%% Math 133 Midterm 3, Fall 2015
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\markright{MATH 133, section 1, Midterm 4}

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\textbf{College of the Holy Cross, Fall Semester, 2015} \\
\textbf{MATH 133, section 1, Midterm 4} \\
\textbf{Thursday, December 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/\  25\hfil \Strut \\
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2  & \hfil/\  20\hfil \Strut \\
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3  & \hfil/\  30\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 $\frac{dy}{dx}$; do not simplify:
\begin{enumerate}
\item[A)] (7.5)  $y = \ln(\sin(x) + \cos(x))$
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\item[B)] (7.5)  $y = \tan^{-1}(e^{3x})$
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\item[C)] (10)  $x^2 y^4 - 4 \sin^{-1}(y^2) + x = 0$  (use implicit differentiation)
\end{enumerate}
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\item{} (20) A stationary observer watches a weather balloon being launched from a point 1000 feet 
away from her position.  The balloon rises at a rate of 20 feet per second.  How
fast is the distance between the balloon and the observer changing when the balloon
is 400 feet above the ground?

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\item{} All parts of this question refer to the plot in Figure 1, \emph{which is} $y = f'(x)$
\emph{for some function} $f(x)$ .  Assume the whole domain
of the functions $f(x)$ and $f'(x)$ is the interval $[-1,8]$ shown. 
\begin{figure}
\begin{center}
  \includegraphics[width=70mm]{M4P3.png}
\caption{$y = f'(x)$ for Problem 3}
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\end{figure}

\begin{enumerate}
\item[(A)]  (10) Give approximate values for the critical points of $f(x)$
in the interval shown:\\
Answer:  $\underline{\qquad\qquad\qquad\qquad\qquad\qquad}$ 
\item[(B)]  (5) Briefly, in your own words, state how the First Derivative Test distinguishes
between local maxima, local minima, and critical points that are neither:
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\item[(C)]  (5) Identify each of the points you found in part (A) as a local maximum,
local minimum, or neither: \\
Answer:  $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$ 
\item[(D)]  (5) Find approximate values for all the inflection points of $f(x)$.\\
Answer:  $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$ 
\item[(E)]  (5) Over which intervals is $y = f(x)$ concave up? concave down?\\
Concave up:  $\underline{\qquad\qquad\qquad\qquad\qquad\qquad}$ Concave down:  
$\underline{\qquad\qquad\qquad\qquad\qquad\qquad}$ 
\end{enumerate}
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\item{} A church window has the shape of a rectangle surmounted by a semicircle (see Figure 2).  The 
total perimeter of the window is 800 cm.  What should the dimensions be to make the 
area of the window be as large as possible (so that it will admit the most light possible)?
Useful information:  The area of a circle of radius $r$ is $A = \pi r^2$ and the circumference
is $C = 2\pi r$.  The area of a rectangle is the product of its length and width.
The parts of this problem will lead you to the answer.
\begin{figure}
\begin{center}
  \includegraphics[width=50mm]{M4P4.png}
\caption{The window for Problem 4.}
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\end{figure}
\begin{enumerate}
\item[(A)] (5) Call the horizontal side of the rectangle $x$ and the vertical side $y$.
Express the total area of the window and the total perimeter in terms of $x$ and $y$.
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\item[(B)] (5) Solve for $y$ in terms of $x$ using the perimeter equation and substitute
into the area equation.
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\item[(C)] (5) Determine a critical point of the area function.
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\item[(D)] (5) How do you know your critical point is a maximum of the area?
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\item[(E)] (5) What are the dimensions $x$ and $y$ of the window of maximum area? 

\end{enumerate}


\end{enumerate}

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