36. Consider all right circular cylinders for which the sum of the height and circumference is 30 centimeters. What is the radius of the one with maximum volume?\\
(A) 3 cm\\
(B) 10 cm\\
(C) 20 cm\\
(D) $\frac { 30 } { \pi ^ { 2 } } \mathrm {~cm}$\\
(E) $\frac { 10 } { \pi } \mathrm {~cm}$

\section*{1993 AP Calculus BC: Section I}
\begin{enumerate}
  \setcounter{enumi}{36}
  \item If $f ( x ) = \left\{ \begin{array} { l l } x & \text { for } x \leq 1 \\ \frac { 1 } { x } & \text { for } x > 1 , \end{array} \right.$ then $\int _ { 0 } ^ { e } f ( x ) d x =$\\
(A) 0\\
(B) $\frac { 3 } { 2 }$\\
(C) 2\\
(D) $e$\\
(E) $e + \frac { 1 } { 2 }$
  \item During a certain epidemic, the number of people that are infected at any time increases at a rate proportional to the number of people that are infected at that time. If 1,000 people are infected when the epidemic is first discovered, and 1,200 are infected 7 days later, how many people are infected 12 days after the epidemic is first discovered?\\
(A) 343\\
(B) 1,343\\
(C) 1,367\\
(D) 1,400\\
(E) 2,057
  \item If $\frac { d y } { d x } = \frac { 1 } { x }$, then the average rate of change of $y$ with respect to $x$ on the closed interval $[ 1,4 ]$ is\\
(A) $- \frac { 1 } { 4 }$\\
(B) $\frac { 1 } { 2 } \ln 2$\\
(C) $\frac { 2 } { 3 } \ln 2$\\
(D) $\frac { 2 } { 5 }$\\
(E) 2
  \item Let $R$ be the region in the first quadrant enclosed by the $x$-axis and the graph of $y = \ln \left( 1 + 2 x - x ^ { 2 } \right)$. If Simpson's Rule with 2 subintervals is used to approximate the area of $R$, the approximation is\\
(A) 0.462\\
(B) 0.693\\
(C) 0.924\\
(D) 0.986\\
(E) 1.850
  \item Let $f ( x ) = \int _ { - 2 } ^ { x ^ { 2 } - 3 x } e ^ { t ^ { 2 } } d t$. At what value of $x$ is $f ( x )$ a minimum?\\
(A) For no value of $x$\\
(B) $\frac { 1 } { 2 }$\\
(C) $\frac { 3 } { 2 }$\\
(D) 2\\
(E) 3
  \item $\lim _ { x \rightarrow 0 } ( 1 + 2 x ) ^ { \csc x } =$\\
(A) 0\\
(B) 1\\
(C) 2\\
(D) $e$\\
(E) $e ^ { 2 }$
\end{enumerate}

\section*{1993 AP Calculus BC: Section I}
\begin{enumerate}
  \setcounter{enumi}{42}
  \item The coefficient of $x ^ { 6 }$ in the Taylor series expansion about $x = 0$ for $f ( x ) = \sin \left( x ^ { 2 } \right)$ is\\
(A) $- \frac { 1 } { 6 }$\\
(B) 0\\
(C) $\frac { 1 } { 120 }$\\
(D) $\frac { 1 } { 6 }$\\
(E) 1
  \item If $f$ is continuous on the interval $[ a , b ]$, then there exists $c$ such that $a < c < b$ and $\int _ { a } ^ { b } f ( x ) d x =$\\
(A) $\frac { f ( c ) } { b - a }$\\
(B) $\frac { f ( b ) - f ( a ) } { b - a }$\\
(C) $f ( b ) - f ( a )$\\
(D) $f ^ { \prime } ( c ) ( b - a )$\\
(E) $f ( c ) ( b - a )$
  \item If $f ( x ) = \sum _ { k = 1 } ^ { \infty } \left( \sin ^ { 2 } x \right) ^ { k }$, then $f ( 1 )$ is\\
(A) 0.369\\
(B) 0.585\\
(C) 2.400\\
(D) 2.426\\
(E) 3.426
\end{enumerate}