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## Guessing the Fourier Expansion of a Function: Instructor's Guide

### Main Ideas

1. Fourier Series
2. Oscillatory Functions

### Students' Task

Estimated Time:15 minutes

The students were assigned a function that was a superposition between two or more harmonic functions and asked to guess the harmonic terms of the series. Student used Mathematica/Maple to verify their guess against the plot of the original function.

### Prerequisite Knowledge

• Superposition
• Basic harmonic functions

### Activity: Introduction

Students were first asked to build any unique superposition function using Fourier: Making Waves. This helped the students to grasp and apply the idea of superposition. Students then were asked to use a Mathematica/Maple worksheet to find the components of a function that is a superposition of several harmonic terms.

### Activity: Student Conversations

• Students have fun building their own unique superposition function using Fourier: Making Waves. This is a good way to start students think about the superposition principle.
• Most students have prior knowledge of superposition and successfully decompose the superposition function without too much trouble.
• Some students simply “guess-and-check” their answers.
• One effective way to guide students is to ask them what they see as the dominant component in the oscillatory nature of the function.

### Activity: Wrap-up

The wrap-up discussion focuses on the application of the superposition principle. The discussion should emphasize that this method of “guess-and-check” is helpful as a learning tool but not practical. This is a good way to introduce Fourier Series.

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