1. What is 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 the function's graph.

2. How Does the Calculator Work?

The calculator uses the ARC formula:

Where:

• \( f \) — The function being analyzed
• \( a \) — The starting point
• \( b \) — The ending point

Explanation: The numerator calculates the change in the function's output, while the denominator calculates the change in input. The ratio gives the average rate of change over the interval.

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 a valid mathematical function in terms of x (e.g., "x^2 + 3*x - 5"), and two distinct points a and b. The function should be evaluable by PHP's eval() function.

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 of change (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 over the interval.

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

Q4: What are common mistakes when calculating ARC?
A: Using identical points (a = b), which makes denominator zero, or misapplying the function values.

Q5: How does ARC relate to real-world applications?
A: It can represent average speed (distance over time), average growth rate (population over time), or average cost change (cost over quantity).