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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
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