TEACHER NOTES

ISSUES RELATED TO THE PROBLEM

Summing functions does not always produce an obvious result, and by adding parameter dependence, interesting mathematical questions can arise.

Prerequisites

Plotting, maximization, derivative, trigonometric functions.

Time allotment - time management

This investigation can be done with Mathematica in 30 minutes.

Expectations

Future payoffs

Students will become more aware with the effects of varying parameters in functions.

The last problem leads nicely into bifurcation theory which is important to understand if interested in studying chaotic dynamical systems.

Extensions

The following questions turn a once easy problem into a full-blown project.

For A = 20, there are two local minima on x > 0. Is there an A value so that there are three local minima on x > 0? How about 4? 5? n?

Create a method for finding the critical values of A for which extreme values arise and use it. Apply your method to other similar functions and see how robust it is.

References and Sources