To determine the average rate of change of the function over the interval [tex]\(12 \leq x \leq 36\)[/tex], we need to follow these steps:
1. Identify the interval boundaries and corresponding function values:
- The interval given is [tex]\([12, 36]\)[/tex].
- For [tex]\(x = 12\)[/tex]: [tex]\(f(12) = 34\)[/tex].
- For [tex]\(x = 36\)[/tex]: [tex]\(f(36) = 28\)[/tex].
2. Apply the formula for the average rate of change:
- The average rate of change of a function [tex]\(f\)[/tex] over the interval [tex]\([x_1, x_2]\)[/tex] is given by:
[tex]\[
\text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\][/tex]
3. Substitute the values into the formula:
- Here, [tex]\(x_1 = 12\)[/tex], [tex]\(x_2 = 36\)[/tex], [tex]\(f(x_1) = f(12) = 34\)[/tex], and [tex]\(f(x_2) = f(36) = 28\)[/tex].
- Substitute these into the formula:
[tex]\[
\text{Average rate of change} = \frac{f(36) - f(12)}{36 - 12} = \frac{28 - 34}{36 - 12}
\][/tex]
4. Simplify the expression:
- First, calculate the difference in the numerator:
[tex]\[
28 - 34 = -6
\][/tex]
- Next, calculate the difference in the denominator:
[tex]\[
36 - 12 = 24
\][/tex]
- Now, divide the results:
[tex]\[
\frac{-6}{24} = -0.25
\][/tex]
Therefore, the average rate of change of the function over the interval [tex]\(12 \leq x \leq 36\)[/tex] is [tex]\(-0.25\)[/tex].
