1. What is Average Rate of Change?

The Average Rate of Change (ARC) 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:

• \( f(b) \) — 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 over the interval [a, b].

3. Importance of Average Rate of Change

Details: Average Rate of Change is fundamental in calculus and real-world applications. It helps determine average velocity, growth rates, and overall trends in data over specific intervals.

4. Using the Calculator

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

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: Can ARC be negative?
A: Yes, if the function decreases over the interval, ARC will be negative, indicating an overall decrease.

Q3: What units does ARC have?
A: Units depend on the function. If f(x) is in meters and x in seconds, ARC would be in meters/second.

Q4: When is ARC equal to the slope?
A: For linear functions, ARC is constant and equals the slope. For non-linear functions, it represents the average slope over the interval.

Q5: What if b = a?
A: The denominator becomes zero, making the calculation undefined. Choose different values for a and b.