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\title{General Mathematics and Computational Science I}
\author{Exercise 15}
\date{November 21, 2006}
\maketitle

\begin{enumerate}

\item \emph{(Extension from Homework Set 14.)}  Use Laplace's method
to find the leading term in the asymptotic behavior of
\[
  \int_{-1}^1 \mathrm{e}^{-s \, \cosh x} \, \mathrm{d} x
\]
as $s \to \infty$.

\item (From Elaydi, p. 28.)  Find the equilibrium points of the
difference equation
\[
  x_{n+1} = \tfrac12 \, x^3_n + \tfrac12 \, x_n
\]
and determine their stability.

\item (From Elaydi, p. 34.)  Suppose that the difference equation
\[
  x_{n+1} = f(x_n)
\]
has a $2$-cycle whose orbit is $\{a,b\}$.  Show that
\begin{enumerate}
\item the $2$-cycle is asymptotically stable if $\lvert f'(a) \,
f'(b)\rvert<1$,
\item the $2$-cycle is unstable if $\lvert f'(a) \,
f'(b)\rvert>1$.
\end{enumerate}

\emph{Hint:} Note that a $2$-cycle is an equilibrium point of
$y_{n+1}=f(f(y_n))$. 

\end{enumerate}
\end{document}

\item Use Stirling's formula to derive Wallis' product:
\begin{equation*}
\frac{\pi}{2}=
\frac{2}{1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\cdot
\frac{6}{5}\cdot\frac{6}{7}\cdot\ldots\cdot\frac{2n}{2n-1}\cdot\frac{2n}{2n+1}\cdot\ldots
\end{equation*}

\item Solve the following linear programming problem.\\
Maximize
\[
  w=x^2+y^2-z^2
\]
subject to:
\begin{eqnarray*}
    y^2-z^2\geq 2\\
    x^2-z^2\leq 2\\
    x^2+y^2\leq 4\\
\end{eqnarray*}

\item Two points $x$ and $y$ are selected at random from the
interval $[0,1]$. What is the probability that their product will be
less than $\frac{1}{2}$, i.e. $xy< \frac{1}{2}$.

\textbf{Hint} It may be helpful to draw a picture of the unit square
and draw the graph of $y=\frac{1}{2x}$.



\item Find all solutions for the underdetermined linear system $Ax=b$,
where
\begin{equation*}
  A =
  \begin{pmatrix}
    2 & 2 & 1 & 0 \\
    1 & 1 & 1 & 1 \\
    1 & 1 & 0 & -1 \\
    3 & 3 & 2 & 1
  \end{pmatrix}
  \qquad \text{and} \qquad
  b =
  \begin{pmatrix}
    -1 \\ 1 \\ -2 \\ 0
  \end{pmatrix} \,.
\end{equation*}

\item Write the following linear programming problem in its standard
form.

Maximize
\[
  z = 2 \, x_1 - x_2 + x_3
\]
subject to
\begin{gather*}
  x_1 - x_2 \leq 1 \,, \\
  x_2 - x_3 \geq 1 \,, \\
  x_3 - x_1 \leq 3 \,, \\
  x_1 \geq -2 \,, \\
  x_2 \leq 1 \,, \\
  x_3 \leq 0 \,.
\end{gather*}

