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\begin{document}

\centerline{\textbf{Sample Midterm Questions}}
\centerline{\textbf{Math 121}, \textbf{Fall 2004}}

\bigskip\bigskip
\noindent
\textbf{1.} Answer the following questions with a brief explanation to justify
your answer.

\smallskip\noindent
(a) What is the period of the function $\sin \left(7 x\right)$?

\smallskip\noindent
(b) What is the value of
\[
\int_0^{700\pi} \sin^2 \left(7x\right)\, dx?
\]

\smallskip\noindent
(c) What is the value of
\[
\int_0^{700\pi} \sin \left(7x\right)\sin \left(70 x\right)\, dx?
\]


\bigskip\bigskip
\noindent
\textbf{2.} Suppose that $f(x)$ is the $2\pi$-periodic function defined by
\[
f(x) = \cases{x & for $-\pi < x \le 0$,\cr
0 & for $0 < x \le \pi$.}
\]
Compute the (real) Fourier series expansion of $f(x)$. What does the Fourier series converge to
at $x=0$, $x=\pi/2$, and $x=\pi$? Why?


\bigskip
\noindent
\textbf{3.} Suppose that
\[
f(x) = 1-x^2\qquad\mbox{$0<x<1$}.
\]
Let $f_p$ be the periodic extension of $f$ (with period $1$), $f_{\mathrm{e}}$ the even periodic extension of
$f$ (with period $2$), and $f_{\mathrm{o}}$ the odd periodic extension of $f$ (with period $2$).

\smallskip\noindent
(a) Sketch
the graphs of $f_p$, $f_{\mathrm{e}}$, and $f_{\mathrm{o}}$ on the interval $-3 < x < 3$.


\smallskip\noindent
(b) Write out the corresponding form of the
Fourier series for these functions, together with expressions for their Fourier coefficients. (Just
write expressions for the coefficients --- don't evaluate any integrals.)

\smallskip\noindent
(c) Which Fourier series converges faster --- the one for $f_{\mathrm{e}}$ or the one for $f_{\mathrm{o}}$? Explain you answer briefly,
but don't do any explicit computations.

\bigskip
\noindent
\textbf{4.} The function
\[
f(x) = \cases{0 & for $-\pi < x \le 0$,\cr
\sin x & for $0 < x \le \pi$.}
\]
Has the Fourier series expansion
\[
f(x) = \frac{1}{\pi} +\frac{1}{2}\sin x - \frac{2}{\pi}\left(\frac{\cos 2x}{2^2-1}
+\frac{\cos 4x}{4^2-1}+\frac{\cos 6x}{6^2-1} + \dots +\frac{\cos 2nx}{(2n)^2-1} + \dots\right)
\]
Apply Parseval's theorem to this function, and use the result to determine the
sum of the infinite series
\[
\sum_{n=1}^\infty \frac{1}{\left[(2n)^2 - 1\right]^2} = \frac{1}{3^2} + \frac{1}{15^2} + \frac{1}{35^2} + \dots.
\]




\bigskip
\noindent
\textbf{5.}
State the orthogonality relation for the functions $e^{inx}$, where $n$ is an integer.

\smallskip\noindent
(a) Use these relations to derive an expression for the Fourier coefficient $c_n$ in the complex
Fourier expansion of a $2\pi$-periodic function $f$,
\[
f(x) = \sum_{n=-\infty}^\infty c_n e^{inx}.
\]

\smallskip\noindent
(b) What can you say about $c_n$ if $f$ is an even function?

\bigskip
\noindent
\textbf{6.} Suppose that $f(x)$ has the complex Fourier series expansion
\[
f(x) = \sum_{n=-\infty}^\infty \frac{1}{1+in} \,e^{inx}.
\]
Find the real Fourier series expansion of $f(x)$ (in terms of sines and cosines).
\end{document}