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%%% Template finexam.tex, August 2007 by Igor Fulman                %%%
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\documentclass[12pt]{article}
\usepackage{FYM3}    % FYM3.sty is a style file for tests, finals

\renewcommand{\topmargin}{0.5in} % THIS MAY NEED TO BE ADJUSTED <=====

%=======================================================================
% adjust for your own test...

\renewcommand{\instructor}{\color{red}Instructor's Name} 
                                                  % Instructor's name 
                                                  %   (Put your name here!)
\renewcommand{\course}{MAT 211}                   % Course number
\renewcommand{\coursetitle}{Mathematics for Business Analysis}
                                                  % Course title
\renewcommand{\examnumber}{Final Exam}            % Test type and number
\renewcommand{\examform}{Form A}                  % Test form
\renewcommand{\examsemester}{Spring}              % Semester
\renewcommand{\examyear}{2007}                    % Year

%=======================================================================

% These don't need to be adjusted
\renewcommand{\pages}{\pageref{last}}
\renewcommand{\problems}{\ref{last}}
\renewcommand{\lastproblem}{\ref{lastpr}}

\def\ds{\displaystyle}

\begin{document}

\thispagestyle{empty}

\exampage % Sets up the cover page of the test

\newpage

\begin{enumerate}

\item  (6 points) Find and classify all the stationary points of the function
$ f(x,y) = 2 x^2 + 3 y^3 - 6 x + 10 y^2 - 15 ,$  i.e. tell whether each point
is a maximum, a minimum, or a saddle point.

\vfil\vfil

\item (8 points) Find the absolute maximum and the absolute minimum of the
function $ f(x)= 3 x y $ on the region bounded by the lines $x=-1$, $x=1$,
$y=-2$, $y=2$.

\vfil

\newpage

\item  Find the maximum and the minimum of the function $ f(x,y) = 3 x y $
under the constraint $ 2x - 5y = 10. $


\vfil\vfil

\item  Given a linear system: $ \cases{ 2 x - 3 y + z = 10 \cr  x + 3y - z = 2
\cr 7x + 3y - 4z = 10 } $

\begin{enumerate}

\item (4 points) Solve the system using the reduced row-echelon form. Give the
answer in the form {\it x=\ldots, y=\ldots, z=\ldots} You may use calculator
features.

\vfil

\newpage

\item  (7 points) Solve the same system using the method of inverse matrix.
You may use calculator features.

\end{enumerate}

\vfil\vfil

\item (6 points) A fair playing die is tossed. Find the probability that the
die shows an odd number, {\bf given that} it's a prime number. (Prime numbers
are: 2, 3, 5, 7, 11, 13... 1 is not a prime number.)

\vfil \vfil

\item (6 points) In a certain pizzeria, a pizza order includes: thin or thick
crust; one cheese out of 3 possible cheeses; and exactly 3 out of 10 possible
toppings. How many different pizza orders are possible?

\vfil

\newpage

\item  A random variable $X$ takes integer values from 7 to 10 inclusive. The
value 7 has the probability of 0.1, the value 8 has the probability 0.2, the
value 9 has the probability of 0.3.

\begin{enumerate}

\item (3 points) Find the probability of the value 10. (Hint: what should be
the total probability?)

\vfil

\item (5 points) Find $ P(X \ge 9).$

[Note to the teacher: make sure the correct answer is NOT 0.5. This will help
to find students' mistakes.]

\vfil

\item (6 points) Find the mathematics expectation of the random variable $X$.
If using a calculator, indicate which features of calculator you used.

\end{enumerate}

\vfil

\item Given un unfair coin with this the probability of the heads being 0.3
and the probability of the tails being 0.7. It is tossed 50 times.

\begin{enumerate}

\item (4 points) Find the probability that there will be exactly 30 heads.

\vfil

\item (6 points) Find the probability that there will be at least 30 heads.

\end{enumerate}

\newpage

\item  Let $X$ be a continuous random variable with the following probability
density function:
 $$  f(x) = \cases{ \vspace{10pt} \ds {7 - x \over 8} & if $ 3 \le x \le 7 $ \cr 0 &
 otherwise} $$

\begin{enumerate}

\item (5 points) Find $ P(4 \le 6 \le 6.5). $ Set up the relevant integral and
evaluate it. You may use calculator features.

\vfil

\item (5 points) Find the mathematical expectation $E(X).$ Again, set up the relevant
integral and evaluate it. You may use calculator features.

\vfil

\item (7 points) Find the cumulative distribution function, c.d.f. $F(x)$ for this
random variable.

\vfil

\newpage

\item (7 points) Find the median for this random variable, i.e. find such a
number $M$ that $ P(X \le M) = \ds {1 \over 2} .$

\end{enumerate}

\vfil

\item Assume that the heights of pine trees across the United States are
normally distributed with the mathematical expectation of 50 feet and the
standard deviation of 6 feet. One tree is chosen randomly.

\begin{enumerate}

\item (5 points) Find the probability that the tree is not higher than 60
feet.

\vfil

\item (5 points) Find such a height $H$ that the probability of the tree being
not higher than $H$ feet is exactly 0.75.

\vfil

\item BONUS QUESTION (10 points) Find such a number $a$ that the probability
of the tree's height being between $(50-a)$ and $(50+a)$ is exactly 0.6.

\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}