To determine the average rate of change of the function over the interval [tex]\(5 \leq x \leq 7\)[/tex], we need to follow these steps:
1. Identify the values of [tex]\(x\)[/tex] at the endpoints of the interval, which are [tex]\(x = 5\)[/tex] and [tex]\(x = 7\)[/tex].
2. Determine the corresponding function values from the table:
- [tex]\(f(5) = 33\)[/tex]
- [tex]\(f(7) = 17\)[/tex]
3. Use the formula for the average rate of change of the function [tex]\(f(x)\)[/tex] over the interval [tex]\([a, b]\)[/tex]:
[tex]\[
\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}
\][/tex]
4. Substitute [tex]\(a = 5\)[/tex], [tex]\(b = 7\)[/tex], [tex]\(f(5) = 33\)[/tex], and [tex]\(f(7) = 17\)[/tex] into the formula:
[tex]\[
\text{Average rate of change} = \frac{f(7) - f(5)}{7 - 5}
\][/tex]
[tex]\[
\text{Average rate of change} = \frac{17 - 33}{7 - 5}
\][/tex]
5. Simplify the numerator and denominator:
[tex]\[
\text{Average rate of change} = \frac{-16}{2}
\][/tex]
6. Divide the numerator by the denominator to find the final answer:
[tex]\[
\text{Average rate of change} = -8
\][/tex]
Thus, the average rate of change of the function over the interval [tex]\(5 \leq x \leq 7\)[/tex] is [tex]\(-8\)[/tex].
