\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=2.5cm\pageannouncements={usual}
                                         \input exam.perm6
                                         \lastversion=0
                                         \date{{\large September 24,
                                             1997}}
                                         \title{{\large\bf Math
                                             108,  Section 1, Test 1}}

\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

Compute the improper integral

$$\int_{1}^{\infty} x^{-7}dx.$$

\correct $\frac{1}{6}$
\wrong $\frac{1}{7}$
\wrong $\frac{1}{8}$
\wrong $-\frac{1}{8}$
\wrong $-\frac{1}{6}$


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\problem

Determine the average value of the function $y=\sin x$ between
$x=0$ and $x=\pi$.

\wrong $2$
\correct $\frac{2}{\pi}$
\wrong $0$
\wrong $-\frac{\pi}{2}$
\wrong $2\pi$


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\problem

Consider the following data for a function $y=y(x)$:\\
$$\displaystyle{\frac{dy}{dx} = x^2 -1, \ y(3) = 20}.$$
Compute the value $y(0)$.

\correct $14$
\wrong $\frac{17}{3}$
\wrong $\frac{20}{3}$
\wrong $3$
\wrong $-\frac{20}{3}$


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\problem

Compute the definite integral $\int_0^1 x^2 e^x \, dx = $

\correct $e - 2$
\wrong $-e -2$
\wrong $e$
\wrong $-e$
\wrong $e-1$




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\problem

Let $ f(x), 2 \le x \le 6.5,  $ be a continuous function whose
known values are given by the table:


$$\vbox{\offinterlineskip \halign{ \strut \vrule # & \vrule #
    &\vrule #&\vrule #&\vrule #&\vrule #&\vrule #&\vrule #
    &\vrule #&\vrule #&\vrule #&\vrule # \vrule height .2 in
    depth .1 in width .4pt \cr \noalign{\hrule} \ $x $&\ \ $2$ &\
    \ $2.5$ &\ \ $3$\ &\ \ $3.5$\ &\ \ $4$\ &\ \ $4.5$\ &\
    \ $5$&\ \ $5.5$\ &\ $6 \ $\vrule\cr \noalign{\hrule}
    $f(x)$&\ \ $4$ &\ \ $4$ &\ \ $5$ &\ \ $6$\ &\ \
    $7$\ &\ \ $6$\ &\ \ $5$\ &\ \ $4$\ &\ $4 $\vrule\cr
    \noalign{\hrule} }}.$$
Estimate $ \int^{6.5}_2 f(x) dx.$

\wrong 20
\wrong 18.5
\wrong 5
\wrong 45
\correct 22.5



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\problem Assume a certain amount of money is in a bank account
with an annual interest rate of 5\% compounded continuously.
Compute the number of years it takes until the amount did double.

\correct $20\ln 2$
\wrong $25$
\wrong $17$
\wrong $20$
\wrong $5\ln 2$



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\problem  Compute the definite integral $\int_1^e \frac{\ln x}{x} \, dx = $

\correct $\frac{1}{2}$
\wrong $\frac{2}{e}$
\wrong $e$
\wrong $e-1$
\wrong $\frac{1-e}{2}$



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\problem

The area of the region bounded by the graphs of $f(x) = e^x$
and $g(x) = e^{-x}$ for $0 \leq x \leq 1$ is

\wrong $e + e^{-1}$
\wrong $e - e^{-1}$
\wrong $e - e^{-1} - 1$
\wrong $e + e^{-1} - 1$
\correct $e + e^{-1} - 2$


\vspace*{4cm}


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\problem

(15 pts) The marginal profit for a certain company is
$MP_1(x)=-x^2+18x-24$. The company expects the daily production
level to rise from $x=10$ to $x=15$ units. The management is
considering a plan that would have the effect of changing the
marginal profit to $MP_2(x)=-x^2+24x-12$. Should the company
adopt the plan. What is the difference in profit between the two
proposed plans.

\vspace*{17cm}


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\problem

(15 pts) Find the present value PV of a perpetual income stream
flowing continuously at a rate of \$3000 per year, and with
interest rate compounded continuously at the rate of 6\%.

\vspace*{17cm}

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\problem

(14 pts) The demand curve of a certain item is $p = D(q) =
-\frac{1}{16}q^2 +10$ and its supply curve is\\ $p = S(q) =
\frac{1}{16}q^2+2$.


A) Find the equilibrium price $p_e$ and equilibrium quantity
  $q_e$.\vspace{8cm}

B) Compute the producer surplus.





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\endtest

