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\markright{Math 135, Section 1, Midterm 2}

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\begin{document}

\begin{center}

\textbf{Math 135, Section 1 -- Midterm Exam 2}\\
\textbf{Friday, October 25}
\end{center}

\vglue 1.0truein

\noindent\textbf{Name:} $\underline{\qquad\qquad\qquad\qquad\qquad\qquad}$
\vskip 0pt plus .3fil

\vglue 1.0truein
\paragraph{Instructions} Please write your answers in the spaces
provided, and show work on the test itself. Use the back of the
preceding page if you need more space for scratch work.

\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/\  15\hfil \Strut \\
\hline
2  & \hfil/\  35\hfil \Strut \\
\hline
3  & \hfil/\  30\hfil \Strut \\
\hline
4  & \hfil/\  20\hfil \Strut \\
\hline\hline
Total   & \hfil/100\hfil \Strut \\
\hline
\end{tabular}
\end{center}
\vfill\eject

\begin{enumerate}

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

  \begin{enumerate}

  \item
    (5) $\ds\lim_{x\to 1}\frac{5x^2 - 3x - 2}{x^2 - 7x + 6}$
\vglue 2.0truein
  \item 
    (5) $\ds\lim_{x\to 2}\frac{5x^2 - 3x - 2}{x^2 - 7x + 6}$
\vglue 2.0truein
  \item 
    (5) $\ds\lim_{x\to\infty}\frac{5x^2 - 3x - 2}{x^2 - 7x + 6}$
\end{enumerate}

\vfil\eject

\item{} The graph of a function~$f$ with $f(-1) = -.2$ and $f(2) = -1$ is shown below.  
  \begin{center}
	\includegraphics[scale = .4]{M2I.png}
  \end{center}

\begin{enumerate}
  \item (10) What are $\ds\lim_{x\to 2^-} f(x)$ and $\ds\lim_{x\to 2^+} f(x)$?

\vglue 1.5truein

  \item (15) Find all $x$ in $(-3,5)$ where $f$ is discontinuous. Explain.
\vglue 1.5truein

  \item (10) Given that $f(x) = x + 3$ for $-1 < x < 0$ and $f(x) = 3 - x - \frac{x^3}{8}$ for $0 \le x < 2$, is
$f$ differentiable at $a = 0$?  Why or why not?  

  \end{enumerate}
\vfill\eject
\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 (5) $\ds\frac{d}{dx}\Bigl(\frac{5}{\sqrt{x}} - e^x + 3\Bigr)$

\vglue 1.0truein
  \item (10) $\ds\frac{d}{du} (u^{5/3} e^u)$
\vglue 1.5truein  
  \item (10) $\ds\frac{d}{dv}\Bigl(\frac{v^3 - 2v}{v^2 + 5v + 1}\Bigr)$
\vglue 2.0truein

  \item (5) $\ds\frac{d}{dx}\Bigl(\frac{e^\pi + \pi^e - x^\pi}{4}\Bigr)$
\end{enumerate}
\vfil\eject


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

  \begin{enumerate}

  \item (5) State the limit definition of the
  derivative~$f'(x)$.

\vglue 1.5truein
  \item (10) Use the definition to
    compute the derivative function of $f(x)=\ds\sqrt{x + 1}$.

\vglue 3.5truein
  \item (5) Find the equation of the line tangent to
    the graph $y=\ds\sqrt{x + 1}$ at $x=8$.

  \end{enumerate}

\end{enumerate}
\end{document}
