%% Math 131 Midterm 1, Fall 2004
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\textbf{College of the Holy Cross, Fall Semester, 2015} \\
\textbf{MATH 133, section 1, Midterm 2} \\
\textbf{Thursday, October 22}
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\vfil

\noindent\textbf{Your Name}: \Blank
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\paragraph{Instructions:} Please show all work necessary to justify your answers, and
write your answers in the spaces provided. Use the back of the
preceding page if you need more space for scratch work.  There are
100 possible points distributed as below.

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

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\begin{tabular}{|l||r|}
\hline
Problem & Points/Poss    \Strut \\
\hline\hline
1  & \hfil/\  15\hfil \Strut \\
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2  & \hfil/\  30\hfil \Strut \\
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3  & \hfil/\  40\hfil \Strut \\
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4  & \hfil/\  15\hfil \Strut \\
\hline\hline
Total   & \hfil/100\hfil \Strut \\
\hline
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\begin{enumerate}

\item{} An object moves along a straight line path with position given by 
$x(t) = 2t^3 + t$, ($t$ in seconds, $x$ in feet).   
\begin{enumerate}
\item[(a)] (5) What is the average velocity of the object on the interval $[1,4]$?

\vskip 3.0truein
\item[ ] Average velocity $=$ $\underline{\qquad\qquad\qquad\qquad}$.
\item[(b)] (10)  Fill in the following table with average velocities computed over the indicated intervals.  Using this information, estimate the \emph{instantaneous velocity} at $t = 1$.  
$$\begin{array}{ccccccccc}
\hline
{\rm interval} &\vline& \qquad[1,2]\qquad &\vline& \quad [1,1.1]\quad &\vline& \quad [1,1.01]\quad &\vline& \quad [1,1.001]\quad \\ \hline
{\rm ave. vel.}&\vline&         &\vline&       &\vline&        &\vline&      \\ \hline 
\end{array}$$
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\item[ ] Estimated instantaneous velocity $=$ $\underline{\qquad\qquad\qquad\qquad}$.
\end{enumerate}
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\eject 
\item{} Answer all parts of this question by referring to the graph $y = f(x)$ above.
\begin{figure}
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\includegraphics[width=.5\linewidth]{FPlot.png}
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\end{figure}

\begin{enumerate}
\item[(a)] (6) $\displaystyle{\lim_{x\to 0^-} f(x)} = \underline{\qquad\qquad\qquad}$ and 
$\displaystyle{\lim_{x\to 0^+} f(x)} = \underline{\qquad\qquad\qquad}$.
\item[(b)] (6) $f(x)$ has an infinite discontinuity at $x = \underline{\qquad\qquad\qquad}$.
\item[(c)] (3)  True/False:  The limit $\displaystyle{\lim_{x\to -1} f(x)}$ does not exist.  
$\underline{\qquad\qquad}$.
\item[(d)] (9)  $\displaystyle{\lim_{x\to 2^-} f(x)} = \underline{\qquad\qquad\qquad}$ and 
$\displaystyle{\lim_{x\to 2^+} f(x)} = \underline{\qquad\qquad\qquad}$.  
Given that $f(2) = 1$, what can we
say about $f$ at $x = 2$?  $f$ is $\underline{\qquad\qquad\qquad\qquad\qquad}$ at $x = 2$.  
\item[(e)] (6)  True/False:  $f(x)$ has a removable discontinuity shown 
in this part of the graph.  $\underline{\qquad\qquad}$.  If so, where is it? 
$x = \underline{\qquad\qquad\qquad}$.  (If not leave this space blank.)  
\end{enumerate} 
\vfil
\eject


\item{}  Compute \emph{any four} of the following limits. (Only the best four will be counted
for your total score.)
\begin{enumerate}
\item[(a)] (10) 
$$\lim_{x\to 3} \frac{x^2 - 5x + 6}{3x^2 - 9x}$$
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Limit = $\underline{\qquad\qquad\qquad\qquad}$
\item[(b)] (10) 
$$\lim_{h \to 4} \frac{\sqrt{12 + h} - \sqrt{16}}{h^2-16}$$
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Limit = $\underline{\qquad\qquad\qquad\qquad}$
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\item[(c)]  (10) 
$$\lim_{t\to 0} \frac{(t^2 + 5)\sin(t)}{t}.$$ 
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Limit = $\underline{\qquad\qquad\qquad\qquad}$ 
\item[(d)]  (10) 
$$\lim_{x\to \infty} \frac{x^3 + 3x + 1}{7x^3 + x^2 + 4x}.$$ 
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Limit = $\underline{\qquad\qquad\qquad\qquad}$ 
\vfil\eject
\item[(e)] (10) 
$$\lim_{x\to 2} \frac{\frac{1}{x^2} - \frac{1}{4}}{x - 2}.$$ 
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Limit = $\underline{\qquad\qquad\qquad\qquad}$ 
\end{enumerate}

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\item{} Let $f(x) = x^2 + 4x + 1$.  
\begin{enumerate}
\item[(a)] (5)  What is the slope of the secant line to the graph through the 
points $(1,6)$ and $(2,13)$?
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Slope = $\underline{\qquad\qquad\qquad\qquad}$ 
\item[(b)] (5)  Give a general formula for the slope of the secant line
through the points $(1,6)$ and $(1+h,(1+h)^2 + 4(1 + h) + 1)$.
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Slope = $\underline{\qquad\qquad\qquad\qquad}$ 
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\item[(c)] (5)  Find the limit as $h \to 0$ of your slope from 
part (b).
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Limit = $\underline{\qquad\qquad\qquad\qquad}$ 
\end{enumerate}
\end{enumerate}

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