Let's find the average rate of change of the function [tex]\( g(x) \)[/tex] on the interval [tex]\( 3 \leq x \leq 6 \)[/tex].
### Step-by-Step Solution:
1. Understanding given intervals and rates:
- The average rate of change on the interval [tex]\(-3 \leq x \leq 0\)[/tex] is 1.
- The average rate of change on the interval [tex]\(0 \leq x \leq 3\)[/tex] is 5.
2. Determine function values at key points:
Let's assume [tex]\( g(-3) = 0 \)[/tex] to simplify our calculations.
- Interval [tex]\(-3\)[/tex] to [tex]\(0\)[/tex]:
[tex]\[
g(0) = g(-3) + \text{rate of change} \times (\text{length of the interval})
\][/tex]
Substituting the values:
[tex]\[
g(0) = 0 + 1 \times (0 - (-3))
\][/tex]
[tex]\[
g(0) = 0 + 1 \times 3 = 3
\][/tex]
- Interval [tex]\(0\)[/tex] to [tex]\(3\)[/tex]:
[tex]\[
g(3) = g(0) + \text{rate of change} \times (\text{length of the interval})
\][/tex]
Substituting the values:
[tex]\[
g(3) = 3 + 5 \times (3 - 0)
\][/tex]
[tex]\[
g(3) = 3 + 5 \times 3 = 3 + 15 = 18
\][/tex]
3. Calculate function value at [tex]\(x = 6\)[/tex]:
To find [tex]\(g(6)\)[/tex] on the interval [tex]\(3 \leq x \leq 6\)[/tex], we assume the rate of change remains constant at 5 (as given).
- Interval [tex]\(3\)[/tex] to [tex]\(6\)[/tex]:
[tex]\[
g(6) = g(3) + \text{rate of change} \times (\text{length of the interval})
\][/tex]
Substituting the values:
[tex]\[
g(6) = 18 + 5 \times (6 - 3)
\][/tex]
[tex]\[
g(6) = 18 + 5 \times 3 = 18 + 15 = 33
\][/tex]
4. Calculate the average rate of change on the interval [tex]\(3 \leq x \leq 6\)[/tex]:
The average rate of change between [tex]\(x = 3\)[/tex] and [tex]\(x = 6\)[/tex] is:
[tex]\[
\text{Average rate of change} = \frac{g(6) - g(3)}{6 - 3}
\][/tex]
Substituting the values:
[tex]\[
\text{Average rate of change} = \frac{33 - 18}{6 - 3}
\][/tex]
[tex]\[
\text{Average rate of change} = \frac{15}{3} = 5
\][/tex]
Therefore, the average rate of change of [tex]\(g\)[/tex] on the interval [tex]\(3 \leq x \leq 6\)[/tex] is [tex]\( \boxed{5} \)[/tex].
