To find the average rate of change of the function [tex]\( f \)[/tex] over the interval [tex]\([-2, 3]\)[/tex], we need to follow these steps:
1. Define the function [tex]\( f \)[/tex] at the specified points [tex]\( x_1 = -2 \)[/tex] and [tex]\( x_2 = 3 \)[/tex].
2. Evaluate the function at these points:
- For [tex]\( x_1 = -2 \)[/tex], since [tex]\(-2 \leq 1\)[/tex], we use the formula [tex]\( f(x) = 3x - 2 \)[/tex]:
[tex]\[
f(-2) = 3(-2) - 2 = -6 - 2 = -8
\][/tex]
- For [tex]\( x_2 = 3 \)[/tex], since [tex]\( 3 > 1 \)[/tex], we use the formula [tex]\( f(x) = x^2 + 1 \)[/tex]:
[tex]\[
f(3) = 3^2 + 1 = 9 + 1 = 10
\][/tex]
3. Find the average rate of change of [tex]\( f \)[/tex] over the interval [tex]\([x_1, x_2] = [-2, 3]\)[/tex] using the formula:
[tex]\[
\text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\][/tex]
4. Substitute the values we found into the formula:
- [tex]\( f(x_1) = f(-2) = -8 \)[/tex]
- [tex]\( f(x_2) = f(3) = 10 \)[/tex]
- [tex]\( x_1 = -2 \)[/tex]
- [tex]\( x_2 = 3 \)[/tex]
[tex]\[
\text{Average rate of change} = \frac{10 - (-8)}{3 - (-2)} = \frac{10 + 8}{3 + 2} = \frac{18}{5} = 3.6
\][/tex]
Summary of results:
- [tex]\( f(-2) = -8 \)[/tex]
- [tex]\( f(3) = 10 \)[/tex]
- Average rate of change = 3.6
