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PROFESSOR: Hi.
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I'm Herb Gross, and welcome
to Calculus Revisited.
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I guess the most difficult
lecture to give with any
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course is probably
the first one.
00:00:48.440 --> 00:00:51.260
And you're sort of tempted to
look at your audience and say
00:00:51.260 --> 00:00:54.120
you're probably wondering why
I called you all here.
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And in this sense, I have
elected to entitle our first
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lecture simply Preface to give a
double overview, an overview
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both of the hardware and the
software that will make up
00:01:07.160 --> 00:01:08.680
this course.
00:01:08.680 --> 00:01:13.240
To begin with, we will have a
series of lectures of which
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this is the first.
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In our lectures, our main aim
will be to give an overview of
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the material being covered, an
insight as to why various
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computations are done, and
insights as to how
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applications of these concepts
will be made.
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The heart of our course will
consist of a regular textbook.
00:01:35.430 --> 00:01:37.630
You see, we have our lectures.
00:01:37.630 --> 00:01:40.390
We have a textbook.
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The textbook is designed to
supply you with deeper
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insights than what we can
give in a lecture.
00:01:48.610 --> 00:01:53.000
In addition, recognizing the
fact that the textbook may
00:01:53.000 --> 00:01:56.430
leave gaps, places where you
may want some additional
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knowledge, we also have
supplementary notes.
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And finally, at the backbone of
our package is what we call
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the study guide.
00:02:06.630 --> 00:02:10.520
The study guide consists of
a breakdown of the course.
00:02:10.520 --> 00:02:14.990
It tells us what the various
lectures will be, the units.
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There are pretests to help you
decide how well prepared you
00:02:19.290 --> 00:02:21.090
are for the topic that's
coming up.
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There is a final examination
at the end of
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each block of material.
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And perhaps most importantly,
especially from an engineer's
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point of view, in each unit that
we study, the study guide
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will consist of exercises
primarily called learning
00:02:37.970 --> 00:02:42.970
exercises, exercises which
hopefully will turn you on
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towards wanting to be able to
apply the material, and at the
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same time, serve as a
springboard by which we can
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highlight why the theory and
many about our lecture points
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are really as important
as they are.
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So much for the hardware
of our course.
00:03:00.140 --> 00:03:03.540
And now let's turn our attention
to the software.
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Just what is calculus?
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In a manner of speaking,
calculus can be viewed as
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being high school mathematics
with one additional concept
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called the limit concept
thrown in.
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If you recall back to your high
school days, remember
00:03:18.770 --> 00:03:21.250
that we're always dealing
with things like
00:03:21.250 --> 00:03:23.390
average rate of speed.
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Notice I say average or constant
rate of speed.
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The old recipe that distance
equals rate times time
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presupposes that the rate is
constant, because if the rate
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is varying, which rate is it
that you use to multiply the
00:03:36.750 --> 00:03:39.200
time by to find the distance?
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You see, in other words, roughly
speaking, we can say
00:03:43.760 --> 00:03:46.660
that at least one branch of
calculus known as differential
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calculus deals with
the subject of
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instantaneous speed.
00:03:50.780 --> 00:03:53.730
And instantaneous speed is a
rather easy thing to talk
00:03:53.730 --> 00:03:55.240
about intuitively.
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Imagine an object moving along
this line and passing the
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point P. And we say to ourselves
how fast was the
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object moving at the instant
that we're at the point P?
00:04:06.260 --> 00:04:09.700
Now, you see, this is some
sort of a problem.
00:04:09.700 --> 00:04:13.560
Because at the instant that
you're at P, you're not in a
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sense moving at all because
you're at P.
00:04:17.010 --> 00:04:20.230
Of course, what we do to reduce
this problem to an old
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one is we say, well, suppose we
have a couple of observers.
00:04:24.190 --> 00:04:26.940
Let's call them O1 and O2.
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Let them be stationed, one on
each side of P. Now, certainly
00:04:30.500 --> 00:04:33.030
what we could do physically
here is we can measure the
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distance between O1 and O2.
00:04:37.910 --> 00:04:43.090
And we can also measure the
time that it takes to
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go from O1 to O2.
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And what we can do is divide
that distance by the time, and
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that, you see, is our old high
school concept of the average
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speed of the particle as
it moves from O1 to O2.
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Now, you see, the question is,
somebody says gee, that's a
00:05:01.330 --> 00:05:03.780
great answer, but it's
the wrong problem.
00:05:03.780 --> 00:05:05.920
We didn't ask what was the
average speed as we
00:05:05.920 --> 00:05:07.420
went from O1 to O2.
00:05:07.420 --> 00:05:09.730
We asked what was the
instantaneous speed.
00:05:09.730 --> 00:05:11.700
And the idea is we say,
well, lookit.
00:05:11.700 --> 00:05:14.880
The average speed and the
instantaneous speed, it seems,
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should be pretty much the same
if the observers were
00:05:17.740 --> 00:05:19.800
relatively close together.
00:05:19.800 --> 00:05:22.670
The next observation is it seems
that if we were to move
00:05:22.670 --> 00:05:26.820
the observers in even closer,
there would be less of a
00:05:26.820 --> 00:05:31.440
discrepancy between O1 and O2
in the sense that-- not a
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discrepancy, but in the sense
that the average speed would
00:05:34.345 --> 00:05:36.900
now seem like a better
approximation to the
00:05:36.900 --> 00:05:39.480
instantaneous speed because
there was less distance for
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something to go wrong in.
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And so we get the idea that
maybe what we should do is
00:05:45.220 --> 00:05:48.480
make the observers gets closer
and closer together.
00:05:48.480 --> 00:05:50.810
That would minimize the
difference between the average
00:05:50.810 --> 00:05:54.380
speed and the instantaneous rate
of speed, and maybe the
00:05:54.380 --> 00:05:56.480
optimal thing would happen
when the two
00:05:56.480 --> 00:05:58.390
observers were together.
00:05:58.390 --> 00:05:59.870
But the strange part is--
00:05:59.870 --> 00:06:02.080
and this is where calculus
really begins.
00:06:02.080 --> 00:06:04.140
This is what calculus
is all about.
00:06:04.140 --> 00:06:07.370
As soon as the observers come
together, notice that what you
00:06:07.370 --> 00:06:10.420
have is that the distance
between them is 0.
00:06:10.420 --> 00:06:13.430
The time that it takes to get
from one to the other is 0.
00:06:13.430 --> 00:06:17.020
And therefore, it appears that
if we divide distance by time,
00:06:17.020 --> 00:06:21.550
we are going to wind
up with 0/0.
00:06:21.550 --> 00:06:25.630
Now, my claim is that 0/0
should be called--
00:06:25.630 --> 00:06:30.190
well, I'll call it undefined,
but actually, I think
00:06:30.190 --> 00:06:36.920
indeterminate would
be a better word.
00:06:36.920 --> 00:06:38.415
Why do I say that?
00:06:38.415 --> 00:06:40.330
Well, here's an interesting
thing.
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When we do arithmetic with small
numbers, observe that if
00:06:43.900 --> 00:06:46.980
you add two small numbers, you
expect the result to be a
00:06:46.980 --> 00:06:47.960
small number.
00:06:47.960 --> 00:06:51.230
If you multiply two small
numbers, you expect the result
00:06:51.230 --> 00:06:52.890
to be a small number.
00:06:52.890 --> 00:06:57.280
Similarly, for division, for
subtraction, the difference of
00:06:57.280 --> 00:06:59.740
two small numbers is
a small number.
00:06:59.740 --> 00:07:02.960
On the other hand, the quotient
of two small numbers
00:07:02.960 --> 00:07:04.450
is rather deceptive.
00:07:04.450 --> 00:07:08.000
Because it's a ratio, if one
of the very small numbers
00:07:08.000 --> 00:07:12.390
happens to be very much larger
compared with the other small
00:07:12.390 --> 00:07:15.200
number, the ratio might
be quite large.
00:07:15.200 --> 00:07:19.960
Well, for example, visualize,
say, 10 to the minus 6,
00:07:19.960 --> 00:07:27.470
1/1,000,000, 0.000001, which
is a pretty small number.
00:07:27.470 --> 00:07:32.310
Now, divide that by 10
to the minus 12th.
00:07:32.310 --> 00:07:35.210
Well, you see, 10 to the minus
12th is a small number, so
00:07:35.210 --> 00:07:38.220
small that it makes 10 to the
minus sixth appear large.
00:07:38.220 --> 00:07:43.440
In fact, the quotient is 10 to
the sixth, which is 1,000,000.
00:07:43.440 --> 00:07:46.210
And here we see that when you're
dealing with the ratio
00:07:46.210 --> 00:07:49.180
of small numbers, you're a
little bit in trouble, because
00:07:49.180 --> 00:07:53.310
we can't tell whether the ratio
will be small, or large,
00:07:53.310 --> 00:07:54.930
or somewhere in between.
00:07:54.930 --> 00:07:57.430
For example, if we reverse
the role of numerator and
00:07:57.430 --> 00:08:00.020
denominator here, we would still
have the quotient of two
00:08:00.020 --> 00:08:03.880
small numbers, but 10 to the
minus 12th divided by 10 to
00:08:03.880 --> 00:08:11.930
the minus sixth is a relatively
small number, 10 to
00:08:11.930 --> 00:08:13.810
the minus 6.
00:08:13.810 --> 00:08:16.600
Of course, this is the physical
way of looking at it.
00:08:16.600 --> 00:08:18.910
Small divided by small
is indeterminate.
00:08:18.910 --> 00:08:21.400
We have a more rigorous way of
looking at this if you want to
00:08:21.400 --> 00:08:24.090
see it from a mathematical
structure point of view.
00:08:24.090 --> 00:08:28.540
Namely, suppose we define a/b
in the traditional way.
00:08:28.540 --> 00:08:33.059
Namely, a/b is that number such
that when we multiply it
00:08:33.059 --> 00:08:36.020
by b we get a.
00:08:36.020 --> 00:08:40.200
Well, what would that say as
far as 0/0 was concerned?
00:08:40.200 --> 00:08:41.419
It would say what?
00:08:41.419 --> 00:08:45.060
That 0/0 is that number such
that when we multiply
00:08:45.060 --> 00:08:48.030
it by 0 we get 0.
00:08:48.030 --> 00:08:51.410
Now, what number has the
property that when we multiply
00:08:51.410 --> 00:08:53.330
it by 0 we get 0?
00:08:53.330 --> 00:08:55.890
And the answer is any number.
00:08:55.890 --> 00:08:58.380
This is why 0/0 is
indeterminate.
00:08:58.380 --> 00:09:01.630
If we say to a person, tell me
the number I must multiply by
00:09:01.630 --> 00:09:05.570
0 to get 0, the answer
is any number.
00:09:05.570 --> 00:09:08.850
Well, the idea then is that we
must avoid the expression 0/0
00:09:08.850 --> 00:09:10.310
at all costs.
00:09:10.310 --> 00:09:14.570
What this means then is that we
say OK, let the observers
00:09:14.570 --> 00:09:17.940
get closer to closer together,
but never touch.
00:09:17.940 --> 00:09:20.410
Now, the point is that as long
as the observers get closer
00:09:20.410 --> 00:09:23.970
and closer together and never
touch, let's ask the question
00:09:23.970 --> 00:09:26.300
how many pairs of observers
do we need?
00:09:26.300 --> 00:09:28.790
And the answer is that
theoretically we need
00:09:28.790 --> 00:09:31.400
infinitely many pairs
of observers.
00:09:31.400 --> 00:09:32.740
Well, why is that?
00:09:32.740 --> 00:09:35.610
Because as long as there's a
distance between a pair of
00:09:35.610 --> 00:09:38.520
observers, we can theoretically
fit in another
00:09:38.520 --> 00:09:40.380
pair of observers.
00:09:40.380 --> 00:09:44.700
This is why in our course we do
not begin with this idea,
00:09:44.700 --> 00:09:48.340
but looking backwards now, we
say ah, we had better find
00:09:48.340 --> 00:09:51.140
some way of giving us the
equivalent of having
00:09:51.140 --> 00:09:53.290
infinitely many pairs
of observers.
00:09:53.290 --> 00:09:59.140
And to do this, the idea that we
come up with is the concept
00:09:59.140 --> 00:10:01.990
called a function.
00:10:01.990 --> 00:10:05.400
Consider the old Galileo freely
falling body problem,
00:10:05.400 --> 00:10:09.520
where the distance that the
body falls s equals 16t
00:10:09.520 --> 00:10:12.950
squared, where t is in seconds
and s is in feet.
00:10:12.950 --> 00:10:17.760
Notice that this apparently
harmless recipe gives us a way
00:10:17.760 --> 00:10:21.250
for finding s for
each given t.
00:10:21.250 --> 00:10:24.650
In other words, to all intents
and purposes, this recipe
00:10:24.650 --> 00:10:28.992
gives us an observer for
each point of time.
00:10:28.992 --> 00:10:32.260
For each time, we can find the
distance, which is physically
00:10:32.260 --> 00:10:36.330
equivalent to knowing an
observer at every point.
00:10:36.330 --> 00:10:40.430
In turn, the study of functions
lends itself to a
00:10:40.430 --> 00:10:43.220
study of graphs, a picture.
00:10:43.220 --> 00:10:47.590
Namely, if we look at s equals
16t squared again, notice that
00:10:47.590 --> 00:10:49.910
we visualize a recipe here.
00:10:49.910 --> 00:10:54.250
t can be viewed as being an
input, s as the output.
00:10:54.250 --> 00:10:58.560
For a given input t, we can
compute the output s.
00:10:58.560 --> 00:11:03.090
In general, if we now elect
to plot the input along a
00:11:03.090 --> 00:11:07.770
horizontal line and the output
at right angles to this, we
00:11:07.770 --> 00:11:11.270
now have a picture of our
relationship, a picture which
00:11:11.270 --> 00:11:14.200
is called a graph.
00:11:14.200 --> 00:11:17.980
You see, we can talk about this
more explicitly as far as
00:11:17.980 --> 00:11:20.830
this particular problem is
concerned, just by taking a
00:11:20.830 --> 00:11:22.650
look at a picture like this.
00:11:22.650 --> 00:11:25.810
In other words, in this
particular problem, the input
00:11:25.810 --> 00:11:30.970
is time t, the output
is distance s.
00:11:30.970 --> 00:11:35.270
For each t, we locate a height
called s by squaring t and
00:11:35.270 --> 00:11:37.450
multiplying by 16.
00:11:37.450 --> 00:11:40.390
And now, what average speed
means in terms of this kind of
00:11:40.390 --> 00:11:42.640
a diagram is the following.
00:11:42.640 --> 00:11:46.410
To find the average speed, all
we have to do is on a given
00:11:46.410 --> 00:11:49.960
time interval find the distance
traveled, which I
00:11:49.960 --> 00:11:53.510
call delta s, the change in
distance, and divide that by
00:11:53.510 --> 00:11:54.860
the change in time.
00:11:54.860 --> 00:11:58.030
That's the average speed, which,
by the way, from a
00:11:58.030 --> 00:12:02.150
geometrical point of view,
becomes known as the slope of
00:12:02.150 --> 00:12:03.820
this particular straight line.
00:12:03.820 --> 00:12:08.130
In other words, average speed is
to functions what slope of
00:12:08.130 --> 00:12:10.830
a straight line is
to geometry.
00:12:10.830 --> 00:12:14.190
At any rate, knowing what the
average rate of speed is, we
00:12:14.190 --> 00:12:18.570
sort of say why couldn't we
define the instantaneous speed
00:12:18.570 --> 00:12:19.500
to be this.
00:12:19.500 --> 00:12:24.040
We will take the change in
distance divided by the change
00:12:24.040 --> 00:12:26.740
in time and see what happens.
00:12:26.740 --> 00:12:28.430
And we write this this way.
00:12:28.430 --> 00:12:30.450
Limit as delta t approaches 0.
00:12:30.450 --> 00:12:33.450
Let's see what happens as that
change in time becomes
00:12:33.450 --> 00:12:37.370
arbitrarily small, but never
equaling 0 because we don't
00:12:37.370 --> 00:12:39.820
want a 0/0 form here.
00:12:39.820 --> 00:12:44.260
You see, this then becomes the
working definition of what we
00:12:44.260 --> 00:12:46.670
call differential calculus.
00:12:46.670 --> 00:12:50.390
The point is that this
particular definition does not
00:12:50.390 --> 00:12:53.240
depend on s equaling
16t squared.
00:12:53.240 --> 00:12:57.430
s could be any function
of t whatsoever.
00:12:57.430 --> 00:13:00.210
We could have a more elaborate
type of situation.
00:13:00.210 --> 00:13:02.170
The important point is what?
00:13:02.170 --> 00:13:05.420
The basic definition
stays the same.
00:13:05.420 --> 00:13:09.980
What changes is the amount of
arithmetic that's necessary to
00:13:09.980 --> 00:13:13.970
handle the particular
relationship between s and t.
00:13:13.970 --> 00:13:18.010
This will be a major part of our
course, the strange thing
00:13:18.010 --> 00:13:20.970
being that even at the very end
of our course when we've
00:13:20.970 --> 00:13:24.200
gone through many, many things,
our basic definition
00:13:24.200 --> 00:13:26.790
of instantaneous rate of
change will have never
00:13:26.790 --> 00:13:27.960
changed from this.
00:13:27.960 --> 00:13:30.440
It will always stay like this.
00:13:30.440 --> 00:13:34.320
But what will change is how much
arithmetic and algebra
00:13:34.320 --> 00:13:38.250
and geometry and trigonometry,
et cetera, we will have to do
00:13:38.250 --> 00:13:40.350
in order to compute these
things from a
00:13:40.350 --> 00:13:42.170
numerical point of view.
00:13:42.170 --> 00:13:45.620
Well, so much for the first
phase of calculus called
00:13:45.620 --> 00:13:47.150
differential calculus.
00:13:47.150 --> 00:13:50.610
A second phase of calculus, one
which was developed by the
00:13:50.610 --> 00:13:54.510
Ancient Greeks by 600 BC, the
subject that ultimately
00:13:54.510 --> 00:13:58.200
becomes known as integral
calculus, concerns problem of
00:13:58.200 --> 00:14:00.954
finding area under a curve.
00:14:00.954 --> 00:14:07.200
Here, I've elected to draw the
parabola y equals x squared on
00:14:07.200 --> 00:14:10.840
the interval from
0, 0 to 1, 0.
00:14:10.840 --> 00:14:17.290
And the question basically is
what is the area bounded by
00:14:17.290 --> 00:14:19.990
this sort of triangular
region?
00:14:19.990 --> 00:14:24.330
Let's call that region R, and
what we would like to find is
00:14:24.330 --> 00:14:26.420
the area of the region R.
00:14:26.420 --> 00:14:30.310
And the Ancient Greeks had a
rather interesting title for
00:14:30.310 --> 00:14:32.300
this type of approach for
finding the area.
00:14:32.300 --> 00:14:35.400
It is both figurative and
literal, I guess.
00:14:35.400 --> 00:14:36.810
It's called the method
of exhaustion.
00:14:40.670 --> 00:14:40.754
What they did was to --
00:14:40.754 --> 00:14:42.610
They would divide the
interval, say,
00:14:42.610 --> 00:14:44.530
into n equal parts.
00:14:44.530 --> 00:14:47.570
And picking the lowest point in
each interval, they would
00:14:47.570 --> 00:14:50.940
inscribe a rectangle.
00:14:50.940 --> 00:14:53.500
Knowing that the area of the
rectangle was the base times
00:14:53.500 --> 00:14:57.110
the height, they would add up
the area of each of these
00:14:57.110 --> 00:15:00.880
rectangles, and know that
whatever that area was, that
00:15:00.880 --> 00:15:04.540
would have to be too small to
be the right answer because
00:15:04.540 --> 00:15:07.040
that region was contained
in R. And that would be
00:15:07.040 --> 00:15:08.780
labeled A sub n--
00:15:08.780 --> 00:15:10.120
lower bar, say--
00:15:10.120 --> 00:15:13.130
to indicate that this was a sum
of rectangles which was
00:15:13.130 --> 00:15:15.780
too small to be the
right answer.
00:15:15.780 --> 00:15:20.210
Similarly, they would then find
the highest point in each
00:15:20.210 --> 00:15:24.430
rectangle, get an
overapproximation by adding up
00:15:24.430 --> 00:15:28.240
the sum of those areas, which
they would call A sub n upper
00:15:28.240 --> 00:15:31.050
bar, and now know that the area
of the regions they were
00:15:31.050 --> 00:15:34.260
looking for was squeezed
in between these two.
00:15:34.260 --> 00:15:37.530
Then what they would do is make
more and more divisions,
00:15:37.530 --> 00:15:40.320
and hopefully, and I think
you can see this sort of
00:15:40.320 --> 00:15:42.840
intuitively happening here,
each of the lower
00:15:42.840 --> 00:15:45.910
approximations gets bigger
and fills out
00:15:45.910 --> 00:15:47.970
the space from inside.
00:15:47.970 --> 00:15:51.730
Each of the upper approximations
gets smaller
00:15:51.730 --> 00:15:54.770
and chops off the space
from outside here.
00:15:54.770 --> 00:16:01.530
And hopefully, if both of these
bounds sort of converge
00:16:01.530 --> 00:16:05.520
to the same value L, we get the
idea that the area of the
00:16:05.520 --> 00:16:08.130
region R must be L.
00:16:08.130 --> 00:16:09.780
This is not anything new.
00:16:09.780 --> 00:16:12.700
In other words, this is a
technique that is some 2,500
00:16:12.700 --> 00:16:16.820
years old, used by the
Ancient Greeks.
00:16:16.820 --> 00:16:19.030
Of course, what happens with
engineering students in
00:16:19.030 --> 00:16:22.620
general is that one frequently
says, but I'm not interested
00:16:22.620 --> 00:16:23.920
in studying area.
00:16:23.920 --> 00:16:25.630
I am not a geometer.
00:16:25.630 --> 00:16:26.720
I am a physicist.
00:16:26.720 --> 00:16:28.370
I am an engineer.
00:16:28.370 --> 00:16:30.990
What good is the area
under a curve?
00:16:30.990 --> 00:16:35.510
And the interesting point here
becomes that if we label the
00:16:35.510 --> 00:16:39.450
coordinate axis rather than x
and y, give them physical
00:16:39.450 --> 00:16:43.490
labels, it turns out that area
under a curve has a physical
00:16:43.490 --> 00:16:44.880
interpretation.
00:16:44.880 --> 00:16:46.390
Consider the same problem.
00:16:46.390 --> 00:16:50.270
Only now, instead of talking
about y equals x squared,
00:16:50.270 --> 00:16:53.340
let's talk about v, the
velocity, equaling the square
00:16:53.340 --> 00:16:54.460
of the time.
00:16:54.460 --> 00:16:57.080
And say that the time
goes to 0 to 1.
00:16:57.080 --> 00:17:00.870
In other words, if we plot
v versus t, we get a
00:17:00.870 --> 00:17:02.380
picture like this.
00:17:02.380 --> 00:17:05.670
And the question that comes up
is what do we mean by the area
00:17:05.670 --> 00:17:07.010
under the curve here?
00:17:07.010 --> 00:17:09.890
And again, without belaboring
this point, not because it's
00:17:09.890 --> 00:17:12.980
not important, but because this
is just an overview and
00:17:12.980 --> 00:17:15.680
we'll come back to all of these
topics later in our
00:17:15.680 --> 00:17:19.849
course, the point I just want
to bring out here is, notice
00:17:19.849 --> 00:17:23.490
that the area under the curve
here is the distance that this
00:17:23.490 --> 00:17:28.820
particle would travel moving at
this speed if the time goes
00:17:28.820 --> 00:17:30.460
from 0 to 1.
00:17:30.460 --> 00:17:32.180
And notice what we're
saying here.
00:17:32.180 --> 00:17:36.670
Again, suppose we divide this
interval into n equal parts
00:17:36.670 --> 00:17:39.760
and inscribe rectangles.
00:17:39.760 --> 00:17:43.100
Notice that each of
these rectangles
00:17:43.100 --> 00:17:44.580
represents a distance.
00:17:44.580 --> 00:17:53.730
Namely, if a particle moved at
the speed over this length of
00:17:53.730 --> 00:17:57.220
time, the area under the curve
would be the distance that it
00:17:57.220 --> 00:17:59.400
traveled during that
time interval.
00:17:59.400 --> 00:18:02.040
In other words, what we're
saying is that if the particle
00:18:02.040 --> 00:18:05.850
moved at this speed from this
time to this time, then moved
00:18:05.850 --> 00:18:10.200
at this speed from this time to
this time, the sum of these
00:18:10.200 --> 00:18:12.830
two areas would give the
distance that the particle
00:18:12.830 --> 00:18:17.110
traveled, which obviously is
less than the distance that
00:18:17.110 --> 00:18:19.640
the particle truly traveled,
because notice that the
00:18:19.640 --> 00:18:22.957
particle was moving at a speed
which at every instance from
00:18:22.957 --> 00:18:26.770
here to here was greater than
this and at every instant from
00:18:26.770 --> 00:18:28.690
here to here was greater
than this.
00:18:28.690 --> 00:18:32.180
In other words, in the same way
as before, that area of
00:18:32.180 --> 00:18:39.580
the region R was whittled in
between A sub n upper bar and
00:18:39.580 --> 00:18:43.250
A sub n lower bar, notice that
the distance traveled by the
00:18:43.250 --> 00:18:48.280
particle can now be limited or
bounded in the same way.
00:18:48.280 --> 00:18:52.470
And in the same way that we
found area as a limit, we can
00:18:52.470 --> 00:18:55.390
now find distance as a limit.
00:18:55.390 --> 00:18:58.740
And these two things,
namely, what?
00:18:58.740 --> 00:19:02.750
Instantaneous speed and area
under a curve are the two
00:19:02.750 --> 00:19:05.960
essential branches of calculus,
differential
00:19:05.960 --> 00:19:10.000
calculus being concerned with
instantaneous rate of speed,
00:19:10.000 --> 00:19:12.790
integral calculus with
area under a curve.
00:19:12.790 --> 00:19:17.560
And the beauty of calculus,
surprisingly enough, in a way
00:19:17.560 --> 00:19:19.530
is only secondary
as far as these
00:19:19.530 --> 00:19:21.020
two topics are concerned.
00:19:21.020 --> 00:19:24.690
The true beauty lies in the fact
that these apparently two
00:19:24.690 --> 00:19:27.430
different branches of calculus,
one of which was
00:19:27.430 --> 00:19:30.850
invented by the Ancient Greeks
as early as 600 BC,
00:19:30.850 --> 00:19:32.030
the other of which--
00:19:32.030 --> 00:19:33.070
differential calculus--
00:19:33.070 --> 00:19:37.370
was not known to man until the
time of Isaac Newton in 1690
00:19:37.370 --> 00:19:41.070
AD are related by a rather
remarkable thing.
00:19:41.070 --> 00:19:44.480
That remarkable thing, which
we will emphasize at great
00:19:44.480 --> 00:19:49.440
length during our course, is
that areas and rates of change
00:19:49.440 --> 00:19:52.245
are related by area
under a curve.
00:19:52.245 --> 00:19:54.600
Now, I don't know how to draw
this so that you see this
00:19:54.600 --> 00:19:58.630
thing as vividly as possible,
but the idea is this.
00:19:58.630 --> 00:20:04.150
Think of area being swept out as
we take a line and move it,
00:20:04.150 --> 00:20:07.400
tracing out the curve this
way towards the right.
00:20:07.400 --> 00:20:13.580
Notice that if we have a certain
amount of area, if we
00:20:13.580 --> 00:20:18.110
now move a little bit further to
the right, notice that the
00:20:18.110 --> 00:20:24.840
new area somehow depends on what
the height of this curve
00:20:24.840 --> 00:20:26.570
is going to be.
00:20:26.570 --> 00:20:30.130
That somehow or other, it seems
that the area under the
00:20:30.130 --> 00:20:35.190
curve must be related to how
fast the height of this line
00:20:35.190 --> 00:20:36.320
is changing.
00:20:36.320 --> 00:20:40.670
Or to look at it inversely, how
fast the area is changing
00:20:40.670 --> 00:20:44.670
should somehow be related to
the height of this line.
00:20:44.670 --> 00:20:47.000
And just what that relationship
is will be
00:20:47.000 --> 00:20:49.730
explored also in great
detail in the course.
00:20:49.730 --> 00:20:53.210
And we will show the beautiful
marriage between this
00:20:53.210 --> 00:20:56.390
differential and integral
calculus through this
00:20:56.390 --> 00:20:59.150
relationship here, which
becomes known as the
00:20:59.150 --> 00:21:02.550
fundamental theorem of
integral calculus.
00:21:02.550 --> 00:21:06.420
At any rate then, what this
should show us is that
00:21:06.420 --> 00:21:07.900
calculus hinges--
00:21:07.900 --> 00:21:12.410
whether it's differential
calculus or integral calculus,
00:21:12.410 --> 00:21:13.960
that calculus hinges
on something
00:21:13.960 --> 00:21:15.730
called the limit concept.
00:21:15.730 --> 00:21:18.930
Again, by way of a very
quick review, one
00:21:18.930 --> 00:21:19.910
of the limit concepts--
00:21:19.910 --> 00:21:22.380
and I think it's easy to see
geometrically rather than
00:21:22.380 --> 00:21:23.300
analytically.
00:21:23.300 --> 00:21:26.410
Imagine that we have a curve,
and we want to find the
00:21:26.410 --> 00:21:30.080
tangent of the curve at the
point P. What we can do is
00:21:30.080 --> 00:21:34.120
take a point Q and draw the
straight line that joins P to
00:21:34.120 --> 00:21:38.690
Q. We could then find the
slope of the line PQ.
00:21:38.690 --> 00:21:41.940
The trouble is that PQ does not
look very much like the
00:21:41.940 --> 00:21:42.930
tangent line.
00:21:42.930 --> 00:21:48.300
So we say OK, let Q move down
so it comes closer to P. We
00:21:48.300 --> 00:21:50.740
can then find the
slopes of PQ1.
00:21:50.740 --> 00:21:53.390
We could find the
slope of PQ2.
00:21:53.390 --> 00:21:57.160
But in each case, we still do
not have the slope of the line
00:21:57.160 --> 00:22:01.370
tangent to the curve at P. But
we get the idea that as Q gets
00:22:01.370 --> 00:22:05.710
closer and closer to P, the
slope, or the secant line that
00:22:05.710 --> 00:22:10.040
joins P to Q, becomes a better
and better approximation to
00:22:10.040 --> 00:22:13.190
the line that would be tangent
to the curve at P.
00:22:13.190 --> 00:22:16.270
In fact, it's rather interesting
that in the 16th
00:22:16.270 --> 00:22:20.280
century, the definition that
was given of a tangent line
00:22:20.280 --> 00:22:25.390
was that a tangent line is a
line which passes through two
00:22:25.390 --> 00:22:27.250
consecutive points on a curve.
00:22:27.250 --> 00:22:29.560
Now, obviously, a curve
does not have
00:22:29.560 --> 00:22:31.270
two consecutive points.
00:22:31.270 --> 00:22:32.650
What they really
meant was what?
00:22:32.650 --> 00:22:37.600
That as Q gets closer and closer
to P, the secant line
00:22:37.600 --> 00:22:40.040
becomes a better and better
approximation for the tangent
00:22:40.040 --> 00:22:43.970
line, and that in a way, if the
two points were allowed to
00:22:43.970 --> 00:22:47.120
coincide, that should give
us the perfect answer.
00:22:47.120 --> 00:22:51.585
The trouble is, just like you
can't divide 0 by 0, if P and
00:22:51.585 --> 00:22:54.560
Q coincide, how many
points do you have?
00:22:54.560 --> 00:22:55.990
Just one point.
00:22:55.990 --> 00:22:59.550
And it takes two points to
determine a straight line.
00:22:59.550 --> 00:23:03.140
No matter how close Q is to P,
we have two distinct points.
00:23:03.140 --> 00:23:06.050
As soon as Q touches
P, we lose this.
00:23:06.050 --> 00:23:10.060
And this is what was meant by
ancient man or medieval man by
00:23:10.060 --> 00:23:12.530
his notion of two consecutive
points.
00:23:12.530 --> 00:23:16.130
And I should put this in double
quotes because I think
00:23:16.130 --> 00:23:19.940
you can see what he's begging
to try to say with the word
00:23:19.940 --> 00:23:22.990
"consecutive," even though from
a purely rigorous point
00:23:22.990 --> 00:23:26.280
of view, this has no
geometric meaning.
00:23:26.280 --> 00:23:30.580
Now, the other form of limit has
to do with adding up areas
00:23:30.580 --> 00:23:32.480
of rectangles under curves.
00:23:32.480 --> 00:23:35.370
Namely, we divided the curve
up into n parts.
00:23:35.370 --> 00:23:38.600
We inscribed n rectangles,
and then we let n
00:23:38.600 --> 00:23:40.320
increase without bound.
00:23:40.320 --> 00:23:44.700
In other words, this is sort of
a discrete type of limit.
00:23:44.700 --> 00:23:49.190
Namely, we must add up a whole
number of areas, but the sum
00:23:49.190 --> 00:23:52.590
is endless in the sense that
the number of rectangles
00:23:52.590 --> 00:23:56.390
becomes greater than any number
we want to preassign.
00:23:56.390 --> 00:24:02.350
And the basic question that we
must contend with here is how
00:24:02.350 --> 00:24:04.230
big is an infinite sum?
00:24:04.230 --> 00:24:07.060
You see, when we say infinite
sum, that just tells you how
00:24:07.060 --> 00:24:08.540
many terms you're combining.
00:24:08.540 --> 00:24:11.800
It doesn't tell you how
big each term, how big
00:24:11.800 --> 00:24:12.930
the sum will be.
00:24:12.930 --> 00:24:15.410
For example, look at
the following sum.
00:24:15.410 --> 00:24:17.310
I will start with 1.
00:24:17.310 --> 00:24:19.640
Then I'll add 1/2 on twice.
00:24:19.640 --> 00:24:23.040
Then I'll add 1/3
on three times.
00:24:23.040 --> 00:24:25.990
And without belaboring this
point, let me then say I'll
00:24:25.990 --> 00:24:30.960
had on 1/4 four times,
1/5 five times, 1/6
00:24:30.960 --> 00:24:33.710
six times, et cetera.
00:24:33.710 --> 00:24:37.890
Notice as I do this that each
time the terms gets smaller,
00:24:37.890 --> 00:24:41.190
yet the sum increases
without any bound.
00:24:41.190 --> 00:24:43.980
Namely, notice that
this adds up to 1.
00:24:43.980 --> 00:24:45.400
This adds up to 1.
00:24:45.400 --> 00:24:47.640
The next four terms
will add up to 1.
00:24:47.640 --> 00:24:51.050
And as I go out further and
further, notice that this sum
00:24:51.050 --> 00:24:53.470
can become as great is
I want, just by me
00:24:53.470 --> 00:24:55.320
adding on enough 1's.
00:24:55.320 --> 00:24:57.850
On the other hand, let's
look at this one.
00:24:57.850 --> 00:25:05.160
1 plus 1/2 plus 1/4 plus 1/8
plus 1/16 plus 1/32.
00:25:05.160 --> 00:25:09.300
In other words, I start with 1
and each time add on half the
00:25:09.300 --> 00:25:10.430
previous number.
00:25:10.430 --> 00:25:13.310
See, 1 plus 1/2 plus
1/4 plus 1/8.
00:25:13.310 --> 00:25:17.320
You may remember this as being
the geometric series whose
00:25:17.320 --> 00:25:20.240
ratio is 1/2.
00:25:20.240 --> 00:25:25.140
The interesting thing is that
now this sum gets as close to
00:25:25.140 --> 00:25:28.230
2 as you want without
ever getting there.
00:25:28.230 --> 00:25:30.790
And rather than prove this right
now, let's just look at
00:25:30.790 --> 00:25:33.690
the geometric interpretation
here.
00:25:33.690 --> 00:25:37.130
Take a line which is
2 inches long.
00:25:37.130 --> 00:25:39.290
Suppose you first go halfway.
00:25:39.290 --> 00:25:40.500
You're now here.
00:25:40.500 --> 00:25:42.620
Now go half the remaining
distance.
00:25:42.620 --> 00:25:43.080
That's what?
00:25:43.080 --> 00:25:44.080
1 plus 1/2.
00:25:44.080 --> 00:25:45.600
That puts you over here.
00:25:45.600 --> 00:25:47.860
Now go half the remaining
distance.
00:25:47.860 --> 00:25:49.990
That means add on 1/4.
00:25:49.990 --> 00:25:51.900
Now go half the remaining
distance.
00:25:51.900 --> 00:25:53.580
That means add on on 1/8.
00:25:53.580 --> 00:25:55.420
Now go half the remaining
distance.
00:25:55.420 --> 00:25:57.720
Add up this on 1/16, you see.
00:25:57.720 --> 00:25:59.260
And ultimately, what happens?
00:25:59.260 --> 00:26:02.150
Well, no matter where you stop,
you've become closer and
00:26:02.150 --> 00:26:04.570
closer to 2 without ever
getting there.
00:26:04.570 --> 00:26:06.970
And as you go further and
further, you can get as close
00:26:06.970 --> 00:26:08.500
to 2 as you want.
00:26:08.500 --> 00:26:11.550
In other words, here are
infinitely many terms whose
00:26:11.550 --> 00:26:13.450
infinite sum is 2.
00:26:13.450 --> 00:26:18.020
Here are infinitely many terms
whose infinite sum is
00:26:18.020 --> 00:26:19.410
infinity, we should
say, because it
00:26:19.410 --> 00:26:20.790
increases without bound.
00:26:20.790 --> 00:26:23.760
And this was the problem that
hung up the Ancient Greek.
00:26:23.760 --> 00:26:26.370
How could you do infinitely
many things in a
00:26:26.370 --> 00:26:27.830
finite amount of time?
00:26:27.830 --> 00:26:31.440
In fact, at the same time that
the Greek was developing
00:26:31.440 --> 00:26:36.020
integral calculus, the famous
greek philosopher Zeno was
00:26:36.020 --> 00:26:39.190
working on things called
Zeno's paradoxes.
00:26:39.190 --> 00:26:42.780
And Zeno's paradoxes are three
in number, of which I only
00:26:42.780 --> 00:26:44.340
want to quote one here.
00:26:44.340 --> 00:26:47.720
But it's a paradox which shows
how Zeno could not visualize
00:26:47.720 --> 00:26:49.500
quite what was happening.
00:26:49.500 --> 00:26:52.810
You see, it's called the
Tortoise and the Hare problem.
00:26:52.810 --> 00:26:56.870
Suppose that you give the
Tortoise a 1 yard head start
00:26:56.870 --> 00:26:58.360
on the Hare.
00:26:58.360 --> 00:27:00.920
And suppose for the sake of
argument, just to mimic the
00:27:00.920 --> 00:27:03.560
problem that we were doing
before, suppose it's a slow
00:27:03.560 --> 00:27:06.840
Hare and a fast Tortoise so that
the Hare only runs twice
00:27:06.840 --> 00:27:08.660
as fast as the Tortoise.
00:27:08.660 --> 00:27:11.680
You see, Zeno's paradox says
that the Hare can never catch
00:27:11.680 --> 00:27:12.480
the Tortoise.
00:27:12.480 --> 00:27:13.360
Why?
00:27:13.360 --> 00:27:16.680
Because to catch the Tortoise,
the Hare must first go the 1
00:27:16.680 --> 00:27:18.960
yard head start that
the Tortoise had.
00:27:18.960 --> 00:27:22.010
Well, by the time the Hare gets
here, the Tortoise has
00:27:22.010 --> 00:27:25.750
gone 1/2 yard because the
Tortoise travels half as fast.
00:27:25.750 --> 00:27:27.650
Now, the Hare must make
up the 1/2 yard.
00:27:27.650 --> 00:27:30.860
But while the Hare makes up
the 1/2 yard, the Tortoise
00:27:30.860 --> 00:27:32.730
goes 1/4 of a yard.
00:27:32.730 --> 00:27:36.300
When the Hare makes up the 1/4
of a yard, the Tortoise goes
00:27:36.300 --> 00:27:37.470
1/8 of a yard.
00:27:37.470 --> 00:27:40.930
And so, Zeno argues, the Hare
gets closer and closer to the
00:27:40.930 --> 00:27:43.830
Tortoise but can't catch him.
00:27:43.830 --> 00:27:46.140
And this, of course, is a rather
strange thing because
00:27:46.140 --> 00:27:49.300
Zeno knew that the Tortoise
would catch the Hare.
00:27:49.300 --> 00:27:50.940
That's it's called a paradox.
00:27:50.940 --> 00:27:54.000
A paradox means something which
appears to be true yet
00:27:54.000 --> 00:27:56.010
is obviously false.
00:27:56.010 --> 00:27:59.380
Now, notice that we can resolve
Zeno's paradox into
00:27:59.380 --> 00:28:01.370
the example we were just
talking about.
00:28:01.370 --> 00:28:03.880
For the sake of argument, notice
what's happening here
00:28:03.880 --> 00:28:04.760
with the time.
00:28:04.760 --> 00:28:07.160
For the sake of argument,
let's suppose that the
00:28:07.160 --> 00:28:09.800
Tortoise travels at
1 yard per second.
00:28:09.800 --> 00:28:10.900
Then what you're saying is--
00:28:10.900 --> 00:28:12.890
I mean, the Hare travels
at 1 yard per second.
00:28:12.890 --> 00:28:15.460
What you're saying is
it takes the Hare 1
00:28:15.460 --> 00:28:17.850
second to go this distance.
00:28:17.850 --> 00:28:22.190
Then it takes him 1/2 a second
to go this distance, then 1/4
00:28:22.190 --> 00:28:24.760
of a second to go
this distance.
00:28:24.760 --> 00:28:27.210
And what you're saying is that
as he's gaining on the
00:28:27.210 --> 00:28:29.630
Tortoise, these are the time
intervals which are
00:28:29.630 --> 00:28:30.910
transpiring.
00:28:30.910 --> 00:28:34.710
And this sum turns
out to be 2.
00:28:34.710 --> 00:28:38.050
Now, of course, those of us who
had eighth grade algebra
00:28:38.050 --> 00:28:40.690
know an easier way of solving
this problem.
00:28:40.690 --> 00:28:44.285
We say lookit, let's solve this
problem algebraically.
00:28:44.285 --> 00:28:49.390
Namely, we say give the Tortoise
a 1 yard head start.
00:28:49.390 --> 00:28:55.170
Now call x the distance of
a point at which the Hare
00:28:55.170 --> 00:28:56.780
catches the Tortoise.
00:28:56.780 --> 00:28:59.610
Now, the Hare is traveling
1 yard per second.
00:28:59.610 --> 00:29:05.380
The Tortoise is traveling
1/2 yard per second, OK?
00:29:05.380 --> 00:29:12.400
So if we take the distance
traveled and divided by the
00:29:12.400 --> 00:29:16.890
rate, that should be the time.
00:29:16.890 --> 00:29:19.600
And since they both are at this
point at the same time,
00:29:19.600 --> 00:29:24.800
we get what? x/1 equals x
minus 1 divided by 1/2.
00:29:24.800 --> 00:29:28.410
And assuming as a prerequisite
that we have had algebra, it
00:29:28.410 --> 00:29:32.270
follows almost trivially
that x equals 2.
00:29:32.270 --> 00:29:36.790
In other words, what this says
is, in reality, that the Hare
00:29:36.790 --> 00:29:39.540
will not overtake the Tortoise
until he catches
00:29:39.540 --> 00:29:41.900
him, which is obvious.
00:29:41.900 --> 00:29:43.740
But what's not so
obvious is what?
00:29:43.740 --> 00:29:46.270
That these infinitely
many terms can add
00:29:46.270 --> 00:29:48.370
up to a finite sum.
00:29:48.370 --> 00:29:52.420
Well, at any rate, this complete
the overview of what
00:29:52.420 --> 00:29:53.430
our course will be like.
00:29:53.430 --> 00:29:58.890
And to help you focus your
attention on what our course
00:29:58.890 --> 00:30:03.450
really says, what we shall do
computationally is this.
00:30:03.450 --> 00:30:06.610
In review, we shall start with
functions, and functions
00:30:06.610 --> 00:30:09.800
involve the modern concept
of sets because they're
00:30:09.800 --> 00:30:12.230
relationships between
sets of objects.
00:30:12.230 --> 00:30:16.680
We'll talk about limits,
derivatives, rate of change,
00:30:16.680 --> 00:30:18.980
integrals, area under curves.
00:30:18.980 --> 00:30:22.080
This will be our fundamental
building block.
00:30:22.080 --> 00:30:25.230
Once this is done, these things
will never change.
00:30:25.230 --> 00:30:28.540
But the remainder of our course
will be to talk about
00:30:28.540 --> 00:30:31.470
applications, which is the name
of the game as far as
00:30:31.470 --> 00:30:33.220
engineering is concerned.
00:30:33.220 --> 00:30:36.300
More elaborate functions,
namely, how do we handle
00:30:36.300 --> 00:30:37.990
tougher relationships.
00:30:37.990 --> 00:30:41.130
Related to the tougher
relationships will come more
00:30:41.130 --> 00:30:43.560
sophisticated techniques.
00:30:43.560 --> 00:30:46.900
And finally, we will conclude
our course with the topic that
00:30:46.900 --> 00:30:50.020
we were just talking about:
infinite series, how do we get
00:30:50.020 --> 00:30:53.690
a hold of what happens when
you add up infinitely many
00:30:53.690 --> 00:30:56.060
things, each of which
gets small.
00:30:56.060 --> 00:31:00.750
At any rate, that concludes
our lecture for today.
00:31:00.750 --> 00:31:04.820
We will have a digression in
the sense that the next few
00:31:04.820 --> 00:31:09.540
lessons will consist of sets,
things that you can read about
00:31:09.540 --> 00:31:12.160
at your leisure in our
supplementary notes.
00:31:12.160 --> 00:31:15.400
Learn to understand these
because the concept of a set
00:31:15.400 --> 00:31:18.970
is the building block, the
fundamental language of modern
00:31:18.970 --> 00:31:20.300
mathematics.
00:31:20.300 --> 00:31:24.470
And then we will return, once we
have sets underway, to talk
00:31:24.470 --> 00:31:26.680
about functions.
00:31:26.680 --> 00:31:29.310
And then we will build
gradually from there.
00:31:29.310 --> 00:31:31.840
Hopefully, when our course ends,
we will have in slow
00:31:31.840 --> 00:31:35.310
motion gone through
today's lesson.
00:31:35.310 --> 00:31:37.620
This completes our presentation
for today.
00:31:37.620 --> 00:31:39.630
And until next time, goodbye.
00:31:45.570 --> 00:31:48.110
NARRATOR: Funding for the
publication of this video is
00:31:48.110 --> 00:31:52.820
provided by the Gabriella and
Paul Rosenbaum Foundation.
00:31:52.820 --> 00:31:56.990
Help OCW continue to provide
free and open access to MIT
00:31:56.990 --> 00:32:01.190
courses by making a donation
at ocw.mit.edu/donate.