1. What is the Average Rate of Change?

The Average Rate of Change (ARC) represents the slope of the secant line between two points on a function. It measures how much a quantity changes on average per unit change in the independent variable over a specific interval.

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 endpoint
• \( a \) — x-coordinate of starting point

Explanation: The formula calculates the slope between two points (a, f(a)) and (b, f(b)) on a function, representing the average rate at which the function changes 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-coordinates b and a. Ensure b ≠ a to avoid division by zero. All values should be numerical.

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 ARC = 0 mean?
A: ARC = 0 indicates no net change over the interval (function values at endpoints are equal).

Q4: How is ARC used in real-world applications?
A: Used in physics for average velocity, economics for average growth rates, and biology for population change rates.

Q5: What are the units for ARC?
A: Units are (output units)/(input units), such as m/s for velocity or $/month for financial growth.