\documentclass[11pt]{article}
\def\Z{\mathbb{Z}}
\def\Q{\mathbb{Q}}
\def\R{\mathbb{R}}
\def\C{\mathbb{C}}
\usepackage{amscd,amssymb,subfigure,hyperref,epsfig,amsthm}
\usepackage[all]{xy}
\setlength{\textwidth}{6.5in}
\setlength{\topmargin}{-0.5in}
\setlength{\oddsidemargin}{0in}
\setlength{\textheight}{9.0in}
\begin{document}
\centerline{Mathematics 241, section 1 -- Multivariable Calculus}
\centerline{Exam 1 -- September 27, 2013}
\vskip 6pt
\noindent
{\it Directions}
\vskip 6pt
Do all work in the blue exam booklet.  Do not place anything you want
considered for credit on this sheet.  There are 100 possible points.
If you have time, you can also try for the 10 Extra Credit points (see back).    
\vskip 6pt
\noindent
I.  In this question, $P = (1,0,1)$, $Q = (1,2,-2)$, and $R = (1,-1,1)$ in $\R^3$.
\begin{enumerate}
\item[A)]  (15) Find the equation of the plane containing $P,Q,R$.
\item[B)]  (15) Give a parametrization of the line segment from $R$ to $Q$
(in that direction), including the proper range of $t$-values.  
Which $t$-value gives the midpoint of the line segment 
from part B (the point equidistant from $R$
and $Q$) in your parametrization?
\item[C)]  (10)  Compute the angle between the vectors $v = Q - P$ and 
$w = R - P$.
\end{enumerate}
\vskip 6pt
\noindent
II. Let $\alpha(t) = (u(t),v(t))$ and $\beta(t) = (w(t),z(t))$ be parametric curves in $\R^2$.
\begin{enumerate}
\item[A)] (10)  Show that if $\alpha(t)$ and $\beta(t)$ are differentiable, then
$${d\over dt} (\alpha(t) \cdot \beta(t)) = \alpha'(t) \cdot \beta(t) + \alpha(t) \cdot \beta'(t),$$
(where $\cdot$ denotes the dot product).  
\item[B)] (10)  Let $\beta(t)$ be a parametric curve in 
$\R^2$.  Suppose that $\beta(t)\cdot \beta(t) = 1$ (dot product)
for all $t$.  Show that the vector $\beta'(t)$ is orthogonal to the vector $\beta(t)$ for all $t$.
\end{enumerate}
\vskip 6pt
\noindent
III.  All parts of this question refer to $\alpha(t) = (\cos(2t)\cos(t),\cos(2t)\sin(t))$
called a {\it 4-leaved rose curve}:
\begin{center}
\includegraphics[width=2in]{Rose.eps}
\end{center}
\begin{enumerate}
\item[A)]  (15) Starting from $t = 0$, what is the first $t$ with
$\alpha(t) = (0,0)$?  At how many different $t$ is $\alpha(t) = (0,0)$
in the range $ 0 \le t < \pi$?
\item[B)]  (15) Find a parametrization of the tangent line to the rose curve 
at $t = \pi/4$. 
\item[C)]  (10) Give a parametrization of a circle that will fit completely inside
the ``leaf'' of $\alpha(t)$ in quadrants I and IV.  (Any circle that does that is OK.)
\end{enumerate}
\vskip 6pt
\noindent
{\bf Extra Credit}  (10) Identify and sketch the slices of the surface
$$\{(x,y,z) \in \R^3 \mid x^2 - y^2 + z^2 = 1\}$$
in the planes $x = 1$ and $x = 2$.
\end{document}
