U1S6V09 The Second Derivative and Concavity.txt

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OK, let's finish this thing up.
So you were thinking about this second half of the graph.
And when we were looking at the first derivative,
hopefully you noticed that on this section f is decreasing.
And so the first derivative here is negative.
And on this stretch, the function is increasing.
So the first derivative is going to be positive on that stretch.
For the second derivative, we want to know
how did the slopes of the tangent lines evolve?
So here's a tangent line.
It has a negative slope.
Then we get one whose slope is not so negative.
Then here's one with a 0 slope, then
a positive slope, then a greater positive slope.
So the slopes increase as we go from one tangent line
to the next.
This graph is bending upwards.
If we drive along this road, our steering wheel
is always pointed to the left.
All of this is telling us that this part of the curve
is concave up.
And so the second derivative is positive
over this entire section.
So that's the full geometric meaning
of the second derivative.
It's telling us how our graph is turning or bending.

Let's work out a numerical example
to see how this can help us.
So I'll erase this here.
And let's say we have a function f
of x equals minus 2x to the fourth plus 3x cubed plus 1.
And we want to know, what does the graph of f
look like near the point x equals 1?
So we can certainly calculate f of 1.
f of 1 is going to be 2.
So this tells us that the point 1, 2 is on the graph.
But that's it.
That's all we have so far.
What else could we do?
Well, we can take a derivative.
Those are always good.
So f prime of x is going to be minus 8x cubed plus 9x
squared plus 0.
So f prime of 1 equals 1.
So that tells us that the tangent line at this point
that we've drawn has slope 1.
So let's draw that in right here.
All right.
Well, not bad, but this doesn't tell us how the graph bends.
It's a polynomial, so it's probably pretty curvy.
For that, we need to go one derivative higher.
What's f double prime?
Well, the second derivative is the derivative of f prime,
so we'll just use the power rule again.
The minus 8x cubed will give us minus 24x squared.
And the 9x squared gives plus 18x for its derivative.
So f double prime of 1 equals minus 6.
And so that's telling us that the second derivative
is negative, at least near 1.
So we see that the graph will be concave down
in the neighborhood of x equals 1.
So we're tangent to this line at this point.
We're concave down.
That means the graph is going to look something like this.

And there we have it.
So hopefully that gives you an idea
of how the second derivative helps us think
about the graph of a function.
Intervals where f double prime is positive are concave up.
Intervals where f double prime is negative are concave down.
What happens if the second derivative is always zero?
We'll let you figure that one out on your own.
Have fun.