Fourier Series: Exploration

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Section 16.9 Fourier Series: Exploration

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

Activity 16.6. Guessing the Fourier Coefficients.

In Figure 16.6 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 16.6. An applet for manipulating the individual Fourier coefficients.

Coming soon.

Activity 16.7. Comparing an exact calculation to your guess.

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 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.

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