1
00:00:00,000 --> 00:00:01,940
FEMALE SPEAKER: The following
content is provided under a
2
00:00:01,940 --> 00:00:03,690
Creative Commons license.
3
00:00:03,690 --> 00:00:06,630
Your support will help MIT
OpenCourseWare continue to
4
00:00:06,630 --> 00:00:09,990
offer high-quality educational
resources for free.
5
00:00:09,990 --> 00:00:12,830
To make a donation or to view
additional materials from
6
00:00:12,830 --> 00:00:16,760
hundreds of MIT courses, visit
MIT OpenCourseWare at
7
00:00:16,760 --> 00:00:18,010
ocw.mit.edu.
8
00:00:18,010 --> 00:00:27,675
9
00:00:27,675 --> 00:00:28,340
PROFESSOR: Hi.
10
00:00:28,340 --> 00:00:31,370
Our subject matter today
concerns that phase of
11
00:00:31,370 --> 00:00:36,080
calculus known as the
'indefinite integral', or the
12
00:00:36,080 --> 00:00:38,010
'antiderivative'.
13
00:00:38,010 --> 00:00:41,260
In terms of the concepts that
we've talked about so far in
14
00:00:41,260 --> 00:00:45,930
our course, and keeping in mind
that one concept with
15
00:00:45,930 --> 00:00:50,050
many applications is
educationally more meaningful
16
00:00:50,050 --> 00:00:54,320
than many concepts each with one
application, I prefer to
17
00:00:54,320 --> 00:00:59,810
call today's lesson the inverse
derivative, or inverse
18
00:00:59,810 --> 00:01:01,340
differentiation.
19
00:01:01,340 --> 00:01:05,530
And again, bring into play a
very well-known feature of our
20
00:01:05,530 --> 00:01:08,830
course, namely, our discussion
of inverse functions.
21
00:01:08,830 --> 00:01:12,750
Now, you see, the only problem
that comes up in this context
22
00:01:12,750 --> 00:01:16,240
is a rather simple one and that
is that all the time that
23
00:01:16,240 --> 00:01:18,820
we've been taking a derivative,
we may not have
24
00:01:18,820 --> 00:01:23,010
realized that we were using the
idea of a function in a
25
00:01:23,010 --> 00:01:24,990
rather different way.
26
00:01:24,990 --> 00:01:28,730
Namely, noticed that by taking
a derivative, we had a rule
27
00:01:28,730 --> 00:01:33,620
which told us how, given a
particular function, to assign
28
00:01:33,620 --> 00:01:37,260
to it a new function called
its 'derivative'.
29
00:01:37,260 --> 00:01:40,130
In other words, what we could
have done is to have
30
00:01:40,130 --> 00:01:43,620
visualized a different function
machine, which I will
31
00:01:43,620 --> 00:01:45,490
call the 'D-machine'.
32
00:01:45,490 --> 00:01:49,050
Hopefully, the 'D' will suggest
differentiation, where
33
00:01:49,050 --> 00:01:51,130
the domain of my the machine--
34
00:01:51,130 --> 00:01:54,410
in other words, the input of
my 'D-machine' will be
35
00:01:54,410 --> 00:01:59,620
differentiable functions and the
output, you see, the image
36
00:01:59,620 --> 00:02:03,190
of my 'D-machine' will
be the derivative of
37
00:02:03,190 --> 00:02:05,290
that particular function.
38
00:02:05,290 --> 00:02:08,310
So in other words then, if we
want to use our typical
39
00:02:08,310 --> 00:02:11,500
notation, what we're
saying is that the
40
00:02:11,500 --> 00:02:13,080
'D-machine' does what?
41
00:02:13,080 --> 00:02:16,480
Given a differentiable function
as its input, the
42
00:02:16,480 --> 00:02:19,830
output will be the derivative
'f prime'.
43
00:02:19,830 --> 00:02:22,710
Now again, we're used to talking
about the derivative
44
00:02:22,710 --> 00:02:25,080
with respect to a
given variable.
45
00:02:25,080 --> 00:02:28,950
Unless otherwise specified,
the variable is 'x'.
46
00:02:28,950 --> 00:02:32,150
And so, perhaps with that in
mind, maybe it would be more
47
00:02:32,150 --> 00:02:37,310
suggestive if I were to do
something like this to
48
00:02:37,310 --> 00:02:41,540
indicate, you see, that the
input is 'f of x' and the
49
00:02:41,540 --> 00:02:44,250
output is 'f prime of x'.
50
00:02:44,250 --> 00:02:48,230
Now, you may recall also that
long before we started dealing
51
00:02:48,230 --> 00:02:51,190
with functions of real
variables, we discussed
52
00:02:51,190 --> 00:02:53,840
functions in general
in terms of circle
53
00:02:53,840 --> 00:02:55,470
diagrams and the like.
54
00:02:55,470 --> 00:02:59,020
In this respect, notice that
we can think of our
55
00:02:59,020 --> 00:03:03,740
'D-machine' as operating on a
particular domain, the domain
56
00:03:03,740 --> 00:03:04,650
being what?
57
00:03:04,650 --> 00:03:08,070
The set of all differentiable
functions.
58
00:03:08,070 --> 00:03:12,600
And the range, or the image,
will also be what?
59
00:03:12,600 --> 00:03:15,640
All functions which
are derivatives of
60
00:03:15,640 --> 00:03:17,780
differentiable functions.
61
00:03:17,780 --> 00:03:20,420
Now, what kind of a
function is 'D'?
62
00:03:20,420 --> 00:03:22,960
Don't lose track of the
fact that things like
63
00:03:22,960 --> 00:03:27,800
one-to-oneness and ontoness are
concepts that transcend
64
00:03:27,800 --> 00:03:29,570
dealing with numbers.
65
00:03:29,570 --> 00:03:31,440
They apply whenever
we're dealing with
66
00:03:31,440 --> 00:03:32,850
any kind of a function.
67
00:03:32,850 --> 00:03:34,590
The idea is something
like this.
68
00:03:34,590 --> 00:03:38,010
If 'x squared' were to go into
the 'D-machine', the output
69
00:03:38,010 --> 00:03:40,020
would be '2x'.
70
00:03:40,020 --> 00:03:43,060
See? 'x squared' is
mapped into '2x'.
71
00:03:43,060 --> 00:03:46,920
'x cubed' is mapped into
'3 x squared'.
72
00:03:46,920 --> 00:03:49,230
See, the derivative of 'x cubed'
with respect to 'x' is
73
00:03:49,230 --> 00:03:50,710
'3 x squared'.
74
00:03:50,710 --> 00:03:55,590
'x squared plus 7', also
mapped into '2x'.
75
00:03:55,590 --> 00:03:58,350
In other words, without going
any further, notice that
76
00:03:58,350 --> 00:04:06,350
whatever 'D' is, 'D'
is not 1 to 1.
77
00:04:06,350 --> 00:04:09,950
You see, two different functions
can have the same
78
00:04:09,950 --> 00:04:10,930
derivative.
79
00:04:10,930 --> 00:04:14,860
In fact, many different
functions can have the same
80
00:04:14,860 --> 00:04:15,730
derivative.
81
00:04:15,730 --> 00:04:18,620
For example, once we know that
the derivative of 'x squared'
82
00:04:18,620 --> 00:04:22,180
is '2x', we certainly know
that the derivative of 'x
83
00:04:22,180 --> 00:04:26,080
squared' plus any constant
is also '2x'.
84
00:04:26,080 --> 00:04:29,050
In other words then, we cannot
make an inverse function
85
00:04:29,050 --> 00:04:31,400
machine as things stand here.
86
00:04:31,400 --> 00:04:32,530
And why is that?
87
00:04:32,530 --> 00:04:36,360
To have an inverse function,
what you must be able to do is
88
00:04:36,360 --> 00:04:39,930
to reverse arrowheads and
still have a function.
89
00:04:39,930 --> 00:04:44,420
Notice that, in a way, 'D
inverse' would be a little bit
90
00:04:44,420 --> 00:04:46,320
of a tricky thing to
talk about here.
91
00:04:46,320 --> 00:04:47,940
But let's pretend that
we could anyway.
92
00:04:47,940 --> 00:04:50,960
In other words, what would go
wrong if we tried to build a
93
00:04:50,960 --> 00:04:53,230
'D' inverse machine?
94
00:04:53,230 --> 00:04:56,340
Well, in terms of the example
that we're talking about, if
95
00:04:56,340 --> 00:05:01,780
the input to the D inverse
machine happened to be '2x',
96
00:05:01,780 --> 00:05:04,390
then there would be infinitely
many different outputs that
97
00:05:04,390 --> 00:05:06,050
we're sure of.
98
00:05:06,050 --> 00:05:06,880
Namely what?
99
00:05:06,880 --> 00:05:09,830
Every function of the form
'x squared plus c'
100
00:05:09,830 --> 00:05:11,410
where 'c' is a constant.
101
00:05:11,410 --> 00:05:14,450
And what do I mean, we can
be sure of that many?
102
00:05:14,450 --> 00:05:17,880
You see, all I know going back
to this diagram here is that
103
00:05:17,880 --> 00:05:21,980
every function of the form 'x
squared plus c' will map into
104
00:05:21,980 --> 00:05:24,550
'2x' under the derivative
function.
105
00:05:24,550 --> 00:05:28,170
The other question that comes
up is, how do you know that
106
00:05:28,170 --> 00:05:32,690
you can find a function whose
derivative is '2x' that comes
107
00:05:32,690 --> 00:05:35,100
from a function which doesn't
have the form 'x
108
00:05:35,100 --> 00:05:36,210
squared plus c'?
109
00:05:36,210 --> 00:05:39,480
How do you know there isn't some
other function 'f of x'
110
00:05:39,480 --> 00:05:42,580
which has the property that
'f of x' maps into '2x'?
111
00:05:42,580 --> 00:05:45,640
In other words, the derivative
of 'f of x' is '2x' even
112
00:05:45,640 --> 00:05:50,040
though 'f of x' does not have
the form 'x squared plus c'.
113
00:05:50,040 --> 00:05:53,810
And this is exactly where the
mean value theorem comes in to
114
00:05:53,810 --> 00:05:55,330
help us over here.
115
00:05:55,330 --> 00:05:59,040
You see, what we're saying here
is, suppose that 'D' of
116
00:05:59,040 --> 00:06:00,590
'f of x' is '2x'.
117
00:06:00,590 --> 00:06:03,300
All we know is that when you
run 'f of x' through the
118
00:06:03,300 --> 00:06:06,970
'D-machine', you wind
up with '2x'.
119
00:06:06,970 --> 00:06:08,050
What does that mean?
120
00:06:08,050 --> 00:06:11,010
'f prime of x' is '2x'.
121
00:06:11,010 --> 00:06:14,700
Well, we know the derivative of
'x squared' is also '2x'.
122
00:06:14,700 --> 00:06:19,360
Therefore, whatever 'f of x'
is by the corollary to the
123
00:06:19,360 --> 00:06:22,690
mean value theorem that we've
studied before, since 'f of x'
124
00:06:22,690 --> 00:06:27,200
and 'x squared' have identical
derivatives, namely, they're
125
00:06:27,200 --> 00:06:31,930
both '2x', it means that they
must differ by a constant.
126
00:06:31,930 --> 00:06:35,580
In other words, 'f of x' minus
'x squared' is a constant.
127
00:06:35,580 --> 00:06:38,700
But to say that 'f of x' minus
'x squared' is a constant is
128
00:06:38,700 --> 00:06:43,000
the same as saying that 'f of
x' belongs to the family 'x
129
00:06:43,000 --> 00:06:44,580
squared plus c'.
130
00:06:44,580 --> 00:06:46,920
In other words, 'f
of x' equals 'x
131
00:06:46,920 --> 00:06:49,420
squared' plus a constant.
132
00:06:49,420 --> 00:06:52,280
What the mean value theorem
tells us is not only does
133
00:06:52,280 --> 00:06:55,430
every function of the form 'x
squared plus c' have its
134
00:06:55,430 --> 00:06:58,120
derivative equal to '2x' but
that every function whose
135
00:06:58,120 --> 00:07:01,980
derivative is '2x' has the
form 'x squared plus c'.
136
00:07:01,980 --> 00:07:04,560
137
00:07:04,560 --> 00:07:06,960
To generalize this, consider
the following.
138
00:07:06,960 --> 00:07:10,570
Suppose we're given
'f of x', OK?
139
00:07:10,570 --> 00:07:16,040
And we now say, look, every
time you differentiate a
140
00:07:16,040 --> 00:07:19,650
function, if you add on a
constant, you don't change
141
00:07:19,650 --> 00:07:21,200
anything, meaning what?
142
00:07:21,200 --> 00:07:22,960
The derivative of
a constant is 0.
143
00:07:22,960 --> 00:07:25,480
In other words, what we're
saying is, if the derivative
144
00:07:25,480 --> 00:07:30,100
of 'f of x' is 'f prime of x',
any function of the form 'f of
145
00:07:30,100 --> 00:07:33,720
x' plus 'c' will have
the same derivative.
146
00:07:33,720 --> 00:07:37,190
And not only that, the only
function whose derivative is
147
00:07:37,190 --> 00:07:40,560
'f prime of x' must
be one of these.
148
00:07:40,560 --> 00:07:43,070
That's what this 'e
sub f' stands for.
149
00:07:43,070 --> 00:07:45,730
You see, what I'm really saying
here is that certainly
150
00:07:45,730 --> 00:07:49,730
when you change the constant,
you change the function.
151
00:07:49,730 --> 00:07:52,220
The point is, with respect
to the thing called the
152
00:07:52,220 --> 00:07:55,670
derivative, you cannot tell
the difference between two
153
00:07:55,670 --> 00:07:58,830
functions just by looking at
their derivatives if they
154
00:07:58,830 --> 00:08:00,290
differ by a constant.
155
00:08:00,290 --> 00:08:03,600
In other words, with respect
to differentiation, two
156
00:08:03,600 --> 00:08:06,750
functions which differ by a
constant are equivalent and
157
00:08:06,750 --> 00:08:10,260
that's why I invented this
notation, 'e of f'.
158
00:08:10,260 --> 00:08:14,020
All I'm saying is that if we
visualize now that the input
159
00:08:14,020 --> 00:08:17,850
of the 'D-machine' is not
individual functions but whole
160
00:08:17,850 --> 00:08:19,890
classes of functions--
161
00:08:19,890 --> 00:08:22,780
in other words, such that they
differ only by a constant,
162
00:08:22,780 --> 00:08:25,780
then you see there is a
one-to-one correspondence
163
00:08:25,780 --> 00:08:28,840
between the input
and the output.
164
00:08:28,840 --> 00:08:30,610
Now that's a subtle point.
165
00:08:30,610 --> 00:08:33,049
It's a point which I'm sure
most of you will grasp.
166
00:08:33,049 --> 00:08:38,240
But more importantly, the
important thing is simply that
167
00:08:38,240 --> 00:08:42,370
once we've seen one function
with a given derivative, in a
168
00:08:42,370 --> 00:08:44,960
manner of speaking, we've
seen them all.
169
00:08:44,960 --> 00:08:47,550
In other words, we only have
enough leeway as to fool
170
00:08:47,550 --> 00:08:48,825
around with an arbitrary
constant.
171
00:08:48,825 --> 00:08:52,410
172
00:08:52,410 --> 00:08:54,790
So let's generalize again.
173
00:08:54,790 --> 00:08:59,250
What do we mean by 'D inverse'
of 'f of x'?
174
00:08:59,250 --> 00:09:00,550
We mean what?
175
00:09:00,550 --> 00:09:04,830
The set of all function 'g of
x' whose derivative with
176
00:09:04,830 --> 00:09:07,955
respect to 'x' is the
given 'f of x'.
177
00:09:07,955 --> 00:09:10,780
See, that's exactly what the
inverse function means.
178
00:09:10,780 --> 00:09:14,530
And by the way, notice that
this tells me my set
179
00:09:14,530 --> 00:09:15,430
implicitly.
180
00:09:15,430 --> 00:09:17,610
Let me put that in parentheses
over here.
181
00:09:17,610 --> 00:09:22,520
Namely, suppose somebody says,
does 'g of x' belong to this
182
00:09:22,520 --> 00:09:24,690
particular set called
'D inverse'?
183
00:09:24,690 --> 00:09:28,880
All I have to do is
differentiate the given 'g'
184
00:09:28,880 --> 00:09:30,530
and see if I get 'f'.
185
00:09:30,530 --> 00:09:33,560
If I do, 'G' belongs
to the set.
186
00:09:33,560 --> 00:09:36,060
If I don't, it doesn't
belong to the set.
187
00:09:36,060 --> 00:09:39,050
But as we've so often stressed
about inverse functions,
188
00:09:39,050 --> 00:09:41,520
notice that the test to see
whether something belongs to
189
00:09:41,520 --> 00:09:45,710
'D inverse', it's sufficient to
know how to differentiate.
190
00:09:45,710 --> 00:09:53,320
By the way, to summarize what
we did before, to list this
191
00:09:53,320 --> 00:09:57,310
thing explicitly, notice that
what we're saying is, to find
192
00:09:57,310 --> 00:10:01,400
the set of all functions whose
derivative is little 'f of x',
193
00:10:01,400 --> 00:10:03,420
all we have to do is what?
194
00:10:03,420 --> 00:10:07,030
Find one function, capital
'F', whose derivative is
195
00:10:07,030 --> 00:10:08,280
little 'f'.
196
00:10:08,280 --> 00:10:11,380
And then our set, explicitly,
is what?
197
00:10:11,380 --> 00:10:14,990
The set of all functions of the
form capital 'F of x' plus
198
00:10:14,990 --> 00:10:18,540
'c' where 'c' is an arbitrary
constant.
199
00:10:18,540 --> 00:10:20,580
Now, I'm sure this
concept is not
200
00:10:20,580 --> 00:10:22,230
difficult for you to grasp.
201
00:10:22,230 --> 00:10:26,140
For those of you who have been
through calculus before, as
202
00:10:26,140 --> 00:10:29,170
our course is intended to be for
this type of person too,
203
00:10:29,170 --> 00:10:32,820
you see, you may not be familiar
with the notation,
204
00:10:32,820 --> 00:10:35,440
the 'D inverse' notation, which
I want to stress here.
205
00:10:35,440 --> 00:10:39,380
But the concept, I hope, is
clear in its own right.
206
00:10:39,380 --> 00:10:44,030
Now of course, you see, the
problem that comes up is that
207
00:10:44,030 --> 00:10:47,840
it's much easier to
differentiate a function than
208
00:10:47,840 --> 00:10:52,020
it is to be given a function and
then have to try to find
209
00:10:52,020 --> 00:10:54,690
what you have to differentiate
to get it.
210
00:10:54,690 --> 00:10:57,310
Without meaning it as
facetiously as it may sound, I
211
00:10:57,310 --> 00:11:01,120
use this in on many occasions,
I prefer to say is much
212
00:11:01,120 --> 00:11:04,910
easier, you see, to scramble an
egg than to unscramble one.
213
00:11:04,910 --> 00:11:07,950
Told what to differentiate,
that's easy enough to do.
214
00:11:07,950 --> 00:11:10,730
Given the derivative, that
may not be so easy.
215
00:11:10,730 --> 00:11:13,780
Let's look in terms
of an example.
216
00:11:13,780 --> 00:11:18,200
But this is, again, a very
nice teachers trick.
217
00:11:18,200 --> 00:11:20,250
To get the right answer, you
start with the answer and then
218
00:11:20,250 --> 00:11:21,120
work to the problem.
219
00:11:21,120 --> 00:11:25,200
I'll start with the function 'h
of x' equals 'x' times the
220
00:11:25,200 --> 00:11:26,970
'square root of 'x
squared plus 1''.
221
00:11:26,970 --> 00:11:31,300
Or written more conveniently in
exponential notation, 'x'
222
00:11:31,300 --> 00:11:34,050
times ''x squared plus
1' to the 1/2'.
223
00:11:34,050 --> 00:11:35,670
Let me differentiate that.
224
00:11:35,670 --> 00:11:36,990
Remember, this is a product.
225
00:11:36,990 --> 00:11:39,430
The derivative of a
product is what?
226
00:11:39,430 --> 00:11:42,690
It's the first factor which is
'x' times the derivative of
227
00:11:42,690 --> 00:11:43,560
the second.
228
00:11:43,560 --> 00:11:44,270
That means what?
229
00:11:44,270 --> 00:11:48,490
I bring the 1/2 down to a
power 1 less, times the
230
00:11:48,490 --> 00:11:51,620
derivative of what's inside
with respect to 'x'.
231
00:11:51,620 --> 00:11:53,110
See, the some old chain
rule again.
232
00:11:53,110 --> 00:11:54,410
That's '2x'.
233
00:11:54,410 --> 00:11:55,160
Then what?
234
00:11:55,160 --> 00:11:59,610
Plus the second factor, 'x
squared plus 1' to the 1/2
235
00:11:59,610 --> 00:12:01,770
times the derivative of the
first with respect to 'x',
236
00:12:01,770 --> 00:12:03,500
which is 1.
237
00:12:03,500 --> 00:12:08,530
At any rate, simplifying,
bringing the minus 1/2 power
238
00:12:08,530 --> 00:12:12,360
into the denominator, then
putting everything over a
239
00:12:12,360 --> 00:12:16,580
common denominator, I wind up
with the fact that 'h prime of
240
00:12:16,580 --> 00:12:21,340
x' is '2 x squared plus 1' over
the 'square root of 'x
241
00:12:21,340 --> 00:12:22,890
squared plus 1''.
242
00:12:22,890 --> 00:12:24,870
I hope I haven't made a
careless error here.
243
00:12:24,870 --> 00:12:27,675
But again, one of the beauties
of the new mathematics is that
244
00:12:27,675 --> 00:12:30,240
it's the method that's
important, OK?
245
00:12:30,240 --> 00:12:32,250
Now, at any rate,
what do I know?
246
00:12:32,250 --> 00:12:35,640
Starting with 'h of x' equaling
'x' times the 'square
247
00:12:35,640 --> 00:12:39,330
root of 'x squared plus 1'', I
now know that its derivative
248
00:12:39,330 --> 00:12:43,160
is '2 x squared plus 1' over
the 'square root of 'x
249
00:12:43,160 --> 00:12:44,480
squared plus 1''.
250
00:12:44,480 --> 00:12:49,190
Let's write that in terms of
our 'D inverse' notation.
251
00:12:49,190 --> 00:12:53,960
Namely, what we're saying is
that 'x' times the 'square
252
00:12:53,960 --> 00:12:57,280
root of 'x squared plus 1''
has the property that its
253
00:12:57,280 --> 00:13:00,620
derivative is '2 x squared plus
1' over the 'square root
254
00:13:00,620 --> 00:13:02,180
of 'x squared plus 1''.
255
00:13:02,180 --> 00:13:05,970
Consequently, every function
which has the property that
256
00:13:05,970 --> 00:13:09,400
its derivative is '2 x squared
plus 1' over the 'square root
257
00:13:09,400 --> 00:13:13,520
of 'x squared plus 1'' must
come from this family.
258
00:13:13,520 --> 00:13:14,440
You see?
259
00:13:14,440 --> 00:13:16,820
Everything of this form
we'll have its
260
00:13:16,820 --> 00:13:18,000
derivative equal to this.
261
00:13:18,000 --> 00:13:21,110
And secondly, any other function
cannot have its
262
00:13:21,110 --> 00:13:23,530
derivative equal to this by
our corollary to the mean
263
00:13:23,530 --> 00:13:24,670
value theorem.
264
00:13:24,670 --> 00:13:27,080
Now you see, what
I'm saying is--
265
00:13:27,080 --> 00:13:30,270
and here's the beauty of what we
mean by inverse operations
266
00:13:30,270 --> 00:13:31,330
and the like.
267
00:13:31,330 --> 00:13:34,310
It's conceivable that you
might not have been
268
00:13:34,310 --> 00:13:37,380
sophisticated enough at this
stage in the game to have been
269
00:13:37,380 --> 00:13:40,650
able to deduce this had
we been given this.
270
00:13:40,650 --> 00:13:44,330
Notice that my cute trick was I
started with this, found out
271
00:13:44,330 --> 00:13:46,160
what the derivative
was, and then just
272
00:13:46,160 --> 00:13:47,800
inverted the emphasis.
273
00:13:47,800 --> 00:13:49,650
A change in emphasis again.
274
00:13:49,650 --> 00:13:52,500
However, notice the following.
275
00:13:52,500 --> 00:13:54,810
Suppose you weren't
able to find this.
276
00:13:54,810 --> 00:13:58,810
And somebody said to you, I
wonder if 'x' times the
277
00:13:58,810 --> 00:14:01,900
'square root of 'x squared plus
1'' is a function whose
278
00:14:01,900 --> 00:14:03,990
derivative is equal to this.
279
00:14:03,990 --> 00:14:06,600
And all I'm saying, without
going through the work again
280
00:14:06,600 --> 00:14:09,670
because I've already done that,
is to simply observe
281
00:14:09,670 --> 00:14:13,520
that even if you did not know
this explicit representation,
282
00:14:13,520 --> 00:14:18,400
by definition, 'D inverse' of '2
x squared plus 1' over the
283
00:14:18,400 --> 00:14:21,310
'square root of 'x squared
plus 1'' is simply what?
284
00:14:21,310 --> 00:14:25,530
The set of all functions 'g of
x' such that 'g prime of x' is
285
00:14:25,530 --> 00:14:28,510
equal to '2 x squared plus 1'
over the 'square root of 'x
286
00:14:28,510 --> 00:14:29,620
squared plus 1''.
287
00:14:29,620 --> 00:14:32,090
In other words, given any
function at all, I could
288
00:14:32,090 --> 00:14:33,210
differentiate it.
289
00:14:33,210 --> 00:14:35,180
If the derivative came
out to be this,
290
00:14:35,180 --> 00:14:36,700
then I have a solution.
291
00:14:36,700 --> 00:14:38,350
It belongs to the
solution set.
292
00:14:38,350 --> 00:14:39,720
Otherwise, it doesn't.
293
00:14:39,720 --> 00:14:44,380
But again, notice that's to
solve any 'D inverse' problem,
294
00:14:44,380 --> 00:14:46,270
it's sufficient to understand a
295
00:14:46,270 --> 00:14:49,170
corresponding derivative property.
296
00:14:49,170 --> 00:14:53,060
In fact, maybe now is a good
time to show how we get
297
00:14:53,060 --> 00:14:57,530
certain recipes for 'D
inverse' type things.
298
00:14:57,530 --> 00:15:00,490
Let me just write down
a typical one.
299
00:15:00,490 --> 00:15:05,060
You see, 'D inverse of 'x to the
n'' is 'x to the 'n + 1''
300
00:15:05,060 --> 00:15:07,770
over 'n + 1' plus a constant.
301
00:15:07,770 --> 00:15:10,920
And of course, observe that as
soon as you see something like
302
00:15:10,920 --> 00:15:14,120
this, you have to beware of
'n' equals negative 1.
303
00:15:14,120 --> 00:15:16,950
Otherwise we have
a 0 denominator.
304
00:15:16,950 --> 00:15:19,700
Now, a person says, this doesn't
look familiar to me.
305
00:15:19,700 --> 00:15:23,870
Again, keep in mind what
D inverse means.
306
00:15:23,870 --> 00:15:28,390
Essentially, to say this is just
a switch in emphasis from
307
00:15:28,390 --> 00:15:30,020
saying what?
308
00:15:30,020 --> 00:15:33,990
That if you run the family of
functions 'x to the 'n + 1''
309
00:15:33,990 --> 00:15:38,670
over 'n + 1' plus a constant
through your 'D-machine', you
310
00:15:38,670 --> 00:15:39,770
get 'x to the n'.
311
00:15:39,770 --> 00:15:41,930
Or more familiarly, what?
312
00:15:41,930 --> 00:15:46,490
The derivative of any member in
this family is 'x to the n'
313
00:15:46,490 --> 00:15:49,750
provided that 'n' is not
equal to minus 1.
314
00:15:49,750 --> 00:15:52,760
Now again, this may look a
little bit abstract to you.
315
00:15:52,760 --> 00:15:56,590
So to avoid this problem,
let's just do a concrete
316
00:15:56,590 --> 00:15:57,670
illustration.
317
00:15:57,670 --> 00:15:59,740
Let's pick a particular
value of 'n' and
318
00:15:59,740 --> 00:16:01,490
work with this thing.
319
00:16:01,490 --> 00:16:04,430
Let's suppose we're told to
determine 'D inverse'
320
00:16:04,430 --> 00:16:05,660
of 'x to the 7th'.
321
00:16:05,660 --> 00:16:07,050
What does that mean?
322
00:16:07,050 --> 00:16:11,420
What it really means is, let's
find a function whose
323
00:16:11,420 --> 00:16:13,660
derivative is 'x to the 7th'.
324
00:16:13,660 --> 00:16:16,150
And why do I say let's
find a function?
325
00:16:16,150 --> 00:16:19,550
Because once I find a function,
all I have to do is
326
00:16:19,550 --> 00:16:22,840
tack on arbitrary constants and
the family that I get that
327
00:16:22,840 --> 00:16:26,460
way is the unique family of
functions which have this
328
00:16:26,460 --> 00:16:27,870
particular derivative.
329
00:16:27,870 --> 00:16:29,730
So I play the detective game.
330
00:16:29,730 --> 00:16:32,680
I know from differential
calculus that if I
331
00:16:32,680 --> 00:16:36,400
differentiate 'x' to the 8th
power, I'll wind up with the
332
00:16:36,400 --> 00:16:37,620
exponent 7, at least.
333
00:16:37,620 --> 00:16:40,950
In other words, the derivative
of 'x' to the 8th power.
334
00:16:40,950 --> 00:16:45,250
'D of 'x to the 8th'' is
'8 'x to the 7th''.
335
00:16:45,250 --> 00:16:47,340
Well, what answer did
I want to get?
336
00:16:47,340 --> 00:16:51,010
I wanted to get 'x to the 7th',
not '8 'x to the 7th''.
337
00:16:51,010 --> 00:16:53,230
So I fudge this thing
a little bit.
338
00:16:53,230 --> 00:16:56,220
I say, evidently what I should
have done was to have started
339
00:16:56,220 --> 00:16:57,890
with 1/8 as much.
340
00:16:57,890 --> 00:17:01,290
In other words, multiplying
equals by equals, I multiply
341
00:17:01,290 --> 00:17:05,950
both sides of this equation by
1/8 and I wind up with '1/8 D
342
00:17:05,950 --> 00:17:08,730
'x to the 8th'' equals
'x to the 7th'.
343
00:17:08,730 --> 00:17:11,240
And now comes a very
crucial step.
344
00:17:11,240 --> 00:17:14,089
And let me write that down
because I think it's something
345
00:17:14,089 --> 00:17:17,329
that we should pay very
close attention to.
346
00:17:17,329 --> 00:17:20,480
And that is that the derivative
has the property
347
00:17:20,480 --> 00:17:23,599
that if you want to
differentiate a constant times
348
00:17:23,599 --> 00:17:28,420
a function, you can take the
constant out and differentiate
349
00:17:28,420 --> 00:17:29,730
just the function.
350
00:17:29,730 --> 00:17:32,210
This is a very crucial point
because you see--
351
00:17:32,210 --> 00:17:35,180
and by the way, notice that in
general, not all functions
352
00:17:35,180 --> 00:17:37,690
have this property.
353
00:17:37,690 --> 00:17:42,100
For example, if you're squaring
something, if you
354
00:17:42,100 --> 00:17:46,810
double the number that's being
squared, the output is 4 times
355
00:17:46,810 --> 00:17:49,290
as much because twice
something squared
356
00:17:49,290 --> 00:17:50,320
is 4 times as much.
357
00:17:50,320 --> 00:17:54,050
In other words, in general,
you do not say that if you
358
00:17:54,050 --> 00:17:56,840
double the input, you're going
to double the output.
359
00:17:56,840 --> 00:17:58,620
Not every function has
that property.
360
00:17:58,620 --> 00:18:01,420
But the function called 'D', the
derivative, does have this
361
00:18:01,420 --> 00:18:02,620
particular property.
362
00:18:02,620 --> 00:18:04,600
And you see, with that
in mind, I can
363
00:18:04,600 --> 00:18:06,460
bring this 1/8 inside.
364
00:18:06,460 --> 00:18:12,330
This is the key step, that 1/8
times the derivative of 'x to
365
00:18:12,330 --> 00:18:16,190
the 8th' is a derivative of
''1/8' x to the 8th'.
366
00:18:16,190 --> 00:18:17,850
That's this key step
over here.
367
00:18:17,850 --> 00:18:20,900
And now, you see, putting all
this together, I find what?
368
00:18:20,900 --> 00:18:23,940
That the derivative of
''1/8' x to the 8th'
369
00:18:23,940 --> 00:18:25,840
is 'x to the 7th'.
370
00:18:25,840 --> 00:18:28,950
And therefore, since I found one
function whose derivative
371
00:18:28,950 --> 00:18:32,560
is 'x to the 7th', I have, in
a sense, found them all,
372
00:18:32,560 --> 00:18:35,830
namely, ''1/8' x to
the 8th' plus 'c'.
373
00:18:35,830 --> 00:18:39,180
In other words, that's what I
call the equivalent class of
374
00:18:39,180 --> 00:18:40,270
''1/8' x to the 8th'.
375
00:18:40,270 --> 00:18:43,030
All the functions that differ
from ''1/8' x to
376
00:18:43,030 --> 00:18:45,350
the 8th' by a constant.
377
00:18:45,350 --> 00:18:48,720
And again, to emphasize this
very important point, let me
378
00:18:48,720 --> 00:18:54,010
again mention, beware of
non-constant factors.
379
00:18:54,010 --> 00:18:57,180
Let me give you a
for instance.
380
00:18:57,180 --> 00:19:00,800
Let's suppose I take almost
the same problem.
381
00:19:00,800 --> 00:19:02,650
And that almost as
a big almost.
382
00:19:02,650 --> 00:19:05,910
Let's suppose I say, let me
find all functions whose
383
00:19:05,910 --> 00:19:09,320
derivative, say, is 'x squared
plus 1' to the 7th power.
384
00:19:09,320 --> 00:19:12,310
In other words, still something
to the 7th power.
385
00:19:12,310 --> 00:19:14,930
So I argue something like,
well, since whenever I
386
00:19:14,930 --> 00:19:18,460
differentiate I reduce the
exponent by 1, to wind up with
387
00:19:18,460 --> 00:19:20,330
a 7th power, maybe
I should have
388
00:19:20,330 --> 00:19:22,490
started with an 8th power.
389
00:19:22,490 --> 00:19:24,850
So I say, OK, that's
what I'll do.
390
00:19:24,850 --> 00:19:26,290
I'll start with an 8th power.
391
00:19:26,290 --> 00:19:28,910
So I say, OK, what is the
derivative of 'x squared plus
392
00:19:28,910 --> 00:19:30,490
1' to the 8th power?
393
00:19:30,490 --> 00:19:33,560
Now notice I know how to
differentiate, hopefully.
394
00:19:33,560 --> 00:19:36,040
And again, let me make this
point very strongly.
395
00:19:36,040 --> 00:19:39,280
There is no sense studying
inverse functions if we don't
396
00:19:39,280 --> 00:19:41,180
know the original
function itself.
397
00:19:41,180 --> 00:19:43,710
Because the whole purpose of the
inverse function, or the
398
00:19:43,710 --> 00:19:47,410
whole strategy behind it, is to
reduce it to the original
399
00:19:47,410 --> 00:19:50,020
function, namely, to switch
the emphasis.
400
00:19:50,020 --> 00:19:52,510
So at any rate, I differentiate
'x squared plus
401
00:19:52,510 --> 00:19:53,760
1' to the 8th power.
402
00:19:53,760 --> 00:19:56,970
I get '8 'x squared plus
1' to the 7th'.
403
00:19:56,970 --> 00:20:01,410
But now, by the chain rule, I
must remember that this part
404
00:20:01,410 --> 00:20:04,330
was only the derivative with
respect to 'x squared plus 1'.
405
00:20:04,330 --> 00:20:05,740
The correction factor is what?
406
00:20:05,740 --> 00:20:07,960
The derivative of what's inside
with respect to 'x'.
407
00:20:07,960 --> 00:20:09,000
That's '2x'.
408
00:20:09,000 --> 00:20:12,180
And so I wind up with that if
I differentiate ''x squared
409
00:20:12,180 --> 00:20:15,960
plus 1' to the 8th', I get '16x'
times ''x squared plus
410
00:20:15,960 --> 00:20:17,100
1' to the 7th'.
411
00:20:17,100 --> 00:20:18,590
Now, how much did
I want to get?
412
00:20:18,590 --> 00:20:21,980
I wanted to get just ''x squared
plus 1' to the 7th'.
413
00:20:21,980 --> 00:20:24,990
That put me off by a
factor of '16x'.
414
00:20:24,990 --> 00:20:28,070
Now I say, OK, I'll
fix that up.
415
00:20:28,070 --> 00:20:32,200
Namely, I'll divide both sides
by '16x', assuming, of course,
416
00:20:32,200 --> 00:20:33,430
that 'x' is not 0.
417
00:20:33,430 --> 00:20:36,450
And this, by the way,
is perfectly valid.
418
00:20:36,450 --> 00:20:41,360
I can now go from here to here
and say, look, '1 over 16x'
419
00:20:41,360 --> 00:20:44,720
times the derivative of ''x
squared plus 1' to the 8th'
420
00:20:44,720 --> 00:20:48,230
with respect to 'x' is ''x
squared plus 1' to the 7th'.
421
00:20:48,230 --> 00:20:52,470
However, notice that I cannot
take this factor and bring it
422
00:20:52,470 --> 00:20:53,560
inside here.
423
00:20:53,560 --> 00:20:57,220
And again, as I so often have
said, also, of course I can
424
00:20:57,220 --> 00:20:58,100
bring it inside here.
425
00:20:58,100 --> 00:20:58,860
I just did.
426
00:20:58,860 --> 00:21:02,260
What I mean is, I don't
get the right answer.
427
00:21:02,260 --> 00:21:04,650
And what's the best proof that
I don't get the right answer?
428
00:21:04,650 --> 00:21:06,600
Very, very simple.
429
00:21:06,600 --> 00:21:13,400
Take this function,
differentiate it, and see if
430
00:21:13,400 --> 00:21:15,980
you get ''x squared plus
1' to the 7th'.
431
00:21:15,980 --> 00:21:18,990
You won't, unless you
differentiate incorrectly.
432
00:21:18,990 --> 00:21:21,520
Don't be like the person who
just differentiates, brings
433
00:21:21,520 --> 00:21:26,050
the 8 down, replaces this by
1 less, multiplies by a
434
00:21:26,050 --> 00:21:29,180
derivative of what's inside,
and cancels everything out.
435
00:21:29,180 --> 00:21:31,930
Notice that the expression
inside the brackets that I've
436
00:21:31,930 --> 00:21:34,490
just circled is a quotient.
437
00:21:34,490 --> 00:21:37,060
And the derivative of a quotient
is obtained in a very
438
00:21:37,060 --> 00:21:37,830
special way.
439
00:21:37,830 --> 00:21:40,750
The denominator times the
derivative of the numerator
440
00:21:40,750 --> 00:21:43,700
minus the numerator times the
derivative of the denominator
441
00:21:43,700 --> 00:21:45,480
over the square of
the denominator.
442
00:21:45,480 --> 00:21:48,970
And all I'm saying is, you won't
get ''x squared plus 1'
443
00:21:48,970 --> 00:21:50,750
to the 7th' power
if you do that.
444
00:21:50,750 --> 00:21:54,820
Again, notice that you do not
have to know the right answer
445
00:21:54,820 --> 00:21:59,150
in order to see what answer
is wrong, OK?
446
00:21:59,150 --> 00:22:00,840
So this would be
a wrong answer.
447
00:22:00,840 --> 00:22:04,350
By the way, this would also be
a wrong answer because we've
448
00:22:04,350 --> 00:22:08,010
already seen that the derivative
of ''1/8' x squared
449
00:22:08,010 --> 00:22:10,130
plus 1' to the 8th
power is what?
450
00:22:10,130 --> 00:22:13,530
You bring the 8 down, which
kills off the 1/8.
451
00:22:13,530 --> 00:22:16,620
You replace this to a power of
one less, which gives you ''x
452
00:22:16,620 --> 00:22:18,390
squared plus 1' to the 7th'.
453
00:22:18,390 --> 00:22:21,960
But the correction factor here
is you must also multiply by a
454
00:22:21,960 --> 00:22:24,220
derivative of what's inside
with respect to 'x'.
455
00:22:24,220 --> 00:22:25,780
And that's to '2x'.
456
00:22:25,780 --> 00:22:27,350
In other words, you do
not get ''x squared
457
00:22:27,350 --> 00:22:29,780
plus 1' to the 7th'.
458
00:22:29,780 --> 00:22:32,970
What you do get is what? ''x
squared plus 1' to the 7th'
459
00:22:32,970 --> 00:22:34,470
times '2x'.
460
00:22:34,470 --> 00:22:37,540
And the question that may now
come up is, how come this
461
00:22:37,540 --> 00:22:41,030
worked when you were raising
'x' to the 8th power but it
462
00:22:41,030 --> 00:22:43,600
didn't work when you were
raising 'x squared plus 1' to
463
00:22:43,600 --> 00:22:46,050
the 8th power?
464
00:22:46,050 --> 00:22:49,580
Again, as always in these
cases, the answer is
465
00:22:49,580 --> 00:22:53,120
immediately available in
terms of derivatives.
466
00:22:53,120 --> 00:22:56,580
We can talk, in fact, about
the inverse chain rule.
467
00:22:56,580 --> 00:22:59,470
That when we really
talked about 'D'--
468
00:22:59,470 --> 00:23:00,880
remember, we mentioned this at
the very beginning of the
469
00:23:00,880 --> 00:23:04,900
lecture, that the variable
inside the parentheses was the
470
00:23:04,900 --> 00:23:08,270
one with respect to which you
were differentiating.
471
00:23:08,270 --> 00:23:16,480
In other words, what we saw was
that if you wanted to get
472
00:23:16,480 --> 00:23:20,110
something to the 7th power, what
you had to differentiate
473
00:23:20,110 --> 00:23:24,690
was 1/8 that same something to
the 8th power if you were
474
00:23:24,690 --> 00:23:27,670
differentiating with respect
to that same variable.
475
00:23:27,670 --> 00:23:31,110
In other words, what would have
been ''x squared plus 1'
476
00:23:31,110 --> 00:23:33,520
to the 7th' would
have been what?
477
00:23:33,520 --> 00:23:38,060
If you would differentiating
not with respect to 'x' but
478
00:23:38,060 --> 00:23:42,070
with respect to 'x
squared plus 1'.
479
00:23:42,070 --> 00:23:45,010
See, what we really wanted when
we wrote this down was
480
00:23:45,010 --> 00:23:47,220
the derivative with
respect to 'x'.
481
00:23:47,220 --> 00:23:50,920
And even though this notation
may look a little bit strange
482
00:23:50,920 --> 00:23:55,130
to you, observe that once you
get used to the notation, this
483
00:23:55,130 --> 00:23:58,160
is just another way of talking
about the chain rule.
484
00:23:58,160 --> 00:24:01,830
Namely, to find the derivative
of this with respect to 'x',
485
00:24:01,830 --> 00:24:05,840
you first take the derivative
with respect to 'x squared
486
00:24:05,840 --> 00:24:10,180
plus 1' and then multiply that
by the derivative of 'x
487
00:24:10,180 --> 00:24:13,150
squared plus 1' with
respect to 'x'.
488
00:24:13,150 --> 00:24:16,140
And if we do that, you see, we
get the answer that we've
489
00:24:16,140 --> 00:24:17,670
talked about before.
490
00:24:17,670 --> 00:24:21,710
In other words, what we could
say is that the function that
491
00:24:21,710 --> 00:24:25,150
you have to differentiate to get
''x squared plus 1' to the
492
00:24:25,150 --> 00:24:31,060
7th' times '2x' is ''1/8' x
squared plus 1' to the 8th'
493
00:24:31,060 --> 00:24:32,250
plus a constant.
494
00:24:32,250 --> 00:24:33,540
And how do I know that?
495
00:24:33,540 --> 00:24:36,830
Well, the way I know that
is simply what?
496
00:24:36,830 --> 00:24:40,740
In terms of inverse functions,
I started with ''1/8' x
497
00:24:40,740 --> 00:24:44,540
squared plus 1' to the 8th',
differentiated it and found
498
00:24:44,540 --> 00:24:48,190
out I got ''x squared plus 1'
to the 7th' times '2x'.
499
00:24:48,190 --> 00:24:51,050
And so this became the recipe.
500
00:24:51,050 --> 00:24:54,170
And again, a rather interesting
aside, if you look
501
00:24:54,170 --> 00:25:04,620
at this, and look at this, it
would appear at first glance
502
00:25:04,620 --> 00:25:07,810
that the top one should be a
more difficult problem than
503
00:25:07,810 --> 00:25:08,810
the bottom one.
504
00:25:08,810 --> 00:25:13,500
The reason being that the input
seems more simple in the
505
00:25:13,500 --> 00:25:14,200
bottom one.
506
00:25:14,200 --> 00:25:17,900
Yet, the interesting point is
that the '2x', which seems to
507
00:25:17,900 --> 00:25:20,770
make this thing more
complicated, is precisely the
508
00:25:20,770 --> 00:25:24,510
factor you need by the chain
rule to make this thing work.
509
00:25:24,510 --> 00:25:27,400
Because when you differentiate
'x squared plus 1' to the 8th
510
00:25:27,400 --> 00:25:30,120
power, you're going to
get a factor of '2x'
511
00:25:30,120 --> 00:25:31,360
by the chain rule.
512
00:25:31,360 --> 00:25:34,570
Now again, the main aim of the
lectures is not to take the
513
00:25:34,570 --> 00:25:37,460
place of the computational
drill supplied in our
514
00:25:37,460 --> 00:25:39,670
exercises and in the
text but to give
515
00:25:39,670 --> 00:25:40,810
you sort of an insight.
516
00:25:40,810 --> 00:25:44,170
And they'll be plenty of drill
on the mechanics of this in
517
00:25:44,170 --> 00:25:46,300
our exercises on this section.
518
00:25:46,300 --> 00:25:49,090
Let me just at least continue
on with the
519
00:25:49,090 --> 00:25:51,000
concept of our recipes.
520
00:25:51,000 --> 00:25:54,270
For example, here's
another one.
521
00:25:54,270 --> 00:25:55,920
And this one says what?
522
00:25:55,920 --> 00:25:59,165
That if you run the sum of two
functions through the 'D
523
00:25:59,165 --> 00:26:03,160
inverse' machine, the output is
the same as if you sent the
524
00:26:03,160 --> 00:26:08,130
functions through separately
and then added them up, OK?
525
00:26:08,130 --> 00:26:11,290
I'll talk about that in
more detail later.
526
00:26:11,290 --> 00:26:15,070
Again, all I want to see is
that this is the analogous
527
00:26:15,070 --> 00:26:18,940
result of, again, a beautiful
property of the derivative.
528
00:26:18,940 --> 00:26:25,870
And that is that the derivative
of a sum is the sum
529
00:26:25,870 --> 00:26:27,660
of the derivatives.
530
00:26:27,660 --> 00:26:29,000
And how does that
work over here?
531
00:26:29,000 --> 00:26:32,440
Again, a very interesting
property throughout advanced
532
00:26:32,440 --> 00:26:34,920
calculus, linear algebra
and the like.
533
00:26:34,920 --> 00:26:37,750
These properties are
very, very special.
534
00:26:37,750 --> 00:26:40,200
And we'll have occasion, as the
course goes on, to talk
535
00:26:40,200 --> 00:26:41,220
about them more.
536
00:26:41,220 --> 00:26:43,840
For the time being, rather than
have you get lost in the
537
00:26:43,840 --> 00:26:47,440
maze of details, let me work
a specific illustration.
538
00:26:47,440 --> 00:26:50,980
Let's suppose I would like to
find the family of functions
539
00:26:50,980 --> 00:26:55,410
whose derivative is 'x to the
5th' plus 'x cubed', OK?
540
00:26:55,410 --> 00:26:57,050
Now, what I'm saying is this.
541
00:26:57,050 --> 00:27:00,030
By my previous result, I
certainly know how to find the
542
00:27:00,030 --> 00:27:03,315
function whose derivative is 'x
to the 5th', namely, ''1/6'
543
00:27:03,315 --> 00:27:06,450
x to the 6th'.
544
00:27:06,450 --> 00:27:09,160
I also know how to find the
function whose derivative is
545
00:27:09,160 --> 00:27:13,200
'x cubed', namely, '1/4'
x to the 4th'.
546
00:27:13,200 --> 00:27:16,810
Putting these two steps together
and say equals added
547
00:27:16,810 --> 00:27:21,250
to equals are equal, I can
conclude that 'D of ''1/6' x
548
00:27:21,250 --> 00:27:25,080
to the sixth'' plus 'D of ''1/4'
x to the fourth'' is 'x
549
00:27:25,080 --> 00:27:27,080
to the 5th' plus 'x cubed'.
550
00:27:27,080 --> 00:27:31,030
Now, the key step is that since
the derivative of a sum
551
00:27:31,030 --> 00:27:34,670
is the sum of the derivatives,
I can say that the sum of
552
00:27:34,670 --> 00:27:37,800
these two derivatives is the
derivative of the sum of the
553
00:27:37,800 --> 00:27:38,480
two functions.
554
00:27:38,480 --> 00:27:40,250
Namely, that this is what?
555
00:27:40,250 --> 00:27:45,470
'D of ''1/6' x to the 6th'' plus
''1/4' x to the 4th', a
556
00:27:45,470 --> 00:27:48,540
very important power and
property of the derivative.
557
00:27:48,540 --> 00:27:52,030
Therefore, have I found one
function whose derivative is
558
00:27:52,030 --> 00:27:53,830
'x to the 5th' plus 'x cubed'?
559
00:27:53,830 --> 00:27:56,750
The answer is yes. ''1/6'
x to the 6th' plus
560
00:27:56,750 --> 00:27:58,420
''1/4' x to the 4th'.
561
00:27:58,420 --> 00:28:02,040
Therefore, what is the family
of all functions whose
562
00:28:02,040 --> 00:28:04,690
derivative is 'x 5th'
plus 'x cubed'?
563
00:28:04,690 --> 00:28:08,030
And again, as before, the
answer is, take your one
564
00:28:08,030 --> 00:28:11,610
solution that you found, tack on
an arbitrary constant, and
565
00:28:11,610 --> 00:28:14,060
I suppose, technically speaking,
I should put the
566
00:28:14,060 --> 00:28:17,490
braces in here to indicate
that my solution is an
567
00:28:17,490 --> 00:28:22,060
infinite set, all belonging
to one family called an
568
00:28:22,060 --> 00:28:24,590
equivalent set of functions
because they have the same
569
00:28:24,590 --> 00:28:25,730
derivative.
570
00:28:25,730 --> 00:28:29,840
Well, let's continue
on and do a little
571
00:28:29,840 --> 00:28:31,490
bit more harder stuff.
572
00:28:31,490 --> 00:28:34,370
Remember, we talked about
implicit differentiation.
573
00:28:34,370 --> 00:28:39,330
Well, is there an analogue
to implicit 'D inverses'?
574
00:28:39,330 --> 00:28:43,470
You see, notice that in every
problem so far, when I wrote
575
00:28:43,470 --> 00:28:47,160
'D inverse', I was essentially
telling you explicitly what
576
00:28:47,160 --> 00:28:49,040
came out of the 'D-machine'.
577
00:28:49,040 --> 00:28:51,480
Now, suppose I twist the
emphasis a little bit.
578
00:28:51,480 --> 00:28:54,670
Suppose I tell you, look, I run
a certain function through
579
00:28:54,670 --> 00:28:55,780
the 'D-machine'.
580
00:28:55,780 --> 00:28:57,650
In other words, I form
its derivative.
581
00:28:57,650 --> 00:29:01,300
What the output is, is the
square of the reciprocal of
582
00:29:01,300 --> 00:29:02,340
the function.
583
00:29:02,340 --> 00:29:06,940
In other words, if 'g of x'
comes in, '1 over 'g of x''
584
00:29:06,940 --> 00:29:08,650
squared comes out.
585
00:29:08,650 --> 00:29:12,050
And now the question is, what
is the function 'g of x'?
586
00:29:12,050 --> 00:29:14,970
And again, notice that if we
don't know what the right 'g
587
00:29:14,970 --> 00:29:18,690
of x', is you can certainly test
a given 'g of x' to see
588
00:29:18,690 --> 00:29:19,580
whether it's right or not.
589
00:29:19,580 --> 00:29:20,930
Namely, what could you do?
590
00:29:20,930 --> 00:29:24,860
You differentiate 'g of x', see
what you get, and if what
591
00:29:24,860 --> 00:29:30,130
you get isn't 1 over the square
of 'g of x', you've got
592
00:29:30,130 --> 00:29:31,580
the wrong answer.
593
00:29:31,580 --> 00:29:34,030
But the question that comes up
is, given this type of a
594
00:29:34,030 --> 00:29:38,000
problem, how do we handle it?
595
00:29:38,000 --> 00:29:38,200
See?
596
00:29:38,200 --> 00:29:42,810
In other words, where does the
inverse idea come in here?
597
00:29:42,810 --> 00:29:46,710
The implicit relationship that
'g of x' is determined by this
598
00:29:46,710 --> 00:29:48,100
particular property.
599
00:29:48,100 --> 00:29:51,410
And again, notice how we use
properties of derivatives.
600
00:29:51,410 --> 00:29:55,370
We're given that 'g prime of x'
is a synonym, identity, for
601
00:29:55,370 --> 00:29:58,310
'1 over 'g squared of x'.
602
00:29:58,310 --> 00:30:02,300
We can cross-multiply and we get
'g squared of x' times 'g
603
00:30:02,300 --> 00:30:05,060
prime of x' is identically
one.
604
00:30:05,060 --> 00:30:08,060
Now, if you're clever about
this-- and remember, notice
605
00:30:08,060 --> 00:30:09,630
this very, very importantly.
606
00:30:09,630 --> 00:30:13,250
For example, ordinary division
is the inverse of ordinary
607
00:30:13,250 --> 00:30:14,440
multiplication.
608
00:30:14,440 --> 00:30:17,410
Notice that to be really cute
in division, you have to be
609
00:30:17,410 --> 00:30:19,320
pretty cute in multiplication.
610
00:30:19,320 --> 00:30:22,510
Since all you're doing is
changing the emphasis, notice
611
00:30:22,510 --> 00:30:28,300
that to handle hard problems in
antiderivatives, you have
612
00:30:28,300 --> 00:30:30,720
to be able to handle tough
derivative problems.
613
00:30:30,720 --> 00:30:33,780
What I'm driving at is, you look
at something like this
614
00:30:33,780 --> 00:30:36,850
and begin to wonder, do you
know a function whose
615
00:30:36,850 --> 00:30:38,630
derivative is this?
616
00:30:38,630 --> 00:30:39,620
See?
617
00:30:39,620 --> 00:30:43,600
The idea is, if you're familiar
with your chain rule,
618
00:30:43,600 --> 00:30:47,740
what is the derivative
of 'g cubed of x'?
619
00:30:47,740 --> 00:30:50,140
To differentiate 'g cubed',
what do you do?
620
00:30:50,140 --> 00:30:54,300
You bring the 3 down to a
power 1 less times the
621
00:30:54,300 --> 00:30:56,500
derivative of 'g of x'
with respect to 'x'.
622
00:30:56,500 --> 00:30:57,610
That's your chain rule.
623
00:30:57,610 --> 00:30:58,920
That's 'g prime of x'.
624
00:30:58,920 --> 00:31:02,140
In other words, if you're clever
enough to see this,
625
00:31:02,140 --> 00:31:07,950
what you say here is, OK, now
I multiply both sides by 3.
626
00:31:07,950 --> 00:31:13,100
The left hand side is just the
derivative of 'g cubed of x'.
627
00:31:13,100 --> 00:31:17,720
The right hand side is the
derivative off '3x'.
628
00:31:17,720 --> 00:31:22,210
Therefore, whatever g of x
is, its cube has the same
629
00:31:22,210 --> 00:31:25,750
derivative with respect
to 'x' as '3x'.
630
00:31:25,750 --> 00:31:28,600
And we've already learned that
if two functions have
631
00:31:28,600 --> 00:31:30,520
identical derivatives,
they differ
632
00:31:30,520 --> 00:31:32,730
by, at most, a constant.
633
00:31:32,730 --> 00:31:38,120
Consequently, 'g cubed
of x' must equal
634
00:31:38,120 --> 00:31:40,590
'3x' plus some constant.
635
00:31:40,590 --> 00:31:47,880
In other words, 'g of x' is the
cube root of '3x plus c'.
636
00:31:47,880 --> 00:31:51,190
Now, time is running short in
terms of other things that I
637
00:31:51,190 --> 00:31:52,990
want to teach you in
today's lesson.
638
00:31:52,990 --> 00:31:56,170
Let me leave this, then,
just for you to check.
639
00:31:56,170 --> 00:32:01,710
Simply differentiate the cube
root of '3x plus c'.
640
00:32:01,710 --> 00:32:05,420
And make sure that when you get
that derivative, it does
641
00:32:05,420 --> 00:32:08,770
turn out to be '1 over
'g of x' squared'.
642
00:32:08,770 --> 00:32:11,310
As I say, it's a straightforward
demonstration.
643
00:32:11,310 --> 00:32:12,940
I leave the details to you.
644
00:32:12,940 --> 00:32:15,980
But the point that's really
important is that whenever you
645
00:32:15,980 --> 00:32:17,060
do get an answer--
646
00:32:17,060 --> 00:32:18,720
the hard part is to
get the answer.
647
00:32:18,720 --> 00:32:22,420
Whenever you do get the answer,
you can check by means
648
00:32:22,420 --> 00:32:25,260
of just taking a derivative.
649
00:32:25,260 --> 00:32:29,090
Now, with all of this talk about
'D inverse' in mind, let
650
00:32:29,090 --> 00:32:32,930
me now go back to the more
traditional notation, the
651
00:32:32,930 --> 00:32:36,510
notation that you'll find in
most textbooks, the notation
652
00:32:36,510 --> 00:32:39,720
that, as I say, if you've had
calculus before, most likely,
653
00:32:39,720 --> 00:32:41,930
you're more familiar with.
654
00:32:41,930 --> 00:32:46,410
And that is the following, that
when we write 'D inverse'
655
00:32:46,410 --> 00:32:50,440
of 'f of x', the average
textbook writes
656
00:32:50,440 --> 00:32:51,880
a symbol like this.
657
00:32:51,880 --> 00:32:54,740
It's called the 'integral'
of 'f of x'.
658
00:32:54,740 --> 00:32:59,690
I will have later lectures to
bemoan this choice of notation
659
00:32:59,690 --> 00:33:00,740
from a different
point of view.
660
00:33:00,740 --> 00:33:03,410
But for the time being, all
we're saying is, instead of
661
00:33:03,410 --> 00:33:04,630
writing 'D inverse'--
662
00:33:04,630 --> 00:33:06,140
again, what's in the name?
663
00:33:06,140 --> 00:33:09,150
Just use this particular
notation.
664
00:33:09,150 --> 00:33:12,830
That when you see this
particular thing, perhaps read
665
00:33:12,830 --> 00:33:14,210
this as what?
666
00:33:14,210 --> 00:33:18,370
That this particular symbol is a
code to tell you to find all
667
00:33:18,370 --> 00:33:21,490
functions whose derivative
is 'f of x'.
668
00:33:21,490 --> 00:33:24,610
And by the way, to summarize
the results that we've
669
00:33:24,610 --> 00:33:28,980
obtained so far, let me just
rewrite some of these basic
670
00:33:28,980 --> 00:33:33,240
results in terms of the more
traditional notation.
671
00:33:33,240 --> 00:33:35,520
When you write this, this is
called the 'indefinite
672
00:33:35,520 --> 00:33:37,780
integral' and what we're saying
is the indefinite
673
00:33:37,780 --> 00:33:41,860
integral of ''x to the n' dx' is
'x to the 'n + 1'' over 'n
674
00:33:41,860 --> 00:33:47,440
+ 1' plus a constant when it
is not equal to minus 1.
675
00:33:47,440 --> 00:33:51,180
The integral of a sum is the
sum of the integrals.
676
00:33:51,180 --> 00:33:55,440
And the integral of a constant
times a function is a constant
677
00:33:55,440 --> 00:33:56,680
times the integral.
678
00:33:56,680 --> 00:33:58,330
All this says is what?
679
00:33:58,330 --> 00:34:02,880
That this is a consequent of the
fact that the derivative
680
00:34:02,880 --> 00:34:04,890
of the sum is the sum
of the derivatives.
681
00:34:04,890 --> 00:34:08,090
This is a consequent of the fact
that the derivative of a
682
00:34:08,090 --> 00:34:11,000
constant times a function is
the constant times the
683
00:34:11,000 --> 00:34:12,590
derivative of the function.
684
00:34:12,590 --> 00:34:15,870
And again, all this is, same
thing we were talking about
685
00:34:15,870 --> 00:34:20,739
before only with the 'D inverse'
notation replaced by
686
00:34:20,739 --> 00:34:25,300
the more common indefinite
integral.
687
00:34:25,300 --> 00:34:28,179
Now, we come to one more problem
which will finish us
688
00:34:28,179 --> 00:34:31,800
up for the day, once we get
through talking about it.
689
00:34:31,800 --> 00:34:32,820
And that's this.
690
00:34:32,820 --> 00:34:37,110
If all this means is find the
functions whose derivative is
691
00:34:37,110 --> 00:34:41,239
'f of x', why write this
notation here?
692
00:34:41,239 --> 00:34:44,000
Why couldn't we have just
written, for example, the
693
00:34:44,000 --> 00:34:47,739
so-called integral sign with
a little 'x' underneath?
694
00:34:47,739 --> 00:34:51,730
You see, we've talked about
misleading notations before.
695
00:34:51,730 --> 00:34:54,460
You see, in terms of our
differential notation, when
696
00:34:54,460 --> 00:34:58,510
you see ''f of x' dx', you have
every right to think of a
697
00:34:58,510 --> 00:35:00,010
differential.
698
00:35:00,010 --> 00:35:04,380
Now, in all fairness, the
chances are that this notation
699
00:35:04,380 --> 00:35:06,710
would not have been invented
if there weren't some
700
00:35:06,710 --> 00:35:10,490
connection between derivatives
and differentials.
701
00:35:10,490 --> 00:35:12,570
So let me mention this point.
702
00:35:12,570 --> 00:35:15,240
Going back as if we were
starting the lecture all over
703
00:35:15,240 --> 00:35:18,900
again, could I have invented a
different machine, which I'll
704
00:35:18,900 --> 00:35:21,910
call the 'script D- machine'?
705
00:35:21,910 --> 00:35:24,410
In other words, I don't want
to call it the same
706
00:35:24,410 --> 00:35:27,230
'D-machine' as before because
now it's going to have a
707
00:35:27,230 --> 00:35:28,660
different set of outputs.
708
00:35:28,660 --> 00:35:30,530
You see, now the input
will still be
709
00:35:30,530 --> 00:35:32,100
differentiable functions.
710
00:35:32,100 --> 00:35:34,780
But the output, instead of being
the derivative of the
711
00:35:34,780 --> 00:35:38,440
function, will be the
differential of the function.
712
00:35:38,440 --> 00:35:41,420
For example, before,
we said what?
713
00:35:41,420 --> 00:35:45,320
If 'x squared' goes in, the
output would be '2x'.
714
00:35:45,320 --> 00:35:50,420
With the script 'D-machine', the
output would be '2x dx'.
715
00:35:50,420 --> 00:35:52,950
Now, even though these machines
are different because
716
00:35:52,950 --> 00:35:55,710
one machine gives you a
differential as an output and
717
00:35:55,710 --> 00:35:57,900
the other gives you a derivative
as an output,
718
00:35:57,900 --> 00:35:59,410
notice that they
are equivalent.
719
00:35:59,410 --> 00:36:04,980
Namely, knowing the
differential, we can pin down
720
00:36:04,980 --> 00:36:08,390
the function the same as when
we knew the derivative.
721
00:36:08,390 --> 00:36:10,630
Now, the question that comes up
is, why should we use this
722
00:36:10,630 --> 00:36:12,440
particular type of notation?
723
00:36:12,440 --> 00:36:16,410
And to use the examples that are
most prevalent in the text
724
00:36:16,410 --> 00:36:19,510
and also in our notes, let me
give you the same problem that
725
00:36:19,510 --> 00:36:21,350
we've done before,
but now from a
726
00:36:21,350 --> 00:36:22,850
different point of view.
727
00:36:22,850 --> 00:36:26,210
Suppose we're still given the
problem 'g prime of x' equals
728
00:36:26,210 --> 00:36:28,480
'1 over 'g of x' squared'.
729
00:36:28,480 --> 00:36:31,670
We say, OK, let 'y'
equal 'g of x'.
730
00:36:31,670 --> 00:36:35,270
When we do that, this problem
now translates into what?
731
00:36:35,270 --> 00:36:40,480
'dy dx' equals '1 over
'y squared''.
732
00:36:40,480 --> 00:36:44,190
Now, if we allow ourselves to
use the differential notation,
733
00:36:44,190 --> 00:36:47,810
which we justified in previous
lectures, this says what?
734
00:36:47,810 --> 00:36:52,000
Cross-multiplying ''y squared'
dy' equals 'dx'.
735
00:36:52,000 --> 00:36:56,050
Notice, by the way, that 'y' is
implicitly a differentiable
736
00:36:56,050 --> 00:36:57,840
function of 'x'.
737
00:36:57,840 --> 00:37:00,780
So what we're saying over here
is, look, here are two
738
00:37:00,780 --> 00:37:05,850
functions which have the same
differential, therefore, if we
739
00:37:05,850 --> 00:37:08,630
integrate them, they differ
by a constant.
740
00:37:08,630 --> 00:37:13,520
Well, to mimic what we were
doing before, let's just say--
741
00:37:13,520 --> 00:37:14,530
let's see, this, I
think, I have a
742
00:37:14,530 --> 00:37:15,070
little bit twisted here.
743
00:37:15,070 --> 00:37:19,300
If ''y squared' dy' equals
'3x', '3 'y squared' dy'
744
00:37:19,300 --> 00:37:20,900
equals '3dx'.
745
00:37:20,900 --> 00:37:24,200
If the differentials are equal,
then the functions
746
00:37:24,200 --> 00:37:25,540
differ by a constant.
747
00:37:25,540 --> 00:37:28,730
But 'y cubed' is the function
whose differential is '3 'y
748
00:37:28,730 --> 00:37:31,240
squared' dy'.
749
00:37:31,240 --> 00:37:34,840
'3x' is the function whose
differential is '3dx'.
750
00:37:34,840 --> 00:37:40,590
And we wind up with 'y cubed'
equals '3x + c', or, 'y'
751
00:37:40,590 --> 00:37:45,710
equals the 'cube root
of '3x + c''.
752
00:37:45,710 --> 00:37:49,120
By the way, let me just pull
this board down so that we can
753
00:37:49,120 --> 00:37:51,440
make a little bit of
a comparison here.
754
00:37:51,440 --> 00:37:55,330
You see, notice that in using
this differential notation,
755
00:37:55,330 --> 00:37:58,770
which allowed us to use some
nice algebraic devices and
756
00:37:58,770 --> 00:38:01,870
didn't seem quite as difficult
for us to recognize certain
757
00:38:01,870 --> 00:38:04,820
things, I hope you notice
that there seems to be a
758
00:38:04,820 --> 00:38:09,200
correspondence between the steps
that we had here and the
759
00:38:09,200 --> 00:38:11,690
steps that took place
over here.
760
00:38:11,690 --> 00:38:15,250
In other words, having done at
the so-called rigorous way,
761
00:38:15,250 --> 00:38:21,620
notice that differentials
give us a very nice,
762
00:38:21,620 --> 00:38:26,140
intuitively-simpler technique
for solving certain types of
763
00:38:26,140 --> 00:38:30,720
problems when we have implicit
relationships
764
00:38:30,720 --> 00:38:33,080
between 'y' and 'x'.
765
00:38:33,080 --> 00:38:35,540
And so, just to illustrate
this in terms of one more
766
00:38:35,540 --> 00:38:38,710
problem, let's take one that's
fairly geometric.
767
00:38:38,710 --> 00:38:41,880
Let's suppose we're given
the following problem.
768
00:38:41,880 --> 00:38:45,790
'dy dx' is 'minus x' over 'y'
and we know that when 'x'
769
00:38:45,790 --> 00:38:48,570
equals 3, 'y' equals 4.
770
00:38:48,570 --> 00:38:51,810
Using the language of
differentials, what we would
771
00:38:51,810 --> 00:38:53,610
do over here is we would
cross-multiply.
772
00:38:53,610 --> 00:38:56,890
773
00:38:56,890 --> 00:38:58,040
Whatever you want to do.
774
00:38:58,040 --> 00:39:03,370
we recognize that, to get '2y',
you must differentiate
775
00:39:03,370 --> 00:39:04,100
'y squared'.
776
00:39:04,100 --> 00:39:05,990
This is a step you don't
really need.
777
00:39:05,990 --> 00:39:07,230
I don't care how you
want to do this.
778
00:39:07,230 --> 00:39:10,550
All I'm saying is that from this
step, I can get to here.
779
00:39:10,550 --> 00:39:15,390
Recognizing that y squared has
'2y dy' as its differential
780
00:39:15,390 --> 00:39:18,550
and that ''minus 'x squared''
has 'minus 2x dx' as its
781
00:39:18,550 --> 00:39:22,520
differential, I know that 'y
squared' is equal to ''minus
782
00:39:22,520 --> 00:39:24,260
'x squared'' plus a constant.
783
00:39:24,260 --> 00:39:25,600
I transpose.
784
00:39:25,600 --> 00:39:28,800
I get that 'x squared' plus
'y squared' is a constant.
785
00:39:28,800 --> 00:39:32,380
Knowing that when 'x' equals
3, 'y' equals 4, I can
786
00:39:32,380 --> 00:39:37,030
determine that the constant must
be 25 when 'x' equals 3
787
00:39:37,030 --> 00:39:38,250
and 'y' is 4.
788
00:39:38,250 --> 00:39:41,870
But since it's a constant, if
it's 25 when 'x' equals 3 and
789
00:39:41,870 --> 00:39:46,050
'y' equals 4, it's
25 everyplace.
790
00:39:46,050 --> 00:39:48,780
And again, if you want a
geometric interpretation of
791
00:39:48,780 --> 00:39:53,550
this, the circle centered at the
origin with radius equal
792
00:39:53,550 --> 00:39:59,550
to 5 is the only curve in the
whole world whose derivative
793
00:39:59,550 --> 00:40:04,070
at any given point is the
negative of the x-coordinate
794
00:40:04,070 --> 00:40:07,050
over the y-coordinate
and passes through
795
00:40:07,050 --> 00:40:09,140
the point (3 , 4).
796
00:40:09,140 --> 00:40:11,690
And if you want to see that
geometrically, let me just
797
00:40:11,690 --> 00:40:13,300
take a second here.
798
00:40:13,300 --> 00:40:19,160
You see, notice that any point
on the circle (x , y), notice
799
00:40:19,160 --> 00:40:22,560
that the tangent line to the
curve, which has slope 'dy
800
00:40:22,560 --> 00:40:24,130
dx', is what?
801
00:40:24,130 --> 00:40:28,370
Perpendicular to the
radius, again.
802
00:40:28,370 --> 00:40:31,610
The radius has slope
'y' over 'x'.
803
00:40:31,610 --> 00:40:32,940
And to be perpendicular--
804
00:40:32,940 --> 00:40:35,000
if two lines are perpendicular,
their slopes
805
00:40:35,000 --> 00:40:37,610
are negative reciprocals.
806
00:40:37,610 --> 00:40:40,200
This is this particular
problem.
807
00:40:40,200 --> 00:40:42,580
And now, all I wanted to show
you is that if you're nervous
808
00:40:42,580 --> 00:40:45,880
about differentials and you
don't like to use them, notice
809
00:40:45,880 --> 00:40:48,150
that the same problem could
have been stated without
810
00:40:48,150 --> 00:40:49,280
differentials.
811
00:40:49,280 --> 00:40:51,600
Namely, we are thinking of a
function which we'll call 'g
812
00:40:51,600 --> 00:40:55,310
of x' such that the derivative
of 'g of x' is 'minus
813
00:40:55,310 --> 00:40:57,370
x' over 'g of x'.
814
00:40:57,370 --> 00:41:00,670
In other words, if the input of
the 'D-machine' is 'g', the
815
00:41:00,670 --> 00:41:03,210
output is 'minus x' over 'g'.
816
00:41:03,210 --> 00:41:06,990
And we also know that when 'x'
equals 3, the output is 4.
817
00:41:06,990 --> 00:41:09,330
So to 'g' of 3 equals 4.
818
00:41:09,330 --> 00:41:12,800
And without going through the
great details here, let's just
819
00:41:12,800 --> 00:41:13,420
notice that we could
820
00:41:13,420 --> 00:41:16,260
cross-multiply the same as before.
821
00:41:16,260 --> 00:41:20,580
We could multiply both sides
by 2, the same as before.
822
00:41:20,580 --> 00:41:23,500
We could recognize that the
left-hand side was the
823
00:41:23,500 --> 00:41:26,770
derivative of ''g of x'
squared', that the right-hand
824
00:41:26,770 --> 00:41:29,350
side is the derivative of
'minus 'x squared''.
825
00:41:29,350 --> 00:41:32,830
Therefore, since they have the
same identical derivatives,
826
00:41:32,830 --> 00:41:35,270
they must differ
by a constant.
827
00:41:35,270 --> 00:41:37,070
OK.
828
00:41:37,070 --> 00:41:41,190
We actually have plus or minus
here, but notice the fact that
829
00:41:41,190 --> 00:41:45,450
when the input is 3, the output
is 4 means that we're
830
00:41:45,450 --> 00:41:48,820
on the positive branch of
the curve, et cetera.
831
00:41:48,820 --> 00:41:52,910
I say et cetera not because
these points aren't important
832
00:41:52,910 --> 00:41:56,410
but because every point that
comes up now has already been
833
00:41:56,410 --> 00:42:00,640
discussed under the heading
of differential calculus.
834
00:42:00,640 --> 00:42:04,370
In other words, that the inverse
of differentiation can
835
00:42:04,370 --> 00:42:07,490
be handled very, very neatly
just by knowing
836
00:42:07,490 --> 00:42:10,190
differentiation with a
switch in emphasis.
837
00:42:10,190 --> 00:42:12,580
Why do we want to know this
particular topic?
838
00:42:12,580 --> 00:42:16,650
Well, because in many cases, to
look at it geometrically,
839
00:42:16,650 --> 00:42:19,280
in the past, we were
given the curve.
840
00:42:19,280 --> 00:42:21,310
We wanted to find out
what the slope was.
841
00:42:21,310 --> 00:42:24,700
In many physical applications,
we are, in a sense, told what
842
00:42:24,700 --> 00:42:27,700
the slope is and have to figure
out what the curve is.
843
00:42:27,700 --> 00:42:29,060
Hence the motivation.
844
00:42:29,060 --> 00:42:32,440
Well, at any rate, there'll be
more about this in the text
845
00:42:32,440 --> 00:42:33,790
and in our exercises.
846
00:42:33,790 --> 00:42:37,890
So until next time, good bye.
847
00:42:37,890 --> 00:42:40,890
MALE SPEAKER: Funding for the
publication of this video was
848
00:42:40,890 --> 00:42:45,610
provided by the Gabriella and
Paul Rosenbaum Foundation.
849
00:42:45,610 --> 00:42:49,780
Help OCW continue to provide
free and open access to MIT
850
00:42:49,780 --> 00:42:53,980
courses by making a donation
at ocw.mit.edu/donate.
851
00:42:53,980 --> 00:42:58,730