%% Math 133 Midterm 3, Fall 2015
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\markright{MATH 133, section 1, Midterm 3}

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\textbf{College of the Holy Cross, Fall Semester, 2017} \\
\textbf{MATH 133, Midterm 3} \\
\textbf{Friday, November 10}
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\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 \\
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1  & \hfil/\  20\hfil \Strut \\
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2  & \hfil/\  20\hfil \Strut \\
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3  & \hfil/\  40\hfil \Strut \\
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4  & \hfil/\  20\hfil \Strut \\
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Total   & \hfil/100\hfil \Strut \\
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\begin{enumerate}

\item{} 
\begin{enumerate}
\item[A)] (5) State the limit definition of the
  derivative~$f'(x)$ as an equation starting with $f'(x) = $.
\vskip 2.0truein
\item[B)] (10) Use the limit definition to
    compute $f'(x)$ for $f(x) = 3x^2 + 4x - 2$.
\vskip 3.0truein
\item[C)] (5) Find the equation of the line tangent to
    the graph $y = 3x^2 + 4x - 2$ at $x = 1$.
\end{enumerate}
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\item{} (20) The bluebook value of a 2013 model car at time $t$ years after
2013 is approximated by $P(t) = 24e^{(-.2) t}$ (in units of \$1000). 
What was the value in 2015, and what was the rate of change of the value then? (Give
your answer with units.) Was the value increasing or decreasing?
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\item{} Compute the indicated derivatives of {\it any four of} the following functions.  If you 
submit answers for all five, the best four will be used to compute your score on this question. You may use any correct
method.  You do not need to simplify your answers unless specifically directed to do so.
But you must show work for full credit.
\begin{enumerate}
\item[(A)]  (10) $f(x) = \displaystyle{(e^x + \sin(x))(\cos(x) + \pi^e)}$.  Compute $f'(x)$.
\vskip 3.0truein
\item[(B)]  (10) $g(x) = \displaystyle{\frac{x^4 - 3x^2}{x^3 + 1}}$.  Compute $g'(x)$ and 
simplify your answer. 
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\item[(C)]  (10) $h(x) = \displaystyle{\frac{1}{\sqrt{x^2 + 4x + 2}}}$.  Compute $h'(x)$.
\vskip 4.0truein
\item[(D)] (10) $j(x) = \displaystyle{(x^2 + 3)e^{x^3 - 5\sin(2x)}}$.  Compute $j'(x)$.
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\item[(E)] (10)  $k(x) = \displaystyle{3x^{7/4} - \frac{3}{x^3} + e^3}$.  Compute both
$k'(x)$ and $k''(x)$.
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\end{enumerate}

\item{}  All parts of this question refer to the plots in Figure 1.  Assume the whole domain
of the functions is the interval $[-2,10]$ shown. 
\begin{figure}
\centering
  \subfloat[$y = A(x)$]{\includegraphics[width=70mm]{M33PA.png}}
  \subfloat[$y = B(x)$]{\includegraphics[width=70mm]{M33PB.png}}
\caption{Plots for Problem 3}
\end{figure}

\begin{enumerate}
\item[(A)] (4) Which has the larger slope:  The secant line to $y = A(x)$ through
$(0,A(0))$ and $(2,A(2))$ or the tangent line to $y = A(x)$ at $(0,A(0))$? \\ 
Answer: $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$
\vskip 5pt
\item[(B)] (4)  Is $A'(4)$ positive or negative?  
Answer: $\underline{\qquad\qquad\qquad}$
\vskip 5pt
\item[(C)] (3) At how many different points is $A'(x) = 0$?  Estimate the 
$x$-values from the graph.  Answer: $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$
\vskip 5pt
\item[(D)] (3) On the $x$-interval $(0,2)$, is $B''(x)$ positive or negative?
Answer: $\underline{\qquad\qquad\qquad}$
\vskip 5pt
\item[(E)] (3) On the $x$-interval $(1,3)$ is $B'(x)$ positive or negative?
Answer: $\underline{\qquad\qquad\qquad}$
\vskip 5pt
\item[(F)] (3)  One of the two functions $A(x)$ and $B(x)$ is the derivative of 
the other.  Which is which?  Answer: 
$\underline{\qquad\qquad\qquad}$ is the derivative of $\underline{\qquad\qquad\qquad\qquad}$.
\end{enumerate}


\end{enumerate}

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