1. What is the Average Rate of Change?

The Average Rate of Change (ARC) measures how a function's output changes on average over a specific input interval. It represents the slope of the secant line between two points on a function's graph and is fundamental in single variable calculus.

2. How Does the Calculator Work?

The calculator uses the Average Rate of Change formula:

Where:

• \( f(x_1) \) — Function value at point x₁
• \( f(x_2) \) — Function value at point x₂
• \( x_1 \) — First input value
• \( x_2 \) — Second input value
• \( ARC \) — Average Rate of Change (slope)

Explanation: The formula calculates the ratio of the change in function values to the change in input values over the interval [x₁, x₂].

3. Importance of Average Rate of Change

Details: Average Rate of Change is crucial for understanding function behavior, approximating instantaneous rates, and serves as the foundation for the derivative concept in calculus.

4. Using the Calculator

Tips: Enter function values f(x₁) and f(x₂), and corresponding input values x₁ and x₂. Ensure x₂ ≠ x₁ to avoid division by zero. All values can be positive, negative, or 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 ARC be negative?
A: Yes, negative ARC indicates the function is decreasing over the interval.

Q3: What does a zero ARC indicate?
A: Zero ARC means the function values are equal at both endpoints (constant over the interval).

Q4: How is ARC related to slope?
A: ARC equals the slope of the secant line connecting the two points on the function's graph.

Q5: What are practical applications of ARC?
A: Used in physics for average velocity, economics for average growth rates, and biology for average reaction rates.