To determine the average rate of change of a function [tex]\( f(x) \)[/tex] over the interval [tex]\([2, 9]\)[/tex], you can use the following steps:
1. Identify the interval endpoints. In this case, the interval is [tex]\([2, 9]\)[/tex], where [tex]\( a = 2 \)[/tex] and [tex]\( b = 9 \)[/tex].
2. Evaluate the function at these endpoints. This means you need to find [tex]\( f(a) \)[/tex] and [tex]\( f(b) \)[/tex]. For our interval, we need [tex]\( f(2) \)[/tex] and [tex]\( f(9) \)[/tex]. Let's assume:
- [tex]\( f(2) = 5 \)[/tex]
- [tex]\( f(9) = 20 \)[/tex]
3. Compute the difference in the function values at the endpoints. This is given by [tex]\( f(b) - f(a) \)[/tex], which for our values is [tex]\( f(9) - f(2) = 20 - 5 = 15 \)[/tex].
4. Compute the difference in the [tex]\( x \)[/tex]-values. This is [tex]\( b - a \)[/tex], which in our case is [tex]\( 9 - 2 = 7 \)[/tex].
5. Determine the average rate of change. It is calculated as the difference in function values divided by the difference in [tex]\( x \)[/tex]-values:
[tex]\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} = \frac{20 - 5}{9 - 2} = \frac{15}{7} \approx 2.143
\][/tex]
So, therefore, the average rate of change for the function [tex]\( f(x) \)[/tex] over the interval [tex]\([2, 9]\)[/tex] is approximately [tex]\( 2.143 \)[/tex].
To summarize, the expression used to find the average rate of change over [tex]\([2, 9]\)[/tex] is:
[tex]\[
\frac{f(9) - f(2)}{9 - 2}
\][/tex]
