\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 2 | MA261 | June 20, 2017}
}
\makebox[\textwidth]{Christina Jamroz, Alden Bradford}

\begin{enumerate}
\ifkey\else
\itemsep2em
\fi

\item \pts{6} Find the length of the curve $\vec{r}(t)=\vec{i}+t^2\vec{j}+t^3\vec{k}$, $0 \leq t \leq 1$. %problem 13.3.5

\key{$\vec{r}'(t)=2t\vec{j}+3t^2\vec{k}$, $|r'(t)|=t\sqrt{4+9t^2}$.

\noindent $L=\int_0^1 t\sqrt{4+9t^2}dt=\frac{1}{27}(13^{3/2}-8)$.} 

\item \pts{6} At what points does the helix $\vec{r}(t)=\langle \sin t, \cos  t, t\rangle$ intersect the sphere $x^2+y^2+z^2=5$? Give the coordinates, not just the $t$-values. %problem 13.1.30, p.846

\key{
Substitute $\vec{r}(t)$ into the equation of the sphere:
\begin{align*}
\sin^2 t + \cos^2 t +t^2 &=5\\
t^2&=4\\
t&=\pm 2
\end{align*}
Now put the $t$-values found back into $\vec{r}(t)$ to get $\langle \sin 2, \cos 2, 2\rangle$ and $\langle \sin(-2),\cos(-2),-2\rangle$.
}

\item \pts{8} Let $\vec{r}(t)= te^t\vec{i}-2\vec{j}+ \sin (t)\vec{k}$.
\begin{enumerate}
	\item Find $\vec{r}'(t)$.
	\item Find a vector equation for the line $\vec{u}(t)$ tangent to the curve at the point where $t=0$.
\end{enumerate}

\key{
\begin{enumerate}
\item $\vec{r}'(t)=(1+t)e^t\vec{i}+\cos(t)\vec{k}$.
\item $\vec{r}(0)=-2\vec{j}$, $\vec{r}'(0)=\vec{i}+\vec{k}$. $\vec{u}(t)=(\vec{i}+\vec{k})t-2\vec{j}$.
\end{enumerate}
}

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

%%UNUSED PROBLEMS
\item Identify the surface described by $36x^2+y^2+36z^2=36$ and give a sketch of the surface. Be sure to provide units on your sketch.%problem 12.6.17 on page 833

\item \pts{4} Find parametric equations for the line through $(5,7,9)$ that is perpendicular to the plane $2x-3y+z=8$.

\key{$x=2t+5$, $y=-3t+7$, $z=t+9$.}

\item \pts{4} Find an equation of the plane through the point $(-1,\frac{1}{2},3)$ and with normal vector $\vec{i}+4\vec{j}+\vec{k}$. %problem 12.5.25 on page 824

\key{$(x+1)+4(y-\frac{1}{2})+(y-3)=0$}