10C - Average rate of change

Variable rate of change

Previously we examined situations with a constant rate of change; however, in may situations the rate of change of one quantity does not remain constant as the other changes. When this is the case it is said to be a variable rate of change. The graphs involved in variable rates of change are non-linear, as such we cannot simply use the gradient to find the rate of change.

Figure 1 - (L) Variable positive rate of change (R) variable negative rate of change.

Average rate of change

For variable rates of change we use the average rate of change over an interval to measure the rate of change of the two quantities. Depending on the shape of the graph, this approximation may or may not be accurate. The average rate of change is found by calculating the gradient of a line segment joining two points on a graph.

Figure 2 - The average rate of change between x = 2 and x = 7.

10C - VIDEO EXAMPLE 1:

Consider the graph below which shows a planes height (meters) with respect to time (minutes) for half of the flight.

What is the average rate of change between:

• t = 0 and t = 10
• t = 10 and t = 90

Average rate of change for a function

For any function, y = f(x), the average rate of change over an interval [a,b] is given by the gradient of a line segment joining the points (a, f(a)) and (b, f(b)):

10C - VIDEO EXAMPLE 2:

Find the average rate of change for the function f(x) = x² - 4 between x = 1 and x = 4.

10C - VIDEO EXAMPLE 3:

The temperature of a room can be modelled by the following function where T is the temperature (in Degrees Celsius) and t is the time after 6:00am:

What is the average rate of change in the temperature between 6:00am and 8:00am?