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Guessing Legendre Coefficients: Instructor's Guide

Main Ideas

1. Maple is used to add orthogonal Legendre functions
2. Students can plot the sum of several functions
3. Students use guess and check method to find the Legendre expansion of a function

Estimated Time: 10-15 minutes

Students try to fit a given function with a linear combination of Legendre Polynomials using the guess and check method. That is, they guess what linear combination they expect and then plot their linear combination with the given function to see how close they are.

This activity can be used as to reinforce students understanding of Legendre Polynomials and as a good introduction to Legendre Series.

Prerequisite Knowledge

• Students must have basic knowledge of the Legendre Polynomials.

Activity: Student Conversations

Students will approach this activity using a variety of problem solving methods:

• Some students look at the number of nodes that the target function has, and start with a Legendre polynomial of that degree.
• Some begin with a guess and then add polynomials based on parity - i.e, adding a positive, odd Legendre polynomial to skew their function to the clockwise.
• more?

Prompting a group that is stuck with one of these methods can be helpful.

Activity: Wrap-up

If students have previous experience with Fourier series, comparing the similarities of Fourier and Legendre decompositions can be a powerful analogy.

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