%% Math 131 Midterm 2, Fall 2004
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%\markright{Math 131, Midterm 2}

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\begin{document}
\centerline{MATH 131, section 1 -- Practice Questions for Exam 2}
\centerline{October 17, 2007}

\begin{enumerate}
\item{}  All parts of this question refer to the parametric
curve $x = 3 \cos(2 t)$, $y = 5 \sin(2 t)$.

\begin{enumerate}
\item Eliminate the parameter $t$ and find a Cartesian equation for
this curve.
\item What portion of the curve is traced out for $0\le t \le \frac{\pi}{2}$, 
and in which direction is the curve being traced?
\item What would change in your answer to part b if the curve above was
replaced by $x = 3 \cos(-2 t)$, $y = 5 \sin(-2 t)$
\end{enumerate}
  
\item{} Use the sum, product, and/or quotient 
  rules to compute the following derivatives. You may use any correct
  method, but must show work and simplify your answers for full
  credit.

\begin{enumerate}
  \item $\ds\frac{d}{dx}\Bigl(5x\sqrt{x}-\frac{2}{x^3}+11x-4\Bigr)$

  
  \item $\ds\frac{d}{dt} (t^2e^{t})$

  
  \item $\ds\frac{d}{dz}\frac{z^2 - 2z + 4}{z^2 + 1}$


  \item $\ds\frac{d}{dx}\Bigl(\frac{\pi^2+\tan(e^\pi)-2x^e}{4}\Bigr)$
\end{enumerate}

\item{} The graph of a function~$f$ is shown
  below with several points marked. Find all the marked points at
  which the following are true, and give explanations for your answers.
  \begin{center}
    \input{midterm2a.eepic}
  \end{center}

\begin{enumerate}
  \item $f$~is discontinuous.

  \item $f$~is continuous, but the graph of $f$~has a vertical
    tangent line. 

  \item $f$~is continuous, but the graph of $f$~has no tangent
    line.

  \end{enumerate}

\vfil\eject
\item{} Compute the indicated limits. Show all work for full credit.

  \begin{enumerate}

  \item
     $\ds\lim_{x\to1}\frac{3x^2-5x-2}{x^2-4x+4}$

  \item 
     $\ds\lim_{x\to 2}\frac{3x^2-5x-2}{x^2-4x+4}$

  \item 
     $\ds\lim_{x\to\infty}\frac{3x^2-5x-2}{x^2-4x+4}$

   \item 
     $\ds\lim_{x\to 2^+}\frac{|x-2|}{x^2 - 5x + 6}$


  \end{enumerate}

\item{} Let $f(x)=x^3-x^2$. 

  \begin{enumerate}
  \item Find all intervals on which~$f$ is
    \textbf{decreasing}.

  \item Find all intervals on which~$f$ is
    \textbf{concave up}.

  \item Find all intervals on which~$f$ is
    both \textbf{increasing}~and \textbf{concave down}.
  \end{enumerate}

\item{} Do not use the differentiation rules from Chapter~3 in this
  question.

  \begin{enumerate}

  \item State the limit definition of the
  derivative~$f'(x)$.

  \item Use the definition to
    compute the derivative function of $f(x)=\ds\frac{1}{3x}$.

  \item Find the equation of the line tangent to
    the graph $y=\ds\frac{1}{3x}$ at $x=2$.

  \end{enumerate}

\item{} The total cost (in $ \$ $) of repaying a car loan at 
interest rate of $r \%$ per year is $C = f(r)$.  

   \begin{enumerate}

   \item What is the meaning of the statement $f(7) = 20000$?
   \item What is the meaning of the statement $f'(7) = 3000$? What are the units
         of $f'(7)$? 
   \end{enumerate}

  \vfil

\end{enumerate}

\end{document}
