Secant lines & average rate of change | Derivatives introduction | AP Calculus AB | Khan Academy

So right over here, we have the graph of ( y ) is equal to ( x^2 ) or at least part of the graph of ( y ) is equal to ( x^2 ). The first thing I'd like to tackle is to think about the average rate of change of ( Y ) with respect to ( X ) over the interval from ( X ) equaling 1 to ( X ) equaling 3.

So, let me write that down. We want to know the average rate of change of ( Y ) with respect to ( X ) over the interval from ( X ) going from 1 to 3, and it's a closed interval where ( X ) could be 1 and ( X ) could be equal to 3.

Well, we could do this even without looking at the graph. If I were to just make a table here where if this is ( X ), and this is ( Y ) is equal to ( x^2 ). When ( X ) is equal to 1, ( Y ) is equal to ( 1^2 ), which is just one. You see that right over there? And when ( X ) is equal to 3, ( Y ) is equal to ( 3^2 ), which is equal to 9.

So you can see when ( X ) is equal to 3, ( Y ) is equal to 9. To figure out the average rate of change of ( Y ) with respect to ( X ), you say, “Okay, well, what's my change in ( X )?” Well, we can see very clearly that our change in ( X ) over this interval is equal to positive 2.

Well, what's our change in ( Y ) over the same interval? Our change in ( Y ) is equal to, when ( X ) increased by 2 from 1 to 3, ( Y ) increases by 8. So it's going to be a positive 8.

So what is our average rate of change? Well, it's going to be our change in ( Y ) over our change in ( X ), which is equal to ( \frac{8}{2} ), which is equal to 4. So that would be our average rate of change over that interval. On average, every time ( X ) increases by 1, ( Y ) is increasing by 4.

And how did we calculate that? We looked at our change in ( X ). Let me draw that here. We looked at our change in ( X ) and we looked at our change in ( Y ), which would be this right over here, and we calculated change in ( Y ) over change in ( X ) for average rate of change.

Now, this might be looking fairly familiar to you because you're used to thinking about change in ( Y ) over change in ( X ) as the slope of a line connecting two points. And that's indeed what we did calculate. If you were to draw a secant line between these two points, we essentially just calculated the slope of that secant line.

The average rate of change between two points, that is the same thing as the slope of the secant line. By looking at the secant line in comparison to the curve over that interval, it hopefully gives you a visual intuition for what even average rate of change means.

Because in the beginning part of the interval, you see that the secant line is actually increasing at a faster rate. But then, as we get closer to 3, it looks like our yellow curve is increasing at a faster rate than the secant line, and then they eventually catch up.

So that's why the slope of the secant line is the average rate of change. Is it the exact rate of change at every point? Absolutely not! The curve's rate of change is constantly changing. It's at a slower rate of change in the beginning part of this interval, and then it's actually increasing at a higher rate as we get closer and closer to 3.

So over the interval, their change in ( Y ) over the change in ( X ) is exactly the same. Now, one question you might be wondering is, why are you learning this in a Calculus class? Couldn't you have learned this in an Algebra class? The answer is yes, but what's going to be interesting, and is really one of the foundational ideas of calculus, is, well, what happens as these points get closer and closer together?

We found the average rate of change between 1 and 3, or the slope of the secant line from (1, 1) to (3, 9). But what instead if you found the slope of the secant line between (2, 4) and (3, 9)? So what if you found this slope? But what if you wanted to get even closer?

Let's say you wanted to find the slope of the secant line between the point (2.5, 6.25) and (3, 9). And what if you just kept getting closer and closer and closer? Well, then the slopes of these secant lines are going to get closer and closer to the slope of the tangent line at ( x ) equal to 3.

And if we can figure out the slope of the tangent line, well then we’re in business! Because then we're not talking about average rate of change; we're going to be talking about instantaneous rate of change, which is one of the central ideas that is the derivative.

And we're going to get there soon, but it's really important to appreciate that the average rate of change between two points is the same thing as the slope of the secant line. As those points get closer and closer together, and as the secant line is connecting two closer and closer points together, as that distance between the points, between the ( X ) values of the points, approach zero, very interesting things are going to happen.