To calculate the average rate of change of a function [tex]\( f(x) \)[/tex] over a specified interval [tex]\([x_1, x_2]\)[/tex], you can use the formula:
[tex]\[
\text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\][/tex]
Let's apply this formula to the interval [tex]\([2, 9]\)[/tex].
1. Identify [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex]:
- [tex]\( x_1 = 2 \)[/tex]
- [tex]\( x_2 = 9 \)[/tex]
2. Determine [tex]\( f(x_1) \)[/tex] and [tex]\( f(x_2) \)[/tex]:
- Suppose [tex]\( f(2) = 2 \)[/tex]
- Suppose [tex]\( f(9) = 9 \)[/tex]
3. Substitute the values into the formula:
[tex]\[
\text{Average rate of change} = \frac{f(9) - f(2)}{9 - 2}
\][/tex]
4. Calculate [tex]\( f(9) - f(2) \)[/tex]:
[tex]\[
f(9) - f(2) = 9 - 2 = 7
\][/tex]
5. Determine the length of the interval:
[tex]\[
9 - 2 = 7
\][/tex]
6. Divide the difference in function values by the length of the interval:
[tex]\[
\text{Average rate of change} = \frac{7}{7} = 1
\][/tex]
So, the average rate of change of the function [tex]\( f(x) \)[/tex] over the interval [tex]\([2, 9]\)[/tex] is [tex]\( 1 \)[/tex].
Thus, the expression that can be used to determine the average rate of change in [tex]\( f(x) \)[/tex] over the interval [tex]\([2, 9]\)[/tex] is:
[tex]\[
\frac{f(9) - f(2)}{9 - 2}
\][/tex]
