1. What is Average Rate of Change?

The average rate of change measures how much a function changes on average between two points. It represents the slope of the secant line between points (x, f(x)) and (x+h, f(x+h)) on the function's graph.

2. How Does the Calculator Work?

The calculator uses the average rate of change formula:

Where:

• \( f(x + h) \) — Function value at point x + h
• \( f(x) \) — Function value at point x
• \( h \) — Interval between the two points

Explanation: This formula calculates the ratio of the change in function values to the change in input values over the interval h.

3. Importance of Average Rate of Change

Details: Average rate of change is fundamental in calculus and real-world applications. It helps determine velocity in physics, growth rates in biology, marginal cost in economics, and slope in geometry.

4. Using the Calculator

Tips: Enter f(x+h) and f(x) values in appropriate units, and the interval h (must be non-zero). The calculator will compute the average rate of change with proper units.

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 h be negative?
A: While mathematically possible, h typically represents a positive interval. Negative h would reverse the direction of measurement.

Q3: What units does the result have?
A: The units are [f(x) units] per [x units]. For example, meters per second if f(x) is position and x is time.

Q4: When is average rate of change zero?
A: When f(x+h) equals f(x), indicating no net change over the interval.

Q5: How does this relate to slope?
A: Average rate of change equals the slope of the secant line between the two points on the function's graph.