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PROFESSOR: Hi.
11
00:00:33,470 --> 00:00:36,680
Our lecture today is called
'Rolle's Theorem and Its
12
00:00:36,680 --> 00:00:37,930
Consequences'.
13
00:00:37,930 --> 00:00:40,960
And I suppose we could've made
a take off on what goes up
14
00:00:40,960 --> 00:00:44,030
must come down, and say that
what Rolle's theorem says
15
00:00:44,030 --> 00:00:47,970
intuitively is that what goes
up smoothly and comes down
16
00:00:47,970 --> 00:00:51,480
smoothly must level
off somewhere.
17
00:00:51,480 --> 00:00:51,880
OK?
18
00:00:51,880 --> 00:00:55,280
Now because that may sound too
easy to understand, let's
19
00:00:55,280 --> 00:00:59,350
cloak that in the language of
more formal mathematics.
20
00:00:59,350 --> 00:01:02,110
Rolle's theorem says this.
21
00:01:02,110 --> 00:01:05,760
Let 'f' be defined and
continuous on the closed
22
00:01:05,760 --> 00:01:08,050
interval from 'a' to 'b'.
23
00:01:08,050 --> 00:01:08,620
In other words, what?
24
00:01:08,620 --> 00:01:09,930
The domain of 'f' is the closed
25
00:01:09,930 --> 00:01:11,140
interval from 'a' to 'b'.
26
00:01:11,140 --> 00:01:14,340
The graph of 'f' is unbroken
on this interval.
27
00:01:14,340 --> 00:01:18,770
And differentiable in the open
interval from 'a' to 'b'.
28
00:01:18,770 --> 00:01:21,840
In other words, you want the
function to be continuous on
29
00:01:21,840 --> 00:01:25,270
the entire interval, but for
differentiability, you only
30
00:01:25,270 --> 00:01:29,000
require that it be smooth,
differentiable, in the
31
00:01:29,000 --> 00:01:32,010
interior of the interval,
in the open interval.
32
00:01:32,010 --> 00:01:38,000
Suppose also that 'f of
a' and 'f of b' are 0.
33
00:01:38,000 --> 00:01:41,990
Then, what Rolle's theorem says
is that 'f prime of c',
34
00:01:41,990 --> 00:01:48,180
the derivative of 'f of x', must
be 0 for some number 'c',
35
00:01:48,180 --> 00:01:51,680
at least one number
'c', in the open
36
00:01:51,680 --> 00:01:54,160
interval from 'a' to 'b'.
37
00:01:54,160 --> 00:01:58,420
Now what this thing says
intuitively is simply this,
38
00:01:58,420 --> 00:02:04,440
suppose you have a curve that is
unbroken for all values of
39
00:02:04,440 --> 00:02:07,130
'x' between 'a' and
'b' inclusively.
40
00:02:07,130 --> 00:02:09,580
Suppose the curve is smooth.
41
00:02:09,580 --> 00:02:14,780
Suppose the curve starts here
and ends here, then what we're
42
00:02:14,780 --> 00:02:18,330
saying is there must be some
point in here where the curve
43
00:02:18,330 --> 00:02:19,030
levels off.
44
00:02:19,030 --> 00:02:21,800
In other words, someplace where
you have a horizontal
45
00:02:21,800 --> 00:02:24,800
tangent, which is what 'f prime
of c' equals 0 means.
46
00:02:24,800 --> 00:02:29,000
In this particular diagram, this
would be the value of 'c'
47
00:02:29,000 --> 00:02:30,510
that we're talking about.
48
00:02:30,510 --> 00:02:35,080
By the way, I think the proof
is intuitively clear.
49
00:02:35,080 --> 00:02:39,510
Namely, if the curve never
leaves the x-axis, then it's
50
00:02:39,510 --> 00:02:42,660
leveled off for the
entire domain.
51
00:02:42,660 --> 00:02:45,937
And if the curve does leave the
x-axis, for example, if
52
00:02:45,937 --> 00:02:50,400
the curve starts to rise, OK,
since it must eventually get
53
00:02:50,400 --> 00:02:55,320
back to the x-axis when 'b' is
0, it must ultimately begin to
54
00:02:55,320 --> 00:02:56,240
fall again.
55
00:02:56,240 --> 00:02:59,860
Well, if the curve goes from
rising to falling,
56
00:02:59,860 --> 00:03:01,040
it must have what?
57
00:03:01,040 --> 00:03:04,010
Since it's coming up and then
going down, it must attain a
58
00:03:04,010 --> 00:03:05,520
maximum value.
59
00:03:05,520 --> 00:03:09,030
Because the curve is unbroken
and smooth, as we saw in our
60
00:03:09,030 --> 00:03:12,690
previous lecture, the maximum
value is characterized by the
61
00:03:12,690 --> 00:03:15,220
derivative at that
point being 0.
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00:03:15,220 --> 00:03:18,920
In fact, the analytic proof is
precisely what we've just
63
00:03:18,920 --> 00:03:22,650
said, only translated into more
mathematical language.
64
00:03:22,650 --> 00:03:25,280
By the way, I would like to
make a slight aside here,
65
00:03:25,280 --> 00:03:28,980
because I think it sometimes
gets confusing to students to
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00:03:28,980 --> 00:03:33,600
see, why do you say that the
function has to be continuous
67
00:03:33,600 --> 00:03:38,000
on the closed interval, but
differentiable only in the
68
00:03:38,000 --> 00:03:39,220
open interval?
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00:03:39,220 --> 00:03:42,580
I thought you might like to see
a contrived example as to
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00:03:42,580 --> 00:03:46,360
what goes wrong if you allow the
curve to be broken at the
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00:03:46,360 --> 00:03:47,220
end points.
72
00:03:47,220 --> 00:03:49,520
See, all I'm thinking of
is something like this.
73
00:03:49,520 --> 00:03:52,880
Suppose I say, look, let's
define a curve as follows.
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00:03:52,880 --> 00:03:56,510
At 'x' equals 'a' and
'x' equals 'b', the
75
00:03:56,510 --> 00:03:59,040
curve will be 0.
76
00:03:59,040 --> 00:04:01,550
So in other words, it'll
cross the x-axis.
77
00:04:01,550 --> 00:04:04,980
Then immediately, for 'x'
greater than 'a', the curve
78
00:04:04,980 --> 00:04:09,090
jumps up to here, comes down
along this line, and then, you
79
00:04:09,090 --> 00:04:13,480
see, when 'x' equals 'b', it
jumps down here again.
80
00:04:13,480 --> 00:04:15,560
In other words, why
not let the curve
81
00:04:15,560 --> 00:04:16,950
be defined as follows?
82
00:04:16,950 --> 00:04:20,170
It will be 0 at these
two endpoints.
83
00:04:20,170 --> 00:04:23,650
It'll be this curve on
the open interval.
84
00:04:23,650 --> 00:04:28,010
Notice in this contrived example
that 'f of a' and 'f
85
00:04:28,010 --> 00:04:31,780
of b' are both equal to 0, but
there is no place in the open
86
00:04:31,780 --> 00:04:35,880
interval where the curve has
a horizontal tangent line.
87
00:04:35,880 --> 00:04:38,790
In other words, the significance
is you've got to
88
00:04:38,790 --> 00:04:41,830
be sure that the curve doesn't
get broken at the ends,
89
00:04:41,830 --> 00:04:43,940
because with these gaps,
all sorts of
90
00:04:43,940 --> 00:04:45,780
crazy things can happen.
91
00:04:45,780 --> 00:04:49,180
Now, just as in our previous
lecture, there are some rather
92
00:04:49,180 --> 00:04:53,960
important cautions that have to
be taken in understanding
93
00:04:53,960 --> 00:04:55,210
Rolle's theorem.
94
00:04:55,210 --> 00:04:59,020
As simple as it is, we have to
be sure that we understand
95
00:04:59,020 --> 00:05:02,070
exactly what's really
happening here.
96
00:05:02,070 --> 00:05:06,260
The first caution is that
Rolle's theorem is what we
97
00:05:06,260 --> 00:05:09,920
mathematicians call an
'existence theorem'.
98
00:05:09,920 --> 00:05:13,800
It says, under certain
conditions, there exists at
99
00:05:13,800 --> 00:05:17,080
least one number 'c' that
has a certain property.
100
00:05:17,080 --> 00:05:19,790
It doesn't tell us how
many "c's" there are.
101
00:05:19,790 --> 00:05:21,870
It doesn't tell us where
to find them.
102
00:05:21,870 --> 00:05:23,040
It just says, what?
103
00:05:23,040 --> 00:05:25,720
There exists at least
one such 'c'.
104
00:05:25,720 --> 00:05:28,770
And the point is, you must
be careful to remember--
105
00:05:28,770 --> 00:05:29,930
so let's take an example.
106
00:05:29,930 --> 00:05:31,970
Here's 'a', here's 'b'.
107
00:05:31,970 --> 00:05:34,826
If the function is continuous
and smooth, in other words, if
108
00:05:34,826 --> 00:05:37,950
the graph is continuous and
smooth, all we're saying is
109
00:05:37,950 --> 00:05:40,710
that at least one number between
'a' and 'b', the
110
00:05:40,710 --> 00:05:43,520
curve, must possess a
horizontal tangent.
111
00:05:43,520 --> 00:05:44,640
There may be more than one.
112
00:05:44,640 --> 00:05:46,480
You see, the curve, for
example, could do
113
00:05:46,480 --> 00:05:48,250
something like this.
114
00:05:48,250 --> 00:05:51,640
See, in other words, here is one
value of 'c', which we'll
115
00:05:51,640 --> 00:05:54,060
call 'c1', horizontal
tangent here.
116
00:05:54,060 --> 00:05:57,730
Here's another value, which
we'll call 'c2', horizontal
117
00:05:57,730 --> 00:05:59,280
tangent here.
118
00:05:59,280 --> 00:06:03,100
See, again, the meaning
of at least one.
119
00:06:03,100 --> 00:06:11,370
Secondly, we must make sure
that we remember that the
120
00:06:11,370 --> 00:06:12,290
curve is smooth.
121
00:06:12,290 --> 00:06:12,780
Meaning what?
122
00:06:12,780 --> 00:06:14,570
That the function is
differentiable.
123
00:06:14,570 --> 00:06:15,980
Now I'm taking the liberty
of drawing
124
00:06:15,980 --> 00:06:17,880
these things in freehand.
125
00:06:17,880 --> 00:06:21,760
There's some mixed emotions
here, if I draw the diagrams
126
00:06:21,760 --> 00:06:24,260
too smoothly all the time, you
lose the significance of
127
00:06:24,260 --> 00:06:26,200
what's going on because
of the picture.
128
00:06:26,200 --> 00:06:28,020
And if I draw them freehand
all the time, you won't
129
00:06:28,020 --> 00:06:29,230
understand what I'm
doing, because I
130
00:06:29,230 --> 00:06:30,380
don't draw very well.
131
00:06:30,380 --> 00:06:32,270
But I think here we can
get away with this.
132
00:06:32,270 --> 00:06:33,650
What I'm driving at is this.
133
00:06:33,650 --> 00:06:36,710
Let's suppose you have
'a' and 'b' here.
134
00:06:36,710 --> 00:06:38,810
Let's suppose that the function,
the curve that we're
135
00:06:38,810 --> 00:06:41,230
drawing, passes through
these two points.
136
00:06:41,230 --> 00:06:44,260
But suppose there happens to
be a sharp corner in here.
137
00:06:44,260 --> 00:06:46,530
Maybe the curve goes like this,
it goes up like this,
138
00:06:46,530 --> 00:06:50,010
then very abruptly comes
down like this.
139
00:06:50,010 --> 00:06:51,550
Notice, what?
140
00:06:51,550 --> 00:06:54,270
That the curve is continuous.
141
00:06:54,270 --> 00:06:57,380
It does reach a local maximum.
142
00:06:57,380 --> 00:07:01,890
But the point is, for this
particular value of 'c', 'f
143
00:07:01,890 --> 00:07:05,320
prime of c' is not
0 by default.
144
00:07:05,320 --> 00:07:10,320
Namely, 'f prime of c'
doesn't even exist.
145
00:07:10,320 --> 00:07:15,230
So in other words, Rolle's
theorem doesn't apply if you
146
00:07:15,230 --> 00:07:16,700
don't have differentiabilities.
147
00:07:16,700 --> 00:07:19,890
I want to make sure you see
where each of the parts of the
148
00:07:19,890 --> 00:07:22,600
hypotheses for the
theorem are used.
149
00:07:22,600 --> 00:07:25,470
By the way, here's another
interesting result, which has
150
00:07:25,470 --> 00:07:27,200
nothing to do with the statement
of the theorem, but
151
00:07:27,200 --> 00:07:31,140
again, another piece of evidence
as to why we like to
152
00:07:31,140 --> 00:07:34,700
shy away from functions which
are not single value.
153
00:07:34,700 --> 00:07:37,700
See, for example, suppose you
allowed the function to be
154
00:07:37,700 --> 00:07:41,520
multivalued, and you say, OK,
I want the curve to pass
155
00:07:41,520 --> 00:07:42,960
through here and here.
156
00:07:42,960 --> 00:07:44,890
And I want it to be smooth.
157
00:07:44,890 --> 00:07:48,180
But the curve does not have
to be single value.
158
00:07:48,180 --> 00:07:49,660
Notice what you can do.
159
00:07:49,660 --> 00:07:52,070
You could have a curve that
does something like this.
160
00:07:52,070 --> 00:07:55,490
161
00:07:55,490 --> 00:07:57,210
I don't know.
162
00:07:57,210 --> 00:08:00,190
Now you see, there will be
a point 'c' where the
163
00:08:00,190 --> 00:08:02,390
derivative will be 0.
164
00:08:02,390 --> 00:08:04,790
Even as badly as I've drawn
this, I think roughly
165
00:08:04,790 --> 00:08:09,430
speaking, we can see that 'c'
would be something like this.
166
00:08:09,430 --> 00:08:12,130
Notice, however, that in
Rolle's theorem, the
167
00:08:12,130 --> 00:08:13,150
statement is what?
168
00:08:13,150 --> 00:08:16,490
That 'c' must be on the open
interval from 'a' to 'b'.
169
00:08:16,490 --> 00:08:20,010
If the function is not single
value, as long as the curve is
170
00:08:20,010 --> 00:08:23,330
smooth, there will be places
where the curve levels off.
171
00:08:23,330 --> 00:08:27,130
But the x-coordinate of the
point at which the curve
172
00:08:27,130 --> 00:08:30,530
levels off may not be
in the interval--
173
00:08:30,530 --> 00:08:34,179
may not be, it could be, but
it might not be, I should
174
00:08:34,179 --> 00:08:37,220
say-- in the open interval
from 'a' to 'b'.
175
00:08:37,220 --> 00:08:39,070
This is a very, very
important result.
176
00:08:39,070 --> 00:08:43,070
In fact, I'll have reason to
make reference to this in a
177
00:08:43,070 --> 00:08:45,700
little while later
in the lecture.
178
00:08:45,700 --> 00:08:47,660
I couldn't make reference to
it later earlier in the
179
00:08:47,660 --> 00:08:49,570
lecture, I guess.
180
00:08:49,570 --> 00:08:54,200
The fourth assumption here is
also an aside, and it's one
181
00:08:54,200 --> 00:08:55,360
that's rather crucial.
182
00:08:55,360 --> 00:08:59,470
In most textbooks in which
Rolle's theorem is stated, the
183
00:08:59,470 --> 00:09:00,350
condition is what?
184
00:09:00,350 --> 00:09:04,180
That 'f of a' equals
'f of b' equals 0.
185
00:09:04,180 --> 00:09:09,340
It turns out that this is too
restrictive, that essentially,
186
00:09:09,340 --> 00:09:13,630
all you need is 'f of
a' equals 'f of b'.
187
00:09:13,630 --> 00:09:17,200
What I mean by that is, let's
suppose 'f of a' is not 0.
188
00:09:17,200 --> 00:09:19,810
Let's suppose this height
represents 'f of a'.
189
00:09:19,810 --> 00:09:24,300
What I'm saying is suppose
that 'f of a' and
190
00:09:24,300 --> 00:09:27,210
'f of b' are equal.
191
00:09:27,210 --> 00:09:29,760
What that means is, if I want to
think of a new axis, which
192
00:09:29,760 --> 00:09:32,330
I call the 'x sub 1' axis--
193
00:09:32,330 --> 00:09:35,980
see, with respect to the 'x sub
1' axis, the curve crosses
194
00:09:35,980 --> 00:09:37,550
the axis at these two points.
195
00:09:37,550 --> 00:09:40,235
In other words, notice that as
long as these two points are
196
00:09:40,235 --> 00:09:43,160
at the same level, the same
argument that we used to prove
197
00:09:43,160 --> 00:09:46,970
Rolle's theorem goes through
unimpeded over here.
198
00:09:46,970 --> 00:09:48,170
Namely, we say what?
199
00:09:48,170 --> 00:09:52,490
What goes up smoothly and
comes down smoothly--
200
00:09:52,490 --> 00:09:54,670
because it has to come down,
because it comes back to the
201
00:09:54,670 --> 00:10:00,540
same level here-- must reach a
point someplace in here where
202
00:10:00,540 --> 00:10:02,280
it levels off.
203
00:10:02,280 --> 00:10:07,100
So these are the four cautions
that I'd like you to look at
204
00:10:07,100 --> 00:10:09,200
when we view Rolle's theorem.
205
00:10:09,200 --> 00:10:12,110
Now, if somebody were to say
to us, what's so important
206
00:10:12,110 --> 00:10:14,290
about Rolle's theorem?
207
00:10:14,290 --> 00:10:17,250
And this happens so often in
mathematics that frequently,
208
00:10:17,250 --> 00:10:21,510
the most important thing about
one particular theorem, is
209
00:10:21,510 --> 00:10:24,420
that it may be the building
block by which a more
210
00:10:24,420 --> 00:10:26,670
important, or more
useful, or more
211
00:10:26,670 --> 00:10:28,910
practical theorem is derived.
212
00:10:28,910 --> 00:10:32,680
And in this respect, I would say
for my own opinion, that
213
00:10:32,680 --> 00:10:35,830
the most important application
of Rolle's theorem is that it
214
00:10:35,830 --> 00:10:39,560
facilitates a very famous result
known as the 'Mean
215
00:10:39,560 --> 00:10:40,590
Value theorem'.
216
00:10:40,590 --> 00:10:43,790
As our course proceeds, from
time to time we will have
217
00:10:43,790 --> 00:10:47,330
ample reason to back track and
make references to the mean
218
00:10:47,330 --> 00:10:48,390
value theorem.
219
00:10:48,390 --> 00:10:51,640
I intend not to make too deep
references to the mean value
220
00:10:51,640 --> 00:10:54,190
theorem now, because what
I would like to do is to
221
00:10:54,190 --> 00:10:57,220
establish the result, give you
enough of an intuitive feeling
222
00:10:57,220 --> 00:11:00,960
so that you can tuck it under
your belt without feeling too
223
00:11:00,960 --> 00:11:04,880
overwhelmed by it, and just
enough applications of it so
224
00:11:04,880 --> 00:11:07,450
that we can get into the next
phase of our calculus course.
225
00:11:07,450 --> 00:11:10,080
But the mean value theorem is
another one of these things
226
00:11:10,080 --> 00:11:12,680
where if you state the thing
analytically, and have no
227
00:11:12,680 --> 00:11:15,490
feeling for what's going on
pictorially, the thing can
228
00:11:15,490 --> 00:11:17,180
become overwhelming.
229
00:11:17,180 --> 00:11:19,940
Let's, in fact, do it in an
overwhelming way, and then
230
00:11:19,940 --> 00:11:22,040
show what the thing
means pictorially.
231
00:11:22,040 --> 00:11:25,520
Notice again how this thing now
starts off the same way as
232
00:11:25,520 --> 00:11:26,560
Rolle's theorem.
233
00:11:26,560 --> 00:11:29,560
Let 'f' be continuous on the
closed interval from 'a' to
234
00:11:29,560 --> 00:11:34,150
'b', and differentiable in the
open interval from 'a' to 'b'.
235
00:11:34,150 --> 00:11:38,110
By the way, this is just an
idiosyncrasy of mine, I don't
236
00:11:38,110 --> 00:11:39,050
know if it's standard.
237
00:11:39,050 --> 00:11:42,070
When I talk about the closed
interval, I have the habit of
238
00:11:42,070 --> 00:11:44,050
saying 'on' the interval.
239
00:11:44,050 --> 00:11:47,280
When I talk about the open
interval, I like to say 'in'
240
00:11:47,280 --> 00:11:49,840
the interval, to sort of
emphasize the interior.
241
00:11:49,840 --> 00:11:53,250
It's just a vocabulary
trait, and don't read
242
00:11:53,250 --> 00:11:54,080
too much into this.
243
00:11:54,080 --> 00:11:55,160
Don't be upset by it.
244
00:11:55,160 --> 00:11:58,060
But it's continuous on
the closed interval,
245
00:11:58,060 --> 00:12:00,050
differentiable in the
open interval.
246
00:12:00,050 --> 00:12:02,360
Now it's again an existence
theorem.
247
00:12:02,360 --> 00:12:04,750
It says then there exists
a number 'c'.
248
00:12:04,750 --> 00:12:06,580
When I say there exists,
it means what?
249
00:12:06,580 --> 00:12:10,190
There is at least one number 'c'
in the open interval from
250
00:12:10,190 --> 00:12:11,080
'a' to 'b'--
251
00:12:11,080 --> 00:12:13,240
and this is the part that
looks kind of tough--
252
00:12:13,240 --> 00:12:19,030
such that ''f of b' minus 'f of
a'' divided by 'b - a' is
253
00:12:19,030 --> 00:12:20,600
'f prime of c'.
254
00:12:20,600 --> 00:12:24,040
And this somehow or other may
seem at first glance to be
255
00:12:24,040 --> 00:12:27,290
more ominous than the intuitive
feeling about
256
00:12:27,290 --> 00:12:28,520
Rolle's theorem.
257
00:12:28,520 --> 00:12:31,590
By the way, as the name implies,
where by 'mean' we
258
00:12:31,590 --> 00:12:34,900
don't mean nasty, we mean
average, if you'd like to see
259
00:12:34,900 --> 00:12:37,240
what this thing means, and I'll
draw you a picture in a
260
00:12:37,240 --> 00:12:44,020
second, all it says is that if
a particle is moving from
261
00:12:44,020 --> 00:12:48,790
point 'a' to point 'b', say, at
at least one point during
262
00:12:48,790 --> 00:12:51,750
its trip, the instantaneous
speed must
263
00:12:51,750 --> 00:12:54,980
equal the average speed.
264
00:12:54,980 --> 00:12:55,210
You see?
265
00:12:55,210 --> 00:12:57,380
Because after all, if you're
always going less than your
266
00:12:57,380 --> 00:13:00,490
average speed, how could you
have had an average speed as
267
00:13:00,490 --> 00:13:02,010
high as your average speed?
268
00:13:02,010 --> 00:13:04,250
And if you're always going less
than your average speed,
269
00:13:04,250 --> 00:13:08,270
how could you have had, you see,
an average speed equaling
270
00:13:08,270 --> 00:13:09,660
this, what it did?
271
00:13:09,660 --> 00:13:13,610
So that somehow or other, all
you're saying is that somehow
272
00:13:13,610 --> 00:13:17,050
or other, the instantaneous
speed at a particular instance
273
00:13:17,050 --> 00:13:21,160
must equal your average speed
someplace along the path.
274
00:13:21,160 --> 00:13:24,550
Now what that means pictorially
is this--
275
00:13:24,550 --> 00:13:27,750
again, I'll chance a
freehand diagram--
276
00:13:27,750 --> 00:13:31,320
suppose our curve is 'y'
equals 'f of x'.
277
00:13:31,320 --> 00:13:35,230
See, I've drawn it
to be smooth.
278
00:13:35,230 --> 00:13:39,380
Suppose it's continuous and
smooth on this open interval
279
00:13:39,380 --> 00:13:41,680
from 'a' to 'b'.
280
00:13:41,680 --> 00:13:45,530
Now what is, if I think of a
particle moving from point 'p'
281
00:13:45,530 --> 00:13:50,270
to point 'q', how do I identify
the average speed?
282
00:13:50,270 --> 00:13:53,200
The average speed is the slope
of the straight line that
283
00:13:53,200 --> 00:13:55,530
joins 'p' to 'q'.
284
00:13:55,530 --> 00:13:57,880
On the other hand, what is
the instantaneous speed?
285
00:13:57,880 --> 00:14:00,280
If we think of it in terms of
the picture, it's the slope of
286
00:14:00,280 --> 00:14:03,620
the line tangent to the curve
at a particular point.
287
00:14:03,620 --> 00:14:06,420
So in other words, what
we're saying is this.
288
00:14:06,420 --> 00:14:11,270
You see, if we were to take the
line 'PQ', and we shift it
289
00:14:11,270 --> 00:14:15,070
parallel to itself, I think you
can sense that the points
290
00:14:15,070 --> 00:14:19,070
'P' and 'Q', if we labeled 'P'
and 'Q' the points at which
291
00:14:19,070 --> 00:14:22,170
this chord intersects the curve,
the points 'P' and 'Q'
292
00:14:22,170 --> 00:14:24,440
will roll in closer and
closer together.
293
00:14:24,440 --> 00:14:29,760
Ultimately, the line will fail
to intersect the curve, and at
294
00:14:29,760 --> 00:14:32,340
the transition point, if the
curve is smooth, all we're
295
00:14:32,340 --> 00:14:36,780
saying is that the last point at
which that line touches the
296
00:14:36,780 --> 00:14:41,460
curve as we move it out, OK,
that the line would be tangent
297
00:14:41,460 --> 00:14:44,060
to the curve at that
particular point.
298
00:14:44,060 --> 00:14:45,430
You see, all we're
saying is what?
299
00:14:45,430 --> 00:14:48,650
That someplace between here and
here there must be a point
300
00:14:48,650 --> 00:14:51,760
where the tangent line to
the curve is parallel
301
00:14:51,760 --> 00:14:53,310
to the chord 'PQ'.
302
00:14:53,310 --> 00:14:56,320
And now, we have all the
ingredients that we need to
303
00:14:56,320 --> 00:14:59,500
see what the mean value theorem
says geometrically.
304
00:14:59,500 --> 00:15:01,550
Let's call the tangent
line 'l'.
305
00:15:01,550 --> 00:15:04,820
First of all, what is the
slope of the line 'PQ'?
306
00:15:04,820 --> 00:15:07,120
Well, it's a straight line.
307
00:15:07,120 --> 00:15:09,610
The slope of a straight
line is 'delta y'
308
00:15:09,610 --> 00:15:11,090
divided by 'delta x'.
309
00:15:11,090 --> 00:15:13,840
Well, notice that this
height here is by
310
00:15:13,840 --> 00:15:15,680
definition 'f of b'.
311
00:15:15,680 --> 00:15:17,880
This height here is 'f of a'.
312
00:15:17,880 --> 00:15:22,820
So this height here is just
'f of b' minus 'f of a'.
313
00:15:22,820 --> 00:15:26,770
This length here is
just 'b - a'.
314
00:15:26,770 --> 00:15:30,710
So the slope, 'delta y' divided
by 'delta x', is just
315
00:15:30,710 --> 00:15:37,180
''f of b' minus 'f of
a'', over 'b - a'.
316
00:15:37,180 --> 00:15:40,010
On the other hand, what is the
slope of the line 'l'?
317
00:15:40,010 --> 00:15:44,340
By definition, it's 'f prime of
x' evaluated at 'x' equals
318
00:15:44,340 --> 00:15:47,100
'c', that's 'f prime of c'.
319
00:15:47,100 --> 00:15:49,240
Now what does it mean in
terms of slopes for
320
00:15:49,240 --> 00:15:50,700
two lines to be parallel?
321
00:15:50,700 --> 00:15:53,630
It means that their
slopes are equal.
322
00:15:53,630 --> 00:15:55,170
And where is 'c'?
323
00:15:55,170 --> 00:16:02,240
'c' is someplace in the open
interval from 'a' to 'b'.
324
00:16:02,240 --> 00:16:07,730
Now the reason I call this
intuitively an extension of
325
00:16:07,730 --> 00:16:09,490
Rolle's theorem--
326
00:16:09,490 --> 00:16:12,270
and by the way, you'll notice
that what I have to say is a
327
00:16:12,270 --> 00:16:16,490
much simpler demonstration than
what's given in the book.
328
00:16:16,490 --> 00:16:19,460
But before you think I'm being
egotistic about this, let me
329
00:16:19,460 --> 00:16:22,610
point out, as is so often the
case that wherever my
330
00:16:22,610 --> 00:16:25,930
demonstrations are easier than
the one in the book, I'm
331
00:16:25,930 --> 00:16:28,520
losing something in
my presentation.
332
00:16:28,520 --> 00:16:31,410
Either I haven't shown the most
analytic representation,
333
00:16:31,410 --> 00:16:34,760
or I'm overlooking a particular
complicated side
334
00:16:34,760 --> 00:16:36,760
effect that might occur.
335
00:16:36,760 --> 00:16:39,330
But disregarding that for the
moment, you see what I'm
336
00:16:39,330 --> 00:16:42,790
saying is this, let's suppose,
for the sake of argument, we
337
00:16:42,790 --> 00:16:47,720
visualize the line 'PQ' as
being our new x-axis.
338
00:16:47,720 --> 00:16:49,710
I'll call that the x1-axis.
339
00:16:49,710 --> 00:16:53,110
And now let's take a line
perpendicular to 'PQ' and call
340
00:16:53,110 --> 00:16:56,740
that our new y-axis,
the y1-axis.
341
00:16:56,740 --> 00:16:58,870
Now look at the curve
that we've drawn.
342
00:16:58,870 --> 00:17:05,079
With respect to the y1- x1-axis,
notice that the curve
343
00:17:05,079 --> 00:17:06,819
is smooth, right?
344
00:17:06,819 --> 00:17:08,329
It's unbroken.
345
00:17:08,329 --> 00:17:13,050
And it cuts the x1-axis
at two points.
346
00:17:13,050 --> 00:17:16,270
Now if we apply Rolle's theorem
with respect to the
347
00:17:16,270 --> 00:17:19,970
x1- y1-axis, we say, look,
here's a curve
348
00:17:19,970 --> 00:17:21,770
which cuts the x-axis--
349
00:17:21,770 --> 00:17:22,660
the x1-axis--
350
00:17:22,660 --> 00:17:23,790
at two points.
351
00:17:23,790 --> 00:17:26,560
It's smooth.
352
00:17:26,560 --> 00:17:28,740
Therefore, it must level
off someplace.
353
00:17:28,740 --> 00:17:31,370
In other words, there must be
some point on this curve where
354
00:17:31,370 --> 00:17:36,480
the tangent line to the curve
is parallel to the x1-axis.
355
00:17:36,480 --> 00:17:39,670
That's exactly, you see, what
this thing here says.
356
00:17:39,670 --> 00:17:42,660
That's another geometric
interpretation that indicates
357
00:17:42,660 --> 00:17:44,620
how Rolle's theorem
might be used.
358
00:17:44,620 --> 00:17:49,000
However, there is a very, very
subtle flaw in what I've said.
359
00:17:49,000 --> 00:17:52,240
One that is so subtle that you
may not even notice it until I
360
00:17:52,240 --> 00:17:54,830
point it out to you, and even
after I point it out, there's
361
00:17:54,830 --> 00:17:56,760
a chance you may not realize
what I've said.
362
00:17:56,760 --> 00:17:59,910
Because it's a point that I know
took me a long, long time
363
00:17:59,910 --> 00:18:01,390
to discover for myself.
364
00:18:01,390 --> 00:18:03,470
And it all hinges on
the concept of
365
00:18:03,470 --> 00:18:05,200
single valuedness again.
366
00:18:05,200 --> 00:18:07,840
The trouble with this
interpretation is the
367
00:18:07,840 --> 00:18:09,380
following--
368
00:18:09,380 --> 00:18:12,770
and by the way, let me point
out, I'm not knocking my
369
00:18:12,770 --> 00:18:15,390
interpretation, I think it's
still a tremendous way of
370
00:18:15,390 --> 00:18:18,110
visualizing the result, but from
an analytical point of
371
00:18:18,110 --> 00:18:20,600
view, why we have
to be careful.
372
00:18:20,600 --> 00:18:23,830
Let's suppose that my curve 'y'
equals 'f of x' happens to
373
00:18:23,830 --> 00:18:27,110
look something like this, OK?
374
00:18:27,110 --> 00:18:30,200
Happens to look something
like this.
375
00:18:30,200 --> 00:18:33,160
Notice, barring any bad drawing
that I've done here,
376
00:18:33,160 --> 00:18:37,130
that this curve is single value,
that no line parallel
377
00:18:37,130 --> 00:18:40,670
to the y-axis cuts this curve
in more than one place.
378
00:18:40,670 --> 00:18:43,150
Now here's my 'a' and
here's my 'b'.
379
00:18:43,150 --> 00:18:45,670
380
00:18:45,670 --> 00:18:49,820
And so I say, OK, by Rolle's
theorem, if I look at this as
381
00:18:49,820 --> 00:18:52,740
being the x-axis and this
as being the y-axis--
382
00:18:52,740 --> 00:18:55,500
in other words, the x1-
y1-axis again--
383
00:18:55,500 --> 00:18:58,930
I say to myself, look, here's
a smooth curve, it cuts the
384
00:18:58,930 --> 00:19:04,620
x-axis in two points, therefore,
someplace between
385
00:19:04,620 --> 00:19:07,210
these two points, there must
be a place where the curve
386
00:19:07,210 --> 00:19:11,120
levels off, et cetera,
et cetera, et cetera.
387
00:19:11,120 --> 00:19:15,370
And the interesting point is to
notice that a given curve,
388
00:19:15,370 --> 00:19:18,850
as to whether it's single valued
or not, is dependent
389
00:19:18,850 --> 00:19:21,830
upon the orientation
of the axes.
390
00:19:21,830 --> 00:19:24,040
In other words, notice that
I've drawn this particular
391
00:19:24,040 --> 00:19:28,100
curve so it is single valued
with respect to the xy-plane.
392
00:19:28,100 --> 00:19:31,580
On the other hand, with
respect to the x1-
393
00:19:31,580 --> 00:19:36,020
y1-coordinate system, this curve
is not single valued.
394
00:19:36,020 --> 00:19:40,920
Namely, observe how a line
parallel to the y1-axis can
395
00:19:40,920 --> 00:19:45,630
intersect this curve at
more than one point.
396
00:19:45,630 --> 00:19:48,590
In other words, whether a curve
is single valued or not
397
00:19:48,590 --> 00:19:51,570
is not an absolute property
independent of
398
00:19:51,570 --> 00:19:52,970
the coordinate system.
399
00:19:52,970 --> 00:19:57,160
So again, if I could be sure
that when I rotated my
400
00:19:57,160 --> 00:20:01,460
coordinate axes the original
single valued curve was still
401
00:20:01,460 --> 00:20:04,010
single valued, then
my above proof
402
00:20:04,010 --> 00:20:05,250
would have been rigorous.
403
00:20:05,250 --> 00:20:07,410
But of course, I can't
be sure of that.
404
00:20:07,410 --> 00:20:11,700
By the way, the technique used
in the book is quite standard,
405
00:20:11,700 --> 00:20:14,270
and what it does is the
following, it still utilizes
406
00:20:14,270 --> 00:20:17,630
Rolle's theorem, but the
technique behind the proof in
407
00:20:17,630 --> 00:20:18,780
the book is this.
408
00:20:18,780 --> 00:20:23,160
The function that we set up is
the vertical distance between
409
00:20:23,160 --> 00:20:27,930
the chord and the curve, as
we move along this way.
410
00:20:27,930 --> 00:20:32,090
And notice that that distance
is 0 at these two endpoints.
411
00:20:32,090 --> 00:20:33,450
OK?
412
00:20:33,450 --> 00:20:37,510
And therefore, Rolle's theorem
applies to that function.
413
00:20:37,510 --> 00:20:39,710
And the whole idea is
something like this.
414
00:20:39,710 --> 00:20:40,950
All we say is--
415
00:20:40,950 --> 00:20:43,780
and the analytic part
proves this--
416
00:20:43,780 --> 00:20:47,190
all we say is look, the point at
which this chord would have
417
00:20:47,190 --> 00:20:50,960
been tangent to the curve is the
place where the vertical
418
00:20:50,960 --> 00:20:55,360
distance between the chord
and the curve is what?
419
00:20:55,360 --> 00:20:56,730
Maximum.
420
00:20:56,730 --> 00:20:58,420
And we won't go into that
right now, that
421
00:20:58,420 --> 00:21:00,250
is done in the text.
422
00:21:00,250 --> 00:21:03,520
All I wanted to do, as I always
will do when possible,
423
00:21:03,520 --> 00:21:06,580
is that whenever the rigorous
proof seems far more
424
00:21:06,580 --> 00:21:10,520
complicated than proofs which
are more intuitive, I will not
425
00:21:10,520 --> 00:21:13,830
take the time in general, in our
lectures, to give the more
426
00:21:13,830 --> 00:21:14,700
rigorous proof.
427
00:21:14,700 --> 00:21:17,590
What I will take the time to
do is to show why the less
428
00:21:17,590 --> 00:21:19,610
rigorous proof has pitfalls.
429
00:21:19,610 --> 00:21:21,640
Well, enough said about
the statement of
430
00:21:21,640 --> 00:21:22,950
the mean value theorem.
431
00:21:22,950 --> 00:21:26,080
Time is getting very short, and
we don't need much more
432
00:21:26,080 --> 00:21:29,260
time to make the home run
ball pitch that we
433
00:21:29,260 --> 00:21:30,520
want to make now.
434
00:21:30,520 --> 00:21:35,040
And the idea is this, that the
most important analytical
435
00:21:35,040 --> 00:21:39,740
reason for having the mean value
theorem is, for those of
436
00:21:39,740 --> 00:21:43,560
us who like to use our geometric
intuition, it turns
437
00:21:43,560 --> 00:21:47,080
out that almost every
geometrically obvious fact
438
00:21:47,080 --> 00:21:52,200
that has a proper analytic
counterpart has the property
439
00:21:52,200 --> 00:21:54,640
that the analytic counterpart
is proven by
440
00:21:54,640 --> 00:21:56,130
the mean value theorem.
441
00:21:56,130 --> 00:21:58,320
See, let me give you a
simple for instance.
442
00:21:58,320 --> 00:22:02,700
In fact, in the text book this
is called the first corollary
443
00:22:02,700 --> 00:22:04,530
to the mean value theorem.
444
00:22:04,530 --> 00:22:07,810
Suppose we have a function
capital 'F of x', and we know
445
00:22:07,810 --> 00:22:11,330
that the derivative is
always equal to 0.
446
00:22:11,330 --> 00:22:14,730
The claim is that 'F of x'
itself must be a constant.
447
00:22:14,730 --> 00:22:17,160
By the way, two cautions here.
448
00:22:17,160 --> 00:22:20,140
Don't say that we've had
this result before.
449
00:22:20,140 --> 00:22:23,030
The result that we had
before was actually
450
00:22:23,030 --> 00:22:24,660
the converse of this.
451
00:22:24,660 --> 00:22:27,790
The result that we had before
was the one that said what?
452
00:22:27,790 --> 00:22:32,570
If 'F of x' is a constant,
then its derivative is 0.
453
00:22:32,570 --> 00:22:34,800
Now we're saying the
opposite--not the opposite,
454
00:22:34,800 --> 00:22:35,400
but the converse.
455
00:22:35,400 --> 00:22:38,000
Now we're saying, look, if the
derivative is always 0--
456
00:22:38,000 --> 00:22:39,980
notice the use of my identity
symbol here--
457
00:22:39,980 --> 00:22:44,680
if the derivative is 0 for all
values of 'x', then the
458
00:22:44,680 --> 00:22:46,480
function must've been
a constant.
459
00:22:46,480 --> 00:22:48,660
Now, you know, geometrically
this is a very
460
00:22:48,660 --> 00:22:50,090
simple thing to visualize.
461
00:22:50,090 --> 00:22:53,160
You say, look, the derivative
is the slope.
462
00:22:53,160 --> 00:22:56,150
And if you're saying that the
slope of the curve is always
463
00:22:56,150 --> 00:23:00,300
horizontal, the curve itself
must be a straight line.
464
00:23:00,300 --> 00:23:02,960
And if the curve is a straight
line, that's exactly what you
465
00:23:02,960 --> 00:23:05,590
mean by saying that the function
is a constant.
466
00:23:05,590 --> 00:23:08,500
How do we prove this using
the mean value theorem?
467
00:23:08,500 --> 00:23:11,250
See, and I just wanted to go
through a proof here once,
468
00:23:11,250 --> 00:23:14,970
just so to get the idea
of what a proof means.
469
00:23:14,970 --> 00:23:16,350
You see, to show that
something is a
470
00:23:16,350 --> 00:23:17,930
constant should mean what?
471
00:23:17,930 --> 00:23:23,430
That if 'a' is unequal to 'b'
for any two values 'a' and
472
00:23:23,430 --> 00:23:25,850
'b', 'F of a'--
473
00:23:25,850 --> 00:23:28,570
well, I'm using capital 'F'
here--capital 'F of a' has to
474
00:23:28,570 --> 00:23:30,210
equal capital 'F of b'.
475
00:23:30,210 --> 00:23:32,500
That's what you mean for a
function to be a constant.
476
00:23:32,500 --> 00:23:36,530
No matter what the input is, the
outputs are always equal.
477
00:23:36,530 --> 00:23:39,310
By the way, if 'a' equals 'b',
it's trivial that 'F of a'
478
00:23:39,310 --> 00:23:40,350
equals 'F of b'.
479
00:23:40,350 --> 00:23:43,500
But essentially, to prove that
capital F is a constant, this
480
00:23:43,500 --> 00:23:44,810
is what I have to prove.
481
00:23:44,810 --> 00:23:47,740
That if 'a' is different from
'b', no matter what 'a' and
482
00:23:47,740 --> 00:23:50,960
'b' I use, that 'F of
a' equals 'F of b'.
483
00:23:50,960 --> 00:23:53,570
And the idea is by the mean
value theorem, we say, look,
484
00:23:53,570 --> 00:23:56,090
what does the mean value
theorem say?
485
00:23:56,090 --> 00:23:59,440
We're assuming now that 'F'
is a continuous and
486
00:23:59,440 --> 00:24:05,420
differentiable function on an
interval, OK, from 'a' to 'b'.
487
00:24:05,420 --> 00:24:08,210
The mean value theorem says
under these conditions, there
488
00:24:08,210 --> 00:24:10,600
exists a number 'c'
between 'a' and
489
00:24:10,600 --> 00:24:12,730
'b' with what property?
490
00:24:12,730 --> 00:24:20,260
That ''F of b' minus 'F of a''
over 'b - a' is equal to 'F
491
00:24:20,260 --> 00:24:22,810
prime of c'.
492
00:24:22,810 --> 00:24:26,260
That's just a statement of
the mean value theorem.
493
00:24:26,260 --> 00:24:29,060
This is always true if the
conditions of the mean value
494
00:24:29,060 --> 00:24:30,150
theorem apply.
495
00:24:30,150 --> 00:24:33,480
Now all we're saying is, in this
particular problem, what
496
00:24:33,480 --> 00:24:36,130
property that capital
'F' have?
497
00:24:36,130 --> 00:24:38,670
It had the property that
its derivative for all
498
00:24:38,670 --> 00:24:40,250
values of 'x' was 0.
499
00:24:40,250 --> 00:24:43,970
In particular then, when 'c'
is the value that we're
500
00:24:43,970 --> 00:24:47,890
talking about, if 'F prime of x'
is 0 for all values of 'x',
501
00:24:47,890 --> 00:24:51,400
in particular, then, it's 0
when 'x' is equal to 'c'.
502
00:24:51,400 --> 00:24:55,920
In other words, by our given
hypothesis, this is 0.
503
00:24:55,920 --> 00:25:00,380
But if a fraction is 0, its
numerator must be 0.
504
00:25:00,380 --> 00:25:03,120
505
00:25:03,120 --> 00:25:04,300
That says what?
506
00:25:04,300 --> 00:25:07,920
'F of b' minus 'F of a' is 0.
507
00:25:07,920 --> 00:25:11,450
See, the only way a quotient
can be 0 is for the
508
00:25:11,450 --> 00:25:12,370
numerator--
509
00:25:12,370 --> 00:25:15,860
or the dividend, the divisor,
I don't know these formal
510
00:25:15,860 --> 00:25:20,210
names, they slipped my mind, but
the top one has to be 0.
511
00:25:20,210 --> 00:25:24,110
And if 'F of b' minus 'F of a'
is 0, that says 'F of b'
512
00:25:24,110 --> 00:25:27,730
equals 'F of a', and that's
precisely what we had to show
513
00:25:27,730 --> 00:25:30,830
to show that 'F'
was a constant.
514
00:25:30,830 --> 00:25:31,490
OK?
515
00:25:31,490 --> 00:25:33,800
So again, notice, it's not that
we're saying that the
516
00:25:33,800 --> 00:25:37,190
mean value theorem is a harder
way of proving what we already
517
00:25:37,190 --> 00:25:40,610
know to be intuitively true,
what we're saying is what?
518
00:25:40,610 --> 00:25:44,890
That we know that many
intuitively obvious results
519
00:25:44,890 --> 00:25:46,380
frequently turn out
to be false.
520
00:25:46,380 --> 00:25:49,760
We would like some analytical
way of knowing which of the
521
00:25:49,760 --> 00:25:52,240
intuitive results are
actually correct.
522
00:25:52,240 --> 00:25:55,125
All I'm saying is the mean value
theorem gives us a big
523
00:25:55,125 --> 00:25:56,080
hint that way.
524
00:25:56,080 --> 00:25:59,750
By the way, let me close by
giving you one more important
525
00:25:59,750 --> 00:26:01,400
illustration of what
we can prove by
526
00:26:01,400 --> 00:26:03,030
the mean value theorem.
527
00:26:03,030 --> 00:26:05,810
And this is called a corollary
of a corollary, as I'll show
528
00:26:05,810 --> 00:26:07,240
you what I mean in a minute.
529
00:26:07,240 --> 00:26:11,900
The next example that I want to
use is what it means to say
530
00:26:11,900 --> 00:26:15,560
suppose I'm given two functions
'f' and 'g', and all
531
00:26:15,560 --> 00:26:17,790
I know about these two
functions is that the
532
00:26:17,790 --> 00:26:19,970
derivatives are identical.
533
00:26:19,970 --> 00:26:23,280
In other ways, that 'f' and 'g'
have the property that for
534
00:26:23,280 --> 00:26:26,950
every value of 'x', 'f
prime of x' is equal
535
00:26:26,950 --> 00:26:28,320
to 'g prime of x'.
536
00:26:28,320 --> 00:26:31,200
By the way, when I say every
value of 'x', again, it's
537
00:26:31,200 --> 00:26:32,890
local versus global.
538
00:26:32,890 --> 00:26:35,850
It's not necessary that this
happens for all 'x', what is
539
00:26:35,850 --> 00:26:37,240
important is what?
540
00:26:37,240 --> 00:26:41,385
That 'x' be defined
on some interval.
541
00:26:41,385 --> 00:26:44,340
In other words, even if I know
that this property is true for
542
00:26:44,340 --> 00:26:47,470
some interval, I don't really
care what happens outside of
543
00:26:47,470 --> 00:26:47,760
that interval.
544
00:26:47,760 --> 00:26:50,660
In terms of local properties,
all I'm saying is, all I know
545
00:26:50,660 --> 00:26:53,300
is that for some interval, maybe
the whole axis, doesn't
546
00:26:53,300 --> 00:26:58,390
make any difference, 'f prime'
is identical to 'g prime'.
547
00:26:58,390 --> 00:27:01,690
Now, you would like to be able
to say, maybe, that if 'f
548
00:27:01,690 --> 00:27:04,300
prime' is equal to 'g prime',
'f' equals 'g'.
549
00:27:04,300 --> 00:27:05,920
But that's not the case.
550
00:27:05,920 --> 00:27:08,510
What is the case is that the
difference between the two
551
00:27:08,510 --> 00:27:10,740
functions must be a constant.
552
00:27:10,740 --> 00:27:14,180
Again, geometrically, what
you're saying is what?
553
00:27:14,180 --> 00:27:20,400
That if you have two curves,
which point by point always
554
00:27:20,400 --> 00:27:21,810
have the same slope--
555
00:27:21,810 --> 00:27:23,410
in other words, for each
'x' value, the
556
00:27:23,410 --> 00:27:24,540
slopes are the same--
557
00:27:24,540 --> 00:27:26,590
is just essentially
saying that the
558
00:27:26,590 --> 00:27:28,530
two curves are parallel.
559
00:27:28,530 --> 00:27:31,130
And if they're parallel curves,
what's a way of
560
00:27:31,130 --> 00:27:32,900
stating that two curves
are parallel?
561
00:27:32,900 --> 00:27:36,910
That one is a constant
displacement of the other.
562
00:27:36,910 --> 00:27:41,300
In other words, the geometric
impact of two curves having
563
00:27:41,300 --> 00:27:44,110
the same derivative is not that
the curves are the same,
564
00:27:44,110 --> 00:27:45,780
but that they're parallel.
565
00:27:45,780 --> 00:27:50,200
And by the way, the proof of
this result is again a
566
00:27:50,200 --> 00:27:52,290
corollary to the mean
value theorem.
567
00:27:52,290 --> 00:27:54,600
Namely, let's look at
the function 'f of
568
00:27:54,600 --> 00:27:55,970
x' minus 'g of x'.
569
00:27:55,970 --> 00:27:58,000
Call that capital 'F of x'.
570
00:27:58,000 --> 00:28:02,100
Let capital 'F of x' be ''f
of x' minus 'g of x''.
571
00:28:02,100 --> 00:28:04,460
Since the derivative of a
difference is the difference
572
00:28:04,460 --> 00:28:08,430
of the derivatives, that would
say the derivative of capital
573
00:28:08,430 --> 00:28:11,970
'F' is the derivative
little 'f' minus the
574
00:28:11,970 --> 00:28:14,360
derivative of 'g'.
575
00:28:14,360 --> 00:28:15,090
OK?
576
00:28:15,090 --> 00:28:18,470
Now what do we know about 'f
prime' and 'g prime of x'?
577
00:28:18,470 --> 00:28:21,250
We know that 'f prime of
x' equals 'g prime
578
00:28:21,250 --> 00:28:22,970
of x' for all 'x'.
579
00:28:22,970 --> 00:28:28,180
Consequently, the difference
between these two must be 0.
580
00:28:28,180 --> 00:28:30,550
Remember, if two functions
are identical, their
581
00:28:30,550 --> 00:28:32,000
difference is 0.
582
00:28:32,000 --> 00:28:35,510
That says, therefore, that
capital 'F prime of x' is
583
00:28:35,510 --> 00:28:36,920
identically 0.
584
00:28:36,920 --> 00:28:38,870
And by our previous theorem--
585
00:28:38,870 --> 00:28:40,430
notice the beautiful
logic of this--
586
00:28:40,430 --> 00:28:44,540
from the mean value theorem,
we proved that if the
587
00:28:44,540 --> 00:28:47,400
derivative of a function is
identically 0, the function
588
00:28:47,400 --> 00:28:49,660
must be a constant.
589
00:28:49,660 --> 00:28:52,260
So we apply that here.
590
00:28:52,260 --> 00:28:54,020
But what was capital 'F'?
591
00:28:54,020 --> 00:28:57,410
It was 'little 'f - g'.
592
00:28:57,410 --> 00:29:00,130
And that proves our
desired result.
593
00:29:00,130 --> 00:29:04,510
Again, what I want you to see
here is that we have not done
594
00:29:04,510 --> 00:29:07,530
anything different with the
mean value theorem.
595
00:29:07,530 --> 00:29:10,140
We're not trying to say we're
going to prove results we
596
00:29:10,140 --> 00:29:11,410
couldn't prove before.
597
00:29:11,410 --> 00:29:12,230
Rather, what?
598
00:29:12,230 --> 00:29:15,970
The mean value theorem is our
way of showing that certain
599
00:29:15,970 --> 00:29:19,540
intuitive results hold true
analytically, that we can talk
600
00:29:19,540 --> 00:29:21,920
about parallel curves,
and things like this.
601
00:29:21,920 --> 00:29:25,250
Most important, in terms of
summarizing this lecture from
602
00:29:25,250 --> 00:29:28,650
a point of view of what's coming
next, it's crucial to
603
00:29:28,650 --> 00:29:33,050
observe that this last example
is what is going to allow us
604
00:29:33,050 --> 00:29:35,080
to enter the study of
something called the
605
00:29:35,080 --> 00:29:36,640
'indefinite integral'.
606
00:29:36,640 --> 00:29:39,820
Or in another manner of
speaking, something called the
607
00:29:39,820 --> 00:29:42,600
inverse of taking
a derivative.
608
00:29:42,600 --> 00:29:46,290
You see, the idea is, notice
that in these two examples we
609
00:29:46,290 --> 00:29:49,490
start with information about
the derivative and deduce
610
00:29:49,490 --> 00:29:51,820
what's true about the
original function.
611
00:29:51,820 --> 00:29:54,820
That's inverting the emphasis
of what we've been doing up
612
00:29:54,820 --> 00:29:57,740
until now, where we've started
with the function and
613
00:29:57,740 --> 00:29:59,800
investigated its derivative.
614
00:29:59,800 --> 00:30:02,910
To see this in more detail,
join me again next time.
615
00:30:02,910 --> 00:30:04,520
And until next time, goodbye.
616
00:30:04,520 --> 00:30:07,500
617
00:30:07,500 --> 00:30:10,700
Funding for the publication of
this video was provided by the
618
00:30:10,700 --> 00:30:14,750
Gabriella and Paul Rosenbaum
Foundation.
619
00:30:14,750 --> 00:30:18,930
Help OCW continue to provide
free and open access to MIT
620
00:30:18,930 --> 00:30:23,130
courses by making a donation
at ocw.mit.edu/donate.
621
00:30:23,130 --> 00:30:27,860