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PROFESSOR: OK, this lecture
is about the slopes, the
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derivatives, of two of the
great functions of
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mathematics: sine
x and cosine x.
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Why do I say great functions?
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What sort of motion do we
see sines and cosines?
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Well, I guess I'm thinking
of oscillations.
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Things go back and forth.
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They go up and down.
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They go round in a circle.
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Your heart beats and
beats and beats.
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Your lungs go in and out.
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The earth goes around the sun.
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So many motions are repeating
motions, and that's when sines
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and cosines show up.
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The opposite is growing
motions.
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That's where we have powers of
x, x cubed, x to the n-th.
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Or if we really want
the motion to get
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going, e to the x.
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Or decaying would be
e to the minus x.
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So there are two kinds here.
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We're talking about the ones
that repeat and stay level,
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and they all involve
sines and cosines.
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And to make that point,
I'm going to have to--
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you know what sines and cosines
are for triangles from
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trigonometry.
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But I have to make those
triangles move.
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So I'm going to put the triangle
in a circle, with one
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corner at the center, and
another corner on the circle,
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and I'm going to move
that point.
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So it's going to be
circular motion.
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It's going to be the
motion that--
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the perfect model of repeating
motion, around
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and around the circle.
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And then the answer we're going
to get is just great.
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The derivative of sine x turns
out to be cosine x.
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And the derivative of cosine x
turns out to be minus sine x.
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You couldn't ask for more.
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So my interest is always to
explain those, but then I want
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to really--
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we're seeing this limit stuff
in taking a derivative, and
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here's a chance for me
to find a limit.
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This turns out to be the crucial
quantity: the sine of
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an angle divided by the angle,
when the angle goes to 0.
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Of course, when it's at 0,
the sine of 0 is 0, so
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we have 0 over 0.
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This is the big problem
of calculus.
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You can't be at the limit,
because it's 0
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over 0 at that point.
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But you can be close to it.
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And then if we drew a graph, had
a calculator, whatever we
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do, we would see that that ratio
is very close to 1, but
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today we're going to actually
prove it from the meaning of
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sine theta.
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Now remember what
that meaning is.
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So back to the start
of the world.
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Actually back to Pythagoras,
way, way back.
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The key fact is what you
remember about right
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triangles, a squared plus b
squared equals c squared.
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That's where everything starts
for a right triangle.
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I don't know if Pythagoras
knew how to prove it.
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I think his friends
helped him.
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A lot of people have
suggested proofs.
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Einstein gave a proof.
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Some US president even
gave a proof.
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So it's a fundamental fact, and
I'm going to divide by c
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squared, because I'd like the
right hand side to be 1.
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So if I divide by c squared, I
just have a squared over c
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squared plus b squared
over c squared is 1.
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And I'm going to make that
hypotenuse in my picture 1.
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So then this will be the a over
c, and that ratio of the
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near side to the hypotenuse
is the cosine.
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So what I have here is
cosine theta squared.
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Let me put theta in there.
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Theta is that angle
at the center.
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And what's b?
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So this is a over c.
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That's cosine theta.
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B over c is this point,
and that's sine theta.
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And they add to 1.
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So that's Pythagoras using
sines and cosines.
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So this is the cosine.
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And this vertical distance
is sine theta.
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OK, so that's the
triangle I like.
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That's the triangle that's
going to move.
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As this point goes around the
circle at a steady speed, this
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triangle is going to move.
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The base will go left and
right, left and right.
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The height will go up and down,
up and down, following
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cosine and sine.
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And we want to know things
about the speed.
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OK, so that's circular motion.
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Now I've introduced
this word radians.
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And let me remind you what they
are and why we need them.
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Why don't we just use 360
degrees for the full circle?
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360 degrees.
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Well, that's a nice
number, 360.
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Somebody must have thought it
was really nice, and chose it
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for measuring angles
around the world.
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It's nice, but it's
not natural.
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Somebody thought of it,
so it's not good.
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What we need is the natural
way to measure the angle.
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If we don't use the natural way,
then this is the sine--
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if I measure this x
in degrees, that
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formula won't be right.
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There will be a miserable factor
that I want to be 1.
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So I have to measure the angles
the right way, and
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here's the idea of radians.
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The measure of that
angle is this
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distance around the circle.
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That distance I'm going to call
theta, and I'm going to
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say this angle is theta
radians when
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that distance is theta.
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So that now, what's
a full circle?
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A full circle would mean the
angle went all the way round.
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I get the whole circumference,
which is 2 pi.
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So 360 degrees is
2 pi radians.
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So the natural number
here is 2 pi.
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This can't be helped, it's
the right one to use.
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Radians are the right way
to measure an angle.
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So now I'm ready to do the job
of finding this derivative.
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OK.
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Let me start at the
key point 0.
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If we get this one, we get
all the rest easily.
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So I'm looking at the graph
of the sine curve.
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I'm starting at 0.
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We know what sine theta looks
like, and I'm interested in
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the slope, the derivative.
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That's what this subject
is about, calculus,
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differentiating.
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So I want to know the
slope at that point.
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And it's 1.
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And how do we show
that it's 1?
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So now I'm coming to the point
where I'm going to give a
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proof that is 1.
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And the proof isn't just for the
sake of formality or rigor
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or something.
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You really have to understand
the sine function, the cosine
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function, and this is
the heart of it.
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OK.
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So we want to show
that slope is 1.
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How am I going to do that?
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That's the slope, right?
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If I go a tiny amount theta,
then I go up sine theta.
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So in this average slope, if
I take a finite step--
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I could have called it delta
theta, but I don't want to
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write deltas all the time.
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So I just go out a little
distance theta and up to the
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sine curve.
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I stopped at the sine
curve by the way.
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The straight line is a little
above the sine curve here.
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And that ratio, up divided by
across, that's the delta sine
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divided by delta theta.
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And because it started at 0,
it's just sine theta is the
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distance up, and theta is
the distance across.
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So this is the average slope.
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And of course you remember
what calculus is doing.
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There's always this limiting
process where you push things
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closer and closer to the point,
and you find the slope
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at that point, sometimes called
the instantaneous
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velocity or slope
or derivative.
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Now here's the way it's
going to work.
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I'm going to show that sine
theta over theta is
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always below 1.
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So two facts I want to prove.
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I want to show that sine theta
over theta is less--
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sorry, sine theta over theta--
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well, let me get this right.
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I might as well put
it the neat way.
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I want to show that sine
theta is below theta.
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This is for theta
greater than 0.
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That's what I'm doing.
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OK.
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So that tells me that this
curve stays under that
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straight line, that 45 degree
line, which I'm claiming is
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the tangent line.
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And it tells me when I divide
it by the theta, it tells me
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that sine theta over
theta is below 1.
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But now how much below 1?
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Right now if I only know this,
I haven't ruled out the
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possibility that the slope
could be much smaller.
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So I need something below it.
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And fantastically, the
cosine is below it.
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So the other thing that I want
to prove is that the cosine--
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and I'll let me do
it this way.
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I'm going to show the tangent of
theta is bigger than theta.
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Again, some range of thetas.
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Positive thetas up
to somewhere.
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I don't know, I think
maybe pi over 2.
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But the main point is near
0, that's the main point.
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So can I just rewrite--
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do you remember what
the tangent is?
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Of course, sine theta
over cosine theta.
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So this is sine theta
over cosine
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theta, bigger than theta.
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We still have to prove this.
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And now I want to bring the
theta down and move the cosine
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up, and that will tell me that
sine theta, when I divide by
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the theta and multiply
by the cosine theta.
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So that was the same as that,
was the same as that.
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And that's what I want.
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That tells that this ratio is
above the cosine curve.
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Do you see that if I can
convince you, and convince
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myself that these are both true,
that this picture is
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right, then--
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I haven't gone into gory
detail about limits.
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If you really insist,
I'll do it later.
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But whatever.
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You can see this has just got to
be true, that if this curve
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is squeezed between the cosine
curve and the 1, then as theta
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gets smaller and smaller,
it's squeezed to
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equal 1 in the limit.
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Allow me to say that that's
pretty darn clear.
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OK.
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Whatever limit meets.
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So these are the main facts
that I need to show.
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And I need to show those
using trig, right?
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I have to draw some graph
that convinces you.
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And this isn't quite good
enough, because I just
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sketched a sine graph.
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I have to say where does
sine theta come from?
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OK.
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So this will be number one, and
this will be number two,
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and when we get those two things
convincing, then we
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know that sine theta over theta
is squeezed between and
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approaches 1.
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00:14:09,380 --> 00:14:14,780
And then we'll know the story
at the start, and you'll see
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that it becomes easy to
find these formulas
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all along the curve.
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OK.
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Ready for these two?
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Number one and number two.
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OK, number one.
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00:14:27,580 --> 00:14:30,200
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Why is sine theta--
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00:14:31,590 --> 00:14:35,240
oh, I can probably see
it on this picture.
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Yeah, I can prove number
one on this picture.
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Look, that piece was
sine theta, right?
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00:14:40,950 --> 00:14:45,060
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00:14:45,060 --> 00:14:48,890
And I want to prove
that this length--
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what am I trying to prove?
254
00:14:49,910 --> 00:14:51,800
That sine theta is
below theta.
255
00:14:51,800 --> 00:14:55,701
Let me write it again
what it is to show.
256
00:14:55,701 --> 00:14:59,000
In math it's always a good
idea to keep reminding
257
00:14:59,000 --> 00:15:01,170
yourself of what it
is you're doing.
258
00:15:01,170 --> 00:15:03,899
Sine theta is below theta.
259
00:15:03,899 --> 00:15:03,958
OK.
260
00:15:03,958 --> 00:15:05,208
So why is it?
261
00:15:05,208 --> 00:15:09,000
262
00:15:09,000 --> 00:15:09,850
And you see it here.
263
00:15:09,850 --> 00:15:12,400
That was sine theta, right?
264
00:15:12,400 --> 00:15:14,260
And where was theta?
265
00:15:14,260 --> 00:15:18,690
Well, because we measured theta
in radians, theta is
266
00:15:18,690 --> 00:15:23,080
this curvy distance that's
clearly longer.
267
00:15:23,080 --> 00:15:28,250
The shortest way from this to
the axis there is straight
268
00:15:28,250 --> 00:15:30,590
down, and that's sine theta.
269
00:15:30,590 --> 00:15:35,560
A slower way is go round and
end up at not the nearest
270
00:15:35,560 --> 00:15:38,310
point, and that was theta.
271
00:15:38,310 --> 00:15:39,570
Is that good?
272
00:15:39,570 --> 00:15:44,570
I could sometimes just to be
even more convincing, you add
273
00:15:44,570 --> 00:15:48,890
a second angle, and you say OK,
there's 2 sine thetas and
274
00:15:48,890 --> 00:15:52,750
here is 2 thetas, and clearly we
all know that the shortest
275
00:15:52,750 --> 00:15:57,210
way from there to there
is the straight way.
276
00:15:57,210 --> 00:16:03,580
So I regard this as done
by that picture.
277
00:16:03,580 --> 00:16:05,960
You see we didn't
just make it up.
278
00:16:05,960 --> 00:16:08,800
It went back to the fundamental
idea of where sine
279
00:16:08,800 --> 00:16:10,390
theta is in a picture.
280
00:16:10,390 --> 00:16:13,920
Now I need another picture.
281
00:16:13,920 --> 00:16:20,100
Yeah, I need another picture
for number two to show that
282
00:16:20,100 --> 00:16:23,340
tan theta is bigger
than theta.
283
00:16:23,340 --> 00:16:24,930
That was our other job.
284
00:16:24,930 --> 00:16:30,370
So essentially, I need that
same picture again.
285
00:16:30,370 --> 00:16:33,900
Whoops, let me draw
that triangle.
286
00:16:33,900 --> 00:16:37,080
287
00:16:37,080 --> 00:16:41,240
Yeah, and it's got a circle.
288
00:16:41,240 --> 00:16:44,320
OK, that's not a bad circle.
289
00:16:44,320 --> 00:16:47,010
It's got an angle theta.
290
00:16:47,010 --> 00:16:48,260
And now I'm going to--
291
00:16:48,260 --> 00:16:51,650
292
00:16:51,650 --> 00:16:54,140
math has always got
some little trick.
293
00:16:54,140 --> 00:16:55,590
So this is it.
294
00:16:55,590 --> 00:17:01,840
Go all the way out, so now the
base is 1, and this is still
295
00:17:01,840 --> 00:17:03,090
the angle theta.
296
00:17:03,090 --> 00:17:06,069
297
00:17:06,069 --> 00:17:08,520
And what else do I know
on that picture?
298
00:17:08,520 --> 00:17:12,060
Now I've scaled the triangle
up from this little one, so
299
00:17:12,060 --> 00:17:13,190
the base is 1.
300
00:17:13,190 --> 00:17:15,589
So what's that height?
301
00:17:15,589 --> 00:17:19,960
Well, the ratio of the opposite
side to the near
302
00:17:19,960 --> 00:17:23,819
side, that's what tangent is.
303
00:17:23,819 --> 00:17:25,619
Tangent is the ratio--
304
00:17:25,619 --> 00:17:27,470
whatever size the triangle--
305
00:17:27,470 --> 00:17:30,310
is the ratio of the opposite
side to the near side.
306
00:17:30,310 --> 00:17:37,180
Sine to cosine, here it's
tan theta is that
307
00:17:37,180 --> 00:17:39,690
distance, and to 1.
308
00:17:39,690 --> 00:17:40,630
Good.
309
00:17:40,630 --> 00:17:43,650
OK, but now how am I
going to see this?
310
00:17:43,650 --> 00:17:48,640
311
00:17:48,640 --> 00:17:51,870
I have to ask you--
and it's OK--
312
00:17:51,870 --> 00:17:55,770
to think about area instead
of distance for a moment.
313
00:17:55,770 --> 00:17:57,230
What about area?
314
00:17:57,230 --> 00:17:59,130
So what do I see of area?
315
00:17:59,130 --> 00:18:06,480
I see right away that the area
of this triangle is smaller
316
00:18:06,480 --> 00:18:08,360
than the area--
317
00:18:08,360 --> 00:18:10,520
sorry, I shouldn't have called
that a triangle.
318
00:18:10,520 --> 00:18:15,020
That's a little piece of pie,
a little sector of a circle.
319
00:18:15,020 --> 00:18:17,080
So the area of this shaded--
320
00:18:17,080 --> 00:18:18,590
did I shade it OK--
321
00:18:18,590 --> 00:18:21,510
322
00:18:21,510 --> 00:18:22,980
is less than--
323
00:18:22,980 --> 00:18:26,330
so this is the area
of the sector.
324
00:18:26,330 --> 00:18:30,470
Can I just call it the pie,
piece of a pie, is less than
325
00:18:30,470 --> 00:18:32,070
the area of the triangle.
326
00:18:32,070 --> 00:18:34,860
327
00:18:34,860 --> 00:18:39,300
But we know what the area
of the triangle is.
328
00:18:39,300 --> 00:18:41,290
What's the area of a triangle?
329
00:18:41,290 --> 00:18:42,750
We can do that.
330
00:18:42,750 --> 00:18:45,200
It's the base half, right?
331
00:18:45,200 --> 00:18:48,010
1/2 times the base
times the height.
332
00:18:48,010 --> 00:18:51,230
So the area of the triangle
is 1/2 times the
333
00:18:51,230 --> 00:18:52,555
base 1 times the height.
334
00:18:52,555 --> 00:18:53,805
OK.
335
00:18:53,805 --> 00:18:56,280
336
00:18:56,280 --> 00:18:58,950
Notice we've got the sign
going the right way.
337
00:18:58,950 --> 00:19:01,680
We want tan theta to be
bigger than something,
338
00:19:01,680 --> 00:19:03,420
so what do I hope?
339
00:19:03,420 --> 00:19:08,620
I hope that the area of this
shaded part, the area of the
340
00:19:08,620 --> 00:19:15,520
circular sector, is
1/2 of theta.
341
00:19:15,520 --> 00:19:18,660
Wouldn't that be wonderful?
342
00:19:18,660 --> 00:19:22,310
If I look at those areas,
nobody's in any doubt that
343
00:19:22,310 --> 00:19:26,390
this piece, this sector that's
inside the triangle, has an
344
00:19:26,390 --> 00:19:28,530
area less than the area
of the triangle.
345
00:19:28,530 --> 00:19:33,560
So now I just have to remember
why is the area of this
346
00:19:33,560 --> 00:19:37,120
sector, half of theta.
347
00:19:37,120 --> 00:19:40,680
You know, there's another reason
why areas come up right
348
00:19:40,680 --> 00:19:44,600
when we use a radians, when we
measure theta with radians.
349
00:19:44,600 --> 00:19:52,050
So remember, just think about
this piece of pie compared to
350
00:19:52,050 --> 00:19:53,300
the whole pie.
351
00:19:53,300 --> 00:19:56,180
352
00:19:56,180 --> 00:19:59,660
What's the area of the
whole piece of pie?
353
00:19:59,660 --> 00:20:03,740
So I'm explaining 1/2 theta.
354
00:20:03,740 --> 00:20:07,320
The area of the whole pie--
355
00:20:07,320 --> 00:20:12,350
I'm going to get some terrible
pun here on the word pie.
356
00:20:12,350 --> 00:20:15,460
Unintended, forgive it.
357
00:20:15,460 --> 00:20:19,170
The area of that whole circle,
the radius is 1, we all know
358
00:20:19,170 --> 00:20:22,030
what the area of a circle
is pi r squared.
359
00:20:22,030 --> 00:20:26,020
r is 1, so the area is pi.
360
00:20:26,020 --> 00:20:29,240
My God, I didn't expect that.
361
00:20:29,240 --> 00:20:30,860
Now what about this?
362
00:20:30,860 --> 00:20:34,280
What fraction is this sector?
363
00:20:34,280 --> 00:20:40,020
Well, the whole angle would be
2 pi, and this part of it is
364
00:20:40,020 --> 00:20:49,320
theta, so I have the sector is
theta over 2 pi, that's the
365
00:20:49,320 --> 00:20:56,420
angle fraction, times the
pi, the whole area.
366
00:20:56,420 --> 00:20:58,190
Do you see it?
367
00:20:58,190 --> 00:21:02,830
This piece of pie, or
pizza, whatever--
368
00:21:02,830 --> 00:21:04,940
yeah, if I'd said pizza, I
wouldn't have had that
369
00:21:04,940 --> 00:21:06,840
terrible pun.
370
00:21:06,840 --> 00:21:08,940
Forget it.
371
00:21:08,940 --> 00:21:14,330
So the area of this piece of
pizza compared to the whole
372
00:21:14,330 --> 00:21:18,050
one is theta over
the whole 2 pi.
373
00:21:18,050 --> 00:21:21,270
374
00:21:21,270 --> 00:21:25,690
Suppose it was a pizza cut
in the usual 6 pieces.
375
00:21:25,690 --> 00:21:28,180
Then this would be a 60
degree angle, but
376
00:21:28,180 --> 00:21:30,290
I don't want degrees.
377
00:21:30,290 --> 00:21:34,010
What would be the angle of that
piece of pizza that's cut
378
00:21:34,010 --> 00:21:36,510
when the whole pizza's
cut in 6?
379
00:21:36,510 --> 00:21:41,750
It would be 1/6 of 360.
380
00:21:41,750 --> 00:21:42,900
That's 60 degrees.
381
00:21:42,900 --> 00:21:44,300
But I don't want degrees.
382
00:21:44,300 --> 00:21:47,580
It's 1/6 of 2 pi.
383
00:21:47,580 --> 00:21:50,130
And this one is theta of 2 pi.
384
00:21:50,130 --> 00:21:52,540
Anyway, the pis cancel.
385
00:21:52,540 --> 00:21:57,090
Theta over 2 is the right
answer, and now we can cancel
386
00:21:57,090 --> 00:21:59,640
the 1/2, and we've
got what we want.
387
00:21:59,640 --> 00:22:03,560
388
00:22:03,560 --> 00:22:08,660
That's pretty nice when you
realize that we were facing
389
00:22:08,660 --> 00:22:12,210
for the first time, more or
less, the sort of tough
390
00:22:12,210 --> 00:22:16,970
problem of calculus when I can't
really divide theta into
391
00:22:16,970 --> 00:22:18,270
sine theta.
392
00:22:18,270 --> 00:22:22,880
Sine theta, I can't
just divide it in.
393
00:22:22,880 --> 00:22:27,770
I have to keep them both
approaching 0 over 0, and see
394
00:22:27,770 --> 00:22:29,910
what that ratio is doing.
395
00:22:29,910 --> 00:22:36,800
And now I said to conclude,
I'll go back and prove the
396
00:22:36,800 --> 00:22:40,830
slopes, find the slopes
at all points.
397
00:22:40,830 --> 00:22:43,140
OK, so at all points--
398
00:22:43,140 --> 00:22:44,920
now let's start with sine x.
399
00:22:44,920 --> 00:22:49,150
400
00:22:49,150 --> 00:22:51,430
So what am I doing now?
401
00:22:51,430 --> 00:22:55,570
I'm looking at the sine curve.
402
00:22:55,570 --> 00:22:58,560
You remember it went up like
this and down like this.
403
00:22:58,560 --> 00:22:59,965
I'm taking any point x.
404
00:22:59,965 --> 00:23:03,220
405
00:23:03,220 --> 00:23:07,870
Suddenly I changed the angle
from theta to x, just because
406
00:23:07,870 --> 00:23:09,830
I'm used to functions of x.
407
00:23:09,830 --> 00:23:12,420
We're just talking
letters there.
408
00:23:12,420 --> 00:23:16,940
X is good, and this is
a graph of sine x.
409
00:23:16,940 --> 00:23:18,327
X is measured in radians still.
410
00:23:18,327 --> 00:23:19,600
OK.
411
00:23:19,600 --> 00:23:23,090
So now what am I doing to find
the derivative at some
412
00:23:23,090 --> 00:23:26,400
particular point?
413
00:23:26,400 --> 00:23:28,740
I look at the sine there.
414
00:23:28,740 --> 00:23:32,550
I go a little distance
delta x.
415
00:23:32,550 --> 00:23:37,120
I go up to here, and
I look to see--
416
00:23:37,120 --> 00:23:42,950
I want to know the change
in sine x divided by
417
00:23:42,950 --> 00:23:46,600
the change in x.
418
00:23:46,600 --> 00:23:48,710
And of course, I'm going
to let the piece
419
00:23:48,710 --> 00:23:49,750
get smaller and smaller.
420
00:23:49,750 --> 00:23:52,320
That's what calculus does.
421
00:23:52,320 --> 00:23:55,700
The main point is my x is
now here instead of
422
00:23:55,700 --> 00:23:57,120
being at the start.
423
00:23:57,120 --> 00:23:59,510
I've done it for the start, but
now I have to do it for
424
00:23:59,510 --> 00:24:00,670
all the other x's.
425
00:24:00,670 --> 00:24:03,170
So there's the x.
426
00:24:03,170 --> 00:24:05,820
There's the x plus delta
x, a little bit long.
427
00:24:05,820 --> 00:24:09,380
428
00:24:09,380 --> 00:24:13,230
In other words, can I write this
in the familiar way, sine
429
00:24:13,230 --> 00:24:20,275
of x plus delta minus sine
there divided by delta x?
430
00:24:20,275 --> 00:24:21,525
OK.
431
00:24:21,525 --> 00:24:23,230
432
00:24:23,230 --> 00:24:28,150
So again, we can't simplify
totally by just dividing the
433
00:24:28,150 --> 00:24:31,000
delta x in.
434
00:24:31,000 --> 00:24:33,920
We've got to go back
to trigonometry.
435
00:24:33,920 --> 00:24:40,500
Trig had a formula for
the sine of a plus b.
436
00:24:40,500 --> 00:24:45,440
Two angles added, then there's
a neat formula for it.
437
00:24:45,440 --> 00:24:46,540
So the sine--
438
00:24:46,540 --> 00:24:50,260
can I remind you of
that formula?
439
00:24:50,260 --> 00:24:55,390
It is the sine of the first
angle times the cosine of the
440
00:24:55,390 --> 00:25:04,960
second minus the cosine of the
first angle times the sine of
441
00:25:04,960 --> 00:25:05,290
the second.
442
00:25:05,290 --> 00:25:06,540
OK?
443
00:25:06,540 --> 00:25:09,760
444
00:25:09,760 --> 00:25:12,095
You remember this,
right, from trig?
445
00:25:12,095 --> 00:25:15,310
The sine of a plus b
is this neat thing.
446
00:25:15,310 --> 00:25:17,620
Now I have to subtract sine x.
447
00:25:17,620 --> 00:25:20,550
So now can I subtract
off sine x?
448
00:25:20,550 --> 00:25:27,650
When I subtract off sine x,
then I need a minus 1.
449
00:25:27,650 --> 00:25:31,920
And now I have to divide
by delta x.
450
00:25:31,920 --> 00:25:37,270
So I divide this by delta x, and
I divide this by delta x.
451
00:25:37,270 --> 00:25:38,520
OK.
452
00:25:38,520 --> 00:25:42,090
453
00:25:42,090 --> 00:25:43,990
This is an expression
I can work with.
454
00:25:43,990 --> 00:25:47,850
That's why I had to remember
this trig formula to get this
455
00:25:47,850 --> 00:25:49,700
expression that I
can work with.
456
00:25:49,700 --> 00:25:51,490
Why do I say I can
work with it?
457
00:25:51,490 --> 00:25:56,430
Because this is exactly what
I've already pinned down.
458
00:25:56,430 --> 00:25:59,080
Delta x is now headed for 0.
459
00:25:59,080 --> 00:26:03,040
This point is going to come
close to this one.
460
00:26:03,040 --> 00:26:07,330
So actually, I've got two terms.
This sine delta x over
461
00:26:07,330 --> 00:26:12,590
delta x, what does that do
as delta x goes to 0?
462
00:26:12,590 --> 00:26:13,960
It goes to 1.
463
00:26:13,960 --> 00:26:16,900
That was the point of that
whole right hand board.
464
00:26:16,900 --> 00:26:21,750
So this thing goes to 1.
465
00:26:21,750 --> 00:26:23,890
Wait a minute.
466
00:26:23,890 --> 00:26:26,380
That's a plus sign.
467
00:26:26,380 --> 00:26:33,530
Everybody watching is going
to think, OK, forgot trig.
468
00:26:33,530 --> 00:26:36,600
The sine of the sum of an angle
is the sine times the
469
00:26:36,600 --> 00:26:41,400
cosine plus the cosine
times the sine.
470
00:26:41,400 --> 00:26:43,190
Sorry about that one too.
471
00:26:43,190 --> 00:26:47,680
OK, so sine of delta x over
delta x goes to 0.
472
00:26:47,680 --> 00:26:54,300
And now finally, this goes to 1,
and actually I need another
473
00:26:54,300 --> 00:26:55,590
little piece.
474
00:26:55,590 --> 00:26:59,030
I need to know this piece, and
I need to know that that
475
00:26:59,030 --> 00:27:02,010
ratio goes to 0.
476
00:27:02,010 --> 00:27:04,520
477
00:27:04,520 --> 00:27:08,750
So I need to go back to that
board and look again at the
478
00:27:08,750 --> 00:27:11,770
cosine curve at 0.
479
00:27:11,770 --> 00:27:17,240
Because this will be a slope
of the cosine curve at 0.
480
00:27:17,240 --> 00:27:20,840
And the slope comes out 0
for the cosine curve.
481
00:27:20,840 --> 00:27:23,030
The slope for the sine
curve came out 1.
482
00:27:23,030 --> 00:27:25,470
Do you see how it's working?
483
00:27:25,470 --> 00:27:29,330
So this is gone because
of the 0.
484
00:27:29,330 --> 00:27:31,570
This is the cosine x times 1.
485
00:27:31,570 --> 00:27:36,590
All together I get
cosine of x.
486
00:27:36,590 --> 00:27:37,230
Hooray.
487
00:27:37,230 --> 00:27:38,430
That's the goal.
488
00:27:38,430 --> 00:27:44,410
That's the predicted plan,
desired formula cos x for the
489
00:27:44,410 --> 00:27:51,070
ratio of delta of sine x over
delta x as delta x goes to 0.
490
00:27:51,070 --> 00:27:53,270
Do you see that?
491
00:27:53,270 --> 00:27:56,760
So we used a trig formula,
and we got the sine
492
00:27:56,760 --> 00:27:59,340
right a little late.
493
00:27:59,340 --> 00:28:01,910
Well, of course the reason I--
494
00:28:01,910 --> 00:28:06,920
one reason I goofed was that the
other example, the other
495
00:28:06,920 --> 00:28:11,640
case we need for the
second formula does
496
00:28:11,640 --> 00:28:13,050
have a minus sign.
497
00:28:13,050 --> 00:28:16,160
And it survives in the end.
498
00:28:16,160 --> 00:28:21,740
So I would do exactly the same
thing for the cosines that I
499
00:28:21,740 --> 00:28:22,930
did for the sines.
500
00:28:22,930 --> 00:28:24,710
If there's another board
underneath here,
501
00:28:24,710 --> 00:28:25,740
I'm going to do it.
502
00:28:25,740 --> 00:28:28,200
Yeah, there is.
503
00:28:28,200 --> 00:28:38,480
Now I want to know the delta
of cosine x over delta x.
504
00:28:38,480 --> 00:28:43,190
That's what we do, we have to
simplify that, then we have to
505
00:28:43,190 --> 00:28:44,930
let delta x go to 0.
506
00:28:44,930 --> 00:28:46,780
So what does this mean?
507
00:28:46,780 --> 00:28:52,570
This means the cosine a little
bit along minus the cosine at
508
00:28:52,570 --> 00:28:55,950
the point divided by delta x.
509
00:28:55,950 --> 00:29:00,810
Again, we can't do that division
just right away, but
510
00:29:00,810 --> 00:29:03,790
we can simplify this
by remembering the
511
00:29:03,790 --> 00:29:06,750
formula that cosign--
512
00:29:06,750 --> 00:29:08,300
now let me try to remember it.
513
00:29:08,300 --> 00:29:17,160
It's a cosine times a cosine
for this guy plus a sine--
514
00:29:17,160 --> 00:29:23,510
no, minus a sine
times the sine.
515
00:29:23,510 --> 00:29:28,610
That's the formula that we all
remember and go to sleep with.
516
00:29:28,610 --> 00:29:30,160
Now divide by delta x.
517
00:29:30,160 --> 00:29:32,460
Oh, first subtract cosine x.
518
00:29:32,460 --> 00:29:35,909
So there was a cosine x, so I
want to subtract one of them.
519
00:29:35,909 --> 00:29:37,370
OK?
520
00:29:37,370 --> 00:29:39,860
And now I have to divide
by the delta x.
521
00:29:39,860 --> 00:29:42,260
So I do that there.
522
00:29:42,260 --> 00:29:44,290
I do it here.
523
00:29:44,290 --> 00:29:50,630
And you see that we're in
the same happy position.
524
00:29:50,630 --> 00:29:54,140
We're in the happy position that
as delta x goes to 0, we
525
00:29:54,140 --> 00:29:55,880
know what this does.
526
00:29:55,880 --> 00:29:57,770
That goes to 1.
527
00:29:57,770 --> 00:30:00,630
We know what this does,
or we soon will.
528
00:30:00,630 --> 00:30:05,000
That goes to 0, just the
way they did on the
529
00:30:05,000 --> 00:30:07,430
board that got raised.
530
00:30:07,430 --> 00:30:10,160
So that term disappeared
just like before.
531
00:30:10,160 --> 00:30:12,340
This term survives.
532
00:30:12,340 --> 00:30:15,620
It's got a 1, it's got now a
sine x, and it's got now a
533
00:30:15,620 --> 00:30:16,950
minus sign.
534
00:30:16,950 --> 00:30:20,940
So that's the final result,
that the limit
535
00:30:20,940 --> 00:30:24,820
is minus sine x.
536
00:30:24,820 --> 00:30:30,740
That's the slope of
the cosine curve.
537
00:30:30,740 --> 00:30:33,050
And you wouldn't want
it any other way.
538
00:30:33,050 --> 00:30:35,030
You want that minus sign.
539
00:30:35,030 --> 00:30:38,660
You'll see it with second
derivatives.
540
00:30:38,660 --> 00:30:42,190
So it's just terrific that
those functions, the
541
00:30:42,190 --> 00:30:45,340
derivative of the sine is the
cosine with a plus, the
542
00:30:45,340 --> 00:30:47,310
derivative of the cosine is
the sine with a minus.
543
00:30:47,310 --> 00:30:48,570
OK.
544
00:30:48,570 --> 00:30:53,310
And we've almost proved it, we
just didn't quite pick up this
545
00:30:53,310 --> 00:30:56,210
point yet, and let me do that.
546
00:30:56,210 --> 00:30:58,190
That will finish this lecture.
547
00:30:58,190 --> 00:31:03,100
Why does that ratio
approach zero?
548
00:31:03,100 --> 00:31:05,750
What is that ratio?
549
00:31:05,750 --> 00:31:11,670
That ratio is coming from
the cosine curve.
550
00:31:11,670 --> 00:31:16,130
The cosine curve at 0, the way
this ratio came from the sine
551
00:31:16,130 --> 00:31:17,440
curve at 0.
552
00:31:17,440 --> 00:31:20,090
Here I'm taking--
553
00:31:20,090 --> 00:31:21,420
this is delta cosine.
554
00:31:21,420 --> 00:31:25,080
555
00:31:25,080 --> 00:31:29,370
There's lots of ways I can do
this, but maybe I'll just do
556
00:31:29,370 --> 00:31:34,580
it the way you see it.
557
00:31:34,580 --> 00:31:37,230
What's the slope of
the cosine at 0?
558
00:31:37,230 --> 00:31:40,340
559
00:31:40,340 --> 00:31:45,060
Yeah, I think I can ask that
without doing limits, without
560
00:31:45,060 --> 00:31:49,220
doing hard work.
561
00:31:49,220 --> 00:31:53,270
I'll just add the rest of the
cosine curve, because we know
562
00:31:53,270 --> 00:31:55,700
it's symmetric.
563
00:31:55,700 --> 00:31:57,410
What's the slope
at that point?
564
00:31:57,410 --> 00:32:01,470
565
00:32:01,470 --> 00:32:04,750
This is actually the most
important application of
566
00:32:04,750 --> 00:32:09,140
calculus, is to locate
the place where a
567
00:32:09,140 --> 00:32:11,470
function has a maximum.
568
00:32:11,470 --> 00:32:14,630
The cosine, its maximum
is right there.
569
00:32:14,630 --> 00:32:19,030
Its maximum value is 1, and it
happens at theta equals 0.
570
00:32:19,030 --> 00:32:21,440
So the slope at a maximum--
571
00:32:21,440 --> 00:32:24,280
all right, I'm going
to put this--
572
00:32:24,280 --> 00:32:30,840
I could get this result by these
pictures, but let me do
573
00:32:30,840 --> 00:32:33,750
it short circuit.
574
00:32:33,750 --> 00:32:39,305
The slope at the maximum is 0.
575
00:32:39,305 --> 00:32:40,555
OK.
576
00:32:40,555 --> 00:32:46,160
577
00:32:46,160 --> 00:32:47,870
Your intuition tells you that.
578
00:32:47,870 --> 00:32:52,960
If the slope was positive,
the function
579
00:32:52,960 --> 00:32:54,370
would still be rising.
580
00:32:54,370 --> 00:32:56,890
It wouldn't be a maximum, it
would be going higher.
581
00:32:56,890 --> 00:33:01,390
If the slope was negative, the
function would be coming down,
582
00:33:01,390 --> 00:33:04,470
and the maximum would
have been earlier.
583
00:33:04,470 --> 00:33:08,870
But here the maximum
is right there.
584
00:33:08,870 --> 00:33:11,840
The slope has to be
0 at that point.
585
00:33:11,840 --> 00:33:16,710
And that's the quantity that we
were after, because this is
586
00:33:16,710 --> 00:33:20,450
the cosine of delta x.
587
00:33:20,450 --> 00:33:23,060
There is the cosine
of delta x.
588
00:33:23,060 --> 00:33:28,970
Here is the 1, here is the delta
x, and this ratio is
589
00:33:28,970 --> 00:33:31,220
height over slope.
590
00:33:31,220 --> 00:33:35,310
It gets to height over slope as
we get closer and closer.
591
00:33:35,310 --> 00:33:39,510
It's the derivative, and
it's 0 at a maximum.
592
00:33:39,510 --> 00:33:47,640
And my notes give another way
to convince yourself that
593
00:33:47,640 --> 00:33:52,855
that's 0 by using these facts
that we've already got.
594
00:33:52,855 --> 00:33:55,040
OK.
595
00:33:55,040 --> 00:34:00,190
End of the-- so let me just
recap one moment, which this
596
00:34:00,190 --> 00:34:02,000
board will do.
597
00:34:02,000 --> 00:34:05,430
We now know the derivatives
of two of the great
598
00:34:05,430 --> 00:34:07,380
functions of calculus.
599
00:34:07,380 --> 00:34:11,600
We already know the derivative
of x to the n-th, and in the
600
00:34:11,600 --> 00:34:16,739
future is coming e to the
x and the logarithm.
601
00:34:16,739 --> 00:34:18,920
Then you've got the big ones.
602
00:34:18,920 --> 00:34:20,469
Thank you.
603
00:34:20,469 --> 00:34:22,280
ANNOUNCER: This has been
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604
00:34:22,280 --> 00:34:24,670
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Gilbert Strang.
605
00:34:24,670 --> 00:34:26,940
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606
00:34:26,940 --> 00:34:28,159
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607
00:34:28,159 --> 00:34:31,290
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608
00:34:31,290 --> 00:34:34,370
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609
00:34:34,370 --> 00:34:35,929
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610
00:34:35,929 --> 00:34:38,069