Question

given the function $f(x) = -x^2 + 5x + 14$, determine the average rate of change of the function over the interval $-2 \leq x \leq 9$.

Explanation:

Step1: Recall average rate of change formula

The average rate of change of $f(x)$ over $[a,b]$ is $\frac{f(b)-f(a)}{b-a}$

Step2: Calculate $f(-2)$

Substitute $x=-2$ into $f(x)$:

$$\begin{align*} f(-2)&=-(-2)^2 + 5(-2) + 14\\ &=-4 -10 +14\\ &=0 \end{align*}$$

Step3: Calculate $f(9)$

Substitute $x=9$ into $f(x)$:

$$\begin{align*} f(9)&=-(9)^2 + 5(9) + 14\\ &=-81 +45 +14\\ &=-22 \end{align*}$$

Step4: Compute average rate of change

Substitute values into the formula, $a=-2, b=9$:

$$ \frac{f(9)-f(-2)}{9-(-2)}=\frac{-22-0}{11}=-2 $$

Answer:

$-2$