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PROFESSOR: Hi, I'm Gilbert
Strang, and this is the very

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first in a series of videos
about highlights of calculus.

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I'm doing these just because
I hope they'll be helpful.

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It seems to me so easy to be
lost in the big calculus

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textbooks and the many, many
problems and in the details.

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But do you see the
big picture?

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Well, I hope this will help.

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For me, calculus is about the

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relation between two functions.

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And one example for those two
functions, one good example,

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is function 1, the distance,
distance traveled, what you

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see on a trip meter in a car.

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And function 2, the one that
goes with distance, is speed,

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how quickly you're going, how
fast you're traveling.

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So that's one pair
of functions.

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Let me give another pair.

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I could get more and more, but
I think if we get these two

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pairs, we can move forward.

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So in this second pair,
height is function 1,

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how high you've climbed.

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If it's a graph, how far the
graph goes above the axis.

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Up, in other words.

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So that's height, and then
the other one tells you

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how fast you climb.

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The height tells how
far you climbed.

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It could be a mountain.

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And then the slope tells you how
quickly you're climbing at

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each point.

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Are you going nearly
straight up?

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Flat?

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Possibly down?

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So distance and speed, height
and slope will serve as good

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examples to start with.

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And let me give you some
letters, some algebra letters

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that you might use.

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Distance, maybe I would
call that f of t.

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So f for how far or for
function, and the idea is that

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t is the input.

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It's the time when you're
asking for the distance.

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The output is the distance.

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Or in the case of height,
maybe y of x would

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be the right one.

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x is how far you go across.

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That's the input.

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And at each x, you have an
output y how far up?

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So f is telling you how far.

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y is telling you the
height of a graph.

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That's function 1, two examples
of function 1.

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Now, what about slope?

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Well, luckily, speed and slope
start with the same letter, so

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I'll often use s for the speed
or the slope for this second--

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oh, it even stands for
a second function.

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But let me tell you
also the right--

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the official--

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letters that make the connection
between function 2

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and function 1.

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If my function is a function of
time, the distance, how far

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I go, then the speed is--

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the right letters are df dt.

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Everybody uses those letters.

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So let me say again how
to pronounce: df dt.

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And Leibniz came up with
that notation, and

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he just got it right.

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And what would this one be?

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Well, corresponding to this, it
looks the same, or dy dx.

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Again, I'll just repeat how
to say that: dy dx.

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And that is the slope, and we
have to understand what those

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symbols mean.

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Right now, I'm just writing
them down as symbols.

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May I begin with the most
important and the simplest

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example of all?

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Let me take that case.

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OK, so the key example here,
the one to get completely

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straight is the case
of constant

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speed, constant slope.

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I'll just graph that.

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So here I'm go to graph.

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Shall I make it the speed?

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Yeah, let's say speed.

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So time is going
along that way.

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Speed is up this way.

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And I'm going to say in this
first example that the speed

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is the same.

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We're traveling at the
constant speed

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of let's say 40.

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So it stays at the
height of 40.

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Oh, properly, I should add units
like miles per hour or

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kilometers per hour or meters
per second or whatever.

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For now, I'll just write 40.

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OK, now if we're traveling at a
speed of 40 miles per hour,

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what's the distance?

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Well, let me start with the
trip meter at zero.

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so this is time again,
and now this is

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going to be the distance.

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After one hour, my
distance is 40.

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So if I mark t equal to
1, I've reached 40.

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That's height of 40.

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At t equal to 2, I've
reached 80.

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At t equal to 1/2, half an
hour, I've reached 20.

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Those points lie on a line.

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The graph of distance covered
when you're just traveling at

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a steady rate, constant rate,
constant speed is just a

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straight line.

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And now I can make
the connection.

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I've been speaking here about
distance and speed.

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But now let me think of
this as the height--

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40 is that height.

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80 is that height--

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and ask about slope.

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What is slope?

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So let's just remember what's
the connection here.

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What's the slope if that's the
distance if I look at my trip

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meter and I know I'm traveling
along at that constant speed,

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how do I find that speed?

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Well, slope, it's the distance
up, which would be 40 after

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one hour, divided
by the distance

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across, 40/1, or 80/2.

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Doesn't matter, because we're
traveling at constant speed,

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so the slope, which is up,
over, across is 40/1,

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80/2, 20 over 1/2.

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I'll put 80/2 as one
example: 40.

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Oh, let me do it--

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that's arithmetic.

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Let me do it with algebra.

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We don't need calculus
yet, by the way.

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Calculus is coming
pretty quickly.

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This is the step we can take.

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Because the speed is constant,
we can just divide the

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distance by the time to find--

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and this slope, let me
right speed also.

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Up, over, across, distance
over time, f/t,

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that gives us s.

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This is s.

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OK, what about--

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calculus goes both ways.

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We can go both ways here.

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We already have practically.

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Here I went in the direction
from 1 to 2.

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Now, I want to go in
the direction--

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suppose I know the speed.

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How do I recover the distance?

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If I know my speed is 40 and I
know I started at zero, what's

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my distance?

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Distance or height,
either one.

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So these are like both.

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Now, I'm just going
the other way.

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Well, you see how.

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How do I find f?

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It's s times t, right?

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Your algebra automatically says
if you see a t there, you

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can put it there.

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So it's s times t.

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It's a straight line.

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s times t, s times
x, y equal sx.

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Let me put another--

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the same idea with
my y, x letters.

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It's that line.

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In other words, if that one
is constant, this one is a

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straight line.

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OK, straightforward, but
very, very fundamental.

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In fact, can I call your
attention to something a

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little more?

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Suppose I measured between
time 2 and time 1.

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So I'm looking between time 2
and time 1, and I look how far

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I went in that time.

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But what I'm trying is--

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I'm going to put in another
little symbol because it's

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going to be really
worth knowing.

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It's really the change in f
divided by the change in t.

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I use that letter delta
to indicate

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a difference between--

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the difference between time 2
and time 1 was 1, and the

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difference between height
2 and height 1 was 40.

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You see, I'm looking at
this little piece.

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And, of course, the
slope is still 40.

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It's still the slope
of that line.

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Yeah, so that really what I'm
measuring in speed there, I

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don't always have to be starting
at t equals 0, and I

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don't always have to be starting
at f equals 0.

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Oh, let me draw that.

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Suppose I started
at f equals 40.

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My trip meter happened
to start at 40.

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After an hour, I'd
be up to 80.

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After another hour,
I'd be up to 120.

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Do you see that this starting
the trip meter, who cares

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where the trip meter started?

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It's the change in the trip
meter that tells how

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long the trip was.

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Clear.

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OK, so that's that example.

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We come back to it because it's
the basic one where the

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speed is constant.

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And even if now I have to move
to a changing speed, you have

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to let me bring calculus
into these lectures.

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OK, I'm going to draw another
picture, and you

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tell me about the--

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yeah, let me draw function 1,
another example of function 1.

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So again I have time.

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I have distance.

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I'm going to start at zero, but
I'm not going to keep the

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speed constant.

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I'm going to start out at
a good speed, but I'm

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going to slow down.

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Do you see me slowing
down there?

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I don't mean slowing down
with the chalk.

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I mean slowing down
with slope.

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The slope started out steep.

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By here, by that point,
the slope was zero.

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What was the car doing here?

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The car is certainly moving
forward because the distance

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is increasing.

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Here it's increasing faster.

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Here it's increasing barely.

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In other words, we're putting
on the brakes.

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The car is slowing down.

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We're coming to a red light.

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In fact, there is the red light
right at that time.

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Now, just stay with it to think
what would the speed

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look like for this problem?

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If that's a picture of
the function, just

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let's get some idea.

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I'm not going to have
a formula yet.

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I'm not putting in
all the details.

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Well, actually, I don't plan to
put in all the details of

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calculus of every possible
step we might take.

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It's the important ones I'm
hoping to show you and I'm

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hoping for you to see that
they are important.

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OK, what is important?

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Roughly, what does the
graph looks like?

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Well, the speed--

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the slope--

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started out somewhere
up there.

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Yeah, it started out at a good
speed and slowed down.

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And by this point, ha!

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Let's mark that time
here on that graph.

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Do you see what is the
speed at that moment?

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The speed at that
moment is zero.

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The car has stopped.

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The speed is decreasing.

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Let me make it decrease,
decrease, decrease, decrease,

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and at that moment, the speed
is zero right there.

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That's that point.

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See, two different pictures,
two different functions.

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but same information.

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So calculus has the job of given
one of those functions,

261
00:16:31,190 --> 00:16:32,300
find the other one.

262
00:16:32,300 --> 00:16:34,690
Given this function,
find that one.

263
00:16:34,690 --> 00:16:36,380
This way is called--

264
00:16:36,380 --> 00:16:41,040
from function one to function
two, that's called

265
00:16:41,040 --> 00:16:43,690
differential calculus.

266
00:16:43,690 --> 00:16:46,230
Big, impressive word anyway.

267
00:16:46,230 --> 00:16:51,800
That's function one to two,
finding the speed.

268
00:16:51,800 --> 00:16:57,860
Going the other direction is
called integral calculus.

269
00:16:57,860 --> 00:17:03,500
The step is called integration
when you take the speed over

270
00:17:03,500 --> 00:17:09,510
that period of time, and you
recover the distance.

271
00:17:09,510 --> 00:17:12,569
So it's differential calculus
in one direction, integral

272
00:17:12,569 --> 00:17:13,930
calculus in the other.

273
00:17:13,930 --> 00:17:15,880
Now, here's a question.

274
00:17:15,880 --> 00:17:18,940

275
00:17:18,940 --> 00:17:22,230
Let me continue that curve
a little longer.

276
00:17:22,230 --> 00:17:24,609
I got it to the red light.

277
00:17:24,609 --> 00:17:32,040
Now imagine that the distance
starts going

278
00:17:32,040 --> 00:17:36,160
down from that point.

279
00:17:36,160 --> 00:17:39,080
What's happening?

280
00:17:39,080 --> 00:17:42,690
The distance is decreasing.

281
00:17:42,690 --> 00:17:45,360
The car is going backwards.

282
00:17:45,360 --> 00:17:48,020
It's going in reverse.

283
00:17:48,020 --> 00:17:51,700
The speed, what's the speed?

284
00:17:51,700 --> 00:17:53,860
Negative.

285
00:17:53,860 --> 00:18:01,430
The speed, because distance is
going from higher to lower,

286
00:18:01,430 --> 00:18:03,320
that counts for negative
speed.

287
00:18:03,320 --> 00:18:08,245
The speed curve would
be going down here.

288
00:18:08,245 --> 00:18:13,730
Do you see that that's a not
brilliantly drawn picture, but

289
00:18:13,730 --> 00:18:15,230
you're seeing the--

290
00:18:15,230 --> 00:18:19,240
that's the farthest it went.

291
00:18:19,240 --> 00:18:24,660
Then the car started backwards,
and the speed curve

292
00:18:24,660 --> 00:18:27,820
reflected that by going
below zero.

293
00:18:27,820 --> 00:18:30,990
You see, two different curves,
but same information.

294
00:18:30,990 --> 00:18:35,950

295
00:18:35,950 --> 00:18:38,290
I'm remembering an old movie.

296
00:18:38,290 --> 00:18:41,250
I don't know if you saw an
old B movie called Ferris

297
00:18:41,250 --> 00:18:43,920
Bueller's Day Off.

298
00:18:43,920 --> 00:18:45,440
Did you see that?

299
00:18:45,440 --> 00:18:49,080
So the kid had borrowed
his father's--

300
00:18:49,080 --> 00:18:55,570
not borrowed, but lifted his
father's good car and drove it

301
00:18:55,570 --> 00:19:00,740
a lot like so and put
on a lot of mileage.

302
00:19:00,740 --> 00:19:03,260
The trip meter was way up, and
he knew his father was going

303
00:19:03,260 --> 00:19:05,390
to notice this.

304
00:19:05,390 --> 00:19:12,620
So he had the idea to put the
car up on a lift, put it in

305
00:19:12,620 --> 00:19:16,820
reverse, and go for a
while, and the trip

306
00:19:16,820 --> 00:19:18,160
meter would go backwards.

307
00:19:18,160 --> 00:19:22,940

308
00:19:22,940 --> 00:19:24,755
I don't know if trip meters
do go backwards.

309
00:19:24,755 --> 00:19:29,380

310
00:19:29,380 --> 00:19:36,000
It's kind of tough to watch them
while going in reverse.

311
00:19:36,000 --> 00:19:43,780
But if whoever made the car
understood calculus, as you

312
00:19:43,780 --> 00:19:47,520
do, the speedometer--

313
00:19:47,520 --> 00:19:49,680
now that I think of it,
speedometers don't have a

314
00:19:49,680 --> 00:19:50,450
below zero.

315
00:19:50,450 --> 00:19:55,540
They should have. And trip
meters should go backwards.

316
00:19:55,540 --> 00:19:59,690
I mean, that movie was just made
for a calculus person.

317
00:19:59,690 --> 00:20:05,830

318
00:20:05,830 --> 00:20:07,950
Maybe I'm remembering more.

319
00:20:07,950 --> 00:20:10,330
I think it didn't work
or something.

320
00:20:10,330 --> 00:20:15,030
And the kid got mad and kicked
the car, and it fell off the

321
00:20:15,030 --> 00:20:20,190
lift, went through
the glass window.

322
00:20:20,190 --> 00:20:23,370
Anyway, calculus would
have saved him if

323
00:20:23,370 --> 00:20:27,480
only the car had been--

324
00:20:27,480 --> 00:20:31,500
or the meters in the car had
been made correctly.

325
00:20:31,500 --> 00:20:36,320
All right, that's one pair.

326
00:20:36,320 --> 00:20:42,650
That's our first real pair in
which the speed changes.

327
00:20:42,650 --> 00:20:44,580
OK.

328
00:20:44,580 --> 00:20:54,460
I thought in this first video,
later, even today, I'll get to

329
00:20:54,460 --> 00:20:59,340
a case where we have formulas.

330
00:20:59,340 --> 00:21:03,200
That's what calculus
moves into.

331
00:21:03,200 --> 00:21:08,420
When f of t is given by some
formula, well, here it's given

332
00:21:08,420 --> 00:21:10,210
by a formula: s times t.

333
00:21:10,210 --> 00:21:13,110

334
00:21:13,110 --> 00:21:15,020
A simple formula.

335
00:21:15,020 --> 00:21:20,350
And then, knowing that, we
know that the speed is s.

336
00:21:20,350 --> 00:21:23,340
Later, we got more functions.

337
00:21:23,340 --> 00:21:29,030
But let me take an example, just
because these pairs of

338
00:21:29,030 --> 00:21:32,030
functions are everywhere.

339
00:21:32,030 --> 00:21:33,970
What could I take?

340
00:21:33,970 --> 00:21:38,920
Maybe height of a person.

341
00:21:38,920 --> 00:21:41,060
Height of a person.

342
00:21:41,060 --> 00:21:47,430
OK, so this is now another
example, just to get practice

343
00:21:47,430 --> 00:21:51,940
in the relation between the
height of a person and the

344
00:21:51,940 --> 00:21:54,470
rate of change of the height.

345
00:21:54,470 --> 00:21:57,120
So this is the height.

346
00:21:57,120 --> 00:21:58,990
Maybe I'll call it y.

347
00:21:58,990 --> 00:22:04,040
Let me write height
of a person.

348
00:22:04,040 --> 00:22:06,840
And what is this going to be?

349
00:22:06,840 --> 00:22:09,950
What is function two?

350
00:22:09,950 --> 00:22:12,510
Well, slope doesn't
seem quite right.

351
00:22:12,510 --> 00:22:18,120
The point about function two is
it tells how fast function

352
00:22:18,120 --> 00:22:20,000
one changes.

353
00:22:20,000 --> 00:22:25,720
It's the rate of change
of the height.

354
00:22:25,720 --> 00:22:28,610
It's the rate of change.

355
00:22:28,610 --> 00:22:34,890
So let me call it s, and it'll
be the rate of change.

356
00:22:34,890 --> 00:22:36,620
Good if I use those words.

357
00:22:36,620 --> 00:22:43,738

358
00:22:43,738 --> 00:22:47,120
Yeah, so I want to think
just how we grow, a

359
00:22:47,120 --> 00:22:49,540
typical person growing.

360
00:22:49,540 --> 00:22:56,910
In fact, as I wrote this on
the board, I thought of

361
00:22:56,910 --> 00:22:58,060
another pair.

362
00:22:58,060 --> 00:23:01,310
Can I just say it in words, this
other pair, and then I'll

363
00:23:01,310 --> 00:23:03,330
come back to this one?

364
00:23:03,330 --> 00:23:05,430
Here's another pair.

365
00:23:05,430 --> 00:23:11,880
This could be money in a bank.

366
00:23:11,880 --> 00:23:15,000

367
00:23:15,000 --> 00:23:16,360
Wealth sounds better.

368
00:23:16,360 --> 00:23:17,610
Let's call it wealth.

369
00:23:17,610 --> 00:23:19,990

370
00:23:19,990 --> 00:23:23,340
That's zippier.

371
00:23:23,340 --> 00:23:25,490
And then what is this one?

372
00:23:25,490 --> 00:23:28,840
If this is your wealth,
your total

373
00:23:28,840 --> 00:23:32,890
assets, what's your worth?

374
00:23:32,890 --> 00:23:38,730
This would be the rate
of change, how

375
00:23:38,730 --> 00:23:41,510
quickly you're saving.

376
00:23:41,510 --> 00:23:42,970
s could be for saving.

377
00:23:42,970 --> 00:23:50,450
Or if you're down here, s
is for spending, right?

378
00:23:50,450 --> 00:23:54,550
If s is positive, that means
you're wealth is increasing,

379
00:23:54,550 --> 00:23:55,800
you're saving.

380
00:23:55,800 --> 00:23:58,010

381
00:23:58,010 --> 00:24:01,940
Negative s means you're
spending, and your wealth goes

382
00:24:01,940 --> 00:24:07,210
whatever, maybe--

383
00:24:07,210 --> 00:24:08,760
I hope-- up.

384
00:24:08,760 --> 00:24:12,470
Height is mostly up, right?

385
00:24:12,470 --> 00:24:14,270
So let me come back to
height of a person.

386
00:24:14,270 --> 00:24:17,010

387
00:24:17,010 --> 00:24:18,360
Now, where--

388
00:24:18,360 --> 00:24:22,780
oh, and this is time in years.

389
00:24:22,780 --> 00:24:28,340
This is t in years, and
this, too, of course.

390
00:24:28,340 --> 00:24:34,620

391
00:24:34,620 --> 00:24:40,340
Actually, I realize
you started at t

392
00:24:40,340 --> 00:24:43,120
equals zero: birth.

393
00:24:43,120 --> 00:24:44,835
You do start at a certain--

394
00:24:44,835 --> 00:24:47,790

395
00:24:47,790 --> 00:24:49,330
actually, what do I know?

396
00:24:49,330 --> 00:24:52,640

397
00:24:52,640 --> 00:24:53,680
You don't say tall.

398
00:24:53,680 --> 00:24:54,530
You say long.

399
00:24:54,530 --> 00:24:56,930
But then as soon as you can
stand up, it's tall,

400
00:24:56,930 --> 00:24:58,180
so let's say tall.

401
00:24:58,180 --> 00:25:02,230

402
00:25:02,230 --> 00:25:03,780
Shall we guessed 20 inches?

403
00:25:03,780 --> 00:25:06,720
If that's way off, I apologize
to everybody.

404
00:25:06,720 --> 00:25:11,060
Let me just say 20, 20 inches.

405
00:25:11,060 --> 00:25:13,270
OK, at year zero.

406
00:25:13,270 --> 00:25:17,270
OK, and then presumably
you grow.

407
00:25:17,270 --> 00:25:19,680
OK, so you grow a little.

408
00:25:19,680 --> 00:25:20,790
What are we headed for?

409
00:25:20,790 --> 00:25:24,450
About 60, 70 inches
or something.

410
00:25:24,450 --> 00:25:25,900
Anyway, you grow.

411
00:25:25,900 --> 00:25:30,230

412
00:25:30,230 --> 00:25:34,230
Let's say that's 10 years old
and here is 20 years old.

413
00:25:34,230 --> 00:25:35,780
OK, so you grow.

414
00:25:35,780 --> 00:25:37,700
Maybe you grow faster
than that.

415
00:25:37,700 --> 00:25:40,990
Let's say you're a healthy
person here.

416
00:25:40,990 --> 00:25:43,970
OK, up you grow.

417
00:25:43,970 --> 00:25:55,200
And then at about maybe age 12
or 13, there's a growth spurt.

418
00:25:55,200 --> 00:25:59,030
And maybe the point is, how do
we see that growth spurt on

419
00:25:59,030 --> 00:26:00,210
the two graphs?

420
00:26:00,210 --> 00:26:03,320
Differently, but it's the
same growth spurt.

421
00:26:03,320 --> 00:26:07,340
OK, so here your height
suddenly jumps up.

422
00:26:07,340 --> 00:26:09,330
Boy, yeah, you catch
up with everybody.

423
00:26:09,330 --> 00:26:12,070

424
00:26:12,070 --> 00:26:16,500
And then at about 12 or 13 well,
then unfortunately, it

425
00:26:16,500 --> 00:26:22,190
doesn't do that forever, and
it kind of levels off here.

426
00:26:22,190 --> 00:26:24,960
It levels off, and actually
you don't

427
00:26:24,960 --> 00:26:28,300
grow a whole lot more.

428
00:26:28,300 --> 00:26:34,930
In fact, I think when you get
to about-- oh, I don't know.

429
00:26:34,930 --> 00:26:35,380
Whatever.

430
00:26:35,380 --> 00:26:37,200
We won't discuss this point.

431
00:26:37,200 --> 00:26:43,900
I say when you get too old,
you probably lose some.

432
00:26:43,900 --> 00:26:47,100
Let's not emphasize that.

433
00:26:47,100 --> 00:26:51,230
OK, so here is the--

434
00:26:51,230 --> 00:26:55,460
now, what's happening
over here?

435
00:26:55,460 --> 00:26:58,950
Well, it's the slope
of that graph.

436
00:26:58,950 --> 00:27:01,080
So the slope might be--

437
00:27:01,080 --> 00:27:07,520
this is time zero, but you're
growing right away.

438
00:27:07,520 --> 00:27:12,230
The s graph, the rate
of growth graph,

439
00:27:12,230 --> 00:27:13,990
doesn't start at zero.

440
00:27:13,990 --> 00:27:17,280
It starts how fast you're
growing, whatever you're

441
00:27:17,280 --> 00:27:21,040
growing, whatever
that slope is.

442
00:27:21,040 --> 00:27:24,710
It's fantastic that when we
draw graphs of things, the

443
00:27:24,710 --> 00:27:27,870
word "slope" is suddenly
the right word.

444
00:27:27,870 --> 00:27:32,370
OK, so you're growing, maybe
at a pretty good rate here.

445
00:27:32,370 --> 00:27:36,960
And let me mark out
10 and 20 years.

446
00:27:36,960 --> 00:27:43,500
And OK, you're doing well,
you're coming along here, and

447
00:27:43,500 --> 00:27:46,050
then the growth spurt.

448
00:27:46,050 --> 00:27:49,295
OK, so then suddenly, your
rate of growth takes off.

449
00:27:49,295 --> 00:27:52,190

450
00:27:52,190 --> 00:27:58,090
But it doesn't stay
that way, right?

451
00:27:58,090 --> 00:28:01,070
Your rate of growth levels
off, in fact,

452
00:28:01,070 --> 00:28:03,990
levels way off, levels--

453
00:28:03,990 --> 00:28:09,650
you'll come down to here, and
you probably don't grow a lot.

454
00:28:09,650 --> 00:28:11,190
Do you see the two?

455
00:28:11,190 --> 00:28:15,000
This was the growth curve.

456
00:28:15,000 --> 00:28:16,940
This was the fast growth.

457
00:28:16,940 --> 00:28:19,090
But then it stopped.

458
00:28:19,090 --> 00:28:21,160
Up here, it slowed down.

459
00:28:21,160 --> 00:28:23,090
Here, it dropped.

460
00:28:23,090 --> 00:28:33,120
And oh, if we allow for this
person who lived too long,

461
00:28:33,120 --> 00:28:36,540
height actually drops.

462
00:28:36,540 --> 00:28:41,640
OK, there is an example
in which I don't--

463
00:28:41,640 --> 00:28:48,550
also I'm sure people have
devised approximate formulas

464
00:28:48,550 --> 00:28:53,480
for average growth rates,
but you see, I'm not--

465
00:28:53,480 --> 00:28:57,520
it's the idea of the relation
between function one and

466
00:28:57,520 --> 00:29:00,770
function two that
I'm emphasizing.

467
00:29:00,770 --> 00:29:11,040
Now, my last example, let me
take one more example, one

468
00:29:11,040 --> 00:29:12,900
more example for this
first lecture.

469
00:29:12,900 --> 00:29:16,050

470
00:29:16,050 --> 00:29:21,990
So let me take a case in
which the speed is--

471
00:29:21,990 --> 00:29:23,483
so here will be my two--

472
00:29:23,483 --> 00:29:29,020

473
00:29:29,020 --> 00:29:30,210
let's use speed.

474
00:29:30,210 --> 00:29:34,230
Let's use this as distance.

475
00:29:34,230 --> 00:29:41,840
This is distance again,
and graph two, as

476
00:29:41,840 --> 00:29:44,045
always, will be speed.

477
00:29:44,045 --> 00:29:49,670

478
00:29:49,670 --> 00:29:53,720
And I'm going to take
a case in which

479
00:29:53,720 --> 00:29:55,630
it's given by a formula.

480
00:29:55,630 --> 00:30:02,030
I'm going to let the speed
be increasing steadily.

481
00:30:02,030 --> 00:30:07,480
OK, so my speed graph this time
is going to go up at a

482
00:30:07,480 --> 00:30:09,810
constant rate.

483
00:30:09,810 --> 00:30:12,690
So this is the speed s.

484
00:30:12,690 --> 00:30:14,780
This is the time t.

485
00:30:14,780 --> 00:30:18,530
So this would be s equals--

486
00:30:18,530 --> 00:30:20,070
s is proportional to t.

487
00:30:20,070 --> 00:30:22,180
That's where you get
a straight line.

488
00:30:22,180 --> 00:30:26,460
s is let's say a times t.

489
00:30:26,460 --> 00:30:30,690
That a, a physicist, if we were
physicists, would say

490
00:30:30,690 --> 00:30:32,570
acceleration.

491
00:30:32,570 --> 00:30:34,260
You're accelerating.

492
00:30:34,260 --> 00:30:40,560
You're keeping your foot on the
gas, steadily speeding up,

493
00:30:40,560 --> 00:30:47,260
and so then s is proportional
to t.

494
00:30:47,260 --> 00:30:54,580
Now, think about the distance.

495
00:30:54,580 --> 00:30:56,600
What's happening
with distance?

496
00:30:56,600 --> 00:31:00,540
If this is accelerating, you're

497
00:31:00,540 --> 00:31:01,780
going faster and faster.

498
00:31:01,780 --> 00:31:05,210
You're covering more and more
speed, more and more distance,

499
00:31:05,210 --> 00:31:08,420
more and more quickly.

500
00:31:08,420 --> 00:31:11,770
If this is slope, the
slope is increasing.

501
00:31:11,770 --> 00:31:12,870
Look, the graph--

502
00:31:12,870 --> 00:31:15,820
let's start the trip
meter at zero.

503
00:31:15,820 --> 00:31:17,920
So you started with
a speed of zero.

504
00:31:17,920 --> 00:31:24,120
You were not really increasing
distance until you got

505
00:31:24,120 --> 00:31:25,820
slightly beyond zero,
and then it

506
00:31:25,820 --> 00:31:27,920
slightly started to increase.

507
00:31:27,920 --> 00:31:36,225
But then it increases faster
and faster, right?

508
00:31:36,225 --> 00:31:47,730
It never gets infinitely fast,
but it keeps going upwards.

509
00:31:47,730 --> 00:31:50,780
And the calculus question
would be

510
00:31:50,780 --> 00:31:52,550
can we give a formula--

511
00:31:52,550 --> 00:31:53,810
an equation--

512
00:31:53,810 --> 00:31:55,060
for the distance?

513
00:31:55,060 --> 00:31:57,820

514
00:31:57,820 --> 00:32:01,970
Because in this case, I guess
I started with function two,

515
00:32:01,970 --> 00:32:05,710
and therefore, it's function
one that I want to look at.

516
00:32:05,710 --> 00:32:08,550
It's always pairs
of functions.

517
00:32:08,550 --> 00:32:17,260
OK, now, let's think where this
would actually happen.

518
00:32:17,260 --> 00:32:22,040
If we were leaning over the
Tower of Pisa or whatever,

519
00:32:22,040 --> 00:32:25,220
like Galileo, and drop
something, or even just drop

520
00:32:25,220 --> 00:32:31,700
something anywhere, that
would be-- we drop it.

521
00:32:31,700 --> 00:32:35,310
At the beginning, it has no
speed, but of course,

522
00:32:35,310 --> 00:32:39,310
instantly it picks up speed.

523
00:32:39,310 --> 00:32:42,580
The a would have something to
do with the gravitational

524
00:32:42,580 --> 00:32:45,820
constant for the Earth,
whatever, and then maybe--

525
00:32:45,820 --> 00:32:47,880
yeah.

526
00:32:47,880 --> 00:32:50,580
And what would be
the distance?

527
00:32:50,580 --> 00:32:56,810
OK, now can I just mention a
small miracle of calculus?

528
00:32:56,810 --> 00:32:59,810
A small miracle.

529
00:32:59,810 --> 00:33:03,270
I'm going this direction now
from speed to distance so I'm

530
00:33:03,270 --> 00:33:05,600
doing integral calculus.

531
00:33:05,600 --> 00:33:11,730
And we'll get to that later.

532
00:33:11,730 --> 00:33:13,500
In the first lectures,
we're almost always

533
00:33:13,500 --> 00:33:15,320
going from one to two.

534
00:33:15,320 --> 00:33:20,190
But here is a neat fact about
going from two to one, that if

535
00:33:20,190 --> 00:33:26,580
this is the time t, then, of
course, this height here will

536
00:33:26,580 --> 00:33:31,600
be a times t.

537
00:33:31,600 --> 00:33:40,710
And the amazing fact is that
this graph tells you the area

538
00:33:40,710 --> 00:33:42,830
under this one.

539
00:33:42,830 --> 00:33:44,550
Graph one--

540
00:33:44,550 --> 00:33:45,630
function one--

541
00:33:45,630 --> 00:33:50,970
tells you the area under
the graph two.

542
00:33:50,970 --> 00:33:56,640
And in this example with a nice
constant acceleration,

543
00:33:56,640 --> 00:34:00,550
steady increase in speed,
we know this.

544
00:34:00,550 --> 00:34:02,000
This is a triangle.

545
00:34:02,000 --> 00:34:03,890
It has a base of t.

546
00:34:03,890 --> 00:34:06,470
It has a height of at.

547
00:34:06,470 --> 00:34:12,610
And the area of a triangle, of
course, the area, and my point

548
00:34:12,610 --> 00:34:16,050
is that the area is
function one--

549
00:34:16,050 --> 00:34:22,130
amazing; that's just
terrific--

550
00:34:22,130 --> 00:34:29,090
will be-- the area of this
triangle here is 1/2 of the

551
00:34:29,090 --> 00:34:31,560
base times the height.

552
00:34:31,560 --> 00:34:36,469

553
00:34:36,469 --> 00:34:39,310
That's the area, and calculus
will tell us

554
00:34:39,310 --> 00:34:41,360
that's function one.

555
00:34:41,360 --> 00:34:50,860
So this function is 1/2
of a times t squared.

556
00:34:50,860 --> 00:34:59,180
So there is a function one,
and here is df dt.

557
00:34:59,180 --> 00:35:03,470
If I go back to the first
letters that I mentioned, if

558
00:35:03,470 --> 00:35:09,250
this is my function f, then
this is my function that--

559
00:35:09,250 --> 00:35:11,650
and notice what kind
of a curve that is.

560
00:35:11,650 --> 00:35:14,250
Do you recognize that
with a square?

561
00:35:14,250 --> 00:35:19,060
That tells me it's a parabola,
a famous and important curve.

562
00:35:19,060 --> 00:35:20,540
And, of course, it's important
because it

563
00:35:20,540 --> 00:35:22,730
has such a neat formula.

564
00:35:22,730 --> 00:35:32,680
OK, so we have found
the function one.

565
00:35:32,680 --> 00:35:36,710
We've recovered the information
in that lost black

566
00:35:36,710 --> 00:35:43,260
box, the distance box, the trip
meter, from what we did

567
00:35:43,260 --> 00:35:50,110
find in black box two, the
speed, the record of speed.

568
00:35:50,110 --> 00:35:52,710
And notice, I'm using the
speed all the way

569
00:35:52,710 --> 00:35:54,560
from here to here.

570
00:35:54,560 --> 00:35:58,770
The speed kind of tells me how
the distance is piled up.

571
00:35:58,770 --> 00:36:03,700
The distance is kind of a
running total, where the speed

572
00:36:03,700 --> 00:36:07,610
at that moment is an
instant thing.

573
00:36:07,610 --> 00:36:11,060
Oh, we have to do that
in future lectures.

574
00:36:11,060 --> 00:36:16,480
The difference between a
running total of total

575
00:36:16,480 --> 00:36:21,780
distance covered and a speed
that's telling me at a moment,

576
00:36:21,780 --> 00:36:25,120
at an instant how distance
is changing.

577
00:36:25,120 --> 00:36:31,050
The slope at this very point
t, that slope is

578
00:36:31,050 --> 00:36:33,440
this height, is at.

579
00:36:33,440 --> 00:36:36,160
OK, so there you have
the first--

580
00:36:36,160 --> 00:36:37,850
well, I'll say the second.

581
00:36:37,850 --> 00:36:42,050
The first pair of calculus
was this one.

582
00:36:42,050 --> 00:36:46,500
f equals st, and it's
derivative was s.

583
00:36:46,500 --> 00:36:50,220
Our second pair is f is this,
and will you allow

584
00:36:50,220 --> 00:36:52,210
me to write df dt?

585
00:36:52,210 --> 00:36:55,100

586
00:36:55,100 --> 00:37:05,134
If f is 1/2 of at squared,
then df dt is at.

587
00:37:05,134 --> 00:37:08,250

588
00:37:08,250 --> 00:37:10,250
You'll see this rule again.

589
00:37:10,250 --> 00:37:15,680
The power two dropped
to a power one.

590
00:37:15,680 --> 00:37:20,790
But the two multiplied the thing
so it canceled the 1/2

591
00:37:20,790 --> 00:37:24,390
and just left the a.

592
00:37:24,390 --> 00:37:27,580
OK, that's a start on the
highlights of calculus.

593
00:37:27,580 --> 00:37:28,810
Thanks.

594
00:37:28,810 --> 00:37:30,610
NARRATOR: This has been
a production of MIT

595
00:37:30,610 --> 00:37:33,000
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Gilbert Strang.

596
00:37:33,000 --> 00:37:35,270
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597
00:37:35,270 --> 00:37:36,490
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598
00:37:36,490 --> 00:37:39,620
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599
00:37:39,620 --> 00:37:42,690
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600
00:37:42,690 --> 00:37:44,250
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601
00:37:44,250 --> 00:37:46,346