To determine the average rate of change of the function [tex]\( g(x) = -x^2 + 5x + 10 \)[/tex] over the interval [tex]\([-1, 5]\)[/tex], let's follow these steps:
1. Evaluate the function at the endpoints of the interval:
First, we need the values of [tex]\( g(x) \)[/tex] at [tex]\( x = -1 \)[/tex] and [tex]\( x = 5 \)[/tex].
For [tex]\( x = -1 \)[/tex]:
[tex]\[
g(-1) = -(-1)^2 + 5(-1) + 10 = -1 - 5 + 10 = 4
\][/tex]
For [tex]\( x = 5 \)[/tex]:
[tex]\[
g(5) = -(5)^2 + 5(5) + 10 = -25 + 25 + 10 = 10
\][/tex]
2. Apply the formula for the average rate of change:
The average rate of change of the function [tex]\( g(x) \)[/tex] over the interval [tex]\([-1, 5]\)[/tex] is given by:
[tex]\[
\text{Average Rate of Change} = \frac{g(x_2) - g(x_1)}{x_2 - x_1}
\][/tex]
where [tex]\( x_1 = -1 \)[/tex] and [tex]\( x_2 = 5 \)[/tex].
Substituting the values we calculated:
[tex]\[
\text{Average Rate of Change} = \frac{g(5) - g(-1)}{5 - (-1)} = \frac{10 - 4}{5 - (-1)} = \frac{10 - 4}{5 + 1} = \frac{6}{6} = 1
\][/tex]
Therefore, the average rate of change of the function [tex]\( g(x) \)[/tex] over the interval [tex]\([-1, 5]\)[/tex] is [tex]\( 1.0 \)[/tex].
So, the values of [tex]\( g(-1) \)[/tex], [tex]\( g(5) \)[/tex], and the average rate of change are:
[tex]\[
g(-1) = 4, \quad g(5) = 10, \quad \text{Average Rate of Change} = 1.0
\][/tex]
