\documentclass[12pt]{article}
\usepackage{physics,graphicx,multicol}
\pagenumbering{gobble}

\renewcommand{\vec}[1]{\ensuremath{\mathbf{#1}}}
\newcommand{\pts}[1]{(#1 points)}

\newif\ifkey
%\keytrue %comment to hide key

\newcommand{\key}[1]{
\ifkey
	\fbox{
	\begin{minipage}{0.875\textwidth}
	#1
	\end{minipage}
	}
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}

\begin{document}
\begin{large}
\noindent
\makebox[\textwidth]{
	\textbf{Quiz 3 | MA261 | June 23, 2017}
}
\makebox[\textwidth]{Christina Jamroz, Alden Bradford}
\begin{enumerate}
\ifkey\else
\itemsep2em
\fi

\item \pts{12} A particle has the velocity function $\vec{v}(t)=\langle 2e^t, \sqrt{8}, 2e^{-t}\rangle$ and initial position $(1,2,3)$. Find the speed, position, and acceleration functions for the particle.

\key{
$\vec{a}(t)=\langle 2e^t,0,-2e^{-t}\rangle$\\
\noindent $\vec{r}(t)=\langle 2e^t-1, \sqrt{8}t+2, -2e^{-t}+5\rangle$\\
\noindent $s(t)=2e^t+2e^{-t}=4\cosh(t)$
}

%problem 14.1.16
\item \pts{8} Find and sketch the domain of $f(x,y)=\sqrt{x^2-y^2}$.

\key{
$x^2-y^2\geq0$ so $x^2\geq y^2$ and $|x|\geq |y|$.\\
\noindent Therefore $-x\leq y\leq x$ or $x\leq y\leq -x$.

\makebox[0.9\textwidth]{\includegraphics[width=0.5\textwidth]{domain}}
}

\end{enumerate}
\end{large}
\end{document}

%%% UNUSED PROBLEMS

%problem 13.4.9
\item A particle has the position function $\vec{r}(t)=\langle t^2+t,t^2-t,t^3\rangle$. Find the velocity, acceleration, and speed of the particle as functions of $t$.

\key{$\vec{v}(t)=\vec{r}'(t)=\langle 2t+1,2t-1,3t^2\rangle$.

\noindent $\vec{a}(t)=\vec{v}'(t)=\langle 2,2,6t\rangle$.

\noindent $s(t)=|\vec{v}(t)|=\sqrt{(2t+1)^2+(2t-1)^2+(3t^2)^2}$.}


%problem 14.1.16
\item \pts{10} Let $f(x,y)=\sqrt{x^2-y^2}$.
\begin{enumerate}
\item Find and sketch the domain of $f(x,y)$.
\item On a separate graph sketch level curves for $f(x,y)=0$, $f(x,y)=1$, and $f(x,y)=2$.
\end{enumerate}


\key{
\begin{multicols}{2}
\begin{enumerate}

\item $|x|\geq |y|$.

\vspace{-\baselineskip}
\makebox[0.4\textwidth]{\includegraphics[width=0.5\textwidth]{domain}}

\item ~\\

\vspace{-1.3\baselineskip}
\makebox[0.4\textwidth]{\includegraphics[width=0.4\textwidth]{contours}}
\end{enumerate}

\end{multicols}
}