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\title{m-3150-001 ~~~ Midterm exam 1}
\author{Andrej Cherkaev}
\date{}
\maketitle
Your name, ...........................................................

\vspace{0.3in}
Solve any three problems: 
\begin{enumerate}
\item Find the period of the following functions:
\begin{eqnarray}
a. & & f_1(x)=\sin(2 x)+ \cos( 4 x) \\
b. && f_3(x)= \sin( 5 x)- 2 \cos(3 x)\\
c. & & f_2(x)= \cos( 3 \pi x)
\end{eqnarray}
\newpage

\item a. Find the Fourier series of the function $f(x)= e^x$, $-\pi<x<
  \pi$. 

b. Differentiate the series and show that $f'(x)=f(x)$. 
\newpage

\item
The function 4-periodic is given by formula
$$
f(x)=\left \{\begin{array}{ll} 3 & \mbox{ if } 0<x\leq 1 \cr
-1 & \mbox{ if } 1<x\leq 2 \cr
0 & \mbox{ if } 2<x\leq 4  .
\end{array} \right.
 $$
a. Find the Fourier series.

b. Skatch the graph of the function and of its
antiderivative, assuming that integral of the antiderivative over the
interval $[0,\,4]$ is zero. Write
the Fourier series for the antiderivative.
\newpage
\item
Find the even and odd half-range expansions of  $f(x)=x^2$,  $0 <x<
1$.  

Skatch the graphs of these expansions in the interval $-1 < x< 1$.

 
\end{enumerate} 

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