Let's go over the answer to the assessment
you just took.
The question was, "which graph represents
the derivative of F of X?"
So let's take a closer look at F of X.
Here it is.
So it's an upward-facing parabola, and it
has a minimum at X equals 2.
So let's use what we know about the relationship
between a graph and the graph of its derivative
to try to write down a few facts for what
we would expect the derivative to look like,
and then try to match up those facts with
our choices.
So what's the first thing I notice about this
function F of X.
Well first of all it starts out decreasing.
So I'm decreasing over here and I continue
to decrease until I reach the slope of minimum.
So what does it tell me about the derivative
if I have a function that's decreasing?
Well when the function is decreasing I know
that the derivative is negative.
So let me out to the side here write down
some properties for what we expect of the
derivative F prime of X.
We expect it to be negative from negative
infinity to 2, because that's where the original
function is decreasing.
Ok so then what happens at 2?
At 2, we reach a local minimum, right here.
Now whenever we're at a local minimum or maximum
we have a horizontal tangent line, and that
tells me my derivative is going to be zero
at that point.
So then I expect my derivative to be zero
when X equals 2.
So F of X should be zero when X equals 2.
And then what happens?
My graph starts to increase, and it continues
to increase for the rest of the time.
So, whenever the original function is increasing,
that means my derivative is positive on that
interval.
So that means the derivative should be positive
from 2 to infinity.
So let's see if any of our graphs match those
characteristics.
Ok so what about choice A?
Choice A is just a horizontal line, it's constant
at the point Y equals 2.
So that doesn't meet any of my criteria.
So I know that my answer is not choice A.
What about choice B?
Well choice B is a line, and it starts out
negative, so that's good.
I see that my line is negative until we get
to 2, and then I see that I cross the axis
so my function is zero when X equals 2.
And then my function is positive for the rest
of the time, so it's positive from 2 to infinity.
So that meets all my criteria, so choice B
is the correct answer in this case.
We can briefly look at choice C. The thing
that's wrong with choice C is that it actually
starts out positive and then goes negative,
but that's opposite of what we want; we want
something that starts out negative and goes
positive, like we saw in choice B. So that's
how you match up a graph to its derivative.
