[MUSIC] Earlier, we saw how the sign, the S I G N, of the derivative, encoded
whether the function was increasing or decreasing.
Thinking back to the graph, here I've just drawn some random graph.
What is the derivative encoding? Well, here at this point a, the slope of this
tangent of negative, the derivative is negative, and yeah, the function's going
down here. At this point B, the slope of this
tangent line's positive, and the function's increasing through here.
All right, the derivative's negative here, and it's positive here.
The function's decreasing here and increasing here.
So that's what the derivative is measuring.
What is the sign of the second derivative really encoding? Maybe we don't have such
a good word for it, so we'll just make up a new word.
The sign, the sign of the second derivative, the sign of the derivative of
the derivative, measures concavity. The word's concavity and here's the two
possibilities. Concave up where the second derivative
was positive and concave down where the second derivative is negative.
And I've drawn sort of cartoony pictures of what the graphs look like in these two
cases. Now note, it's not just increasing or
decreasing but this concavity is recording sort of the shape Of the graph
in some sense. Positive second derivative makes it look
like this, negative second derivative makes the graph look like this and I'm
just labeling these two things, concave up and concave down.
And this makes sense, if we think of the second derivative as measuring the
change, in the derivative. So let's think back to this graph again,
here's this graph of some random function.
Look at this part of the graph right here.
That looks like the concave up shape from before where the second derivative was
positive. So we might think the second derivative
is positive here. That would mean that the derivative is
increasing. What that really means is that the slope
of a tangent line through this region is increasing,
and that's exactly what's happening. The slope is negative here.
And as I move this tangent line over, the slope of that tangent line is increasing.
The second derivative is positive here. You can tell yourself the same story for
concave down. So look over here in our sample graph.
That part of the graph. Looks like this concave down picture,
where the second derivative is negative. Now if the second derivative is negative,
that means the derivative is decreasing. And yeah, the slope of the tangent line
through this region is going down, right? The slope starts off pretty
positive over here and as I move this tangent line over the slope is zero, and
now getting more and more negative. So in this part of the graph, the 2nd
derivative is negative. What happens inbetween? Where does the
regime change take place? So over here, the second derivative is negative.
Over here, the second derivative is positive.
There's a point in between, maybe it's right here, and at that point, the second
derivative is equal to zero and on one side, it's concave down and on the other
side, it's concave up. A point where the concavity actually
changes is called an inflection point. All right it's concave down over here and
it's concave up over here and the place where the change is taking place, we're
just going to call those points inflection points.
It's not that the terminology itself is so important but we want words to
describe the qualitative phenomenon that we're seeing in these graphs.
Inflection points are something you can really feel.
I mean, if you're driving in a car, you're braking.
Right? That means the second derivative is negative.
You're slowing down. And then, suddenly, you step on the gas.
Now you're accelerating, and your second derivative is positive.
What happened, right? Something big happened.
[MUSIC] You're changing regimes from concave down to concave up.
And you want to denote that change somehow.
We're going to call that change an inflection point.
[MUSIC]