U1S3V16 Graphing the derivative of a function.txt

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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, a horizontal tangent line.
But there's one third place where
the slope of the tangent line is 0.
And that's kind of hidden right in here.
And actually I have drawn and 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 could even sort of draw them lightly through here.
You have three horizontal tangent lines.
So at those points we know that the derivatives
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 keep track of what
the signs of the derivative are.
So let me just draw--
this will 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.
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 really we can divide up the x values
as left of whatever that x value is, in between these two
x values, in between these two x values,
and to the right of this x value.
That's 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 has a 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 is 0 and here it's 0 for the x values,
and I take a point, we notice the sign is negative there
also.
So in fact, the sign of f prime changed at this 0 of f prime,
but it stays the same around this 0 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 looked to the right of this x value
and I take a point, I see that the slope of the tangent
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 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.
The 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--

I'll 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.
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.
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 derivatives values have to be headed closer to 0.
Now what happens when a 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 is negative.
And then the slope of these tangent lines
start to get shallower.
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 steep
as it ever gets in that region between these two zeros,
and then 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, we have to be careful.
We shouldn't go through 0 here, because the derivative's
output, the sign is negative.
Notice, so the tangent line, it was negative, negative,
negative, 0.
Oop, it's still negative.
So the outputs are still negative.
And they're going to be negative all the way to this 0.
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 line starts 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 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 0 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 gets more positive.
So it's going to do something like that roughly.
So let me fill in the dotted line so we can see it clearly.

Now this is not exact, but this is
a fairly good drawing, I think we can say, of f prime of x,
y equals x 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 the 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 cubeds.
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 you 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.
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 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.