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given the function ( h(x) = -x^2 - 10x + 29 ), determine the average rate of change of the function over the interval ( -6 leq x leq -1 ).
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Explanation:

Step1: Recall the average rate of change formula

The average rate of change of a function \( h(x) \) over the interval \([a, b]\) is given by \(\frac{h(b)-h(a)}{b - a}\). Here, \( a=-6 \) and \( b = - 1\).

Step2: Calculate \( h(-6) \)

Substitute \( x=-6 \) into \( h(x)=-x^{2}-10x + 29 \):
\[

$$\begin{align*} h(-6)&=-(-6)^{2}-10\times(-6)+29\\ &=-36 + 60+29\\ &=24 + 29\\ &=53 \end{align*}$$

\]

Step3: Calculate \( h(-1) \)

Substitute \( x = - 1\) into \( h(x)=-x^{2}-10x + 29 \):
\[

$$\begin{align*} h(-1)&=-(-1)^{2}-10\times(-1)+29\\ &=-1 + 10+29\\ &=9 + 29\\ &=38 \end{align*}$$

\]

Step4: Calculate the average rate of change

Using the formula \(\frac{h(b)-h(a)}{b - a}\) with \( a=-6 \), \( b=-1 \), \( h(-6) = 53\) and \( h(-1)=38\):
\[

$$\begin{align*} \frac{h(-1)-h(-6)}{-1-(-6)}&=\frac{38 - 53}{-1 + 6}\\ &=\frac{-15}{5}\\ &=- 3 \end{align*}$$

\]

Answer:

\(-3\)