1. What is Average Rate of Change?

The average rate of change measures how much a function changes on average between two points. It represents the slope of the secant line connecting the points (a, f(a)) and (b, f(b)) on the function's graph.

2. How Does the Calculator Work?

The calculator uses the average rate of change formula:

Where:

• \( f(a) \) — Function value at point a
• \( f(b) \) — Function value at point b
• \( a \) — Starting point of the interval
• \( b \) — Ending point of the interval

Explanation: This formula calculates the ratio of the change in function values to the change in input values over the interval [a, b].

3. Importance of Average Rate of Change

Details: The average rate of change is fundamental in calculus and real-world applications. It helps understand how quantities change over time or distance, such as velocity, growth rates, and economic trends.

4. Using the Calculator

Tips: Enter the function values f(a) and f(b), and the interval endpoints a and b. Ensure that a and b are different values (b ≠ a) to avoid division by zero.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between average and instantaneous rate of change?
A: Average rate measures change over an interval, while instantaneous rate measures change at a single point (derivative).

Q2: Can the average rate of change be negative?
A: Yes, it can be negative if the function is decreasing over the interval [a, b].

Q3: What does a zero average rate of change indicate?
A: A zero value indicates no net change in the function over the interval, though the function may have fluctuated.

Q4: How is this used in real-world applications?
A: Used in physics for average velocity, economics for growth rates, biology for population changes, and many other fields.

Q5: What if the interval endpoints are the same?
A: The formula becomes undefined due to division by zero. The interval must have distinct endpoints.