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[MUSIC].

Â Earlier we saw how the sign, the S-I-G-N, of

Â the derivative encoded whether the function was increasing or decreasing.

Â Thinking back to the graph, here I've just drawn some random graph.

Â What is the derivative encoding?

Â Well here this point a, the slope of this tangent line is

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

Â At this point b, the slope of this tangent

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

Â All right?

Â The derivative is negative here, and it's positive here.

Â The function's decreasing here, and increasing here.

Â So that's what the derivative is measuring.

Â What is the sign of the second derivative really encoding?

Â Maybe we don't have such a good word for it so we'll just make up a new word.

Â The S-I-G-N, the sign of the second derivative, the

Â sign of the derivative of the derivative measures concavity.

Â The word's concavity, and here's the two possibilities, concave up where the second

Â derivative is positive, and concave down where the second derivative is negative.

Â And I've drawn sort of cartoony pictures of

Â what the graphs look like in these two cases.

Â Now, note it's not just increasing or decreasing, but this concavity

Â is recording sort of the shape of the graph in some sense.

Â Positive second derivative makes it look like

Â this, negative second derivative makes the graph

Â look like this, and I'm just labeling

Â these two things concave up and concave down.

Â 1:33

And this makes sense if we think of the

Â second derivative as measuring the change in the derivative.

Â So let's think back to this graph again.

Â Here's this graph of some random function.

Â Look at this part of the graph right here.

Â That looks like the concave up shape

Â from before, where the second derivative was positive.

Â So we might think that the second derivative is positive here.

Â That would mean that the derivative is increasing.

Â What that really means is that the slope

Â of a tangent line through this region is increasing.

Â And that's exactly what's happening.

Â The slope is negative here, and as I move this

Â tangent line over, the slope of that tangent line is increasing.

Â The second derivative is positive here.

Â You can tell yourself the same story for concave down.

Â So look over here in our sample graph.

Â That part of the graph looks like this

Â concave down picture where the second derivative's negative.

Â Now, if the second derivative is negative, that means the derivative is decreasing.

Â And yeah, the slope of the tangent line through this region is going down, right?

Â The slope starts off pretty positive over here, and as I move this

Â tangent line over, the slope is zero, and now getting more and more negative.

Â 2:56

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

Â What happens in between?

Â Where does the regime change take place?

Â So over here, the second derivative is negative.

Â Over here, the second derivative is positive.

Â There's a point in between, maybe it's right here.

Â And at that point the second derivative is equal to zero.

Â And on one side it's concave down, and on the other side it's concave up.

Â A point where the concavity actually changes is called an inflection point.

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

Â place where the change is taking place, we're

Â going to, just going to call those points inflection points.

Â It's not that the terminology itself is so important, but we want

Â words to describe the qualitative phenomena

Â that we're seeing in these graphs.

Â Inflection points are something you can really feel.

Â I mean, if you're driving in a car, you're braking, right?

Â That means the second derivative's negative.

Â You're slowing down.

Â And then suddenly you step on the gas.

Â Now you're accelerating.

Â Your second derivative's positive.

Â What happened, right?

Â Something big happened.

Â You're changing regimes from concave down to concave

Â up and you want to denote that change somehow.

Â We're going to call that change an inflection point.

Â [MUSIC]

Â