\input LaTest.tex
\documentstyle{latest}
\textwidth 6.3in\textheight 8.5in\parindent 0cm
\topmargin -1cm\oddsidemargin 0.7cm

\begin{document}
\baselineskip 0.5cm\everyproblem={\mathsurround=4pt}
\normalafterproblemskip=2.8cm\pageannouncements={usual}
                                         \input exam.perm8
                                         \lastversion=3
                                         \date{{\large December 3,
                                             1997}}
                                         \title{{\large\bf Math
                                             108,  Section 1, Test 3}}

\comment{\large
\begin{minipage}[c]{6in}
  1. Please cross \fbox{$\times $} the correct answers. \\
  2. This test will be exactly 50 minutes in length.  When you
  are told to begin, but not before, glance through the entire
  test and put your name on each page.  It is YOUR RESPONSIBILITY
  to make sure your test consists of 7 PAGES with 11 PROBLEMS.
  The point value for each multiple choice problem is 7 points.
  Problems 9 and 10 and 11 are worth 44 points total.
  Use the back of the test pages  for scratch work.\\
  3. On all partial credit problems, show your work, indicating
  clearly how you arrived at your answer. The points you receive
  for problems 9, 10 and 11 will depend on the extent to which
  the work you show convinces the grader (at the time of grading)
  that you did all or part of the problem correctly.
\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 Which of the points $A = (1.3, -2.7, 0)$, $B =
(-1.9,0,3.2)$, $C = (2.5, 0.1,-0.3)$, $D = (2.9, 0.1,-0.3)$ $E =
(-0.1, 3.1,-4.3)$, is closest to the $yz$-plane?
\correct $E$
\wrong $A$
\wrong $B$
\wrong $C$
\wrong $D$



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\problem If $f(x,y)=x^2+y^2+9$ then find the equation of the
tangent plane at $(x_0,y_0)=(1, 2)$.


\correct $2x+4y-z+4=0$
\wrong $4x+2y-z+4=0$
\wrong $2x+2y-z+8=0$
\wrong $x+2y-z+2=0$
\wrong $2x+2y-z+9=0$

\vspace{1cm}

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\problem Find the line of least squares $y=ax+b$ which best fits
the data points $(-2,-7),$ $(-1,-2)$ and $(3,18)$.

\correct $y=5x+3$
\wrong  $y=4x+4$
\wrong  $y=4x+6$
\wrong  $y=5x+4$
\wrong  $y=5x+6$

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\problem Compute $\frac{\partial f}{\partial x}(0,1)$ for the
function $f(x,y)=x^2e^{3x+y}+5ye^y+2xy$.

\correct 2
\wrong 3
\wrong 5
\wrong 0
\wrong 4

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\problem Consider the function $f(x,y)=x^2+3xy+2y^2$. Then one has:

\correct $(0,0)$ is a saddle point.
\wrong  $(0,0)$ is a relative minimum.
\wrong  $(0,0)$ is a relative maximum.
\wrong The second derivative test is inconclusive.
\wrong There are no critical points.

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\problem Determine the expected value for the probability density
function $f(x)=\frac{1}{4}$, for $2\leq x\leq 6$.

\correct 4
\wrong 2
\wrong 6
\wrong $\frac{1}{4}$
\wrong 0

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\problem Determine the cumulative distribution function $F(x)$
for the probability density function\\
$f(x)=\frac{1}{8}x$, for $0\leq x\leq 4$.

\correct $F(x)=\frac{1}{16}x^2$
\wrong $F(x)=\frac{1}{8}$
\wrong $F(x)=\frac{1}{8}x^2$
\wrong $F(x)=8x^2$
\wrong $F(x)=\frac{1}{4}x^2$

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\problem Let $Z$ have the standard normal distribution. Compute
Pr$(-1\leq Z\leq 0)$.

\correct 0.3413
\wrong 0.500
\wrong 0.1587
\wrong 0.8413
\wrong 0.6742

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\problem Suppose that the amount of time (in minutes) required to
serve a customer at a bank is described by the probability
density function $f(x)=\smallfrac{1}{5}e^{-\smallfrac{1}{5}x},$
$0\leq x<\infty.$

(8 pts) Compute the probability that a customer will have to wait
more than 10 minutes.

\vspace*{8cm}

(7 pts) Determine the expected waiting time. (Show your work/reasoning).

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\problem Let $X$ be a discrete random variable whose distribution is given
by
$$
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x& 1&2&3&4&5&6 \\ \hline
f_X(x)& \frac{4}{7}&\frac{2}{7}&0&0&0&\frac{1}{7}\\ \hline
\end{array}
$$
 (6 pts) Compute the expected value E$(X)$.

\vspace*{6cm}

 (6 pts) Compute the Variance Var$(X)$.

\vspace*{6cm}

 (2 pts) Compute the standard deviation $\sigma(X)$.



\vspace*{5cm}

%\vspace*{17cm}


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\problem (15pt) The Cobb-Douglas production function for a
certain product is given by $f(x,y) = 5x^{\smallfrac{4}{5}}
y^{\smallfrac{1}{5}}$ where $f(x,y)$ is the quantity produced,
$x$ denotes units of labor force and $y$ denotes units of
capital.  Assume that each unit of labor costs \$300, each unit
of capital costs \$100, and the total budget is \$15,000. Find the
amounts of labor and capital which will maximize the company's
production while keeping  within the constraints of the budget.

\vspace*{15cm}



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\endtest
