To calculate the average rate of change of a function [tex]\( f(x) \)[/tex] over the interval [tex]\([2, 9]\)[/tex], we'll use the following formula for the average rate of change:
[tex]\[ \frac{f(b) - f(a)}{b - a} \][/tex]
where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the endpoints of the interval.
In this specific case:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 9 \)[/tex]
So, substituting [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the formula:
- The difference in [tex]\( x \)[/tex]-values over the interval is [tex]\( b - a = 9 - 2 = 7 \)[/tex].
- Thus, the expression that can be used to determine the average rate of change over the interval [tex]\([2, 9]\)[/tex] is:
[tex]\[ \frac{f(9) - f(2)}{9 - 2} \][/tex]
This expression simplifies to:
[tex]\[ \frac{f(9) - f(2)}{7} \][/tex]
Thus, the expression needed to determine the average rate of change of [tex]\( f(x) \)[/tex] over the interval [tex]\([2, 9]\)[/tex] is:
[tex]\[ \frac{f(9) - f(2)}{7} \][/tex]
