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PROFESSOR: OK, hi.

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This is the second in my videos
about the main ideas,

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the big picture of calculus.

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And this is an important one,
because I want to introduce

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and compute some derivatives.

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And you'll remember the overall
situation is we have

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pairs of functions, distance
and speed, function 1 and

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function 2, height of a graph,
slope of the graph, height of

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a mountain, slope
of a mountain.

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And it's the connection
between those two that

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calculus is about.

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And so our problem today is, you
could imagine we have an

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airplane climbing.

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Its height is y as it
covers a distance x.

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And its flight recorder will--

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Well, probably it has two
flight recorders.

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Let's suppose it has.

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Or your car has two recorders.

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One records the distance, the
height, the total amount

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achieved up to that moment,
up to that time, t,

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or that point, x.

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The second recorder would tell
you at every instant

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what the speed is.

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So it would tell you the
speed at all times.

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Do you see the difference?

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The speed is like what's
happening at an instant.

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The distance or the height, y,
is the total accumulation of

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how far you've gone, how
high you've gone.

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And now I'm going to suppose
that this speed, this second

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function, the recorder is lost.
But the information is

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there, and how to recover it.

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So that's the question.

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How, if I have a total record,
say of height--

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I'll say mostly with y of x.

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I write these two so that you
realize that letters are not

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what calculus is about.

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It's ideas.

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And here is a central idea.

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if I know the height--

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as I go along, I know the
height, it could go down--

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how can I recover from that
height what the slope is at

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each point?

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So here's something rather
important, that's the notation

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that Leibniz created, and it was
a good, good idea for the

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derivative.

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And you'll see where
it comes from.

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But somehow I'm dividing
distance up by distance

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across, and that ratio of
up to across is a slope.

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So let me develop what
we're doing.

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So the one thing we can do and
now will do is, for the great

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functions of calculus,
a few very special,

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very important functions.

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We will actually figure
out what the slope is.

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These are given by formulas,
and I'll find a formula for

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the slope, dy/dx equals.

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And I won't write it in yet.

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Let me keep a little suspense.

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But this short list of the
great functions is

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tremendously valuable.

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The process that we go through
takes a little time, but once

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we do it it's done.

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Once we write in the answer
here, we know it.

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And the point is that other
functions of science, of

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engineering, of economics,
of life, come from these

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functions by multiplying--

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I could multiply that
times that--

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and then I need a product
rule, the rule for the

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derivative, the slope
of a product.

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I could divide that by that.

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So I need a quotient rule.

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I could do a chain.

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And you'll see that's maybe the
best and most valuable.

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e to the sine x, so I'm putting
e to the x together

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with sine x in a chain of
functions, e to the sine of x.

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Then we need a chain rule.

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That's all coming.

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Let me start--

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Well, let me even start by
giving away the main facts for

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these three examples, because
they're three you want to

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remember, and might
as well start now.

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The x to the nth, so that's
something if n is positive.

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x to the nth climbs up.

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Let me draw a graph here of y
equals x squared, because

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that's one we'll work out in
detail, y equals x squared.

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So this direction is x.

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This direction is y.

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And I want to know the slope.

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And the point is that that slope
is changing as I go up.

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So the slope depends on x.

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The slope will be
different here.

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So it's getting steeper
and steeper.

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I'll figure out that slope.

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For this example, x
squared went in

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as 2, and it's climbing.

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If n was minus 2, just because
that's also on our list, n

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could be negative, the function
would be dropping.

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You remember x to the minus 2,
that negative exponent means

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divide by x squared.

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x to the minus 2 is 1 divided
by x squared,

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and it'll be dropping.

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So n could be positive
or negative here.

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So I tell you the derivative.

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The derivative is easy to
remember, the set number n.

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It's another power of
x, and that power is

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one less, one down.

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You lose one power.

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I'm going to go through the
steps here for n is equal to

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2, so I hope to get the answer
2 times x to the 2 minus 1

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will just be 1, 2x.

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But what does the slope mean?

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That's what this lecture
is really telling you.

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I'll tell you the answer for
if it's sine x going in,

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beautifully, the derivative of
sine x is cos x, the cosine.

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The derivative of the sine curve
is the cosine curve.

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You couldn't hope for
more than that.

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And then we'll also, at the same
time, find the derivative

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of the cosine curve, which
is minus the sine curve.

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It turns out a minus sine comes
in because the cosine

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curve drops at the start.

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And would you like
to know this one?

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e to the x, which I will
introduce in a lecture coming

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very soon, because it's the
function of calculus.

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And the reason it's so terrific
is, the connection

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between the function, whatever
it is, whatever this number e

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is to whatever the xth power
means, the slope is the same

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as the function, e to the x.

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That's amazing.

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As the function changes,
the slope changes

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and they stay equal.

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Really, my help is just to say,
if you know those three,

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you're really off to a
good start, plus the

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rules that you need.

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All right, now I'll tackle this
particular one and say,

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what does slope mean?

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So I'm given the recorder
that I have. This

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is function 1 here.

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This is function 1,
the one I know.

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And I know it at every point.

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If I only had the trip meter
after an hour or two hours or

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three hours, well, calculus
can't do the impossible.

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It can't know, if I only knew
the distance reached after

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each hour, I couldn't tell what
it did over that hour,

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how often you had to break,
how often you accelerated.

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I could only find an average
speed over that hour.

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That would be easy.

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So averages don't
need calculus.

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It's instant stuff, what happens
at a moment, what is

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that speedometer reading at the
moment x, say, x equal 1.

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What is the slope?

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Yeah, let me put in x equals 1
on this graph and x equals 2.

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And now x squared is going
to be at height 1.

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If x is 1, then x squared
is also 1.

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If x is 2, x squared
will be 4.

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What's the average?

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Let me just come back
one second to

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that average business.

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The average slope there would
be, in a distance across of 1,

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I went up by how much?

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3.

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I went up from 1 to 4.

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I have to do a subtraction.

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Differences, derivatives, that's
the connection here.

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So it's 4 minus 1.

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That is 3.

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So I would say the
average is 3/1.

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But that's not calculus.

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Calculus is looking at
the instant thing.

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And let me begin at this
instant, at that point.

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What does the slope look like
to you at that point?

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At at x equals 0, here's
x equals 0, and

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here's y equals 0.

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We're very much 0.

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You see it's climbing, but at
that moment, it's like it just

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started from a red
light, whatever.

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The speed is 0 at that point.

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And I want to say
the slope is 0.

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That's flat right there.

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That's flat.

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If I continued the curve, if I
continued the x squared curve

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for x negative, it would be the
same as for x positive.

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Well, it doesn't look
very the same.

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Let me improve that.

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It would start up the same way
and be completely symmetric.

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Everybody sees that, at
that 0 position, the

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curve has hit bottom.

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Actually, this will
be a major, major

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application of calculus.

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You identify the bottom of a
curve by the fact that the

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slope is 0.

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It's not going up.

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It's not going down.

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It's 0 at that point.

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But now, what do I mean
by slope at a point?

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Here comes the new idea.

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If I go way over to 1,
that's too much.

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I just want to stay
near this point.

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I'll go over a little
bit, and I call that

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little bit delta x.

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So that letter, delta,
signals to our

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minds small, some small.

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And actually, probably smaller
than I drew it there.

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And then, so what's
the average?

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I'd like to just find
the average

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speed, or average slope.

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If I go over by delta x, and
then how high do I go up?

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Well, the curve is y
equals x squared.

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So how high is this?

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So the average is up first,
divided by across.

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Across is our usual delta x.

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How far did it go up?

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Well, if our curve is x squared
and I'm at the point,

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delta x, then it's
delta x squared.

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00:13:32,830 --> 00:13:35,360

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That's the average over the
first piece, over short, over

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the first piece of the curve.

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Out is-- from here
out to delta x.

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OK.

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00:13:50,592 --> 00:13:52,160

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Now, again, that's still only
an average, because delta x

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might have been short.

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I want to push it to 0.

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That's where calculus comes
in, taking the limit of

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shorter and shorter and shorter
pieces in order to

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zoom in on that instant, that
moment, that spot where we're

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looking at the slope, and where
we're expecting the

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answer is 0, in this case.

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And you see that the average,
it happens to

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be especially simple.

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Delta x squared over delta
x is just delta x.

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So the average slope
is extremely small.

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And I'll just complete
that thought.

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So the instant slope-- instant
slope at 0, at x equals 0, I

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let this delta x get smaller
and smaller.

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00:14:56,860 --> 00:15:01,440
I get the answer is 0, which
is just what I expected it.

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And you could say, well,
not too exciting.

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But it was an easy one to do.

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It was the first time that we
actually went through the

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steps of computing.

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00:15:12,260 --> 00:15:15,180
This is a, like, a delta y.

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This is the delta x.

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Instead of 3/1, starting
here I had delta x

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squared over delta x.

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That was easy to see.

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It was delta x.

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And if I move in closer, that
average slope is smaller and

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smaller, and the slope
at that instant is 0.

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00:15:35,670 --> 00:15:37,970
No problem.

255
00:15:37,970 --> 00:15:43,340
The travel, the climbing
began from rest, but

256
00:15:43,340 --> 00:15:45,220
it picked up speed.

257
00:15:45,220 --> 00:15:48,300
The slope here is
certainly not 0.

258
00:15:48,300 --> 00:15:50,200
We'll find that slope.

259
00:15:50,200 --> 00:15:52,740
We need now to find the
slope at every point.

260
00:15:52,740 --> 00:15:55,210
OK.

261
00:15:55,210 --> 00:15:57,810
That's a good start.

262
00:15:57,810 --> 00:16:04,170
Now I'm ready to find the
slope at any point.

263
00:16:04,170 --> 00:16:09,890
Instead of just x equals 0,
which we've now done, I better

264
00:16:09,890 --> 00:16:18,920
draw a new graph of the same
picture, climbing up.

265
00:16:18,920 --> 00:16:23,900
Now I'm putting in a little
climb at some point x here.

266
00:16:23,900 --> 00:16:26,085
I'm up at a height, x squared.

267
00:16:26,085 --> 00:16:30,590

268
00:16:30,590 --> 00:16:33,310
I'm at that point
on the climb.

269
00:16:33,310 --> 00:16:37,410
I'd like to know the slope
there, at that point.

270
00:16:37,410 --> 00:16:40,110
How am I going to do it?

271
00:16:40,110 --> 00:16:45,880
I will follow this as the
central dogma of calculus, of

272
00:16:45,880 --> 00:16:50,810
differential calculus, function
1 to function 2.

273
00:16:50,810 --> 00:16:56,990
Take a little delta x, go as
far as x plus delta x.

274
00:16:56,990 --> 00:16:59,860
That will take you to a higher
point on the curve.

275
00:16:59,860 --> 00:17:05,230
That's now the point x plus
delta x squared, because our

276
00:17:05,230 --> 00:17:09,730
curve is still y equals
x squared in this

277
00:17:09,730 --> 00:17:11,919
nice, simple parabola.

278
00:17:11,919 --> 00:17:13,810
OK.

279
00:17:13,810 --> 00:17:19,829
So now I you look at distance
across and distance up.

280
00:17:19,829 --> 00:17:25,210
So delta y is the change up.

281
00:17:25,210 --> 00:17:30,360
Delta x is the across.

282
00:17:30,360 --> 00:17:35,500
And I have to put
what is delta y.

283
00:17:35,500 --> 00:17:38,310
I have to write in,
what is delta y?

284
00:17:38,310 --> 00:17:40,280
It's this distance up.

285
00:17:40,280 --> 00:17:44,700
It's x plus delta x squared.

286
00:17:44,700 --> 00:17:49,120
That's this height minus
this height.

287
00:17:49,120 --> 00:17:51,610
I'm not counting this
bit, of course.

288
00:17:51,610 --> 00:17:53,190
It's that that I want.

289
00:17:53,190 --> 00:17:58,040
That's the delta y,
is this piece.

290
00:17:58,040 --> 00:18:03,600
So it's up to this, subtract
x squared.

291
00:18:03,600 --> 00:18:05,780
That's delta y.

292
00:18:05,780 --> 00:18:07,470
That's important.

293
00:18:07,470 --> 00:18:11,520
Now I divide by delta x.

294
00:18:11,520 --> 00:18:13,820
This is all algebra now.

295
00:18:13,820 --> 00:18:18,320
Calculus is going to come in
a moment, but not yet.

296
00:18:18,320 --> 00:18:20,160
For algebra, what do I do?

297
00:18:20,160 --> 00:18:21,870
I multiply this out.

298
00:18:21,870 --> 00:18:23,520
I see that thing squared.

299
00:18:23,520 --> 00:18:26,160
I remember that x is squared.

300
00:18:26,160 --> 00:18:29,190
And then I have this
times this twice.

301
00:18:29,190 --> 00:18:34,133
2x delta x's, and then I
have delta x squared.

302
00:18:34,133 --> 00:18:36,750

303
00:18:36,750 --> 00:18:38,820
And then I'm subtracting
x squared.

304
00:18:38,820 --> 00:18:47,080
So that's delta y written
out in full glory.

305
00:18:47,080 --> 00:18:53,080
I wrote it out because
now I can simplify by

306
00:18:53,080 --> 00:18:55,400
canceling the x squared.

307
00:18:55,400 --> 00:18:58,200
I'm not surprised.

308
00:18:58,200 --> 00:19:01,840
Now, in this case, I can
actually do the division.

309
00:19:01,840 --> 00:19:04,420
Delta x is just there.

310
00:19:04,420 --> 00:19:06,520
So it leaves me with a 2x.

311
00:19:06,520 --> 00:19:08,810
Delta x over delta x is 1.

312
00:19:08,810 --> 00:19:11,810
And then here's a delta x
squared over a delta x, so

313
00:19:11,810 --> 00:19:13,940
that leaves me with
one delta x.

314
00:19:13,940 --> 00:19:17,930

315
00:19:17,930 --> 00:19:21,400
As you get the hang of
calculus, you see the

316
00:19:21,400 --> 00:19:27,590
important things is like this
first order, delta x to the

317
00:19:27,590 --> 00:19:28,680
first power.

318
00:19:28,680 --> 00:19:34,370
Delta x squared, that, when
divided by delta x, gives us

319
00:19:34,370 --> 00:19:38,130
this, which is going
to disappear.

320
00:19:38,130 --> 00:19:39,490
That's the point.

321
00:19:39,490 --> 00:19:48,940
This was the average over a
short but still not instant

322
00:19:48,940 --> 00:19:52,080
range, distance.

323
00:19:52,080 --> 00:19:54,870
Now, what happens?

324
00:19:54,870 --> 00:19:56,130
Now dy/dx.

325
00:19:56,130 --> 00:20:00,180

326
00:20:00,180 --> 00:20:09,470
So if this is short, short over
short, this is darn short

327
00:20:09,470 --> 00:20:11,160
over darn short.

328
00:20:11,160 --> 00:20:18,610
That d is, well, it's
too short to see.

329
00:20:18,610 --> 00:20:23,880
So I don't actually now try
to separate a distance dy.

330
00:20:23,880 --> 00:20:31,350
This isn't a true division,
because it's effectively 0/0.

331
00:20:31,350 --> 00:20:35,360
And you might say, well, 0/0,
what's the meaning?

332
00:20:35,360 --> 00:20:40,620
Well, the meaning of 0/0, in
this situation, is, I take the

333
00:20:40,620 --> 00:20:54,140
limit of this one, which does
have a meaning, because those

334
00:20:54,140 --> 00:20:55,070
are true numbers.

335
00:20:55,070 --> 00:20:58,320
They're little numbers
but they're numbers.

336
00:20:58,320 --> 00:21:07,440
And this was this, so now here's
the big step, leaving

337
00:21:07,440 --> 00:21:11,380
algebra behind, going to
calculus in order to get

338
00:21:11,380 --> 00:21:13,780
what's happening at a point.

339
00:21:13,780 --> 00:21:16,230
I let delta x go to 0.

340
00:21:16,230 --> 00:21:17,510
And what is that?

341
00:21:17,510 --> 00:21:19,900
So delta y over delta
x is this.

342
00:21:19,900 --> 00:21:22,380
What is the dy/dx?

343
00:21:22,380 --> 00:21:28,960
So in the limit, ah,
it's not hard.

344
00:21:28,960 --> 00:21:31,270
Here's the 2x.

345
00:21:31,270 --> 00:21:34,030
It's there.

346
00:21:34,030 --> 00:21:35,570
Here's the delta x.

347
00:21:35,570 --> 00:21:39,280
In the limit, it disappears.

348
00:21:39,280 --> 00:21:44,460
So the conclusion is that
the derivative is 2x.

349
00:21:44,460 --> 00:21:48,330

350
00:21:48,330 --> 00:21:50,000
So that's function two.

351
00:21:50,000 --> 00:21:51,660
That's function two here.

352
00:21:51,660 --> 00:21:53,450
That's the slope function.

353
00:21:53,450 --> 00:21:55,060
That's the speed function.

354
00:21:55,060 --> 00:21:56,390
Maybe I should draw it.

355
00:21:56,390 --> 00:21:59,370
Can I draw it above and then
I'll put the board back up?

356
00:21:59,370 --> 00:22:06,740
So here's a picture of function
2, the derivative, or

357
00:22:06,740 --> 00:22:08,910
the slope, which I
was calling s.

358
00:22:08,910 --> 00:22:12,070
So that's the s function,
against x.

359
00:22:12,070 --> 00:22:16,320
x is still the thing that's
varying, or it could be t, or

360
00:22:16,320 --> 00:22:20,270
it could be whatever
letter we've got.

361
00:22:20,270 --> 00:22:23,710
And the answer was 2x
for this function.

362
00:22:23,710 --> 00:22:28,970
So if I graph it, it starts
at 0, and it climbs

363
00:22:28,970 --> 00:22:32,540
steadily with slope 2.

364
00:22:32,540 --> 00:22:37,184
So that's a graph of s of x.

365
00:22:37,184 --> 00:22:40,460
And for example--

366
00:22:40,460 --> 00:22:45,490
yeah, so take a couple of
points on that graph--

367
00:22:45,490 --> 00:22:50,520
at x equals 0, the slope is 0.

368
00:22:50,520 --> 00:22:55,520
And we did that first. And
we actually got it right.

369
00:22:55,520 --> 00:22:59,680
The slope is 0 at the start,
at the bottom of the curve.

370
00:22:59,680 --> 00:23:04,430
At some other point on the
curve, what's the slope here?

371
00:23:04,430 --> 00:23:08,240
Ha, yeah, tell me
the slope there.

372
00:23:08,240 --> 00:23:12,100
At that point on the curve, an
average slope was 3/1, but

373
00:23:12,100 --> 00:23:15,180
that was the slope of this,
like, you know--

374
00:23:15,180 --> 00:23:19,490

375
00:23:19,490 --> 00:23:21,660
sometimes called a chord.

376
00:23:21,660 --> 00:23:26,040
That's over a big jump of 1.

377
00:23:26,040 --> 00:23:29,190
Then I did it over a small jump
of delta x, and then I

378
00:23:29,190 --> 00:23:34,790
let delta x go to 0, so it was
an instant infinitesimal jump.

379
00:23:34,790 --> 00:23:39,630
So the actual slope, the way to
visualize it is that it's

380
00:23:39,630 --> 00:23:41,560
more like that.

381
00:23:41,560 --> 00:23:45,170
That's the line that's
really giving the

382
00:23:45,170 --> 00:23:47,860
slope at that point.

383
00:23:47,860 --> 00:23:49,240
That's my best picture.

384
00:23:49,240 --> 00:23:54,920
It's not Rembrandt,
but it's got it.

385
00:23:54,920 --> 00:23:57,040
And what is the slope
at that point?

386
00:23:57,040 --> 00:24:00,540
Well, that's what our
calculation was.

387
00:24:00,540 --> 00:24:02,410
It found the slope
at that point.

388
00:24:02,410 --> 00:24:11,250
And at the particular point, x
equals 1, the height was 2.

389
00:24:11,250 --> 00:24:12,570
The slope is 2.

390
00:24:12,570 --> 00:24:17,720
The actual tangent line
is only-- is there.

391
00:24:17,720 --> 00:24:18,400
You see?

392
00:24:18,400 --> 00:24:20,730
It's up.

393
00:24:20,730 --> 00:24:24,350
Oh, wait a minute.

394
00:24:24,350 --> 00:24:27,950
Yeah, well, the slope is 2.

395
00:24:27,950 --> 00:24:31,230
I don't know.

396
00:24:31,230 --> 00:24:34,510
This goes up to 3.

397
00:24:34,510 --> 00:24:41,300
It's not Rembrandt, but
the math is OK.

398
00:24:41,300 --> 00:24:43,070
So what have we done?

399
00:24:43,070 --> 00:24:49,130
We've taken the first small step
and literally I could say

400
00:24:49,130 --> 00:24:52,290
small step, almost a play on
words because that's the

401
00:24:52,290 --> 00:24:55,180
point, the step is so small--

402
00:24:55,180 --> 00:24:59,170
to getting these great
functions.

403
00:24:59,170 --> 00:25:06,110
Before I close this lecture, can
I draw this pair, function

404
00:25:06,110 --> 00:25:14,630
1 and function 2, and just see
that the movement of the

405
00:25:14,630 --> 00:25:16,650
curves is what we
would expect.

406
00:25:16,650 --> 00:25:23,710
So let me, just for one more
good example, great example,

407
00:25:23,710 --> 00:25:26,510
actually, is let me draw.

408
00:25:26,510 --> 00:25:28,990
Here goes x.

409
00:25:28,990 --> 00:25:32,830
In fact, maybe I already drew
in the first letter, lecture

410
00:25:32,830 --> 00:25:37,890
that bit out to 90 degrees.

411
00:25:37,890 --> 00:25:41,140
Only if we want a nice formula,
we better call that

412
00:25:41,140 --> 00:25:45,000
pi over 2 radians.

413
00:25:45,000 --> 00:25:46,870
And here's a graph of sine x.

414
00:25:46,870 --> 00:25:48,450
This is y.

415
00:25:48,450 --> 00:25:52,380
This is the function
1, sine x.

416
00:25:52,380 --> 00:25:54,530
And what's function 2?

417
00:25:54,530 --> 00:25:57,210
What can we see about
function 2?

418
00:25:57,210 --> 00:26:00,400

419
00:26:00,400 --> 00:26:01,650
Again, x.

420
00:26:01,650 --> 00:26:04,560

421
00:26:04,560 --> 00:26:06,410
We see a slope.

422
00:26:06,410 --> 00:26:09,160
This is not the same
as x squared.

423
00:26:09,160 --> 00:26:12,230
This starts with a
definite slope.

424
00:26:12,230 --> 00:26:17,690
And it turns out this will be
one of the most important

425
00:26:17,690 --> 00:26:19,480
limits we'll find.

426
00:26:19,480 --> 00:26:24,760
We'll discover that the first
little delta x, which goes up

427
00:26:24,760 --> 00:26:30,610
by sine of delta x, has a slope
that gets closer and

428
00:26:30,610 --> 00:26:34,120
closer to 1.

429
00:26:34,120 --> 00:26:36,440
Good.

430
00:26:36,440 --> 00:26:42,210
Luckily, cosine does start
at 1, so we're OK so far.

431
00:26:42,210 --> 00:26:45,120
Now the slope is dropping.

432
00:26:45,120 --> 00:26:49,330
And what's the slope at the
top of the sine curve?

433
00:26:49,330 --> 00:26:51,810
It's a maximum.

434
00:26:51,810 --> 00:26:56,710
But we identify that by the
fact that the slope is 0,

435
00:26:56,710 --> 00:27:01,690
because we know the thing is
going to go down here and go

436
00:27:01,690 --> 00:27:02,780
somewhere else.

437
00:27:02,780 --> 00:27:04,410
The slope there is 0.

438
00:27:04,410 --> 00:27:07,400
The tangent line
is horizontal.

439
00:27:07,400 --> 00:27:12,980
And that is that point.

440
00:27:12,980 --> 00:27:14,260
It passes through 0.

441
00:27:14,260 --> 00:27:15,980
The slope is dropping.

442
00:27:15,980 --> 00:27:18,030
So this is the slope curve.

443
00:27:18,030 --> 00:27:22,200
And the great thing is that
it's the cosine of x.

444
00:27:22,200 --> 00:27:26,090
And what I'm doing now is
not proving this fact.

445
00:27:26,090 --> 00:27:28,610
I'm not doing my delta x's.

446
00:27:28,610 --> 00:27:33,160
That's the job I have to do
once, and it won't be today,

447
00:27:33,160 --> 00:27:34,445
but I only have to do it once.

448
00:27:34,445 --> 00:27:37,820

449
00:27:37,820 --> 00:27:40,330
But today, I'm just saying
it makes sense

450
00:27:40,330 --> 00:27:42,630
the slope is dropping.

451
00:27:42,630 --> 00:27:46,700
In that first part, I'm going
up, so the slope is positive

452
00:27:46,700 --> 00:27:49,910
but the slope is dropping.

453
00:27:49,910 --> 00:27:52,650
And then, at this point,
it hits 0.

454
00:27:52,650 --> 00:27:54,500
And that's this point.

455
00:27:54,500 --> 00:27:58,730
And then the slope
turns negative.

456
00:27:58,730 --> 00:28:00,110
I'm falling.

457
00:28:00,110 --> 00:28:02,830
So the slope goes negative,
and actually

458
00:28:02,830 --> 00:28:04,420
it follows the cosine.

459
00:28:04,420 --> 00:28:09,240
So I go along here to that
point, and then I can continue

460
00:28:09,240 --> 00:28:13,840
on to this point where
it bottoms out again

461
00:28:13,840 --> 00:28:16,760
and then starts up.

462
00:28:16,760 --> 00:28:19,140
So where is that
on this curve?

463
00:28:19,140 --> 00:28:23,300
Well, I'd better draw a
little further out.

464
00:28:23,300 --> 00:28:27,730
This bottom here
would be the--

465
00:28:27,730 --> 00:28:28,980
This is our pi/2.

466
00:28:28,980 --> 00:28:31,220

467
00:28:31,220 --> 00:28:36,270
This is our pi, 180 degrees,
everybody would say.

468
00:28:36,270 --> 00:28:39,280

469
00:28:39,280 --> 00:28:41,510
So what's happening
on that curve?

470
00:28:41,510 --> 00:28:47,760
The function is dropping, and
actually it's dropping its

471
00:28:47,760 --> 00:28:51,510
fastest. It's dropping its
fastest at that point, which

472
00:28:51,510 --> 00:28:53,640
is the slope is minus 1.

473
00:28:53,640 --> 00:28:57,180
And then the slope is still
negative, but it's not so

474
00:28:57,180 --> 00:29:02,670
negative, and it comes back up
to 0 at 3 pi/2 So this is the

475
00:29:02,670 --> 00:29:05,500
point, 3 pi/2.

476
00:29:05,500 --> 00:29:10,530
And this has come back
to 0 at that point.

477
00:29:10,530 --> 00:29:14,500
And then it finishes the
whole thing at 2 pi.

478
00:29:14,500 --> 00:29:17,920
This finishes up here
back at 1 again.

479
00:29:17,920 --> 00:29:19,170
It's climbing.

480
00:29:19,170 --> 00:29:22,050

481
00:29:22,050 --> 00:29:26,850
All right, climbing, dropping,
faster, slower, maximum,

482
00:29:26,850 --> 00:29:33,070
minimum, those are the words
that make derivatives

483
00:29:33,070 --> 00:29:36,150
important and useful to learn.

484
00:29:36,150 --> 00:29:40,930
And we've done, in detail,
the first of our

485
00:29:40,930 --> 00:29:45,580
great list of functions.

486
00:29:45,580 --> 00:29:46,720
Thanks.

487
00:29:46,720 --> 00:29:48,480
FEMALE SPEAKER: This has been
a production of MIT

488
00:29:48,480 --> 00:29:50,870
OpenCourseWare and
Gilbert Strang.

489
00:29:50,870 --> 00:29:53,140
Funding for this video was
provided by the Lord

490
00:29:53,140 --> 00:29:54,360
Foundation.

491
00:29:54,360 --> 00:29:57,490
To help OCW continue to provide
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492
00:29:57,490 --> 00:30:00,570
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make a donation at

493
00:30:00,570 --> 00:30:02,130
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494
00:30:02,130 --> 00:30:04,223