Question

given the function $h(x)=-x^{2}+8x + 20$, determine the average rate of change of the function over the interval $0leq xleq5$.

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = h(x)$ over the interval $[a,b]$ is $\frac{h(b)-h(a)}{b - a}$, where $a = 0$ and $b = 5$.

Step2: Calculate $h(0)$

Substitute $x = 0$ into $h(x)=-x^{2}+8x + 20$.
$h(0)=-(0)^{2}+8(0)+20=20$.

Step3: Calculate $h(5)$

Substitute $x = 5$ into $h(x)=-x^{2}+8x + 20$.
$h(5)=-(5)^{2}+8(5)+20=-25 + 40+20=35$.

Step4: Calculate the average rate of change

Use the formula $\frac{h(5)-h(0)}{5 - 0}$.
$\frac{35 - 20}{5}=\frac{15}{5}=3$.

Answer:

$3$