https://file.aiquickdraw.com/mathbot/file/1729527654578n7p2hge4.jpg

To solve the problem of finding the average rate of change (ARC) of the function \( f(x) = 9x - 8 \) over the interval \([2,3]\), we can use the formula for the average rate of change:

\[
\text{ARC} = \frac{f(b) - f(a)}{b - a}
\]
where \( a = 2 \) and \( b = 3 \).

### Step-by-Step Solution:
1. First, calculate \( f(2) \):
\[
f(2) = 9(2) - 8 = 18 - 8 = 10
\]

2. Now, calculate \( f(3) \):
\[
f(3) = 9(3) - 8 = 27 - 8 = 19
\]

3. Apply these values to the ARC formula:
\[
\text{ARC} = \frac{f(3) - f(2)}{3 - 2} = \frac{19 - 10}{1} = 9
\]

Thus, the average rate of change is \( 9 \), which is not listed as an option in the problem (the choices are \( 10 \), \( 6 \), and \( 12 \)). This suggests the problem might have a typo or incorrect answer choices.

Would you like further clarification or details on the steps?

Here are 5 related questions to expand your understanding:
1. What is the geometric meaning of the average rate of change?
2. How does the ARC formula relate to the derivative of a function?
3. Can the average rate of change be negative? Under what conditions?
4. What is the significance of linearity in functions like \( f(x) = 9x - 8 \) when calculating the ARC?
5. How would the ARC formula change if the function was non-linear, such as a quadratic function?

**Tip:** When dealing with linear functions, the average rate of change over any interval is always the slope of the line.

Find the average rate of change of the function f(x) = 9x - 8 on the interval [2, 3].

Learn how to find the average rate of change of the linear function f(x) = 9x - 8 on the interval [2, 3]. This step-by-step solution involves calculating f(2), f(3), and applying the formula for average rate of change. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation