\documentclass[12pt]{article}
\usepackage{fullpage,amsmath,amssymb,graphicx,enumitem,color,txfonts,lastpage,fancyhdr}

\pagestyle{fancy}
\renewcommand{\headrulewidth}{0pt}
\cfoot{\thepage/\pageref{LastPage}}

\newcommand{\bbQ}{{\mathbb Q}}
\newcommand{\bbR}{{\mathbb R}}
\newcommand{\calO}{{\mathcal O}}

\begin{document}

\begin{center}
  {\Large MAT257 Term Test 1 Rejects}
\end{center}

The following questions were a part of a question pool for the 2020-21 MAT257 Term Test 1, but at the end, they were not included.

\begin{enumerate}

\item If $A$ is an interval, its closure is equal to the closure of its interior. Is this true for all subsets of $\bbR$?

\item Prove that if $C\subset\bbR^n$ is a compact set and $f\colon C\to\bbR^m$ is continuous, then the graph of $f$, the set $\Gamma(f)\coloneqq\{(x,f(x))\colon x\in C\}$, is compact.

\item Let $A=(a_{ij})$ be an $n\times n$ matrix, let $f\colon\bbR^n\to\bbR$ be defined by $f(x)=\sum a_{ij}x_ix_j$, and let $y\in\bbR^n$. Find $Df(y)$.

\item We will say that a function $f\colon\bbR^n\to\bbR^n$ is ``uniformly $\epsilon$-near $\lambda$'', where $\lambda$ is an invertible linear transformation, if $f(0)=0$ and
    \[ \forall x_1,x_2\in\bbR^n,\qquad |\lambda(x_1-x_2)-(f(x_1)-f(x_2))|\leq\epsilon|x_1-x_2|. \]
    Prove that if $\epsilon>0$ is sufficiently small, then for every $y\in\bbR^n$ there is some $x\in\bbR^n$ such that $f(x)=y$.

\item A function $f\colon\bbR^n\to\bbR^n$ is tiny; namely, $f\in o(h)$. Show that $f$ is differentiable at $0$. What is $f'(0)$?

\item Define $f\colon\bbR^2\to\bbR^2$ by $f(x,y)=(e^x+e^y,e^x+e^{-y})$.
    \begin{enumerate}
    \item Explain why every point $(a,b)\in\bbR^2$ has a neighborhood on which $f$ is invertible.
    \item If $f(a,b)=(c,d)$, compute the differential of $f^{-1}$ at $(c,d)$.
    \end{enumerate}

\item We consider the equation $x+y+z=\sin xyz$ very near $x=y=z=0$, and we want to solve for $z$ as a function of $x$ and $y$. Show that this can be done and compute the partial derivatives of the solution with respect to $x$ and to $y$.

\item Determine the interiors, exteriors, and boundaries of the subsets of $\bbR^2$ given below:
    \begin{enumerate}
    \item $A=\{(x,y)\colon x,y\in\bbQ\}$.
    \item $B=\{(x,y)\colon 0<x^2+y^2<1\}$.
    \item $C=\{(x,y)\colon y<x^2\}$.
    \item $D=\{(x,y)\colon y\leq x^2\}$.
    \end{enumerate}

\item Show that there is no neighborhood of $0$ in which the function $f\colon\bbR^2\to\bbR$ given by $f(x,y)=|xy|$ has continuous partial derivatives. Yet show that this $f$ is differentiable at $0$.

\item A function $f\colon\bbR^3\to\bbR^2$ satisfies $f(0)=(1,2)$ and $f'(0)=\begin{pmatrix}1&2&3\\0&0&1\end{pmatrix}$, and a function $g\colon\bbR^2\to\bbR^2$ is given by $g(x,y)=(x+2y+1,3xy)$. Compute $(g\circ f)'(0)$.

\item For some values of $n$ and $m$, give an example of a continuously differentiable function $\gamma\colon\bbR^n\to\bbR^m$ whose differential is 1-1 for every $a\in\bbR^n$, yet such that $\gamma$ itself is not 1-1.

\item Determine the interiors, exteriors, and boundaries of the subsets of $\bbR^2$ given below:
    \begin{enumerate}
    \item $A=\{(x,y)\colon x=0\}$.
    \item $B=\{(x,y)\colon 0\leq x<1\}$.
    \item $C=\{(x,y)\colon 0\leq x<1\text{ and }0\leq y<1\}$.
    \item $D=\{(x,y)\colon x\in\bbQ\text{ and }y>0\}$.
    \end{enumerate}

\item A function $f\colon{\mathbb R}^2\to{\mathbb R}$ is called ``rotation invariant'' if for every $\theta$, $f\circ T_\theta=f$, where $T_\theta\colon{\mathbb R}^2\to{\mathbb R}^2$ is the linear transformation given (in the standard basis) by the matrix $\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}$. If a given $f\colon{\mathbb R}^2\to{\mathbb R}$ is differentiable and rotation invariant, show that $Df(x,y)(-y,x)=0$ for every $(x,y)\in{\mathbb R}^2$.

\end{enumerate}

\end{document}