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Section 10.8 Fourier Series: Small Group Activity

Make sure to complete the activity in Section 10.2 before attempting this one.

Activity 10.8.1. Calculating Fourier Coefficients.

Refer to Section 10.5 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 10.8.1 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 10.8.1. An applet for manipulating the individual Fourier coefficients.
Solution

(Add wrap-up.)