Average Rate of Change Calculator
Calculate the average rate of change between two points on a function
Calculate Average Rate of Change
Coordinates of the first point
Coordinates of the second point
Results
Analysis
Example Calculation
Train Speed Example
Problem: A train travels from Paris to Rome (1420.6 km) in 12.5 hours.
Point 1: (0 hours, 0 km) - Starting point
Point 2: (12.5 hours, 1420.6 km) - End point
Solution
A = (1420.6 - 0) / (12.5 - 0)
A = 1420.6 / 12.5
A = 113.648 km/h
The train's average speed was 113.648 km/h.
Formula Reference
Average Rate of Change
A: Average rate of change
(x₁, y₁): First point coordinates
(x₂, y₂): Second point coordinates
Alternative Forms
• Δy / Δx (change in y over change in x)
• Rise over run
• Slope of secant line
Key Concepts
Positive rate = function increasing on average
Negative rate = function decreasing on average
Zero rate = no net change between points
Undefined rate = vertical line (x₁ = x₂)
Understanding Average Rate of Change
What is Average Rate of Change?
The average rate of change describes how one quantity changes in relation to another over a specific interval. It represents the slope of the line connecting two points on a graph, also known as the secant line.
Real-World Applications
- •Speed: Change in distance over change in time
- •Population growth: Change in population over time
- •Economics: Change in cost per unit produced
- •Temperature: Rate of heating or cooling
Key Differences
Average vs. Instantaneous Rate
Average rate of change gives the overall rate between two points, while instantaneous rate (derivative) gives the rate at a specific point.
Slope vs. Rate of Change
For linear functions, slope and average rate of change are identical. For nonlinear functions, the average rate of change varies depending on the interval chosen.