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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.

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And you're sort of tempted to
look at your audience and say

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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

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this course.

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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.

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You see, we have our lectures.

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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.

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In addition, recognizing the
fact that the textbook may

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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.

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The study guide consists of
a breakdown of the course.

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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

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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

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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.

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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

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that we're always dealing
with things like

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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

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time by to find the distance?

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You see, in other words, roughly
speaking, we can say

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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.

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And instantaneous speed is a
rather easy thing to talk

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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?

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Now, you see, this is some
sort of a problem.

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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.

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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.

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Let's call them O1 and O2.

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Let them be stationed, one on
each side of P. Now, certainly

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what we could do physically
here is we can measure the

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distance between O1 and O2.

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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

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great answer, but it's
the wrong problem.

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We didn't ask what was the
average speed as we

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went from O1 to O2.

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We asked what was the
instantaneous speed.

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And the idea is we say,
well, lookit.

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The average speed and the
instantaneous speed, it seems,

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should be pretty much the same
if the observers were

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relatively close together.

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The next observation is it seems
that if we were to move

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the observers in even closer,
there would be less of a

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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

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now seem like a better
approximation to the

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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

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make the observers gets closer
and closer together.

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That would minimize the
difference between the average

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speed and the instantaneous rate
of speed, and maybe the

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optimal thing would happen
when the two

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observers were together.

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But the strange part is--

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and this is where calculus
really begins.

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This is what calculus
is all about.

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As soon as the observers come
together, notice that what you

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have is that the distance
between them is 0.

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The time that it takes to get
from one to the other is 0.

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And therefore, it appears that
if we divide distance by time,

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we are going to wind
up with 0/0.

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Now, my claim is that 0/0
should be called--

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well, I'll call it undefined,
but actually, I think

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indeterminate would
be a better word.

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Why do I say that?

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Well, here's an interesting
thing.

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When we do arithmetic with small
numbers, observe that if

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you add two small numbers, you
expect the result to be a

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small number.

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If you multiply two small
numbers, you expect the result

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to be a small number.

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Similarly, for division, for
subtraction, the difference of

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two small numbers is
a small number.

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On the other hand, the quotient
of two small numbers

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is rather deceptive.

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Because it's a ratio, if one
of the very small numbers

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happens to be very much larger
compared with the other small

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number, the ratio might
be quite large.

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Well, for example, visualize,
say, 10 to the minus 6,

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1/1,000,000, 0.000001, which
is a pretty small number.

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Now, divide that by 10
to the minus 12th.

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Well, you see, 10 to the minus
12th is a small number, so

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small that it makes 10 to the
minus sixth appear large.

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In fact, the quotient is 10 to
the sixth, which is 1,000,000.

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And here we see that when you're
dealing with the ratio

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of small numbers, you're a
little bit in trouble, because

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we can't tell whether the ratio
will be small, or large,

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or somewhere in between.

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For example, if we reverse
the role of numerator and

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denominator here, we would still
have the quotient of two

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small numbers, but 10 to the
minus 12th divided by 10 to

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the minus sixth is a relatively
small number, 10 to

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the minus 6.

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Of course, this is the physical
way of looking at it.

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Small divided by small
is indeterminate.

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We have a more rigorous way of
looking at this if you want to

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see it from a mathematical
structure point of view.

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Namely, suppose we define a/b
in the traditional way.

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Namely, a/b is that number such
that when we multiply it

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by b we get a.

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Well, what would that say as
far as 0/0 was concerned?

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It would say what?

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That 0/0 is that number such
that when we multiply

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it by 0 we get 0.

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Now, what number has the
property that when we multiply

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it by 0 we get 0?

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And the answer is any number.

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This is why 0/0 is
indeterminate.

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If we say to a person, tell me
the number I must multiply by

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0 to get 0, the answer
is any number.

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Well, the idea then is that we
must avoid the expression 0/0

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at all costs.

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What this means then is that we
say OK, let the observers

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get closer to closer together,
but never touch.

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Now, the point is that as long
as the observers get closer

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and closer together and never
touch, let's ask the question

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how many pairs of observers
do we need?

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00:09:26,300 --> 00:09:28,790
And the answer is that
theoretically we need

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infinitely many pairs
of observers.

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Well, why is that?

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Because as long as there's a
distance between a pair of

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observers, we can theoretically
fit in another

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pair of observers.

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This is why in our course we do
not begin with this idea,

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but looking backwards now, we
say ah, we had better find

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some way of giving us the
equivalent of having

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00:09:51,140 --> 00:09:53,290
infinitely many pairs
of observers.

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And to do this, the idea that we
come up with is the concept

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called a function.

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00:10:01,990 --> 00:10:05,400
Consider the old Galileo freely
falling body problem,

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00:10:05,400 --> 00:10:09,520
where the distance that the
body falls s equals 16t

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00:10:09,520 --> 00:10:12,950
squared, where t is in seconds
and s is in feet.

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00:10:12,950 --> 00:10:17,760
Notice that this apparently
harmless recipe gives us a way

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for finding s for
each given t.

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00:10:21,250 --> 00:10:24,650
In other words, to all intents
and purposes, this recipe

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gives us an observer for
each point of time.

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For each time, we can find the
distance, which is physically

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equivalent to knowing an
observer at every point.

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00:10:36,330 --> 00:10:40,430
In turn, the study of functions
lends itself to a

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study of graphs, a picture.

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00:10:43,220 --> 00:10:47,590
Namely, if we look at s equals
16t squared again, notice that

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00:10:47,590 --> 00:10:49,910
we visualize a recipe here.

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00:10:49,910 --> 00:10:54,250
t can be viewed as being an
input, s as the output.

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For a given input t, we can
compute the output s.

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00:10:58,560 --> 00:11:03,090
In general, if we now elect
to plot the input along a

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00:11:03,090 --> 00:11:07,770
horizontal line and the output
at right angles to this, we

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00:11:07,770 --> 00:11:11,270
now have a picture of our
relationship, a picture which

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is called a graph.

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00:11:14,200 --> 00:11:17,980
You see, we can talk about this
more explicitly as far as

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00:11:17,980 --> 00:11:20,830
this particular problem is
concerned, just by taking a

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00:11:20,830 --> 00:11:22,650
look at a picture like this.

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In other words, in this
particular problem, the input

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is time t, the output
is distance s.

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00:11:30,970 --> 00:11:35,270
For each t, we locate a height
called s by squaring t and

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multiplying by 16.

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00:11:37,450 --> 00:11:40,390
And now, what average speed
means in terms of this kind of

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a diagram is the following.

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00:11:42,640 --> 00:11:46,410
To find the average speed, all
we have to do is on a given

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00:11:46,410 --> 00:11:49,960
time interval find the distance
traveled, which I

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00:11:49,960 --> 00:11:53,510
call delta s, the change in
distance, and divide that by

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00:11:53,510 --> 00:11:54,860
the change in time.

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00:11:54,860 --> 00:11:58,030
That's the average speed, which,
by the way, from a

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00:11:58,030 --> 00:12:02,150
geometrical point of view,
becomes known as the slope of

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00:12:02,150 --> 00:12:03,820
this particular straight line.

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00:12:03,820 --> 00:12:08,130
In other words, average speed is
to functions what slope of

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00:12:08,130 --> 00:12:10,830
a straight line is
to geometry.

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00:12:10,830 --> 00:12:14,190
At any rate, knowing what the
average rate of speed is, we

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00:12:14,190 --> 00:12:18,570
sort of say why couldn't we
define the instantaneous speed

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00:12:18,570 --> 00:12:19,500
to be this.

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00:12:19,500 --> 00:12:24,040
We will take the change in
distance divided by the change

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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.

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00:12:28,430 --> 00:12:30,450
Limit as delta t approaches 0.

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00:12:30,450 --> 00:12:33,450
Let's see what happens as that
change in time becomes

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00:12:33,450 --> 00:12:37,370
arbitrarily small, but never
equaling 0 because we don't

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00:12:37,370 --> 00:12:39,820
want a 0/0 form here.

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00:12:39,820 --> 00:12:44,260
You see, this then becomes the
working definition of what we

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00:12:44,260 --> 00:12:46,670
call differential calculus.

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00:12:46,670 --> 00:12:50,390
The point is that this
particular definition does not

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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.

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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

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00:13:09,980 --> 00:13:13,970
handle the particular
relationship between s and t.

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00:13:13,970 --> 00:13:18,010
This will be a major part of our
course, the strange thing

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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.