%% MATH 131-01 Final Exam, Fall 2004
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\begin{document}

\begin{center}

\textbf{Holy Cross College, Fall Semester, 2004} \\
\textbf{MATH 131, Section 01, Final Exam} \\
\textbf{Monday, December~13, 2:30~PM} \\
Professor Little
\end{center}

\vfil

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

\noindent
{\it Instructions.} Clearly mark your answers, and show work on
the test itself. Use the back of the preceding page if you need more
space for scratch work. {\it You must show all work for full
credit}, but please place answers in the boxes provided where appropriate.

\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/\  20\hfil \Strut \\
\hline
3       & \hfil/\  20\hfil \Strut \\
\hline
4       & \hfil/\  15\hfil \Strut \\
\hline
5       & \hfil/\  25\hfil \Strut \\
\hline
6       & \hfil/\  20\hfil \Strut \\
\hline
7       & \hfil/\  30\hfil \Strut \\
\hline
8       & \hfil/\  15\hfil \Strut \\
\hline
9       & \hfil/\  20\hfil \Strut \\
\hline
10       & \hfil/\  15\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 ``extra''
graph that does not match any of these functions.


\begin{itemize}
  \item{(a)} $f(x) = e^{-x} + 2$\hfill
    ~I\quad ~II\quad ~III\quad
    ~IV\quad ~V

    \vskip 0.2truein


  \item{(b)} $f(x) = x^3 - 4x$\hfill
    ~I\quad ~II\quad ~III\quad
    ~IV\quad ~V

    \vskip 0.2truein


  \item{(c)} $f(x) = 2\cos(2\pi x)$\hfill 
    ~I\quad ~II\quad
    ~III\quad ~IV\quad ~V

    \vskip 0.2truein


  \item{(d)} $f(x) = \frac{1}{x^2 - 9}$\hfill
    ~I\quad ~II\quad ~III\quad
    ~IV\quad ~V

    \vskip 0.2truein

  \item{(e)} Give a formula of a function that matches the graph you 
     did not circle in parts (a) - (d).

\end{itemize}





\vfil\eject
\noindent 
2. [20~points] One of the functions given in the following table is linear
and the other is exponential.  Find formulas of the appropriate
type for each.

\[
\begin{array}{|c||r|r|r|r|r|}
\hline
x          & 1 & 2 & 3& 4& 5\Strut \\
\hline
f(x)       & 1.2 & 0.6 & 0.3 & 0.15 & 0.075 \Strut \\
\hline
g(x)       & -2.3& -0.6& 1.1 & 2.8 & 4.5\Strut \\
\hline
\end{array}
\]

\vfill
\Answer{$f(x) = $}
\Answer{$g(x) = $}

\eject
\noindent
3.  
\begin{itemize}
\item{(a)} [15~points]  The depth of water in a tank oscillates sinusoidally
once every 4 hours.  The smallest depth is 2 feet and the maximum depth
is 5 feet, which occurs at $t = 0$.  Find a formula for the depth $d(t)$
if $t$ is the time in hours.
\vskip 3.0truein
\Answer{$d(t) = $}

\sep

\item{(b)} [5~points]  How fast is the depth changing at $t = 1.3$ hours?
Is it increasing or decreasing?
\vfill
\Answer{Rate of change $=$}
\end{itemize}
\eject

\noindent
4.  Compute the following limits [5~points each].  Any legal method is OK.
\begin{itemize}
\item{(a)}  $\ds \lim_{x\to 1} \frac{x^2 + 1}{x - 3}$
\vskip 1.7truein
\Answer{Limit $=$}
\sep
\item{(b)}  $\ds \lim_{x\to 2} \frac{x^2 - 4x + 4}{\cos(x-2) - 1}$
\vskip 1.7truein
\Answer{Limit $=$}
\sep
\item{(c)}  $\ds \lim_{x\to \infty} \frac{5x^2 - x + 21}{3x^2 + x + 1}$
\vskip 1.7truein
\Answer{Limit $=$}
\end{itemize}
\eject

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

\Answer{$f'(x)=$}

\sep

\item{(b)} [10~points] Use the definition to compute $f'(x)$ for
$f(x) = \sqrt{x}$.
\vskip 3.0truein

\sep

\item{(c)} [10~points] Find the equation of the tangent line
to the graph $y = \sqrt{x}$ at the point $[4,2]$.
\vskip 2.0truein
\Answer{Tangent line $=$}
\end{itemize}
\eject

\noindent
6.  Compute the following derivatives using the derivative rules.
You need not simplify. [5~points each]
\begin{itemize} 
\item{(a)}  $f(t) = t^3 - \frac {1}{{\root 4\of t}} + \pi^t$.
\vskip 1.5truein
\Answer{$f'(t) = $}

\sep

\item{(b)}  $\ds g(x) = \frac{x^2 - 2}{\cos(x) + 1}$

\vskip 1.5truein
\Answer{$g'(x) = $}

\sep

\item{(c)} $\ds h(z) = \ln(4z^2 + 2e^{\arctan(z)})$

\vskip 1.5truein
\Answer{$h'(z) = $}

\sep
\eject

\item{(d)} Find $\frac{dy}{dx}$ if $x^2 y - 2y^3 = 3$.

\vskip 1.5truein
\Answer{$\frac{dy}{dx} = $}
\end{itemize}

\sep
\noindent
7.  Consider the family of curves defined by $y = f(x) = x^4 + 2ax^2$, where
$a$ is any fixed real number.  

\begin{itemize}
\item{(a)}  [10~points]  Find the {\it critical points} of $f$, construct 
a sign diagram for $f'(x)$ in the case that $a < 0$.  Which of your 
critical points are local maxima and which are local minima?

\vskip 2.5truein

\sep 
\item{(b)}  [10~points] Repeat part a, but assume now that $a > 0$.  

\end{itemize}
\eject

\begin{itemize}
\item{(c)} [10~points] How many different {\it inflection points} does
the graph $y = f(x)$ have if $a < 0$?  Explain.
\end{itemize}

\vskip 2.0truein

\sep

\noindent
8. [15~points]  A cubical block of dry ice (solid $CO_2$) is 
evaporating and losing volume at the rate of $10~cm^3/min$.  How 
fast are the sides of cube shrinking when the block has volume 
$125~cm^3$?  Give the units of your answer.

\vfill
\Answer{Answer:}

\eject
\noindent
9.  [20~points] Ship A travels along the path given by the parametric
curve $x = t, y = t^2$.  At the same time, ship B travels along
the curve $x = t, y = 4t-5$.  At what time are the two ships
{\it closest to one another}?
\vfill
\Answer{$t = $}
\eject

\noindent
10.  
\begin{itemize}
\item{(a)} [7.5~points]  Compute the left- and right-hand sums
for the function $f(t) = e^{-t^2}$ on the interval $[0,1]$, using
$n = 4$ equal subdivisions.
\vskip 4.0truein
\Answer{LHS $=$} $\quad\quad$ \Answer{RHS $=$}

\eject
\item{(b)} [7.5~points] Consider the graph $y = f(t)$ given below.
Compute the average value $\overline{y}$ of $f$ on the 
interval $0\le t \le 5$, given that the area marked $A$ 
is 25 square units, and the areas marked $B$ and $C$ are both 16 square units.

\vskip .5truein

\centerline{\psfig{figure=10.ps,height=3.0in,width=3.0in,angle=-90}}

\vskip .5truein
\Answer{$\overline{y} = $}
\end{itemize}
\end{document}
