%% Math 131-01 Final Exam, Fall 2003

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\usepackage{math131}

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\markright{Math 131-01, Final Exam}

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\def\Answer#1{$\phantom{.}$\hfill
        #1\quad\fbox{$\phantom{\Big|Put answer here}$}}
\def\ShortAnswer#1{$\phantom{.}$\hfill
        #1\quad\fbox{$\phantom{\Big|answer}$}}

\def\Blank{$\underline{\phantom{Please write your name in this
        space}}$} 
\def\Strut{$\vphantom{\Big|}$}

\begin{document}

\begin{center}

\textbf{Holy Cross College, Fall Semester, 2003} \\
\textbf{Math 131, Section 01, Final Exam} \\
\textbf{Tuesday, December~16, 8:30~AM} \\
Professor Hwang
\end{center}

\vfil

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

\paragraph{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. \textbf{You must show all work for full
  credit}.

\vfil

\noindent\textbf{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/\  20\hfil \Strut \\
\hline
5       & \hfil/\  20\hfil \Strut \\
\hline
6       & \hfil/\  20\hfil \Strut \\
\hline
7       & \hfil/\  15\hfil \Strut \\
\hline
8       & \hfil/\  20\hfil \Strut \\
\hline
9       & \hfil/\  20\hfil \Strut \\
\hline\hline
Total   & \hfil/175\hfil \Strut \\
\hline
\end{tabular}
\end{center}

\vfil

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

\eject

\be

%% Differentiation freebie

\item{} [10~points each]\quad Let 
  $f(x)=e^{-x}\cos x=\ds\frac{\cos x}{e^x}$.

\bd

\item{(a)} Use the product rule to compute~$f'(x)$. Simplify your
  answer.

  \vfil\sep
  
\item{(b)} Use the quotient rule to compute~$f'(x)$. Simplify your
  answer.

\ed

\vfil\eject  

%% Linear approximation

\item{} [20 points]\quad A small portion of a graph $y=f(x)$ (with
  horizontal and vertical scales as shown) is given below. Determine
  (with justification) which of the following functions might be
  depicted, and circle your choice(s).
  \[
  f_1(x)=-\sin x,\qquad\qquad
  f_2(x)=\ln (1-x),\qquad\qquad
  f_3(x)=1-e^x
  \]
\vfil\vfil

  \begin{center}
    \input{final-01f.eepic}
  \end{center}

\eject

%% Left- and right-hand sums, approximation, accuracy

\item{} [10~points each]\quad The graph of $f(t)=2-e^{-t^2/2}$ on the
  interval $[0,2]$ is shown.

  \begin{center}
    \input{final-01a.eepic}
  \end{center}

\bd

\item{(a)} On the graph above, sketch the left- and right-hand
  rectangles for 4~intervals of equal length, and shade the area
  representing the difference of the upper bound and the lower bound.

  \sep
  
\item{(b)} Suppose left- and right-hand sums with $n$~intervals are
  used to compute upper and lower bounds for $\int_0^2 f(t)\,dt$. Find
  the smallest number,~$n$, of intervals so that
  \[
  \textrm{upper}-\textrm{lower}\leq0.02=\textstyle\frac{2}{100}
  \]
  Your answer to part~(a) should help you recall the formula to use.
  Do not attempt to compute the corresponding sums!

\ed

\vfil

\eject

%% Optimization

\item{} [20~points]\quad A rectangular poster will consist of a
  printed message, of area 150~square inches, surrounded by a border.
  The border is to be 1~inch wide along the sides and bottom, and
  2~inches wide along the top. What dimensions of the poster minimize
  its total area?

  \begin{center}
    \input{final-03g.eepic}
  \end{center}

  \vfil

  \ShortAnswer{Total width:}  \ShortAnswer{Total height:}
  \eject

%% Limits, continuity, l'Hopital's rule

\item{} Let $\ds f(x)=\frac{1-\cos x}{x}$ for $x\neq0$.

\bd

\item{(a)} [5~points]\quad What value for~$f(0)$ makes~$f$ continuous
  at $x=0$?

  \vfil

  \ShortAnswer{$f(0)=$}
  \sep

\item{(b)} [10~points]\quad Compute $f'(x)$ for $x\neq0$.

  \vfil\vfil

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

\item{(c)} [5~points]\quad Find $\lim\limits_{x\to0}f'(x)$.

  \vfil

  \ShortAnswer{$\lim\limits_{x\to0}f'(x)=$}

  \ed

\eject

%% Definite integrals and signed areas, average value

\item{} [10 points each]\quad Each part of this question refers to the
  function pictured below. The areas enclosed are $A=25$, $B=16$, and
  $C=16$.

  \begin{center}
    \input{final-03b.eepic}
  \end{center}

\bd

\item{(a)} Find $\ds\int_0^3 f(x)\,dx$ (show all work for full credit)

  \vfil\vfil

  \ShortAnswer{$\ds\int_0^3 f(x)\,dx=$}
  \sep

\item{(b)} Find the average value of~$f$ on the interval~$[0,5]$ \\
  Suggestion: Start by writing down the average value in terms of an
  integral. 

  \ed

  \vfil

  \ShortAnswer{Average value for $0\leq x\leq5$:}
  \eject

%% Related rates

\item{} The top of a 13-foot ladder rests against a vertical wall, and
  the foot of the ladder slides away from the wall at 3~ft/sec. Let
  $x$~be the distance from the wall to the foot of the ladder in feet,
  and $y$~the height of the top of the ladder in feet. Both parts of
  this question concern the instant at which the base of the ladder is
  6~ft from the wall.

  \begin{center}
    \input{final-01c.eepic}
  \end{center}

  \bd
  
\item{(a)} [5~points]\quad How high is the top of the ladder? Express
  your answer \textbf{symbolically}.

  \vfil

  \ShortAnswer{Height of ladder:}
  \sep
  
\item{(b)} [10~points]\quad How rapidly is the top of the ladder
  dropping? For full credit, express your answer 
  \textbf{both symbolically and numerically}, including units. No
  credit for answers off by more than a factor of~10.

  \ed

  \vfil\vfil

  \ShortAnswer{Speed (symbolic):}  \ShortAnswer{Speed (numerical):}

  \eject

%% Derivatives and monotonicity
  
\item{} [10~points each]\quad Let $a>0$ be constant, and define
  $f(x)=(x-a)\sqrt[3]{x}$ for $x$~real.  

\bd

\item{(a)} Compute $f'(x)$ and make a sign diagram. Find the critical
  point(s) of~$f$, and classify as local maximum, local minimum, or
  neither. (Please see hints below.)

  \vfil\sep

\item{(b)} Compute $f''(x)$ and make a sign diagram. Find the
  inflection point(s) of~$f$, if any.

  \ed

  \vfil

  \sep

  \parbox[b]{3.25in}{Hints: Much of this problem is algebra management.
    Unless you expand or factor when appropriate, you are likely to
    get bogged down.  You will need to use properties of exponents,
    since your calculator may not be able to handle all the tasks you
    require.}  
  \input{final-01d.eepic}
  
  For reference in checking your answers, the graph for a typical
  value of~$a$ is shown.  Your answers should be deduced from
  calculation, not from the appearance of the graph.

\eject

%% Implicit differentiation

\item{} [10~points each]\quad A curve is defined implicitly by the
  equation $x^3-3xy^2=2$.

\bd

\item{(a)} Find $\ds\frac{dy}{dx}$ in terms of $x$~and $y$

  \vfil\vfil
  \Answer{$\ds\frac{dy}{dx}=$}
  \sep

  \vspace*{-2in}  
\item{(b)} \parbox[t]{3.25in}{Use part~(a) to find the equation
    $y=mx+b$ of the line tangent to the curve at the point
    $(x_0,y_0)=(2,-1)$, and sketch this line.

    \bigskip
  
    Hints: The horizontal and vertical scales in the figure are the
    same.  You must determine the location of the origin and the size
    of the grid squares. For this, find the coordinates of the points
    on the curve where the tangent line is horizontal, then locate
    these points visually.}  
\input{final-03e.eepic}

  \ed

  \vfil\vfil\vfil
  \Answer{Tangent line at $(2,-1)$:}

\ee

\end{document}

