PROFESSOR: Welcome
to recitation.
Today in this video
what we're going to do
is look at how we can
determine the graph
of a derivative of a
function from the graph
of the function itself.
So I've given a function here.
We're calling it just y equals
f of x-- or this is the curve,
y equals f of x.
So we're thinking about
a function f of x.
I'm not giving you the
equation for the function.
I'm just giving you the graph.
And what I'd like
you to do, what
I'd like us to do
in this time, is
to figure out what
the curve y equals
f prime of x will look like.
So that's our objective.
So what we'll do first is
try and figure out the things
that we know about f prime of x.
So what I want to
remind you is that when
you think about a
function's derivative,
remember its
derivative's output is
measuring the slope of the
tangent line at each point.
So that's what we're
interested in finding,
is understanding the
slope of the tangent line
of this curve at each x-value.
So it's always
easiest when you're
thinking about a derivative
to find the places where
the slope of the
tangent line is 0.
Because those are
the only places
where you can hope to change
the sign on the derivative.
So what we'd like to do is
first identify, on this curve,
where the tangent line
has slope equal to 0.
And I think there are two places
we can find it fairly easily.
That would be at whatever this
x value is, that slope there
is 0.
It's going to be a
horizontal tangent line.
And then whatever
this x value is.
The slope there is also 0.
Horizontal tangent line.
But there's a third place where
the slope of the tangent line
is 0, and that's kind
of hidden right in here.
And actually, I've
drawn in-- maybe you
think there are a
few more-- but we're
going to assume that this
function is always continuing
down through this region.
So there are three places where
the tangent line is horizontal.
So I can even sort of draw
them lightly through here.
You have three
horizontal tangent lines.
So at those points, we know
that the derivative's value is
equal to 0, the
output is equal to 0.
And now what we can determine
is, between those regions,
where are the values of
the derivative positive and
negative?
So what I'm going
to do is below here,
I'm just going to
make a line and we're
going to sort of
keep track of what
the signs of the derivative are.
So let me just draw.
This would be sort of
our sign on f prime.
OK.
So that's going to tell
us what our signs are.
So right below,
we'll keep track.
So here, this, I'll
just come straight down.
Here we know the sign of
f prime is equal to 0.
OK?
We know it's equal to 0 there.
We know it's also
equal to 0 here,
and we know it's
also equal to 0 here.
OK?
And now the question
is, what is the sign
of f prime in this region?
So to the left of
whatever that x value is.
What is the sign of f prime in
this region, in this region,
and then to the right?
So there are really-- we
can divide up the x-values
as left of whatever that
x-value is, in between these two
values, in between
these two values,
and to the right
of this x-value.
That's really,
really what we need
to do to determine what
the signs of f prime are.
So again, what we want
to do to understand
f prime is we look at the
slope of the tangent line
of the curve y equals f of x.
So let's pick a place in this
region left of where it's 0,
say right here, and let's
look at the tangent line.
The tangent line has
what kind of slope?
Well, it has a positive slope.
And in fact, if you look along
here, you see all of the slopes
are positive.
So f prime is
bigger than 0 here.
And now I'm just
going to record that.
I'm going to keep that
in mind as a plus.
The sign is positive there.
Now, if I look right of
where f prime equals 0,
if I look for
x-values to the right,
I see that as I
move to the right,
the tangent line
is curving down.
So let me do it with the chalk.
You see the tangent line looks,
has a slope negative slope.
If I draw one point in, it
looks something like that.
So the slope is negative there.
So here I can record that.
The sign of f prime
is a minus sign there.
Now, if I look between
these two x-values, which
I'm saying here it's 0 and
here it's 0 for the x values,
and I take a take a point, we
notice the sign is negative
there, also.
So in fact, the sign
of f prime changed
at this zero of f prime,
but it stays the same
around this zero of f prime.
So it's negative and then
it goes to negative again.
It's negative, then
0, then negative.
And then if I look to
the right of this x-value
and I take a point, I see that
the slope of the tangent line
is positive.
And so the sign
there is positive.
So we have the derivative is
positive, and then 0, and then
negative, and then 0, and then
negative, and then 0, and then
positive.
So there's a lot going on.
But I, if I want to plot,
now, y equals f prime of x, I
have some sort of launching
point by which to do that.
So what I can do is, I know
that the derivative 0--
I'm going to draw the
derivative in blue,
here-- the derivative is 0, its
output is 0 at these places.
So I'm going to put
those points on.
And then if I were just trying
to get a rough idea of what
happens, the derivative is
positive left of this x value.
So it's certainly coming down.
It's coming down.
Oops, let me make
these a little darker.
It's coming down
because it's positive.
It's coming down to 0-- it
has to stay above the x-axis,
but it has to head towards 0.
Right?
What does that
actually correspond to?
Well, look at what
the slopes are doing.
The slopes of these
tangent lines,
as I move in the x-direction,
the slope-- let me just
keep my hand, watch what my hand
is doing-- the slope is always
positive, but it's becoming
less and less vertical, right?
It's headed towards horizontal.
So the slope that
was steeper over here
is becoming less steep.
The steepness is really the
magnitude of the derivative.
That's really measuring how far
it is, the output is, from 0.
So as the derivative
becomes less steep,
the derivative's values have
to be headed closer to 0.
Now, what happens when the
derivative is equal to 0 here?
Well, all of a sudden the
slopes are becoming negative.
So the outputs of the
derivative are negative.
It's going down.
But then once it hits here,
again, notice what happens.
The derivative is 0 again,
and notice how I get there.
The derivative's
negative, and then it
starts to-- the slopes of
these tangent lines start
to get shallower.
Right?
They were steep and
then somewhere they
start to get shallower.
So there's someplace sort of
in the x-values between here
and here where the
derivative is as
steep as it gets in this region,
and then gets less steep.
The steepest point
is that point where
you have the biggest magnitude
in that region for f prime.
So that's where it's going
to be furthest from 0.
So if I'm guessing,
it looks like right
around here the tangent
line is as steep as it ever
gets in that region,
between these two zeros,
and then it gets less steep.
So I'd say, right
around there we
should say, OK, that's
as low as it goes
and now it's going
to come back up.
OK?
So hopefully that makes sense.
We'll get to see it again, here.
Between these two zeros the
same kind of thing happens.
But notice-- this is, we have
to be careful-- we shouldn't
go through 0 here because
the derivative's output,
the sign is negative.
Right?
Notice, so the
tangent line, it was
negative, negative, negative,
0, oh, it's still negative.
So the outputs are
still negative,
and they're going to be negative
all the way to this zero.
And what we need to see again
is the same kind of thing
happens as happened
in this region
will happen in this region.
The point being that,
again, we're 0 here.
We're 0 here.
So somewhere in the
middle, we start at 0,
the tangent lines start to get
steeper, then at some point
they stop getting steeper,
they start getting shallower.
That place looks maybe
right around here.
That's the sort of
steepest tangent line,
then it gets less steep.
So that's the place where
the derivative's magnitude
is going to be the
biggest in this region.
And actually, I've
sort of drawn it,
they look like they're about
the same steepness at those two
places, so I should probably
put the outputs about the same
down here.
Their magnitudes
are about the same.
So this has to bounce
off, come up here.
I made that a little
sharper than I meant to.
OK?
So that's the place.
That's the output here--
or the tangent line, sorry.
The tangent line at this
x value is the steepest
that we get in this region,
so the output at that x-value
is the lowest we get.
And then, when
we're to the right
of this zero for
the derivative, we
start seeing the
tangent lines positive--
we pointed that out already--
and it gets more positive.
So it starts at 0, it
starts to get positive,
and then it gets more positive.
It's going to do something
like that, roughly.
So let me fill in the dotted
lines so we can see it clearly.
Well, this is not
exact, but this
is a fairly good drawing,
I think we can say,
of f prime of x.
y equals f prime of x.
And now I'm going to
ask you a question.
I'm going to write
it on the board,
and then I'm going to give you
a moment to think about it.
So let me write the question.
It's, find a function
y equals-- or sorry--
find a function g of x so that
y equals g prime of x looks
like y equals f prime of x.
OK, let me be clear
about that, and then I'll
give you a moment
to think about it.
So I want you to
find a function g
of x so that its derivative's
graph, y equals g prime of x,
looks exactly like the graph
we've drawn in blue here,
y equals f prime of x.
Now, I don't want you to
find something in terms
of x squareds and x cubes.
I don't want you to find an
actual g of x equals something
in terms of x.
I want you to just try
and find a relationship
that it must have with f.
So I'm going to give me a
moment to think about it
and work out your answer,
and I'll be back to tell you.
OK.
Welcome back.
So what we're looking
for is a function
g of x so that its
derivative, when I graph it,
y equals g prime of x, I
get exactly the same curve
as the blue one.
The blue one.
And the point is
that if you thought
about it for a little
bit, what you really
need is a function that looks
exactly like this function,
y equals f of x, at all
the x-values in terms
of its slopes, but
those slopes can happen
shifted up or down anywhere.
So the point is that if I
take the function y equals
f of x and I add a constant
to it, which shifts
the whole graph up or
down, the tangent lines
are unaffected by that shift.
And so I get exactly
the same picture
when I take the
derivative of that graph.
When I look at that the tangent
line slopes of that graph.
So you could draw
another picture
and check it for yourself if
you didn't feel convinced,
shift this, shift this
curve up, and then look
at what the tangent
lines do on that curve.
But then you'll see its
derivative's outputs
are exactly the same.
So we'll stop there.
