%m175 s02f.tex
\documentclass[10pt,twoside]{article}
% =========================================================================
% document style changes
% =========================================================================

\usepackage{amsmath}                    % AMS math packages
\usepackage{amssymb}                    %
\usepackage{epsfig, pstricks}                     % To include postscript figures 
%\usepackage{subfigure}
%\usepackage{wrapfig}
%\usepackage{shadow}

\setlength{\parindent}{0in}             % Control margins and amount of text
\setlength{\topmargin}{0in}            %%  Changed from {-.75in} 
\setlength{\oddsidemargin}{-.2in}  %  Changed from {-.15in} 
\setlength{\evensidemargin}{-.2in}  %  Changed from {-.15in} 
\setlength{\textheight}{10in} 
\setlength{\textwidth}{7.0in} 
\pagestyle{empty}                       % No page numbers

\newcommand{\hl}{\hline}                % Shortcut commands
\newcommand{\hsp}{\hspace{0.25in}}
\newcommand{\vsp}{\vspace{0.25in}}
\newcommand{\ds}{\displaystyle}         %% \newcommand{\ds}[1]{\displaystyle{#1}}
% Shortcuts
\newcommand{\bdoc}{\begin{document}}
\newcommand{\edoc}{\end{document}}
\newcommand{\ben}{\begin{enumerate}}
\newcommand{\een}{\end{enumerate}}
\newcommand{\bet}{\begin{tabular}}
\newcommand{\eet}{\end{tabular}}
\newcommand{\bef}{\begin{figure}}
\newcommand{\eef}{\end{figure}}

% Symbols
\newcommand{\lra}{\Longleftrightarrow}
\newcommand{\re}{\mbox{I\!R}}
\newcommand{\dydx}{\ds\frac{dy}{dx}}
\newcommand{\ui}{\hat{\imath}}
\newcommand{\uj}{\hat{\jmath}}
\newcommand{\uk}{\hat{k}}
\newcommand{\tri}{\ds{\int\!\!\int\!\!\int}}
\newcommand{\doub}{\ds{\int\!\!\int}}

\pagestyle{myheadings}
\markright{Math 175 \hfill May 6, 2002 \hfill Final Exam \hfill
  Name:\underline{\hspace{2in}} }

\begin{document}
% 
\begin{center}
  \textbf{Final Exam} \\[3pt]
   \begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c||c|}
   \hline
   Prob. & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 &10 
          & 11 & 12 & 13 & 14 & 15 & 16 &  \\[2pt]
   \hline
   Value & 24 & 10 &9& 17 &8 & 5 & 10 & 5 & 10 & 21
         & 9 & 18 & 12 & 10&12 & 20 &   200 \\
   \hline
   Points\rule[-4pt]{0pt}{18pt} &   &&&  &  &   &  &&  & & & &  &  &  & &   
   \hspace{24pt} \\
   \hline
  \end{tabular}
\end{center}

% =========================================================================

\begin{enumerate}
  \item Find derivatives of the following functions:
    \begin{enumerate}
      \item \(\ds{ u= 4 \sqrt{x} - \frac{3}{x}+\sqrt 3 }\)
        \vfill
      \item \( g(x)=\ds\frac{3}{(5-2x^2)^{3/4}}-4  \)
        \vfill
%        \vspace{1in}
      \item \( h(y)=\ds\frac{3y}{1-5y} \)
        \vfill
      \item \( f(x)= (x-1)^3(x^2-2)^4 \)
        \vfill
    \end{enumerate}
  \item Find the {\bf second} derivative of \(\ds{y = \frac{1-3x}{1+4x} }\) .
	 \vfill
	 \vfill
\newpage
  \item Find the equation of the line tangent to the curve 
    \( y= 3\root 3\of{3-8x} \) at  $(-3,9)$ .
    \vspace{2in}

  \item Evaluate the following limits :
    \begin{enumerate}
      \item \(\ds{\lim_{x\to -3} \frac{2x+5}{x-1}   }\)
          \vspace{1in}
      \item \(\ds{\lim_{x\to 2} \frac{5x-10}{x^2-4}   }\)
          \vspace{1in}
      \item \(\ds{\lim_{x\to 0} \frac{x}{2x+3}   }\)
          \vspace{1in}
      \item  \(\ds{\lim_{\Delta x\to 0} 
          \frac{(x+\Delta x)^2-5-(x^2-5)}{\Delta x}   }\)
    \end{enumerate}
\newpage
  \item Find $\dydx$ for \( x^2+y^2=5x +2xy \)
    \vspace{2in}

  \item During each cycle, the velocity $v$ (in mm/s) of a piston is 
    \( v=6t-6t^2\), where $t$ is the time (in s).  Find the
    acceleration $a$ of the piston after 1 second.
    \vspace{1.2in}

  \item Find the $x$-coordinates of relative maximums and relative minimums, if
    any, of \(\ds{ y=\frac{x^4}{4} -\frac{2x^3}{3}+8 }\).  Label each answer as
    either a maximum, a minimum or neither.  {\bf Please label completely any
    figures you draw as part of your work} \vfill

\newpage
  \item Find the vertical asymptote(s) for 
    \( y=\ds\frac{3}{\sqrt{4-x^2}}\)
    \vspace{.6in}


   \item Find the $x$ coordinate of the inflection point, if any, 
	for \(\ds{y=\frac{x^5}{10}-\frac{2x^4}{3}+\frac{4x^3}{3}+7 }\).
     \vspace {1.8in}

   \item Evaluate the indefinite integrals:
    \begin{enumerate}
    \item \(\ds{ \int \sqrt{u}(u^2+2)  \,du}\)
      \vfill
    \item \(\ds\int (4x-18x^2)(x^2-3x^3-4)^2\,dx\)
      \vfill
    \item \(\ds{\int 3t^2 \root3\of{1-3t^3}  \,dt}\)
      \vfill
   \end{enumerate}
\newpage
  \item In a physics experiment, the the rate of change
    of the pressure is given by \( \ds\frac{dP}{dt}=100(t+1)^{1/4} \) it is also
    known that \(P(0)=0\).  Find a formula for \(P(t)\).
    \vspace{3in}
 \item Evaluate the following definite integrals:
    \ben
    \item \(\ds\int_1^3 x(3x-4) \,dx\)
      \vfill
    \item \(\ds \int_0^1 (t^3+4t)\root 4\of{t^4+8t^2+16} \,dt \)
      \vfill
    \een
\newpage
   \item Find the area of the region between the following functions:\\
    \( y^2=x,\hsp y=x-2\)
\begin{figure}[h]
\pspicture(0,0)(18,5)
%%%%   \psgrid[subgriddiv=1,griddots=10,gridlabels=7pt](0,0)(18,5)
        \rput[bl](5,-4){\includegraphics[angle=-90,width=20cm]{../figs/s02fa.ps}}
        \rput[r](13.7,5.1){$y^2=x$}
        \rput[l](14.6,3.8){$y=x-2$}
\endpspicture
\end{figure}
        \vfill

\item Given \( f(x)=\sqrt[3]{3x-1}\).
  \begin{enumerate}
    \item Find the linearization, $L(x)$, of $f(x)$ at $a=3$.
      \vspace{1.8in}
    \item Use your answer to (a) to approximate $\sqrt[3]{8.3}$ ( {\small notice
that  $f(3.1) =\sqrt[3]{8.3} $} )
      \vspace{1in}
  \end{enumerate}

\newpage
\item For the following Optimization problem
   be sure you answer the question asked. Also be sure to show that
  your answer is a maximum or a minimum.  (That is use the First
  Derivative Test or the Second Derivative Test in your answer.) 

  A rancher has 2400ft of fencing and wants to fence off a rectangular field
  that borders a straight river (need no fence along river). What are the
  dimensions of the field that has the largest area?
\begin{figure}[h]
\pspicture(0,0)(18,5.6)
%%%  \psgrid[subgriddiv=1,griddots=10,gridlabels=7pt](0,0)(18,5)
        \rput[bl](1,-6.3){\includegraphics[angle=-90,width=25cm]{../figs/s02f.ps}}
        \rput[r](13.6,5.55){$y$}
        \rput[l](16.5,4.2){$x$}
\endpspicture
\end{figure}
      \vfill

\newpage
\item Find the volume of the solid generated by revolving the region
  bounded by the given curves about the given axis.
  \begin{enumerate}
    \item \( y^2=x, y=4, x=0; \) about the $y$-axis (disks)

\begin{figure}[h]
\pspicture(0,0)(18,5.6)
%%%   \psgrid[subgriddiv=1,griddots=10,gridlabels=7pt](0,0)(18,5)
        \rput[bl](5,-4){\includegraphics[angle=-90,width=20cm]{../figs/s02fb.ps}}
        \rput[r](14.3,5.2){$y=4$}
        \rput[l](14.4,3.5){$y^2=x$}
\endpspicture
\end{figure}
      \vfill
    \item $y=\sqrt{x^2-1}, y=0, x=3;$ about the $y$-axis (shells)

\begin{figure}[h]
\pspicture(0,0)(18,5.7)
%%%   \psgrid[subgriddiv=1,griddots=10,gridlabels=7pt](0,0)(18,5)
        \rput[bl](5,-4){\includegraphics[angle=-90,width=20cm]{../figs/s02fc.ps}}
        \rput[l](16.6,2.2){$x=3$}
        \rput[l](13.5,3.8){$y=\sqrt{x^2-1}$}
\endpspicture
\end{figure}
      \vfill
  \end{enumerate}


\end{enumerate}
%{\tiny \jobname .dvi}
\end{document}