\documentclass[12pt]{article}
\usepackage{fullpage,amsmath,amsfonts,amsthm,html}

\newcommand{\x}{\boldsymbol{x}}
\newcommand{\y}{\boldsymbol{y}}
\newcommand{\0}{\boldsymbol{0}}
\renewcommand{\a}{\boldsymbol{a}}
\renewcommand{\b}{\boldsymbol{b}}
\renewcommand{\c}{\boldsymbol{c}}
\renewcommand{\r}{\boldsymbol{r}}
\renewcommand{\v}{\boldsymbol{v}}
\renewcommand{\u}{\boldsymbol{u}}
\newcommand{\h}{\boldsymbol{h}}

\renewcommand{\d}{\mathrm{d}}

\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\V}{\mathcal{V}}
\newcommand{\W}{\mathcal{W}}
\newcommand{\U}{\mathcal{U}}


\begin{document}
\title{Perspectives of Mathematics II}
\author{Quiz 1}
\date{March 1, 2010}
\maketitle

Each problem is worth 5 points.  

\begin{enumerate}
\item Let $\V=\R^2$ and let $A$ be the non-symmetric matrix
\[
  A =
  \begin{pmatrix}
  \tfrac12 & 1 \\ 0 & \tfrac12
  \end{pmatrix} \,.
\]
Find the maxima and minima of 
\[
  f(\v) = \v^T A \v
\]
subject to $\lVert \v \rVert = 1$.

\item Let $\V = \{ u \in C^2([-1,1]) \colon u(-1)=u(1)=0 \}$.  
Find the minimum of
\[
  f(u) = \int_{-1}^1 u'(x)^2 \, \d x
\]
on $\V$ subject to the constraint $g(u)=0$ with
\[
  g(u) = \int_{-1}^1 u(x) \, \d x - 2 \,.
\]

\item You have a set of experimentally obtained data points
$(x_1,y_1), \dots (x_n,y_n)$ of the plane and suspect that they might
be related by a power law of the type
\[
  y = c \, x^\alpha \,.
\]
How would you analyze this data and obtain the exponent $\alpha$?

(Note: a qualitative description is sufficient.)

\end{enumerate}

\end{document}
