U1S6V08 Second derivative, part 2.txt

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We're now thinking about this section of the graph of f
right here.
If we want to think about f double prime,
then we need to think about how f
prime changes, how the slopes of the tangent lines change.
Right now, we have a tangent line whose slope is zero.
And then we get a negative slope.
And then an even bigger negative slope.
So if we graph f prime, it looks like this-- where we start off
at zero and then we get more and more negative as we go along.
So that's the graph of the first derivative f prime.
And if we want f prime's derivative,
we need to look at this and realize
that f prime is going downward, it's decreasing.
So its derivative, which is f double prime,
is going to be negative in this section.
Now you might remember that f double prime
was also negative in this section of the graph.
So we can actually extend this arrow here.
f double prime is negative over this entire interval.
Is there a commonality that this entire stretch
has that corresponds with f double prime being negative?
The answer is yes, there is a commonality.
And to see it, let's play with these tangent lines
a little more.
It'll be easier if I take off half of the tangent line
so I only have this right half.
But notice what happens.
Look at how it turns.
As we go from left to right, it's
always turning in the same direction-- clockwise.
The graph is curving downward.
Another way to think about this--
imagine this is a bird's eye view of a road.
And you're driving from this point
to this point along the road.
Which way would your steering wheel be turned?
Well, you'd always be steering to the right
as we go from here to here.
So that's what's going on with this section of the graph of f.
We're always turning in the same direction.
And this bending downward, or turning to the right, that's
called concave down.
And concave down graphs corresponds
to negative second derivatives and decreasing
first derivatives.

So that finishes our analysis of the left half of our function.
We're going to ask you work on the right half on your own.
And then we'll wrap things up and be on our way.