Calculus - Average Value of a Function - An Exploration

As part of the AP Calculus course description students need to understand what it means to find the average value of a function. This activity incorporates the graphing calculator to develop an understanding for the average value of a function.

Students draw graph for four functions to view all the function values between two x values. Since students understand that they find an average for a set of numbers by dividing the sum of the numbers by the total number of numbers, this lesson begins by averaging a finite number of functions values. This is extended to finding the area of the bounded region and dividing that area by the difference between the two x-value limits to find the average value that the function takes on.

The four functions studied are

• y = 2x + 5 between x = 0 and x = 5

• y = 0.5x^2 between x = 0 and x = 5

• y = - 2x^2 + 10x between x = 0 and x = 5 and

• The piecewise function y = 1.5x^2 between x = 0 and x = 5 and y = -2x + 10 between x = 2 and x = 5.

By the end of the lesson or activity students are writing that the average value of a function is found by integrating the function between the two x values and then dividing that integral by the difference between the two x values.

Students are then given five exercises to complete. Two involve simple nonlinear functions, one is based only on a graph of a piecewise function, and then two other problems are based on algebraic descriptions of two functions.

A complete set of solutions is included.

A powerpoint, by the same name can be integrated with this activity.

Comments from Buyers:

• Thank you

• This is a good exploration that guides the students right to the graphical meaning of average value.

• Thank you - very helpful.