\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.perm4
                                         \lastversion=3
                                         \date{{\large December 16,
                                             1997}}
                                         \title{{\large\bf Math
                                             108,  Final Exam}}

\comment{\large
\begin{minipage}[c]{6in}
  1. Please cross \fbox{$\times $} the correct answers. \\
  2. This test will be exactly 120 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 11 PAGES with 18 PROBLEMS.
  The point value for each multiple choice problem is 7 points.
  Problems 15. 16, 17, 18 are worth 13 points each resulting in a
  total of 52 points.
  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 the partial credit problems 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{10}\noprofessor \nosection \vertmultchoice=2.2in
\multiplechoiceskip=0.8cm \let\ansborder=\boxborder \boxwidth=1cm
\boxdepth=0.4cm\pagenumberstyle=1
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\problem Compute the definite integral $\int_0^1 x e^x \, dx = $

\correct $1$
\wrong $e -1$
\wrong $e$
\wrong $-e$
\wrong $-1$



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\problem

Assume a certain amount of money is in a bank account
with an annual interest rate of 10\% compounded continuously.
Compute the number of years it takes until the amount triples.

\correct $10\ln 3$
\wrong $10$
\wrong $17$
\wrong $20$
\wrong $5\ln 3$



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\problem  Compute the improper integral

$$\int_{2}^{\infty} x^{-4}\,dx.$$

\correct $\frac{1}{24}$
\wrong $\frac{1}{3}$
\wrong $\frac{1}{8}$
\wrong $-\frac{1}{12}$
\wrong $-\frac{1}{6}$


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\problem

A company determines that its marginal profit function is
$MP(x)=-300x^2+600x+400$.  If the company makes \$300 when it sells 2
items, what is the total profit function?

\correct  $-100x^3+300x^2+400x-900$
\wrong  $-100x^3+300x^2+400x$
\wrong  $-100x^3+300x^2+400x+300$
\wrong  $-100x^3+300x^2+400x+150$
\wrong  $-100x^3+300x^2+400x-750$

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\problem

Solve the following differential equation with given initial
condition:
$$y^{\prime}=\frac{1}{3}y-1,\ \ y(0)=1.$$


\correct $f(t)=3-2e^{\smallfrac{1}{3}t}$
\wrong $f(t)=4-2e^{\smallfrac{1}{3}t}$
\wrong $f(t)=3-\frac{1}{2}e^{\smallfrac{1}{3}t}$
\wrong $f(t)=3-e^{\smallfrac{1}{2}t}$
\wrong $f(t)=5-e^{3t}$

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\problem  Assume $y^{\prime}=t^3$ and  $y(0)=1$. Compute $y(2)$.

\correct 5
\wrong 3
\wrong 9
\wrong 0
\wrong 12

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\problem  Find the equation of the plane through the points
$(3,0,0)$,$(0,4,0)$ and $(0,0,2)$.


\correct $4x+3y+6z=12$
\wrong $x+\smallfrac{3}{4}y+2z=3$
\wrong $8x+4y+2z=8$
\wrong $\smallfrac{1}{3}x+\smallfrac{1}{4}y+\smallfrac{1}{2}z=0$
\wrong $\smallfrac{1}{6}x+\smallfrac{1}{3}y+\smallfrac{1}{2}z=4$



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\problem  For what value of $k$ does the following system of
equations have {\bf no solution}?
$$
\begin{array}{rcl}
9x-6y&=&-31\\
-15x+ky&=&13
\end{array}
$$

\correct $k=10$
\wrong $k=-10$
\wrong $k=\smallfrac{2}{31}$
\wrong $k=90$
\wrong $k=\smallfrac{2}{3}$


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\problem Determine which of the following functions is a solution
of the differential equation $y''-6y'+8y=0$.


\correct $e^{4t}$
\wrong $e^{8t}$
\wrong $e^{5t}$
\wrong $e^{6t}$
\wrong $e^{-5t}$


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\problem

A continuous random variable $X$ has a probability density function
$f(x)=\frac{3}{8}x^2$ for $0\leq x \leq 2$.  What is the expected value
$E(X)$?

\correct  $\frac{3}{2}$
\wrong  $\frac{3}{16}$
\wrong  $1$
\wrong  $\frac{5}{2}$
\wrong  $\frac{4}{3}$

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\problem Find the line of least squares $y=ax+b$ which best fits
the data points $(-3,-5),$ $(1,3)$ and $(2,5)$.

\correct $y=2x+1$
\wrong  $y=3x+2$
\wrong  $y=x+1$
\wrong  $y=5x+4$
\wrong  $y=3x$

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\problem Consider the function $f(x,y)=-6x^2-7xy-2y^2$. 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 Let $X$ be a discrete random variable whose distribution is given
by
$$
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline x& 2&4&6&8&10&12&14 \\
 \hline f_X(x)& 0&\frac{1}{5}&0&0&\frac{1}{5}&\frac{3}{5}&0\\
 \hline
\end{array}
$$

Compute the Variance Var$(X)$.

\correct $\frac{48}{5}$
\wrong $\frac{27}{5}$
\wrong $\frac{36}{5}$
\wrong $\frac{12}{5}$
\wrong $\frac{32}{5}$

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

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

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\problem
 (13 pts) The demand curve of a certain item is $p = D(q)
=\frac{100}{q+1}$ and its supply curve is
$p = S(q) =q+1$.


(6 pts) Find the equilibrium price $p_e$ and equilibrium quantity
  $q_e$.
\vspace*{8cm}

(7 pts) Compute the producer surplus.

\vspace*{8cm}

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\problem (13 pts) A person opens an Individual Retirement Account
(IRA) with the initial amount of \$50,000.  Then \$6,000 per
year is deposited in this IRA in a uniform and continuous
manner.  Assume that the interest rate is 7.5\% compounded
continuously.
\medskip


(4 pts) Model this problem as a Calculus problem by finding a
{\bf differential equation} and an {\bf initial condition}
describing the amount of money, $M(t)$, in the IRA at any time
$t$.

\vspace*{4cm}

(5 pts) Solve the obtained differential equation, i.e.  find
$M(t)$ at any time $t$.

\vspace*{8cm}

(4 pts) Compute the balance in the IRA after 20 years. (If you do
not have a calculator you can approximate $e^{1.5}$ with 4.5. Simplify
your answer as much as possible.)

%\vspace*{4cm}


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\problem (13 pts) The Cobb-Douglas production function for a
certain product is given by $f(x,y) = 5x^{\smallfrac{3}{5}}
y^{\smallfrac{2}{5}}$ 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 \$400, each unit
of capital costs \$100, and the total budget is \$20,000. Find the
amounts of labor and capital which will maximize the company's
production while keeping  within the constraints of the budget.


\vspace*{18cm}

{\bf Answer:} \fbox{\rule[-3mm]{0cm}{8mm}$x=$ \hspace{2cm}
  $y=$\hspace{2cm}}
\vspace*{-4cm}

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\problem (13 pts) A bank has 1,000,000 credit card holders. During
the preceding year the average billing to each card holder was
\$195 and the standard deviation was \$60. It is assumed that the
billings are normally distributed. Compute the number of
customers whose bill exceeds \$300.

\vspace*{19cm}

{\bf Answer:} \fbox{\rule[-3mm]{0cm}{8mm}Number=\hspace{2cm}}
\vspace*{-5cm}

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\endtest
