Review average rate of change and how to apply it to solve problems.

What is average rate of change?

The average rate of change of function ff over the interval axba\leq x\leq b is given by this expression:
It is a measure of how much the function changed per unit, on average, over that interval.
It is derived from the slope of the straight line connecting the interval's endpoints on the function's graph.
Want to learn more about average rate of change? Check out this video.

Finding average rate of change

Example 1: Average rate of change from graph

Let's find the average rate of change of ff over the interval 0x90\leq x\leq 9:
We can see from the graph that f(0)=7f(0)=-7 and f(9)=3f(9)=3.
Average rate of change=f(9)f(0)90=3(7)9=109\begin{aligned} \text{Average rate of change}&=\dfrac{f(9)-f(0)}{9-0} \\\\ &=\dfrac{3-(-7)}{9} \\\\ &=\dfrac{10}{9} \end{aligned}

Example 2: Average rate of change from equation

Let's find the rate of change of g(x)=x39xg(x)= x^3 - 9x over the interval 1x61\leq x\leq 6.
g(6)=6396=162g(6)=6^3-9\cdot 6=162
Average rate of change=g(6)g(1)61=162(8)5=34\begin{aligned} \text{Average rate of change}&=\dfrac{g(6)-g(1)}{6-1} \\\\ &=\dfrac{162-(-8)}{5} \\\\ &=34 \end{aligned}
Problem 1
What is the average rate of change of gg over the interval 8x2-8\leq x\leq -2?
Want to try more problems like this? Check out this exercise.