PROFESSOR: Hi, I'm Gilbert
Strang, and this is the very
first in a series of videos
about highlights of calculus.
I'm doing these just because
I hope they'll be helpful.
It seems to me so easy to be
lost in the big calculus
textbooks and the many, many
problems and in the details.
But do you see the
big picture?
Well, I hope this will help.
For me, calculus is about the
relation between two functions.
And one example for those two
functions, one good example,
is function 1, the distance,
distance traveled, what you
see on a trip meter in a car.
And function 2, the one that
goes with distance, is speed,
how quickly you're going, how
fast you're traveling.
So that's one pair
of functions.
Let me give another pair.
I could get more and more, but
I think if we get these two
pairs, we can move forward.
So in this second pair,
height is function 1,
how high you've climbed.
If it's a graph, how far the
graph goes above the axis.
Up, in other words.
So that's height, and then
the other one tells you
how fast you climb.
The height tells how
far you climbed.
It could be a mountain.
And then the slope tells you how
quickly you're climbing at
each point.
Are you going nearly
straight up?
Flat?
Possibly down?
So distance and speed, height
and slope will serve as good
examples to start with.
And let me give you some
letters, some algebra letters
that you might use.
Distance, maybe I would
call that f of t.
So f for how far or for
function, and the idea is that
t is the input.
It's the time when you're
asking for the distance.
The output is the distance.
Or in the case of height,
maybe y of x would
be the right one.
x is how far you go across.
That's the input.
And at each x, you have an
output y how far up?
So f is telling you how far.
y is telling you the
height of a graph.
That's function 1, two examples
of function 1.
Now, what about slope?
Well, luckily, speed and slope
start with the same letter, so
I'll often use s for the speed
or the slope for this second--
oh, it even stands for
a second function.
But let me tell you
also the right--
the official--
letters that make the connection
between function 2
and function 1.
If my function is a function of
time, the distance, how far
I go, then the speed is--
the right letters are df dt.
Everybody uses those letters.
So let me say again how
to pronounce: df dt.
And Leibniz came up with
that notation, and
he just got it right.
And what would this one be?
Well, corresponding to this, it
looks the same, or dy dx.
Again, I'll just repeat how
to say that: dy dx.
And that is the slope, and we
have to understand what those
symbols mean.
Right now, I'm just writing
them down as symbols.
May I begin with the most
important and the simplest
example of all?
Let me take that case.
OK, so the key example here,
the one to get completely
straight is the case
of constant
speed, constant slope.
I'll just graph that.
So here I'm go to graph.
Shall I make it the speed?
Yeah, let's say speed.
So time is going
along that way.
Speed is up this way.
And I'm going to say in this
first example that the speed
is the same.
We're traveling at the
constant speed
of let's say 40.
So it stays at the
height of 40.
Oh, properly, I should add units
like miles per hour or
kilometers per hour or meters
per second or whatever.
For now, I'll just write 40.
OK, now if we're traveling at a
speed of 40 miles per hour,
what's the distance?
Well, let me start with the
trip meter at zero.
so this is time again,
and now this is
going to be the distance.
After one hour, my
distance is 40.
So if I mark t equal to
1, I've reached 40.
That's height of 40.
At t equal to 2, I've
reached 80.
At t equal to 1/2, half an
hour, I've reached 20.
Those points lie on a line.
The graph of distance covered
when you're just traveling at
a steady rate, constant rate,
constant speed is just a
straight line.
And now I can make
the connection.
I've been speaking here about
distance and speed.
But now let me think of
this as the height--
40 is that height.
80 is that height--
and ask about slope.
What is slope?
So let's just remember what's
the connection here.
What's the slope if that's the
distance if I look at my trip
meter and I know I'm traveling
along at that constant speed,
how do I find that speed?
Well, slope, it's the distance
up, which would be 40 after
one hour, divided
by the distance
across, 40/1, or 80/2.
Doesn't matter, because we're
traveling at constant speed,
so the slope, which is up,
over, across is 40/1,
80/2, 20 over 1/2.
I'll put 80/2 as one
example: 40.
Oh, let me do it--
that's arithmetic.
Let me do it with algebra.
We don't need calculus
yet, by the way.
Calculus is coming
pretty quickly.
This is the step we can take.
Because the speed is constant,
we can just divide the
distance by the time to find--
and this slope, let me
right speed also.
Up, over, across, distance
over time, f/t,
that gives us s.
This is s.
OK, what about--
calculus goes both ways.
We can go both ways here.
We already have practically.
Here I went in the direction
from 1 to 2.
Now, I want to go in
the direction--
suppose I know the speed.
How do I recover the distance?
If I know my speed is 40 and I
know I started at zero, what's
my distance?
Distance or height,
either one.
So these are like both.
Now, I'm just going
the other way.
Well, you see how.
How do I find f?
It's s times t, right?
Your algebra automatically says
if you see a t there, you
can put it there.
So it's s times t.
It's a straight line.
s times t, s times
x, y equal sx.
Let me put another--
the same idea with
my y, x letters.
It's that line.
In other words, if that one
is constant, this one is a
straight line.
OK, straightforward, but
very, very fundamental.
In fact, can I call your
attention to something a
little more?
Suppose I measured between
time 2 and time 1.
So I'm looking between time 2
and time 1, and I look how far
I went in that time.
But what I'm trying is--
I'm going to put in another
little symbol because it's
going to be really
worth knowing.
It's really the change in f
divided by the change in t.
I use that letter delta
to indicate
a difference between--
the difference between time 2
and time 1 was 1, and the
difference between height
2 and height 1 was 40.
You see, I'm looking at
this little piece.
And, of course, the
slope is still 40.
It's still the slope
of that line.
Yeah, so that really what I'm
measuring in speed there, I
don't always have to be starting
at t equals 0, and I
don't always have to be starting
at f equals 0.
Oh, let me draw that.
Suppose I started
at f equals 40.
My trip meter happened
to start at 40.
After an hour, I'd
be up to 80.
After another hour,
I'd be up to 120.
Do you see that this starting
the trip meter, who cares
where the trip meter started?
It's the change in the trip
meter that tells how
long the trip was.
Clear.
OK, so that's that example.
We come back to it because it's
the basic one where the
speed is constant.
And even if now I have to move
to a changing speed, you have
to let me bring calculus
into these lectures.
OK, I'm going to draw another
picture, and you
tell me about the--
yeah, let me draw function 1,
another example of function 1.
So again I have time.
I have distance.
I'm going to start at zero, but
I'm not going to keep the
speed constant.
I'm going to start out at
a good speed, but I'm
going to slow down.
Do you see me slowing
down there?
I don't mean slowing down
with the chalk.
I mean slowing down
with slope.
The slope started out steep.
By here, by that point,
the slope was zero.
What was the car doing here?
The car is certainly moving
forward because the distance
is increasing.
Here it's increasing faster.
Here it's increasing barely.
In other words, we're putting
on the brakes.
The car is slowing down.
We're coming to a red light.
In fact, there is the red light
right at that time.
Now, just stay with it to think
what would the speed
look like for this problem?
If that's a picture of
the function, just
let's get some idea.
I'm not going to have
a formula yet.
I'm not putting in
all the details.
Well, actually, I don't plan to
put in all the details of
calculus of every possible
step we might take.
It's the important ones I'm
hoping to show you and I'm
hoping for you to see that
they are important.
OK, what is important?
Roughly, what does the
graph looks like?
Well, the speed--
the slope--
started out somewhere
up there.
Yeah, it started out at a good
speed and slowed down.
And by this point, ha!
Let's mark that time
here on that graph.
Do you see what is the
speed at that moment?
The speed at that
moment is zero.
The car has stopped.
The speed is decreasing.
Let me make it decrease,
decrease, decrease, decrease,
and at that moment, the speed
is zero right there.
That's that point.
See, two different pictures,
two different functions.
but same information.
So calculus has the job of given
one of those functions,
find the other one.
Given this function,
find that one.
This way is called--
from function one to function
two, that's called
differential calculus.
Big, impressive word anyway.
That's function one to two,
finding the speed.
Going the other direction is
called integral calculus.
The step is called integration
when you take the speed over
that period of time, and you
recover the distance.
So it's differential calculus
in one direction, integral
calculus in the other.
Now, here's a question.
Let me continue that curve
a little longer.
I got it to the red light.
Now imagine that the distance
starts going
down from that point.
What's happening?
The distance is decreasing.
The car is going backwards.
It's going in reverse.
The speed, what's the speed?
Negative.
The speed, because distance is
going from higher to lower,
that counts for negative
speed.
The speed curve would
be going down here.
Do you see that that's a not
brilliantly drawn picture, but
you're seeing the--
that's the farthest it went.
Then the car started backwards,
and the speed curve
reflected that by going
below zero.
You see, two different curves,
but same information.
I'm remembering an old movie.
I don't know if you saw an
old B movie called Ferris
Bueller's Day Off.
Did you see that?
So the kid had borrowed
his father's--
not borrowed, but lifted his
father's good car and drove it
a lot like so and put
on a lot of mileage.
The trip meter was way up, and
he knew his father was going
to notice this.
So he had the idea to put the
car up on a lift, put it in
reverse, and go for a
while, and the trip
meter would go backwards.
I don't know if trip meters
do go backwards.
It's kind of tough to watch them
while going in reverse.
But if whoever made the car
understood calculus, as you
do, the speedometer--
now that I think of it,
speedometers don't have a
below zero.
They should have. And trip
meters should go backwards.
I mean, that movie was just made
for a calculus person.
Maybe I'm remembering more.
I think it didn't work
or something.
And the kid got mad and kicked
the car, and it fell off the
lift, went through
the glass window.
Anyway, calculus would
have saved him if
only the car had been--
or the meters in the car had
been made correctly.
All right, that's one pair.
That's our first real pair in
which the speed changes.
OK.
I thought in this first video,
later, even today, I'll get to
a case where we have formulas.
That's what calculus
moves into.
When f of t is given by some
formula, well, here it's given
by a formula: s times t.
A simple formula.
And then, knowing that, we
know that the speed is s.
Later, we got more functions.
But let me take an example, just
because these pairs of
functions are everywhere.
What could I take?
Maybe height of a person.
Height of a person.
OK, so this is now another
example, just to get practice
in the relation between the
height of a person and the
rate of change of the height.
So this is the height.
Maybe I'll call it y.
Let me write height
of a person.
And what is this going to be?
What is function two?
Well, slope doesn't
seem quite right.
The point about function two is
it tells how fast function
one changes.
It's the rate of change
of the height.
It's the rate of change.
So let me call it s, and it'll
be the rate of change.
Good if I use those words.
Yeah, so I want to think
just how we grow, a
typical person growing.
In fact, as I wrote this on
the board, I thought of
another pair.
Can I just say it in words, this
other pair, and then I'll
come back to this one?
Here's another pair.
This could be money in a bank.
Wealth sounds better.
Let's call it wealth.
That's zippier.
And then what is this one?
If this is your wealth,
your total
assets, what's your worth?
This would be the rate
of change, how
quickly you're saving.
s could be for saving.
Or if you're down here, s
is for spending, right?
If s is positive, that means
you're wealth is increasing,
you're saving.
Negative s means you're
spending, and your wealth goes
whatever, maybe--
I hope-- up.
Height is mostly up, right?
So let me come back to
height of a person.
Now, where--
oh, and this is time in years.
This is t in years, and
this, too, of course.
Actually, I realize
you started at t
equals zero: birth.
You do start at a certain--
actually, what do I know?
You don't say tall.
You say long.
But then as soon as you can
stand up, it's tall,
so let's say tall.
Shall we guessed 20 inches?
If that's way off, I apologize
to everybody.
Let me just say 20, 20 inches.
OK, at year zero.
OK, and then presumably
you grow.
OK, so you grow a little.
What are we headed for?
About 60, 70 inches
or something.
Anyway, you grow.
Let's say that's 10 years old
and here is 20 years old.
OK, so you grow.
Maybe you grow faster
than that.
Let's say you're a healthy
person here.
OK, up you grow.
And then at about maybe age 12
or 13, there's a growth spurt.
And maybe the point is, how do
we see that growth spurt on
the two graphs?
Differently, but it's the
same growth spurt.
OK, so here your height
suddenly jumps up.
Boy, yeah, you catch
up with everybody.
And then at about 12 or 13 well,
then unfortunately, it
doesn't do that forever, and
it kind of levels off here.
It levels off, and actually
you don't
grow a whole lot more.
In fact, I think when you get
to about-- oh, I don't know.
Whatever.
We won't discuss this point.
I say when you get too old,
you probably lose some.
Let's not emphasize that.
OK, so here is the--
now, what's happening
over here?
Well, it's the slope
of that graph.
So the slope might be--
this is time zero, but you're
growing right away.
The s graph, the rate
of growth graph,
doesn't start at zero.
It starts how fast you're
growing, whatever you're
growing, whatever
that slope is.
It's fantastic that when we
draw graphs of things, the
word "slope" is suddenly
the right word.
OK, so you're growing, maybe
at a pretty good rate here.
And let me mark out
10 and 20 years.
And OK, you're doing well,
you're coming along here, and
then the growth spurt.
OK, so then suddenly, your
rate of growth takes off.
But it doesn't stay
that way, right?
Your rate of growth levels
off, in fact,
levels way off, levels--
you'll come down to here, and
you probably don't grow a lot.
Do you see the two?
This was the growth curve.
This was the fast growth.
But then it stopped.
Up here, it slowed down.
Here, it dropped.
And oh, if we allow for this
person who lived too long,
height actually drops.
OK, there is an example
in which I don't--
also I'm sure people have
devised approximate formulas
for average growth rates,
but you see, I'm not--
it's the idea of the relation
between function one and
function two that
I'm emphasizing.
Now, my last example, let me
take one more example, one
more example for this
first lecture.
So let me take a case in
which the speed is--
so here will be my two--
let's use speed.
Let's use this as distance.
This is distance again,
and graph two, as
always, will be speed.
And I'm going to take
a case in which
it's given by a formula.
I'm going to let the speed
be increasing steadily.
OK, so my speed graph this time
is going to go up at a
constant rate.
So this is the speed s.
This is the time t.
So this would be s equals--
s is proportional to t.
That's where you get
a straight line.
s is let's say a times t.
That a, a physicist, if we were
physicists, would say
acceleration.
You're accelerating.
You're keeping your foot on the
gas, steadily speeding up,
and so then s is proportional
to t.
Now, think about the distance.
What's happening
with distance?
If this is accelerating, you're
going faster and faster.
You're covering more and more
speed, more and more distance,
more and more quickly.
If this is slope, the
slope is increasing.
Look, the graph--
let's start the trip
meter at zero.
So you started with
a speed of zero.
You were not really increasing
distance until you got
slightly beyond zero,
and then it
slightly started to increase.
But then it increases faster
and faster, right?
It never gets infinitely fast,
but it keeps going upwards.
And the calculus question
would be
can we give a formula--
an equation--
for the distance?
Because in this case, I guess
I started with function two,
and therefore, it's function
one that I want to look at.
It's always pairs
of functions.
OK, now, let's think where this
would actually happen.
If we were leaning over the
Tower of Pisa or whatever,
like Galileo, and drop
something, or even just drop
something anywhere, that
would be-- we drop it.
At the beginning, it has no
speed, but of course,
instantly it picks up speed.
The a would have something to
do with the gravitational
constant for the Earth,
whatever, and then maybe--
yeah.
And what would be
the distance?
OK, now can I just mention a
small miracle of calculus?
A small miracle.
I'm going this direction now
from speed to distance so I'm
doing integral calculus.
And we'll get to that later.
In the first lectures,
we're almost always
going from one to two.
But here is a neat fact about
going from two to one, that if
this is the time t, then, of
course, this height here will
be a times t.
And the amazing fact is that
this graph tells you the area
under this one.
Graph one--
function one--
tells you the area under
the graph two.
And in this example with a nice
constant acceleration,
steady increase in speed,
we know this.
This is a triangle.
It has a base of t.
It has a height of at.
And the area of a triangle, of
course, the area, and my point
is that the area is
function one--
amazing; that's just
terrific--
will be-- the area of this
triangle here is 1/2 of the
base times the height.
That's the area, and calculus
will tell us
that's function one.
So this function is 1/2
of a times t squared.
So there is a function one,
and here is df dt.
If I go back to the first
letters that I mentioned, if
this is my function f, then
this is my function that--
and notice what kind
of a curve that is.
Do you recognize that
with a square?
That tells me it's a parabola,
a famous and important curve.
And, of course, it's important
because it
has such a neat formula.
OK, so we have found
the function one.
We've recovered the information
in that lost black
box, the distance box, the trip
meter, from what we did
find in black box two, the
speed, the record of speed.
And notice, I'm using the
speed all the way
from here to here.
The speed kind of tells me how
the distance is piled up.
The distance is kind of a
running total, where the speed
at that moment is an
instant thing.
Oh, we have to do that
in future lectures.
The difference between a
running total of total
distance covered and a speed
that's telling me at a moment,
at an instant how distance
is changing.
The slope at this very point
t, that slope is
this height, is at.
OK, so there you have
the first--
well, I'll say the second.
The first pair of calculus
was this one.
f equals st, and it's
derivative was s.
Our second pair is f is this,
and will you allow
me to write df dt?
If f is 1/2 of at squared,
then df dt is at.
You'll see this rule again.
The power two dropped
to a power one.
But the two multiplied the thing
so it canceled the 1/2
and just left the a.
OK, that's a start on the
highlights of calculus.
Thanks.
NARRATOR: This has been
a production of MIT
OpenCourseWare and
Gilbert Strang.
Funding for this video was
provided by the Lord
Foundation.
To help OCW continue to provide
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