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

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\begin{document}

\begin{center}
\textbf{College of the Holy Cross}\\
\textbf{Math 135, Section 1 -- Midterm Exam 2}\\
\textbf{Friday, November 1}
\end{center}

\vglue 1.0truein

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

\vglue 1.0truein
\paragraph{Instructions} For full credit, you must show 
{\it all work} on the test pages.  Use the back of the
preceding page if you need more space for scratch work.  The numbers
next to each part of the questions are their point values.  I will
be liberal with partial credit, so even if you don't see how to complete
a problem, do as much as you can and be sure to write what you know.
\vglue 0.5truein

\vfil

\noindent\textbf{Please do not write in the space below}

\begin{center}
\begin{tabular}{|l||r|}
\hline
Problem & Points/Poss    \Strut \\
\hline\hline
I  & \hfil/\  20\hfil \Strut \\
\hline
II  & \hfil/\  25\hfil \Strut \\
\hline
III  & \hfil/\  30\hfil \Strut \\
\hline
IV  & \hfil/\  25\hfil \Strut \\
\hline\hline
Total   & \hfil/100\hfil \Strut \\
\hline
\end{tabular}
\end{center}
\vfill\eject

\begin{enumerate}

\item[I.] Compute the indicated limits. You must show all necessary work to justify your answer to 
receive full credit.

  \begin{enumerate}

  \item
    (5) $\ds\lim_{x\to 1}\frac{x^2 - 5x + 2}{2x^2 - 2x + 3}$
\vglue 1.8truein
  \item 
    (5) $\ds\lim_{x\to 1}\frac{x^2 - 7x + 6}{2x^2 - 2x}$
\vglue 1.8truein
  \item 
    (5) $\ds\lim_{x\to\infty}\frac{x^2 - 5x + 2}{2x^2 - 2x + 3}$
\vglue 1.8truein
  \item 
    (5) $\ds\lim_{\theta\to 0} \frac{\sin(4\theta)}{\sin(5\theta)}$
\end{enumerate}

\vfil\eject

\item[II.] The graph of the function 
$$f(x) = \begin{cases}
               \ds\frac{1}{x+1} & x < -1\\
&\\
               -x + 2 & -1 \le x \le 0\\
&\\
               \ds\frac{4x^3 - 24x^2 + 44x - 24}{x^3 - 5x^2 + 2x + 8} & 0 < x < 3
\end{cases}$$
 is shown above. 
\begin{figure}  
  \begin{center}
	\includegraphics[scale = .4]{M2II.png}
  \end{center}
\end{figure}

\begin{enumerate}
  \item (10) What are $\ds\lim_{x\to 0^-} f(x)$ and $\ds\lim_{x\to 0^+} f(x)$?  (In your answer say clearly which is which.)

\vglue 1.0truein

  \item (15) Find all $x$ in $(-2,3)$ where $f$ is discontinuous. Give the types of each of the discontinuities.


  \end{enumerate}


\vfill\eject
\item[III.] Do not use the short-cut 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) Estimate the derivative of $f(x) = \ds\sqrt{x + 2}$ at $a = 7$ \emph{numerically}
  by computing difference quotients of $f$ with $h = \pm .1$, then $h = \pm .01$.  Enter 
  your values in the table below, and then state what your estimate of $f'(7)$ is.  

\vglue 3.5truein

$$\begin{array}{ccccccccccc}
\hline
h &\vline& -.1 \qquad &\vline& -.01 \qquad &\vline& .01 \qquad &\vline& .1 \qquad&\vline\\ \hline
{\rm difference\ quotient\ value}&\vline&         &\vline&       &\vline&        &\vline&   &\vline \\ \hline 
\end{array}$$
\vskip 10pt
\noindent
$f'(7) \doteq \underline{\qquad\qquad\qquad\qquad}$
\vfil\eject

  \item (10) Use the definition to
    compute the derivative function of $f(x)=\ds\sqrt{x + 2}$.

\vglue 3.5truein
  \item (5) Find the equation of the line tangent to
    the graph $y=\ds\sqrt{x + 2}$ at $a=7$.

  \end{enumerate}


\vfil\eject
\item[IV.] Use the short-cut 
  rules to compute the following derivatives; it is {\it not necessary} to simplify
your answers for this. Show all work.

\begin{enumerate}
  \item (5) $\ds\frac{d}{dx}\left(\frac{3}{\sqrt{x}} - e^x + 3x\right)$

\vglue 2.4truein
  \item (10) $\ds\frac{d}{dv}\left((v^2 - 2v)(v^3 + 1)\right)$
\vglue 2.4truein

  \item (10) $\ds\frac{d}{dx}\left(\frac{x^2 - 4x + 3}{x^4 - x}\right)$
\end{enumerate}
\end{enumerate}
\end{document}
