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Section 10.7 Fourier Series: Exploration

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

Activity 10.7.1. Guessing the Fourier Coefficients.

In Figure 10.7.1 below, use the sliders to match the given function (shown in blue) exactly. Use only graphical reasoning. Hint: Only three of the sliders need to be set to nonzero values.

When you are done, make a note of any relationship you see between the values of the coefficients and the shape of the graph.

Figure 10.7.1. An applet for manipulating the individual Fourier coefficients.
Answer

Coming soon.

Activity 10.7.2. Comparing an exact calculation to your guess.

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 for the function \(f(x)=-\frac12+\sin(2\pi x)\sin(4\pi x)\) used in the previous activity. Check that your calculated coefficients agree with your earlier guess.

Hint

The Sage code calculates \(a_0\text{.}\) You will need to make minor changes in order to calculate the remaining coefficients.