because
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the scale gets small incredibly
fast, is that if you go right
29
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up from that, and you do
this same trick over again.
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00:01:34 --> 00:01:38
That is, this is your second
guess, x1, and now you draw
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the second tangent line.
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Which is going to
come down this way.
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That's really close.
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You can see here on the
chalkboard, it's practically
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the same as the dot of x.
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So that's the next guess.
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00:01:54 --> 00:01:56
Which is x2.
38
00:01:56 --> 00:02:03
And I want to analyze,
now, how close it gets.
39
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And just describe to
you how it works.
40
00:02:05 --> 00:02:09
So let me just remind you
of the formulas, too.
41
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It's worth having
them in your head.
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So the formula for the
next one is this.
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And then the idea is just
to repeat this process.
44
00:02:23 --> 00:02:28
Which has a fancy name, which
is in algorithms, which is
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00:02:28 --> 00:02:31
to iterate, if you like.
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00:02:31 --> 00:02:32
So we repeat the process.
47
00:02:32 --> 00:02:35
And that means, for example,
we generate x2 from x1
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by the same formula.
49
00:02:41 --> 00:02:43
And we did this last time.
50
00:02:43 --> 00:02:47
And, more generally, the n +
1st is generated from the nth
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guess, by this formula here.
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So what I'd like to do is just
draw the picture of one step
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a little bit more closely.
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So I want to blow up
the picture, which
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is above me there.
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That's a little too high.
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Where are my erasers?
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Got to get it a little lower
than that, since I'm going
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to depict everything
above the line here.
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So here's my curve coming down.
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And suppose that x1 is here,
so this is directly above
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it is this point here.
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And then as I drew it,
this green tangent
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coming down like that.
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It's a little bit closer, and
this was the place, x2, and
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then here was x, our target,
which is where the curve
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crosses as opposed to the
straight tangent line crossing.
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So that's the picture that I
want you to keep in mind.
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And now, we're just going
to do a very qualitative
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kind of error analysis.
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So here's our error analysis.
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And we're starting out, the
distance between x1 and x is
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what we want to measure.
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In other words, how close we
are to where we're heading.
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And so I've called that, I'm
going to call that Error 1.
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That's x - x1.
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In absolute value.
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And then, E2 would be x -
x2, in absolute value.
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And so forth.
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And, last time, when I was
estimating the size of this, so
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En would be whatever it was.
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Last time, remember, we did
it for a specific case.
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So last time, I actually
wrote down the numbers.
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And they were these numbers,
maybe you could call them En,
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which was the absolute value
of square root of 5 - xn.
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These are the sizes that I
was writing down last time.
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And I just want to talk about
in general what to expect.
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That worked amazingly well, and
I want to show you that that's
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true pretty much in general.
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00:05:16 --> 00:05:23
So the first distance, again,
is E1, is this distance here.
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That's the E1.
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And then this shorter distance,
here, this little bit,
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which I'll mark maybe
in green, is E2.
94
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So how much shorter
is E1 than E2?
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00:05:40 --> 00:05:44
Well, the idea is
pretty simple.
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00:05:44 --> 00:05:47
It's that if this distance in
this vertical, distances are
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probably about the same as
the perpendicular distance.
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And this is basically the
situation of a curve
99
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touching a tangent line.
100
00:05:54 --> 00:05:58
Then the separation is
going to be quadratic.
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00:05:58 --> 00:06:00
And that's basically
what's going to happen.
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00:06:00 --> 00:06:04
So, in other words the distance
E2 is going to be about the
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square of the distance E1.
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And that's really the only
feature of this that
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I want to point out.
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So, approximately, this
is the situation that
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we're going to get.
108
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And so what that means is, and
maybe thinking from last time,
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00:06:23 --> 00:06:25
what we had was something
roughly like this.
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You have an E0, you have an
E1, you have an E2, you
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have an E3, and so forth.
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00:06:31 --> 00:06:34
Maybe I'll write down E4 here.
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And last time, this
was about 10 ^ - 1.
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So the expectation based on
this rule is that the next
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error's the square of
the previous one.
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So that's 10 ^ - 2.
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The next one is the square
of the previous one.
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So that's 10 ^ - 4.
119
00:06:49 --> 00:06:52
And the next one is the square
of that, that's 10 ^ - 8.
120
00:06:52 --> 00:06:55
And this one is 10 ^ - 16.
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00:06:55 --> 00:06:59
So the thing that's impressive
about this list of numbers is
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you can see that the number
of digits of accuracy
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00:07:02 --> 00:07:09
doubles at each stage.
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Accuracy doubles at each step.
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00:07:20 --> 00:07:23
The number of digits of
accuracy doubles at each step.
126
00:07:23 --> 00:07:28
So, very, very quickly you get
past the accuracy of your
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00:07:28 --> 00:07:31
calculator, as you saw
on your problem set.
128
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And this thing
works beautifully.
129
00:07:34 --> 00:07:40
So, let me just summarize
by saying that Newton's
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00:07:40 --> 00:07:49
method works very well.
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By which I mean
this kind of rate.
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And I want to be just
slightly specific.
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If, there's really two
conditions disguised in
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00:07:59 --> 00:08:01
this, that are going on.
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One is that f' has
to be, not to be.
136
00:08:04 --> 00:08:09
To be not small.
137
00:08:09 --> 00:08:19
And f'' has to be not too big.
138
00:08:19 --> 00:08:21
That's roughly speaking what's
going on. i'll explain
139
00:08:21 --> 00:08:23
these in just a second.
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00:08:23 --> 00:08:34
And x0 starts nearby.
141
00:08:34 --> 00:08:37
Nearby the target value x.
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00:08:37 --> 00:08:42
So that's really
what's going on here.
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00:08:42 --> 00:08:44
So let me just
illustrate to you.
144
00:08:44 --> 00:08:48
So I'm not going to explain
this, except to say the reason
145
00:08:48 --> 00:08:52
why this second derivative gets
involved is that it's how
146
00:08:52 --> 00:08:55
curved the curve is, that
how far away you get.
147
00:08:55 --> 00:08:57
If the second derivative
were 0, that would be
148
00:08:57 --> 00:08:58
the best possible case.
149
00:08:58 --> 00:09:00
Then we would get
it on the nose.
150
00:09:00 --> 00:09:03
If the second derivative is
not too big, that means the
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00:09:03 --> 00:09:05
quadratic part is not too big.
152
00:09:05 --> 00:09:07
So we don't get away very
far from the green line
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00:09:07 --> 00:09:14
to the curve itself.
154
00:09:14 --> 00:09:17
The other thing to say
is, as I said, that x0
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00:09:17 --> 00:09:19
needs to start nearby.
156
00:09:19 --> 00:09:22
So I'll explain that by
explaining what maybe
157
00:09:22 --> 00:09:23
could go wrong.
158
00:09:23 --> 00:09:39
So the ways the method can
fail, and one example which
159
00:09:39 --> 00:09:45
actually would have happened in
our example from last time,
160
00:09:45 --> 00:09:50
which was y = x^2 - 5, is
suppose we'd started
161
00:09:50 --> 00:09:54
x0 over here.
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00:09:54 --> 00:09:57
Then this thing would have gone
off to the left, and we would
163
00:09:57 --> 00:10:03
have landed on not the square
root of 5 but the other root.
164
00:10:03 --> 00:10:11
So if it's too far away,
then we get the wrong root.
165
00:10:11 --> 00:10:15
So that's an example,
explaining that the x0 needs
166
00:10:15 --> 00:10:18
to start near the root
that we're talking about.
167
00:10:18 --> 00:10:22
Otherwise, the method
doesn't know which root
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00:10:22 --> 00:10:22
you're asking for.
169
00:10:22 --> 00:10:24
It only knows where
you started.
170
00:10:24 --> 00:10:27
So it may go off to
the wrong place.
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OK, it can't read your mind.
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00:10:34 --> 00:10:35
Yes, question.
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00:10:35 --> 00:10:42
STUDENT: [INAUDIBLE]
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PROFESSOR: Oh, good question.
175
00:10:44 --> 00:10:48
So the question was,
what if the first error
176
00:10:48 --> 00:10:49
is larger than 1.
177
00:10:49 --> 00:10:51
Are you in trouble?
178
00:10:51 --> 00:10:56
And the answer is,
absolutely, yes.
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00:10:56 --> 00:10:58
If you have quadratic
behavior, you can see.
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00:10:58 --> 00:11:01
If you have a quadratic
nearby, it's pretty close
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00:11:01 --> 00:11:02
to the straight line.
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00:11:02 --> 00:11:05
But far away, a parabola is
miles from a straight line.
183
00:11:05 --> 00:11:08
It's way, way, way far away.
184
00:11:08 --> 00:11:14
So if you're foolish enough to
start over here, you may have
185
00:11:14 --> 00:11:17
some trouble making progress.
186
00:11:17 --> 00:11:20
Actually, it isn't - when I,
that little wiggle there
187
00:11:20 --> 00:11:22
just meant proportional to.
188
00:11:22 --> 00:11:24
In fact, in the particular
case of a parabola, it
189
00:11:24 --> 00:11:25
manages to get back.
190
00:11:25 --> 00:11:27
It saves itself.
191
00:11:27 --> 00:11:30
But there's no guarantee
of that sort of thing.
192
00:11:30 --> 00:11:32
You really do want to
start reasonably close.
193
00:11:32 --> 00:11:33
Yep.
194
00:11:33 --> 00:11:40
STUDENT: [INAUDIBLE]
195
00:11:40 --> 00:11:42
PROFESSOR: What you have to
do is you have to watch out.
196
00:11:42 --> 00:11:46
That is, it's hard to know what
assumptions to make about x0.
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00:11:46 --> 00:11:50
You plug it into the machine
and you see what you get.
198
00:11:50 --> 00:11:53
And either it works
or it doesn't.
199
00:11:53 --> 00:11:55
You can tell that it's marching
toward a specific place, and
200
00:11:55 --> 00:11:58
you can tell that that place
probably is a 0, usually.
201
00:11:58 --> 00:12:00
But maybe it's not the one
you were looking for.
202
00:12:00 --> 00:12:01
So in other words, you
have to use your head.
203
00:12:01 --> 00:12:05
You run the program and
then you see what it does.
204
00:12:05 --> 00:12:07
And if you're lucky - the
problem is, if you have no
205
00:12:07 --> 00:12:12
idea where the 0 is, you may
just wander around forever.
206
00:12:12 --> 00:12:14
As we'll see in a second.
207
00:12:14 --> 00:12:20
So the next example
here is the following.
208
00:12:20 --> 00:12:24
I said that f' has to
be not too small.
209
00:12:24 --> 00:12:27
There's a real catastrophe
hiding just inside
210
00:12:27 --> 00:12:28
this picture.
211
00:12:28 --> 00:12:30
Which is the transition between
when you find the positive root
212
00:12:30 --> 00:12:32
and when you find the
negative root here.
213
00:12:32 --> 00:12:35
Which is, if you're
right down here.
214
00:12:35 --> 00:12:38
If you were foolish enough to
get 0, then what's going to
215
00:12:38 --> 00:12:41
happen is your tangent
line is horizontal.
216
00:12:41 --> 00:12:44
It doesn't even meet the axis.
217
00:12:44 --> 00:12:47
So in the formula, you can
see that's a catastrophe.
218
00:12:47 --> 00:12:53
Because there's an f'
in the denominator.
219
00:12:53 --> 00:12:53
So that's 0.
220
00:12:53 --> 00:12:55
That's undefined.
221
00:12:55 --> 00:12:57
It's not surprising, it's
consistent that the parallel
222
00:12:57 --> 00:12:59
line doesn't meet the axis.
223
00:12:59 --> 00:13:01
And you have no x1.
224
00:13:01 --> 00:13:09
So you had, so if you like,
another point here is that
225
00:13:09 --> 00:13:12
f' = 0 is a disaster.
226
00:13:12 --> 00:13:24
A disaster for the method.
227
00:13:24 --> 00:13:34
Because the next, so say, if f
(x0) = 0, then x1 is undefined.
228
00:13:34 --> 00:13:39
And finally, there's one other
weird thing that can happen.
229
00:13:39 --> 00:13:43
Which is, which I'll just draw
a picture of schematically.
230
00:13:43 --> 00:13:47
Which you can get
from a wiggle.
231
00:13:47 --> 00:13:49
So this wiggle has three roots.
232
00:13:49 --> 00:13:52
The way I've drawn it.
233
00:13:52 --> 00:13:56
And it can be that you can
start over here with your x0.
234
00:13:56 --> 00:14:01
And draw your tangent line
and go over here to an x1.
235
00:14:01 --> 00:14:05
And then that tangent line will
take you right back to the x0.
236
00:14:05 --> 00:14:09
I didn't draw it quite right,
but that's about right.
237
00:14:09 --> 00:14:11
So it goes over like this.
238
00:14:11 --> 00:14:13
So let me draw the two
tangent lines, so that
239
00:14:13 --> 00:14:14
you can see it properly.
240
00:14:14 --> 00:14:16
Sorry, I messed it up.
241
00:14:16 --> 00:14:18
So here are the two
tangent lines.
242
00:14:18 --> 00:14:20
This guy and this guy.
243
00:14:20 --> 00:14:25
And it just goes back and
forth. x0 cycles to x1,
244
00:14:25 --> 00:14:27
and x1 goes back to x0.
245
00:14:27 --> 00:14:30
We have a cycle.
246
00:14:30 --> 00:14:32
And it never goes anywhere.
247
00:14:32 --> 00:14:35
This is, the grass
is always greener.
248
00:14:35 --> 00:14:37
It's over here, it thinks, oh,
I really would prefer to go to
249
00:14:37 --> 00:14:40
this 0 and then it thinks
oh, I want to go back.
250
00:14:40 --> 00:14:43
And it goes back and forth,
and back and forth.
251
00:14:43 --> 00:14:47
Grass is always greener on
the other side of the fence.
252
00:14:47 --> 00:14:50
Never, never gets anywhere.
253
00:14:50 --> 00:14:52
So those are the sorts of
things that can go wrong
254
00:14:52 --> 00:14:53
with Newton's method.
255
00:14:53 --> 00:14:55
Nevertheless, it's fantastic.
256
00:14:55 --> 00:14:59
It works very well, in
a lot of situations.
257
00:14:59 --> 00:15:02
And solves basically any
equation that you can
258
00:15:02 --> 00:15:11
imagine, numerically.
259
00:15:11 --> 00:15:12
Next we're going to move on.
260
00:15:12 --> 00:15:13
We're going to move on
to something which is
261
00:15:13 --> 00:15:15
a little theoretical.
262
00:15:15 --> 00:15:18
Which is the mean
value theorem.
263
00:15:18 --> 00:15:24
And that will allow us in just
a day or so to launch into the
264
00:15:24 --> 00:15:27
ideas of integration, which
is the whole second
265
00:15:27 --> 00:15:31
half of the course.
266
00:15:31 --> 00:15:50
So let's get started with that.
267
00:15:50 --> 00:15:57
The mean value theorem will
henceforth be abbreviated MVT.
268
00:15:57 --> 00:15:59
So I don't have to write
quite as much every
269
00:15:59 --> 00:16:03
time I refer to it.
270
00:16:03 --> 00:16:07
The mean values theorem,
colloquially, says
271
00:16:07 --> 00:16:09
the following.
272
00:16:09 --> 00:16:23
If you go from Boston to LA,
which I think a lot of Red Sox
273
00:16:23 --> 00:16:31
fans are going to want to do
soon, so that's 3,000 miles.
274
00:16:31 --> 00:16:51
In 6 hours, then at some
time you are going
275
00:16:51 --> 00:16:55
at a certain speed.
276
00:16:55 --> 00:17:01
The average of this speed.
277
00:17:01 --> 00:17:08
Average, so speed, which
in this case is what?
278
00:17:08 --> 00:17:10
So we're going at
the average speed.
279
00:17:10 --> 00:17:17
That's 3,000 miles times
6 hours, so that's
280
00:17:17 --> 00:17:21
500 miles per hour.
281
00:17:21 --> 00:17:23
Exactly.
282
00:17:23 --> 00:17:26
So sometime on your journey, of
course, some of the time you're
283
00:17:26 --> 00:17:28
going more than 500 miles an
hour, sometimes you
284
00:17:28 --> 00:17:29
are going less.
285
00:17:29 --> 00:17:32
And some time you must've
been going 500 miles
286
00:17:32 --> 00:17:35
an hour exactly.
287
00:17:35 --> 00:17:37
That's the mean value theorem.
288
00:17:37 --> 00:17:41
The reason why it's called mean
value theorem is that word mean
289
00:17:41 --> 00:17:55
is the same as the
word average.
290
00:17:55 --> 00:18:08
So now I'm going to state it in
math symbols, the same theorem.
291
00:18:08 --> 00:18:10
And it's a formula.
292
00:18:10 --> 00:18:19
It says that the difference
quotient, so this is the
293
00:18:19 --> 00:18:23
distance traveled divided
by the time elapsed.
294
00:18:23 --> 00:18:28
That's the average speed, is
equal to the infinitesimal
295
00:18:28 --> 00:18:35
speed for some time in between.
296
00:18:35 --> 00:18:46
So some c, which is
in between a and b.
297
00:18:46 --> 00:18:48
I'm not quite done.
298
00:18:48 --> 00:18:53
It's a real theorem,
it has hypotheses.
299
00:18:53 --> 00:18:57
I've told you the conclusion
first, but there are some
300
00:18:57 --> 00:18:59
hypotheses, they're
straightforward hypotheses.
301
00:18:59 --> 00:19:09
Provided f is differentiable;
that is, it has a derivative
302
00:19:09 --> 00:19:12
in the interval a < x < b.
303
00:19:12 --> 00:19:29
And continuous in a < or =
x, less than or equal to.
304
00:19:29 --> 00:19:32
There has to be a sense that
you can make out of the speed,
305
00:19:32 --> 00:19:36
or the rate of change of f
at each intermediate point.
306
00:19:36 --> 00:19:40
And in order for the values
at the ends to make sense,
307
00:19:40 --> 00:19:41
it has to be continuous.
308
00:19:41 --> 00:19:46
There has to be a link between
the values at the ends and
309
00:19:46 --> 00:19:47
what's going on in between.
310
00:19:47 --> 00:19:50
If it were discontinuous, there
would be no relation between
311
00:19:50 --> 00:19:55
the left and right values and
the rest of the function.
312
00:19:55 --> 00:20:00
So here's the theorem,
conclusion and its hypothesis.
313
00:20:00 --> 00:20:11
And it means what I said
more colloquially up above.
314
00:20:11 --> 00:20:14
Now, I'm going to prove
this theorem immediately.
315
00:20:14 --> 00:20:18
At least, give a geometric
intuitive argument, which is
316
00:20:18 --> 00:20:22
not very different from the
one that's given in a very
317
00:20:22 --> 00:20:26
systematic treatment.
318
00:20:26 --> 00:20:34
So here's the proof of
the mean value theorem.
319
00:20:34 --> 00:20:36
It's really just a picture.
320
00:20:36 --> 00:20:42
So here's a place, and here's
another place on the graph.
321
00:20:42 --> 00:20:47
And the graph is going along
like this, let's say.
322
00:20:47 --> 00:20:50
And this line here
is the secant line.
323
00:20:50 --> 00:20:55
So this is (a, f (
a )) down here.
324
00:20:55 --> 00:20:59
And this is (b, f
( b )) up there.
325
00:20:59 --> 00:21:03
And this segment is the secant,
its slope is the slope
326
00:21:03 --> 00:21:04
that we're aiming for.
327
00:21:04 --> 00:21:08
The slope of that line is
the left-hand side of
328
00:21:08 --> 00:21:11
this formula here.
329
00:21:11 --> 00:21:14
So we need to find
something with that slope.
330
00:21:14 --> 00:21:16
And what we need to find is a
tangent line with that slope,
331
00:21:16 --> 00:21:18
because what's on the
right-hand side is the
332
00:21:18 --> 00:21:20
slip of a tangent line.
333
00:21:20 --> 00:21:22
So here's how we construct it.
334
00:21:22 --> 00:21:27
We take a parallel
line, down here.
335
00:21:27 --> 00:21:30
And then we just translate
it up, leaving it
336
00:21:30 --> 00:21:32
parallel, we move it up.
337
00:21:32 --> 00:21:34
Towards this one.
338
00:21:34 --> 00:21:38
Until it touches.
339
00:21:38 --> 00:21:43
And where it touches, at this
point of tangency, down there,
340
00:21:43 --> 00:21:47
I've just found my value of c.
341
00:21:47 --> 00:21:49
And you can see that if the
tangent line is parallel to
342
00:21:49 --> 00:21:53
this line, that's exactly
the equation we want.
343
00:21:53 --> 00:21:59
So this thing has slope f' (c).
344
00:21:59 --> 00:22:07
And this other one has slope
equal to this complicated
345
00:22:07 --> 00:22:15
expression, f ( b)
- f (a) / (b - a).
346
00:22:15 --> 00:22:19
That is almost the
end of the proof.
347
00:22:19 --> 00:22:25
There's one problem.
348
00:22:25 --> 00:22:29
So, again, we move a
parallel line up.
349
00:22:29 --> 00:22:43
Move up the parallel
line until it touches.
350
00:22:43 --> 00:22:46
There's a little subtlety here,
which I just want to emphasize.
351
00:22:46 --> 00:22:49
Which is that that dotted
line keeps on going here.
352
00:22:49 --> 00:22:52
But when we bring it up, we're
going to ignore what's
353
00:22:52 --> 00:22:54
happening outside of a.
354
00:22:54 --> 00:22:56
And beyond b, alright?
355
00:22:56 --> 00:23:01
So we're just going to ignore
the rest of the graph.
356
00:23:01 --> 00:23:06
But there is one thing
that can go wrong.
357
00:23:06 --> 00:23:16
So if it does not touch, then
the picture looks likes this.
358
00:23:16 --> 00:23:18
Here are the same two points.
359
00:23:18 --> 00:23:20
And the graph is all above.
360
00:23:20 --> 00:23:22
And we brought up our thing.
361
00:23:22 --> 00:23:23
And it went like that.
362
00:23:23 --> 00:23:27
So we didn't construct
a tangent line.
363
00:23:27 --> 00:23:29
If this happens.
364
00:23:29 --> 00:23:31
So we're in trouble,
in that point.
365
00:23:31 --> 00:23:37
In this situation, sorry.
366
00:23:37 --> 00:23:40
But there's a trick, which
is a straightforward trick.
367
00:23:40 --> 00:23:55
Then bring the tangent
lines down from the top.
368
00:23:55 --> 00:23:58
So parallel lines, sorry,
not tangent lines.
369
00:23:58 --> 00:24:06
Parallel lines.
370
00:24:06 --> 00:24:11
From above.
371
00:24:11 --> 00:24:16
So, that's the whole story.
372
00:24:16 --> 00:24:22
That's how we cook up
this point c, with
373
00:24:22 --> 00:24:43
the right properties.
374
00:24:43 --> 00:24:46
I want to point out just one
more theoretical thing.
375
00:24:46 --> 00:24:50
And then the rest, we're going
to be drawing conclusions.
376
00:24:50 --> 00:24:53
So there's one more theoretical
remark about the proof, which
377
00:24:53 --> 00:24:59
is something that is fairly
important to understand.
378
00:24:59 --> 00:25:02
When you understand a proof,
you should always be thinking
379
00:25:02 --> 00:25:06
about why the hypotheses
are necessary.
380
00:25:06 --> 00:25:08
Where do I use the hypothesis.
381
00:25:08 --> 00:25:10
And I want to give you an
example where the proof doesn't
382
00:25:10 --> 00:25:15
work to show you that the
hypothesis is an important one.
383
00:25:15 --> 00:25:17
So the example is
the following.
384
00:25:17 --> 00:25:21
I'll just take a function
which is two straight
385
00:25:21 --> 00:25:22
lines like this.
386
00:25:22 --> 00:25:28
And if you try to perform this
trick with these things, then
387
00:25:28 --> 00:25:32
it's going to come up and
it's going to touch here.
388
00:25:32 --> 00:25:35
But the problem is that
the tangent line is
389
00:25:35 --> 00:25:36
not defined here.
390
00:25:36 --> 00:25:39
There are lots of tangents,
and there's no derivative
391
00:25:39 --> 00:25:40
at this point.
392
00:25:40 --> 00:25:44
So the derivative
doesn't exist here.
393
00:25:44 --> 00:25:57
So this is the claim that one
bad point ruins the proof.
394
00:25:57 --> 00:26:07
We need f' to exist at all
so, f' ( x ) to exist
395
00:26:07 --> 00:26:14
at all x in between.
396
00:26:14 --> 00:26:30
Can't get away even with
one defective point.
397
00:26:30 --> 00:26:40
Now it's time to draw
some consequences.
398
00:26:40 --> 00:26:48
And the main consequence is
going to have to do with
399
00:26:48 --> 00:26:57
applications to graphing.
400
00:26:57 --> 00:27:01
But we'll see tomorrow and for
the rest of the course that
401
00:27:01 --> 00:27:03
this is even more significance.
402
00:27:03 --> 00:27:09
It's significant to all
the rest of Calculus.
403
00:27:09 --> 00:27:12
I'm going to list three
consequences which you're
404
00:27:12 --> 00:27:14
quite familiar with already.
405
00:27:14 --> 00:27:27
So, the first one is if f' is
positive, then f is increasing.
406
00:27:27 --> 00:27:40
And the second one is if f' is
negative, then f is decreasing.
407
00:27:40 --> 00:27:44
And the last one seems
like the simplest.
408
00:27:44 --> 00:27:48
But even this one alone is
the key to everything.
409
00:27:48 --> 00:28:03
If f' = 0, then f is constant.
410
00:28:03 --> 00:28:13
These are three consequences,
now, of the mean value theorem.
411
00:28:13 --> 00:28:17
And let me show you
how they're proved.
412
00:28:17 --> 00:28:22
I just told you that they were
true, maybe a while ago.
413
00:28:22 --> 00:28:27
And certainly I mentioned
the first two.
414
00:28:27 --> 00:28:29
The last one was so simple
that we maybe just
415
00:28:29 --> 00:28:30
swept it under the rug.
416
00:28:30 --> 00:28:36
You did use it on a problem
set, once or twice.
417
00:28:36 --> 00:28:39
But it turns out that this
actually requires proof,
418
00:28:39 --> 00:28:48
and we're going to give
the proof right now.
419
00:28:48 --> 00:28:51
The way that the proof
goes is simply to write
420
00:28:51 --> 00:28:54
down, to rewrite star.
421
00:28:54 --> 00:28:59
Rewrite our formula.
422
00:28:59 --> 00:29:10
Which says that f (b) - f
(a) / (b - a) = f' (c).
423
00:29:10 --> 00:29:14
And you see I've written it
from left to right here to say
424
00:29:14 --> 00:29:16
that the right-hand side
information about the
425
00:29:16 --> 00:29:18
derivative is going to be
giving the information
426
00:29:18 --> 00:29:19
about the function.
427
00:29:19 --> 00:29:22
That's the way I'm
going to read it.
428
00:29:22 --> 00:29:27
In order to express this,
though, I'm going to just
429
00:29:27 --> 00:29:30
rewrite it a couple
of times here.
430
00:29:30 --> 00:29:37
So here's f ( a ), multiplying
through by the denominator.
431
00:29:37 --> 00:29:40
And now I'm going to write it
in another customary form
432
00:29:40 --> 00:29:42
for the mean value theorem.
433
00:29:42 --> 00:29:51
Which is f ( b ) = f (
a ) + f' (c) ( b - a).
434
00:29:51 --> 00:29:53
So here's another version.
435
00:29:53 --> 00:29:55
I should probably have
put this one in the box
436
00:29:55 --> 00:29:59
to begin with anyway.
437
00:29:59 --> 00:30:03
And, just changing it
around algebraically,
438
00:30:03 --> 00:30:07
it's this fact here.
439
00:30:07 --> 00:30:13
They're the same thing.
440
00:30:13 --> 00:30:18
And now with the formula
written in this form, I
441
00:30:18 --> 00:30:24
claim that I can check
these three facts.
442
00:30:24 --> 00:30:26
Let's start with the first one.
443
00:30:26 --> 00:30:33
I'm going to set things
up always so that a < b.
444
00:30:33 --> 00:30:36
And that's the setup
of the theorem.
445
00:30:36 --> 00:30:42
And so that means that
b - a is positive.
446
00:30:42 --> 00:30:48
Which means that this factor
over here is a positive number.
447
00:30:48 --> 00:30:56
If f' is positive, which is
what happens in the first case,
448
00:30:56 --> 00:30:59
that's the assumption that
we're making, then this
449
00:30:59 --> 00:31:01
is a positive number.
450
00:31:01 --> 00:31:08
And so f( b ) > f( a ).
451
00:31:08 --> 00:31:09
Which means that
it's increasing.
452
00:31:09 --> 00:31:13
It goes up as the
value goes up.
453
00:31:13 --> 00:31:20
Similarly, if f' (c) is
negative, then this is a
454
00:31:20 --> 00:31:22
positive times a negative
number, this is negative.
455
00:31:22 --> 00:31:25
f ( b ) < f(a).
456
00:31:25 --> 00:31:37
So it goes the other way.
457
00:31:37 --> 00:31:39
Maybe I'll write this way.
458
00:31:39 --> 00:31:49
And finally, if f' (c) =
0, then f ( b ) = f(a).
459
00:31:49 --> 00:31:53
Which if you apply it to all
possible ends means if you can
460
00:31:53 --> 00:31:56
do it for every interval,
which you can't, then that
461
00:31:56 --> 00:31:57
means that f is constant.
462
00:31:57 --> 00:32:12
It never gets to change values.
463
00:32:12 --> 00:32:17
Well you might have believed
these facts already.
464
00:32:17 --> 00:32:20
But I just want to emphasize
to you that this turns out to
465
00:32:20 --> 00:32:25
be the one key link between
infinitesimals, between limits
466
00:32:25 --> 00:32:27
and these actual differences.
467
00:32:27 --> 00:32:30
Before, we were saying that
the difference quotient
468
00:32:30 --> 00:32:32
was approximately equal
to the derivative.
469
00:32:32 --> 00:32:35
Now we're saying that it's
exactly equal to a derivative.
470
00:32:35 --> 00:32:38
Although we don't know
exactly which point to use.
471
00:32:38 --> 00:32:47
It's some point in between.
472
00:32:47 --> 00:32:49
I'm going to be deducing some
other consequences in a
473
00:32:49 --> 00:32:52
second, but let me stop for
second to make sure that
474
00:32:52 --> 00:32:53
everybody's on board.
475
00:32:53 --> 00:32:56
Especially since I've finished
the blackboards here.
476
00:32:56 --> 00:32:59
Before we, everybody happy?
477
00:32:59 --> 00:33:00
One question.
478
00:33:00 --> 00:33:09
STUDENT: [INAUDIBLE]
479
00:33:09 --> 00:33:10
PROFESSOR: I'm just going to
repeat your question first.
480
00:33:10 --> 00:33:13
I'm a little bit confused,
you said, about what
481
00:33:13 --> 00:33:16
guarantees that there's
a point of tangency.
482
00:33:16 --> 00:33:19
That's what you said.
483
00:33:19 --> 00:33:21
So do you want to elaborate, or
do you want to want to stop
484
00:33:21 --> 00:33:23
with what you just send?
485
00:33:23 --> 00:33:24
What is it that confuses you?
486
00:33:24 --> 00:33:29
STUDENT: [INAUDIBLE]
487
00:33:29 --> 00:33:29
PROFESSOR: Yeah.
488
00:33:29 --> 00:33:43
STUDENT: [INAUDIBLE]
489
00:33:43 --> 00:33:46
PROFESSOR: So I'm not claiming
that there's only one point.
490
00:33:46 --> 00:33:48
This could wiggle a lot
of times and it maybe
491
00:33:48 --> 00:33:49
touches at ten places.
492
00:33:49 --> 00:33:54
In other words, it's OK with me
if it touches more than once.
493
00:33:54 --> 00:33:56
Then I just have more,
the more the merrier.
494
00:33:56 --> 00:33:59
In other words, I don't
want there necessarily
495
00:33:59 --> 00:34:00
only to be one.
496
00:34:00 --> 00:34:02
It could come down like this.
497
00:34:02 --> 00:34:05
And touch a second time.
498
00:34:05 --> 00:34:09
Is that what was
concerning you?
499
00:34:09 --> 00:34:11
So in mathematics, when we
claim that this is true
500
00:34:11 --> 00:34:14
for some point, we don't
necessarily mean that it
501
00:34:14 --> 00:34:16
doesn't work for others.
502
00:34:16 --> 00:34:19
In fact, if the function is
constant, this is 0 and
503
00:34:19 --> 00:34:25
in fact this equation
is true for every c.
504
00:34:25 --> 00:34:30
That satisfies your question?
505
00:34:30 --> 00:34:33
The fact that this point exists
actually is a touchy point.
506
00:34:33 --> 00:34:35
I just convinced you
of it visually.
507
00:34:35 --> 00:34:39
It's a geometric issue, whether
you're allowed to do this.
508
00:34:39 --> 00:34:42
Indeed, it has to do with the
existence of tangent lines and
509
00:34:42 --> 00:34:46
more analysis then we
can do in this class.
510
00:34:46 --> 00:34:46
Yeah.
511
00:34:46 --> 00:34:47
Another question.
512
00:34:47 --> 00:34:48
STUDENT: [INAUDIBLE]
513
00:34:48 --> 00:34:51
PROFESSOR: Pardon me.
514
00:34:51 --> 00:34:51
STUDENT: [INAUDIBLE]
515
00:34:51 --> 00:34:53
PROFESSOR: The question
is, what's the difference
516
00:34:53 --> 00:34:56
between this and the
linear approximation.
517
00:34:56 --> 00:35:11
And I think, let me see
if I can describe that.
518
00:35:11 --> 00:35:12
I'll leave the theorem
on the board.
519
00:35:12 --> 00:35:14
I'm going to get rid of
the colloquial version
520
00:35:14 --> 00:35:19
of the theorem.
521
00:35:19 --> 00:35:26
And I'll try to describe to you
the difference between this
522
00:35:26 --> 00:35:32
and the linear approximation.
523
00:35:32 --> 00:35:35
I was planning to do that in
a while, but we'll do it
524
00:35:35 --> 00:35:36
right now since that's
what you're asking.
525
00:35:36 --> 00:35:37
That's fine.
526
00:35:37 --> 00:35:45
So here's the situation.
527
00:35:45 --> 00:35:52
The linear approximation, so
let's say comparison with
528
00:35:52 --> 00:35:57
linear approximation.
529
00:35:57 --> 00:35:59
They're very closely related.
530
00:35:59 --> 00:36:02
The linear approximation says
the change in f over the change
531
00:36:02 --> 00:36:06
in x, that's the left-hand
side of this thing, is
532
00:36:06 --> 00:36:09
approximately f' (a).
533
00:36:09 --> 00:36:21
For b near a, and
b - a = delta x.
534
00:36:21 --> 00:36:23
This statement, which is in the
box, which is sitting right up
535
00:36:23 --> 00:36:29
there, is the statement that
this change in f is actually
536
00:36:29 --> 00:36:31
equal to something.
537
00:36:31 --> 00:36:33
Not approximately equal to it.
538
00:36:33 --> 00:36:37
It's equal to f' of some c.
539
00:36:37 --> 00:36:41
And the problem here is that we
don't know exactly which c.
540
00:36:41 --> 00:36:43
This is for some c.
541
00:36:43 --> 00:36:53
Between a and b.
542
00:36:53 --> 00:36:54
Right, so.
543
00:36:54 --> 00:36:59
That's the difference
between the two.
544
00:36:59 --> 00:37:19
And let me elaborate
a little bit.
545
00:37:19 --> 00:37:27
If you're trying to understand
what f ( b ) - f ( a ) / (b -
546
00:37:27 --> 00:37:32
a) is, the mean value theorem
is telling you for sure that
547
00:37:32 --> 00:37:35
it's equal to this f' (c).
548
00:37:35 --> 00:37:39
So that means it's less than or
equal to the largest possible
549
00:37:39 --> 00:37:44
value on the largest value
you can get, for sure.
550
00:37:44 --> 00:37:48
And this is on the
whole interval.
551
00:37:48 --> 00:37:50
And I'm going to include
the ends, because when you
552
00:37:50 --> 00:37:54
take a max it's sometimes
achieved at the ends.
553
00:37:54 --> 00:37:58
And similarly, because it's f'
(c), it's definitely bigger
554
00:37:58 --> 00:38:07
than the min on this
same interval here.
555
00:38:07 --> 00:38:13
This is all you can say based
on the mean value theorem.
556
00:38:13 --> 00:38:14
All you know is this.
557
00:38:14 --> 00:38:19
And colloquially, what that
means is that the average
558
00:38:19 --> 00:38:29
speed is between the
maximum and the minimum.
559
00:38:29 --> 00:38:31
Not very surprising.
560
00:38:31 --> 00:38:34
The mean value theorem
is supposed to be very
561
00:38:34 --> 00:38:36
intuitively obvious.
562
00:38:36 --> 00:38:40
It's saying the average speed
is trapped between the maximum
563
00:38:40 --> 00:38:42
speed and the minimum speed.
564
00:38:42 --> 00:38:46
For sure, that's something,
that's why, incidentally this
565
00:38:46 --> 00:38:50
wasn't really proved when
Newton and Leibniz were around.
566
00:38:50 --> 00:38:52
But, let's write this so
that you can read it.
567
00:38:52 --> 00:39:01
Average speed is between
the max and the min.
568
00:39:01 --> 00:39:04
But nobody had any trouble,
they didn't disbelieve it
569
00:39:04 --> 00:39:09
because it's a very
natural thing.
570
00:39:09 --> 00:39:16
Now if, for example, I take any
kind of linear approximation;
571
00:39:16 --> 00:39:25
say, for instance, e ^ x
is approximately 1 + x.
572
00:39:25 --> 00:39:30
Then I'm making the guess, now,
don't want to say this yet.
573
00:39:30 --> 00:39:35
That's not going to explain
it to you well enough.
574
00:39:35 --> 00:39:38
What we're saying, so this
is the mean value here.
575
00:39:38 --> 00:39:40
This is what the mean
value theorem says.
576
00:39:40 --> 00:39:47
And here's the linear
approximation.
577
00:39:47 --> 00:39:52
The linear approximation is
saying that the average speed
578
00:39:52 --> 00:39:59
is approximately the initial
speed, or possibly
579
00:39:59 --> 00:40:01
the final speed.
580
00:40:01 --> 00:40:07
So if a is the left endpoint,
then it's the initial speed.
581
00:40:07 --> 00:40:09
If it happens to be the right
endpoint, if the value of x is
582
00:40:09 --> 00:40:13
to the left then it's
the final speed.
583
00:40:13 --> 00:40:16
So those are the - so you can
see it's approximately right.
584
00:40:16 --> 00:40:19
Because the speed, when you're
on a short interval, shouldn't
585
00:40:19 --> 00:40:20
be varying very much.
586
00:40:20 --> 00:40:22
The max and the min should
be pretty close together.
587
00:40:22 --> 00:40:25
So that's why the linear
approximation is reasonable.
588
00:40:25 --> 00:40:30
And this is telling you
absolutely, it's no less
589
00:40:30 --> 00:40:34
than the min and no
more than the max.
590
00:40:34 --> 00:40:34
Yeah.
591
00:40:34 --> 00:40:41
STUDENT: [INAUDIBLE]
592
00:40:41 --> 00:40:42
PROFESSOR: The little kink?
593
00:40:42 --> 00:40:46
STUDENT: [INAUDIBLE]
594
00:40:46 --> 00:40:47
PROFESSOR: If you
approach from the top.
595
00:40:47 --> 00:40:50
So if it's still under here
I can show you it again.
596
00:40:50 --> 00:40:51
Oh yeah, it's still there.
597
00:40:51 --> 00:40:51
Good.
598
00:40:51 --> 00:40:54
STUDENT: [INAUDIBLE]
599
00:40:54 --> 00:40:56
PROFESSOR: Oh, the one
with the wiggle on top?
600
00:40:56 --> 00:40:58
Yeah, this one you can't.
601
00:40:58 --> 00:41:00
Because there's nothing to
touch and it also fails
602
00:41:00 --> 00:41:02
from the bottom because
there's this bad point.
603
00:41:02 --> 00:41:04
From the top, it could work.
604
00:41:04 --> 00:41:05
It can certainly
work both ways.
605
00:41:05 --> 00:41:07
So, for example.
606
00:41:07 --> 00:41:10
See if you're a machine,
you maybe don't have
607
00:41:10 --> 00:41:11
a way of doing this.
608
00:41:11 --> 00:41:14
But if you're a human being
you can spot all the places.
609
00:41:14 --> 00:41:17
There are a bunch of spots
where the slope is right.
610
00:41:17 --> 00:41:20
And it's perfectly OK.
611
00:41:20 --> 00:41:21
All of them work.
612
00:41:21 --> 00:41:25
STUDENT: [INAUDIBLE]
613
00:41:25 --> 00:41:26
PROFESSOR: It's not that
the c is the same.
614
00:41:26 --> 00:41:30
It's just we've now found
one, two, three, four, five
615
00:41:30 --> 00:41:31
c's for which it works.
616
00:41:31 --> 00:41:37
STUDENT: [INAUDIBLE]
617
00:41:37 --> 00:41:41
PROFESSOR: If you're asked to
find a c, so first of all
618
00:41:41 --> 00:41:44
that's kind of a
phony question.
619
00:41:44 --> 00:41:46
There are some questions
on your problem set which
620
00:41:46 --> 00:41:48
ask you to find a c.
621
00:41:48 --> 00:41:51
That actually is struggling to
get you to understand what the
622
00:41:51 --> 00:41:56
statement of the mean value
theorem is, but you should not
623
00:41:56 --> 00:41:58
pay a lot of attention
to those questions.
624
00:41:58 --> 00:42:01
They're not very impressive.
625
00:42:01 --> 00:42:04
But, of course, you would have
to find all the - if it asked
626
00:42:04 --> 00:42:06
you to find one, you find one.
627
00:42:06 --> 00:42:10
If you can find
some more, fine.
628
00:42:10 --> 00:42:13
You can pick whichever
one you want.
629
00:42:13 --> 00:42:16
Mean value theorem
just doesn't care.
630
00:42:16 --> 00:42:19
The mean value theorem doesn't
care because actually, the mean
631
00:42:19 --> 00:42:25
value theorem is never used
except in real life, except
632
00:42:25 --> 00:42:28
in this context here.
633
00:42:28 --> 00:42:32
You can never nail down which c
it is, so the only thing you
634
00:42:32 --> 00:42:35
can say is that you're going
slower than the maximum speed
635
00:42:35 --> 00:42:40
and faster than the
minimum speed.
636
00:42:40 --> 00:42:41
Sorry, say that again?
637
00:42:41 --> 00:42:47
STUDENT: [INAUDIBLE]
638
00:42:47 --> 00:42:49
PROFESSOR: If you're asked
for a specific c, you have
639
00:42:49 --> 00:42:51
to find a specific c.
640
00:42:51 --> 00:42:53
And it has to be in the range.
641
00:42:53 --> 00:43:04
In between, it has
to be in here.
642
00:43:04 --> 00:43:07
So now I want to tell you about
another kind of application,
643
00:43:07 --> 00:43:11
which is really just a
consequence of what
644
00:43:11 --> 00:43:22
I've described here.
645
00:43:22 --> 00:43:26
I should emphasize, by
the way, this, probably,
646
00:43:26 --> 00:43:27
should be doing this.
647
00:43:27 --> 00:43:32
I guess we've never
used this color here.
648
00:43:32 --> 00:43:32
This popular.
649
00:43:32 --> 00:43:33
This is pink.
650
00:43:33 --> 00:43:35
So this one is so good.
651
00:43:35 --> 00:43:47
So since we're
going to do this.
652
00:43:47 --> 00:43:51
So the reason why the
exclamation points are
653
00:43:51 --> 00:43:54
temporary, this is
such an obvious fact.
654
00:43:54 --> 00:43:58
But this is the way that you're
going to want to use the mean
655
00:43:58 --> 00:44:01
value theorem, and this is the
only way you need to understand
656
00:44:01 --> 00:44:02
the mean value theorem.
657
00:44:02 --> 00:44:06
On your test, or ever
in your whole life.
658
00:44:06 --> 00:44:10
So this is the way
it will be used.
659
00:44:10 --> 00:44:17
As I will make very clear
when we review for the exam.
660
00:44:17 --> 00:44:19
In practice what happens is you
even forget about the mean
661
00:44:19 --> 00:44:22
value theorem, and what you
remember is these three
662
00:44:22 --> 00:44:24
properties here.
663
00:44:24 --> 00:44:26
Which are themselves
consequences of the
664
00:44:26 --> 00:44:27
mean value theorem.
665
00:44:27 --> 00:44:31
So these are the ones that
I want to illustrate now.
666
00:44:31 --> 00:44:35
In my next discussion here.
667
00:44:35 --> 00:44:43
I'm just going to talk about
inequalities. inequalities
668
00:44:43 --> 00:44:46
are relationships
between functions.
669
00:44:46 --> 00:44:50
And I'm going to prove a couple
of them using the properties
670
00:44:50 --> 00:44:52
over there, the properties
that functions with positive
671
00:44:52 --> 00:44:56
derivatives are increasing.
672
00:44:56 --> 00:45:08
Here's an example. e ^ x
> 1 + x, where x > 0.
673
00:45:08 --> 00:45:10
The proof is the following.
674
00:45:10 --> 00:45:16
I consider, so here's a proof.
675
00:45:16 --> 00:45:21
I consider the function f ( x
), which is the difference.
676
00:45:21 --> 00:45:27
e ^ x - (1 + x).
677
00:45:27 --> 00:45:35
I observe that it starts at f (
0 ) equal to, well, that's e
678
00:45:35 --> 00:45:42
^ 0 - (1 + 0), which is 0.
679
00:45:42 --> 00:45:48
And, it keeps on going.
f' (x) = e ^ x.
680
00:45:48 --> 00:45:50
If I differentiate
here, the 1 goes away.
681
00:45:50 --> 00:45:54
I get - 1.
682
00:45:54 --> 00:45:55
That's the derivative
of the function.
683
00:45:55 --> 00:45:58
And this function,
because e ^ x > 1, for
684
00:45:58 --> 00:46:03
x positive is positive.
685
00:46:03 --> 00:46:07
As x gets bigger and bigger,
this rate of increase
686
00:46:07 --> 00:46:08
is positive.
687
00:46:08 --> 00:46:15
And therefore, three dots,
that's therefore, f ( x ) is
688
00:46:15 --> 00:46:18
bigger than its starting place.
689
00:46:18 --> 00:46:23
For x > 0.
690
00:46:23 --> 00:46:27
If it's increasing, then
that's, in particular, it's
691
00:46:27 --> 00:46:28
increasing starting from 0.
692
00:46:28 --> 00:46:30
So this is true.
693
00:46:30 --> 00:46:36
Now, all I have to do is read
what this inequality says.
694
00:46:36 --> 00:46:40
And what it says is that e ^ x,
just plug in for f ( x ), which
695
00:46:40 --> 00:46:45
is right here. -( 1 + x) is
greater than the starting
696
00:46:45 --> 00:46:48
value, which was 0.
697
00:46:48 --> 00:46:52
Now, I put the thing that's
negative on the other side.
698
00:46:52 --> 00:47:01
So that's the same
thing as e^x > 1 + x.
699
00:47:01 --> 00:47:04
That's a typical inequality.
700
00:47:04 --> 00:47:11
And now, we'll use
this principle again.
701
00:47:11 --> 00:47:12
Oh gee, I erased
the wrong thing.
702
00:47:12 --> 00:47:15
I erased the statement
and not the proof.
703
00:47:15 --> 00:47:23
Well, hide the proof.
704
00:47:23 --> 00:47:25
The next thing I want to
prove to you is that e ^
705
00:47:25 --> 00:47:33
x > 1 + x + (x^2 / 2).
706
00:47:33 --> 00:47:34
So, how do I do that?
707
00:47:34 --> 00:47:42
I introduce a function g (x),
which is e^x minus this.
708
00:47:42 --> 00:47:44
And now, I'm just going
to do exactly the same
709
00:47:44 --> 00:47:45
thing I did before.
710
00:47:45 --> 00:47:49
Which is, I get
started with g ( 0 ).
711
00:47:49 --> 00:47:51
Which is 1 - 1.
712
00:47:51 --> 00:47:53
Which is 0.
713
00:47:53 --> 00:48:00
And g' ( x ) is e ^ x minus
- now, look at what happens
714
00:48:00 --> 00:48:03
when I differentiate this.
715
00:48:03 --> 00:48:04
The 1 goes away.
716
00:48:04 --> 00:48:10
The x gives me a 1, and the
x^2 / 2 gives me a + x.
717
00:48:10 --> 00:48:18
And this one is positive for
x > 0, because of step 1.
718
00:48:18 --> 00:48:21
Because of the previous
one that I did.
719
00:48:21 --> 00:48:28
So this one is increasing.
g is increasing.
720
00:48:28 --> 00:48:33
Which says that g
( x ) > g ( 0 ).
721
00:48:33 --> 00:48:36
And if you just read that off,
it's exactly the same as our
722
00:48:36 --> 00:48:48
inequality here. e^x
> 1 + x + (x^2 / 2).
723
00:48:48 --> 00:48:54
Now, you can keep on going with
this essentially forever.
724
00:48:54 --> 00:48:58
And let me just write
down what you get.
725
00:48:58 --> 00:49:04
You get e ^ x > 1
+ x + (x^2 / 2).
726
00:49:04 --> 00:49:09
The next one turns out
to be (x^3 / 3 * 2)
727
00:49:09 --> 00:49:14
+ (x^4 / 4 * 3 * 2).
728
00:49:14 --> 00:49:16
And you can do
whatever you want.
729
00:49:16 --> 00:49:19
You can do others.
730
00:49:19 --> 00:49:22
And this is like the
tortoise and the hare.
731
00:49:22 --> 00:49:27
This is the tortoise, and this
is the hare, it's always ahead.
732
00:49:27 --> 00:49:31
But eventually, if you go
infinitely far, it catches up.
733
00:49:31 --> 00:49:38
So this turns out to be exactly
equal to e ^ x in the limit.
734
00:49:38 --> 00:49:41
And we'll talk about that maybe
at the end of the course.
735
00:49:41 --> 00:49:41