Fourier Series: Small Group Activity

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Section 16.10 Fourier Series: Small Group Activity

Make sure to complete the activity in Section 16.2 before attempting this one.

Activity 16.8. Calculating Fourier Coefficients.

Refer to Section 16.5 to find the formulas for the coefficients in a Fourier series. Use the Sage code below to calculate the coefficients $a_m$ and $b_m$ for $m=0,1,2,3$ for the function $-\frac12+\sin(2\pi x)\sin(4\pi x)\text{.}$

The applet in Figure 16.7 shows the function $-\frac12+\sin(2\pi x)\sin(4\pi x)$ (in blue). As you move the sliders, the corresponding Fourier series is also shown (in green). Set the sliders to the values that you calculated. Write as many statements as you can about the relationships between the values of the coefficients and the shape of the graph.

Figure 16.7. An applet for manipulating the individual Fourier coefficients.

Solution.

(Add wrap-up.)