Enter the coordinate points at two different points along a function or line to determine the average rate of change (AROC).

AROC Formula

The following formula is used to calculate the average rate of change of a function:

AROC = ( fx(x2) - fx(x1) ) / ( X2-X1) 
  • Where AROC is the average rate of change
  • X2 and X1 are coordinate points along the line
  • fx(x2) and fx(x1) are the values of the function at those x-coordinates

This formula can be more simply written as:

AROC = (Y2 -Y1) / (X2 - X1)

While this is the same formula as slope, the reason the term AROC is used instead is that it’s used to describe the average rates of changes of non-linear lines, while slopes are typically used with linear lines.

AROC Definition

AROC, or Average Rate of Change, is a mathematical concept that measures the average rate at which a quantity changes over a specific interval. It provides valuable information about the rate of change of a variable, allowing us to analyze and understand patterns, trends, and relationships in data.

In essence, AROC represents the slope of a line connecting two points on a graph. It is calculated by dividing the change in the variable’s value over the interval by the corresponding change in time or another independent variable.

By determining the average rate of change, we can gain insights into how a quantity changes over time or in relation to another variable.

In economics, AROC is used to study financial trends. By calculating the average rate of change in variables such as GDP, inflation, or employment rates, economists can evaluate the health of an economy, identify growth patterns, and make informed predictions.
AROC is also widely employed in the natural and social sciences. For instance, in biology, it helps measure population growth rates or the speed at which chemical reactions occur.

Example Problem

How to calculate AROC?

First, determine the two x coordinate points along the function line. For this example, we will use 3 and 5, respectively.

Next, evaluate the function at those points. For the point located at x=3, the function is equal to 30. For the point located at x=5, the function evaluates to 75.

Finally, calculate the average rate of change using the formula above:

AROC = (Y2 -Y1) / (X2 – X1)

AROC = (75-30) / (5-3)

AROC = 22.5


What is the difference between AROC and instantaneous rate of change?
AROC measures the average rate of change between two points on a function, giving an overall view of the change over an interval. Instantaneous rate of change, on the other hand, measures the rate of change at a specific point, similar to the concept of the slope of the tangent line at that point.

Can AROC be negative?
Yes, the average rate of change (AROC) can be negative. A negative AROC indicates that the function is decreasing over the selected interval. This means that as the independent variable increases, the dependent variable decreases.

How is AROC used in economics?
In economics, AROC is used to analyze trends and changes in various economic indicators over time, such as GDP growth rate, inflation rate, and employment rates. It helps economists and policymakers understand the overall direction of economic changes and make informed decisions.

Why is AROC important in understanding non-linear functions?
AROC is particularly important for non-linear functions because it allows us to approximate the rate of change between any two points on the function, providing a simple measure of change over that interval. This is crucial for understanding the behavior of non-linear functions, where the rate of change is not constant across intervals.