\documentclass[11 pt]{article}

%pagestyle{empty} uncomment to eliminate page numbering

\setlength{\oddsidemargin}{-.45in}
%\setlength{\evensidemargin}{-.5in}
\setlength{\textwidth}{7.1in}
\setlength{\topmargin}{-0.95in}
\setlength{\textheight}{9.7in}

%Change second above dimensions to below in order to widen page for margin labels:
%\setlength{\oddsidemargin}{36pt}
	\setlength{\evensidemargin}{74pt}
%\setlength{\textwidth}{350pt}
%\setlength{\topmargin}{27 pt}
%\setlength{\textheight}{522 pt}
%\setlength{\marginparwidth}{80 pt}
%\setlength{\marginparsep}{7pt}

 \pagestyle{empty} %(no page numbering} ; below is for adding footer


%Exam header and page total footer
\usepackage{fancyhdr}
\pagestyle{fancy}
%\lfoot{}
%\cfoot{}
%\rfoot{\rfoot{{\bf{Page Total}} \dotfill  \pagescorebox}}
%\lhead{}
\chead{MTH 402}
%\rhead{}

\usepackage{graphics}

% The epsfig package clashes with maple2e below. Use one or the other.
%\usepackage{epsfig}
%\usepackage{wrapfig}
%\input epsf.sty

%\usepackage{maple2e}
% \def\emptyline{\vspace{12pt}}
%\DefineParaStyle{Maple Output}
%\DefineParaStyle{Warning}
%\DefineCharStyle{2D Math}
%\DefineCharStyle{2D Output}

%\usepackage{bbding}
% Package is incompatible with marvosym and wasysym
%\usepackage{pstcol,pst-node,pstricks,pst-text,pst-grad,pst-3d}
\usepackage{pstcol,pstricks,pst-text,pst-grad,pst-node}
\usepackage{graphicx}
\usepackage{latexsym}
\usepackage{amssymb}
\usepackage{amsfonts}
\usepackage[intlimits]{amsmath}
%\usepackage{wasysym}
%\usepackage{marvosym}
%\usepackage{pifont}

\usepackage{amsthm}
\usepackage{multicol}
\usepackage{psfrag}
%\usepackage{pxfonts} % combine with \varointctrclockwise\limits_{C_2} for counterclockwise line integral
%comment out wasysym, amstheorem, and theoremstyle[plain]

%***********************************
%**********THEOREM-LIKE ENVIRONMENTS
%***********************************
\theoremstyle{definition}
\newtheorem{definition}{Definition}
\newtheorem{theorem}{Theorem}

\newtheorem{example}{Example}
\newtheorem{activity}{Sample Application }
\newtheorem{project}{Project}

%\newtheorem*{weak_duality}{Weak Duality}
%\newtheorem*{strong_duality}{Strong Duality}
%\newtheorem*{duality_theorem}{The Duality Theorem}

\newtheorem{waypoint}{Waypoint}[definition]

\newenvironment{progress_ck}[1][.9\linewidth]
{ \vspace{.1in} \noindent \begin{center}\begin{minipage}[c]{#1}
$\Diamondblack$ \hrulefill $\Diamondblack$
\begin{waypoint}}
{\end{waypoint}  $\Diamondblack$ \hrulefill $\Diamondblack$ \end{minipage} \end{center}}


\newcounter{progress_ck_counter}
%**********************************

%COMMANDS
\newcommand{\be}{\begin{enumerate}}
\newcommand{\ee}{\end{enumerate}}

\newcommand{\bi}{\begin{itemize}}
\newcommand{\ei}{\end{itemize}}

%\newcommand{\Re}{\mathfrak{Re}}
%\newcommand{\Im}{\mathfrak{Im}}
\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\Nat}{\mathbb N}
\newcommand{\Q}{\mathbb Q}
\newcommand{\Z}{\mathbb Z}

\newcommand{\vecc}{\mathbf c}
\newcommand{\vecx}{\mathbf x}
\newcommand{\vecs}{\mathbf s}
\newcommand{\vece}{\mathbf e}
\newcommand{\veca}{\mathbf a}
\newcommand{\vecd}{\mathbf d}


\newcommand{\vecxs}{{\mathbf {x}}_s}
\newcommand{\vecxsh}{{\mathbf {x}}_{s,h}}
\newcommand{\vecxsp}{{\mathbf {x}}_{s,p}}
\newcommand{\vecb}{\mathbf b}
\newcommand{\vecu}{\mathbf u}
\newcommand{\vecv}{\mathbf v}
\newcommand{\vecw}{\mathbf w}
\newcommand{\vecn}{\mathbf n}
\newcommand{\vecy}{\mathbf y}
\newcommand{\vecz}{\mathbf z}
\newcommand{\onevec}{\ensuremath{\mathbf{1}}}
\newcommand{\zerovec}{\ensuremath{\mathbf{0}}}
\newcommand{\deltavec}{\boldsymbol{\delta}}

\newcommand{\bs}{\blacksquare}

\newcommand{\F}{\mathbf F}
\newcommand{\T}{\mathbf T}
\newcommand{\N}{\mathbf N}
\newcommand{\B}{\mathbf B}
\newcommand{\vecR}{\mathbf R}
\newcommand{\V}{\mathbf V}

\newcommand{\tor}{\boldsymbol{\tau}}


\newcommand{\dr}{d\mathbf r}
\newcommand{\dS}{d\mathbf S}

\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}

\newcommand{\x}{\phantom{ +x_1}}

\newcommand{\bv}{\mbox{BV}}
\newcommand{\BV}{\mbox{BV }}
\newcommand{\NBV}{\mbox{NBV }}

\newcommand{\LPC}{\emph{FuelPro Petroleum Company }}
\newcommand{\LP}{\emph{FuelPro }}

\newcommand{\CMC}{\emph{ConPro Manufacturing Company }}
\newcommand{\CM}{\emph{ConPro }}

\newcommand{\newpar}{\vspace{.15in}\noindent}
\newcommand{\noi}{\noindent}

\newcommand{\ds}{\displaystyle}
\newcommand{\cbox}{\framebox{\parbox[c][.5 in][c]{.5 in}{\makebox[.5in][l]{}} } \hspace{.025in}}
\newcommand{\pagescorebox}{\framebox{\parbox[c][.35 in][c]{.8 in}{\makebox[2in][l]{}} } \hspace{.025in}}

%MARGIN LABEL
\newcommand{\marginlabel}[1]{\mbox{}\marginpar{\raggedleft\hspace{0pt}#1}}

%MAPLE FONT
\newcommand{\mf}[1]{ {\ttfamily{\upshape{#1}}}}

%REAL AND IMAGINARY PARTS
\newcommand{\re}[1]{{\mathfrak{Re}}(#1)}
\newcommand{\im}[1]{{\mathfrak{Im}}(#1)}
\newcommand{\conj}[1]{{\overline{#1}}}
\newcommand{\reaxis}{{\mathfrak{Re}}}
\newcommand{\imaxis}{{\mathfrak{Im}}}

%**********************************
%MAPLE CUSTOMIZED LIST COMMAND

\newenvironment{maplelist}{\renewcommand{\labelitemi}{$>$}\begin{itemize}}{\end{itemize}}

%\begin{list}{$\diamondsuit$}{\setlength{\leftmargin}{.5in}
%\setlength{\rightmargin}{.5in}}
%\item something here
%\item something else here
%\end{list}

\newcommand{\mo}[1]{{\rmfamily{\upshape {\begin{equation*} #1  \end{equation*}}}}}

%MAPLE ARGUMENT FONT
\newcommand{\ma}[1]{{\ttfamily{\slshape{#1}}}}

%MAPLE OUTPUT FONT FOR MULTILINE
\newcommand{\MultiLineMapleOutput}[1]{{\rmfamily{\upshape {\begin{multline*} #1  \end{multline*}}}}}

%***********************************
% LP WITH USE OF PHANTOM
%\begin{align}
% \mbox{maximize } z &=10x_1+8x_2  \\
%	\mbox{ subject to} \notag\\
%	x_1 \phantom{ss20x_2} &\leq 10 \notag\\
%	x_1+2x_2 &\leq 20  \notag\\
%	10x_1+2x_2 &\leq 120 \notag \\
%	x_1,x_2 &\geq 0.\notag
%	\end{align}	


%TABLEAU
%\begin{table}[h]
%\centering
%\begin{tabular}{|c||c|c|c|c|c||c|}
%\hline
%$z$ & $x_1$ & $x_2$ & $s_1$ & $s_2$ & $s_3$ & RHS \\
%\hline \hline
%1 & 0 & -8 & 10 & 0 & 0 & 100 \\ \hline
%0 &  1 & 0 & 1 & 0 & 0 & 10 \\ \hline
%0 &  0 & 2 & -1 & 1 & 0 & 10 \\ \hline
%0 &  0 & 7 & -10 & 0 & 1 & 20 \\ \hline
%\end{tabular}
%\caption{Tableau after first iteration}\label{Ta:petroleum_second_tableau}
%\end{table}


%LABELS
%Section: \label{S:Name}
%Theorem/Explore: \label{Th:Name}
%Table: \label{Ta:Name}
%Reference \cite{Burch}

%CUSTOMIZED ITEMIZED LIST SYMBOL.
%\begin{list}{default_label}{\setlength{\leftmargin}{.5in} \setlength{\rightmargin}{.5in}}

%CHANGE TO DOUBLE AND 1.5 SPACING
%\renewcommand{\baselinestretch}{1.5}

%****************************************
%MATH SYMBOLS AND SHORTCUTS
%****************************************

%LABELED EQUATION. TO REFER TO THE EQUATION LATER IN TEXT TYPE "(\ref{E:quadratic})".
%begin{equation} \label{E:quadratic}
% f(x)=x^2 \text{hi}
%\end{equation}

%USE OF SPLIT
%begin{equation} \label{E:quadratic}
% \begin{split} &f(x)=long formula \\
% & broken over two lines \end{split}
%\end{equation}

%LABELLED ALIGN
%\item Consider the system of equations (\ref{E:system1})	
%	\begin{align}\label{E:system1}
%		3x_1-x_2+2x_3 & =-4  \notag \\
%		2x_1+x_2+x_3 & =-1  \notag \\
%		x_1+3x_2& =2.  \notag
%	\end{align}

%ALIGNED FORMULA
%\begin{align*}
%f(x)&= x^2+2x+1 \\ \intertext{because..}
%   &=(x+1)^2
%\end{align*}

%TWO ALIGNED FORMULA COLUMNS
%\begin{alignat}{2}
%P_0(x)&=1, & \qquad P_1(x)&=x \notag \\
%P_2(x)&=\frac{1}{2}(3x^2-1), & \qquad P_3(x)&=\frac{1}{2}(5x^3-3x) \notag
 % \end{alignat}

% ANNOTATED PROOF
% \begin{alignat*}{2}
%\vecc \vecx_0 &\leq \left(\vecy_0 A \right) \vecx_0 & & \quad\text{($\vecy_0$ is dual-feasible)} \\
% &= \vecy_0 \left(  A  \vecx_0 \right) & & \quad\text{(associativity)} \\
% &\leq \vecy_0 \vecb. & &\quad\text{($\vecx_0$ is primal-feasible).}
% \end{alignat*}


%GATHERING FORMULAS: TWO EQUATIONS, WITH NO ALIGNMENT
% \begin{gather*}
% x^2+2x+1
%  (x+1)^2
% \end{gather*}

%SPLITTING A LONG FORMULA WITH MULTLINE
%\begin{multline*}
% x^5+x^4+x^3 \\
% +x^2+x+1
%\end{multline*}

%AUGMENTED MATRIX
%\left[  \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix}
%\left| \begin{matrix} 1 \\ 2 \end{matrix} \right. \right]

%MATRIX WITH ELLIPSES
%\begin{bmatrix}
%f_{11}^{-1} & 0 & \ldots & 0 \\
%0 & f_{22}^{-1} & \ldots  & 0  \\
%\vdots & & \ddots & \vdots  \\
%0 & 0 & \ldots & f_{nn}^{-1}
%\end{bmatrix}\nabla f(\vecx_k)

%PIECEWISE DEFINED FUNCTION
%p(t)=\begin{cases}
%		\frac{1}{2},	&\text{if $t \in [-1,1]$};\\
%		0,	&\text{otherwise,}
%	\end{cases}

%DOUBLE INTEGRAL
%\iint\limits_D \frac {\partial Q}{\partial x} - \frac {\partial P}{\partial y} \, dx \, dy

%CONTOUR AND DOUBLE INTEGRALS WITH CONTOUR/REGION BELOW INTEGRAL SIGN
% \oint\limits_C
% \iint\limits_D

% SCALED MATH SYMBOLS
%\mbox{ \raisebox{-.2in}{ \scalebox{2}[6]{$\int$}}}_{\scalebox{1.5}[1.5]{$\Real$}}

%EVALUATION OF DERIVATIVE VIA LEIBNIZ NOTATION (INCLUDES ONE-SIDED PARENTHESIS)
% \left. \frac{\partial z}{\partial x}\right|_{(x,y)=(1,-2)}

%OVERSET DEFINITION
%\overset{\text{def}}{=}

%DELIMITERS
% \left\{   \right\}

%SCALED, CENTERED EQUATION
%\begin{equation}\label{E:convexity_FOC_unconstrained_ConPro_Hessian}
%\mbox{
%{\scalebox{1.5}{ $\ds
%H_f(x_1,x_2)=\begin{bmatrix} \frac{-350x_2^{\frac{1}{3}}}{x_1^{\frac{3}{2}}} & \frac{700}{3\sqrt{x_1}x_2^{\frac{2}{3}}} \\
%\frac{700}{3\sqrt{x_1}x_2^{\frac{2}{3}}} & \frac{-2800\sqrt{x_1}}{9x_2^{\frac{5}{3}} } \end{bmatrix}$
%}}
%}
%\end{equation}

%GENERAL NLP
%\begin{equation}\label{E:diff_Gradient_general_NLP}
%\mbox{minimize (or maximize) } f(\vecx) \mbox{ where }\vecx \in S.
%\end{equation}

%GENERAL LP
%\begin{align}\label{E:simplex_FuelPro_standard_form}
% \mbox{maximize } z &=4x_1+3x_2  \\
%	\mbox{  subject to} \notag \\
 %  x_1 \phantom{+ 2x_2} + s_1 \ph \ph &= 8  \notag \\
 %   2x_1 + 2x_2 \ph +s_2 \ph &= 28  \notag \\
 %   3x_1 + 2x_2 \ph \ph +s_3 &= 32  \notag \\
%	x_1,x_2, s_1, s_2, s_3  &\geq 0.  \notag
%	\end{align}

%********************************************************
%BOXES
%********************************************************

%BOX
%The second argument in the parbox (1in} controls the box width; first argument controls height
%\framebox{\parbox[c][.2in][c]{1in}{\makebox[.5in][l]{}} }

%DOTFILL TO CHECKBOX
%\dotfill \cbox

%SAMPLE TRUE-FALSE QUESTION
%	\item \cbox
%	If $A$ and $B$ are matrices and $AB=0$, then $A=0$ or $B=0$
%	\vspace{2.3in}

%********************************************************
%FIGURES
%********************************************************

%FIGURE WITH GRAPHICS PACKAGE
%\begin{figure}[h]
%\hfil {  \includegraphics{c:/images/OR/blending_feasible_region.eps} }  \hfil
%\caption{Feasible region for Fishback Petroleum LP}\label{Fi:blending_feasible_region}
%\end{figure}
%scale with \scalebox{.5}{ }

%FIGURE WITH PSFRAG TAGS: Make sure Illustrator text format is 12 pt. Times New Roman, regular font
%\begin{figure}[h]
%\centering
%\psfrag{x1_axis_label}[Bc][Bc]{$x_1$}
%\psfrag{x2_axis_label}[Bc][Bc]{$x_2$}
%\psfrag{f_axis_label}[Bc][Bc]{$f$}
%\scalebox{.5}{\includegraphics{c:/images/OR/nonlinear/optimality_sufficient_SOC_quad_form_INDEF.eps} }
%\caption{Quadratic form $\ds f(x_1,x_2)=x_1^2+4x_1x_2+x_2^2$}
%\label{Fi:optimality_sufficient_SOC_quad_form_INDEF}
%\end{figure}

%FIGURE WITH EPSFIG PACKAGE: height SCALES PROPORTIONALY (SEE LATEX COMPANION, P. 317)
% \epsfig{file=c:/images/Lanczos/roof_function.eps,height=1cm.}

%TWO FIGURES SIDE BY SIDE, SCALED, AND LABELLED SEPARATELY
%\begin{figure}[h]
%\centering
%\raisebox{1.4in}{I)}
%\scalebox{.6}{\includegraphics{c:/mth300/Exams/exam2_gradientfield3.eps} }
%\raisebox{1.4in}{II)}
%\scalebox{.6}{\includegraphics{c:/mth300/Exams/exam2_gradientfield1.eps} }
%\end{figure}

%ARRAY OF FIGURES
%\begin{tabular}{ccc}
%\includegraphics{<1>}&\includegraphics{<2>}&\includegraphics{<3>}\\
%\includegraphics{<4>}&\includegraphics{<5>}&\includegraphics{<6>}
%\end{tabular}

% COMPLEX PLANE
%\begin{figure}[h]
%\centering
%\psfrag{horizontalaxis}[Bc][Bc]{$\reaxis$}
%\psfrag{verticalaxis}[Bc][Bc]{$\imaxis$}
%\includegraphics[width=.3\linewidth]{open_unit_disc.eps}
%\caption{The open unit disk, $|z|<1$.}
%\label{Fi:open_unit_disk}
%\end{figure}

%
%**********TEXT AND FIGURE SIDE BY SIDE*******************
%
%\begin{figure}[h]
%\begin{tabular}{lr}

%\raisebox{.8in}{\parbox[b]{3.3in}{1. Suppose $f$ is the function whose contour diagram is shown to the right. Determine
%whether the following partial derivatives are
%positive, negative, or zero. Briefly justify each of your responses. (12 points)

%\vspace{.4in}
%\noindent
%(a) $f_x(P)$

%}}

%&

%\centering
%\psfrag{xaxislabel}[Bc][Bc]{$x$}
%\psfrag{yaxislabel}[Bc][Bc]{$y$}
%\psfrag{z1}[Bc][Bc]{$z=6$}
%\psfrag{z2}[Bc][Bc]{$z=5$}

%\scalebox{.7}{\includegraphics{exam2_W2007_partial_signguess_lost_part.eps} }

%\end{tabular}
%\end{figure}

%******************************************
%MATRICES AND TABLES
%******************************************

%MATRIX COMMANDS
%\begin{bmatrix}a & b \\ c & d \end{bmatrix}

%TABLE COMMANDS
%\begin{tabular}{c|c} $x$ & $y$ \\ \hline a & b \\ c & d \\ e & f \ \end{tabular}

%MULTICOLUMN TABLE
%\begin{table}[h]
%\begin{center}
%\begin{tabular}{|c|c|c|c|c|c|c|}
%\hline
%& \multicolumn{6}{c|}{Genotype of Parents}\\ \hline
%Genotype of Offspring & AA-AA & AA-Aa & AA-aa & Aa-Aa & Aa-aa & aa-aa \\ \hline
%AA &  &  &  & &  &    \\ \hline
%Aa &  &  &  & &  &    \\ \hline
%aa &  &  &  & &  &    \\ \hline
%\end{tabular}
%\end{center}
%\end{table}

%MULTIVARIABLE CALCULUS 2-WAY TABLE
%\begin{table}[h]
%	\begin{center}
%	\begin{tabular}{|c||c|c|c|}
%	\hline
%	$x \diagdown y$ & $.8$ & $1$ & $1.2$ \\ \hline \hline
%	$1.8$ & $5.53$ & $5.85$ & $5.60$\\ \hline
%	$2$   & $6.11$ & $6.14$ & $6.16$ \\ \hline
%	$2.2$ & $6.69$ & $6.42$ & $6.73$ \\ \hline
%	\end{tabular}
%	\end{center}
%	\caption{Table for function $z=f(x,y)$.}
%	\end{table}

%VARIABLE SPACED COLUMNS IN A TABLE
%\begin{tabular}{p{2.5in} p{2.5in}}
%	$\det(BA)$ &   $\det(A^{-1})$ \vspace{2in} \\
%	$\det(3B)$ &   $\displaystyle \det \left( \begin{bmatrix} 1 & 2 & 4 \\ 2 & 0 & -2 \\ 1 & 2 & 2 \end{bmatrix} \right)$
%	\end{tabular}

%TABLE/GRAPH ON ONE LINE COMMANDS
%\begin{figure}[h]
%\centering
%\begin{tabular}{cc}
%	\raisebox{.75in}{\begin{tabular}[b]{c|c} $x$ & $y$ \\ \hline a & b \\ c & d \\ e & f  \end{tabular} } &	
%	\hspace{1in} \includegraphics[2in,2in]{c:/mth310/project/Hamilton.bmp}
%\end{tabular}
%\caption{{\scriptsize Table and graph for $f$.}}
%\end{figure}

%***********************************************
%BOX ENVIRONMENTS
%******************************************

%TEXT IN A BOX VIA \framebox[width][position]{text} WHERE position = l,r,c
%\framebox[2in][l]{Name}

%\framebox{\parbox[c][.2in][c]{3 in}{\makebox[.5in][l]{}} }

%TEXT IN A BOX VIA \fbox
%\setlength{\fboxrule}{2pt}\setlength{\fboxsep}{2mm}
%\fbox{Text in a box}

%RAISEBOX / ROTATEBOX
%\raisebox{lift}{contents}  \rotatebox{-90}{contents}

%SCALEBOX
%\scalebox{fraction}{contents}
%\scalebox{horz fraction }[vert fraction]{contents}

%TRUE-FALSE CHECKBOX
%\newcommand{\checkbox}{\framebox{\parbox[c][.125 in][c]{.125 in}{\makebox[.125in][l]{}} } }

%PARBOX: POSITIONED RELATIVE TO BASELINE c, t, b
%\parbox[pos]{width}{text}

%MINIPAGE: CREATES A MINIVERSION OF THE PAGE WITHIN THE PAGE; CAN BE USED INSIDE FIGURE AND TABLE ENVS
%\begin{minipage}[pos]{width} text
%\end{minipage}

%**************************************************
%MISC.
%**************************************************

%\footnote{text}

%\marginpar[left-text]{right-text} USE SCRIPTSIZE AND CHANGE TEXTWIDTH AT START OF FILE

%WEB URL
%\texttt{http://www.phy.syr.edu/courses/java-suite/crosspro.html}

%MAPLE FILE NAME
%\textbf{extrema.mws}

% POSITIONING
% TO CREATE ENVIRONOMENT: \begin{  center, flushleft, flushright,}
% WITHIN ANOTHER ENVIRONMENT \centering, \raggedright, \raggedleft

%***************************************************
%CONFIDENTIALITY AGREEMENT
%***************************************************

%By signing my name below, I promise that I have not collaborated in any
%way with any other person on this exam and that all the work is entirely
%my own.  In addition, I have not used any written resources other than
%my text or course notes.\\

%\noindent Signature: \underline{\hspace{7.8cm}}\\

%***************************************************

%COOL SYMBOLS   Use with \scalebox{scale factor}{symbol} All but Smiley and last are in wasysym, marvosym, and % %bbding package

\usepackage{wasysym}
\usepackage{marvosym}
\usepackage{pifont}


%TELEPHONE
%{\large{\Telefon}}

%FAX
%{\large{\FAX}

%CELLULAR
%{\large{\Mobilefone}}

%POINTING HAND:
\newcommand{\handpointright}{ \scalebox{2}{$\mbox{\ding{43}}$}}

%PENCIL: Use above with \ding{46}

%CHECKMARK
% \ding{52} %(looks nicer) OR
%\newcommand{\ckmark}{ \mbox{\scalebox{2}{$\ensuremath{\mathbf{\ckmark}}$}}}

%SMILEY FACE: \Smiley or
\newcommand{\smileyface}{ \scalebox{1.5}{$\mbox{\smiley}$}}
\newcommand{\frownieface}{ \raisebox{-.03in}{\scalebox{1.5}{$\mbox{\frownie}$}}}

%CIRCLED NUMBERS (1)-(10): \ding{n} n=172-181 (in pifont}


%***************************************************
%************END OF PREAMBLE************************
%***************************************************


%***************************************************
%************EXAM HEADER****************************
%***************************************************
%\begin{document}
%\noindent{{\LARGE {{\textbf {Final Exam}} }}}
%\hspace{2in} {\Large {{\textbf {Name: }} }}\\
%\noindent{{  {\bfseries{ Fall 2007, Dr. Fishback}} }} \\
%\vspace{.3in}
%\noindent
%Total Score (100 Points Possible): \dotfill
%	\framebox{\parbox[c][.4 in][c]{.7 in}{\makebox[.4in][l]{}} }
%\vspace{-.1in}
%\hrule

%\vspace{.1in}
%\noindent {\bfseries {\underbar{Directions:}}}
%{\small{
%\bi
%\item Show your work! If a question asks for an explanation, keep your response brief and to the point.
%\item Unless otherwise stated, you may use a graphing calculator to assist you on any question, provided you state explicitly
%how you are doing so. However, using symbolic manipulating capabilities of calculators
%such as the TI-89 or TI-92 is not permitted.
%\item If a question asks for an {\underbar{exact}} value, then a calculator approximation is insufficient to receive full
%credit. (For example, $\sqrt{2}$ is exact, whereas $1.414\ldots$ is not.)
%\item The use of a note sheet or external aid of any other type on this quiz is not permitted.
%\item If you have a cell phone {\large{\Mobilefone}}, please turn it completely off at this time.
%\ei
%}}

%***************************************************
%************ACTIVITY HEADER****************************
%***************************************************

\begin{document}

\begin{center}
{\Large {{\textbf {The Basics of Complex Numbers}} }} \\
\end{center}
{\bf{Activity Group Members:
}}
\vspace{1in}
\hrule
\noindent
\setlength{\fboxrule}{1pt}\fbox{\parbox{\linewidth}{
 {\bf {Directions:}}
Work in groups of two to four students on this activity. Each group will submit one solution
consisting of a copy of this handout with solutions neatly handwritten in the space provided. This writeup should adhere to the \emph{Guidelines for Written Assignments} documents posted at the course web site. (Extra copies of this handout
 are also available there.) You may seek assistance on this activity only from other members of your group and from your instructor.\\
}}


\begin{table}[h]
\caption{Sandwich nutritional data}\label{Ta:Maple_Diet_Problem_Nutritional_Data}
\begin{tabular}{|l|c|c|c|c|}
\hline
Sandwich &
Turkey Breast
& Club
& Veggie Sub
& $\ds \begin{matrix} \mbox{Breakfast} \\ \mbox{Sandwich} \end{matrix}$
\\ \hline
Calories &  280   &  320        &  230      &    470    \\ \hline
Fat (gm.)   &   4.5  &   6          &  3        &    19    \\ \hline
Cholesterol (gm.)& 20  &  35         &   0      &     200  \\ \hline
%Sodium (gm.)     &  1000   &  1290    &  500      &     1500   \\ \hline
Carbohydrates (gm.)  &  46   &  47    &  44       &     53   \\ \hline
Fiber (gm.)     &   5  & 5            &   5       &    5    \\ \hline
Protein (gm.)     &  18   &  24       &  9        &    28    \\ \hline
Vitamin A (pct. RDA)    &   8  &  8       &   8       &    10    \\ \hline
Vitamin C (pct. RDA)     &  35   &  35     &   35      &   15     \\ \hline
Calcium (pct. RDA)     &  6   &   8        &  6        &    25    \\ \hline
Iron (pct. RDA)      &  25   &  30         &   25      &   25     \\
\hline
\end{tabular}
\end{table}


\newpar
Here are examples involving Maple exponentiation
\begin{maplelist}
\item \mf{n:=2:}
\item \mf{x:=7:}
\item \mf{x\^{}n;}
\mo{49}
\end{maplelist}



All of us have encountered complex numbers at one time or another. For most of
us, the first time occurs when solving quadratic equations. As an example, suppose
that $x^2-6x+13=0$. Then, by the quadratic formula,
\begin{align*}
x &= \frac{-(-6) \pm \sqrt{(-6)^2-4\cdot 1\cdot 13}}{2\cdot 1} \\
&= \frac{6 \pm \sqrt{-16}}{2} \\
&= \frac{ 6 \pm 4\sqrt{-1}}{2} \\
&= 3 \pm 2i.
\end{align*}

\noindent
Typically in the past, we have ignored such answers, deeming them ``second class citizens,''
so to speak, as numbers go. However, in this course, complex numbers \emph{reign supreme}!

\newpar
A complex number is any number of the form $a+bi$, where $i=\sqrt{1}$, and $a$ and $b$ are real numbers. Of course, $i^2=-1$. Typically, we use letters like $z$ and $w$ to denote complex numbers, e.g.,
\begin{equation*}
z = 3 + 2i.
\end{equation*}
If $z=a+bi$, then $b$ is called the \emph{imaginary part} of $z$, denoted
$\im{z}$, and $a$ is called the \emph{real part}, denoted $\re{z}$. The \emph{complex conjugate}
of a complex number, $z$, is defined as $\conj{z}=a-bi$. Thus for $z=3+2i$,
$\re{z}=3$, $\im{z}=2$, and $\conj{z}=3-2i$.

\newpar
Of course, every real number may be
viewed as a complex number having imaginary part equal to zero. The set of complex numbers is denoted
by $\C$. In terms of sets of numbers,
\begin{equation*}
\Nat \subset \Z \subset \Q \subset \R \subset \C,
\end{equation*}
\noi
where $\Nat$, $\Z$, $\Q$ and $\R$ denote the sets of natural numbers, integers, rational numbers, and real numbers, respectively.
\newpar
Complex numbers are added together by adding their respective real and imaginary parts. For example, if $z=3+2i$ and $w=-1+5i$, then $z+w=2+7i$. Complex numbers be multiplied by real scalars, e.g., $4z=4(3+2i)=12+8i$, and can also be multiplied by one another using a distributive property for multiplication:
\begin{align*}
z\cdot w &= (3+2i)(-1+5i) \\
&= -3 +15i -2i +10i^2 \\
&= -3+13i -10 \\
&= -13+13i.
\end{align*}

\newpage
\be
\item Assume $z=3+2i$ and $w=-1+5i$. Express each of the following quantities
in the form $a+bi$.
\be
\item $\re{z}$+ $\re{w}$
\vspace{.75in}
\item $\re{z+w}$
\vspace{.75in}
\item $2z+3w$
\vspace{.75in}
\item $z(2-i)$
\vspace{.75in}
\item $\conj{z}+\conj{w}$
\vspace{.75in}
\item $\conj{z+w}$
\vspace{.75in}
\item $\conj{zw}$
\vspace{.75in}
\item $\ds \frac{1}{z}$ (Hint: Multiply the numerator and denominator by $\overline{z}$.)
\newpage
\item $\ds \conj{\left(\frac{z}{w+z}\right)}$
\vspace{1.75in}
\ee

\item Assume $z=a+bi$ and $w=c+di$, where $a$,$b$,$c$, and $d$ belong to $\R$. For each statement below, determine whether the statement is true or false. Justify your reasoning with a brief proof or a counterexample. (By a proof, we mean we start with one side of the equation and manipulate it to obtain the other side of the equation. We do not work with both sides at the same time.\footnote{Students who have completed MTH 210 should be familiar with this idea.})
\be
\item $\conj{zw}=\conj{z}\cdot\conj{w}$
\vspace{1.75in}
\item $\re{zw}=\re{z}\cdot\re{w}$
\vspace{1.75in}
\ee

\vfil
Because the complex numbers can be added and multiplied together in the ways one would expect and because of other basic properties, they form what students who have completed MTH 310 know to be
a \emph{field}. Moreover, if we view scalars as real numbers, the set of complex numbers is a \emph{vector space}.

\newpage
\noindent
That $\C$ forms a vector space gives us a graphical perspective of complex numbers.
This is accomplished by means of what we refer to as
the \emph{complex plane}. We identify the horizonal axis with the real part
of the complex number and the vertical axis with the imaginary part. A complex number $z=a+bi$ is then
plotted as the point $(a,b)$ shown in Figure \ref{Fi:axes}.

%\begin{figure}[h]
%\centering
%\psfrag{horizontalaxis}[Bc][Bc]{$\reaxis$}
%\psfrag{verticalaxis}[Bc][Bc]{$\imaxis$}
%\psfrag{point}[Bc][Bc]{$a+bi$}
%\psfrag{a}[Bc][Bc]{$a$}
%\psfrag{b}[Bc][Bc]{$b$}
%\includegraphics[width=.7\linewidth]{complex_plane_axes.eps}
%\caption{The complex plane.}
%\label{Fi:axes}
%\end{figure}


Using this vector representation of a complex number, we can speak of its ``length.'' The \emph{modulus} or \emph{absolute value} or \emph{norm} of a complex number $z=a+bi$ is its distance to the origin
in the complex plane. We use the notation $|z|$ to denote the modulus of $z$. By the Pythagorean Theorem, $|z|= \sqrt{a^2+b^2}$. For example, if $z=1+2i$, then $|z|=\sqrt{1^2+2^2}=\sqrt{5}$.

\item Assume that $z=1+2i$ and $w=-2+3i$. Plot each of these Figure \ref{Fi:axes}. Then calculate each quantity below and graph the result in Figure \ref{Fi:axes}.
\be
\item $z+w$. (Observe how the result compares with the ``parallelogram'' or ``tip-to-tail'' method for
adding vectors.)
\vspace{.4in}
\item $\overline{z}$
\vspace{.4in}
\item $-w$.
\vspace{.4in}
\item $\im{z}$
\ee

\newpage
\item Based upon the preceding examples and other similar ones you may wish to consider, explain the graphical relationship that exists between $z$ and $\conj{z}$. Then do the same for $z$
    and $-z$. For each pair of numbers, describe the relationship in terms of the concept of
    \emph{reflections}.
\vspace{1.75in}
\item Show that $|z+w| \ne |z|+|w|$.
\vspace{1.25in}
\item Suppose $z=a+bi$. Show that $z\cdot \conj{z}=|z|^2$.
\vspace{1.75in}

\noindent
The modulus provides a convenient means of describing certain sets in the complex plane. For example, the set of numbers $S=\left\{z \mid z \in \C \mbox{ and } |z|=1 \right\}$, which we can abbreviate simply as $|z|=1$, is the set of all complex numbers that are one unit away from the origin in the complex plane. This of course is really nothing more than the unit circle. The set $|z|<1$ is the set of complex numbers less than one unit away from this origin, a set we refer to as the \emph{open unit disk}. The word ``open'' emphasizes the fact that the boundary, i.e., the unit circle itself, is not included. Graphically, the open unit disk is shown in Figure \ref{Fi:open_unit_disk}.

%\begin{figure}[h]
%\centering
%\psfrag{horizontalaxis}[Bc][Bc]{$\reaxis$}
%\psfrag{verticalaxis}[Bc][Bc]{$\imaxis$}
%\includegraphics[width=.3\linewidth]{open_unit_disc.eps}
%\caption{The open unit disk, $|z|<1$.}
%\label{Fi:open_unit_disk}
%\end{figure}

\noi
Disks of different radii and centered at other complex numbers can be expressed using similar
notation. For example, the set $\ds \left| z-(1+2i) \right| < 3$ consists of the open disk
of radius 3 having center at the point $z_0=1+2i$.

\newpar
\item Read Example 2 on p. 9 of the text. Then sketch each of the following sets in the complex plane.
Support each answer with appropriate calculations and/or a brief explanation.
\be
\item $| z+1+2i| < 2$
\vspace{2in}
\item $|z| \geq 2$
\newpage
\item $\im{z+i} < 3$
\vspace{3in}
\item $|z-1|^2+|z-i|^2=2$
\ee


\ee

\end{document}

