U1S6V07 Second derivative, part 1.txt

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This is our function f, and f prime at this point
is just the slope of this tangent line right here.
However, in this section, we're interested in f double prime,
which is the derivative of f prime,
or the rate of change of f prime.
So let's just move along the curve
and see how f prime changes.
What's going to happen to these tangent lines?
How do they evolve?
Right now, we have a tangent line whose slope
is positive and fairly large.
As we move along, we're getting a slope
that's still positive, but not quite as large.
And then when we get over here, we have a slope that's 0.
So if we graph f prime, it would look something like this.
We start off with a very positive value.
And as we move along, we have still positive but not
quite as large.
And then we get here, and we get a value of 0.

So that's the graph of f prime.
What, then, can we say about f double prime?
Well, that's the derivative of f prime.
And as we can see, f prime is decreasing.
And a decreasing function ought to have a negative derivative,
so that's f double prime-- it's negative.
And it's because of this evolution of the tangent lines
up above.
The tangent lines change-- they evolve,
they turn as we go from here to here,
and it's this turning that make f double prime negative.
So you can think of the second derivative
as in some way measuring the turning of the graph of f,
and that's a really useful concept.
So next up is this section of the function, and as usual,
we have some questions for you to work through.
We'll discuss when you're done.