1
00:00:00,040 --> 00:00:01,940
NARRATOR: 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
00:00:39,230 --> 00:00:40,020
PROFESSOR: Hi.
9
00:00:40,020 --> 00:00:44,140
I'm Herb Gross, and welcome
to Calculus Revisited.
10
00:00:44,140 --> 00:00:46,280
I guess the most difficult
lecture to give with any
11
00:00:46,280 --> 00:00:48,440
course is probably
the first one.
12
00:00:48,440 --> 00:00:51,260
And you're sort of tempted to
look at your audience and say
13
00:00:51,260 --> 00:00:54,120
you're probably wondering why
I called you all here.
14
00:00:54,120 --> 00:00:57,740
And in this sense, I have
elected to entitle our first
15
00:00:57,740 --> 00:01:03,150
lecture simply Preface to give a
double overview, an overview
16
00:01:03,150 --> 00:01:07,160
both of the hardware and the
software that will make up
17
00:01:07,160 --> 00:01:08,680
this course.
18
00:01:08,680 --> 00:01:13,240
To begin with, we will have a
series of lectures of which
19
00:01:13,240 --> 00:01:15,080
this is the first.
20
00:01:15,080 --> 00:01:19,150
In our lectures, our main aim
will be to give an overview of
21
00:01:19,150 --> 00:01:22,880
the material being covered, an
insight as to why various
22
00:01:22,880 --> 00:01:26,260
computations are done, and
insights as to how
23
00:01:26,260 --> 00:01:29,710
applications of these concepts
will be made.
24
00:01:29,710 --> 00:01:35,430
The heart of our course will
consist of a regular textbook.
25
00:01:35,430 --> 00:01:37,630
You see, we have our lectures.
26
00:01:37,630 --> 00:01:40,390
We have a textbook.
27
00:01:40,390 --> 00:01:44,810
The textbook is designed to
supply you with deeper
28
00:01:44,810 --> 00:01:48,610
insights than what we can
give in a lecture.
29
00:01:48,610 --> 00:01:53,000
In addition, recognizing the
fact that the textbook may
30
00:01:53,000 --> 00:01:56,430
leave gaps, places where you
may want some additional
31
00:01:56,430 --> 00:02:00,030
knowledge, we also have
supplementary notes.
32
00:02:00,030 --> 00:02:05,360
And finally, at the backbone of
our package is what we call
33
00:02:05,360 --> 00:02:06,630
the study guide.
34
00:02:06,630 --> 00:02:10,520
The study guide consists of
a breakdown of the course.
35
00:02:10,520 --> 00:02:14,990
It tells us what the various
lectures will be, the units.
36
00:02:14,990 --> 00:02:19,290
There are pretests to help you
decide how well prepared you
37
00:02:19,290 --> 00:02:21,090
are for the topic that's
coming up.
38
00:02:21,090 --> 00:02:23,640
There is a final examination
at the end of
39
00:02:23,640 --> 00:02:25,300
each block of material.
40
00:02:25,300 --> 00:02:28,640
And perhaps most importantly,
especially from an engineer's
41
00:02:28,640 --> 00:02:33,450
point of view, in each unit that
we study, the study guide
42
00:02:33,450 --> 00:02:37,970
will consist of exercises
primarily called learning
43
00:02:37,970 --> 00:02:42,970
exercises, exercises which
hopefully will turn you on
44
00:02:42,970 --> 00:02:46,060
towards wanting to be able to
apply the material, and at the
45
00:02:46,060 --> 00:02:50,100
same time, serve as a
springboard by which we can
46
00:02:50,100 --> 00:02:54,050
highlight why the theory and
many about our lecture points
47
00:02:54,050 --> 00:02:56,780
are really as important
as they are.
48
00:02:56,780 --> 00:03:00,140
So much for the hardware
of our course.
49
00:03:00,140 --> 00:03:03,540
And now let's turn our attention
to the software.
50
00:03:03,540 --> 00:03:05,920
Just what is calculus?
51
00:03:05,920 --> 00:03:09,870
In a manner of speaking,
calculus can be viewed as
52
00:03:09,870 --> 00:03:13,770
being high school mathematics
with one additional concept
53
00:03:13,770 --> 00:03:16,150
called the limit concept
thrown in.
54
00:03:16,150 --> 00:03:18,770
If you recall back to your high
school days, remember
55
00:03:18,770 --> 00:03:21,250
that we're always dealing
with things like
56
00:03:21,250 --> 00:03:23,390
average rate of speed.
57
00:03:23,390 --> 00:03:26,400
Notice I say average or constant
rate of speed.
58
00:03:26,400 --> 00:03:29,800
The old recipe that distance
equals rate times time
59
00:03:29,800 --> 00:03:33,200
presupposes that the rate is
constant, because if the rate
60
00:03:33,200 --> 00:03:36,750
is varying, which rate is it
that you use to multiply the
61
00:03:36,750 --> 00:03:39,200
time by to find the distance?
62
00:03:39,200 --> 00:03:43,760
You see, in other words, roughly
speaking, we can say
63
00:03:43,760 --> 00:03:46,660
that at least one branch of
calculus known as differential
64
00:03:46,660 --> 00:03:49,170
calculus deals with
the subject of
65
00:03:49,170 --> 00:03:50,780
instantaneous speed.
66
00:03:50,780 --> 00:03:53,730
And instantaneous speed is a
rather easy thing to talk
67
00:03:53,730 --> 00:03:55,240
about intuitively.
68
00:03:55,240 --> 00:03:59,740
Imagine an object moving along
this line and passing the
69
00:03:59,740 --> 00:04:03,520
point P. And we say to ourselves
how fast was the
70
00:04:03,520 --> 00:04:06,260
object moving at the instant
that we're at the point P?
71
00:04:06,260 --> 00:04:09,700
Now, you see, this is some
sort of a problem.
72
00:04:09,700 --> 00:04:13,560
Because at the instant that
you're at P, you're not in a
73
00:04:13,560 --> 00:04:17,010
sense moving at all because
you're at P.
74
00:04:17,010 --> 00:04:20,230
Of course, what we do to reduce
this problem to an old
75
00:04:20,230 --> 00:04:24,190
one is we say, well, suppose we
have a couple of observers.
76
00:04:24,190 --> 00:04:26,940
Let's call them O1 and O2.
77
00:04:26,940 --> 00:04:30,500
Let them be stationed, one on
each side of P. Now, certainly
78
00:04:30,500 --> 00:04:33,030
what we could do physically
here is we can measure the
79
00:04:33,030 --> 00:04:37,910
distance between O1 and O2.
80
00:04:37,910 --> 00:04:43,090
And we can also measure the
time that it takes to
81
00:04:43,090 --> 00:04:46,420
go from O1 to O2.
82
00:04:46,420 --> 00:04:50,270
And what we can do is divide
that distance by the time, and
83
00:04:50,270 --> 00:04:54,670
that, you see, is our old high
school concept of the average
84
00:04:54,670 --> 00:04:58,285
speed of the particle as
it moves from O1 to O2.
85
00:04:58,285 --> 00:05:01,330
Now, you see, the question is,
somebody says gee, that's a
86
00:05:01,330 --> 00:05:03,780
great answer, but it's
the wrong problem.
87
00:05:03,780 --> 00:05:05,920
We didn't ask what was the
average speed as we
88
00:05:05,920 --> 00:05:07,420
went from O1 to O2.
89
00:05:07,420 --> 00:05:09,730
We asked what was the
instantaneous speed.
90
00:05:09,730 --> 00:05:11,700
And the idea is we say,
well, lookit.
91
00:05:11,700 --> 00:05:14,880
The average speed and the
instantaneous speed, it seems,
92
00:05:14,880 --> 00:05:17,740
should be pretty much the same
if the observers were
93
00:05:17,740 --> 00:05:19,800
relatively close together.
94
00:05:19,800 --> 00:05:22,670
The next observation is it seems
that if we were to move
95
00:05:22,670 --> 00:05:26,820
the observers in even closer,
there would be less of a
96
00:05:26,820 --> 00:05:31,440
discrepancy between O1 and O2
in the sense that-- not a
97
00:05:31,440 --> 00:05:34,345
discrepancy, but in the sense
that the average speed would
98
00:05:34,345 --> 00:05:36,900
now seem like a better
approximation to the
99
00:05:36,900 --> 00:05:39,480
instantaneous speed because
there was less distance for
100
00:05:39,480 --> 00:05:41,725
something to go wrong in.
101
00:05:41,725 --> 00:05:45,220
And so we get the idea that
maybe what we should do is
102
00:05:45,220 --> 00:05:48,480
make the observers gets closer
and closer together.
103
00:05:48,480 --> 00:05:50,810
That would minimize the
difference between the average
104
00:05:50,810 --> 00:05:54,380
speed and the instantaneous rate
of speed, and maybe the
105
00:05:54,380 --> 00:05:56,480
optimal thing would happen
when the two
106
00:05:56,480 --> 00:05:58,390
observers were together.
107
00:05:58,390 --> 00:05:59,870
But the strange part is--
108
00:05:59,870 --> 00:06:02,080
and this is where calculus
really begins.
109
00:06:02,080 --> 00:06:04,140
This is what calculus
is all about.
110
00:06:04,140 --> 00:06:07,370
As soon as the observers come
together, notice that what you
111
00:06:07,370 --> 00:06:10,420
have is that the distance
between them is 0.
112
00:06:10,420 --> 00:06:13,430
The time that it takes to get
from one to the other is 0.
113
00:06:13,430 --> 00:06:17,020
And therefore, it appears that
if we divide distance by time,
114
00:06:17,020 --> 00:06:21,550
we are going to wind
up with 0/0.
115
00:06:21,550 --> 00:06:25,630
Now, my claim is that 0/0
should be called--
116
00:06:25,630 --> 00:06:30,190
well, I'll call it undefined,
but actually, I think
117
00:06:30,190 --> 00:06:36,920
indeterminate would
be a better word.
118
00:06:36,920 --> 00:06:38,415
Why do I say that?
119
00:06:38,415 --> 00:06:40,330
Well, here's an interesting
thing.
120
00:06:40,330 --> 00:06:43,900
When we do arithmetic with small
numbers, observe that if
121
00:06:43,900 --> 00:06:46,980
you add two small numbers, you
expect the result to be a
122
00:06:46,980 --> 00:06:47,960
small number.
123
00:06:47,960 --> 00:06:51,230
If you multiply two small
numbers, you expect the result
124
00:06:51,230 --> 00:06:52,890
to be a small number.
125
00:06:52,890 --> 00:06:57,280
Similarly, for division, for
subtraction, the difference of
126
00:06:57,280 --> 00:06:59,740
two small numbers is
a small number.
127
00:06:59,740 --> 00:07:02,960
On the other hand, the quotient
of two small numbers
128
00:07:02,960 --> 00:07:04,450
is rather deceptive.
129
00:07:04,450 --> 00:07:08,000
Because it's a ratio, if one
of the very small numbers
130
00:07:08,000 --> 00:07:12,390
happens to be very much larger
compared with the other small
131
00:07:12,390 --> 00:07:15,200
number, the ratio might
be quite large.
132
00:07:15,200 --> 00:07:19,960
Well, for example, visualize,
say, 10 to the minus 6,
133
00:07:19,960 --> 00:07:27,470
1/1,000,000, 0.000001, which
is a pretty small number.
134
00:07:27,470 --> 00:07:32,310
Now, divide that by 10
to the minus 12th.
135
00:07:32,310 --> 00:07:35,210
Well, you see, 10 to the minus
12th is a small number, so
136
00:07:35,210 --> 00:07:38,220
small that it makes 10 to the
minus sixth appear large.
137
00:07:38,220 --> 00:07:43,440
In fact, the quotient is 10 to
the sixth, which is 1,000,000.
138
00:07:43,440 --> 00:07:46,210
And here we see that when you're
dealing with the ratio
139
00:07:46,210 --> 00:07:49,180
of small numbers, you're a
little bit in trouble, because
140
00:07:49,180 --> 00:07:53,310
we can't tell whether the ratio
will be small, or large,
141
00:07:53,310 --> 00:07:54,930
or somewhere in between.
142
00:07:54,930 --> 00:07:57,430
For example, if we reverse
the role of numerator and
143
00:07:57,430 --> 00:08:00,020
denominator here, we would still
have the quotient of two
144
00:08:00,020 --> 00:08:03,880
small numbers, but 10 to the
minus 12th divided by 10 to
145
00:08:03,880 --> 00:08:11,930
the minus sixth is a relatively
small number, 10 to
146
00:08:11,930 --> 00:08:13,810
the minus 6.
147
00:08:13,810 --> 00:08:16,600
Of course, this is the physical
way of looking at it.
148
00:08:16,600 --> 00:08:18,910
Small divided by small
is indeterminate.
149
00:08:18,910 --> 00:08:21,400
We have a more rigorous way of
looking at this if you want to
150
00:08:21,400 --> 00:08:24,090
see it from a mathematical
structure point of view.
151
00:08:24,090 --> 00:08:28,540
Namely, suppose we define a/b
in the traditional way.
152
00:08:28,540 --> 00:08:33,059
Namely, a/b is that number such
that when we multiply it
153
00:08:33,059 --> 00:08:36,020
by b we get a.
154
00:08:36,020 --> 00:08:40,200
Well, what would that say as
far as 0/0 was concerned?
155
00:08:40,200 --> 00:08:41,419
It would say what?
156
00:08:41,419 --> 00:08:45,060
That 0/0 is that number such
that when we multiply
157
00:08:45,060 --> 00:08:48,030
it by 0 we get 0.
158
00:08:48,030 --> 00:08:51,410
Now, what number has the
property that when we multiply
159
00:08:51,410 --> 00:08:53,330
it by 0 we get 0?
160
00:08:53,330 --> 00:08:55,890
And the answer is any number.
161
00:08:55,890 --> 00:08:58,380
This is why 0/0 is
indeterminate.
162
00:08:58,380 --> 00:09:01,630
If we say to a person, tell me
the number I must multiply by
163
00:09:01,630 --> 00:09:05,570
0 to get 0, the answer
is any number.
164
00:09:05,570 --> 00:09:08,850
Well, the idea then is that we
must avoid the expression 0/0
165
00:09:08,850 --> 00:09:10,310
at all costs.
166
00:09:10,310 --> 00:09:14,570
What this means then is that we
say OK, let the observers
167
00:09:14,570 --> 00:09:17,940
get closer to closer together,
but never touch.
168
00:09:17,940 --> 00:09:20,410
Now, the point is that as long
as the observers get closer
169
00:09:20,410 --> 00:09:23,970
and closer together and never
touch, let's ask the question
170
00:09:23,970 --> 00:09:26,300
how many pairs of observers
do we need?
171
00:09:26,300 --> 00:09:28,790
And the answer is that
theoretically we need
172
00:09:28,790 --> 00:09:31,400
infinitely many pairs
of observers.
173
00:09:31,400 --> 00:09:32,740
Well, why is that?
174
00:09:32,740 --> 00:09:35,610
Because as long as there's a
distance between a pair of
175
00:09:35,610 --> 00:09:38,520
observers, we can theoretically
fit in another
176
00:09:38,520 --> 00:09:40,380
pair of observers.
177
00:09:40,380 --> 00:09:44,700
This is why in our course we do
not begin with this idea,
178
00:09:44,700 --> 00:09:48,340
but looking backwards now, we
say ah, we had better find
179
00:09:48,340 --> 00:09:51,140
some way of giving us the
equivalent of having
180
00:09:51,140 --> 00:09:53,290
infinitely many pairs
of observers.
181
00:09:53,290 --> 00:09:59,140
And to do this, the idea that we
come up with is the concept
182
00:09:59,140 --> 00:10:01,990
called a function.
183
00:10:01,990 --> 00:10:05,400
Consider the old Galileo freely
falling body problem,
184
00:10:05,400 --> 00:10:09,520
where the distance that the
body falls s equals 16t
185
00:10:09,520 --> 00:10:12,950
squared, where t is in seconds
and s is in feet.
186
00:10:12,950 --> 00:10:17,760
Notice that this apparently
harmless recipe gives us a way
187
00:10:17,760 --> 00:10:21,250
for finding s for
each given t.
188
00:10:21,250 --> 00:10:24,650
In other words, to all intents
and purposes, this recipe
189
00:10:24,650 --> 00:10:28,992
gives us an observer for
each point of time.
190
00:10:28,992 --> 00:10:32,260
For each time, we can find the
distance, which is physically
191
00:10:32,260 --> 00:10:36,330
equivalent to knowing an
observer at every point.
192
00:10:36,330 --> 00:10:40,430
In turn, the study of functions
lends itself to a
193
00:10:40,430 --> 00:10:43,220
study of graphs, a picture.
194
00:10:43,220 --> 00:10:47,590
Namely, if we look at s equals
16t squared again, notice that
195
00:10:47,590 --> 00:10:49,910
we visualize a recipe here.
196
00:10:49,910 --> 00:10:54,250
t can be viewed as being an
input, s as the output.
197
00:10:54,250 --> 00:10:58,560
For a given input t, we can
compute the output s.
198
00:10:58,560 --> 00:11:03,090
In general, if we now elect
to plot the input along a
199
00:11:03,090 --> 00:11:07,770
horizontal line and the output
at right angles to this, we
200
00:11:07,770 --> 00:11:11,270
now have a picture of our
relationship, a picture which
201
00:11:11,270 --> 00:11:14,200
is called a graph.
202
00:11:14,200 --> 00:11:17,980
You see, we can talk about this
more explicitly as far as
203
00:11:17,980 --> 00:11:20,830
this particular problem is
concerned, just by taking a
204
00:11:20,830 --> 00:11:22,650
look at a picture like this.
205
00:11:22,650 --> 00:11:25,810
In other words, in this
particular problem, the input
206
00:11:25,810 --> 00:11:30,970
is time t, the output
is distance s.
207
00:11:30,970 --> 00:11:35,270
For each t, we locate a height
called s by squaring t and
208
00:11:35,270 --> 00:11:37,450
multiplying by 16.
209
00:11:37,450 --> 00:11:40,390
And now, what average speed
means in terms of this kind of
210
00:11:40,390 --> 00:11:42,640
a diagram is the following.
211
00:11:42,640 --> 00:11:46,410
To find the average speed, all
we have to do is on a given
212
00:11:46,410 --> 00:11:49,960
time interval find the distance
traveled, which I
213
00:11:49,960 --> 00:11:53,510
call delta s, the change in
distance, and divide that by
214
00:11:53,510 --> 00:11:54,860
the change in time.
215
00:11:54,860 --> 00:11:58,030
That's the average speed, which,
by the way, from a
216
00:11:58,030 --> 00:12:02,150
geometrical point of view,
becomes known as the slope of
217
00:12:02,150 --> 00:12:03,820
this particular straight line.
218
00:12:03,820 --> 00:12:08,130
In other words, average speed is
to functions what slope of
219
00:12:08,130 --> 00:12:10,830
a straight line is
to geometry.
220
00:12:10,830 --> 00:12:14,190
At any rate, knowing what the
average rate of speed is, we
221
00:12:14,190 --> 00:12:18,570
sort of say why couldn't we
define the instantaneous speed
222
00:12:18,570 --> 00:12:19,500
to be this.
223
00:12:19,500 --> 00:12:24,040
We will take the change in
distance divided by the change
224
00:12:24,040 --> 00:12:26,740
in time and see what happens.
225
00:12:26,740 --> 00:12:28,430
And we write this this way.
226
00:12:28,430 --> 00:12:30,450
Limit as delta t approaches 0.
227
00:12:30,450 --> 00:12:33,450
Let's see what happens as that
change in time becomes
228
00:12:33,450 --> 00:12:37,370
arbitrarily small, but never
equaling 0 because we don't
229
00:12:37,370 --> 00:12:39,820
want a 0/0 form here.
230
00:12:39,820 --> 00:12:44,260
You see, this then becomes the
working definition of what we
231
00:12:44,260 --> 00:12:46,670
call differential calculus.
232
00:12:46,670 --> 00:12:50,390
The point is that this
particular definition does not
233
00:12:50,390 --> 00:12:53,240
depend on s equaling
16t squared.
234
00:12:53,240 --> 00:12:57,430
s could be any function
of t whatsoever.
235
00:12:57,430 --> 00:13:00,210
We could have a more elaborate
type of situation.
236
00:13:00,210 --> 00:13:02,170
The important point is what?
237
00:13:02,170 --> 00:13:05,420
The basic definition
stays the same.
238
00:13:05,420 --> 00:13:09,980
What changes is the amount of
arithmetic that's necessary to
239
00:13:09,980 --> 00:13:13,970
handle the particular
relationship between s and t.
240
00:13:13,970 --> 00:13:18,010
This will be a major part of our
course, the strange thing
241
00:13:18,010 --> 00:13:20,970
being that even at the very end
of our course when we've
242
00:13:20,970 --> 00:13:24,200
gone through many, many things,
our basic definition
243
00:13:24,200 --> 00:13:26,790
of instantaneous rate of
change will have never
244
00:13:26,790 --> 00:13:27,960
changed from this.
245
00:13:27,960 --> 00:13:30,440
It will always stay like this.
246
00:13:30,440 --> 00:13:34,320
But what will change is how much
arithmetic and algebra
247
00:13:34,320 --> 00:13:38,250
and geometry and trigonometry,
et cetera, we will have to do
248
00:13:38,250 --> 00:13:40,350
in order to compute these
things from a
249
00:13:40,350 --> 00:13:42,170
numerical point of view.
250
00:13:42,170 --> 00:13:45,620
Well, so much for the first
phase of calculus called
251
00:13:45,620 --> 00:13:47,150
differential calculus.
252
00:13:47,150 --> 00:13:50,610
A second phase of calculus, one
which was developed by the
253
00:13:50,610 --> 00:13:54,510
Ancient Greeks by 600 BC, the
subject that ultimately
254
00:13:54,510 --> 00:13:58,200
becomes known as integral
calculus, concerns problem of
255
00:13:58,200 --> 00:14:00,954
finding area under a curve.
256
00:14:00,954 --> 00:14:07,200
Here, I've elected to draw the
parabola y equals x squared on
257
00:14:07,200 --> 00:14:10,840
the interval from
0, 0 to 1, 0.
258
00:14:10,840 --> 00:14:17,290
And the question basically is
what is the area bounded by
259
00:14:17,290 --> 00:14:19,990
this sort of triangular
region?
260
00:14:19,990 --> 00:14:24,330
Let's call that region R, and
what we would like to find is
261
00:14:24,330 --> 00:14:26,420
the area of the region R.
262
00:14:26,420 --> 00:14:30,310
And the Ancient Greeks had a
rather interesting title for
263
00:14:30,310 --> 00:14:32,300
this type of approach for
finding the area.
264
00:14:32,300 --> 00:14:35,400
It is both figurative and
literal, I guess.
265
00:14:35,400 --> 00:14:36,810
It's called the method
of exhaustion.
266
00:14:40,670 --> 00:14:40,754
What they did was to --
267
00:14:40,754 --> 00:14:42,610
They would divide the
interval, say,
268
00:14:42,610 --> 00:14:44,530
into n equal parts.
269
00:14:44,530 --> 00:14:47,570
And picking the lowest point in
each interval, they would
270
00:14:47,570 --> 00:14:50,940
inscribe a rectangle.
271
00:14:50,940 --> 00:14:53,500
Knowing that the area of the
rectangle was the base times
272
00:14:53,500 --> 00:14:57,110
the height, they would add up
the area of each of these
273
00:14:57,110 --> 00:15:00,880
rectangles, and know that
whatever that area was, that
274
00:15:00,880 --> 00:15:04,540
would have to be too small to
be the right answer because
275
00:15:04,540 --> 00:15:07,040
that region was contained
in R. And that would be
276
00:15:07,040 --> 00:15:08,780
labeled A sub n--
277
00:15:08,780 --> 00:15:10,120
lower bar, say--
278
00:15:10,120 --> 00:15:13,130
to indicate that this was a sum
of rectangles which was
279
00:15:13,130 --> 00:15:15,780
too small to be the
right answer.
280
00:15:15,780 --> 00:15:20,210
Similarly, they would then find
the highest point in each
281
00:15:20,210 --> 00:15:24,430
rectangle, get an
overapproximation by adding up
282
00:15:24,430 --> 00:15:28,240
the sum of those areas, which
they would call A sub n upper
283
00:15:28,240 --> 00:15:31,050
bar, and now know that the area
of the regions they were
284
00:15:31,050 --> 00:15:34,260
looking for was squeezed
in between these two.
285
00:15:34,260 --> 00:15:37,530
Then what they would do is make
more and more divisions,
286
00:15:37,530 --> 00:15:40,320
and hopefully, and I think
you can see this sort of
287
00:15:40,320 --> 00:15:42,840
intuitively happening here,
each of the lower
288
00:15:42,840 --> 00:15:45,910
approximations gets bigger
and fills out
289
00:15:45,910 --> 00:15:47,970
the space from inside.
290
00:15:47,970 --> 00:15:51,730
Each of the upper approximations
gets smaller
291
00:15:51,730 --> 00:15:54,770
and chops off the space
from outside here.
292
00:15:54,770 --> 00:16:01,530
And hopefully, if both of these
bounds sort of converge
293
00:16:01,530 --> 00:16:05,520
to the same value L, we get the
idea that the area of the
294
00:16:05,520 --> 00:16:08,130
region R must be L.
295
00:16:08,130 --> 00:16:09,780
This is not anything new.
296
00:16:09,780 --> 00:16:12,700
In other words, this is a
technique that is some 2,500
297
00:16:12,700 --> 00:16:16,820
years old, used by the
Ancient Greeks.
298
00:16:16,820 --> 00:16:19,030
Of course, what happens with
engineering students in
299
00:16:19,030 --> 00:16:22,620
general is that one frequently
says, but I'm not interested
300
00:16:22,620 --> 00:16:23,920
in studying area.
301
00:16:23,920 --> 00:16:25,630
I am not a geometer.
302
00:16:25,630 --> 00:16:26,720
I am a physicist.
303
00:16:26,720 --> 00:16:28,370
I am an engineer.
304
00:16:28,370 --> 00:16:30,990
What good is the area
under a curve?
305
00:16:30,990 --> 00:16:35,510
And the interesting point here
becomes that if we label the
306
00:16:35,510 --> 00:16:39,450
coordinate axis rather than x
and y, give them physical
307
00:16:39,450 --> 00:16:43,490
labels, it turns out that area
under a curve has a physical
308
00:16:43,490 --> 00:16:44,880
interpretation.
309
00:16:44,880 --> 00:16:46,390
Consider the same problem.
310
00:16:46,390 --> 00:16:50,270
Only now, instead of talking
about y equals x squared,
311
00:16:50,270 --> 00:16:53,340
let's talk about v, the
velocity, equaling the square
312
00:16:53,340 --> 00:16:54,460
of the time.
313
00:16:54,460 --> 00:16:57,080
And say that the time
goes to 0 to 1.
314
00:16:57,080 --> 00:17:00,870
In other words, if we plot
v versus t, we get a
315
00:17:00,870 --> 00:17:02,380
picture like this.
316
00:17:02,380 --> 00:17:05,670
And the question that comes up
is what do we mean by the area
317
00:17:05,670 --> 00:17:07,010
under the curve here?
318
00:17:07,010 --> 00:17:09,890
And again, without belaboring
this point, not because it's
319
00:17:09,890 --> 00:17:12,980
not important, but because this
is just an overview and
320
00:17:12,980 --> 00:17:15,680
we'll come back to all of these
topics later in our
321
00:17:15,680 --> 00:17:19,849
course, the point I just want
to bring out here is, notice
322
00:17:19,849 --> 00:17:23,490
that the area under the curve
here is the distance that this
323
00:17:23,490 --> 00:17:28,820
particle would travel moving at
this speed if the time goes
324
00:17:28,820 --> 00:17:30,460
from 0 to 1.
325
00:17:30,460 --> 00:17:32,180
And notice what we're
saying here.
326
00:17:32,180 --> 00:17:36,670
Again, suppose we divide this
interval into n equal parts
327
00:17:36,670 --> 00:17:39,760
and inscribe rectangles.
328
00:17:39,760 --> 00:17:43,100
Notice that each of
these rectangles
329
00:17:43,100 --> 00:17:44,580
represents a distance.
330
00:17:44,580 --> 00:17:53,730
Namely, if a particle moved at
the speed over this length of
331
00:17:53,730 --> 00:17:57,220
time, the area under the curve
would be the distance that it
332
00:17:57,220 --> 00:17:59,400
traveled during that
time interval.
333
00:17:59,400 --> 00:18:02,040
In other words, what we're
saying is that if the particle
334
00:18:02,040 --> 00:18:05,850
moved at this speed from this
time to this time, then moved
335
00:18:05,850 --> 00:18:10,200
at this speed from this time to
this time, the sum of these
336
00:18:10,200 --> 00:18:12,830
two areas would give the
distance that the particle
337
00:18:12,830 --> 00:18:17,110
traveled, which obviously is
less than the distance that
338
00:18:17,110 --> 00:18:19,640
the particle truly traveled,
because notice that the
339
00:18:19,640 --> 00:18:22,957
particle was moving at a speed
which at every instance from
340
00:18:22,957 --> 00:18:26,770
here to here was greater than
this and at every instant from
341
00:18:26,770 --> 00:18:28,690
here to here was greater
than this.
342
00:18:28,690 --> 00:18:32,180
In other words, in the same way
as before, that area of
343
00:18:32,180 --> 00:18:39,580
the region R was whittled in
between A sub n upper bar and
344
00:18:39,580 --> 00:18:43,250
A sub n lower bar, notice that
the distance traveled by the
345
00:18:43,250 --> 00:18:48,280
particle can now be limited or
bounded in the same way.
346
00:18:48,280 --> 00:18:52,470
And in the same way that we
found area as a limit, we can
347
00:18:52,470 --> 00:18:55,390
now find distance as a limit.
348
00:18:55,390 --> 00:18:58,740
And these two things,
namely, what?
349
00:18:58,740 --> 00:19:02,750
Instantaneous speed and area
under a curve are the two
350
00:19:02,750 --> 00:19:05,960
essential branches of calculus,
differential
351
00:19:05,960 --> 00:19:10,000
calculus being concerned with
instantaneous rate of speed,
352
00:19:10,000 --> 00:19:12,790
integral calculus with
area under a curve.
353
00:19:12,790 --> 00:19:17,560
And the beauty of calculus,
surprisingly enough, in a way
354
00:19:17,560 --> 00:19:19,530
is only secondary
as far as these
355
00:19:19,530 --> 00:19:21,020
two topics are concerned.
356
00:19:21,020 --> 00:19:24,690
The true beauty lies in the fact
that these apparently two
357
00:19:24,690 --> 00:19:27,430
different branches of calculus,
one of which was
358
00:19:27,430 --> 00:19:30,850
invented by the Ancient Greeks
as early as 600 BC,
359
00:19:30,850 --> 00:19:32,030
the other of which--
360
00:19:32,030 --> 00:19:33,070
differential calculus--
361
00:19:33,070 --> 00:19:37,370
was not known to man until the
time of Isaac Newton in 1690
362
00:19:37,370 --> 00:19:41,070
AD are related by a rather
remarkable thing.
363
00:19:41,070 --> 00:19:44,480
That remarkable thing, which
we will emphasize at great
364
00:19:44,480 --> 00:19:49,440
length during our course, is
that areas and rates of change
365
00:19:49,440 --> 00:19:52,245
are related by area
under a curve.
366
00:19:52,245 --> 00:19:54,600
Now, I don't know how to draw
this so that you see this
367
00:19:54,600 --> 00:19:58,630
thing as vividly as possible,
but the idea is this.
368
00:19:58,630 --> 00:20:04,150
Think of area being swept out as
we take a line and move it,
369
00:20:04,150 --> 00:20:07,400
tracing out the curve this
way towards the right.
370
00:20:07,400 --> 00:20:13,580
Notice that if we have a certain
amount of area, if we
371
00:20:13,580 --> 00:20:18,110
now move a little bit further to
the right, notice that the
372
00:20:18,110 --> 00:20:24,840
new area somehow depends on what
the height of this curve
373
00:20:24,840 --> 00:20:26,570
is going to be.
374
00:20:26,570 --> 00:20:30,130
That somehow or other, it seems
that the area under the
375
00:20:30,130 --> 00:20:35,190
curve must be related to how
fast the height of this line
376
00:20:35,190 --> 00:20:36,320
is changing.
377
00:20:36,320 --> 00:20:40,670
Or to look at it inversely, how
fast the area is changing
378
00:20:40,670 --> 00:20:44,670
should somehow be related to
the height of this line.
379
00:20:44,670 --> 00:20:47,000
And just what that relationship
is will be
380
00:20:47,000 --> 00:20:49,730
explored also in great
detail in the course.
381
00:20:49,730 --> 00:20:53,210
And we will show the beautiful
marriage between this
382
00:20:53,210 --> 00:20:56,390
differential and integral
calculus through this
383
00:20:56,390 --> 00:20:59,150
relationship here, which
becomes known as the
384
00:20:59,150 --> 00:21:02,550
fundamental theorem of
integral calculus.
385
00:21:02,550 --> 00:21:06,420
At any rate then, what this
should show us is that
386
00:21:06,420 --> 00:21:07,900
calculus hinges--
387
00:21:07,900 --> 00:21:12,410
whether it's differential
calculus or integral calculus,
388
00:21:12,410 --> 00:21:13,960
that calculus hinges
on something
389
00:21:13,960 --> 00:21:15,730
called the limit concept.
390
00:21:15,730 --> 00:21:18,930
Again, by way of a very
quick review, one
391
00:21:18,930 --> 00:21:19,910
of the limit concepts--
392
00:21:19,910 --> 00:21:22,380
and I think it's easy to see
geometrically rather than
393
00:21:22,380 --> 00:21:23,300
analytically.
394
00:21:23,300 --> 00:21:26,410
Imagine that we have a curve,
and we want to find the
395
00:21:26,410 --> 00:21:30,080
tangent of the curve at the
point P. What we can do is
396
00:21:30,080 --> 00:21:34,120
take a point Q and draw the
straight line that joins P to
397
00:21:34,120 --> 00:21:38,690
Q. We could then find the
slope of the line PQ.
398
00:21:38,690 --> 00:21:41,940
The trouble is that PQ does not
look very much like the
399
00:21:41,940 --> 00:21:42,930
tangent line.
400
00:21:42,930 --> 00:21:48,300
So we say OK, let Q move down
so it comes closer to P. We
401
00:21:48,300 --> 00:21:50,740
can then find the
slopes of PQ1.
402
00:21:50,740 --> 00:21:53,390
We could find the
slope of PQ2.
403
00:21:53,390 --> 00:21:57,160
But in each case, we still do
not have the slope of the line
404
00:21:57,160 --> 00:22:01,370
tangent to the curve at P. But
we get the idea that as Q gets
405
00:22:01,370 --> 00:22:05,710
closer and closer to P, the
slope, or the secant line that
406
00:22:05,710 --> 00:22:10,040
joins P to Q, becomes a better
and better approximation to
407
00:22:10,040 --> 00:22:13,190
the line that would be tangent
to the curve at P.
408
00:22:13,190 --> 00:22:16,270
In fact, it's rather interesting
that in the 16th
409
00:22:16,270 --> 00:22:20,280
century, the definition that
was given of a tangent line
410
00:22:20,280 --> 00:22:25,390
was that a tangent line is a
line which passes through two
411
00:22:25,390 --> 00:22:27,250
consecutive points on a curve.
412
00:22:27,250 --> 00:22:29,560
Now, obviously, a curve
does not have
413
00:22:29,560 --> 00:22:31,270
two consecutive points.
414
00:22:31,270 --> 00:22:32,650
What they really
meant was what?
415
00:22:32,650 --> 00:22:37,600
That as Q gets closer and closer
to P, the secant line
416
00:22:37,600 --> 00:22:40,040
becomes a better and better
approximation for the tangent
417
00:22:40,040 --> 00:22:43,970
line, and that in a way, if the
two points were allowed to
418
00:22:43,970 --> 00:22:47,120
coincide, that should give
us the perfect answer.
419
00:22:47,120 --> 00:22:51,585
The trouble is, just like you
can't divide 0 by 0, if P and
420
00:22:51,585 --> 00:22:54,560
Q coincide, how many
points do you have?
421
00:22:54,560 --> 00:22:55,990
Just one point.
422
00:22:55,990 --> 00:22:59,550
And it takes two points to
determine a straight line.
423
00:22:59,550 --> 00:23:03,140
No matter how close Q is to P,
we have two distinct points.
424
00:23:03,140 --> 00:23:06,050
As soon as Q touches
P, we lose this.
425
00:23:06,050 --> 00:23:10,060
And this is what was meant by
ancient man or medieval man by
426
00:23:10,060 --> 00:23:12,530
his notion of two consecutive
points.
427
00:23:12,530 --> 00:23:16,130
And I should put this in double
quotes because I think
428
00:23:16,130 --> 00:23:19,940
you can see what he's begging
to try to say with the word
429
00:23:19,940 --> 00:23:22,990
"consecutive," even though from
a purely rigorous point
430
00:23:22,990 --> 00:23:26,280
of view, this has no
geometric meaning.
431
00:23:26,280 --> 00:23:30,580
Now, the other form of limit has
to do with adding up areas
432
00:23:30,580 --> 00:23:32,480
of rectangles under curves.
433
00:23:32,480 --> 00:23:35,370
Namely, we divided the curve
up into n parts.
434
00:23:35,370 --> 00:23:38,600
We inscribed n rectangles,
and then we let n
435
00:23:38,600 --> 00:23:40,320
increase without bound.
436
00:23:40,320 --> 00:23:44,700
In other words, this is sort of
a discrete type of limit.
437
00:23:44,700 --> 00:23:49,190
Namely, we must add up a whole
number of areas, but the sum
438
00:23:49,190 --> 00:23:52,590
is endless in the sense that
the number of rectangles
439
00:23:52,590 --> 00:23:56,390
becomes greater than any number
we want to preassign.
440
00:23:56,390 --> 00:24:02,350
And the basic question that we
must contend with here is how
441
00:24:02,350 --> 00:24:04,230
big is an infinite sum?
442
00:24:04,230 --> 00:24:07,060
You see, when we say infinite
sum, that just tells you how
443
00:24:07,060 --> 00:24:08,540
many terms you're combining.
444
00:24:08,540 --> 00:24:11,800
It doesn't tell you how
big each term, how big
445
00:24:11,800 --> 00:24:12,930
the sum will be.
446
00:24:12,930 --> 00:24:15,410
For example, look at
the following sum.
447
00:24:15,410 --> 00:24:17,310
I will start with 1.
448
00:24:17,310 --> 00:24:19,640
Then I'll add 1/2 on twice.
449
00:24:19,640 --> 00:24:23,040
Then I'll add 1/3
on three times.
450
00:24:23,040 --> 00:24:25,990
And without belaboring this
point, let me then say I'll
451
00:24:25,990 --> 00:24:30,960
had on 1/4 four times,
1/5 five times, 1/6
452
00:24:30,960 --> 00:24:33,710
six times, et cetera.
453
00:24:33,710 --> 00:24:37,890
Notice as I do this that each
time the terms gets smaller,
454
00:24:37,890 --> 00:24:41,190
yet the sum increases
without any bound.
455
00:24:41,190 --> 00:24:43,980
Namely, notice that
this adds up to 1.
456
00:24:43,980 --> 00:24:45,400
This adds up to 1.
457
00:24:45,400 --> 00:24:47,640
The next four terms
will add up to 1.
458
00:24:47,640 --> 00:24:51,050
And as I go out further and
further, notice that this sum
459
00:24:51,050 --> 00:24:53,470
can become as great is
I want, just by me
460
00:24:53,470 --> 00:24:55,320
adding on enough 1's.
461
00:24:55,320 --> 00:24:57,850
On the other hand, let's
look at this one.
462
00:24:57,850 --> 00:25:05,160
1 plus 1/2 plus 1/4 plus 1/8
plus 1/16 plus 1/32.
463
00:25:05,160 --> 00:25:09,300
In other words, I start with 1
and each time add on half the
464
00:25:09,300 --> 00:25:10,430
previous number.
465
00:25:10,430 --> 00:25:13,310
See, 1 plus 1/2 plus
1/4 plus 1/8.
466
00:25:13,310 --> 00:25:17,320
You may remember this as being
the geometric series whose
467
00:25:17,320 --> 00:25:20,240
ratio is 1/2.
468
00:25:20,240 --> 00:25:25,140
The interesting thing is that
now this sum gets as close to
469
00:25:25,140 --> 00:25:28,230
2 as you want without
ever getting there.
470
00:25:28,230 --> 00:25:30,790
And rather than prove this right
now, let's just look at
471
00:25:30,790 --> 00:25:33,690
the geometric interpretation
here.
472
00:25:33,690 --> 00:25:37,130
Take a line which is
2 inches long.
473
00:25:37,130 --> 00:25:39,290
Suppose you first go halfway.
474
00:25:39,290 --> 00:25:40,500
You're now here.
475
00:25:40,500 --> 00:25:42,620
Now go half the remaining
distance.
476
00:25:42,620 --> 00:25:43,080
That's what?
477
00:25:43,080 --> 00:25:44,080
1 plus 1/2.
478
00:25:44,080 --> 00:25:45,600
That puts you over here.
479
00:25:45,600 --> 00:25:47,860
Now go half the remaining
distance.
480
00:25:47,860 --> 00:25:49,990
That means add on 1/4.
481
00:25:49,990 --> 00:25:51,900
Now go half the remaining
distance.
482
00:25:51,900 --> 00:25:53,580
That means add on on 1/8.
483
00:25:53,580 --> 00:25:55,420
Now go half the remaining
distance.
484
00:25:55,420 --> 00:25:57,720
Add up this on 1/16, you see.
485
00:25:57,720 --> 00:25:59,260
And ultimately, what happens?
486
00:25:59,260 --> 00:26:02,150
Well, no matter where you stop,
you've become closer and
487
00:26:02,150 --> 00:26:04,570
closer to 2 without ever
getting there.
488
00:26:04,570 --> 00:26:06,970
And as you go further and
further, you can get as close
489
00:26:06,970 --> 00:26:08,500
to 2 as you want.
490
00:26:08,500 --> 00:26:11,550
In other words, here are
infinitely many terms whose
491
00:26:11,550 --> 00:26:13,450
infinite sum is 2.
492
00:26:13,450 --> 00:26:18,020
Here are infinitely many terms
whose infinite sum is
493
00:26:18,020 --> 00:26:19,410
infinity, we should
say, because it
494
00:26:19,410 --> 00:26:20,790
increases without bound.
495
00:26:20,790 --> 00:26:23,760
And this was the problem that
hung up the Ancient Greek.
496
00:26:23,760 --> 00:26:26,370
How could you do infinitely
many things in a
497
00:26:26,370 --> 00:26:27,830
finite amount of time?
498
00:26:27,830 --> 00:26:31,440
In fact, at the same time that
the Greek was developing
499
00:26:31,440 --> 00:26:36,020
integral calculus, the famous
greek philosopher Zeno was
500
00:26:36,020 --> 00:26:39,190
working on things called
Zeno's paradoxes.
501
00:26:39,190 --> 00:26:42,780
And Zeno's paradoxes are three
in number, of which I only
502
00:26:42,780 --> 00:26:44,340
want to quote one here.
503
00:26:44,340 --> 00:26:47,720
But it's a paradox which shows
how Zeno could not visualize
504
00:26:47,720 --> 00:26:49,500
quite what was happening.
505
00:26:49,500 --> 00:26:52,810
You see, it's called the
Tortoise and the Hare problem.
506
00:26:52,810 --> 00:26:56,870
Suppose that you give the
Tortoise a 1 yard head start
507
00:26:56,870 --> 00:26:58,360
on the Hare.
508
00:26:58,360 --> 00:27:00,920
And suppose for the sake of
argument, just to mimic the
509
00:27:00,920 --> 00:27:03,560
problem that we were doing
before, suppose it's a slow
510
00:27:03,560 --> 00:27:06,840
Hare and a fast Tortoise so that
the Hare only runs twice
511
00:27:06,840 --> 00:27:08,660
as fast as the Tortoise.
512
00:27:08,660 --> 00:27:11,680
You see, Zeno's paradox says
that the Hare can never catch
513
00:27:11,680 --> 00:27:12,480
the Tortoise.
514
00:27:12,480 --> 00:27:13,360
Why?
515
00:27:13,360 --> 00:27:16,680
Because to catch the Tortoise,
the Hare must first go the 1
516
00:27:16,680 --> 00:27:18,960
yard head start that
the Tortoise had.
517
00:27:18,960 --> 00:27:22,010
Well, by the time the Hare gets
here, the Tortoise has
518
00:27:22,010 --> 00:27:25,750
gone 1/2 yard because the
Tortoise travels half as fast.
519
00:27:25,750 --> 00:27:27,650
Now, the Hare must make
up the 1/2 yard.
520
00:27:27,650 --> 00:27:30,860
But while the Hare makes up
the 1/2 yard, the Tortoise
521
00:27:30,860 --> 00:27:32,730
goes 1/4 of a yard.
522
00:27:32,730 --> 00:27:36,300
When the Hare makes up the 1/4
of a yard, the Tortoise goes
523
00:27:36,300 --> 00:27:37,470
1/8 of a yard.
524
00:27:37,470 --> 00:27:40,930
And so, Zeno argues, the Hare
gets closer and closer to the
525
00:27:40,930 --> 00:27:43,830
Tortoise but can't catch him.
526
00:27:43,830 --> 00:27:46,140
And this, of course, is a rather
strange thing because
527
00:27:46,140 --> 00:27:49,300
Zeno knew that the Tortoise
would catch the Hare.
528
00:27:49,300 --> 00:27:50,940
That's it's called a paradox.
529
00:27:50,940 --> 00:27:54,000
A paradox means something which
appears to be true yet
530
00:27:54,000 --> 00:27:56,010
is obviously false.
531
00:27:56,010 --> 00:27:59,380
Now, notice that we can resolve
Zeno's paradox into
532
00:27:59,380 --> 00:28:01,370
the example we were just
talking about.
533
00:28:01,370 --> 00:28:03,880
For the sake of argument, notice
what's happening here
534
00:28:03,880 --> 00:28:04,760
with the time.
535
00:28:04,760 --> 00:28:07,160
For the sake of argument,
let's suppose that the
536
00:28:07,160 --> 00:28:09,800
Tortoise travels at
1 yard per second.
537
00:28:09,800 --> 00:28:10,900
Then what you're saying is--
538
00:28:10,900 --> 00:28:12,890
I mean, the Hare travels
at 1 yard per second.
539
00:28:12,890 --> 00:28:15,460
What you're saying is
it takes the Hare 1
540
00:28:15,460 --> 00:28:17,850
second to go this distance.
541
00:28:17,850 --> 00:28:22,190
Then it takes him 1/2 a second
to go this distance, then 1/4
542
00:28:22,190 --> 00:28:24,760
of a second to go
this distance.
543
00:28:24,760 --> 00:28:27,210
And what you're saying is that
as he's gaining on the
544
00:28:27,210 --> 00:28:29,630
Tortoise, these are the time
intervals which are
545
00:28:29,630 --> 00:28:30,910
transpiring.
546
00:28:30,910 --> 00:28:34,710
And this sum turns
out to be 2.
547
00:28:34,710 --> 00:28:38,050
Now, of course, those of us who
had eighth grade algebra
548
00:28:38,050 --> 00:28:40,690
know an easier way of solving
this problem.
549
00:28:40,690 --> 00:28:44,285
We say lookit, let's solve this
problem algebraically.
550
00:28:44,285 --> 00:28:49,390
Namely, we say give the Tortoise
a 1 yard head start.
551
00:28:49,390 --> 00:28:55,170
Now call x the distance of
a point at which the Hare
552
00:28:55,170 --> 00:28:56,780
catches the Tortoise.
553
00:28:56,780 --> 00:28:59,610
Now, the Hare is traveling
1 yard per second.
554
00:28:59,610 --> 00:29:05,380
The Tortoise is traveling
1/2 yard per second, OK?
555
00:29:05,380 --> 00:29:12,400
So if we take the distance
traveled and divided by the
556
00:29:12,400 --> 00:29:16,890
rate, that should be the time.
557
00:29:16,890 --> 00:29:19,600
And since they both are at this
point at the same time,
558
00:29:19,600 --> 00:29:24,800
we get what? x/1 equals x
minus 1 divided by 1/2.
559
00:29:24,800 --> 00:29:28,410
And assuming as a prerequisite
that we have had algebra, it
560
00:29:28,410 --> 00:29:32,270
follows almost trivially
that x equals 2.
561
00:29:32,270 --> 00:29:36,790
In other words, what this says
is, in reality, that the Hare
562
00:29:36,790 --> 00:29:39,540
will not overtake the Tortoise
until he catches
563
00:29:39,540 --> 00:29:41,900
him, which is obvious.
564
00:29:41,900 --> 00:29:43,740
But what's not so
obvious is what?
565
00:29:43,740 --> 00:29:46,270
That these infinitely
many terms can add
566
00:29:46,270 --> 00:29:48,370
up to a finite sum.
567
00:29:48,370 --> 00:29:52,420
Well, at any rate, this complete
the overview of what
568
00:29:52,420 --> 00:29:53,430
our course will be like.
569
00:29:53,430 --> 00:29:58,890
And to help you focus your
attention on what our course
570
00:29:58,890 --> 00:30:03,450
really says, what we shall do
computationally is this.
571
00:30:03,450 --> 00:30:06,610
In review, we shall start with
functions, and functions
572
00:30:06,610 --> 00:30:09,800
involve the modern concept
of sets because they're
573
00:30:09,800 --> 00:30:12,230
relationships between
sets of objects.
574
00:30:12,230 --> 00:30:16,680
We'll talk about limits,
derivatives, rate of change,
575
00:30:16,680 --> 00:30:18,980
integrals, area under curves.
576
00:30:18,980 --> 00:30:22,080
This will be our fundamental
building block.
577
00:30:22,080 --> 00:30:25,230
Once this is done, these things
will never change.
578
00:30:25,230 --> 00:30:28,540
But the remainder of our course
will be to talk about
579
00:30:28,540 --> 00:30:31,470
applications, which is the name
of the game as far as
580
00:30:31,470 --> 00:30:33,220
engineering is concerned.
581
00:30:33,220 --> 00:30:36,300
More elaborate functions,
namely, how do we handle
582
00:30:36,300 --> 00:30:37,990
tougher relationships.
583
00:30:37,990 --> 00:30:41,130
Related to the tougher
relationships will come more
584
00:30:41,130 --> 00:30:43,560
sophisticated techniques.
585
00:30:43,560 --> 00:30:46,900
And finally, we will conclude
our course with the topic that
586
00:30:46,900 --> 00:30:50,020
we were just talking about:
infinite series, how do we get
587
00:30:50,020 --> 00:30:53,690
a hold of what happens when
you add up infinitely many
588
00:30:53,690 --> 00:30:56,060
things, each of which
gets small.
589
00:30:56,060 --> 00:31:00,750
At any rate, that concludes
our lecture for today.
590
00:31:00,750 --> 00:31:04,820
We will have a digression in
the sense that the next few
591
00:31:04,820 --> 00:31:09,540
lessons will consist of sets,
things that you can read about
592
00:31:09,540 --> 00:31:12,160
at your leisure in our
supplementary notes.
593
00:31:12,160 --> 00:31:15,400
Learn to understand these
because the concept of a set
594
00:31:15,400 --> 00:31:18,970
is the building block, the
fundamental language of modern
595
00:31:18,970 --> 00:31:20,300
mathematics.
596
00:31:20,300 --> 00:31:24,470
And then we will return, once we
have sets underway, to talk
597
00:31:24,470 --> 00:31:26,680
about functions.
598
00:31:26,680 --> 00:31:29,310
And then we will build
gradually from there.
599
00:31:29,310 --> 00:31:31,840
Hopefully, when our course ends,
we will have in slow
600
00:31:31,840 --> 00:31:35,310
motion gone through
today's lesson.
601
00:31:35,310 --> 00:31:37,620
This completes our presentation
for today.
602
00:31:37,620 --> 00:31:39,630
And until next time, goodbye.
603
00:31:45,570 --> 00:31:48,110
NARRATOR: Funding for the
publication of this video is
604
00:31:48,110 --> 00:31:52,820
provided by the Gabriella and
Paul Rosenbaum Foundation.
605
00:31:52,820 --> 00:31:56,990
Help OCW continue to provide
free and open access to MIT
606
00:31:56,990 --> 00:32:01,190
courses by making a donation
at ocw.mit.edu/donate.