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\title{Midterm 2. M 3150-001, Spring 2015}
%\author{Andrej Cherkaev}
\date{}
\maketitle
Your name (please print) ..................................................

\begin{enumerate}
\item Are the following functions piece-wise differentiable? (if they are
not, state the reason)
\begin{enumerate}
\item $$ f_1(x)= |x|, \quad x \in [-1, 1];  $$

\vspace{0.3in}
\item  $$  f_2(x)= x \log x \quad x \in [0, 1]; $$

\vspace{0.3in}
\item   $$  f_3(x)= \left\{ \begin{array}{ll}
x +1& \quad x \in [0, 1] \\
-x &\quad x \in [1,2].
\end{array} \right. $$

\vspace{0.3in}
\item  $$ f_4(x)=  x^{2/3},\quad x \in [0, 2];  $$

\vspace{0.3in}
\item  $$f_5(x)=  \frac{1+x}{1-x}, \quad x \in [0, 2].  $$

\end{enumerate}
\newpage

\item Consider function $g(x)= e^x$, $x\in [0, 1]$. Its Fourier series
expansion represents a periodic function $\hat{g}(x)$, $x\in (-\infty,
\infty)$.

Find the series and sketch the graphs of  $\hat{g}(x)$ for:
\begin{enumerate}
\item General (sin and cosine) Fourier series  expansion,

\vspace{0.1in}
\item Sine Fourier series  expansion,

\vspace{0.1in}

\item Cosine  Fourier series expansion.

\vspace{0.1in}



Do not evaluate the integrals.
\end{enumerate}



\newpage
\item

\begin{enumerate}
\item Find the sine and cosine Fourier series of
 $$  f(x)= \left\{ \begin{array}{ll}
1& \quad x \in [0, 1] \\
0 &\quad x \in [1,2]
\end{array} \right. .$$

\item Find first two nonzero coefficients of both series.
\end{enumerate}

\vspace{3in}





\newpage
\item A string vibrates according to the equation $u_{tt}=4 u_{xx} $, where
$u(x, t)$ is the deflection, $x\in [0, 1]$, $t\in [0, \infty)$.
\newline
The ends are fixed: $u(0, t)=0$, $ u(1, t)=0$,
\newline
initial shape $u(x,t)$ is
$u(x, 0) =3 \sin(2 \pi x)- 2\sin(5 \pi x)$, \newline
initial speed is zero, $\frac{\partial u}{\partial t}|_{t=0}
=0. $\newline
Find $u(x, t)$.



\end{enumerate}



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