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PROFESSOR: Welcome
to recitation.

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Today in this video what we're
going to do is look at how we

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can determine the graph of a
derivative of a function from

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the graph of the function
itself.

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So I've given a function here.

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We're calling it just y equals f
of x-- or this is the curve,

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y equals f of x.

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So we're thinking about
a function f of x.

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I'm not giving you the equation
for the function.

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I'm just giving you the graph.

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And what I'd like you to do,
what I'd like us to do in this

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time, is to figure out what the
curve, y equals f prime of

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x will look like.

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So that's our objective.

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So what we'll do first is try
and figure out the things that

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we know about f prime of x.

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So what I want to remind you is
that when you think about a

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function's derivative, remember
its derivative's

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output is measuring the
slope of the tangent

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line at each point.

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So that's what we're interested
in finding, is

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understanding the slope of the
tangent line of this curve at

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each x value.

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So it's always easiest when
you're thinking about a

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derivative to find the places
where the slope of the

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tangent line is 0.

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Because those are the only
places where you can hope to

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change the sign on
the derivative.

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So what we'd like to do is first
identify, on this curve,

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where the tangent line
has slope equal to 0.

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And I think there are
two places we can

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find it fairly easily.

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That would be at whatever
this x value is, that

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slope there is 0.

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It's going to be a horizontal
tangent line.

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And then whatever
this x value is.

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The slope there is also 0
horizontal tangent line.

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But there's a third place where
the slope of the tangent

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line is 0, and that's kind
of hidden right in here.

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And actually, I've drawn in--
maybe you think there are a

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few more-- but we're going to
assume that this function is

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always continuing down
through this region.

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So there are three places where
the horizontal, tangent

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line is horizontal.

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So I can even sort of draw them
lightly through here.

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You have three horizontal
tangent lines.

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So at those points, we know that
the derivative's value is

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equal to 0, the output
is equal to 0.

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And now what we can determine
is, between those regions,

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where are the values
of the derivative

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positive and negative?

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So what I'm going to do is below
here, I'm just going to

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make a line and we're going to
sort of keep track of what the

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signs of the derivative are.

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So let me just draw.

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This would be sort of
our sign on f prime.

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OK.

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So that's going to tell
us what our signs are.

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So right below, we'll
keep track.

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So here, this, I'll just
come straight down.

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Here we know the sign of f
prime is equal to 0.OK?

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We know it's equal to 0 there.

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We know it's also equal to 0
here, and we know it's also

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equal to 0 here.

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OK?

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And now the question is,
what is the sign of f

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prime in this region?

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So to the left of whatever
that x value is.

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What is the sign of f prime in
this region, in this region,

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and then to the right?

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So there really, we can divide
up the x values as left of

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whatever that x value is, in
between these two values, in

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between these two values, and to
the right of this x value.

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That's really, really what we
need to do to determine what

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the signs of f prime are.

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So again, what we want to do
to understand f prime is we

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look at the slope of the tangent
line of the curve, y

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equals f of x.

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So let's pick a place in this
region left of where it's 0,

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say right here, and let's look
at the tangent line.

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The tangent line has
what kind of slope?

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Well, it has a positive slope.

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And in fact, if you look along
here, you see all of the

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slopes are positive.

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So f prime is bigger
than 0 here.

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And now I'm just going
to record that.

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I'm going to keep that
in mind as a plus.

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The sign is positive there.

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Now, if I look right of where
f prime equals 0, if I look

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for x values to the right, I
see that as I move to the

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right, the tangent line
is curving down.

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So let me do it with
the chalk.

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You see the tangent
line looks, has a

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slope negative slope.

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If I draw one point in, it looks
something like that.

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So the slope is negative
there.

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So here I can record that.

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The sign of f prime is
a minus sign there.

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Now, if I look between these two
x values, which I'm saying

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here is 0 and here it's 0 for
the x values, and I take a

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take a point, we notice the sign
is negative there, also.

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So in fact, the sign of f prime
changed at this 0 of f

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prime, but it stays the same
around this 0 of f prime.

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So it's negative and then it
goes to negative again.

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It's a negative, then
0, then negative.

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And then if I look to the right
of this x value and I

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take a point, I see that
the slope of the

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tangent line is positive.

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And so the sign there
is positive.

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So we have the derivative is
positive, and then 0, and then

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negative, and then 0, and
then negative, and

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then 0, and then positive.

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So there's a lot going on.

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But I, if I want to plot, now,
y equals f prime of x, I have

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some sort of launching point
by which to do that.

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So what I can do is, I know
that the derivative 0--

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I'm going to draw the derivative
in blue, here-- the

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derivative is 0, its output
is 0 at these places.

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So I'm going to put
those points on.

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And then if I were just trying
to get a rough idea of what

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happens, the derivative is
positive left of this x value.

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So it's certainly coming down.

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It's coming down.

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Oops, let me make these
a little darker.

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It's coming down because
it's positive.

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It's coming down to 0-- it has
to stay above the x-axis, but

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it has to head towards 0.

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Right?

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What does that actually
correspond to?

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Well, look at what the
slopes are doing.

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The slopes of these tangent
lines, as I move in the x

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direction, the slope--

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let me just keep my hand, watch
what my hand is doing--

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the slope is always positive,
but it's becoming less and

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less vertical, right?

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It's headed towards
horizontal.

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So the slope that was
steeper over here is

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becoming less steep.

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The steepness is really the
magnitude of the derivative.

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That's really measuring
how far it is, the

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output is from 0.

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So as the derivative becomes
less steep, the derivative's

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values have to be headed
closer to 0.

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Now, what happens when the
derivative is equal to 0 here?

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Well, all of a sudden the slopes
are becoming negative.

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So the outputs of the derivative
are negative.

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It's going down.

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But then once it hits here,
again, notice what happens.

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The derivative is 0 again, and
notice how I get there.

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The derivative's negative, and
then it starts to, the slopes

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of these tangent lines start
to get shallower.

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Right?

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They were steep and then
somewhere they

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start to get shallower.

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So there's someplace sort of in
the x values between here

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and here where the derivative is
as steep as it gets in this

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region, and then gets
less steep.

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The steepest point is that
point where you have the

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biggest magnitude in that
region for f prime.

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So that's where it's going
to be furthest from 0.

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So if I'm guessing, it looks
like right around here the

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tangent line is as steep as it
ever gets in that region,

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between these two 0s, and
then it gets less steep.

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So I'd say, right around there
we should say, OK, that's as

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low as it goes and now it's
going to come back up.

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OK?

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So hopefully that makes sense.

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We'll get to see
it again, here.

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Between these two 0s the same
kind of thing happens.

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But notice--

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this is, we have
to be careful--

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we shouldn't go through 0 here
because the derivative's

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output, the sign is negative.

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Right?

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Notice, so the tangent line,
it was negative, negative,

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negative, 0, oh, it's
still negative.

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So the outputs are still
negative, and they're going to

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be negative all the
way to this 0.

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And what we need to see again
is the same kind of thing

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happenes as happened
in this region will

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happen in this region.

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The point being that,
again, we're 0 here.

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We're 0 here.

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So somewhere in the middle, we
start at 0, the tangent lines

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start to get steeper, then at
some point they stop getting

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steeper, they start
getting shallower.

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That place looks maybe
right around here.

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That's the sort of steepest
tangent line, then it gets

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less steep.

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So that's the place where the
derivative's magnitude is

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going to be the biggest
in this region.

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And actually, I've sort of
drawn it, they look like

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they're about the same steepness
at those two places,

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so I should probably put
the outputs about

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the same down here.

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Their magnitudes are
about the same.

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So this has to bounce
off, come up here.

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I made that a little sharper
than I meant to.

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OK?

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So that's the place.

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That's the output here--

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or the tangent line, sorry.

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The tangent line at this x value
is the steepest that we

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get in this region, so the
output at that x value is the

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lowest we get.

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And then, when we're to the
right of this 0 for the

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derivative, we start seeing the
tangent lines positive--

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we pointed that out already--
and it gets more positive.

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So it starts at 0, it starts to
get positive, and then it

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gets more positive.

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It's going to do something
like that, roughly.

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So let me fill in the
dotted lines so

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we can see it clearly.

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Well, this is not exact, but
this is a fairly good drawing,

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I think we can say,
of f prime of x.

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y equals f prime of x.

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And now I'm going to
ask you a question.

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I'm going to write it on the
board, and then I'm going to

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give you a moment to
think about it.

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So let me write the question.

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It's, find a function y equals--
or sorry-- find a

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function g of x so that y equals
g prime of x looks like

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y equals f prime of x.

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OK, let me be clear about that,
and then I'll give you a

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moment to think about it.

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So I want you to find a function
g of x so that its

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derivative's graph, y equals g
prime of x, looks exactly like

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the graph we've drawn in blue
here, y equals f prime of x.

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Now, I don't want you to find
something in terms of x

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squared's and x cubes.

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I don't want you to find an
actual g of x equals something

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in terms of x.

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I want you to just try and find
a relationship that it

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must have with f.

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So I'm going to give me a moment
to think about it and

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work out your answer, and I'll
be back to tell you.

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OK.

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Welcome back.

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So what we're looking for is a
function g of x so that its

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00:10:58,070 --> 00:11:01,270
derivative, when I graph it, y
equals g prime of x, I get

246
00:11:01,270 --> 00:11:03,500
exactly the same curve
as the blue one.

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00:11:03,500 --> 00:11:04,970
The blue one.

248
00:11:04,970 --> 00:11:07,480
And the point is that if you
thought about it for a little

249
00:11:07,480 --> 00:11:12,270
bit, what you really need is a
function that looks exactly

250
00:11:12,270 --> 00:11:17,190
like this function, y equals f
of x, at all the x values in

251
00:11:17,190 --> 00:11:21,910
terms of its slopes, but those
slopes can happen shifted up

252
00:11:21,910 --> 00:11:23,190
or down anywhere.

253
00:11:23,190 --> 00:11:26,780
So the point is that if I take
the function y equals f of x

254
00:11:26,780 --> 00:11:29,670
and I add a constant to it,
which shifts the whole graph

255
00:11:29,670 --> 00:11:34,570
up or down, the tangent lines
are unaffected by that shift.

256
00:11:34,570 --> 00:11:37,030
And so I get exactly the same
picture when I take the

257
00:11:37,030 --> 00:11:39,070
derivative of that graph.

258
00:11:39,070 --> 00:11:40,650
When I look at that
the tangent line

259
00:11:40,650 --> 00:11:43,070
slopes of that graph.

260
00:11:43,070 --> 00:11:45,570
So you could draw another
picture and check it for

261
00:11:45,570 --> 00:11:49,326
yourself if you didn't feel
convinced, shift this, shift

262
00:11:49,326 --> 00:11:52,570
this curve up, and then look at
what the tangent lines do

263
00:11:52,570 --> 00:11:53,400
on that curve.

264
00:11:53,400 --> 00:11:55,720
But then you'll see its
derivative's outputs are

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00:11:55,720 --> 00:11:57,350
exactly the same.

266
00:11:57,350 --> 00:11:59,210
So we'll stop there.

267
00:11:59,210 --> 00:11:59,319