To determine the average rate of change of the function [tex]\( f(x) = 17 - x^2 \)[/tex] over the interval [tex]\([1, 5]\)[/tex], we can use the formula for the average rate of change of a function over an interval [tex]\([a, b]\)[/tex]:
[tex]\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\][/tex]
In this case, [tex]\( a = 1 \)[/tex] and [tex]\( b = 5 \)[/tex].
Let's start by calculating the values of the function at these points:
1. Calculate [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = 17 - 1^2 = 17 - 1 = 16
\][/tex]
2. Calculate [tex]\( f(5) \)[/tex]:
[tex]\[
f(5) = 17 - 5^2 = 17 - 25 = -8
\][/tex]
Now, we can use these values to find the average rate of change:
[tex]\[
\text{Average Rate of Change} = \frac{f(5) - f(1)}{5 - 1} = \frac{-8 - 16}{5 - 1} = \frac{-24}{4} = -6
\][/tex]
Thus, the average rate of change of the function [tex]\( f(x) = 17 - x^2 \)[/tex] over the interval [tex]\([1,5]\)[/tex] is [tex]\(\boxed{-6}\)[/tex].
