What is the average rate of change of a function f over an interval [a, b] for which a 6= b? (i) f(b) − f(a) (ii) f(b) − f(a) a − b (iii) f(b) − f(a) b − a (iv) none of the above

To find the average rate of change of a function \( f \) over an interval \([a, b]\), where \( a \neq b \), we use the formula:

\[
\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}
\]

This formula essentially measures the change in the function's value (i.e., \( f(b) - f(a) \)) over the change in the input (i.e., \( b - a \)).

Now, let's compare this with the given options:

1. \( f(b) - f(a) \): This expression represents the change in the function values only and does not take into account the interval length \( b - a \). This is not the average rate of change.
2. \( \frac{f(b) - f(a)}{a - b} \): This expression is close, but the denominator is \( a - b \), which is negative if \( a < b \). The correct formula should have \( b - a \) in the denominator. Thus, this is not the correct expression either.
3. \( \frac{f(b) - f(a)}{b - a} \): This expression matches the formula for the average rate of change. It correctly accounts for the change in function value over the change in input. 

4. None of the above: Since option (iii) matches the correct formula, this option is incorrect.

Therefore, the correct answer is:

**(iii) \(\frac{f(b) - f(a)}{b - a}\)**

Would you like more details or have any questions? Here are five related questions to consider:

1. How does the average rate of change relate to the derivative of a function?
2. What is the average rate of change for a linear function over any interval \([a, b]\)?
3. How can the concept of average rate of change be applied to real-world problems?
4. What does the sign (positive/negative) of the average rate of change indicate about the function?
5. How does the average rate of change differ from the instantaneous rate of change?

**Tip:** The average rate of change of a function over an interval can be thought of as the slope of the secant line that passes through the points \((a, f(a))\) and \((b, f(b))\) on the function's graph.

Explore how to calculate the average rate of change of a function over an interval [a, b]. Learn the formula (f(b) - f(a)) / (b - a) and its application in mathematics. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation