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\documentclass[12pt]{article}
\usepackage{FYM2}      % Style file for tests
\usepackage{epsf}      % Package for inserting postscript files

\renewcommand{\topmargin}{0.25in} % THIS MAY NEED TO BE ADJUSTED <=====

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\renewcommand{\instructor}{Fulman}             % Instructor's name
\renewcommand{\course}{MAT 211}                % Course number
\renewcommand{\coursetitle}{Mathematics for Business Analysis}
                                               % Course title
\renewcommand{\examnumber}{Test \#1}           % Test type and number
\renewcommand{\examform}{Form A}               % Test form 
\renewcommand{\examsemester}{Spring}           % Semester
\renewcommand{\examyear}{2007}                 % year

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\renewcommand{\problems}{\ref{last}}
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\def\ds{\displaystyle}

\begin{document}

\thispagestyle{plain}
\exampage    % Sets up the cover page of the test

\begin{center}{\bf Formulas to use}\end{center}

\vspace{10pt}

\noindent
If $ f(x) = x^n $ then $ f'(x) = nx^{n-1} $ \\[5pt]
If $ f(x) = e^x $ then $ f'(x) = e^x $ \\[5pt]
If $ f(x) = a^x $ then $ f'(x) = a^x \ln a $ \\[5pt]
If $ f(x) = \ln x $ then $ f'(x) = \ds {1 \over x} $ \\[15pt]
If $ k(x) = c f(x) $ then $ k'(x) = c f'(x) $ \\[15pt]
If $ k(x) = f(x) \pm g(x) $ then $ k'(x) = f(x) \pm g(x) $ \\[15pt]
If $ k(x) = f(x) \cdot g(x) $ then $ k'(x) = f'(x) g(x) + f(x) g'(x) $ \\[15pt]
If $ k(x) = \ds {f(x) \over g(x)} $ then $ k'(x) = 
\ds { {f'(x) g(x) - f(x) g'(x)} \over g(x)^2} $ \\[15pt]
If $ k(x) = f(g(x)) = f \circ g (x) $ then $ k'(x) = f'(g(x)) \cdot g'(x) $ 
                                                                  \\[30pt]
$ x^2 + y^2 = r^2 $ --- circle centered at the origin with radius $r.$\\[30pt]
$ (a \pm b) ^2 = a^2 \pm 2 a b + b^2 $
$ \hspace{1in} a^2 - b^2 = (a-b)(a+b) $ \\[30pt]
Classification of stationary points: $ D = f_{xx} f_{yy} - f_{xy}^2 $ 
\begin{enumerate}
\item $ f_{xx} > 0 $, $ D > 0 $ --- local minimum
\item $ f_{xx} < 0 $, $ D > 0 $ --- local maximum
\item $ D < 0 $ --- saddle point
\end{enumerate}

\newpage

\begin{enumerate}

\item Draw 3 level curves (contours) for the following functions. Show your 
work, i.e. show how you obtained the equations of the level curves. Use an 
appropriate scale (appropriate window dimensions if you use calculator). Label 
each level curve with the corresponding $z$-value. Show the scale on the $x$- 
and $y$-axes, too.

\begin{enumerate}
\item (5 points) $ f(x,y) = 3 x - 5 y $

\newpage

\item (7 points) $ g(x,y) = 2 \sqrt{ x^2 + y^2 } $

\end{enumerate}

\newpage

\item The following picture shows some level curves (contours) for a certain 
function $f(x,y).$ The numbers on the curves denote the values of the function
($z$-values) on each level curve. \\[10pt]
\vbox to 3.985in{\hbox to 6 in{\hfil\epsffile{contour.eps}\hfil\hfil\hfil}}

\vspace{-.25in}

\begin{enumerate}
\item (4 points)\ Estimate the values $f(1,3)$ and $f(1,4.5).$ Show the 
corresponding points on the picture. 


\vspace{1 in}

\item (6 points)\ Find all (local and global) maximum, minimum, and saddle 
points, and show them on the picture. List the points here, indicate the type 
of each point, its coordinates and the value of the function $f(x,y).$

\newpage

\item (6 points)\ Find (approximately) the partial derivatives $f_x(1,3)$ and 
$f_y(1,3).$  Show your work.

\end{enumerate}

\vfil\vfil

\item Find the first-order partial derivatives for the following functions:

\begin{enumerate}
\item (5 points)\ $ f(x,y) = 5 x^2 y^3 - 3 x y^{10} - 7 \sqrt{x} + 3 \ln y $

\vfil

\newpage

\item (8 points)\ $ g(x,y) = 10 x y \cdot \ln (5xy - 3x + 5y) $

\vfil\vfil

\item (8 points)\ $ \ds h(x,y) = { e^{x+2y} \over xy} $ 

\end{enumerate}

\vfil 

\newpage

\item Find all the second-order derivatives of the following functions:

\begin{enumerate}

\item  (10 points)\  $ \ds f(x,y) = {2x - 3y \over 5x + 4y} $ \\[5pt]
(Hint: You may find it useful to expand the denominator of the first order 
derivatives.)

\vfil\vfil

\item  (10 points)\  $ \ds f(x,y) = \ln (5x - 7y) $
\end{enumerate}

\vfil 

\newpage

\item Find and {\bf classify} stationary points for the following functions, 
i.e. tell whether each point is a maximum, a minimum, or a saddle point.

\begin{enumerate}

\item (8 points)\ $ f(x,y) = 2x - 3y - x^2 - 5 y^2 + 10 $

\vfil\vfil

\item (9 points)\ $ g(x,y) = x^2 - 4xy + 3y^2 + 6x + 2y - 30 $

\vfil 

\newpage

\item (10 points)\ $ h(x,y) = x^2 - y^3 + 6xy - 10 $

\label{lastpr} 
% must be BEFORE THE FIRST "/end{enumerate}" - counting problems' parts

\end{enumerate}

\label{last}   
% must be BEFORE THE LAST "/end{enumerate}" - counting problems and pages

\end{enumerate}

\end{document}