1. What is the Average Rate of Change?

The Average Rate of Change (ARC) of a function between two points measures how much the function's output changes per unit change in input. It represents the slope of the secant line between two points on a graph of the function.

2. How Does the Calculator Work?

The calculator uses the ARC formula:

Where:

• \( f(a) \) — Function value at point a
• \( f(b) \) — Function value at point b
• \( a \) — First point (input value)
• \( b \) — Second point (input value)

Explanation: The formula calculates the ratio of the change in function values to the change in input values between two points.

3. Importance of Average Rate of Change

Details: ARC is fundamental in calculus and real-world applications. It helps understand how quantities change relative to each other, appearing in physics (velocity), economics (marginal cost), and biology (growth rates).

4. Using the Calculator

Tips: Enter the function values at points a and b, then enter the points themselves. Points a and b must be different (b - a ≠ 0).

5. Frequently Asked Questions (FAQ)

Q1: How is ARC different from instantaneous rate of change?
A: ARC measures change over an interval, while instantaneous rate (derivative) measures change at a single point.

Q2: What does a negative ARC indicate?
A: A negative ARC means the function is decreasing on average between the two points.

Q3: Can ARC be zero?
A: Yes, when f(a) = f(b), indicating no net change between the points.

Q4: What units does ARC have?
A: ARC units are (function output units) per (input units), e.g., m/s for position vs. time.

Q5: How does ARC relate to linear functions?
A: For linear functions, ARC is constant and equals the slope.