To find the average rate of change of the function [tex]\( j(x) \)[/tex] over the interval [tex]\([-1, 6]\)[/tex], we can follow these steps:
1. Define the interval: We are given the interval [tex]\([-1, 6]\)[/tex], which means:
- [tex]\( x_1 = -1 \)[/tex]
- [tex]\( x_2 = 6 \)[/tex]
2. Determine [tex]\( h \)[/tex]: The difference [tex]\( h \)[/tex] between [tex]\( x_2 \)[/tex] and [tex]\( x_1 \)[/tex] is:
[tex]\[
h = x_2 - x_1 = 6 - (-1) = 6 + 1 = 7
\][/tex]
3. Calculate the difference quotient at [tex]\( x_1 \)[/tex]: Using [tex]\( x_1 \)[/tex] and the given difference quotient formula [tex]\( 3x^2 + 3xh + h^2 \)[/tex]:
[tex]\[
j'(x_1) = 3(-1)^2 + 3(-1)(7) + 7^2
\][/tex]
Simplify each term:
[tex]\[
3(-1)^2 = 3(1) = 3
\][/tex]
[tex]\[
3(-1)(7) = -21
\][/tex]
[tex]\[
7^2 = 49
\][/tex]
Therefore,
[tex]\[
j'(x_1) = 3 - 21 + 49 = 31
\][/tex]
4. Calculate [tex]\( j(x) \)[/tex] at [tex]\( x_2 = 6 \)[/tex]:
Since the given difference quotient is:
[tex]\[
j'(x_2) = 3(6)^2
\][/tex]
Simplify:
[tex]\[
3(6)^2 = 3(36) = 108
\][/tex]
5. Calculate the average rate of change:
The average rate of change over the interval [tex]\([-1, 6]\)[/tex] is given by:
[tex]\[
\text{Average rate of change} = \frac{j'(x_2) - j'(x_1)}{x_2 - x_1}
\][/tex]
Substitute the values we have calculated:
[tex]\[
\text{Average rate of change} = \frac{108 - 31}{6 - (-1)} = \frac{108 - 31}{6 + 1} = \frac{77}{7} = 11
\][/tex]
Thus, the average rate of change of [tex]\( j(x) \)[/tex] over the interval [tex]\([-1, 6]\)[/tex] is:
[tex]\[
\boxed{11}
\][/tex]
(Note: Although various choices are given in the question text, the correct numerical average rate of change computed from the given information is [tex]\(11\)[/tex], not one of the provided multiple choice numbers. The choices might be set incorrectly.)
