Connecting average rates of change with the concavity of a graph

0:00- [Instructor] The table gives values of f of x. Use average rate of change, or using average rate of change, identify the concavity of f of x over the interval from 1 to 4, including the boundaries. So pause this video, have a go at that before we do it together.

0:14Okay, so one way to think about it: average rate of change is just, over some interval, you're gonna see your change in the value of the function divided by the change in the value of the input. So one way to think about it, you could view it, if we're looking at the average rate of change between x equals a and x equals b it would be f of b minus f of a minus f of a over b minus a.

0:42Now, what we wanna do is look at the interval they're talking about. So from 1 to 4, so that's this interval right over here. And we wanna look at small changes in x. And the easiest one to compute is just changes of 1. And then think about what is the average rate of change over those intervals.

0:59If that average rate of change increases as our xs increase over the interval, that means we're concave up. That means that the graph is getting steeper and steeper. If our average rate of change stays the same, that means that that could very well be a linear situation.

1:18And if our average rate of change starts to decline, that means we are getting less steep and less steep as our xs increase. And this right over here would be concave downward. So let's see what happens. So let's first look at the average rate of change from 1 to 2.

1:34So what's the change in the value of our function? It's going to be negative 7 minus negative 1.5, and then that's going to be that over 2 minus 1. So this is going to be equal to, this is the same thing as negative 7 plus 1.5. So that would be equal to 5.5 or negative 5.5.

1:58And then it's negative 5.5 over 1, this is just 1 here. So I could write over 1. Let me just write negative 5.5. So that is our average rate of change going from that point to this point. Now let's go to the next point. So here, our change in our value of our function is negative 3.5 minus 7, and then it's over, it's still gonna be over 1 because it's gonna be over 3 minus 2.

2:26I'll just write that down just so you see it. So this is going to be equal to negative 3.5. Oh, this is minus negative 7. Let me be very careful. Negative 3.5 minus negative 7 is equal to positive 3.5 over 1, which is equal to 3.5. So it looks like our average rate of change is increasing.

2:47But let's validate that by just looking at this last interval right over here. So now we're gonna go from this point to this point. So our ending point is 33, our ending f function value, minus negative 3.5, all of that over 4 minus 3. So this is going to be equal to 36.5 in the numerator.

3:11Did I do that right? Yep. All of that over 1. So that's increasing even more, increasing a lot. So our average rate of change is definitely increasing over this interval. And so we can say that the concavity is positive, or we're concave upwards may be a better way of saying it.

3:29We are concave upwards. We are in this scenario. At least over that interval, we're getting steeper and steeper and steeper as our x is increasing.