0:00

[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 this point a, the slope of this tangent line is

negative, the derivative is negative, and yeah, the function's going down here.

At this point b, the slope of this tangent

line is positive and the function's increasing through here.

All right?

The derivative is 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 S-I-G-N, 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 is 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.

1:33

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 that 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's 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.

2:56

So in this part of the graph, the second derivative is negative.

What happens in between?

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.

Alright, the, it's concave down over here, and it's concave up over here and the

place where the change is taking place, we're

going to, 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 phenomena

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's negative.

You're slowing down.

And then suddenly you step on the gas.

Now you're accelerating.

Your second derivative's positive.

What happened, right?

Something big happened.

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]