What is Average Rate of Change (AROC)?
The Average Rate of Change (AROC) measures how much a function changes on average between two points. It represents the slope of the secant line connecting two points on a curve.
Formula
The AROC formula is:
$$\text{AROC} = \frac{f(x_{2}) - f(x_{1})}{x_{2} - x_{1}} = \frac{Y_{2} - Y_{1}}{X_{2} - X_{1}}$$
Where:
- (x₁, f(x₁)) is the first point
- (x₂, f(x₂)) is the second point
- The result represents the average slope between these points
Example Calculation
Let's calculate the AROC between two points:
Given:
- X₁ = 2, X₂ = 6
- f(X₁) = 10, f(X₂) = 50
Solution:
$$\text{AROC} = \frac{50 - 10}{6 - 2} = \frac{40}{4} = 10$$
The average rate of change is 10, meaning the function increases by an average of 10 units for each 1-unit increase in x over this interval.
Applications
AROC is used in various fields:
- Physics: Average velocity (distance over time)
- Economics: Average cost or revenue change
- Biology: Average population growth rate
- Finance: Average return on investment
AROC vs Instantaneous Rate of Change
- AROC: Measures average change over an interval
- Instantaneous Rate of Change: Measures change at a specific point (derivative)
As the interval becomes smaller (as x₂ approaches x₁), the AROC approaches the instantaneous rate of change.
Key Properties
- Units: AROC has units of "output per input" (e.g., meters per second, dollars per year)
- Sign: Positive AROC indicates increase, negative indicates decrease
- Magnitude: Larger absolute value indicates faster change
- Linearity: For linear functions, AROC is constant everywhere