05.04 Looking at Rates of Change

unit 5

Looking at Rates of Change

 Goal

You will calculate the average rate of change of a function over a specified interval and identify how well given rates of change reflect the changes in an exponential function. Then, apply that knowledge by answering questions. 

  Estimated completion time: 40 minutes

 Watch

When we calculate the average rate of change for a linear function, no matter what interval we pick, the value of the rate of change is the same. A constant rate of change is an important feature of linear functions! When a linear function is represented by a graph, the slope of the line is the rate of change of the function.

Exponential functions also have important features. We've learned about exponential growth and exponential decay, both of which are characterized by a constant quotient over equal intervals. But what does this mean for the value of the average rate of change for an exponential function over a specific interval?

Let's look at an exponential function we studied earlier. Let A be the function that models the area A(t), in square yards, of algae covering a pond weeks after beginning treatment to control the algae bloom. Here is a table showing about how many square yards of algae remain during the first 5 weeks of treatment.

The average rate of change of A from the start of treatment to week 2 is about -107 square yards per week since . The average rate of change of A from week 2 to week 4, however, is only about -12 square yards per week since .

These calculations show that A is decreasing over both intervals, but the average rate of change is less from weeks 0 to 2 than from weeks 2 to 4, which is due to the effect of the decay factor. If we had looked at an exponential growth function instead, the values for the average rate of change of each interval would be positive with the second interval having a greater value than the first, which is due to the effect of the growth factor.

Watch the following video about rates of change.

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