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\title{Brief Article}
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                                         \input exam.perm 
                                         \lastversion=3
                                         \date{{\large December 3, 1997}}
                                         \title{{\large\bf Math 108, Exam 3}}
\comment{\large 
\begin{minipage}[c]{6in}
This exam is worth a total of 100 points.  There are 11 problems -- 8 multiple choice 
and 3 partial credit.  Each multiple choice problem is worth 7 points.  Point values are
assigned next to the partial credit problems. Please show all work on the partial
credit section of the test inside the test booklet. Use the front sheet to answer 
the multiple choice section by putting a \fbox{$\times $} in the appropriate box.  
You will have until 9:20am to complete the exam. Good luck! 
\end{minipage}}

\answersheetfootline{\large\bf\fbox{\rule[-2mm]{0mm}{8mm}Sign
your name\hspace{6cm}}
\hspace{6cm}}\twocolumn{6}\noprofessor \nosection \vertmultchoice=2.5in
\multiplechoiceskip=0.8cm \let\ansborder=\boxborder \boxwidth=1cm
\boxdepth=0.4cm\pagenumberstyle=1


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\problem  

Find the value of k which makes $f(x)=3x^2+kx$ a probability density function
on the interval $0\leq x\leq 2$.

\wrong $k=\frac{3}{2}$
\wrong $k=\frac{-5}{8}$
\correct $k=\frac{-7}{2}$
\wrong $k=2$
\wrong $k=-3$

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\problem  

It is given that $(3,3)$ is a critical point of a function $f(x,y)$.  If $\pderx f(x,y)$=6,
$\pdery f(x,y) = 6y$ and $\mixpd f(x,y)=-6$, then what can be said of $(3,3)$?

\correct $(3,3)$ is a relative minimum.
\wrong $(3,3)$ is a relative maximum.
\wrong $(3,3)$ is a saddle point.
\wrong The test is inconclusive.
\wrong There is not enough information to determine this.

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\problem  

Suppose that the amount of time required to serve a customer at a bank has an
exponential density function with $\mu = 5$.  What is the probability that the
customer will be served in one minute or less?  (i.e. determine $P(0 \leq X \leq 1)$.)

\wrong  $e^{\frac{-1}{5}}$
\wrong $1+e^{\frac{-1}{5}}$
\correct $1-e^{\frac{-1}{5}}$
\wrong $\frac{1}{5}e^{\frac{-1}{5}}$
\wrong $1-\frac{1}{5}e^{\frac{-1}{5}}$

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\problem  

Find the equation of the plane which has x-intercept (3,0,0), y-intercept (0,-2,0) and
z-intercept (0,0,6).

\wrong $z=6-3x+2y$
\correct $z=6-2x+3y$
\wrong $z=6+\frac{1}{2}x-\frac{1}{3}y$
\wrong $z=6-\frac{1}{2}x+\frac{1}{3}y$
\wrong $z=6+3x-2y$

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\problem  

Consider the following probability table.

\begin{tabular}{|c|c|c|c|c|c|c|}
\hline Outcome & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline Probability & $\frac{1}{9}$ & $\frac{1}{18}$ & $\frac{1}{6}$ & 
$\frac{5}{18}$ & $\frac{2}{9}$ & $\frac{1}{6}$ \\
\hline
\end{tabular}

What is the expected value E(X)?

\wrong $\frac{31}{57}$
\wrong $\frac{55}{18}$
\wrong $\frac{1200}{104976}$
\wrong $\frac{152}{74}$
\correct $\frac{53}{18}$

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\problem  

Determine the cumulative distribution function for the probability density
function $f(x)=\frac{3\sqrt{x}}{16}$ for $0 \leq x \leq 4$.

\wrong $F(x)=\frac{3}{16}x^{\frac{1}{2}}$
\wrong $F(x)=\frac{3}{32}x^{\frac{-1}{2}}$
\correct $F(x)=\frac{1}{8}x^{\frac{3}{2}}$
\wrong $F(x)=\frac{1}{12}x^2$
\wrong $F(x)=\frac{1}{3}x^{\frac{-1}{2}}$

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\problem  

Compute $\derx f$ for the function $f(x,y)=x^2ye^{2xy-x^3}$.

\correct $2xye^{2xy-x^3}+x^2y(2y-3x^2)(e^{2xy-x^3})$
\wrong $2xye^{2xy-x^3}+x^2ye^{2xy-x^3}$
\wrong $2xye^{2xy-x^3}(2y-3x^2)$
\wrong $2xy+(2y-3x^2)(e^{2xy-x^3})$
\wrong $2xye^{2xy-x^3}$

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\problem  

The time in minutes required to complete an assembly on a production line
is a random variable X with probability density function $f(x)=2x$ for
$0 \leq x \leq 1$.  It is given that $E(X) = \mu = \frac{2}{3}$.  Compute the
variance Var(X).

\wrong $\frac{1}{9}$
\wrong $\frac{4}{9}$
\wrong $\frac{2}{13}$
\wrong $\frac{1}{3\sqrt2}$
\correct $\frac{1}{18}$

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\problem

Find the critical points of the function $f(x,y)=x^2+4xy+2y^4$ and determine 
whether they are relative maxima, relative minima, or saddle points.  (Hint:  You
should find three critical points.) (22 points) 

\vspace*{15cm}

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\problem 

Suppose that you are being asked to analyze the income for a company whose
total profit fits into the following chart.

\begin{tabular} {|c|c|c|c|}
\hline Year & 1 & 2 & 3 \\
\hline Profit & 2 & 5 & 7 \\
\hline
\end{tabular}

What is the regression line for these data points?  (Use the least squares
method.) (22 points) 

\vspace{12cm}

What would be your prediction for the profits for the fourth year? 
\vspace*{3cm} 

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\problem  

A company determines that its production function is given by
 $f(x,y)=72x^{\frac{1}{4}}y^{\frac{3}{4}}$ where x is the amount of capital and
y is the amount of labor.  Suppose capital costs \$9 per unit and labor costs
\$18 per unit.  The company has a budget of \$12,000.  Find the amounts of labor
and capital which will maximize the company's production while keeping within the
constraints of the budget. (22 points)


\vspace*{15cm}

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