## Section10.7Fourier Series: Exploration

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

###### Activity10.7.1.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.

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