WEBVTT

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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,

00:13:30.530 --> 00:13:32.830
delta x, then it's
delta x squared.

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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.

00:13:46.760 --> 00:13:49.342
Out is-- from here
out to delta x.

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OK.

00:13:52.160 --> 00:13:57.920
Now, again, that's still only
an average, because delta x

00:13:57.920 --> 00:13:59.480
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

00:14:06.680 --> 00:14:09.930
shorter and shorter and shorter
pieces in order to

00:14:09.930 --> 00:14:17.120
zoom in on that instant, that
moment, that spot where we're

00:14:17.120 --> 00:14:19.630
looking at the slope, and where
we're expecting the

00:14:19.630 --> 00:14:22.500
answer is 0, in this case.

00:14:22.500 --> 00:14:26.270
And you see that the average,
it happens to

00:14:26.270 --> 00:14:28.300
be especially simple.

00:14:28.300 --> 00:14:31.180
Delta x squared over delta
x is just delta x.

00:14:31.180 --> 00:14:38.710
So the average slope
is extremely small.

00:14:38.710 --> 00:14:43.140
And I'll just complete
that thought.

00:14:43.140 --> 00:14:55.040
So the instant slope-- instant
slope at 0, at x equals 0, I

00:14:55.040 --> 00:14:56.860
let this delta x get smaller
and smaller.

00:14:56.860 --> 00:15:01.440
I get the answer is 0, which
is just what I expected it.

00:15:01.440 --> 00:15:04.520
And you could say, well,
not too exciting.

00:15:04.520 --> 00:15:07.110
But it was an easy one to do.

00:15:07.110 --> 00:15:10.400
It was the first time that we
actually went through the

00:15:10.400 --> 00:15:12.260
steps of computing.

00:15:12.260 --> 00:15:15.180
This is a, like, a delta y.

00:15:15.180 --> 00:15:18.440
This is the delta x.

00:15:18.440 --> 00:15:23.140
Instead of 3/1, starting
here I had delta x

00:15:23.140 --> 00:15:24.290
squared over delta x.

00:15:24.290 --> 00:15:25.520
That was easy to see.

00:15:25.520 --> 00:15:26.730
It was delta x.

00:15:26.730 --> 00:15:31.180
And if I move in closer, that
average slope is smaller and

00:15:31.180 --> 00:15:35.670
smaller, and the slope
at that instant is 0.

00:15:35.670 --> 00:15:37.970
No problem.

00:15:37.970 --> 00:15:43.340
The travel, the climbing
began from rest, but

00:15:43.340 --> 00:15:45.220
it picked up speed.

00:15:45.220 --> 00:15:48.300
The slope here is
certainly not 0.

00:15:48.300 --> 00:15:50.200
We'll find that slope.

00:15:50.200 --> 00:15:52.740
We need now to find the
slope at every point.

00:15:52.740 --> 00:15:55.210
OK.

00:15:55.210 --> 00:15:57.810
That's a good start.

00:15:57.810 --> 00:16:04.170
Now I'm ready to find the
slope at any point.

00:16:04.170 --> 00:16:09.890
Instead of just x equals 0,
which we've now done, I better

00:16:09.890 --> 00:16:18.920
draw a new graph of the same
picture, climbing up.

00:16:18.920 --> 00:16:23.900
Now I'm putting in a little
climb at some point x here.

00:16:23.900 --> 00:16:26.085
I'm up at a height, x squared.

00:16:30.590 --> 00:16:33.310
I'm at that point
on the climb.

00:16:33.310 --> 00:16:37.410
I'd like to know the slope
there, at that point.

00:16:37.410 --> 00:16:40.110
How am I going to do it?

00:16:40.110 --> 00:16:45.880
I will follow this as the
central dogma of calculus, of

00:16:45.880 --> 00:16:50.810
differential calculus, function
1 to function 2.

00:16:50.810 --> 00:16:56.990
Take a little delta x, go as
far as x plus delta x.

00:16:56.990 --> 00:16:59.860
That will take you to a higher
point on the curve.

00:16:59.860 --> 00:17:05.230
That's now the point x plus
delta x squared, because our

00:17:05.230 --> 00:17:09.730
curve is still y equals
x squared in this

00:17:09.730 --> 00:17:11.919
nice, simple parabola.

00:17:11.919 --> 00:17:13.810
OK.

00:17:13.810 --> 00:17:19.829
So now I you look at distance
across and distance up.

00:17:19.829 --> 00:17:25.210
So delta y is the change up.

00:17:25.210 --> 00:17:30.360
Delta x is the across.

00:17:30.360 --> 00:17:35.500
And I have to put
what is delta y.

00:17:35.500 --> 00:17:38.310
I have to write in,
what is delta y?

00:17:38.310 --> 00:17:40.280
It's this distance up.

00:17:40.280 --> 00:17:44.700
It's x plus delta x squared.

00:17:44.700 --> 00:17:49.120
That's this height minus
this height.

00:17:49.120 --> 00:17:51.610
I'm not counting this
bit, of course.

00:17:51.610 --> 00:17:53.190
It's that that I want.

00:17:53.190 --> 00:17:58.040
That's the delta y,
is this piece.

00:17:58.040 --> 00:18:03.600
So it's up to this, subtract
x squared.

00:18:03.600 --> 00:18:05.780
That's delta y.

00:18:05.780 --> 00:18:07.470
That's important.

00:18:07.470 --> 00:18:11.520
Now I divide by delta x.

00:18:11.520 --> 00:18:13.820
This is all algebra now.

00:18:13.820 --> 00:18:18.320
Calculus is going to come in
a moment, but not yet.

00:18:18.320 --> 00:18:20.160
For algebra, what do I do?

00:18:20.160 --> 00:18:21.870
I multiply this out.

00:18:21.870 --> 00:18:23.520
I see that thing squared.

00:18:23.520 --> 00:18:26.160
I remember that x is squared.

00:18:26.160 --> 00:18:29.190
And then I have this
times this twice.

00:18:29.190 --> 00:18:34.133
2x delta x's, and then I
have delta x squared.

00:18:36.750 --> 00:18:38.820
And then I'm subtracting
x squared.

00:18:38.820 --> 00:18:47.080
So that's delta y written
out in full glory.

00:18:47.080 --> 00:18:53.080
I wrote it out because
now I can simplify by

00:18:53.080 --> 00:18:55.400
canceling the x squared.

00:18:55.400 --> 00:18:58.200
I'm not surprised.

00:18:58.200 --> 00:19:01.840
Now, in this case, I can
actually do the division.

00:19:01.840 --> 00:19:04.420
Delta x is just there.

00:19:04.420 --> 00:19:06.520
So it leaves me with a 2x.

00:19:06.520 --> 00:19:08.810
Delta x over delta x is 1.

00:19:08.810 --> 00:19:11.810
And then here's a delta x
squared over a delta x, so

00:19:11.810 --> 00:19:13.940
that leaves me with
one delta x.

00:19:17.930 --> 00:19:21.400
As you get the hang of
calculus, you see the

00:19:21.400 --> 00:19:27.590
important things is like this
first order, delta x to the

00:19:27.590 --> 00:19:28.680
first power.

00:19:28.680 --> 00:19:34.370
Delta x squared, that, when
divided by delta x, gives us

00:19:34.370 --> 00:19:38.130
this, which is going
to disappear.

00:19:38.130 --> 00:19:39.490
That's the point.

00:19:39.490 --> 00:19:48.940
This was the average over a
short but still not instant

00:19:48.940 --> 00:19:52.080
range, distance.

00:19:52.080 --> 00:19:54.870
Now, what happens?

00:19:54.870 --> 00:19:56.130
Now dy/dx.

00:20:00.180 --> 00:20:09.470
So if this is short, short over
short, this is darn short

00:20:09.470 --> 00:20:11.160
over darn short.

00:20:11.160 --> 00:20:18.610
That d is, well, it's
too short to see.

00:20:18.610 --> 00:20:23.880
So I don't actually now try
to separate a distance dy.

00:20:23.880 --> 00:20:31.350
This isn't a true division,
because it's effectively 0/0.

00:20:31.350 --> 00:20:35.360
And you might say, well, 0/0,
what's the meaning?

00:20:35.360 --> 00:20:40.620
Well, the meaning of 0/0, in
this situation, is, I take the

00:20:40.620 --> 00:20:54.140
limit of this one, which does
have a meaning, because those

00:20:54.140 --> 00:20:55.070
are true numbers.

00:20:55.070 --> 00:20:58.320
They're little numbers
but they're numbers.

00:20:58.320 --> 00:21:07.440
And this was this, so now here's
the big step, leaving

00:21:07.440 --> 00:21:11.380
algebra behind, going to
calculus in order to get

00:21:11.380 --> 00:21:13.780
what's happening at a point.

00:21:13.780 --> 00:21:16.230
I let delta x go to 0.

00:21:16.230 --> 00:21:17.510
And what is that?

00:21:17.510 --> 00:21:19.900
So delta y over delta
x is this.

00:21:19.900 --> 00:21:22.380
What is the dy/dx?

00:21:22.380 --> 00:21:28.960
So in the limit, ah,
it's not hard.

00:21:28.960 --> 00:21:31.270
Here's the 2x.

00:21:31.270 --> 00:21:34.030
It's there.

00:21:34.030 --> 00:21:35.570
Here's the delta x.

00:21:35.570 --> 00:21:39.280
In the limit, it disappears.

00:21:39.280 --> 00:21:44.460
So the conclusion is that
the derivative is 2x.

00:21:48.330 --> 00:21:50.000
So that's function two.

00:21:50.000 --> 00:21:51.660
That's function two here.

00:21:51.660 --> 00:21:53.450
That's the slope function.

00:21:53.450 --> 00:21:55.060
That's the speed function.

00:21:55.060 --> 00:21:56.390
Maybe I should draw it.

00:21:56.390 --> 00:21:59.370
Can I draw it above and then
I'll put the board back up?

00:21:59.370 --> 00:22:06.740
So here's a picture of function
2, the derivative, or

00:22:06.740 --> 00:22:08.910
the slope, which I
was calling s.

00:22:08.910 --> 00:22:12.070
So that's the s function,
against x.

00:22:12.070 --> 00:22:16.320
x is still the thing that's
varying, or it could be t, or

00:22:16.320 --> 00:22:20.270
it could be whatever
letter we've got.

00:22:20.270 --> 00:22:23.710
And the answer was 2x
for this function.

00:22:23.710 --> 00:22:28.970
So if I graph it, it starts
at 0, and it climbs

00:22:28.970 --> 00:22:32.540
steadily with slope 2.

00:22:32.540 --> 00:22:37.184
So that's a graph of s of x.

00:22:37.184 --> 00:22:40.460
And for example--

00:22:40.460 --> 00:22:45.490
yeah, so take a couple of
points on that graph--

00:22:45.490 --> 00:22:50.520
at x equals 0, the slope is 0.

00:22:50.520 --> 00:22:55.520
And we did that first. And
we actually got it right.

00:22:55.520 --> 00:22:59.680
The slope is 0 at the start,
at the bottom of the curve.

00:22:59.680 --> 00:23:04.430
At some other point on the
curve, what's the slope here?

00:23:04.430 --> 00:23:08.240
Ha, yeah, tell me
the slope there.

00:23:08.240 --> 00:23:12.100
At that point on the curve, an
average slope was 3/1, but

00:23:12.100 --> 00:23:15.180
that was the slope of this,
like, you know--

00:23:19.490 --> 00:23:21.660
sometimes called a chord.

00:23:21.660 --> 00:23:26.040
That's over a big jump of 1.

00:23:26.040 --> 00:23:29.190
Then I did it over a small jump
of delta x, and then I

00:23:29.190 --> 00:23:34.790
let delta x go to 0, so it was
an instant infinitesimal jump.

00:23:34.790 --> 00:23:39.630
So the actual slope, the way to
visualize it is that it's

00:23:39.630 --> 00:23:41.560
more like that.

00:23:41.560 --> 00:23:45.170
That's the line that's
really giving the

00:23:45.170 --> 00:23:47.860
slope at that point.

00:23:47.860 --> 00:23:49.240
That's my best picture.

00:23:49.240 --> 00:23:54.920
It's not Rembrandt,
but it's got it.

00:23:54.920 --> 00:23:57.040
And what is the slope
at that point?

00:23:57.040 --> 00:24:00.540
Well, that's what our
calculation was.

00:24:00.540 --> 00:24:02.410
It found the slope
at that point.

00:24:02.410 --> 00:24:11.250
And at the particular point, x
equals 1, the height was 2.

00:24:11.250 --> 00:24:12.570
The slope is 2.

00:24:12.570 --> 00:24:17.720
The actual tangent line
is only-- is there.

00:24:17.720 --> 00:24:18.400
You see?

00:24:18.400 --> 00:24:20.730
It's up.

00:24:20.730 --> 00:24:24.350
Oh, wait a minute.

00:24:24.350 --> 00:24:27.950
Yeah, well, the slope is 2.

00:24:27.950 --> 00:24:31.230
I don't know.

00:24:31.230 --> 00:24:34.510
This goes up to 3.

00:24:34.510 --> 00:24:41.300
It's not Rembrandt, but
the math is OK.

00:24:41.300 --> 00:24:43.070
So what have we done?

00:24:43.070 --> 00:24:49.130
We've taken the first small step
and literally I could say

00:24:49.130 --> 00:24:52.290
small step, almost a play on
words because that's the

00:24:52.290 --> 00:24:55.180
point, the step is so small--

00:24:55.180 --> 00:24:59.170
to getting these great
functions.

00:24:59.170 --> 00:25:06.110
Before I close this lecture, can
I draw this pair, function

00:25:06.110 --> 00:25:14.630
1 and function 2, and just see
that the movement of the

00:25:14.630 --> 00:25:16.650
curves is what we
would expect.

00:25:16.650 --> 00:25:23.710
So let me, just for one more
good example, great example,

00:25:23.710 --> 00:25:26.510
actually, is let me draw.

00:25:26.510 --> 00:25:28.990
Here goes x.

00:25:28.990 --> 00:25:32.830
In fact, maybe I already drew
in the first letter, lecture

00:25:32.830 --> 00:25:37.890
that bit out to 90 degrees.

00:25:37.890 --> 00:25:41.140
Only if we want a nice formula,
we better call that

00:25:41.140 --> 00:25:45.000
pi over 2 radians.

00:25:45.000 --> 00:25:46.870
And here's a graph of sine x.

00:25:46.870 --> 00:25:48.450
This is y.

00:25:48.450 --> 00:25:52.380
This is the function
1, sine x.

00:25:52.380 --> 00:25:54.530
And what's function 2?

00:25:54.530 --> 00:25:57.210
What can we see about
function 2?

00:26:00.400 --> 00:26:01.650
Again, x.

00:26:04.560 --> 00:26:06.410
We see a slope.

00:26:06.410 --> 00:26:09.160
This is not the same
as x squared.

00:26:09.160 --> 00:26:12.230
This starts with a
definite slope.

00:26:12.230 --> 00:26:17.690
And it turns out this will be
one of the most important

00:26:17.690 --> 00:26:19.480
limits we'll find.

00:26:19.480 --> 00:26:24.760
We'll discover that the first
little delta x, which goes up

00:26:24.760 --> 00:26:30.610
by sine of delta x, has a slope
that gets closer and

00:26:30.610 --> 00:26:34.120
closer to 1.

00:26:34.120 --> 00:26:36.440
Good.

00:26:36.440 --> 00:26:42.210
Luckily, cosine does start
at 1, so we're OK so far.

00:26:42.210 --> 00:26:45.120
Now the slope is dropping.

00:26:45.120 --> 00:26:49.330
And what's the slope at the
top of the sine curve?

00:26:49.330 --> 00:26:51.810
It's a maximum.

00:26:51.810 --> 00:26:56.710
But we identify that by the
fact that the slope is 0,

00:26:56.710 --> 00:27:01.690
because we know the thing is
going to go down here and go

00:27:01.690 --> 00:27:02.780
somewhere else.

00:27:02.780 --> 00:27:04.410
The slope there is 0.

00:27:04.410 --> 00:27:07.400
The tangent line
is horizontal.

00:27:07.400 --> 00:27:12.980
And that is that point.

00:27:12.980 --> 00:27:14.260
It passes through 0.

00:27:14.260 --> 00:27:15.980
The slope is dropping.

00:27:15.980 --> 00:27:18.030
So this is the slope curve.

00:27:18.030 --> 00:27:22.200
And the great thing is that
it's the cosine of x.

00:27:22.200 --> 00:27:26.090
And what I'm doing now is
not proving this fact.

00:27:26.090 --> 00:27:28.610
I'm not doing my delta x's.

00:27:28.610 --> 00:27:33.160
That's the job I have to do
once, and it won't be today,

00:27:33.160 --> 00:27:34.445
but I only have to do it once.

00:27:37.820 --> 00:27:40.330
But today, I'm just saying
it makes sense

00:27:40.330 --> 00:27:42.630
the slope is dropping.

00:27:42.630 --> 00:27:46.700
In that first part, I'm going
up, so the slope is positive

00:27:46.700 --> 00:27:49.910
but the slope is dropping.

00:27:49.910 --> 00:27:52.650
And then, at this point,
it hits 0.

00:27:52.650 --> 00:27:54.500
And that's this point.

00:27:54.500 --> 00:27:58.730
And then the slope
turns negative.

00:27:58.730 --> 00:28:00.110
I'm falling.

00:28:00.110 --> 00:28:02.830
So the slope goes negative,
and actually

00:28:02.830 --> 00:28:04.420
it follows the cosine.

00:28:04.420 --> 00:28:09.240
So I go along here to that
point, and then I can continue

00:28:09.240 --> 00:28:13.840
on to this point where
it bottoms out again

00:28:13.840 --> 00:28:16.760
and then starts up.

00:28:16.760 --> 00:28:19.140
So where is that
on this curve?

00:28:19.140 --> 00:28:23.300
Well, I'd better draw a
little further out.

00:28:23.300 --> 00:28:27.730
This bottom here
would be the--

00:28:27.730 --> 00:28:28.980
This is our pi/2.

00:28:31.220 --> 00:28:36.270
This is our pi, 180 degrees,
everybody would say.

00:28:39.280 --> 00:28:41.510
So what's happening
on that curve?

00:28:41.510 --> 00:28:47.760
The function is dropping, and
actually it's dropping its

00:28:47.760 --> 00:28:51.510
fastest. It's dropping its
fastest at that point, which

00:28:51.510 --> 00:28:53.640
is the slope is minus 1.

00:28:53.640 --> 00:28:57.180
And then the slope is still
negative, but it's not so

00:28:57.180 --> 00:29:02.670
negative, and it comes back up
to 0 at 3 pi/2 So this is the

00:29:02.670 --> 00:29:05.500
point, 3 pi/2.

00:29:05.500 --> 00:29:10.530
And this has come back
to 0 at that point.

00:29:10.530 --> 00:29:14.500
And then it finishes the
whole thing at 2 pi.

00:29:14.500 --> 00:29:17.920
This finishes up here
back at 1 again.

00:29:17.920 --> 00:29:19.170
It's climbing.

00:29:22.050 --> 00:29:26.850
All right, climbing, dropping,
faster, slower, maximum,

00:29:26.850 --> 00:29:33.070
minimum, those are the words
that make derivatives

00:29:33.070 --> 00:29:36.150
important and useful to learn.

00:29:36.150 --> 00:29:40.930
And we've done, in detail,
the first of our

00:29:40.930 --> 00:29:45.580
great list of functions.

00:29:45.580 --> 00:29:46.720
Thanks.

00:29:46.720 --> 00:29:48.480
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00:29:48.480 --> 00:29:50.870
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00:29:50.870 --> 00:29:53.140
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00:29:53.140 --> 00:29:54.360
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00:29:54.360 --> 00:29:57.490
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