WEBVTT
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PROFESSOR: Welcome
to recitation.
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Today in this video
what we're going to do
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is look at how we can
determine the graph
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of a derivative of a
function from the graph
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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
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I'd like us to do
in this time, is
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to figure out what
the curve y equals
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f prime of 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
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that we know about f prime of x.
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So what I want to
remind you is that when
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you think about a
function's derivative,
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remember its
derivative's output is
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measuring the slope of the
tangent line at each point.
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So that's what we're
interested in finding,
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is understanding the
slope of the tangent line
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of this curve at each x-value.
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So it's always
easiest when you're
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thinking about a derivative
to find the places where
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the slope of the
tangent line is 0.
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Because those are
the only places
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where you can hope to 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 find it fairly easily.
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That would be at whatever this
x value is, that slope there
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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.
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Horizontal tangent line.
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But there's a third place where
the slope of the tangent line
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is 0, and that's kind
of hidden right in here.
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And actually, I've
drawn in-- maybe you
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think there are a
few more-- but we're
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going to assume that this
function is always continuing
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down through this region.
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So there are three places where
the tangent 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 positive and
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negative?
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So what I'm going
to do is below here,
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I'm just going to
make a line and we're
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going to sort of
keep track of what
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the 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.
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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,
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and we know it's
also equal to 0 here.
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OK?
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And now the question
is, what is the sign
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of f 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 are really-- we
can divide up the x-values
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as left of whatever that
x-value is, in between these two
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values, in between
these two values,
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and to the right
of this x-value.
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That's really,
really what we need
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to do to determine what
the signs of f prime are.
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So again, what we want
to do to understand
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f prime is we look at the
slope of the tangent line
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of the curve y 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 slopes
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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,
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if I look for
x-values to the right,
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I see that as I
move to the right,
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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 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
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I'm saying here it's 0 and
here it's 0 for the x values,
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and I take a take a point, we
notice the sign is negative
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there, also.
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So in fact, the sign
of f prime changed
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at this zero of f prime,
but it stays the same
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around this zero of f prime.
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So it's negative and then
it goes to negative again.
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It's negative, then
0, then negative.
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And then if I look to
the right of this x-value
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and I take a point, I see that
the slope of the tangent line
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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 then 0, and then
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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
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have 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,
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here-- the 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,
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but 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,
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as I move in the x-direction,
the slope-- let me just
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keep my hand, watch what my hand
is doing-- the slope is always
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positive, but it's becoming
less and less vertical, right?
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It's headed towards horizontal.
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So the slope that
was steeper over here
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is 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 output is, from 0.
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So as the derivative
becomes less steep,
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the derivative's 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
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starts to-- the slopes of
these tangent lines start
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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
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steep as it gets in this region,
and then gets less steep.
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The steepest point
is that point where
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you have the 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
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around here the tangent
line is as steep as it ever
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gets in that region,
between these two zeros,
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and then it gets less steep.
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So I'd say, right
around there we
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should say, OK, that's
as low as it goes
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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 zeros the
same kind of thing happens.
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But notice-- this is, we have
to be careful-- we shouldn't
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go through 0 here because
the derivative's output,
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the sign is negative.
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Right?
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Notice, so the
tangent line, it was
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negative, negative, negative,
0, oh, it's still negative.
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So the outputs are
still negative,
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and they're going to be negative
all the way to this zero.
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And what we need to see again
is the same kind of thing
00:07:53.530 --> 00:07:55.040
happens as happened
in this region
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will 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,
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the tangent lines start to get
steeper, then at some point
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they stop getting 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,
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then it gets less steep.
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So that's the place where
the derivative's magnitude
00:08:19.480 --> 00:08:22.621
is going to be the
biggest in this region.
00:08:22.621 --> 00:08:24.120
And actually, I've
sort of drawn it,
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they look like they're about
the same steepness at those two
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places, so I should probably
put the outputs about the same
00:08:30.370 --> 00:08:30.870
down here.
00:08:30.870 --> 00:08:33.150
Their magnitudes
are about the same.
00:08:33.150 --> 00:08:36.686
So this has to bounce
off, come up here.
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I made that a little
sharper than I meant to.
00:08:38.560 --> 00:08:40.720
OK?
00:08:40.720 --> 00:08:41.710
So that's the place.
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That's the output here--
or the tangent line, sorry.
00:08:45.100 --> 00:08:47.690
The tangent line at this
x value is the steepest
00:08:47.690 --> 00:08:52.210
that we get in this region,
so the output at that x-value
00:08:52.210 --> 00:08:53.830
is the lowest we get.
00:08:53.830 --> 00:08:55.370
And then, when
we're to the right
00:08:55.370 --> 00:08:57.680
of this zero for
the derivative, we
00:08:57.680 --> 00:08:59.610
start seeing the
tangent lines positive--
00:08:59.610 --> 00:09:02.706
we pointed that out already--
and it gets more positive.
00:09:02.706 --> 00:09:04.580
So it starts at 0, it
starts to get positive,
00:09:04.580 --> 00:09:06.220
and then it gets more positive.
00:09:06.220 --> 00:09:10.730
It's going to do something
like that, roughly.
00:09:10.730 --> 00:09:13.580
So let me fill in the dotted
lines so we can see it clearly.
00:09:23.020 --> 00:09:25.430
Well, this is not
exact, but this
00:09:25.430 --> 00:09:29.130
is a fairly good drawing,
I think we can say,
00:09:29.130 --> 00:09:30.260
of f prime of x.
00:09:30.260 --> 00:09:32.740
y equals f prime of x.
00:09:32.740 --> 00:09:34.670
And now I'm going to
ask you a question.
00:09:34.670 --> 00:09:36.444
I'm going to write
it on the board,
00:09:36.444 --> 00:09:38.860
and then I'm going to give you
a moment to think about it.
00:09:38.860 --> 00:09:40.068
So let me write the question.
00:09:43.590 --> 00:09:54.085
It's, find a function
y equals-- or sorry--
00:09:54.085 --> 00:10:06.200
find a function g of x so that
y equals g prime of x looks
00:10:06.200 --> 00:10:11.566
like y equals f prime of x.
00:10:11.566 --> 00:10:13.440
OK, let me be clear
about that, and then I'll
00:10:13.440 --> 00:10:15.010
give you a moment
to think about it.
00:10:15.010 --> 00:10:16.525
So I want you to
find a function g
00:10:16.525 --> 00:10:20.910
of x so that its derivative's
graph, y equals g prime of x,
00:10:20.910 --> 00:10:23.500
looks exactly like the graph
we've drawn in blue here,
00:10:23.500 --> 00:10:25.230
y equals f prime of x.
00:10:25.230 --> 00:10:27.470
Now, I don't want you to
find something in terms
00:10:27.470 --> 00:10:29.460
of x squareds and x cubes.
00:10:29.460 --> 00:10:34.620
I don't want you to find an
actual g of x equals something
00:10:34.620 --> 00:10:35.390
in terms of x.
00:10:35.390 --> 00:10:38.260
I want you to just try
and find a relationship
00:10:38.260 --> 00:10:40.770
that it must have with f.
00:10:40.770 --> 00:10:42.929
So I'm going to give me a
moment to think about it
00:10:42.929 --> 00:10:45.220
and work out your answer,
and I'll be back to tell you.
00:10:53.780 --> 00:10:54.280
OK.
00:10:54.280 --> 00:10:55.170
Welcome back.
00:10:55.170 --> 00:10:57.160
So what we're looking
for is a function
00:10:57.160 --> 00:10:59.840
g of x so that its
derivative, when I graph it,
00:10:59.840 --> 00:11:02.330
y equals g prime of x, I
get exactly the same curve
00:11:02.330 --> 00:11:03.500
as the blue one.
00:11:03.500 --> 00:11:04.970
The blue one.
00:11:04.970 --> 00:11:06.780
And the point is
that if you thought
00:11:06.780 --> 00:11:08.840
about it for a little
bit, what you really
00:11:08.840 --> 00:11:13.670
need is a function that looks
exactly like this function,
00:11:13.670 --> 00:11:17.510
y equals f of x, at all
the x-values in terms
00:11:17.510 --> 00:11:21.230
of its slopes, but
those slopes can happen
00:11:21.230 --> 00:11:23.190
shifted up or down anywhere.
00:11:23.190 --> 00:11:25.650
So the point is that if I
take the function y equals
00:11:25.650 --> 00:11:28.920
f of x and I add a constant
to it, which shifts
00:11:28.920 --> 00:11:32.500
the whole graph up or
down, the tangent lines
00:11:32.500 --> 00:11:34.570
are unaffected by that shift.
00:11:34.570 --> 00:11:36.400
And so I get exactly
the same picture
00:11:36.400 --> 00:11:39.070
when I take the
derivative of that graph.
00:11:39.070 --> 00:11:43.070
When I look at that the tangent
line slopes of that graph.
00:11:43.070 --> 00:11:45.010
So you could draw
another picture
00:11:45.010 --> 00:11:48.790
and check it for yourself if
you didn't feel convinced,
00:11:48.790 --> 00:11:51.390
shift this, shift this
curve up, and then look
00:11:51.390 --> 00:11:53.400
at what the tangent
lines do on that curve.
00:11:53.400 --> 00:11:55.590
But then you'll see its
derivative's outputs
00:11:55.590 --> 00:11:57.350
are exactly the same.
00:11:57.350 --> 00:11:58.819
So we'll stop there.