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

\def\d{{\mathrm d}}
\def\e{{\mathrm e}}
\def\R{{\mathbb R}}
\def\v{\boldsymbol{v}}

\newtheorem*{inft}{Inverse Function Theorem}
\newtheorem*{imft}{Implicit Function Theorem}

\begin{document}

\title{Advanced Calculus and \\Methods of Mathematical Physics}
\author{Take-Home Midterm}
\date{Tuesday, March 17, 2020}
\maketitle

\textbf{Please use a separate answer sheet for each problem!}


\begin{enumerate}

\item Consider the power series
\[
  \sum_{n=0}^\infty 2^{n+1} \, (x-1)^n \,.
\]
\begin{enumerate}
\item What is the center and the radius of the ball of convergence?
\item Within its ball of convergence, what does the series converge
to?
\item Does the series converge anywhere on the boundary of the ball of
convergence? 
\end{enumerate}
\hfill(5+5+5)

\item Let $a>0$, $x_0>0$ and define a sequence by
\[
  x_{n+1} = F(x_n) \equiv \frac12 \, x_n + \frac{a}{2 \, x_n} \,.
\]

\begin{enumerate}
\item Show that if $x_*$ is a fixed point of $F$, then $x_* = \sqrt
a$.
\item Show that $F$ is a strict contraction on $[\sqrt a, \infty)$ and
that it maps any interval $[\sqrt a, b]$, $b>\sqrt a$, into itself.
Conclude that $F$ has a unique fixed point on this interval.

\item What happens if $x_0 \in (0,\sqrt a)$?
\end{enumerate}
\hfill(5+5+5)

\item Let $f \colon \R^2 \to \R$ be given by
\[
  f(x,y) = \ln \frac{1+x^2}{1-y} \,.
\]
\begin{enumerate}
\item Compute the Taylor polynomial (truncated Taylor series) about
$(x,y)=(0,0)$ of degree $2$.
\item Give a bound on second order Taylor remainder for $\lVert (x,y)
\rVert \leq \tfrac12$.
\item On which (open) set does the Taylor series converge?  (No need
to test convergence on the boundaries.)
\end{enumerate}
\hfill(5+5+5)

\item For $A \in \operatorname{Mat}(n\times n)$, define
\[
  f(A) = (I - A)^{-1} \,,
\]
where $I$ is the $n \times n$-identity matrix.
\begin{enumerate}
\item Show that $I-A$ is invertible when $\lVert A \rVert <1$, so that
$f(A)$ is well-defined for such matrices.

\emph{Hint:} A square matrix $M$ is invertible if the homogeneous
linear equation $Mx=0$ has only the trivial solution $x=0$.

\item Show that the derivative of $f$ is given by
\[
  \d f(A) B = (I-A)^{-1} B (I-A)^{-1} \,.
\]
\end{enumerate}
\hfill(5+5)

\item Let $S=\operatorname{SL}(2,\R)$ denote the set of $2 \times 2$
matrices with real entries that have determinant~$1$.\footnote{Recall from
Linear Algebra that the determinant of a $2 \times 2$ matrix is given
by
\[
  \det
  \begin{pmatrix}
    a & b \\ c & d
  \end{pmatrix}
  = ad - bc \,.
\]}
Use the implicit function theorem to show that locally, $S$ can be
parameterized smoothly by a function of three variables.

\emph{Hint:} Observe that the determinant condition implies that at
least one of the entries of the matrix must be non-zero.  Pick that
variable and use the IFT to express it locally as a function of the
other three.
\hfill(10)

\item Let $p,q \geq 1$ with
\[
  \frac1p + \frac1q = 1 \,.
\]
\begin{enumerate}
\item Fix $c>0$ and maximize
\[
  f(x,y) = x \, y
\]
subject to $x, y \geq 0$ and
\[
  \frac1p \, x^p + \frac1q \, y^q = c \,.
\]
\item Conclude that
\[
  x \, y \leq \frac1p \, x^p + \frac1q \, y^q \,.
\]

\end{enumerate}
\hfill(5+5)


\end{enumerate}
\end{document}




