question
given the function ( r(x) = -x^3 - 8x + 18 ), determine the average rate of change of the function over the interval ( -8 leq x leq 2 ).
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# Explanation:
## Step1: Recall the average rate of change formula
The average rate of change of a function $ r(x) $ over the interval $[a, b]$ is given by $\frac{r(b)-r(a)}{b - a}$. Here, $ a=-8 $ and $ b = 2 $.

## Step2: Calculate $ r(-8) $
Substitute $ x=-8 $ into $ r(x)=-x^{2}-8x + 18 $:
$$
\begin{align*}
r(-8)&=-(-8)^{2}-8\times(-8)+18\
&=-64 + 64+18\
&=18
\end{align*}
$$

## Step3: Calculate $ r(2) $
Substitute $ x = 2 $ into $ r(x)=-x^{2}-8x + 18 $:
$$
\begin{align*}
r(2)&=- (2)^{2}-8\times2+18\
&=-4-16 + 18\
&=-2
\end{align*}
$$

## Step4: Calculate the average rate of change
Using the formula $\frac{r(b)-r(a)}{b - a}$ with $ a=-8 $, $ b = 2 $, $ r(-8)=18 $ and $ r(2)=-2 $:
$$
\begin{align*}
\frac{r(2)-r(-8)}{2-(-8)}&=\frac{-2 - 18}{2 + 8}\
&=\frac{-20}{10}\
&=-2
\end{align*}
$$

# Answer:
$-2$

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the average rate of change formula

Step2: Calculate \( r(-8) \)

Step3: Calculate \( r(2) \)

Step4: Calculate the average rate of change

Answer: