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