U1S3V10 Graphing the Derivative of f.txt

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OK, so here we are.
We've got our original function f on the top.
Here it is.
And we've put the entire derivative, f prime,
on the bottom.
We had to squeeze them a little bit to fit them on the screen.
Hopefully that's OK.
But we're just going to walk through from left to right
and see how all of this fits together.
So we're starting with the tangent line
here at x equals 0.
And that's given by this dotted blue line.
And the slope of this line is 3.
So that's telling us that f prime of 0 equals 3.
And that's why down below on this graph
we have a point at 0, 3.
It's at height 3.
That's really the thing to keep in mind,
is every tangent line up above corresponds to a single point
down below.
And for the derivative, the thing that matters
is the slope of that tangent line.
That slope is the y-coordinate of the corresponding point
on the bottom graph.

So if we start moving, we're going
to start getting different tangent lines.
So over here we've got a different tangent line
and it's not quite as steep as what we had before.
And that tells us that f prime is not
going to be quite as big as it was before.
So that's why down below we have a point that is lower
than what came previously.
So we've gone down on the graph of f prime.
So even though the point of tangency is moving this way--
it's moving upwards-- the slopes of the tangent lines
are doing something else.
The slope of this one is not as big as the slope of this one.

Ultimately, we're going to get to this point at x equals 3.
And here, even though we're pretty high
up on the graph of f, the tangent line is horizontal.
And that's a slope of 0.
So down below, that's why we have the graph
of f prime hitting the y-axis.
It's because f prime of 3 equals 0.
What happens after this?
Well, we start going downhill on the graph of f.
f is starting to decrease here, and that
means our tangent lines will have negative slope.
And f prime is now in negative territory.
We hit the tangent line with the most negative slope right
around here.
So you can see down below this is
where f prime is bottoming out.
If we go a little bit beyond, then the point
is a little bit lower on the graph of f
but the slope of the tangent line
is not as negative as it once was.
So f prime is going from most negative here to less
negative to here at x equals 6.
Well, we have a horizontal tangent line again.
And that means that f prime is back to 0.

And then after this, well, now we have positive slopes again.
So f prime is back in positive territory.

So that's the story of our derivative.
We've graphed f prime.
We have another somewhat harder example for you,
so why don't you look at those problems
and I'll talk to you later.