Question

given the function ( g(x) = -x^2 + 3x + 5 ), determine the average rate of change of the function over the interval ( -4 leq x leq 6 ).

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( g(x) \) over the interval \([a, b]\) is given by \(\frac{g(b)-g(a)}{b - a}\). Here, \( a=-4 \) and \( b = 6 \).

Step2: Calculate \( g(-4) \)

Substitute \( x=-4 \) into \( g(x)=-x^{2}+3x + 5 \):
\[

$$\begin{align*} g(-4)&=-(-4)^{2}+3(-4)+5\\ &=-16-12 + 5\\ &=-23 \end{align*}$$

\]

Step3: Calculate \( g(6) \)

Substitute \( x = 6 \) into \( g(x)=-x^{2}+3x + 5 \):
\[

$$\begin{align*} g(6)&=-6^{2}+3(6)+5\\ &=-36 + 18+5\\ &=-13 \end{align*}$$

\]

Step4: Calculate the average rate of change

Using the formula \(\frac{g(b)-g(a)}{b - a}\) with \( a=-4 \), \( b = 6 \), \( g(-4)=-23 \) and \( g(6)=-13 \):
\[

$$\begin{align*} \frac{g(6)-g(-4)}{6-(-4)}&=\frac{-13-(-23)}{6 + 4}\\ &=\frac{-13 + 23}{10}\\ &=\frac{10}{10}\\ &=1 \end{align*}$$

\]

Answer:

1