Students use prepared Sage code or a prepared Mathematica notebook to plot \(\sin\theta\) simultaneously with several terms of a power series expansion to judge how well the approximation fits. Students can alter the worksheet to change the number of terms in the expansion and even to change the function that is being considered. Students should have already calculated the coefficients for the power series expansion in a previous activity, Calculating Coefficients for a Power Series.
Use the Sage code in the activity at this link or the attached Mathematica Notebook to plot the function \(\sin\theta\) and power series approximations to the function to explore how well the approximations work.
You will first need to calculate the coefficients of the power series. For the first part of the worksheet, calculate these coefficients for the power series around \(\theta=0\) and for the second part of the worksheet, calculate the coefficients around \(\theta=\frac{\pi}{6}\). You may have done these calculations in Calculating Coefficients for a Power Series.
You will need to know a few things about Mathematica Notebooks:
Students should be able to calculate coefficients for a power series expansion and they need to have the series expansion for \(\sin(\theta)\) available, for example from having completed the activity Calculating Coefficients for a Power Series.
This activity is designed to be a soft introduction to Mathematica. The notebook 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 notebook with Shift/Enter.
An alternative version of the notebook using graphs with sliders, which does NOT require the students to enter any information is also provided. This version works well as a review of the geometeric meaning of power series.