## Section16.9Fourier Series: Exploration

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

### Activity16.6.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.

Coming soon.

### Activity16.7.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.

L=1
m=0
f=-0.5+sin(2*pi*x)*sin(4*pi*x)
2/L*integrate(f*cos(2*pi*m*x/L),x,0,L)

Hint.

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