The average rate of change is a fundamental concept in mathematics and physics that describes how a quantity changes over a specific interval. It essentially measures the "steepness" of a function between two points. For a function, say f(x), the average rate of change between two points (x₁, y₁) and (x₂, y₂) is calculated as the difference in the y-values (the change in the dependent variable) divided by the difference in the x-values (the change in the independent variable).
(x₁, y₁) are the coordinates of the starting point.
(x₂, y₂) are the coordinates of the ending point.
y₂ - y₁ represents the total change in the y-value (often denoted as Δy).
x₂ - x₁ represents the total change in the x-value (often denoted as Δx).
A positive average rate of change indicates that the function is increasing over the interval, while a negative rate indicates the function is decreasing. A rate of zero suggests the function is constant or has balanced increases and decreases over the interval.
Applications:
Physics: Calculating average velocity or acceleration between two points in time. For instance, if a car's position changes from 10 meters to 100 meters over 10 seconds, its average velocity is (100 – 10) / (10 – 0) = 9 meters per second.
Economics: Analyzing average changes in stock prices, inflation rates, or economic growth over a period.
Calculus: It forms the basis for understanding instantaneous rates of change (derivatives) by considering smaller and smaller intervals.
Example:
Let's say we are analyzing the growth of a plant. We record its height at two different times:
At x₁ = 2 days, the height was y₁ = 5 cm.
At x₂ = 7 days, the height was y₂ = 20 cm.
To find the average rate of change in the plant's height per day, we use the formula: