%% MATH 131-01 Final Exam, Fall 2007
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\begin{document}

\begin{center}

\textbf{College of the Holy Cross, Fall Semester, 2007} \\
\textbf{MATH 131, Section 01, Final Exam} \\
\textbf{Saturday, December~8, 2:30~PM} \\
Professor Little
\end{center}


\noindent{\bf Your Name}: \Name
\vskip 0pt plus .3fil

\noindent
{\it Directions.} {\it You must show all work required to derive
each answer for full credit}.  Please place final answers in the boxes provided. 
Use the back of the preceding page if you need more
space for scratch work.  There are 200 possible points, distributed as shown below.

\vfil

\noindent
{\it Please do not write in the space below}

\begin{center}
\begin{tabular}{|l||r|}
\hline
Problem & Points/Poss    \Strut \\
\hline\hline
1       & \hfil/\  20\hfil \Strut \\
\hline
2       & \hfil/\  25\hfil \Strut \\
\hline
3       & \hfil/\  15\hfil \Strut \\
\hline
4       & \hfil/\  25\hfil \Strut \\
\hline
5       & \hfil/\  15\hfil \Strut \\
\hline
6       & \hfil/\  25\hfil \Strut \\
\hline
7       & \hfil/\  25\hfil \Strut \\
\hline
8       & \hfil/\  20\hfil \Strut \\
\hline
9       & \hfil/\  20\hfil \Strut \\
\hline
10       & \hfil/\  10\hfil \Strut \\
\hline\hline
Total   & \hfil/200\hfil \Strut \\
\hline
\end{tabular}
\end{center}

\vfil

\[
\mathscr{H\!A\!P\!P\!Y\ \  H\!O\!L\!I\!D\!A\!Y\!S}!
\]

\eject



\noindent
1. [5~points each]\quad Circle the number of the graph
showing each of the following functions.  Note that there is one more
graph than there are formulas.  


\begin{itemize}
  \item[(a)] $\ds f(x) = 2 - 2\sin(\pi x)$\hfill
    ~I\quad ~II\quad ~III\quad
    ~IV\quad ~V

    \vskip 0.2truein


  \item[(b)] $\ds f(x) = \frac{x^2}{x^2-1}$\hfill
    ~I\quad ~II\quad ~III\quad
    ~IV\quad ~V

    \vskip 0.2truein


  \item[(c)] $\ds f(x) = \ln(\sqrt{x})$\hfill 
    ~I\quad ~II\quad
    ~III\quad ~IV\quad ~V

    \vskip 0.2truein


  \item[(d)] $\ds f(x) = e^x + e^{-x}$\hfill
    ~I\quad ~II\quad ~III\quad
    ~IV\quad ~V

    \vskip 0.2truein

\end{itemize}

\begin{center}
I: \includegraphics[width=1.5truein,height=1.5truein]{../1I.ps} 
II: \includegraphics[width=1.5truein,height=1.5truein]{../1II.ps}
\end{center}
\begin{center}
III: \includegraphics[width=1.5truein,height=1.5truein]{../1III.ps} 
IV: \includegraphics[width=1.5truein,height=1.5truein]{../1IV.ps}
\end{center}
\begin{center}
V: \includegraphics[width=1.5truein,height=1.5truein]{../1V.ps} 
\end{center}


\vfil\eject
\noindent 
2.  
\begin{itemize}
\item[(a)] [5~points] What is the domain of the function $f(x) = \sqrt{x^2 - 4}$?

\MyAnswer{Domain: }{$\phantom{A very very very very long answer here}$}%
\vskip 1.0truein
\item[(b)] [10~points] Find a formula for the inverse function of $f(x) = \frac{x}{3} - 1$ and
plot $y = f(x)$ and $y = f^{-1}(x)$ on the same coordinate axes below.

\MyAnswer{Inverse function:}{$\phantom{A very very very very long answer here}$}%
\vskip 3.0truein
%\begin{center}
%\includegraphics[width=2.0truein,height=2.0truein]{../2b.ps} 
%\end{center}
\item[(c)] [5~points] Find the interval of all real 
$x$ satisfying $-\pi/4 < \tan^{-1}(x) < \pi/3$. 

\MyAnswer{Interval:}{$\phantom{A very very very very long answer here}$}%
\vskip 1.0truein
\item[(d)] [5~points] Express as a single logarithm: $\ln(6) - \ln(12) + 5\ln(2)$. 
  
\MyAnswer{Single log:}{$\phantom{A very very very very long answer here}$}%
\vskip 1.0truein
\end{itemize}

\vfil
\eject
\noindent
3.  Compute the following limits [5~points each].  Any legal method is OK.
\begin{itemize}
\item[(a)]  $\ds \lim_{x\to 3^-} \frac{|x-3|}{x^2 - 2x - 3}$

\MyAnswer{}{$\phantom{A very very very very long answer here}$}%
\vskip 2.0truein

\item[(b)]  $\ds \lim_{x\to -1} \frac{x^3-x}{x^2 + 5x + 4}$

\MyAnswer{}{$\phantom{A very very very very long answer here}$}%
\vskip 2.0truein

\item[(c)]  $\ds \lim_{x\to +\infty} \frac{\ln(x^2)}{x^{3/2}}$

\MyAnswer{}{$\phantom{A very very very very long answer here}$}%
\vskip 2.0truein
\end{itemize}

\vfil 
\eject
\noindent
4. 
\begin{itemize}
\item[(a)] [5~points] State the limit definition of the derivative.

\MyAnswer{$f'(x)$ = }{$\phantom{A very very very very long answer here}$}%
\vskip 1.0truein

\item[(b)] [10~points] Use the definition to compute $f'(x)$ for
$\ds f(x) = \frac{1}{x-1}$.

\vskip 3.0truein
\MyAnswer{$f'(x)$ = }{$\phantom{A very very very very long answer here}$}%

\item[(c)] [10~points] Find the equation of the tangent line
to the graph $y = \frac{1}{x-1}$ at the point $(5,1/4)$.


\MyAnswer{Tangent line:}{$\phantom{A very very very very long answer here}$}%
\vskip 2.0truein
\end{itemize}
\vfil\eject

\noindent
5.  The following graph shows $y = g'(x)$, the derivative
of some function $g(x)$.  
\begin{center}
\includegraphics[width=2.0truein,height=2.0truein]{../7.ps} 
\end{center}
\centerline{\it Note: this is $y=g'(x)$, not $y = g(x)$.}
\begin{itemize}
\item[(a)] [5~points] At which of the points $x = 0,a,b,c,d,e,f$
does the function $g(x)$ have local maxima? 

\MyAnswer{Local maxima:}{$\phantom{A very very very very long answer here}$}%
\vskip 1.0truein

\item[(b)] [5~points] At which of the points $x = 0,a,b,c,d,e,f$
does the function $g(x)$ have local minima? 

\MyAnswer{Local minima:}{$\phantom{A very very very very long answer here}$}%
\vskip 1.0truein

\item[(c)] [5~points] On the interval $(a,b)$, is $y = g(x)$ concave up or down?

\MyAnswer{Concave:}{$\phantom{An answer here}$}%
\vskip 1.0truein

\end{itemize}
\vfil\eject

\noindent
6.  Compute the following derivatives using the derivative rules.
You need {\it not} simplify. [5~points each]
\begin{itemize} 
\item[(a)]  $f(x) = 3x^5 + \frac {5}{x^3} + e^x$.

\MyAnswer{$f'(x) = $}{$\phantom{A very very very very long answer here}$}%
\vskip 1.0truein

\item[(b)]  $\ds g(x) = \frac{\sin(x) + 1}{\sin^{-1}(x) + 1}$.

\MyTallAnswer{$g'(x) = $}{$\phantom{A very very very very long answer here}$}%
\vskip 1.0truein

\item[(c)] $\ds h(x) = \sqrt{\ln(x) + 4\cos(\tan(x))}$.

\MyTallAnswer{$h'(x) = $}{$\phantom{A very very very very long answer here}$}%
\vskip 1.0truein

\item[(d)] $\ds j(x) = (x^2 + 3)^{(x + 2)}$.   

\MyTallAnswer{$j'(x) = $}{$\phantom{A very very very very long answer here}$}%
\vskip 1.0truein

\vfil\eject

\item[(e)] Find $\ds \frac{dy}{dx}$ if $x^3 + y^3 = 18xy$.

\MyTallAnswer{$\ds \frac{dy}{dx}=$}{$\phantom{A very very very very long answer here}$}%
\vskip 2.0truein

\end{itemize}
\vskip 10pt
\noindent
7.  All parts of this question refer to the function 
$$f(x) = \frac{(x+1)^2}{x^2 + 1}.$$
\begin{itemize}
\item[(a)] [5~points] Compute $f'(x)$ and simplify.

\MyTallAnswer{$f'(x) = $}{$\phantom{A very very very very long answer here}$}%
\vskip 2.0truein

\rightline{\it Problem 7 continues on next page}
\vfil\eject

\item[(b)] [10~points] Find all critical numbers of $f$ and determine the
intervals on which $f$ is increasing and decreasing.

\MyAnswer{Critical numbers:}{$\phantom{A very very very very long answer here}$}%

\MyAnswer{Increasing: }{$\phantom{A very very very very long answer here}$}%

\MyAnswer{Decreasing: }{$\phantom{A very very very very long answer here}$}%
\vskip 1.5truein

\item[(c)] [5~points] {\bf Given:} the second derivative in simplified
form is
$$f''(x) = \frac{4x(x^2 - 3)}{(x^2+1)^3}.$$ 
Determine all points of inflection and the intervals where $y = f(x)$ is concave
up and concave down.

\MyAnswer{Inflections:}{$\phantom{A very very very very long answer here}$}%

\MyAnswer{Concave up: }{$\phantom{A very very very very long answer here}$}%

\MyAnswer{Concave down: }{$\phantom{A very very very very long answer here}$}%
\vskip 1.5truein

\item[(d)] [5~points] Find the absolute maximum and minimum of $f(x)$ on the interval $[0,5]$.   
\end{itemize}
\MyAnswer{Absolute max:}{$\phantom{A long answer here}$}%

\MyAnswer{Absolute min: }{$\phantom{A long answer here}$}%
\vskip 1.5truein
\vfil
\eject
\noindent
8. [20~points]  A potter forms 20 cm${}^3$ of clay into 
a cylindrical tube.  As she rolls the tube on a flat surface, 
its length $L$ increases and its radius $r$ decreases but the quantity
of clay stays constant.  If the length is
increasing at the rate of $.1$ cm/sec, find the rate at which
the radius is changing when the radius is $1$cm.  ({\it Note:} the 
volume of a cylinder of radius $r$ and height $h$ is $V = \pi r^2 h$.)   

\vskip 7.0truein
\MyAnswer{Rate:}{$\phantom{A very very very very long answer here}$}%
\vfil\eject

\noindent
9.  [20~points] A rectangular swimming pool with sides $x,y$
is to be built with an area of 1800 square feet.  The owner
wants 5-foot wide decks along either side and 10-foot wide
decks at the two ends.  Find the dimensions of the rectangular
lot of smallest area on which such a swimming pool can be built.
Be sure you say how you know your answer gives a minimum of the area of the lot.
\vskip 7.0truein
\MyAnswer{Dimensions of lot:}{$\phantom{A very very very very long answer here}$}%
\vfil\eject
\noindent
10.  On and near the surface of the moon, the acceleration of gravity is $-1.66$ meters
per second per second.

\begin{itemize}
\item[(a)] [5~points]  An astronaut throws a moon rock upward
from a height of 1.5 meters above the surface 
with an initial velocity of $3$ meters per second.
Find the height of the rock above the surface as a function of time $t$
in seconds. 

\vskip 1.8truein
\MyAnswer{Height:}{$\phantom{A very very very very long answer here}$}%

\item[(b)] [3~points]  How long does it take for the rock from part 
a to fall back to the surface of the moon?

\vskip 1.8truein
\MyAnswer{Time:}{$\phantom{A very very very very long answer here}$}%

\item[(c)] [2~points]  What is the velocity of the rock from part a when 
it strikes the surface of the moon?  

\vskip 1.5truein
\MyAnswer{Velocity:}{$\phantom{A very very very very long answer here}$}%
\end{itemize}

\end{document}
