Rates of Change

The rate of change shows how one quantity varies in response to another.

For example, the temperature inside a room varies throughout the day—lower in the early morning, rising during midday, and falling again at night.

This variation in temperature can be represented by a mathematical function, where the temperature is expressed as a function of time.

The average rate of change over a time interval is calculated by dividing the change in temperature by the change in time.

Mathematically, this matches the slope of a secant line connecting two points on the graph.

To find how fast the temperature is changing at a specific instant, the instantaneous rate of change is used.

The limit of the difference quotient as the time interval shrinks to zero gives the exact instantaneous rate, which is the slope of the tangent line at that point.

In real life, the concept of rate of change is used to calculate the rate of concentration change in chemical reactions. The average rate of change gives the change in concentration over a period, and the instantaneous rate of change is the rate of change at a specific moment.