WeBWorK Standalone Renderer

(a) Suppose you're given the following Fourier coefficients for a function on the interval $[-\pi,\pi]$ : $\displaystyle a_0 = \frac{2}{2}$ , $\displaystyle a_1 = \frac{4}{\pi}$ , $\displaystyle a_3 = -\frac{4}{3 \pi}$ , $\displaystyle a_5 = \frac{4}{5 \pi}$ , and $a_2, a_4, b_1, b_2, b_3, b_4, b_5$ are all zero. Find the following Fourier approximations to the Fourier series $\displaystyle a_0 + \sum_{n=1}^{\infty} (a_n\cos(nx) + b_n\sin(nx))$ .

$F_1(x) =$
$F_3(x) =$
$F_5(x) =$

(b) Using a calculator, graph the Fourier approximation $F_5(x)$ and then select the letter of the graph which most closely resembles your graph.



A	B	C	D

(Click on a graph to enlarge it.)

(c) Which function given below could the Fourier approximation $F_5(x)$ model?

$\displaystyle f(x) = \left\lbrace \begin{array}{rrr} -2, && -\pi \leq x < 0, \\ 2, && 0 \leq x \leq \pi. \end{array} \right.$		$\displaystyle g(x) = \left\lbrace \begin{array}{rrr} 0, && -\pi \leq x < -\pi/2, \\ 2, && -\pi/2 \leq x \leq \pi/2, \\ 0, && \pi/2 < x \leq \pi. \end{array} \right.$

$\displaystyle h(x) = \left\lbrace \begin{array}{rrr} 2, && -\pi \leq x < 0, \\ -2, && 0 \leq x \leq \pi. \end{array} \right.$		$\displaystyle j(x) = \left\lbrace \begin{array}{rrr} 0, && -\pi \leq x < -\pi/2, \\ -2, && -\pi/2 \leq x \leq \pi/2, \\ 0, && \pi/2 < x \leq \pi. \end{array} \right.$

You can earn partial credit on this problem.