determine the average rate of change of the function on the given interval. express your answer in exact…

Explanation:

Step1: Recall average - rate - of - change formula

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

Step2: Calculate for part (a)

For $h(x)=x^{3}+1$ on $[-4,-3]$, $a=-4$, $b = - 3$. First, find $h(-3)$ and $h(-4)$: $h(-3)=(-3)^{3}+1=-27 + 1=-26$. $h(-4)=(-4)^{3}+1=-64 + 1=-63$. Then, $\frac{h(-3)-h(-4)}{-3-(-4)}=\frac{-26-(-63)}{-3 + 4}=\frac{-26 + 63}{1}=37$.

Step3: Calculate for part (b)

For $h(x)=x^{3}+1$ on $[-2,1]$, $a=-2$, $b = 1$. $h(1)=1^{3}+1=2$. $h(-2)=(-2)^{3}+1=-8 + 1=-7$. Then, $\frac{h(1)-h(-2)}{1-(-2)}=\frac{2-(-7)}{1 + 2}=\frac{2 + 7}{3}=3$.

Step4: Calculate for part (c)

For $h(x)=x^{3}+1$ on $[2,5]$, $a = 2$, $b = 5$. $h(5)=5^{3}+1=125+1=126$. $h(2)=2^{3}+1=8 + 1=9$. Then, $\frac{h(5)-h(2)}{5-2}=\frac{126 - 9}{3}=\frac{117}{3}=39$.

Answer:

(a) 37 (b) 3 (c) 39