Understanding Rates of Change

A rate of change is a measure that describes how one quantity changes in relation to another quantity. In the context of functions, it often refers to how the output value (dependent variable) changes as the input value (independent variable) changes. Rates of change can be constant or variable:

• Constant Rate of Change: This occurs in linear functions, where the rate of change remains the same across the domain of the function.
• Variable Rate of Change: This is observed in nonlinear functions, where the rate of change varies at different points in the domain.

Example of Rates of Change

To illustrate the concept of rates of change, let's consider a simple example involving distance and time, which models a linear relationship.

Example

Imagine a car travels at a constant speed of 60 miles per hour. The distance d (in miles) that the car travels can be represented as a function of time t (in hours), such as d(t) = 60t.

Calculating the Rate of Change

In this scenario, the rate of change of distance with respect to time is constant, as the car travels at a steady speed. This rate of change can be calculated by taking any two points on the function and dividing the change in distance by the change in time. For instance, from t = 1 hour to t = 2 hours, the car travels from d(1) = 60(1) = 60 miles to d(2) = 60(2) = 120 miles.

The rate of change, R, can be calculated as:

R = Δd / Δt = (d(2) - d(1)) / (2 - 1) = (120 - 60) / 1 = 60 miles per hour

Graphical Interpretation

Graphically, for a linear function, the rate of change corresponds to the slope of the line. In our example, a graph of d(t) versus t would yield a straight line with a slope of 60, indicating that for each hour of travel, the distance increases by 60 miles.