%% Math 135 Midterm 3, Fall 2016
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\markright{MATH 135, section 1, Makeup Midterm 3}

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\begin{center}

\textbf{College of the Holy Cross, Fall Semester, 2016} \\
\textbf{MATH 135, section 1, Midterm 3} \\
\textbf{Monday, December 5}
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\vfil

\noindent\textbf{Your Name}: \Blank
\vskip 0pt plus .3fil

\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/\  40\hfil \Strut \\
\hline
2  & \hfil/\  20\hfil \Strut \\
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3  & \hfil/\  15\hfil \Strut \\
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4  & \hfil/\  25\hfil \Strut \\
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Total   & \hfil/100\hfil \Strut \\
\hline
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\begin{enumerate}

\item{} Find $\frac{dy}{dx}$ and simplify:
\begin{enumerate}
\item[A)] (10) $\displaystyle y = \frac{e^x + x^2}{\sqrt{x} + x + 1}$
\vskip 3.0truein
\item[B)] (10)  $y = \sin(3 \ln(x) - 4 \sec(x))$
\vskip 3.0truein
\item[C)] (10)  $y = \tan^{-1}(e^{-4x})$
\vskip 3.0truein
\item[D)] (10)  $x^3 y^2 - 4 \sin^{-1}(x^2) + y = x$  (use implicit differentiation)
\vskip 3.0truein
\end{enumerate}
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\item{} (20) A stationary observer watches a weather balloon being launched 
straight upward from a point 
450 feet away from her position.  The distance between her and the balloon is
increasing at a rate of 18 feet per second.  How fast is the balloon's height
changing when it 
is 600 feet above the ground?

\vfil\eject
\item{}  All parts of this question refer to the plots in Figure 1.  
Assume the whole domain of the functions is the interval $[-2,8]$ shown (don't
try to extrapolate what might happen on a larger interval).
\begin{figure}
\centering
  \subfloat[$y = A(x)$]{\includegraphics[width=70mm]{135P3IIIA.png}}
  \subfloat[$y = B(x)$]{\includegraphics[width=70mm]{135P3IIIB.png}}
\caption{Plots for Problem 3}
\end{figure}

\begin{enumerate}
\item[(A)] (3)  Is $A'(4)$ positive or negative?  
Answer: $\underline{\qquad\qquad\qquad}$
\vskip 20pt
\item[(B)] (3) At how many different points is $B'(x) = 0$?  Estimate the 
$x$-values from the graph.  Answer: $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$
\vskip 20pt
\item[(C)] (3) On the $x$-interval $(2,4)$, is $A''(x)$ positive or negative?
Answer: $\underline{\qquad\qquad\qquad}$
\vskip 20pt
\item[(D)] (3) On the $x$-interval $(2,4)$ is $B'(x)$ positive or negative?
Answer: $\underline{\qquad\qquad\qquad}$
\vskip 20pt
\item[(E)] (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 $\underline{\qquad\qquad\qquad\qquad}$.
\vskip 5pt
\end{enumerate}
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\item{} All parts of this problem refer to $f(x) = x^4 - 4 x^2 - 5$. 

\begin{enumerate}
\item[(A)] (10) Find all critical points of $f(x)$. 
\vskip 3truein
\item[(B)] (10) Find the absolute maximum and minimum values of $f(x)$ 
on the interval $[0,3]$.  
\vskip 2truein
\item[(C)]  (5) Which of the critical points you found in part A are local maximum
points and which are local minimum points?  (Any correct method for this is OK.)
\end{enumerate}
\end{enumerate}
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