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PROFESSOR: Hi.

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Well, if you're ready,
this will be the

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other big side of calculus.

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We still have two functions,
as before.

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Let me call them the height and
the slope: y of x and the

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slope, s of x.

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Function one and function two.

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That's what calculus is about.

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And earlier, we figured out how
to go from function one if

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we knew that, how to
find the slope.

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Of course, it was easy when
function one was just a

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straight line.

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Then the slope was just
up divided by across.

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But when function
one was curving,

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we had to do something.

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We had to take just a small
distance across, a small

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distance up and divide,
and then let small

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get smaller and smaller.

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The result of that was function
two, the derivative.

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Today, we're going
the other way.

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We know the slope.

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We know s of x, and we want
to find the height, y

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of x, of the graph.

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We know how the graph is sloping
at every point, and we

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have to put all that information
together to find

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its height.

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OK.

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Let me say the easiest way to
do it, when it works, is to

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recognize if we are given a
formula for the slope, to

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recognize maybe we know a height
that goes with it.

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Let me take an example.

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So the one, and most important
case of a height

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is x to some power.

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x to n.

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That's one that everybody
learns.

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When y is x to the nth, the
result of this process that

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produces function two going in
that direction is dy/dx is

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that n comes down and we
have one lower power.

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So if we happen to have that
as our slope, that would be

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our height.

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Nothing more to do.

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But many other slopes
are possible.

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Let's just stay with this
another minute.

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What if the slope was
x to the nth itself?

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What if we started with function
two as x to the nth.

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Suppose dy/dx is x to the nth
power, like x squared.

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Where does x squared
come from?

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Well, look at this rule.

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That rule said that for these
power functions, the power

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drops by one.

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So if I want to end up with x
to the nth, going backwards,

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function one had better be,
involve an x to the n plus 1.

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But that's not perfect, of
course, because when I take

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the slope of that one, this
factor, this power, this

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exponent, n plus n
will come down.

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Just as n came down to here, n
plus 1 will come down, and

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therefore I'd better divide by
n plus 1 so that when I do

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take the slope, when I do go to
number two, the n plus 1's

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will cancel.

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The power will drop by one, and
I'll have x to the nth.

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OK, there's an example--

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quite a useful one--

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of going from step two to
step one just by kind of

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recognizing what
you want there.

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So this is one set that
everybody learns.

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Another is sines and cosines,
if you can fit it into that.

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Another is either
the x or logx.

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That's pretty much the list.
And then you learn, in the

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future, ways to change things
around to fit into one of

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those forms. But then,
of course, there

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are many, many cases--

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many, many functions, too-- that
you don't fit, can't fit

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into a form where you can
recognize from some list which

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you either have learned or you
find on the web or you find in

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the calculator.

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A lot of lists have been made
to help you go from two to

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one, but today we have
to understand what

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is the actual process.

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What is the, what is the the
reverse process to this one?

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And of course, this one involved
a limit as delta x

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went to 0, because always I have
to remember that things

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can be curving, things can be
changing, I can't assume that

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they're saying the same.

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And then, the reverse
direction.

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Maybe I just tell you first
what the symbol is.

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If I have this s of x and I want
to get back to y of x--

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so this is from two to one--

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one will be y of x.

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The symbol for y
of x will be--

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it's, I'm just really wanting to
draw that integral symbol.

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The integral, I would say the
integral of s of x dx.

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And you'll see why--

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this is, that's an s--

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you'll see why that's a
reasonable way to write it.

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But of course, first, we need
the idea behind it.

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OK, so how am I going
to proceed?

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Step one, I'll take steps, I
won't try to get immediately

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to the case of continuous
change.

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I'll take single, individual
steps.

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Let me do that.

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And then, I'm going to
take smaller steps.

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And then in the limit, I'm
taking continuous steps.

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OK, so first, big steps.

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So let me put down, for example,
suppose I have y's.

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Suppose the y's stepped up like
0, 1, 4, 9, 16, whatever.

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So those are heights, and, of a
graph that's sort of pieces

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of straight lines, only
changing a few times.

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What would be the slopes?

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Going now, this is
one going to two.

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The slope is s.

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Well, the slope, if the step
size is one and I go up by

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one, the slope will be 1.

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Here I go up by 3.

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Here I go up by 5.

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Here I go up by 7.

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So if I, to go from there to
there, I'm taking differences.

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I'm taking delta y's.

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These s's are delta y's.

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How do I go backwards?

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Suppose I gave you 1, 3, 5, 7,
and I started y off at 0.

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How could you recover
the rest of the y's?

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Well, that I would, that 1 plus
3 is the 4, 1 plus 3 plus

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5 is the 9, 1 plus
3 adds 7 more.

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You're up to 16 and onwards.

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So if going this way is a
subtraction at each step,

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going this way is a kind
of running addition.

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I add up all the slopes to
see how far I've climbed.

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Let me do a second example,
just to make that point.

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Let me start you this time,
let me start you with the

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slopes, because that's
today's job.

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Suppose the slopes are
4, 3, 2, 1, 0.

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So if these were speeds,
I would say,

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OK, I'm slowing down.

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I'm slowing down, but I'm
still moving forward.

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Positive speeds, but putting
the brakes on.

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What would be the distances?

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If the trip meter starts, start
the trip meter at 0.

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OK.

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Then in the first second or
hour, the first delta x we

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would go 4.

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And what goes there?

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7, right?

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I'm doing, I'm accumulating,
adding up the distances.

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Here I was at 4.

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I went another three
to 7, to 9, 10.

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Here's, I didn't move.

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No speed, zero slope, stays
flat, hit the top at 10.

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OK, I could do this with any
bunch of numbers and I can do

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it with letters.

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So now I move from arithmetic
to algebra.

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Algebra just means I'll do it
with any letters, but I'm not

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yet doing it continuously,
which is what

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calculus will do.

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So with letters, I have here y0,
y1, y2, y3, y4, let's say.

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So those are the y's.

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Now what are the slopes between
them if the step

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across is 1?

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So what are the steps upward,
what are the delta y's?

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Well, y1 minus y0, y2 minus y1,
y3 minus y2, y4 minus y3.

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And then my question is these
are the s's, these are the

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delta y's, you could say.

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These are the y's.

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What happens if I add
all those delta y's?

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Do you see what happens?

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What happens if I add those
four changes to

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get the total change?

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Well, when I add those, do you
see that y1 will cancel minus

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y1, y2 will cancel minus y2,
y3 will cancel minus y3.

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So the sum--

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and I use just a sigma symbol,
but I'll just say sum and you

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know what I mean--

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of these delta y's is what?

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What happened after all
those cancellations?

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Did everything cancel?

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No way.

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y4 is still there, minus
y0 is still there.

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So it was y4 minus y0.

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The last y minus the first y.

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I'll just write y last minus
y 0, y first. y end minus y

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start is the sum of
the delta y's.

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Simple algebra.

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Reminding us again and again and
again that the opposite,

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the inverse to go the other way
from two to one, we add

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pieces to get back to the y's.

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Now I'm coming closer to what I
want, but I'm moving toward

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calculus now.

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So calculus, I got there by
delta y's over delta x's.

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So in moving toward calculus,
what am I doing?

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I'm thinking of the changes
delta y over

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smaller steps delta x.

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So I just want to
take this step.

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I want to divide by delta x
and multiply by delta x.

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Why do I do that?

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That's because it's this delta
y over delta x that is--

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it's those ratios, whatever
the size of delta x is.

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And it's going to get
smaller and smaller.

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I'm going to look at the change
over very short steps.

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Then it's that ratio
that make sense.

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Delta y over delta x is
a reasonable number.

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It's close to the s.

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It's close to dy dx, but it
hasn't got there yet.

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I'm multiplying by the delta x,
the small step that's going

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to 0 but hasn't got there yet.

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And I'm adding and I get the
last one minus the first one.

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Now here comes the
limiting step.

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So the limiting step will be
the limit of this left hand

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side, this sum.

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So in the limit, I'll have more
and more and more things.

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As delta x gets smaller, if
I'm thinking of some fixed

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total change in x, I'm chopping
that up into smaller

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and smaller pieces, more
and more pieces.

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00:15:24,560 --> 00:15:30,690
So more and more pieces of the
slopes at different points

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00:15:30,690 --> 00:15:39,640
along times the size of the
piece give this answer.

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So now can I jump to the
way that I would

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00:15:43,420 --> 00:15:46,090
write this in the limit?

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00:15:49,260 --> 00:15:53,570
So now let delta x go to 0.

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And I ask the right hand
side, y last minus y

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first is not changing.

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y at the end--

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I'll write something different,
y end minus y

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start, just to make that
same point again.

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But it's this that's changing.

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As delta x goes to 0,
this becomes dy dx.

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The little delta x's
are going to 0.

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Here's the way I write it.

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So in that limit, I can't
legally write that sigma, so

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this integral symbol is kind
of copied from that sigma.

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But it's telling you that
a limit has happened.

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And in that limit, this
is dy dx and this, the

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00:17:05,589 --> 00:17:08,560
notation, is dx.

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00:17:08,560 --> 00:17:18,809
I've got what I predicted
here, with s of x there.

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00:17:24,670 --> 00:17:30,070
So what I hope this discussion
has, by starting with numbers,

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by going to algebra, by looking
at the sum of those

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things, which was simple, and
then by going to the limit,

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which was not simple.

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So a whole lot of limit has been
not fully explained, and

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I think the right thing to do
now is to do an example.

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00:17:53,180 --> 00:17:54,570
So let me move to an example.

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00:17:57,250 --> 00:18:02,350
So I'll take a particular
function and follow this

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00:18:02,350 --> 00:18:06,070
process, this limiting process
and see what it gives.

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00:18:06,070 --> 00:18:13,380
And it will give us function one
and, as a bonus, it will

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00:18:13,380 --> 00:18:18,550
give us a new meaning
for function one.

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00:18:18,550 --> 00:18:19,430
Let's do it.

244
00:18:19,430 --> 00:18:24,770
So now I'm going to take
a particular s of x.

245
00:18:24,770 --> 00:18:26,400
So here's x.

246
00:18:26,400 --> 00:18:34,670
Let me take s of x
to be 2 minus 2x.

247
00:18:37,790 --> 00:18:41,370
I didn't want to take one
totally simple that I already

248
00:18:41,370 --> 00:18:47,830
had started the lecture with,
but it's not difficult either.

249
00:18:47,830 --> 00:18:51,310
We'll be able to see what's
happening here.

250
00:18:51,310 --> 00:18:54,410
OK, so let me graph it, because
I want to do this now

251
00:18:54,410 --> 00:18:55,690
with the graph.

252
00:18:55,690 --> 00:19:05,960
So at x equal 1, s of x has
dropped to 0, where when x was

253
00:19:05,960 --> 00:19:07,540
0, it started at 2.

254
00:19:07,540 --> 00:19:10,850
So it started somewhere here.

255
00:19:10,850 --> 00:19:13,240
Here was 1, halfway down.

256
00:19:13,240 --> 00:19:16,470
It's going to come down
in a straight line.

257
00:19:16,470 --> 00:19:17,750
And let me stop there.

258
00:19:17,750 --> 00:19:24,690
It could continue, but let me
call y end is going to be 0

259
00:19:24,690 --> 00:19:27,650
and y start is going to be--

260
00:19:27,650 --> 00:19:29,350
well, we'll see about
that, sorry.

261
00:19:29,350 --> 00:19:30,840
s at the end is 0.

262
00:19:30,840 --> 00:19:32,150
S at the end is 0.

263
00:19:32,150 --> 00:19:34,640
I don't yet know what
y at the end is.

264
00:19:34,640 --> 00:19:37,750
It's not 0.

265
00:19:37,750 --> 00:19:40,000
So what's my idea?

266
00:19:40,000 --> 00:19:42,980
Well, not mine.

267
00:19:42,980 --> 00:19:48,210
Newton and Leibniz and a lot
of people had these ideas.

268
00:19:48,210 --> 00:19:51,170
It's kind of interesting.

269
00:19:51,170 --> 00:19:54,130
So Archimedes.

270
00:19:54,130 --> 00:19:58,260
He goes way, way back, before
Newton or Leibniz or anybody

271
00:19:58,260 --> 00:19:59,510
conceived of them.

272
00:20:02,430 --> 00:20:05,770
Archimedes figured out
how to deal with a

273
00:20:05,770 --> 00:20:11,000
curve with a parabola.

274
00:20:11,000 --> 00:20:15,880
Archimedes got from a parabola,
he got from x

275
00:20:15,880 --> 00:20:25,540
squared, the parabola, back to
a height by special ideas.

276
00:20:25,540 --> 00:20:28,700
He was one of the great
mathematicians of all time.

277
00:20:28,700 --> 00:20:35,700
But even Archimedes didn't see
what you now see, this

278
00:20:35,700 --> 00:20:39,190
connection between function
one and function two.

279
00:20:39,190 --> 00:20:42,950
If he had seen that, he would
have gone further.

280
00:20:42,950 --> 00:20:44,410
All right, now let's see it.

281
00:20:52,490 --> 00:20:56,420
Let me take a delta
x equal to 1/4.

282
00:20:56,420 --> 00:21:02,000
So this is delta x here, and
this one is two delta x's, and

283
00:21:02,000 --> 00:21:06,370
this one is three delta x's, and
the one is four delta x's

284
00:21:06,370 --> 00:21:14,440
in my original delta x, which is
1/4, which is small but not

285
00:21:14,440 --> 00:21:16,550
really small.

286
00:21:16,550 --> 00:21:19,700
So now what do I do?

287
00:21:19,700 --> 00:21:24,540
Look at this first period.

288
00:21:24,540 --> 00:21:29,610
The slope, the s function,
function two--

289
00:21:29,610 --> 00:21:33,520
see, over here is going
to be function one.

290
00:21:33,520 --> 00:21:39,010
This is going to be the y,
the integral of that.

291
00:21:39,010 --> 00:21:41,680
But I don't know what
it is yet, so it's

292
00:21:41,680 --> 00:21:44,750
pretty open to question.

293
00:21:44,750 --> 00:21:47,190
OK, so now let's get there.

294
00:21:47,190 --> 00:21:52,150
So the point is that
over this interval,

295
00:21:52,150 --> 00:21:55,200
the slope is changing.

296
00:21:55,200 --> 00:21:57,920
It's changing a significant
amount.

297
00:21:57,920 --> 00:21:59,990
Not too much, but
it's changing.

298
00:21:59,990 --> 00:22:02,880
And I don't know, from the
algebra, I don't know how to

299
00:22:02,880 --> 00:22:04,130
deal with that.

300
00:22:08,340 --> 00:22:12,320
I'm just going to take a value
within this sum value and stay

301
00:22:12,320 --> 00:22:15,110
with it within that interval,
and I'll take

302
00:22:15,110 --> 00:22:16,750
the starting value.

303
00:22:16,750 --> 00:22:25,070
So over this first delta x, I'm
going to pretend that the

304
00:22:25,070 --> 00:22:27,330
slope stays at 2.

305
00:22:27,330 --> 00:22:31,820
So I'm pretending that this
is my slope function.

306
00:22:31,820 --> 00:22:37,050
Then over my next delta x, I'm
going to pretend that it stays

307
00:22:37,050 --> 00:22:39,660
at probably 1 1/2.

308
00:22:39,660 --> 00:22:42,550
And then I'm going to pretend
that it stays at 1, and I'm

309
00:22:42,550 --> 00:22:44,110
going to pretend that
it stays at 1/2.

310
00:22:50,770 --> 00:22:55,590
If you allow me to go back to
distance and speed, I'm

311
00:22:55,590 --> 00:23:03,320
chopping up the full time, the
day, let's say, into four

312
00:23:03,320 --> 00:23:11,350
pieces and in each piece, the
speed is changing, which I'm

313
00:23:11,350 --> 00:23:13,990
not ready to deal with,
which algebra isn't

314
00:23:13,990 --> 00:23:14,830
ready to deal with.

315
00:23:14,830 --> 00:23:22,000
So the best I could do was say
OK, so suppose the speed is

316
00:23:22,000 --> 00:23:28,550
constant at what it was at the
start of that short time.

317
00:23:28,550 --> 00:23:32,090
So those would be delta t's
rather than delta x's.

318
00:23:32,090 --> 00:23:36,560
The s would be representing
speed, but no difference in

319
00:23:36,560 --> 00:23:39,000
the picture.

320
00:23:39,000 --> 00:23:42,270
So now let me do these things.

321
00:23:46,520 --> 00:23:49,650
Now I'm going to do this
addition, which won't give me

322
00:23:49,650 --> 00:23:55,790
exactly the right y because
those rectangles are not

323
00:23:55,790 --> 00:23:57,670
exactly right.

324
00:23:57,670 --> 00:24:03,940
But I'll get them better by
taking smaller delta x's.

325
00:24:03,940 --> 00:24:05,540
Let me see, what
do I have here?

326
00:24:08,540 --> 00:24:13,870
Over this first time, I
have my slope, which

327
00:24:13,870 --> 00:24:16,130
I'm taking to b 2.

328
00:24:16,130 --> 00:24:22,290
So that's the delta y over delta
x, that's the s, and

329
00:24:22,290 --> 00:24:23,630
then times the delta x.

330
00:24:28,960 --> 00:24:30,700
And what is that?

331
00:24:30,700 --> 00:24:33,840
We might as well just face it,
that that 2 times that delta

332
00:24:33,840 --> 00:24:38,330
x, we can think of that
as the area in

333
00:24:38,330 --> 00:24:40,340
that tall, thin rectangle.

334
00:24:40,340 --> 00:24:43,256
Well, I've introduced the word
area for the first time.

335
00:24:43,256 --> 00:24:47,940
It never showed up on
the previous board.

336
00:24:47,940 --> 00:24:52,570
It's the extra insight
that's coming today.

337
00:24:52,570 --> 00:24:58,680
Now over the second short
period, I'm going to keep fix

338
00:24:58,680 --> 00:25:01,000
my speed at 1 1/2.

339
00:25:01,000 --> 00:25:10,540
1 1/2 times delta x, because my
speed I'm setting at 1 1/2,

340
00:25:10,540 --> 00:25:14,980
and this is how long I go so
this is a distance or a change

341
00:25:14,980 --> 00:25:17,450
in height, a change in y.

342
00:25:17,450 --> 00:25:18,820
And you see what's coming.

343
00:25:18,820 --> 00:25:23,740
The next one will be a 1 times
the delta x, and the last will

344
00:25:23,740 --> 00:25:26,676
be 1/2 times the delta x.

345
00:25:30,020 --> 00:25:35,480
This is adding the way
I did in the algebra.

346
00:25:35,480 --> 00:25:36,720
And what do I get?

347
00:25:36,720 --> 00:25:42,810
Well, this, again, is the
area of that piece.

348
00:25:42,810 --> 00:25:46,310
This one is the area of that
piece, this one is the area of

349
00:25:46,310 --> 00:25:46,950
that piece.

350
00:25:46,950 --> 00:25:55,200
I get an overestimate because
the true slopes dropped a

351
00:25:55,200 --> 00:25:56,670
little within each piece.

352
00:25:59,510 --> 00:26:05,440
I get some quantity which I can
figure out, but it's not

353
00:26:05,440 --> 00:26:06,110
the right answer.

354
00:26:06,110 --> 00:26:07,380
It's not the final answer.

355
00:26:07,380 --> 00:26:11,910
And what is now the main
step to get there?

356
00:26:11,910 --> 00:26:16,790
Chop delta x into half,
you could say.

357
00:26:16,790 --> 00:26:20,150
Why not cut it in half?

358
00:26:20,150 --> 00:26:22,450
Now I'll have a different
picture.

359
00:26:22,450 --> 00:26:24,790
Can you see what this
picture is doing?

360
00:26:24,790 --> 00:26:32,770
Now over the first little half
of the old step I am up here,

361
00:26:32,770 --> 00:26:34,220
but then I drop to here.

362
00:26:41,100 --> 00:26:43,060
Can I do this with an eraser?

363
00:26:43,060 --> 00:26:47,755
A little bit got chopped away, a
little bit got chopped away.

364
00:26:47,755 --> 00:26:49,005
Where has it gone?

365
00:26:52,030 --> 00:26:58,790
I'm going to have
this zig zag.

366
00:26:58,790 --> 00:27:00,040
That wasn't too bad.

367
00:27:09,880 --> 00:27:15,680
I'm replacing that with a sum of
eight pieces, because delta

368
00:27:15,680 --> 00:27:17,190
x is now down to 1/8.

369
00:27:20,410 --> 00:27:24,090
This is what we said
about that sum.

370
00:27:24,090 --> 00:27:29,690
That sum has got more and more
terms because it has a term

371
00:27:29,690 --> 00:27:36,860
for every little delta x, and
the size of that term is about

372
00:27:36,860 --> 00:27:39,660
like delta x multiplied
by an s.

373
00:27:39,660 --> 00:27:47,030
So what I getting in the limit
is a kind of running sum, a

374
00:27:47,030 --> 00:27:50,980
running counter, a mileage
meter, a trip mileage, that's

375
00:27:50,980 --> 00:27:54,160
adding up distance
based on speed.

376
00:27:58,432 --> 00:28:04,870
Do you see what I'm getting
in the graph picture?

377
00:28:04,870 --> 00:28:09,790
What happens to the shaded part
as delta x gets smaller

378
00:28:09,790 --> 00:28:11,400
and smaller?

379
00:28:11,400 --> 00:28:19,940
This shaded part is going
to be the curve.

380
00:28:19,940 --> 00:28:26,310
These little long pieces are
going to get reduced, reduced,

381
00:28:26,310 --> 00:28:31,740
reduced, and in the end
the total height at

382
00:28:31,740 --> 00:28:34,680
1 is going to be--

383
00:28:34,680 --> 00:28:36,800
ta-da, this is the moment--

384
00:28:36,800 --> 00:28:42,160
the area under the
slope curve.

385
00:28:42,160 --> 00:28:49,750
This y turns out to be
the area under the s

386
00:28:49,750 --> 00:28:54,340
of x curve, or y.

387
00:28:54,340 --> 00:28:57,640
So at x equal 1, what is it?

388
00:28:57,640 --> 00:28:59,860
What's the area out to 1?

389
00:28:59,860 --> 00:29:03,260
Well, we've got a
triangle there.

390
00:29:03,260 --> 00:29:07,735
Its base is 1, its
height is 2.

391
00:29:07,735 --> 00:29:10,870
The area of a triangle is 1/2
the base times the height, so

392
00:29:10,870 --> 00:29:12,530
I have 1/2 times 1 times 2.

393
00:29:12,530 --> 00:29:13,780
I've got 1.

394
00:29:18,150 --> 00:29:20,610
The area at the end is 1.

395
00:29:20,610 --> 00:29:21,942
But--

396
00:29:21,942 --> 00:29:23,250
well, I shouldn't say but.

397
00:29:23,250 --> 00:29:31,960
I should let you applaud first.
What if I only went

398
00:29:31,960 --> 00:29:33,210
that far, halfway?

399
00:29:35,960 --> 00:29:41,830
What if that was s end?

400
00:29:41,830 --> 00:29:46,750
What if I want to know what
is y at x equal 1/2?

401
00:29:46,750 --> 00:29:50,720
Then it'll be of course, just
the area up to that point.

402
00:29:50,720 --> 00:29:56,590
Can I remove this part of the
picture for a moment?

403
00:29:56,590 --> 00:29:59,050
I'm always looking at area.

404
00:29:59,050 --> 00:30:02,440
And the area of that, do we know
what that area would be?

405
00:30:02,440 --> 00:30:06,140
It's not a triangle anymore,
it's some kind of a trapezoid.

406
00:30:06,140 --> 00:30:13,270
As delta x goes to 0, I'm going
to get the correct area,

407
00:30:13,270 --> 00:30:15,820
which will be what?

408
00:30:15,820 --> 00:30:21,250
Let's see, I have 1/2 as the
base and the average height is

409
00:30:21,250 --> 00:30:22,700
about 1 1/2.

410
00:30:22,700 --> 00:30:24,490
Can I do that little
calculation?

411
00:30:24,490 --> 00:30:30,760
The base is 1/2 and the average
height is 1 1/2.

412
00:30:30,760 --> 00:30:32,800
I think I get 3/4.

413
00:30:32,800 --> 00:30:37,110
So halfway along, it's
got up to 3/4.

414
00:30:37,110 --> 00:30:38,020
Where is 3/4?

415
00:30:38,020 --> 00:30:43,430
So this is 1, this is 3/4,
this is 1/2, this is 1/4.

416
00:30:46,500 --> 00:30:48,550
So at 1, it's at 3/4.

417
00:30:53,680 --> 00:30:57,180
Halfway along, its at 3/4.

418
00:30:57,180 --> 00:31:03,640
I would like to know
that graph now.

419
00:31:03,640 --> 00:31:07,960
I'm ready to jump
to the limit.

420
00:31:07,960 --> 00:31:13,230
Let me do it the way I said at
the very start of the lecture.

421
00:31:13,230 --> 00:31:16,370
Let me take this and
try to guess.

422
00:31:16,370 --> 00:31:18,320
So, I'm taking a shortcut.

423
00:31:18,320 --> 00:31:23,210
Because do we go through this
horrible process every time we

424
00:31:23,210 --> 00:31:24,150
want to do an integral?

425
00:31:24,150 --> 00:31:26,150
Of course we don't.

426
00:31:26,150 --> 00:31:32,470
The best way is, can we find a
y function, a function, one,

427
00:31:32,470 --> 00:31:35,220
that has that derivative?

428
00:31:35,220 --> 00:31:38,010
Let's just try it.

429
00:31:38,010 --> 00:31:40,130
I'm allowed to take it in
two pieces, that's a

430
00:31:40,130 --> 00:31:42,000
very valuable fact.

431
00:31:42,000 --> 00:31:46,720
So what has the derivative 2?

432
00:31:46,720 --> 00:31:49,730
If the slope is 2, what's
the function?

433
00:31:49,730 --> 00:31:52,850
If the speed is 2, what's
the distance?

434
00:31:52,850 --> 00:32:01,420
It's constant speed, 2, times
the total distance.

435
00:32:01,420 --> 00:32:06,770
The slope of the 2x line
is 2, clearly.

436
00:32:06,770 --> 00:32:08,460
What about the 2x?

437
00:32:11,190 --> 00:32:14,120
Which function has
the slope 2x?

438
00:32:14,120 --> 00:32:16,360
Well, we saw it over here.

439
00:32:16,360 --> 00:32:23,330
The function that has the slope
2x is x squared, because

440
00:32:23,330 --> 00:32:26,685
when I take the slope of x
squared, the 2 comes down.

441
00:32:29,850 --> 00:32:33,130
The 2 shows up, I have
one smaller power, x

442
00:32:33,130 --> 00:32:34,230
to the first power.

443
00:32:34,230 --> 00:32:41,830
This is the correct y, and I
hope that my graph gets those

444
00:32:41,830 --> 00:32:42,940
points right.

445
00:32:42,940 --> 00:32:48,020
At x equal to 1, this is
2 minus 1, this is the

446
00:32:48,020 --> 00:32:51,190
correct height, 1.

447
00:32:51,190 --> 00:32:57,270
At x equal to 1/2, all right,
here is the moment of truth.

448
00:32:57,270 --> 00:33:04,860
Now set x equal to 1/2, and what
do you get for this y?

449
00:33:04,860 --> 00:33:08,580
You get 2 times 1/2--
that's 1--

450
00:33:08,580 --> 00:33:14,336
minus 1/2 squared, 1/4.

451
00:33:14,336 --> 00:33:16,080
Hey, miracle.

452
00:33:16,080 --> 00:33:18,360
3/4.

453
00:33:18,360 --> 00:33:24,100
This area I figured
to be 3/4 and this

454
00:33:24,100 --> 00:33:27,370
approach also gave 3/4.

455
00:33:27,370 --> 00:33:30,470
Either way, multiplying
those is 3/4,

456
00:33:30,470 --> 00:33:33,810
subtracting those is 3/4.

457
00:33:33,810 --> 00:33:37,080
What does my graph look like?

458
00:33:37,080 --> 00:33:38,850
What does the graph
of that look like?

459
00:33:45,320 --> 00:33:46,740
What's the slope at the start?

460
00:33:46,740 --> 00:33:50,000
The slope at the start
is s at the start.

461
00:33:50,000 --> 00:33:53,040
And s at the start, when
x is 0, the slope is 2.

462
00:33:53,040 --> 00:33:56,340
So it starts out with
a slope of 2.

463
00:33:56,340 --> 00:34:04,370
But it's slowing down, it's a
little bit like this one where

464
00:34:04,370 --> 00:34:08,260
the car was slowing down, we're
not picking up distance

465
00:34:08,260 --> 00:34:11,080
so quickly, we're not picking
up height so quickly.

466
00:34:11,080 --> 00:34:13,429
But we're still going
forward, we're still

467
00:34:13,429 --> 00:34:14,880
picking up some height.

468
00:34:14,880 --> 00:34:20,260
So it starts with a slope of 2,
bends around to there, and

469
00:34:20,260 --> 00:34:22,350
I guess maybe that is--

470
00:34:22,350 --> 00:34:23,909
yes.

471
00:34:23,909 --> 00:34:26,760
That picture is almost
good, but not great.

472
00:34:30,469 --> 00:34:35,760
So the slope is 2 and there.

473
00:34:35,760 --> 00:34:39,040
And what is the slope
at this point?

474
00:34:39,040 --> 00:34:42,760
You can't tell from my picture,
which isn't perfect.

475
00:34:42,760 --> 00:34:44,520
The slope, I'm told
what it is.

476
00:34:44,520 --> 00:34:47,650
When x is 1, the slope
is 2 minus 2.

477
00:34:47,650 --> 00:34:49,600
Slope 0.

478
00:34:49,600 --> 00:34:50,409
The slope is 0.

479
00:34:50,409 --> 00:34:53,909
We're not picking up any more
height, any more area.

480
00:34:53,909 --> 00:34:56,780
And of course, that's right.

481
00:34:56,780 --> 00:35:01,500
At this point, we're not
picking up more area.

482
00:35:01,500 --> 00:35:06,400
If I continue beyond here,
we're losing area because

483
00:35:06,400 --> 00:35:11,820
below the axis, I'll count as
negative area just because if

484
00:35:11,820 --> 00:35:15,500
it was speed, I'd be
going backwards.

485
00:35:15,500 --> 00:35:16,730
That's what will happen here.

486
00:35:16,730 --> 00:35:17,980
I'll start down.

487
00:35:20,610 --> 00:35:23,710
If that continued, this
would still be the

488
00:35:23,710 --> 00:35:26,290
correct thing to graph.

489
00:35:26,290 --> 00:35:30,750
If I do graph it, that's
actually the top at x equal 1,

490
00:35:30,750 --> 00:35:34,460
and then it starts down and
probably by, I don't know

491
00:35:34,460 --> 00:35:39,090
where, x equal something,
maybe by x equal 2.

492
00:35:39,090 --> 00:35:40,230
Oh yeah, you can see.

493
00:35:40,230 --> 00:35:44,760
By x equal to 2, it's
got down to 0 again.

494
00:35:44,760 --> 00:35:51,750
When x is 2, this is now 0 and
you can see that when x is 2,

495
00:35:51,750 --> 00:35:55,860
we'll have the bad area--

496
00:35:55,860 --> 00:35:58,590
the car going backwards--

497
00:35:58,590 --> 00:36:02,030
will be identical to
the forward area.

498
00:36:02,030 --> 00:36:06,655
The total area is 0, and I'll be
at this point when x is 2.

499
00:36:10,840 --> 00:36:12,235
Let me just recap a moment.

500
00:36:14,760 --> 00:36:17,240
Today was about going
from function two

501
00:36:17,240 --> 00:36:19,940
back to function one.

502
00:36:19,940 --> 00:36:23,020
The quickest way to do it is
to find a function one that

503
00:36:23,020 --> 00:36:26,710
gives that function two
and then you're in.

504
00:36:26,710 --> 00:36:30,080
But if you can't do that or if
you want to understand what

505
00:36:30,080 --> 00:36:35,410
the real, behind it, limiting
process is, it's like the

506
00:36:35,410 --> 00:36:43,060
algebra but it's this expression
here that's

507
00:36:43,060 --> 00:36:45,900
concealing so much
mathematics.

508
00:36:45,900 --> 00:36:50,710
Delta x going to 0, these ratios
going to the actual

509
00:36:50,710 --> 00:36:57,430
function, and the delta x I
replaced by the symbol dx,

510
00:36:57,430 --> 00:37:01,200
indicating an infinitesimal.

511
00:37:01,200 --> 00:37:01,930
We'll see it more.

512
00:37:01,930 --> 00:37:03,105
Thank you.

513
00:37:03,105 --> 00:37:05,350
NARRATOR: This has been
a production of MIT

514
00:37:05,350 --> 00:37:07,740
OpenCourseWare and
Gilbert Strang.

515
00:37:07,740 --> 00:37:10,010
Funding for this video was
provided by the Lord

516
00:37:10,010 --> 00:37:11,230
Foundation.

517
00:37:11,230 --> 00:37:14,360
To help OCW continue to provide
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518
00:37:14,360 --> 00:37:17,440
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519
00:37:17,440 --> 00:37:19,000
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