Average Rate of Change Calculator
Calculation Result:
What is the Average Rate of Change?
The average rate of change measures how much a function changes per unit over a specific interval. In geometry, it represents the slope of the secant line that passes through two points on a curve. In physics, it often represents average velocity or average acceleration.
The Formula
To calculate the average rate of change of a function f(x) over the interval [a, b], we use the following formula:
Average Rate of Change = [f(b) – f(a)] / (b – a)
Where:
- f(b) is the value of the function at the end of the interval.
- f(a) is the value of the function at the start of the interval.
- b – a represents the total change in the independent variable (x).
How to Use This Calculator
1. Enter your starting x-value (x₁) and the corresponding function value (y₁).
2. Enter your ending x-value (x₂) and the corresponding function value (y₂).
3. Click "Calculate" to see the average rate of change over that specific interval.
Practical Example:
Suppose an object's position is 10 meters at 2 seconds and 40 meters at 5 seconds. To find the average velocity (rate of change of position):
- x₁ = 2, y₁ = 10
- x₂ = 5, y₂ = 40
- Calculation: (40 – 10) / (5 – 2) = 30 / 3 = 10 m/s.
Why It Matters
Understanding the rate of change is fundamental in calculus, economics, and data science. It helps professionals determine growth rates, speed, and how variables interact over time. While the "instantaneous" rate of change requires derivatives, the average rate of change provides a reliable summary of behavior across a distance or time frame.