Rates of change

What is average rate of change?

The average rate of change of function $f$ over the interval $a\leq x\leq b$ is given by this expression:

$\dfrac{f(b)-f(a)}{b-a}$

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.

Let's find the average rate of change of $f$ over the interval $0\leq x\leq 9$:

[[☃ image 1]]

We can see from the graph that $f(0)=-7$ and $f(9)=3$.

$\begin{align}
\text{Average rate of change}&=\dfrac{f(9)-f(0)}{9-0}
\\
&=\dfrac{3-(-7)}{9}
\\
&=\dfrac{10}{9}
\end{align}$

Let's find the rate of change of $g(x)= x^3 - 9x$ over the interval $1\leq x\leq 6$.

$g(1)=1^3-9\cdot1=-8$

$g(6)=6^3-9\cdot 6=162$

$\begin{align}
\text{Average rate of change}&=\dfrac{g(6)-g(1)}{6-1}
\\
&=\dfrac{162-(-8)}{5}
\\
&=34
\end{align}$

[[☃ graded-group-set 1]]

Want to try more problems like this? Check out this exercise.