Question

given the function $h(x) = -x^2 + 5x + 8$, determine the average rate of change of the function over the interval $0 \leq x \leq 7$.

Explanation:

Step1: Find $h(0)$

Substitute $x=0$ into $h(x)$:
$h(0) = -(0)^2 + 5(0) + 8 = 8$

Step2: Find $h(7)$

Substitute $x=7$ into $h(x)$:
$h(7) = -(7)^2 + 5(7) + 8 = -49 + 35 + 8 = -6$

Step3: Apply average rate formula

Use $\frac{h(b)-h(a)}{b-a}$ for $[a,b]=[0,7]$:
$\frac{h(7)-h(0)}{7-0} = \frac{-6 - 8}{7-0} = \frac{-14}{7}$

Step4: Simplify the expression

$\frac{-14}{7} = -2$

Answer:

$-2$