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PROFESSOR: To begin today
I want to remind you,
9
00:00:24,120 --> 00:00:27,170
I need to write it down on
the board at least twice,
10
00:00:27,170 --> 00:00:33,230
of the fundamental
theorem of calculus.
11
00:00:33,230 --> 00:00:38,496
We called it FTC 1 because
it's the first version
12
00:00:38,496 --> 00:00:39,620
of the fundamental theorem.
13
00:00:39,620 --> 00:00:42,340
We'll be talking about
another version, called
14
00:00:42,340 --> 00:00:44,940
the second version, today.
15
00:00:44,940 --> 00:00:53,610
And what it says
is this: If F' = f,
16
00:00:53,610 --> 00:01:03,330
then the integral from a to b
of f(x) dx is equal to F(b) -
17
00:01:03,330 --> 00:01:03,830
F(a).
18
00:01:06,410 --> 00:01:10,750
So that's the fundamental
theorem of calculus.
19
00:01:10,750 --> 00:01:15,600
And the way we used
it last time was,
20
00:01:15,600 --> 00:01:22,090
this was used to
evaluate integrals.
21
00:01:22,090 --> 00:01:28,110
Not surprisingly,
that's how we used it.
22
00:01:28,110 --> 00:01:35,670
But today, I want to
reverse that point of view.
23
00:01:35,670 --> 00:01:38,900
We're going to read
the equation backwards,
24
00:01:38,900 --> 00:01:49,240
and we're going to
write it this way.
25
00:01:49,240 --> 00:02:01,010
And we're going to use f
to understand capital F.
26
00:02:01,010 --> 00:02:04,370
Or in other words,
the derivative
27
00:02:04,370 --> 00:02:07,700
to understand the function.
28
00:02:07,700 --> 00:02:13,380
So that's the reversal of point
of view that I'd like to make.
29
00:02:13,380 --> 00:02:16,310
And we'll make this
point in various ways.
30
00:02:16,310 --> 00:02:28,230
So information about
f, about F', gives us
31
00:02:28,230 --> 00:02:37,750
information about F. Now,
since there were questions
32
00:02:37,750 --> 00:02:39,740
about the mean
value theorem, I'm
33
00:02:39,740 --> 00:02:42,380
going to illustrate
this first by making
34
00:02:42,380 --> 00:02:46,710
a comparison between the
fundamental theorem of calculus
35
00:02:46,710 --> 00:02:50,600
and the mean value theorem.
36
00:02:50,600 --> 00:02:56,460
So we're going to compare this
fundamental theorem of calculus
37
00:02:56,460 --> 00:03:01,489
with what we call the
mean value theorem.
38
00:03:01,489 --> 00:03:02,905
And in order to
do that, I'm going
39
00:03:02,905 --> 00:03:05,070
to introduce a
couple of notations.
40
00:03:05,070 --> 00:03:08,870
I'll write delta
F as F(b) - F(a).
41
00:03:11,820 --> 00:03:17,770
And another highly imaginative
notation, delta x = b - a.
42
00:03:17,770 --> 00:03:21,760
So here's the change in F,
there's the change in x.
43
00:03:21,760 --> 00:03:25,690
And then, this
fundamental theorem
44
00:03:25,690 --> 00:03:30,800
can be written, of course, right
up above there is the formula.
45
00:03:30,800 --> 00:03:36,430
And it's the
formula for delta F.
46
00:03:36,430 --> 00:03:39,240
So this is what we call
the fundamental theorem
47
00:03:39,240 --> 00:03:44,090
of calculus.
48
00:03:44,090 --> 00:03:51,810
I'm going to divide
by delta x, now.
49
00:03:51,810 --> 00:03:56,510
And If I divide by delta x,
that's the same thing as 1 /
50
00:03:56,510 --> 00:04:02,880
(b-a) times the integral
from a to b of f(x) dx.
51
00:04:02,880 --> 00:04:05,110
So I've just rewritten
the formula here.
52
00:04:05,110 --> 00:04:11,750
And this expression here,
on the right-hand side,
53
00:04:11,750 --> 00:04:13,650
is a fairly important one.
54
00:04:13,650 --> 00:04:23,370
This is the average of f.
55
00:04:23,370 --> 00:04:29,490
That's the average value of f.
56
00:04:29,490 --> 00:04:32,560
Now, so this is
going to permit me
57
00:04:32,560 --> 00:04:35,980
to make the comparison between
the mean value theorem, which
58
00:04:35,980 --> 00:04:38,630
we don't have stated yet here.
59
00:04:38,630 --> 00:04:41,010
And the fundamental theorem.
60
00:04:41,010 --> 00:04:49,130
And I'll do it in the
form of inequalities.
61
00:04:49,130 --> 00:04:51,079
So right in the
middle here, I'm going
62
00:04:51,079 --> 00:04:52,370
to put the fundamental theorem.
63
00:04:52,370 --> 00:04:56,580
It says that delta F in
this notation is equal to,
64
00:04:56,580 --> 00:04:59,150
well if I multiply
by delta x again, I
65
00:04:59,150 --> 00:05:01,850
can write it as
the average of f--
66
00:05:01,850 --> 00:05:04,870
So I'm going to write it
as the average of F' here.
67
00:05:04,870 --> 00:05:07,100
Times delta x.
68
00:05:07,100 --> 00:05:09,120
So we have this
factor here, which
69
00:05:09,120 --> 00:05:11,870
is the average of F', or
the average of little f,
70
00:05:11,870 --> 00:05:13,810
it's the same thing.
71
00:05:13,810 --> 00:05:15,430
And then I multiplied
through again.
72
00:05:15,430 --> 00:05:20,810
So I put the thing
in the red box, here.
73
00:05:20,810 --> 00:05:29,970
STUDENT: [INAUDIBLE]
74
00:05:29,970 --> 00:05:37,190
PROFESSOR: Isn't what
the average of big F?
75
00:05:37,190 --> 00:05:39,990
So the question is,
why is this the average
76
00:05:39,990 --> 00:05:44,990
of little f rather than
the average of big F.
77
00:05:44,990 --> 00:05:50,130
So the average of a function
is the typical value.
78
00:05:50,130 --> 00:05:54,680
If, for example,
little f were constant,
79
00:05:54,680 --> 00:05:59,770
little f were constant, then
this integral would be--
80
00:05:59,770 --> 00:06:03,920
So the question is why
is this the average.
81
00:06:03,920 --> 00:06:08,970
And I'll take a little
second to explain that.
82
00:06:08,970 --> 00:06:13,050
But I think I'll
explain it over here.
83
00:06:13,050 --> 00:06:19,010
Because I'm going to erase it.
84
00:06:19,010 --> 00:06:27,270
So the idea of an
average is the following.
85
00:06:27,270 --> 00:06:32,281
For example, imagine
that a = 0 and b = n,
86
00:06:32,281 --> 00:06:33,600
let's say for example.
87
00:06:33,600 --> 00:06:46,850
And so we might sum the
function from 1 to n.
88
00:06:46,850 --> 00:06:49,520
Now, that would be the sum
of the values from 1 to n.
89
00:06:49,520 --> 00:06:54,960
But the average is,
we divide by n here.
90
00:06:54,960 --> 00:06:56,140
So this is the average.
91
00:06:56,140 --> 00:07:00,180
And this is a kind
of Riemann sum,
92
00:07:00,180 --> 00:07:06,290
representing the integral
from 0 to n, of f(x) dx.
93
00:07:06,290 --> 00:07:10,700
Where the increment,
delta x, is 1.
94
00:07:10,700 --> 00:07:13,470
So this is the notion of
an average value here,
95
00:07:13,470 --> 00:07:15,760
but in the continuum
setting, as opposed
96
00:07:15,760 --> 00:07:20,570
to the discrete setting.
97
00:07:20,570 --> 00:07:24,270
Whereas what's on
the left-hand side
98
00:07:24,270 --> 00:07:28,250
is the change in
F. The capital F.
99
00:07:28,250 --> 00:07:31,790
And this is the average
of the little f.
100
00:07:31,790 --> 00:07:33,860
So an average is a sum.
101
00:07:33,860 --> 00:07:39,520
And it's like an integral.
102
00:07:39,520 --> 00:07:42,720
So, in other words what I have
here is that the change in F
103
00:07:42,720 --> 00:07:45,620
is the average of its
infinitesimal change
104
00:07:45,620 --> 00:07:50,000
times the amount of time
elapsed, if you like.
105
00:07:50,000 --> 00:07:56,100
So this is the statement
of the fundamental theorem.
106
00:07:56,100 --> 00:07:56,960
Just rewritten.
107
00:07:56,960 --> 00:07:58,450
Exactly what I wrote there.
108
00:07:58,450 --> 00:08:01,780
But I multiplied
back by delta x.
109
00:08:01,780 --> 00:08:07,530
Now, let me compare this
with the mean value theorem.
110
00:08:07,530 --> 00:08:13,590
The mean value theorem
also is an equation.
111
00:08:13,590 --> 00:08:17,790
The mean value theorem says that
this is equal to F'(c) delta x.
112
00:08:21,560 --> 00:08:23,270
Now, I pulled a fast one on you.
113
00:08:23,270 --> 00:08:26,560
I used capital F's here
to make the analogy clear.
114
00:08:26,560 --> 00:08:30,710
But the role of the letter
is important to make
115
00:08:30,710 --> 00:08:32,200
the transition to
this comparison.
116
00:08:32,200 --> 00:08:35,010
We're talking about the
function capital F here.
117
00:08:35,010 --> 00:08:36,620
And its derivative.
118
00:08:36,620 --> 00:08:38,510
Now, this is true.
119
00:08:38,510 --> 00:08:42,510
So now I claim that this
thing is fairly specific.
120
00:08:42,510 --> 00:08:47,750
Whereas this, unfortunately,
is a little bit vague.
121
00:08:47,750 --> 00:08:51,240
And the reason why it's vague
is that c is just somewhere
122
00:08:51,240 --> 00:08:52,790
in the interval.
123
00:08:52,790 --> 00:09:01,010
So some c-- Sorry, this is
some c, in between a and b.
124
00:09:01,010 --> 00:09:04,475
So really, since we don't
know where this thing is,
125
00:09:04,475 --> 00:09:05,850
we don't know
which of the values
126
00:09:05,850 --> 00:09:07,690
it is, we can't say what it is.
127
00:09:07,690 --> 00:09:11,700
All we can do is
say, well for sure
128
00:09:11,700 --> 00:09:13,280
it's less than
the largest value,
129
00:09:13,280 --> 00:09:18,090
say, the maximum of
F', times delta x.
130
00:09:18,090 --> 00:09:20,720
And the only thing we can
say for sure on the other end
131
00:09:20,720 --> 00:09:23,720
is that it's less than
or equal to-- sorry,
132
00:09:23,720 --> 00:09:25,700
it's greater than or
equal to the minimum
133
00:09:25,700 --> 00:09:27,410
of F' times delta x.
134
00:09:27,410 --> 00:09:29,090
Over that same interval.
135
00:09:29,090 --> 00:09:39,650
This is over 0 less
than-- sorry, a < x < b.
136
00:09:39,650 --> 00:09:43,220
So that means that the
fundamental theorem of calculus
137
00:09:43,220 --> 00:09:45,950
is a much more specific thing.
138
00:09:45,950 --> 00:09:48,272
And indeed it gives
the same conclusion.
139
00:09:48,272 --> 00:09:50,230
It's much stronger than
the mean value theorem.
140
00:09:50,230 --> 00:09:52,330
It's way better than
the mean value theorem.
141
00:09:52,330 --> 00:09:55,100
In fact, as soon as
we have integrals,
142
00:09:55,100 --> 00:09:57,280
we can abandon the
mean value theorem.
143
00:09:57,280 --> 00:09:58,550
We don't want it.
144
00:09:58,550 --> 00:10:00,810
It's too simple-minded.
145
00:10:00,810 --> 00:10:03,700
And what we have is something
much more sophisticated,
146
00:10:03,700 --> 00:10:04,940
which we can use.
147
00:10:04,940 --> 00:10:05,760
Which is this.
148
00:10:05,760 --> 00:10:08,400
So it's obvious that
if this is the average,
149
00:10:08,400 --> 00:10:09,960
the average is less
than the maximum.
150
00:10:09,960 --> 00:10:14,610
So it's obvious that
it works just as well
151
00:10:14,610 --> 00:10:15,950
to draw this conclusion.
152
00:10:15,950 --> 00:10:20,010
And similarly over
here with the minimum.
153
00:10:20,010 --> 00:10:22,240
OK, the average is always
bigger than the minimum
154
00:10:22,240 --> 00:10:25,430
and smaller than the max.
155
00:10:25,430 --> 00:10:28,680
So this is the
connection, if you like.
156
00:10:28,680 --> 00:10:33,050
And I'm going to elaborate just
one step further by talking
157
00:10:33,050 --> 00:10:36,340
about the problem that
you had on the exam.
158
00:10:36,340 --> 00:10:39,640
So there was an Exam 2 problem.
159
00:10:39,640 --> 00:10:43,590
And I'll show you how it works
using the mean value theorem
160
00:10:43,590 --> 00:10:45,602
and how it works
using integrals.
161
00:10:45,602 --> 00:10:47,810
But I'm going to have to
use this notation capital F.
162
00:10:47,810 --> 00:10:51,850
So capital F', as
opposed to the little f,
163
00:10:51,850 --> 00:10:55,270
which was what was the
notation that was on your exam.
164
00:10:55,270 --> 00:10:57,620
So we had this situation here.
165
00:10:57,620 --> 00:11:01,380
These were the givens
of the problem.
166
00:11:01,380 --> 00:11:11,620
And then the question was,
the mean value theorem says,
167
00:11:11,620 --> 00:11:14,160
or implies, if you
like, it doesn't say it,
168
00:11:14,160 --> 00:11:25,370
but it implies it - implies
A is less than capital F of 4
169
00:11:25,370 --> 00:11:38,690
is less than B,
for which A and B?
170
00:11:38,690 --> 00:11:43,330
So let's take a look
at what it says.
171
00:11:43,330 --> 00:11:48,840
Well, the mean value
theorem says that F( F(4) -
172
00:11:48,840 --> 00:11:53,620
F(0) = F'(c) (4 - 0).
173
00:11:57,240 --> 00:12:03,770
This is this F' times delta
x, this is the change in x.
174
00:12:03,770 --> 00:12:08,100
And that's the same
thing as 1/(1+c) times 4.
175
00:12:12,750 --> 00:12:21,780
And so the range of values
of this number here is from /
176
00:12:21,780 --> 00:12:24,630
1/(1+0) times 4, that's 4.
177
00:12:24,630 --> 00:12:28,000
To, that's the largest value,
to the smallest that it gets,
178
00:12:28,000 --> 00:12:32,510
which is 1/(1+4) times 4.
179
00:12:32,510 --> 00:12:41,020
That's the range.
180
00:12:41,020 --> 00:12:54,410
And so the conclusion is that
F(4) - f(0) is between, well,
181
00:12:54,410 --> 00:12:55,090
let's see.
182
00:12:55,090 --> 00:12:59,040
It's between 4 and 4/5.
183
00:12:59,040 --> 00:13:01,850
Which are those two
numbers down there.
184
00:13:01,850 --> 00:13:03,920
And if you remember
that F(0) was 1,
185
00:13:03,920 --> 00:13:15,680
this is the same F(4)
is between 5 and 9/5.
186
00:13:15,680 --> 00:13:19,650
So that's the way that
you were supposed to solve
187
00:13:19,650 --> 00:13:22,570
the problem on the exam.
188
00:13:22,570 --> 00:13:25,760
On the other hand,
let's compare to what
189
00:13:25,760 --> 00:13:27,940
you would do with the
fundamental theorem
190
00:13:27,940 --> 00:13:31,140
of calculus.
191
00:13:31,140 --> 00:13:33,090
With the fundamentals
theorem of calculus,
192
00:13:33,090 --> 00:13:35,090
we have the following formula.
193
00:13:35,090 --> 00:13:41,710
F(4) - F(0) is equal to the
integral from 0 to 4 of dx /
194
00:13:41,710 --> 00:13:42,210
(1+x).
195
00:13:46,100 --> 00:13:52,700
That's what the
fundamental theorem says.
196
00:13:52,700 --> 00:13:58,550
And now I claim that we can
get these same types of results
197
00:13:58,550 --> 00:14:00,780
by a very elementary
observation.
198
00:14:00,780 --> 00:14:03,470
It's really the same
observation that I made up here,
199
00:14:03,470 --> 00:14:06,110
that the average is less
than or equal to the maximum.
200
00:14:06,110 --> 00:14:11,810
Which is that the biggest this
can ever be is, let's see.
201
00:14:11,810 --> 00:14:15,280
The biggest it is
when x is 0, that's 1.
202
00:14:15,280 --> 00:14:20,360
So the biggest it
ever gets is this.
203
00:14:20,360 --> 00:14:25,110
And that's equal to 4.
204
00:14:25,110 --> 00:14:25,660
Right?
205
00:14:25,660 --> 00:14:30,910
On the other hand, the
smallest it ever gets to be,
206
00:14:30,910 --> 00:14:36,770
it's equal to this.
207
00:14:36,770 --> 00:14:39,620
The smallest it ever gets
to be is the integral
208
00:14:39,620 --> 00:14:42,400
from 0 to 4 of 1/5 dx.
209
00:14:42,400 --> 00:14:46,110
Because that's the lowest
value that the integrand takes.
210
00:14:46,110 --> 00:14:48,390
When x = 4, it's 1/5.
211
00:14:48,390 --> 00:14:54,270
And that's equal to 4/5.
212
00:14:54,270 --> 00:14:56,440
Now, there's a
little tiny detail
213
00:14:56,440 --> 00:14:58,350
which is that really
we know that this
214
00:14:58,350 --> 00:15:01,560
is the area of some rectangle
and this is strictly smaller.
215
00:15:01,560 --> 00:15:03,990
And we know that these
inequalities are actually
216
00:15:03,990 --> 00:15:05,210
strict.
217
00:15:05,210 --> 00:15:07,620
But that's a minor point.
218
00:15:07,620 --> 00:15:12,270
And certainly not one that
we'll pay close attention to.
219
00:15:12,270 --> 00:15:17,910
But now, let me show you what
this looks like geometrically.
220
00:15:17,910 --> 00:15:22,270
So geometrically, we interpret
this as the area under a curve.
221
00:15:22,270 --> 00:15:31,850
Here's a piece of the
curve y = 1/(1+x).
222
00:15:31,850 --> 00:15:37,130
And it's going up to 4
and starting at 0 here.
223
00:15:37,130 --> 00:15:41,510
And the first
estimate that we made
224
00:15:41,510 --> 00:15:46,030
- that is, the upper bound
- was by trapping this
225
00:15:46,030 --> 00:15:53,200
in this big rectangle here.
226
00:15:53,200 --> 00:15:55,610
We compared it to the
constant function,
227
00:15:55,610 --> 00:15:57,990
which was 1 all the way across.
228
00:15:57,990 --> 00:16:00,300
This is y = 1.
229
00:16:00,300 --> 00:16:05,780
And then we also trapped
it from underneath
230
00:16:05,780 --> 00:16:08,600
by the function which
was at the bottom.
231
00:16:08,600 --> 00:16:14,430
And this was y = 1/5.
232
00:16:14,430 --> 00:16:18,410
And so what this really
is is, these things
233
00:16:18,410 --> 00:16:21,060
are the simplest
possible Riemann sum.
234
00:16:21,060 --> 00:16:22,860
Sort of a silly Riemann sum.
235
00:16:22,860 --> 00:16:34,390
This is a Riemann sum
with one rectangle.
236
00:16:34,390 --> 00:16:36,420
This is the simplest
possible one.
237
00:16:36,420 --> 00:16:38,580
And so this is a very,
very crude estimate.
238
00:16:38,580 --> 00:16:40,970
You can see it misses by a mile.
239
00:16:40,970 --> 00:16:42,840
The larger and
the smaller values
240
00:16:42,840 --> 00:16:46,370
are off by a factor of 5.
241
00:16:46,370 --> 00:16:50,660
But this one is called
the-- this one is the lower
242
00:16:50,660 --> 00:16:53,390
Riemann sum.
243
00:16:53,390 --> 00:16:59,560
And that one is less
than our actual integral.
244
00:16:59,560 --> 00:17:14,100
Which is less than
the upper Riemann sum.
245
00:17:14,100 --> 00:17:19,070
And you should, by now, have
looked at those upper and lower
246
00:17:19,070 --> 00:17:20,670
sums on your homework.
247
00:17:20,670 --> 00:17:22,460
So it's just the
rectangles underneath
248
00:17:22,460 --> 00:17:25,390
and the rectangles on top.
249
00:17:25,390 --> 00:17:27,370
So at this point,
we can literally
250
00:17:27,370 --> 00:17:28,662
abandon the mean value theorem.
251
00:17:28,662 --> 00:17:30,953
Because we have a much better
way of getting at things.
252
00:17:30,953 --> 00:17:32,770
If we chop things up
into more rectangles,
253
00:17:32,770 --> 00:17:35,550
we'll get much better
numerical approximations.
254
00:17:35,550 --> 00:17:38,587
And if we use simpleminded
expressions with integrals,
255
00:17:38,587 --> 00:17:40,670
we'll be able to figure
out any bound we could get
256
00:17:40,670 --> 00:17:42,610
using the mean value theorem.
257
00:17:42,610 --> 00:17:45,790
So that's not the relevance
of the mean value theorem.
258
00:17:45,790 --> 00:17:48,810
I'll explain to you why
we talked about it, even,
259
00:17:48,810 --> 00:17:51,310
in a few minutes.
260
00:17:51,310 --> 00:17:59,790
OK, are there any
questions before we go on?
261
00:17:59,790 --> 00:18:00,290
Yeah.
262
00:18:00,290 --> 00:18:07,000
STUDENT: [INAUDIBLE]
263
00:18:07,000 --> 00:18:09,860
PROFESSOR: I knew that the
range of c was from 0 to 4,
264
00:18:09,860 --> 00:18:11,980
I should have said
that right here.
265
00:18:11,980 --> 00:18:13,715
This is true for this theorem.
266
00:18:13,715 --> 00:18:16,720
The mean value theorem comes
with an extra statement,
267
00:18:16,720 --> 00:18:17,950
which I missed.
268
00:18:17,950 --> 00:18:21,960
Which is that this is for
some c between 0 and 4.
269
00:18:21,960 --> 00:18:23,690
So I know the range
is between 0 and 4.
270
00:18:23,690 --> 00:18:25,148
The reason why it's
between 0 and 4
271
00:18:25,148 --> 00:18:27,560
is that's part of the
mean value theorem.
272
00:18:27,560 --> 00:18:29,610
We started at 0, we ended at 4.
273
00:18:29,610 --> 00:18:32,420
So the c has to be
somewhere in between.
274
00:18:32,420 --> 00:18:42,057
That's part of the
mean value theorem.
275
00:18:42,057 --> 00:18:42,890
STUDENT: [INAUDIBLE]
276
00:18:42,890 --> 00:18:43,730
PROFESSOR: The
question is, do you
277
00:18:43,730 --> 00:18:45,820
exclude any values that
are above 4 and below 0.
278
00:18:45,820 --> 00:18:46,884
Yes, absolutely.
279
00:18:46,884 --> 00:18:48,550
The point is that in
order to figure out
280
00:18:48,550 --> 00:18:51,942
how F changes, capital F
changes, between 0 and 4,
281
00:18:51,942 --> 00:18:54,150
you need only pay attention
to the values in between.
282
00:18:54,150 --> 00:18:55,608
You don't have to
pay any attention
283
00:18:55,608 --> 00:18:59,740
to what the function is
doing below 0 or above 4.
284
00:18:59,740 --> 00:19:07,090
Those things are
strictly irrelevant.
285
00:19:07,090 --> 00:19:16,569
STUDENT: [INAUDIBLE]
286
00:19:16,569 --> 00:19:18,110
PROFESSOR: Yeah, I
mean it's strictly
287
00:19:18,110 --> 00:19:19,950
in between these two numbers.
288
00:19:19,950 --> 00:19:23,420
I have to understand what the
lowest and the highest one is.
289
00:19:23,420 --> 00:19:24,700
STUDENT: [INAUDIBLE]
290
00:19:24,700 --> 00:19:35,031
PROFESSOR: It's
approaching that, so.
291
00:19:35,031 --> 00:19:35,530
OK.
292
00:19:35,530 --> 00:19:39,650
So now, the next thing that
we're going to talk about
293
00:19:39,650 --> 00:19:41,820
is, since I've got
that 1 up there,
294
00:19:41,820 --> 00:19:44,180
that Fundamental
Theorem of Calculus 1, I
295
00:19:44,180 --> 00:20:05,070
need to talk about version 2.
296
00:20:05,070 --> 00:20:15,530
So here is the Fundamental
Theorem of Calculus version 2.
297
00:20:15,530 --> 00:20:20,660
I'm going to start out
with a function little f,
298
00:20:20,660 --> 00:20:28,002
and I'm going to assume
that it's continuous.
299
00:20:28,002 --> 00:20:30,210
And then I'm going to define
a new function, which is
300
00:20:30,210 --> 00:20:33,030
defined as a definite integral.
301
00:20:33,030 --> 00:20:40,310
G(x) is the integral
from a to x of f(t) dt.
302
00:20:40,310 --> 00:20:42,829
Now, I want to emphasize here
because it's the first time
303
00:20:42,829 --> 00:20:44,370
that I'm writing
something like this,
304
00:20:44,370 --> 00:20:47,300
that this is a fairly
complicated gadget.
305
00:20:47,300 --> 00:20:52,110
It plays a very basic and very
fundamental but simple role,
306
00:20:52,110 --> 00:20:54,490
but it nevertheless is
a little complicated.
307
00:20:54,490 --> 00:20:58,370
What's happening here is
that the upper limit I've now
308
00:20:58,370 --> 00:21:05,840
called x, and the variable t
is ranging between a and x,
309
00:21:05,840 --> 00:21:08,030
and that the a and
the x are fixed
310
00:21:08,030 --> 00:21:12,290
when I calculate the integral.
311
00:21:12,290 --> 00:21:14,730
And the t is what's
called the dummy variable.
312
00:21:14,730 --> 00:21:16,290
It's the variable
of integration.
313
00:21:16,290 --> 00:21:21,420
You'll see a lot of people who
will mix this x with this t.
314
00:21:21,420 --> 00:21:25,190
And if you do that,
you will get confused,
315
00:21:25,190 --> 00:21:28,150
potentially hopelessly
confused, in this class.
316
00:21:28,150 --> 00:21:32,310
In 18.02 you will be
completely lost if you do that.
317
00:21:32,310 --> 00:21:33,720
So don't do it.
318
00:21:33,720 --> 00:21:38,410
Don't mix these two guys up.
319
00:21:38,410 --> 00:21:42,089
It's actually done by
many people in textbooks,
320
00:21:42,089 --> 00:21:43,130
and it's fairly careless.
321
00:21:43,130 --> 00:21:45,230
Especially in
old-fashioned textbooks.
322
00:21:45,230 --> 00:21:48,610
But don't do it.
323
00:21:48,610 --> 00:21:50,570
So here we have this G(x).
324
00:21:50,570 --> 00:21:56,250
Now, remember, this G(x)
really does make sense.
325
00:21:56,250 --> 00:21:59,650
If you give me an a,
and you give me an x,
326
00:21:59,650 --> 00:22:01,960
I can figure out what this
is, because I can figure out
327
00:22:01,960 --> 00:22:02,830
the Riemann sum.
328
00:22:02,830 --> 00:22:05,090
So of course I need to know
what the function is, too.
329
00:22:05,090 --> 00:22:07,960
But anyway, we have a numerical
procedure for figuring out
330
00:22:07,960 --> 00:22:09,920
what the function G is.
331
00:22:09,920 --> 00:22:13,010
Now, as is suggested by this
mysterious letter x being
332
00:22:13,010 --> 00:22:16,350
in the place where it is,
I'm actually going to vary x.
333
00:22:16,350 --> 00:22:19,240
So the conclusion is
that if this is true,
334
00:22:19,240 --> 00:22:21,870
and this is just a parenthesis,
not part of the theorem.
335
00:22:21,870 --> 00:22:25,770
It's just an indication of
what the notation means.
336
00:22:25,770 --> 00:22:40,430
Then G' = f.
337
00:22:40,430 --> 00:22:43,030
Let me first explain what the
significance of this theorem
338
00:22:43,030 --> 00:22:48,280
is, from the point of view
of differential equations.
339
00:22:48,280 --> 00:23:04,420
G(x) solves the differential
equation y' = f(x).
340
00:23:04,420 --> 00:23:09,390
So y' = f, I shouldn't put
the x in if I got it here,
341
00:23:09,390 --> 00:23:13,240
with the condition y(a) = 0.
342
00:23:13,240 --> 00:23:19,000
So it solves this pair
of conditions here.
343
00:23:19,000 --> 00:23:21,860
The rate of change, and
the initial position
344
00:23:21,860 --> 00:23:23,280
is specified here.
345
00:23:23,280 --> 00:23:29,660
Because when you integrate
from a to a, you get 0 always.
346
00:23:29,660 --> 00:23:34,100
And what this theorem
says is you can always
347
00:23:34,100 --> 00:23:35,700
solve that equation.
348
00:23:35,700 --> 00:23:38,130
When we did differential
equations, I said that already.
349
00:23:38,130 --> 00:23:39,990
I said we'll treat
these as always solved.
350
00:23:39,990 --> 00:23:41,530
Well, here's the reason.
351
00:23:41,530 --> 00:23:45,080
We have a numerical procedure
for computing things like this.
352
00:23:45,080 --> 00:23:49,330
We could always
solve this equation.
353
00:23:49,330 --> 00:23:52,290
And the formula is a
fairly complicated gadget,
354
00:23:52,290 --> 00:23:58,400
but so far just associated
with Riemann sums.
355
00:23:58,400 --> 00:24:01,000
Alright, now.
356
00:24:01,000 --> 00:24:13,820
Let's just do one example.
357
00:24:13,820 --> 00:24:17,900
Unfortunately, not a complicated
example and maybe not
358
00:24:17,900 --> 00:24:21,230
persuasive as to why you would
care about this just yet.
359
00:24:21,230 --> 00:24:23,670
But nevertheless very important.
360
00:24:23,670 --> 00:24:26,470
Because this is the quiz
question which everybody gets
361
00:24:26,470 --> 00:24:29,530
wrong until they practice it.
362
00:24:29,530 --> 00:24:35,960
So the integral from,
say 1 to x, of dt / t^2.
363
00:24:38,710 --> 00:24:45,420
Let's try this one here.
364
00:24:45,420 --> 00:24:56,650
So here's an example of
this theorem, I claim.
365
00:24:56,650 --> 00:25:00,850
Now, this is a question
which challenges your ability
366
00:25:00,850 --> 00:25:04,740
to understand what
the question means.
367
00:25:04,740 --> 00:25:06,800
Because it's got
a lot of symbols.
368
00:25:06,800 --> 00:25:09,730
It's got the integration and
it's got the differentiation.
369
00:25:09,730 --> 00:25:15,700
However, what it really is
is an exercise in recopying.
370
00:25:15,700 --> 00:25:20,660
You look at it and you
write down the answer.
371
00:25:20,660 --> 00:25:24,570
And the reason is
that, by definition,
372
00:25:24,570 --> 00:25:29,190
this function in here is
a function of the form
373
00:25:29,190 --> 00:25:34,320
G(x) of the theorem over here.
374
00:25:34,320 --> 00:25:35,920
So this is the G(x).
375
00:25:35,920 --> 00:25:42,980
And by definition, we
said that G'(x) = f(x).
376
00:25:42,980 --> 00:25:46,170
Well, what's the f(x)?
377
00:25:46,170 --> 00:25:47,010
Look inside here.
378
00:25:47,010 --> 00:25:48,760
It's what's called
the integrand.
379
00:25:48,760 --> 00:25:53,640
This is the integral from
0 to x of f(t) dt, right?
380
00:25:53,640 --> 00:26:00,710
Where the f(t) is
equal to 1 / t^2.
381
00:26:00,710 --> 00:26:02,910
So your ability is challenged.
382
00:26:02,910 --> 00:26:06,060
You have to take that 1 /
t^2 and you have to plug
383
00:26:06,060 --> 00:26:09,960
in the letter x,
instead of t, for it.
384
00:26:09,960 --> 00:26:11,480
And then write it down.
385
00:26:11,480 --> 00:26:18,120
As I say, this is an exercise
in recopying what's there.
386
00:26:18,120 --> 00:26:19,917
So this is quite
easy to do, right?
387
00:26:19,917 --> 00:26:21,750
I mean, you just look
and you write it down.
388
00:26:21,750 --> 00:26:28,260
But nevertheless, it looks like
a long, elaborate object here.
389
00:26:28,260 --> 00:26:28,760
Pardon me?
390
00:26:28,760 --> 00:26:30,590
STUDENT: [INAUDIBLE]
391
00:26:30,590 --> 00:26:32,940
PROFESSOR: So the question
was, why did I integrate.
392
00:26:32,940 --> 00:26:34,000
STUDENT: [INAUDIBLE]
393
00:26:34,000 --> 00:26:36,990
PROFESSOR: Why did
I not integrate?
394
00:26:36,990 --> 00:26:37,720
Ah.
395
00:26:37,720 --> 00:26:38,870
Very good question.
396
00:26:38,870 --> 00:26:41,010
Why did I not integrate.
397
00:26:41,010 --> 00:26:45,300
The reason why I didn't
integrate is I didn't need to.
398
00:26:45,300 --> 00:26:47,665
Just as when you take the
antiderivative-- sorry,
399
00:26:47,665 --> 00:26:50,040
the derivative of something,
you take the antiderivative,
400
00:26:50,040 --> 00:26:51,750
you get back to the thing.
401
00:26:51,750 --> 00:26:54,947
So, in this case, we're taking
the antiderivative of something
402
00:26:54,947 --> 00:26:56,030
and we're differentiating.
403
00:26:56,030 --> 00:26:58,380
So we end back in the same
place where we started.
404
00:26:58,380 --> 00:27:01,780
We started with f(t),
we're ending with f.
405
00:27:01,780 --> 00:27:04,990
Little f.
406
00:27:04,990 --> 00:27:06,950
So you integrate, and
then differentiate.
407
00:27:06,950 --> 00:27:09,410
And you get back
to the same place.
408
00:27:09,410 --> 00:27:12,330
Now, the only difference between
this and the other version
409
00:27:12,330 --> 00:27:15,610
is, in this case when you
differentiate and integrate
410
00:27:15,610 --> 00:27:18,195
you could be off by a constant.
411
00:27:18,195 --> 00:27:19,570
That's what that
shift, why there
412
00:27:19,570 --> 00:27:21,420
are two pieces to this one.
413
00:27:21,420 --> 00:27:23,020
But there's never
an extra piece here.
414
00:27:23,020 --> 00:27:24,919
There's no plus c here.
415
00:27:24,919 --> 00:27:26,460
When you integrate
and differentiate,
416
00:27:26,460 --> 00:27:28,080
you kill whatever
the constant is.
417
00:27:28,080 --> 00:27:31,920
Because the derivative
of a constant is 0.
418
00:27:31,920 --> 00:27:36,220
So no matter what the constant
is, hiding inside of G,
419
00:27:36,220 --> 00:27:41,140
you're getting the same result.
So this is the basic idea.
420
00:27:41,140 --> 00:27:46,220
Now, I just want
to double-check it,
421
00:27:46,220 --> 00:27:52,124
using the Fundamental
Theorem of Calculus 1 here.
422
00:27:52,124 --> 00:27:53,790
So let's actually
evaluate the integral.
423
00:27:53,790 --> 00:27:54,880
So now I'm going
to do what you've
424
00:27:54,880 --> 00:27:56,185
suggested, which
is I'm just going
425
00:27:56,185 --> 00:27:57,370
to check whether it's true.
426
00:27:57,370 --> 00:27:59,560
No, no I am because I'm
going just double-check
427
00:27:59,560 --> 00:28:01,120
that it's consistent.
428
00:28:01,120 --> 00:28:03,447
It certainly is slower
this way, and we're not
429
00:28:03,447 --> 00:28:05,030
going to want to do
this all the time,
430
00:28:05,030 --> 00:28:06,810
but we might as well check one.
431
00:28:06,810 --> 00:28:09,370
So this is our integral.
432
00:28:09,370 --> 00:28:10,860
And we know how to do it.
433
00:28:10,860 --> 00:28:13,260
No, I need to do it.
434
00:28:13,260 --> 00:28:17,820
And this is -t^(-1),
evaluated at 1 and x.
435
00:28:17,820 --> 00:28:21,670
Again, there's something
subliminally here
436
00:28:21,670 --> 00:28:23,300
for you to think about.
437
00:28:23,300 --> 00:28:28,737
Which is that, remember, it's t
is ranging between 1 and t = x.
438
00:28:28,737 --> 00:28:30,820
And this is one of the big
reasons why this letter
439
00:28:30,820 --> 00:28:32,860
t has to be different from x.
440
00:28:32,860 --> 00:28:35,370
Because here it's
1 and there it's x.
441
00:28:35,370 --> 00:28:37,210
It's not x.
442
00:28:37,210 --> 00:28:38,890
So you can't put an x here.
443
00:28:38,890 --> 00:28:44,130
Again, this is t = 1 and
this is t = x over there.
444
00:28:44,130 --> 00:28:48,630
And now if I plug
that in, I get what?
445
00:28:48,630 --> 00:28:55,860
I get -1/x, and
then I get -(-1).
446
00:28:55,860 --> 00:28:59,090
So this is, let me get rid
of those little t's there.
447
00:28:59,090 --> 00:29:05,690
This is a little easier to read.
448
00:29:05,690 --> 00:29:07,030
And so now let's check it.
449
00:29:07,030 --> 00:29:07,730
It's d/dx.
450
00:29:07,730 --> 00:29:09,500
So here's what G(x) is.
451
00:29:09,500 --> 00:29:12,260
G(x) = 1 - 1/x.
452
00:29:12,260 --> 00:29:14,860
That's what G(x) is.
453
00:29:14,860 --> 00:29:20,500
And if I differentiate
that, I get +1 / x^2.
454
00:29:20,500 --> 00:29:26,170
That's it.
455
00:29:26,170 --> 00:29:40,810
You see the constant
washed away.
456
00:29:40,810 --> 00:29:42,070
So now, here's my job.
457
00:29:42,070 --> 00:29:44,440
My job is to prove
these theorems.
458
00:29:44,440 --> 00:29:45,809
I never did prove them for you.
459
00:29:45,809 --> 00:29:47,725
So, I'm going to prove
the Fundamental Theorem
460
00:29:47,725 --> 00:29:49,340
of Calculus.
461
00:29:49,340 --> 00:29:51,860
But I'm going to do 2 first.
462
00:29:51,860 --> 00:29:53,420
And then I'm going to do 1.
463
00:29:53,420 --> 00:29:56,490
And it's just going to take
me just one blackboard.
464
00:29:56,490 --> 00:30:00,310
It's not that hard.
465
00:30:00,310 --> 00:30:03,150
The proof is by picture.
466
00:30:03,150 --> 00:30:08,310
And, using the interpretation
as area under the curve.
467
00:30:08,310 --> 00:30:12,170
So if here's the
value of a, and this
468
00:30:12,170 --> 00:30:22,550
is the graph of the
function y equals f of x.
469
00:30:22,550 --> 00:30:26,380
Then I want to draw
three vertical lines.
470
00:30:26,380 --> 00:30:29,330
One of them is going to be at x.
471
00:30:29,330 --> 00:30:33,790
And one of them is going
to be at x + delta x.
472
00:30:33,790 --> 00:30:35,950
So here I have the
interval from 0
473
00:30:35,950 --> 00:30:39,510
to x, and next I have the
interval from x to delta
474
00:30:39,510 --> 00:30:42,900
x more than that.
475
00:30:42,900 --> 00:30:50,360
And now the pieces that I've
got are the area of this part.
476
00:30:50,360 --> 00:30:53,430
So this has area
which has a name.
477
00:30:53,430 --> 00:30:55,860
It's called G(x).
478
00:30:55,860 --> 00:31:00,790
By definition, G(x), which
is sitting right over here
479
00:31:00,790 --> 00:31:03,940
in the fundamental theorem,
is the integral from a
480
00:31:03,940 --> 00:31:06,100
to x of this function.
481
00:31:06,100 --> 00:31:07,890
So it's the area
under the curve.
482
00:31:07,890 --> 00:31:10,400
So that area is G(x).
483
00:31:10,400 --> 00:31:17,770
Now this other
chunk here, I claim
484
00:31:17,770 --> 00:31:23,430
that this is delta G. This
is the change in G. It's
485
00:31:23,430 --> 00:31:26,260
the value of G(x) that is the
area of the whole business all
486
00:31:26,260 --> 00:31:30,530
the way up to x + delta x
minus the first part, G(x).
487
00:31:30,530 --> 00:31:31,640
So it's what's left over.
488
00:31:31,640 --> 00:31:39,780
It's the incremental
amount of area there.
489
00:31:39,780 --> 00:31:45,830
And now I am going to carry out
a pretty standard estimation
490
00:31:45,830 --> 00:31:46,520
here.
491
00:31:46,520 --> 00:31:48,850
This is practically a rectangle.
492
00:31:48,850 --> 00:31:51,700
And it's got a base of delta
x, and so we need to figure out
493
00:31:51,700 --> 00:31:55,320
what its height is.
494
00:31:55,320 --> 00:32:02,120
This is delta G, and it's
approximately its base
495
00:32:02,120 --> 00:32:05,340
times its height.
496
00:32:05,340 --> 00:32:06,820
But what is the height?
497
00:32:06,820 --> 00:32:10,740
Well, the height is maybe either
this segment or this segment
498
00:32:10,740 --> 00:32:11,920
or something in between.
499
00:32:11,920 --> 00:32:13,720
But they're all about the same.
500
00:32:13,720 --> 00:32:17,070
So I'm just going to put in
the value at the first point.
501
00:32:17,070 --> 00:32:19,430
That's the left end there.
502
00:32:19,430 --> 00:32:25,550
So that's this
height here, is f(x).
503
00:32:25,550 --> 00:32:28,200
So this is f(x), and so
really I approximate it
504
00:32:28,200 --> 00:32:30,050
by that rectangle there.
505
00:32:30,050 --> 00:32:33,760
And now if I divide
and take the limit,
506
00:32:33,760 --> 00:32:38,510
as delta x goes to 0,
of delta G / delta x,
507
00:32:38,510 --> 00:32:40,270
it's going to equal f(x).
508
00:32:43,120 --> 00:32:48,920
And this is where I'm using
the fact that f is continuous.
509
00:32:48,920 --> 00:32:51,270
Because I need the
values nearby to be
510
00:32:51,270 --> 00:32:59,690
similar to the
value in the limit.
511
00:32:59,690 --> 00:33:00,650
OK, that's the end.
512
00:33:00,650 --> 00:33:05,020
This the end of the proof, so
I'll put a nice little Q.E.D.
513
00:33:05,020 --> 00:33:10,800
here.
514
00:33:10,800 --> 00:33:14,360
So we've done Fundamental
Theorem of Calculus 2,
515
00:33:14,360 --> 00:33:19,440
and now we're ready for
Fundamental Theorem of Calculus
516
00:33:19,440 --> 00:33:38,420
1.
517
00:33:38,420 --> 00:33:44,580
So now I still have it on
the blackboard to remind you.
518
00:33:44,580 --> 00:33:48,200
It says that the integral
of the derivative
519
00:33:48,200 --> 00:33:50,980
is the function, at least the
difference between the values
520
00:33:50,980 --> 00:33:54,830
of the function at two places.
521
00:33:54,830 --> 00:34:08,040
So the place where we start is
with this property that F' = f.
522
00:34:08,040 --> 00:34:10,600
That's the starting--
that's the hypothesis.
523
00:34:10,600 --> 00:34:12,710
Now, unfortunately,
I'm going to have
524
00:34:12,710 --> 00:34:14,420
to assume something
extra in order
525
00:34:14,420 --> 00:34:18,660
to use the Fundamental
Theorem of Calculus 2,
526
00:34:18,660 --> 00:34:27,980
which is I'm going to
assume that f is continuous.
527
00:34:27,980 --> 00:34:30,490
That's not really
necessary, but that's
528
00:34:30,490 --> 00:34:32,170
just a very minor
technical point,
529
00:34:32,170 --> 00:34:34,280
which I'm just going to ignore.
530
00:34:34,280 --> 00:34:40,610
So we're going to
start with F' = f.
531
00:34:40,610 --> 00:34:46,530
And then I'm going
to go somewhere else.
532
00:34:46,530 --> 00:34:52,620
I'm going to define
a new function, G(x),
533
00:34:52,620 --> 00:35:00,130
which is the integral
from a to x of f(t) dt.
534
00:35:00,130 --> 00:35:04,560
This is where we needed all
of the labor of Riemann sums.
535
00:35:04,560 --> 00:35:07,170
Because otherwise we don't
have a way of making sense out
536
00:35:07,170 --> 00:35:10,340
of what this even means.
537
00:35:10,340 --> 00:35:13,350
So hiding behind
this one sentence
538
00:35:13,350 --> 00:35:16,050
is the fact that we
actually have a number.
539
00:35:16,050 --> 00:35:18,430
We have a formula
for such functions.
540
00:35:18,430 --> 00:35:20,550
So there is a
function G(x) which,
541
00:35:20,550 --> 00:35:22,220
once you've produced
a little f for me,
542
00:35:22,220 --> 00:35:27,420
I can cook up a function
capital G for you.
543
00:35:27,420 --> 00:35:31,390
Now, we're going to apply this
Fundamental Theorem of Calculus
544
00:35:31,390 --> 00:35:34,650
2, the one that we've
already checked.
545
00:35:34,650 --> 00:35:36,030
So what does it say?
546
00:35:36,030 --> 00:35:46,510
It says that G' = f.
547
00:35:46,510 --> 00:35:49,820
And so now we're in the
following situation.
548
00:35:49,820 --> 00:35:54,030
We know that F'(x) = G'(x).
549
00:35:58,080 --> 00:36:00,360
That's what we've got so far.
550
00:36:00,360 --> 00:36:07,210
And now we have one last step to
get a good connection between F
551
00:36:07,210 --> 00:36:10,120
and G. Which is that we
can conclude that F(x)
552
00:36:10,120 --> 00:36:12,060
is G(x) plus a constant.
553
00:36:20,440 --> 00:36:26,520
Now, this little step
may seem innocuous
554
00:36:26,520 --> 00:36:32,610
but I remind you that this
is the spot that requires
555
00:36:32,610 --> 00:36:36,030
the mean value theorem.
556
00:36:36,030 --> 00:36:39,650
So in order not to lie
to you, we actually
557
00:36:39,650 --> 00:36:42,680
tell you what the underpinnings
of all of calculus are.
558
00:36:42,680 --> 00:36:45,152
And they're this: the
fact, if you like,
559
00:36:45,152 --> 00:36:47,110
that if two functions
have the same derivative,
560
00:36:47,110 --> 00:36:48,430
they differ by a constant.
561
00:36:48,430 --> 00:36:50,750
Or that if a function
has derivative 0,
562
00:36:50,750 --> 00:36:53,960
it's a constant itself.
563
00:36:53,960 --> 00:36:57,850
Now, that is the fundamental
step that's needed,
564
00:36:57,850 --> 00:36:59,740
the underlying
step that's needed.
565
00:36:59,740 --> 00:37:02,500
And, unfortunately, there
aren't any proofs of it
566
00:37:02,500 --> 00:37:06,050
that are less complicated than
using the mean value theorem.
567
00:37:06,050 --> 00:37:08,560
And so that's why we talk a
little bit about the mean value
568
00:37:08,560 --> 00:37:10,934
theorem, because we don't want
to lie to you about what's
569
00:37:10,934 --> 00:37:11,731
really going on.
570
00:37:11,731 --> 00:37:12,230
Yes.
571
00:37:12,230 --> 00:37:19,070
STUDENT: [INAUDIBLE]
572
00:37:19,070 --> 00:37:24,210
PROFESSOR: The question is how
did I get from here, to here.
573
00:37:24,210 --> 00:37:27,440
And the answer is that
if G' is little f,
574
00:37:27,440 --> 00:37:32,950
and we also know that F'
is little f, then F' is G'.
575
00:37:32,950 --> 00:37:37,090
OK.
576
00:37:37,090 --> 00:37:50,710
Other questions?
577
00:37:50,710 --> 00:37:52,750
Alright, so we're almost done.
578
00:37:52,750 --> 00:37:57,400
I just have to work out
the arithmetic here.
579
00:37:57,400 --> 00:38:02,030
So I start with F(b) - F(a).
580
00:38:04,810 --> 00:38:12,330
And that's equal to
(G(b) + c) - (G(a) + c).
581
00:38:18,130 --> 00:38:20,970
And then I cancel the c's.
582
00:38:20,970 --> 00:38:23,310
So I have here G(b) - G(a).
583
00:38:29,960 --> 00:38:32,665
And now I just have to
check what each of these is.
584
00:38:32,665 --> 00:38:35,470
So remember the
definition of G here.
585
00:38:35,470 --> 00:38:38,100
G(b) is just what we want.
586
00:38:38,100 --> 00:38:42,460
The integral from
a to b of f(x) dx.
587
00:38:42,460 --> 00:38:46,430
Well I called it f(t) dt,
that's the same as f(x) dx
588
00:38:46,430 --> 00:38:48,665
now, because I have
the limit being b
589
00:38:48,665 --> 00:38:52,900
and I'm allowed to use
x as the dummy variable.
590
00:38:52,900 --> 00:38:55,780
Now the other one,
I claim, is 0.
591
00:38:55,780 --> 00:38:59,680
Because it's the
integral from a to a.
592
00:38:59,680 --> 00:39:03,490
This one is the
integral from a to a.
593
00:39:03,490 --> 00:39:06,510
Which gives us 0.
594
00:39:06,510 --> 00:39:09,370
So this is just this minus
0, and that's the end.
595
00:39:09,370 --> 00:39:13,460
That's it.
596
00:39:13,460 --> 00:39:20,271
I started with F(b) - F(a),
I got to the integral.
597
00:39:20,271 --> 00:39:20,770
Question?
598
00:39:20,770 --> 00:39:27,370
STUDENT: [INAUDIBLE]
599
00:39:27,370 --> 00:39:32,610
PROFESSOR: How did I get from
F(b) - F(a), is (G(b) + c) -
600
00:39:32,610 --> 00:39:35,430
(G(a) + c), that's the question.
601
00:39:35,430 --> 00:39:40,190
STUDENT: [INAUDIBLE]
602
00:39:40,190 --> 00:39:44,160
PROFESSOR: Oh, sorry
this is an equals sign.
603
00:39:44,160 --> 00:39:47,090
Sorry, the second
line didn't draw.
604
00:39:47,090 --> 00:39:48,640
OK, equals.
605
00:39:48,640 --> 00:39:53,141
Because we're plugging in,
for F(x), the formula for it.
606
00:39:53,141 --> 00:39:53,640
Yes.
607
00:39:53,640 --> 00:39:57,310
STUDENT: [INAUDIBLE]
608
00:39:57,310 --> 00:39:59,630
PROFESSOR: This step here?
609
00:39:59,630 --> 00:40:04,730
Or this one?
610
00:40:04,730 --> 00:40:09,392
STUDENT: [INAUDIBLE]
611
00:40:09,392 --> 00:40:10,100
PROFESSOR: Right.
612
00:40:10,100 --> 00:40:12,090
So that was a good question.
613
00:40:12,090 --> 00:40:15,120
But the answer is that
that's the statement
614
00:40:15,120 --> 00:40:16,246
that we're aiming for.
615
00:40:16,246 --> 00:40:18,120
That's the Fundamental
Theorem of Calculus 1,
616
00:40:18,120 --> 00:40:19,614
which we don't know yet.
617
00:40:19,614 --> 00:40:22,030
So we're trying to prove it,
and that's why we haven't, we
618
00:40:22,030 --> 00:40:25,950
can't assume it.
619
00:40:25,950 --> 00:40:33,270
OK, so let me just notice that
in the example that we had,
620
00:40:33,270 --> 00:40:36,410
before we go on to
something else here.
621
00:40:36,410 --> 00:40:48,530
In the example above, what we
had was the following thing.
622
00:40:48,530 --> 00:40:58,070
We had, say, F(x) = -1/x.
623
00:40:58,070 --> 00:41:01,890
So F'(x) = 1 / x^2.
624
00:41:01,890 --> 00:41:08,850
And, say, G(x) = 1 - 1/x.
625
00:41:08,850 --> 00:41:11,970
And you can see that
either way you do that,
626
00:41:11,970 --> 00:41:14,310
if you integrate from
1 to 2, let's say,
627
00:41:14,310 --> 00:41:18,670
which is what we had
over there, dt / t^2,
628
00:41:18,670 --> 00:41:28,140
you're going to get either -1/t,
1 to 2, or, if you like, 1 -
629
00:41:28,140 --> 00:41:31,200
1/t, 1 to 2.
630
00:41:31,200 --> 00:41:33,930
So this is the F version,
this is the G version.
631
00:41:33,930 --> 00:41:45,410
And that's what plays itself
out here, in this general proof.
632
00:41:45,410 --> 00:41:49,640
Alright.
633
00:41:49,640 --> 00:41:54,910
So now I want to go back to
the theme for today, which
634
00:41:54,910 --> 00:42:01,290
is using little f to understand
capital F. In other words,
635
00:42:01,290 --> 00:42:05,890
using the derivative of F
to understand capital F.
636
00:42:05,890 --> 00:42:22,630
And I want to illustrate it by
some more complicated examples.
637
00:42:22,630 --> 00:42:27,510
So I guess I just erased it, but
we just took the antiderivative
638
00:42:27,510 --> 00:42:29,740
of 1 / t^2.
639
00:42:29,740 --> 00:42:34,340
And there's-- all of the
powers work easily, but one.
640
00:42:34,340 --> 00:42:39,190
And the tricky one
is the power 1 / x.
641
00:42:39,190 --> 00:42:43,280
So let's consider the
differential equation L' (x) =
642
00:42:43,280 --> 00:42:44,550
1 / x.
643
00:42:44,550 --> 00:42:54,490
And say, with the
initial value L(1) = 0.
644
00:42:54,490 --> 00:42:58,840
The solution, so the Fundamental
Theorem of Calculus 2
645
00:42:58,840 --> 00:43:07,330
tells us the solution
is this function here.
646
00:43:07,330 --> 00:43:14,070
L(x) equals the integral
from 1 to x, dt / t.
647
00:43:14,070 --> 00:43:16,490
That's how we solve
all such equations.
648
00:43:16,490 --> 00:43:19,290
We just integrate, take
the definite integral.
649
00:43:19,290 --> 00:43:27,400
And I'm starting at 1 because
I insisted that L(1) be 0.
650
00:43:27,400 --> 00:43:31,100
So that's the solution
to the problem.
651
00:43:31,100 --> 00:43:33,900
And now the thing
that's interesting here
652
00:43:33,900 --> 00:43:35,720
is that we started
from a polynomial.
653
00:43:35,720 --> 00:43:37,990
Or we started from a rational,
a ratio of polynomials;
654
00:43:37,990 --> 00:43:40,460
that is, 1 / t or 1 / x.
655
00:43:40,460 --> 00:43:42,490
And we get to a function
which is actually
656
00:43:42,490 --> 00:43:44,990
what's known as a
transcendental function.
657
00:43:44,990 --> 00:43:46,455
It's not an algebraic function.
658
00:43:46,455 --> 00:43:47,080
Yeah, question.
659
00:43:47,080 --> 00:43:57,000
STUDENT: [INAUDIBLE]
660
00:43:57,000 --> 00:44:04,970
PROFESSOR: The question is
why is this equal to that.
661
00:44:04,970 --> 00:44:08,210
And the answer is, it's
for the same reason
662
00:44:08,210 --> 00:44:10,200
that this is equal to that.
663
00:44:10,200 --> 00:44:14,030
It's the same reason as this.
664
00:44:14,030 --> 00:44:16,760
It's that the 1's cancel.
665
00:44:16,760 --> 00:44:19,800
We've taken the value of
something at 2 minus the value
666
00:44:19,800 --> 00:44:20,300
at 1.
667
00:44:20,300 --> 00:44:22,500
The value at 2 minus
the value at 1.
668
00:44:22,500 --> 00:44:24,720
And you'll get a 1 in the
one case, and you get a 1
669
00:44:24,720 --> 00:44:25,470
in the other case.
670
00:44:25,470 --> 00:44:27,580
And you subtract them
and they will cancel.
671
00:44:27,580 --> 00:44:28,620
They'll give you 0.
672
00:44:28,620 --> 00:44:31,380
These two things
really are equal.
673
00:44:31,380 --> 00:44:33,507
This is not a function
evaluated at one place,
674
00:44:33,507 --> 00:44:35,590
it's the difference between
the function evaluated
675
00:44:35,590 --> 00:44:37,025
at 2 and the value at 1.
676
00:44:37,025 --> 00:44:39,570
And whenever you subtract
two things like that,
677
00:44:39,570 --> 00:44:40,900
constants drop out.
678
00:44:40,900 --> 00:44:44,370
STUDENT: [INAUDIBLE]
PROFESSOR: That's right.
679
00:44:44,370 --> 00:44:46,030
If I put 2 here,
if I put c here,
680
00:44:46,030 --> 00:44:47,370
it would have been the same.
681
00:44:47,370 --> 00:44:49,010
It would just have dropped out.
682
00:44:49,010 --> 00:44:50,640
It's not there.
683
00:44:50,640 --> 00:44:53,260
And that's exactly this
arithmetic right here.
684
00:44:53,260 --> 00:44:55,520
It doesn't matter which
antiderivative you take.
685
00:44:55,520 --> 00:44:57,850
When you take the differences,
the c's will cancel.
686
00:44:57,850 --> 00:45:03,130
You always get the
same answer in the end.
687
00:45:03,130 --> 00:45:04,850
That's exactly why
I wrote this down,
688
00:45:04,850 --> 00:45:06,290
so that you would see that.
689
00:45:06,290 --> 00:45:12,890
It doesn't matter
which one you do.
690
00:45:12,890 --> 00:45:21,540
So, we still have a couple
of minutes left here.
691
00:45:21,540 --> 00:45:23,630
This is actually--
So let me go back.
692
00:45:23,630 --> 00:45:29,720
So here's the antiderivative
of 1 / x, with value 1 at 0.
693
00:45:29,720 --> 00:45:32,180
Now, in disguise, we know
what this function is.
694
00:45:32,180 --> 00:45:35,410
We know this function is
the logarithm function.
695
00:45:35,410 --> 00:45:37,580
But this is actually
a better way
696
00:45:37,580 --> 00:45:42,030
of deriving all of the
formulas for the logarithm.
697
00:45:42,030 --> 00:45:44,850
This is a much quicker and
more efficient way of doing it.
698
00:45:44,850 --> 00:45:47,280
We had to do it by very
laborious processes.
699
00:45:47,280 --> 00:45:51,690
This will allow us
to do it very easily.
700
00:45:51,690 --> 00:45:56,040
And so, I'm going to
do that next time.
701
00:45:56,040 --> 00:45:58,395
But rather than do
that now, I'm going
702
00:45:58,395 --> 00:46:10,280
to point out to you that we can
also get truly new functions.
703
00:46:10,280 --> 00:46:12,130
OK, so there are all
kinds of new functions.
704
00:46:12,130 --> 00:46:15,690
So this is the first example
of this kind would be,
705
00:46:15,690 --> 00:46:21,630
for example, to solve the
equation y' = e^(-x^2) with
706
00:46:21,630 --> 00:46:25,140
y(0) = 0, let's say.
707
00:46:25,140 --> 00:46:28,047
Now, the solution to
that is a function
708
00:46:28,047 --> 00:46:30,380
which again I can write down
by the fundamental theorem.
709
00:46:30,380 --> 00:46:48,160
It's the integral from
0 to x of e^(-t^2) dt.
710
00:46:48,160 --> 00:46:52,370
This is a very famous function.
711
00:46:52,370 --> 00:46:55,820
This shape here is
known as the bell curve.
712
00:46:55,820 --> 00:46:59,170
And it's the thing that comes
up in probability all the time.
713
00:46:59,170 --> 00:47:01,880
This shape e^(-x^2).
714
00:47:01,880 --> 00:47:04,590
And our function is
geometrically just the area
715
00:47:04,590 --> 00:47:06,310
under the curve here.
716
00:47:06,310 --> 00:47:09,080
This is F(x).
717
00:47:09,080 --> 00:47:12,350
If this place is x.
718
00:47:12,350 --> 00:47:13,860
So I have a
geometric definition,
719
00:47:13,860 --> 00:47:16,620
I have a way of constructing
what it is by Riemann sums.
720
00:47:16,620 --> 00:47:18,810
And I have a function here.
721
00:47:18,810 --> 00:47:26,850
But the curious thing about
F(x) is that F(x) cannot be
722
00:47:26,850 --> 00:47:34,470
expressed in terms of
any function you've seen
723
00:47:34,470 --> 00:47:35,360
previously.
724
00:47:35,360 --> 00:47:44,210
So logs, exponentials,
trig functions, cannot be.
725
00:47:44,210 --> 00:47:51,600
It's a totally new function.
726
00:47:51,600 --> 00:47:53,730
Nevertheless, we'll
be able to get
727
00:47:53,730 --> 00:47:56,030
any possible piece of
information we would want to,
728
00:47:56,030 --> 00:47:56,910
out of this function.
729
00:47:56,910 --> 00:47:58,770
It's perfectly
acceptable function,
730
00:47:58,770 --> 00:48:00,250
it will work just great for us.
731
00:48:00,250 --> 00:48:01,700
Just like any other function.
732
00:48:01,700 --> 00:48:03,540
Just like the log.
733
00:48:03,540 --> 00:48:08,890
And what this is analogous to
is the following kind of thing.
734
00:48:08,890 --> 00:48:12,650
If you take the circle, the
ancient Greeks, if you like,
735
00:48:12,650 --> 00:48:18,030
already understood that if you
have a circle of radius 1, then
736
00:48:18,030 --> 00:48:23,290
its area is pi.
737
00:48:23,290 --> 00:48:24,820
So that's a geometric
construction
738
00:48:24,820 --> 00:48:31,430
of what you could
call a new number.
739
00:48:31,430 --> 00:48:34,280
Which is outside of the realm
of what you might expect.
740
00:48:34,280 --> 00:48:37,840
And the weird thing
about this number pi
741
00:48:37,840 --> 00:48:49,780
is that it is not the root
of an algebraic equation
742
00:48:49,780 --> 00:48:57,260
with rational coefficients.
743
00:48:57,260 --> 00:49:00,540
It's what's called
transcendental.
744
00:49:00,540 --> 00:49:02,680
Meaning, it's just completely
outside of the realm
745
00:49:02,680 --> 00:49:04,050
of algebra.
746
00:49:04,050 --> 00:49:06,660
And, indeed, the
logarithm function
747
00:49:06,660 --> 00:49:08,790
is called a
transcendental function,
748
00:49:08,790 --> 00:49:11,350
because it's completely out
of the realm of algebra.
749
00:49:11,350 --> 00:49:14,710
It's only in calculus
that you come up
750
00:49:14,710 --> 00:49:16,190
with this kind of thing.
751
00:49:16,190 --> 00:49:20,030
So these kinds of
functions will have access
752
00:49:20,030 --> 00:49:23,130
to a huge class of new
functions here, all of which
753
00:49:23,130 --> 00:49:26,730
are important tools in
science and engineering.
754
00:49:26,730 --> 00:49:29,537
So, see you next time.