Question

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:

\(-2\)