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## Approximating Functions with a Power Series : Instructor's Guide

### Main Ideas

1. Visualizing the fit of a power series approximation to a goven function.
2. Convergence for a power series.

### Students' Task

Estimated Time: 20 minutes

Students have already calculated the coefficients for a power series expansion. Students plot several terms of the expansion against the original function in order to judge how well the approximation fits the original function.

### Prerequisite Knowledge

This activity is a good followup to Calculating Coefficients for a Power Series.

This activity is designed to be a soft introduction to Mathematica or Maple. The notebook/worksheet is already prepared, but is missing some information which students will need to fill in. They will also need to learn how to step through a notebooks/spreadsheet. Students should be able to calculate coefficients for a power series expansion and they need to have the series expansion for $\sin(\theta)$ available.

### Activity: Introduction

No introduction is needed - students can jump right in!

### Activity: Student Conversations

1. Students have to modify the worksheet in order to plot approximations better than 3rd order. Students who are uncomfortable with Maple (or equivalent) may have a little trouble.
2. Students are asked to determine how many terms are needed in the approximation in order to fit the $\sin{\theta}$ function from $0$ to $\pi$. Students should be encouraged to explore higher order approximations.

### Activity: Wrap-up

1. This activity leads into a nice discussion of idealizations and making approximations. The question of “How many terms do I need to keep in my approximation?” is related to the question of “What domain do I care about?”
2. Most students at the middle division level are familiar with small-angle approximations and the example of simple harmonic motion of a pendulum. This activity illustrates nicely how small your angle must be in order for the approximation $\sin{\theta}\approx \theta$ to make sense.
3. You can also discuss some nice sense-making activities. One such example is being able to tell if you've got the sign wrong for a particular term - if it makes the approximation worse (the approximation diverges from the original function faster than it did with fewer terms), then you may have made a sign error.

### Extensions

This activity is part of a sequence of activities addressing Power Series and their application to physics. The following activities are part of this sequence.

• Preceding activities:
• Recall the Electrostatic Potential due to a Point Charge: This small whiteboard question has students recall the basic expression for the electrostatic potential due to a point charge which is used to begin a classroom conversation regarding what is meant by $\frac{1}{r}$.
• The Distance Between Two Points - Star Trek: This kinesthetic activity has students work together to resolve a given problem using geometry which opens the class to a discussion about position vectors and how to generalize the $\frac{1}{r}$ factor to $\dfrac{1}{|\vec{r}-\vec{r}'|}$.
• Calculating Coefficients for a Power Series: This small group activity has students work out the expansion coefficients of a familiar function, $\sin(\theta)$, which gives them more experience working with power series.
• Follow-up activity:
• Electrostatic Potential Due to Two Point Charges: This small group activity has students apply what they know about power series approximations to find a general expression and asymptotic solution to the electrostatic potential due to two point charges.

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