\documentstyle{letter}
\textwidth 6in
\textheight 8.7in  \parindent 0cm
\topmargin -1cm
\oddsidemargin 0.7cm

\begin{document}

\baselineskip 0.5cm
\everyproblem={\mathsurround=4pt}
\normalafterproblemskip=3cm
\pageannouncements={usual}

\lastversion=4 
\input TestII
.perm

\title{{\large\bf Math 105, Test 2}}
\comment{ \large 
\begin{minipage}[c]{5.8in}
1. Please cross \fbox{$\times $} the correct answers.   \\
2.  This test will be exactly 60 minutes in length.  
It consists of 12 multiple choice problems worth 5 points each with no partial credit and 2
problems where partial credit is possible.    \\
3. {\bf  On all partial credit problems, show your work, indicating clearly 
 how you arrived at your answer. 
 The points you receive for problems 13 and 14 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} Signature:\hspace{6cm}}
\hspace{6cm}}
\twocolumn{8} 
\date{{\large October 15, 1992}}
%\noprofessor 
%\nosection 
\vertmultchoice=2.5in
%\multiplechoiceskip=0.8cm \let\ansborder=\boxborder \boxwidth=1cm\boxdepth=0.4cm
\multiplechoiceskip=0.8cm \let\ansborder=\boxborder \boxwidth=0.7cm\boxdepth=0.2cm


\pagenumberstyle=1 
 
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\problem (5 pts.)  If $v = a^2r^{-1} + r^2a$, calculate ${d^2v \over dr^2}$.



\correct $2a^2r^{-3} + 2a$
\wrong  $2r^{-1}$
\wrong  $-2ar^{-2} + 2r$
\wrong  $-a^2r^{-2} + 2a$
\wrong  $2r^{-3}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problem (5 pts.)  The following drawing shows the graph of a function $f(x)$ near
 $x = 2$.
\vskip 4.5truecm
 
What can you say about the values of $f'(2)$ and $f''(2)$ ?


\correct $f'(2) > 0, f''(2) > 0$
\wrong $f'(2) > 0, f''(2) < 0$
\wrong $f'(2) > 0, f''(2) = 0 $
\wrong $f'(2) < 0, f''(2) > 0$
\wrong $f'(2) < 0, f''(2) < 0$


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\problem (5 pts.)  Which approximate value do you get for $8.012^{2/3}$ by
applying the formula
 
$f(a + h) - f(a) \approx f'(a) h$
 to the function $f(x) = x ^{2/3}$ ?


\correct $4.004$
\wrong  $4.006$
\wrong  $4.008$
\wrong  $4.002$
\wrong  $4.016$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\problem (5 pts.) For which values of $x$ does the function $f(x) = x^4 - 6x^2 + 5$
 have  relative maximum points?


\correct $0$
\wrong $\sqrt 3, -\sqrt 3$
\wrong $1, -1$
\wrong $0, \sqrt 3, -\sqrt 3$
\wrong none

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\problem (5 pts.)  For which values of $x$ does the function $f(x) = x^3 - 3x^2 - 9x + 2$
have inflection points?



\correct $1$
\wrong  $3$
\wrong  $-1, 3$
\wrong  $-1$
\wrong  none

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\problem (5 pts.)  Which of the following lines is an asymptote of the graph of the
function
 
$f(x) = 2x + 3 - {5 \over x}$ ?  



\correct $y = 2x + 3$
\wrong $y = 2$
\wrong $y = 0$
\wrong $y = 2x - 5$ 
\wrong $y = 3x - 5$



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\problem (5 pts.)   The height over the ground of a ball thrown into the air is
given by $s(t) = 96t - 16t^2$. Determine the maximum height the ball will reach.


\correct $144$
\wrong  $112$
\wrong  $96$
\wrong  $64$
\wrong  $80$

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\problem (5 pts.)  A rectangular plot of farmland will be bounded on one side by a river
and on the other three sides by a single-strand electric fence. If $800$ square feet
are to be enclosed, what is the minimum amount of wire needed ?


\correct $80$ feet
\wrong   $70$ feet
\wrong   $40$ feet
\wrong   $802$ feet
\wrong   $100$ feet
 

 

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\problem (5 pts.)  A tour service offers the following rates
\par
a) \$ 200 each if 50 people go on tour
\par
b) For each additional person everyones charge is reduced by \$ 2.
\par
How many people should go on tour in order to maximize the tour service's revenue?
 

\correct $75$
\wrong  $50$
\wrong $100$
\wrong  $125$
\wrong  $150$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\problem (5 pts.) Compute the derivative of 
$f(x) = (x + 3)^{25}(x^2 + 1)$.

\correct $(x + 3)^{24}(27x^2 + 6x + 25)$
\wrong   $50x(x + 3)^{24}$
\wrong   $(x + 3)^{24}(23x^2 - 6x + 25)$
\wrong   $25(x + 3)^{24}(2x + 1)$
\wrong   $(x + 3)^{24}(x^3 + 3x^2 + 51x + 3)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problem (5 pts.)  Compute the derivative of $f(x) = \displaystyle{{x^2 + x \over x^2 - 2}}$

\correct $\displaystyle{{-x^2 - 4x - 2 \over (x^2 - 2)^2}}$
\wrong   $\displaystyle{{4x^3 + 3x^2 - 4x - 2 \over (x^2 - 2)^2}}$
\wrong   $\displaystyle{2x + 1 \over 2x}$ 
\wrong   $\displaystyle{-x^2 - 4x - 2 \over x^2 - 2}$
\wrong   $\displaystyle{4x^2 + 2x \over (x^2 - 2)^2}$

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\problem (5 pts.) Let $f(x)$ and $g(x)$ be functions with
$$f(2) =1,\ f'(2) = {1 \over 3},\ f(3) = 2,\ f'(3) = 1,\ g(2) = 3,\ g'(2) = -3$$
Compute the value of the derivative of $f(g(x))$ at $x = 2$.

\correct $-3$
\wrong   $-1$
\wrong   $-6$
\wrong   $3$
\wrong   $1$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 



\problem (20 pts. - Partial Credit) \hskip 0.2cm


Sketch the graph of the function $f(x) = x^3 - 6x^2 + 9x + 1$, indicating all
relative extreme points and inflection points.

\vskip 15cm
\vfill\eject
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\problem (20 pts. -Partial Credit)  \hskip 0.2cm
 

 An open rectangular box is to be made from an 8-in $\times$ 15-in piece of
cardboard by cutting squares from the corners and folding up the sides. What are
the dimensions of the box of largest volume that can be made this way? 

\vskip 15cm
\vfill\eject
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\endtest


