TEACHER NOTES

ISSUES RELATED TO THE PROBLEM

This is a fun problem to do in class. It gets noisy, but since there are infinitely many frequency choices, the students can each find their own unique tunes.

Prerequisites

Students should be familiar with frequency in trigonometric functions.

Time allotment - time management

If done in class (recommended), this can be one of two examples of exponential functions or the sole problem of the day depending on how much you want the students to explore and how many parts of the problem you wish to complete.

Expectations

Expect the students to have fun!

Expect a noisy class!

There is typically some difficulty in finding a and b for the exponential function from the data. As such, this should not be the only example that you use.

It is unlikely that students will know how to make the frequency vary as desired in number 6 especially since knowledge of derivatives is NOT a prerequisite for this problem. However, take the opportunity to give the students another reason to learn about derivatives. (The frequency is a rate of oscillation -- and thus the derivative of the input to oscillatory functions.)

Future payoffs

Students are given a reason to like exponential functions.

This problem provides a nice opportunity to review trigonometry even though its main focus is on introducing exponential functions.

Extensions

This could be extended to a project or extra credit by having students explore various types of musical scales and tones. For example, find a relationship between tones that sound good together and tones that do not.

References and Sources

Hughes-Hallett, Gleason, et al. 1994. Calculus. Wiley, New York, NY, p. 18.