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PROFESSOR: Hi.
I'm Herb Gross, and welcome
to Calculus Revisited.
I guess the most difficult
lecture to give with any
course is probably
the first one.
And you're sort of tempted to
look at your audience and say
you're probably wondering why
I called you all here.
And in this sense, I have
elected to entitle our first
lecture simply Preface to give a
double overview, an overview
both of the hardware and the
software that will make up
this course.
To begin with, we will have a
series of lectures of which
this is the first.
In our lectures, our main aim
will be to give an overview of
the material being covered, an
insight as to why various
computations are done, and
insights as to how
applications of these concepts
will be made.
The heart of our course will
consist of a regular textbook.
You see, we have our lectures.
We have a textbook.
The textbook is designed to
supply you with deeper
insights than what we can
give in a lecture.
In addition, recognizing the
fact that the textbook may
leave gaps, places where you
may want some additional
knowledge, we also have
supplementary notes.
And finally, at the backbone of
our package is what we call
the study guide.
The study guide consists of
a breakdown of the course.
It tells us what the various
lectures will be, the units.
There are pretests to help you
decide how well prepared you
are for the topic that's
coming up.
There is a final examination
at the end of
each block of material.
And perhaps most importantly,
especially from an engineer's
point of view, in each unit that
we study, the study guide
will consist of exercises
primarily called learning
exercises, exercises which
hopefully will turn you on
towards wanting to be able to
apply the material, and at the
same time, serve as a
springboard by which we can
highlight why the theory and
many about our lecture points
are really as important
as they are.
So much for the hardware
of our course.
And now let's turn our attention
to the software.
Just what is calculus?
In a manner of speaking,
calculus can be viewed as
being high school mathematics
with one additional concept
called the limit concept
thrown in.
If you recall back to your high
school days, remember
that we're always dealing
with things like
average rate of speed.
Notice I say average or constant
rate of speed.
The old recipe that distance
equals rate times time
presupposes that the rate is
constant, because if the rate
is varying, which rate is it
that you use to multiply the
time by to find the distance?
You see, in other words, roughly
speaking, we can say
that at least one branch of
calculus known as differential
calculus deals with
the subject of
instantaneous speed.
And instantaneous speed is a
rather easy thing to talk
about intuitively.
Imagine an object moving along
this line and passing the
point P. And we say to ourselves
how fast was the
object moving at the instant
that we're at the point P?
Now, you see, this is some
sort of a problem.
Because at the instant that
you're at P, you're not in a
sense moving at all because
you're at P.
Of course, what we do to reduce
this problem to an old
one is we say, well, suppose we
have a couple of observers.
Let's call them O1 and O2.
Let them be stationed, one on
each side of P. Now, certainly
what we could do physically
here is we can measure the
distance between O1 and O2.
And we can also measure the
time that it takes to
go from O1 to O2.
And what we can do is divide
that distance by the time, and
that, you see, is our old high
school concept of the average
speed of the particle as
it moves from O1 to O2.
Now, you see, the question is,
somebody says gee, that's a
great answer, but it's
the wrong problem.
We didn't ask what was the
average speed as we
went from O1 to O2.
We asked what was the
instantaneous speed.
And the idea is we say,
well, lookit.
The average speed and the
instantaneous speed, it seems,
should be pretty much the same
if the observers were
relatively close together.
The next observation is it seems
that if we were to move
the observers in even closer,
there would be less of a
discrepancy between O1 and O2
in the sense that-- not a
discrepancy, but in the sense
that the average speed would
now seem like a better
approximation to the
instantaneous speed because
there was less distance for
something to go wrong in.
And so we get the idea that
maybe what we should do is
make the observers gets closer
and closer together.
That would minimize the
difference between the average
speed and the instantaneous rate
of speed, and maybe the
optimal thing would happen
when the two
observers were together.
But the strange part is--
and this is where calculus
really begins.
This is what calculus
is all about.
As soon as the observers come
together, notice that what you
have is that the distance
between them is 0.
The time that it takes to get
from one to the other is 0.
And therefore, it appears that
if we divide distance by time,
we are going to wind
up with 0/0.
Now, my claim is that 0/0
should be called--
well, I'll call it undefined,
but actually, I think
indeterminate would
be a better word.
Why do I say that?
Well, here's an interesting
thing.
When we do arithmetic with small
numbers, observe that if
you add two small numbers, you
expect the result to be a
small number.
If you multiply two small
numbers, you expect the result
to be a small number.
Similarly, for division, for
subtraction, the difference of
two small numbers is
a small number.
On the other hand, the quotient
of two small numbers
is rather deceptive.
Because it's a ratio, if one
of the very small numbers
happens to be very much larger
compared with the other small
number, the ratio might
be quite large.
Well, for example, visualize,
say, 10 to the minus 6,
1/1,000,000, 0.000001, which
is a pretty small number.
Now, divide that by 10
to the minus 12th.
Well, you see, 10 to the minus
12th is a small number, so
small that it makes 10 to the
minus sixth appear large.
In fact, the quotient is 10 to
the sixth, which is 1,000,000.
And here we see that when you're
dealing with the ratio
of small numbers, you're a
little bit in trouble, because
we can't tell whether the ratio
will be small, or large,
or somewhere in between.
For example, if we reverse
the role of numerator and
denominator here, we would still
have the quotient of two
small numbers, but 10 to the
minus 12th divided by 10 to
the minus sixth is a relatively
small number, 10 to
the minus 6.
Of course, this is the physical
way of looking at it.
Small divided by small
is indeterminate.
We have a more rigorous way of
looking at this if you want to
see it from a mathematical
structure point of view.
Namely, suppose we define a/b
in the traditional way.
Namely, a/b is that number such
that when we multiply it
by b we get a.
Well, what would that say as
far as 0/0 was concerned?
It would say what?
That 0/0 is that number such
that when we multiply
it by 0 we get 0.
Now, what number has the
property that when we multiply
it by 0 we get 0?
And the answer is any number.
This is why 0/0 is
indeterminate.
If we say to a person, tell me
the number I must multiply by
0 to get 0, the answer
is any number.
Well, the idea then is that we
must avoid the expression 0/0
at all costs.
What this means then is that we
say OK, let the observers
get closer to closer together,
but never touch.
Now, the point is that as long
as the observers get closer
and closer together and never
touch, let's ask the question
how many pairs of observers
do we need?
And the answer is that
theoretically we need
infinitely many pairs
of observers.
Well, why is that?
Because as long as there's a
distance between a pair of
observers, we can theoretically
fit in another
pair of observers.
This is why in our course we do
not begin with this idea,
but looking backwards now, we
say ah, we had better find
some way of giving us the
equivalent of having
infinitely many pairs
of observers.
And to do this, the idea that we
come up with is the concept
called a function.
Consider the old Galileo freely
falling body problem,
where the distance that the
body falls s equals 16t
squared, where t is in seconds
and s is in feet.
Notice that this apparently
harmless recipe gives us a way
for finding s for
each given t.
In other words, to all intents
and purposes, this recipe
gives us an observer for
each point of time.
For each time, we can find the
distance, which is physically
equivalent to knowing an
observer at every point.
In turn, the study of functions
lends itself to a
study of graphs, a picture.
Namely, if we look at s equals
16t squared again, notice that
we visualize a recipe here.
t can be viewed as being an
input, s as the output.
For a given input t, we can
compute the output s.
In general, if we now elect
to plot the input along a
horizontal line and the output
at right angles to this, we
now have a picture of our
relationship, a picture which
is called a graph.
You see, we can talk about this
more explicitly as far as
this particular problem is
concerned, just by taking a
look at a picture like this.
In other words, in this
particular problem, the input
is time t, the output
is distance s.
For each t, we locate a height
called s by squaring t and
multiplying by 16.
And now, what average speed
means in terms of this kind of
a diagram is the following.
To find the average speed, all
we have to do is on a given
time interval find the distance
traveled, which I
call delta s, the change in
distance, and divide that by
the change in time.
That's the average speed, which,
by the way, from a
geometrical point of view,
becomes known as the slope of
this particular straight line.
In other words, average speed is
to functions what slope of
a straight line is
to geometry.
At any rate, knowing what the
average rate of speed is, we
sort of say why couldn't we
define the instantaneous speed
to be this.
We will take the change in
distance divided by the change
in time and see what happens.
And we write this this way.
Limit as delta t approaches 0.
Let's see what happens as that
change in time becomes
arbitrarily small, but never
equaling 0 because we don't
want a 0/0 form here.
You see, this then becomes the
working definition of what we
call differential calculus.
The point is that this
particular definition does not
depend on s equaling
16t squared.
s could be any function
of t whatsoever.
We could have a more elaborate
type of situation.
The important point is what?
The basic definition
stays the same.
What changes is the amount of
arithmetic that's necessary to
handle the particular
relationship between s and t.
This will be a major part of our
course, the strange thing
being that even at the very end
of our course when we've
gone through many, many things,
our basic definition
of instantaneous rate of
change will have never
changed from this.
It will always stay like this.
But what will change is how much
arithmetic and algebra
and geometry and trigonometry,
et cetera, we will have to do
in order to compute these
things from a
numerical point of view.
Well, so much for the first
phase of calculus called
differential calculus.
A second phase of calculus, one
which was developed by the
Ancient Greeks by 600 BC, the
subject that ultimately
becomes known as integral
calculus, concerns problem of
finding area under a curve.
Here, I've elected to draw the
parabola y equals x squared on
the interval from
0, 0 to 1, 0.
And the question basically is
what is the area bounded by
this sort of triangular
region?
Let's call that region R, and
what we would like to find is
the area of the region R.
And the Ancient Greeks had a
rather interesting title for
this type of approach for
finding the area.
It is both figurative and
literal, I guess.
It's called the method
of exhaustion.
What they did was to --
They would divide the
interval, say,
into n equal parts.
And picking the lowest point in
each interval, they would
inscribe a rectangle.
Knowing that the area of the
rectangle was the base times
the height, they would add up
the area of each of these
rectangles, and know that
whatever that area was, that
would have to be too small to
be the right answer because
that region was contained
in R. And that would be
labeled A sub n--
lower bar, say--
to indicate that this was a sum
of rectangles which was
too small to be the
right answer.
Similarly, they would then find
the highest point in each
rectangle, get an
overapproximation by adding up
the sum of those areas, which
they would call A sub n upper
bar, and now know that the area
of the regions they were
looking for was squeezed
in between these two.
Then what they would do is make
more and more divisions,
and hopefully, and I think
you can see this sort of
intuitively happening here,
each of the lower
approximations gets bigger
and fills out
the space from inside.
Each of the upper approximations
gets smaller
and chops off the space
from outside here.
And hopefully, if both of these
bounds sort of converge
to the same value L, we get the
idea that the area of the
region R must be L.
This is not anything new.
In other words, this is a
technique that is some 2,500
years old, used by the
Ancient Greeks.
Of course, what happens with
engineering students in
general is that one frequently
says, but I'm not interested
in studying area.
I am not a geometer.
I am a physicist.
I am an engineer.
What good is the area
under a curve?
And the interesting point here
becomes that if we label the
coordinate axis rather than x
and y, give them physical
labels, it turns out that area
under a curve has a physical
interpretation.
Consider the same problem.
Only now, instead of talking
about y equals x squared,
let's talk about v, the
velocity, equaling the square
of the time.
And say that the time
goes to 0 to 1.
In other words, if we plot
v versus t, we get a
picture like this.
And the question that comes up
is what do we mean by the area
under the curve here?
And again, without belaboring
this point, not because it's
not important, but because this
is just an overview and
we'll come back to all of these
topics later in our
course, the point I just want
to bring out here is, notice
that the area under the curve
here is the distance that this
particle would travel moving at
this speed if the time goes
from 0 to 1.
And notice what we're
saying here.
Again, suppose we divide this
interval into n equal parts
and inscribe rectangles.
Notice that each of
these rectangles
represents a distance.
Namely, if a particle moved at
the speed over this length of
time, the area under the curve
would be the distance that it
traveled during that
time interval.
In other words, what we're
saying is that if the particle
moved at this speed from this
time to this time, then moved
at this speed from this time to
this time, the sum of these
two areas would give the
distance that the particle
traveled, which obviously is
less than the distance that
the particle truly traveled,
because notice that the
particle was moving at a speed
which at every instance from
here to here was greater than
this and at every instant from
here to here was greater
than this.
In other words, in the same way
as before, that area of
the region R was whittled in
between A sub n upper bar and
A sub n lower bar, notice that
the distance traveled by the
particle can now be limited or
bounded in the same way.
And in the same way that we
found area as a limit, we can
now find distance as a limit.
And these two things,
namely, what?
Instantaneous speed and area
under a curve are the two
essential branches of calculus,
differential
calculus being concerned with
instantaneous rate of speed,
integral calculus with
area under a curve.
And the beauty of calculus,
surprisingly enough, in a way
is only secondary
as far as these
two topics are concerned.
The true beauty lies in the fact
that these apparently two
different branches of calculus,
one of which was
invented by the Ancient Greeks
as early as 600 BC,
the other of which--
differential calculus--
was not known to man until the
time of Isaac Newton in 1690
AD are related by a rather
remarkable thing.
That remarkable thing, which
we will emphasize at great
length during our course, is
that areas and rates of change
are related by area
under a curve.
Now, I don't know how to draw
this so that you see this
thing as vividly as possible,
but the idea is this.
Think of area being swept out as
we take a line and move it,
tracing out the curve this
way towards the right.
Notice that if we have a certain
amount of area, if we
now move a little bit further to
the right, notice that the
new area somehow depends on what
the height of this curve
is going to be.
That somehow or other, it seems
that the area under the
curve must be related to how
fast the height of this line
is changing.
Or to look at it inversely, how
fast the area is changing
should somehow be related to
the height of this line.
And just what that relationship
is will be
explored also in great
detail in the course.
And we will show the beautiful
marriage between this
differential and integral
calculus through this
relationship here, which
becomes known as the
fundamental theorem of
integral calculus.
At any rate then, what this
should show us is that
calculus hinges--
whether it's differential
calculus or integral calculus,
that calculus hinges
on something
called the limit concept.
Again, by way of a very
quick review, one
of the limit concepts--
and I think it's easy to see
geometrically rather than
analytically.
Imagine that we have a curve,
and we want to find the
tangent of the curve at the
point P. What we can do is
take a point Q and draw the
straight line that joins P to
Q. We could then find the
slope of the line PQ.
The trouble is that PQ does not
look very much like the
tangent line.
So we say OK, let Q move down
so it comes closer to P. We
can then find the
slopes of PQ1.
We could find the
slope of PQ2.
But in each case, we still do
not have the slope of the line
tangent to the curve at P. But
we get the idea that as Q gets
closer and closer to P, the
slope, or the secant line that
joins P to Q, becomes a better
and better approximation to
the line that would be tangent
to the curve at P.
In fact, it's rather interesting
that in the 16th
century, the definition that
was given of a tangent line
was that a tangent line is a
line which passes through two
consecutive points on a curve.
Now, obviously, a curve
does not have
two consecutive points.
What they really
meant was what?
That as Q gets closer and closer
to P, the secant line
becomes a better and better
approximation for the tangent
line, and that in a way, if the
two points were allowed to
coincide, that should give
us the perfect answer.
The trouble is, just like you
can't divide 0 by 0, if P and
Q coincide, how many
points do you have?
Just one point.
And it takes two points to
determine a straight line.
No matter how close Q is to P,
we have two distinct points.
As soon as Q touches
P, we lose this.
And this is what was meant by
ancient man or medieval man by
his notion of two consecutive
points.
And I should put this in double
quotes because I think
you can see what he's begging
to try to say with the word
"consecutive," even though from
a purely rigorous point
of view, this has no
geometric meaning.
Now, the other form of limit has
to do with adding up areas
of rectangles under curves.
Namely, we divided the curve
up into n parts.
We inscribed n rectangles,
and then we let n
increase without bound.
In other words, this is sort of
a discrete type of limit.
Namely, we must add up a whole
number of areas, but the sum
is endless in the sense that
the number of rectangles
becomes greater than any number
we want to preassign.
And the basic question that we
must contend with here is how
big is an infinite sum?
You see, when we say infinite
sum, that just tells you how
many terms you're combining.
It doesn't tell you how
big each term, how big
the sum will be.
For example, look at
the following sum.
I will start with 1.
Then I'll add 1/2 on twice.
Then I'll add 1/3
on three times.
And without belaboring this
point, let me then say I'll
had on 1/4 four times,
1/5 five times, 1/6
six times, et cetera.
Notice as I do this that each
time the terms gets smaller,
yet the sum increases
without any bound.
Namely, notice that
this adds up to 1.
This adds up to 1.
The next four terms
will add up to 1.
And as I go out further and
further, notice that this sum
can become as great is
I want, just by me
adding on enough 1's.
On the other hand, let's
look at this one.
1 plus 1/2 plus 1/4 plus 1/8
plus 1/16 plus 1/32.
In other words, I start with 1
and each time add on half the
previous number.
See, 1 plus 1/2 plus
1/4 plus 1/8.
You may remember this as being
the geometric series whose
ratio is 1/2.
The interesting thing is that
now this sum gets as close to
2 as you want without
ever getting there.
And rather than prove this right
now, let's just look at
the geometric interpretation
here.
Take a line which is
2 inches long.
Suppose you first go halfway.
You're now here.
Now go half the remaining
distance.
That's what?
1 plus 1/2.
That puts you over here.
Now go half the remaining
distance.
That means add on 1/4.
Now go half the remaining
distance.
That means add on on 1/8.
Now go half the remaining
distance.
Add up this on 1/16, you see.
And ultimately, what happens?
Well, no matter where you stop,
you've become closer and
closer to 2 without ever
getting there.
And as you go further and
further, you can get as close
to 2 as you want.
In other words, here are
infinitely many terms whose
infinite sum is 2.
Here are infinitely many terms
whose infinite sum is
infinity, we should
say, because it
increases without bound.
And this was the problem that
hung up the Ancient Greek.
How could you do infinitely
many things in a
finite amount of time?
In fact, at the same time that
the Greek was developing
integral calculus, the famous
greek philosopher Zeno was
working on things called
Zeno's paradoxes.
And Zeno's paradoxes are three
in number, of which I only
want to quote one here.
But it's a paradox which shows
how Zeno could not visualize
quite what was happening.
You see, it's called the
Tortoise and the Hare problem.
Suppose that you give the
Tortoise a 1 yard head start
on the Hare.
And suppose for the sake of
argument, just to mimic the
problem that we were doing
before, suppose it's a slow
Hare and a fast Tortoise so that
the Hare only runs twice
as fast as the Tortoise.
You see, Zeno's paradox says
that the Hare can never catch
the Tortoise.
Why?
Because to catch the Tortoise,
the Hare must first go the 1
yard head start that
the Tortoise had.
Well, by the time the Hare gets
here, the Tortoise has
gone 1/2 yard because the
Tortoise travels half as fast.
Now, the Hare must make
up the 1/2 yard.
But while the Hare makes up
the 1/2 yard, the Tortoise
goes 1/4 of a yard.
When the Hare makes up the 1/4
of a yard, the Tortoise goes
1/8 of a yard.
And so, Zeno argues, the Hare
gets closer and closer to the
Tortoise but can't catch him.
And this, of course, is a rather
strange thing because
Zeno knew that the Tortoise
would catch the Hare.
That's it's called a paradox.
A paradox means something which
appears to be true yet
is obviously false.
Now, notice that we can resolve
Zeno's paradox into
the example we were just
talking about.
For the sake of argument, notice
what's happening here
with the time.
For the sake of argument,
let's suppose that the
Tortoise travels at
1 yard per second.
Then what you're saying is--
I mean, the Hare travels
at 1 yard per second.
What you're saying is
it takes the Hare 1
second to go this distance.
Then it takes him 1/2 a second
to go this distance, then 1/4
of a second to go
this distance.
And what you're saying is that
as he's gaining on the
Tortoise, these are the time
intervals which are
transpiring.
And this sum turns
out to be 2.
Now, of course, those of us who
had eighth grade algebra
know an easier way of solving
this problem.
We say lookit, let's solve this
problem algebraically.
Namely, we say give the Tortoise
a 1 yard head start.
Now call x the distance of
a point at which the Hare
catches the Tortoise.
Now, the Hare is traveling
1 yard per second.
The Tortoise is traveling
1/2 yard per second, OK?
So if we take the distance
traveled and divided by the
rate, that should be the time.
And since they both are at this
point at the same time,
we get what? x/1 equals x
minus 1 divided by 1/2.
And assuming as a prerequisite
that we have had algebra, it
follows almost trivially
that x equals 2.
In other words, what this says
is, in reality, that the Hare
will not overtake the Tortoise
until he catches
him, which is obvious.
But what's not so
obvious is what?
That these infinitely
many terms can add
up to a finite sum.
Well, at any rate, this complete
the overview of what
our course will be like.
And to help you focus your
attention on what our course
really says, what we shall do
computationally is this.
In review, we shall start with
functions, and functions
involve the modern concept
of sets because they're
relationships between
sets of objects.
We'll talk about limits,
derivatives, rate of change,
integrals, area under curves.
This will be our fundamental
building block.
Once this is done, these things
will never change.
But the remainder of our course
will be to talk about
applications, which is the name
of the game as far as
engineering is concerned.
More elaborate functions,
namely, how do we handle
tougher relationships.
Related to the tougher
relationships will come more
sophisticated techniques.
And finally, we will conclude
our course with the topic that
we were just talking about:
infinite series, how do we get
a hold of what happens when
you add up infinitely many
things, each of which
gets small.
At any rate, that concludes
our lecture for today.
We will have a digression in
the sense that the next few
lessons will consist of sets,
things that you can read about
at your leisure in our
supplementary notes.
Learn to understand these
because the concept of a set
is the building block, the
fundamental language of modern
mathematics.
And then we will return, once we
have sets underway, to talk
about functions.
And then we will build
gradually from there.
Hopefully, when our course ends,
we will have in slow
motion gone through
today's lesson.
This completes our presentation
for today.
And until next time, goodbye.
NARRATOR: Funding for the
publication of this video is
provided by the Gabriella and
Paul Rosenbaum Foundation.
Help OCW continue to provide
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courses by making a donation
at ocw.mit.edu/donate.
