for the following exercises, find the average rate of change of each function on the interval specified. 28…

Explanation:

Step1: Recall the average - rate - of - change formula

The average rate of change of a function $y = f(x)$ on the interval $[a,b]$ is given by $\frac{f(b)-f(a)}{b - a}$.

Step2: Solve for $f(x)=x^{2}$ on $[1,5]$

First, find $f(5)$ and $f(1)$. $f(5)=5^{2}=25$, $f(1)=1^{2}=1$. Then, calculate the average rate of change: $\frac{f(5)-f(1)}{5 - 1}=\frac{25 - 1}{4}=\frac{24}{4}=6$.

Step3: Solve for $h(x)=5-2x^{2}$ on $[-2,4]$

Find $h(4)$ and $h(-2)$. $h(4)=5-2\times4^{2}=5 - 32=-27$, $h(-2)=5-2\times(-2)^{2}=5 - 8=-3$. Then, calculate the average rate of change: $\frac{h(4)-h(-2)}{4-(-2)}=\frac{-27-(-3)}{6}=\frac{-27 + 3}{6}=\frac{-24}{6}=-4$.

Answer:

For $f(x)=x^{2}$ on $[1,5]$, the average rate of change is $6$. For $h(x)=5 - 2x^{2}$ on $[-2,4]$, the average rate of change is $-4$.