1. What is Average Rate of Change?

The Average Rate of Change (ARC) measures how much a quantity changes on average between two points. In mathematics, it represents the slope of the secant line between two points on a function's graph.

2. How Does the Calculator Work?

The calculator uses the Average Rate of Change formula:

Where:

• \( f(b) \) — Function value at point b
• \( f(a) \) — Function value at point a
• \( b \) — x-coordinate of the second point
• \( a \) — x-coordinate of the first point

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

3. Importance of Average Rate of Change

Details: Average Rate of Change is fundamental in calculus and real-world applications. It helps determine average speed, growth rates, and overall trends between two points in time or space.

4. Using the Calculator

Tips: Enter the function values f(b) and f(a), and their corresponding x-values b and a. Ensure b ≠ a to avoid division by zero. All values can be positive or negative real numbers.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between average and instantaneous rate of change?
A: Average rate of change 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 indicates the function is decreasing on average over the interval from a to b.

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

Q4: What are common applications of ARC?
A: Used in physics for average velocity, economics for average growth rates, and biology for average population change.

Q5: How is ARC related to slope?
A: ARC represents the slope of the secant line connecting points (a, f(a)) and (b, f(b)) on the function's graph.