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Activities

Computer Simulation

30 min.

Visualization of Power Series Approximations
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.
  • Taylor series power series approximation
    Found in: Theoretical Mechanics, Static Fields, Central Forces, AIMS Maxwell, Problem-Solving, None course(s) Found in: Power Series Sequence (Mechanics), Power Series Sequence (E&M) sequence(s)

Small Group Activity

30 min.

Calculating Coefficients for a Power Series

This activity starts with a brief lecture introduction to power series and a short derivation of the formula for calculating the power series coefficients.

\[c_n={1\over n!}\, f^{(n)}(z_0)\]

Students use this formula to compute the power series coefficients for a \(\sin\theta\) (around both the origin and (if time allows) \(\frac{\pi}{6}\)). The meaning of these coefficients and the convergence behavior for each approximation is discussed in the whole-class wrap-up and in the follow-up activity: Visualization of Power Series Approximations.

  • Taylor Series Coefficents Power Series
    Found in: Theoretical Mechanics, AIMS Maxwell, Static Fields, Problem-Solving, None course(s) Found in: Power Series Sequence (Mechanics), Power Series Sequence (E&M) sequence(s)

Computational Activity

120 min.

Kinetic energy
Students implement a finite-difference approximation for the kinetic energy operator as a matrix, and then use numpy to solve for eigenvalues and eigenstates, which they visualize.