\documentclass[12pt]{article}
\usepackage{amssymb}
\topmargin -.7in
\textheight 9in
\textwidth 6in
\oddsidemargin .25in 
\evensidemargin .25in
\begin{document}

\noindent {\bf CARNEGIE MELLON UNIVERSITY} \hfill21-259 Calculus in 3-D\\
\noindent{\bf Department of Mathematics} \hfill Spring 2003\\

\vspace{.25in} 

\begin{center}
\textbf{Review Problems for Test 1} 
\end{center}


\begin{enumerate}
\item  Let $\stackrel{\rightharpoonup }{a}=<1,3,-1>,\stackrel{\rightharpoonup}{b}=<0,4,2>,\stackrel{\rightharpoonup }{c}=<5,-1,3>.$
Compute each of the following:
 
\begin{enumerate} 
\item[(a)]  $\left| \stackrel{\rightharpoonup }{a}\right| $


\item[(b)]  $\stackrel{\rightharpoonup }{a}\cdot \stackrel{\rightharpoonup }{%
c}$


\item[(c)]  the angle between $\stackrel{\rightharpoonup }{a}$ and $%
\stackrel{\rightharpoonup }{c}$

\item[(d)]  the vector projection of $\stackrel{\rightharpoonup }{b}$ onto $%
\stackrel{\rightharpoonup }{a}$

\item[(e)]  the direction cosines of $\stackrel{\rightharpoonup }{c}$

\item[(f)]  $\stackrel{\rightharpoonup}{a} \cdot (\stackrel{\rightharpoonup}{b}\times \stackrel{\rightharpoonup}{c})$
\end{enumerate}

\item[2.]\begin{enumerate}
\item[(a)] Write an equation for the plane which passes through $(3,-6,8)$ and is normal to $<7,-2,-3>.$



\item[(b)]  Write the symmetric and parametric equations of the line which passes through $(1,3,-5)$ and is parallel to $<7,-2,-3>.$\end{enumerate}


\item[3.]  Find the line of intersection of the planes $x+y+z=4$ and $2x-y+5z=-1.
$

\item[4.]  Find a number $d$ such that the distance from the point $P=(4,3,0)$
to the plane $x-2y+z=d$ is $3.$

\item[5.]  Find the distance from the point $(1,0,-1)$ to the line $x=t,\ y=-1-t,\ z = 1+2t$.  

\item[6.]  Let $\stackrel{\rightharpoonup }{a}=<2,0,-1>$ and $\stackrel{\rightharpoonup}{b}=<1,2,0>.$ Write $\stackrel{\rightharpoonup }{b}$ as the sum of a vector $\stackrel{\rightharpoonup}{b}_1$ parallel to $\stackrel{\rightharpoonup}{a}$ and a vector $\stackrel{\rightharpoonup}{b}_2$ perpendicular to $\stackrel{\rightharpoonup }{a}.$

\item[7.]  Find the points of intersection of the line $\frac{ x}{-2}=\frac{y-1} {3}=\frac {z}{\sqrt{3}}$ and the sphere $x^2+3x+y^2+z^2=2.$ 

\item[8.]  Find the plane that contains the line $x=y=z$ and the point $%
P_0=(1,2,3).$

\item[9.]  Given that $\stackrel{\rightharpoonup}{a}$ and $\stackrel{ \rightharpoonup }{b}$ are vectors with $\left| \stackrel{\rightharpoonup }{a} \right| =1$ and $\left| \stackrel{\rightharpoonup}{a}-\stackrel{ \rightharpoonup }{b}\right| =\left| \stackrel{\rightharpoonup }{b}\right| ,$ find $\stackrel{\rightharpoonup }{a}\cdot \stackrel{\rightharpoonup }{b}.$

\item[10.]  Let $\stackrel{\rightharpoonup }{a}=<1,1,1>,\stackrel{\rightharpoonup}{b}=<2,-3,1>$ and $\stackrel{\rightharpoonup }{c}=<1,0,2>.$ Find a vector $\stackrel{\rightharpoonup }{v}$ satisfying 
\[
\stackrel{\rightharpoonup }{a}\cdot \stackrel{\rightharpoonup }{v}=\stackrel{ \rightharpoonup}{b}\cdot \stackrel{\rightharpoonup }{v}=0
\]
\[
\stackrel{\rightharpoonup }{c}\cdot \stackrel{\rightharpoonup }{v}=2.
\]

\item[11.]  Let $\stackrel{\rightharpoonup }{v}$ be a vector satisfying $\left| \stackrel{\rightharpoonup }{v}\times \stackrel{\rightharpoonup }{i}\right| =A,\left| \stackrel{\rightharpoonup }{v}\times \stackrel{\rightharpoonup }{j}%
\right| =B,\left| \stackrel{\rightharpoonup }{v}\times \stackrel{ \rightharpoonup }{k}\right| =C$ Express $\left| \stackrel{\rightharpoonup }{v }\right| $ in terms of $A,B,C.$

\item[12.]\begin{enumerate}\item[(a)] 
A space curve is described by a vector function $\stackrel{ \rightharpoonup }{r}(t)$ satisfying 
\[
\stackrel{\rightharpoonup }{r^{\prime }}(t)=<e^t\cos 2t,e^t\sin 2t,e^t>.
\]
Find the unit tangent vector $\stackrel{\rightharpoonup }{T}(t),$ the principal unit normal vector $\stackrel{\rightharpoonup }{N}(t)$, the binormal vector $\vec{B}(t)$, and the curvature $\kappa (t).$

\item[(b)] The torsion $\tau$ of a space curve is defined by $\tau = \ - \vec{N} \cdot 
{\displaystyle \frac{d \vec{B}}{dS} }$.

Find the torsion of the curve from part (a) when $t=0$.
\end{enumerate} 

\item[13.]  Find the length of the curve described by 
\[
\stackrel{\rightharpoonup }{r}(t)=<\sin 3t,\cos 3t,2t^{3/2}>
\]
for $1\leq t\leq 3.$

\item[14.]  Find the limit, or show that the limit does not exist.

\begin{enumerate}
\item[(a)]  ${\displaystyle \lim_{(x,y) \rightarrow (0,0)}} \ {\displaystyle \frac{x^2+y^2+2x^4}{x^2+y^2}}$

\item[(b)]  $\displaystyle{\lim_{\left( x,y\right) \rightarrow \left( 0,0\right) }}
\  \frac{x^3}{xy^2+y^3}$
\end{enumerate}

\item[15.]  Suppose that $f\left( x,y\right) =\left( y-x^2\right) ^2.$ Sketch the level curves of $f$ for $c=0$ and $c=1.$

\item[16.]  Suppose that $f\left( x,y\right) =\sin \left( x+3y\right)
+x^2y^3+2y^2.$ Compute $f_x,f_y,f_{xx},f_{xy},$ and $f_{yy}.$

\end{enumerate}



\end{document}
