25. Consider the curve in the $x y$-plane represented by $x = e ^ { t }$ and $y = t e ^ { - t }$ for $t \geq 0$. The slope of the line tangent to the curve at the point where $x = 3$ is\\
(A) 20.086\\
(B) 0.342\\
(C) - 0.005\\
(D) - 0.011\\
(E) - 0.033

\section*{1993 AP Calculus BC: Section I}
\begin{enumerate}
  \setcounter{enumi}{25}
  \item If $y = \arctan \left( e ^ { 2 x } \right)$, then $\frac { d y } { d x } =$\\
(A) $\frac { 2 e ^ { 2 x } } { \sqrt { 1 - e ^ { 4 x } } }$\\
(B) $\frac { 2 e ^ { 2 x } } { 1 + e ^ { 4 x } }$\\
(C) $\frac { e ^ { 2 x } } { 1 + e ^ { 4 x } }$\\
(D) $\frac { 1 } { \sqrt { 1 - e ^ { 4 x } } }$\\
(E) $\frac { 1 } { 1 + e ^ { 4 x } }$
  \item The interval of convergence of $\sum _ { n = 0 } ^ { \infty } \frac { ( x - 1 ) ^ { n } } { 3 ^ { n } }$ is\\
(A) $- 3 < x \leq 3$\\
(B) $- 3 \leq x \leq 3$\\
(C) $- 2 < x < 4$\\
(D) $- 2 \leq x < 4$\\
(E) $0 \leq x \leq 2$
  \item If a particle moves in the $x y$-plane so that at time $t > 0$ its position vector is $\left( \ln \left( t ^ { 2 } + 2 t \right) , 2 t ^ { 2 } \right)$, then at time $t = 2$, its velocity vector is\\
(A) $\left( \frac { 3 } { 4 } , 8 \right)$\\
(B) $\left( \frac { 3 } { 4 } , 4 \right)$\\
(C) $\left( \frac { 1 } { 8 } , 8 \right)$\\
(D) $\left( \frac { 1 } { 8 } , 4 \right)$\\
(E) $\left( - \frac { 5 } { 16 } , 4 \right)$
  \item $\int x \sec ^ { 2 } x d x =$\\
(A) $\quad x \tan x + C$\\
(B) $\frac { x ^ { 2 } } { 2 } \tan x + C$\\
(C) $\sec ^ { 2 } x + 2 \sec ^ { 2 } x \tan x + C$\\
(D) $\quad x \tan x - \ln | \cos x | + C$\\
(E) $\quad x \tan x + \ln | \cos x | + C$
  \item What is the volume of the solid generated by rotating about the $x$-axis the region enclosed by the curve $y = \sec x$ and the lines $x = 0 , y = 0$, and $x = \frac { \pi } { 3 }$ ?\\
(A) $\frac { \pi } { \sqrt { 3 } }$\\
(B) $\pi$\\
(C) $\pi \sqrt { 3 }$\\
(D) $\frac { 8 \pi } { 3 }$\\
(E) $\quad \pi \ln \left( \frac { 1 } { 2 } + \sqrt { 3 } \right)$
  \item If $s _ { n } = \left( \frac { ( 5 + n ) ^ { 100 } } { 5 ^ { n + 1 } } \right) \left( \frac { 5 ^ { n } } { ( 4 + n ) ^ { 100 } } \right)$, to what number does the sequence $\left\{ s _ { n } \right\}$ converge?\\
(A) $\frac { 1 } { 5 }$\\
(B) 1\\
(C) $\frac { 5 } { 4 }$\\
(D) $\left( \frac { 5 } { 4 } \right) ^ { 100 }$\\
(E) The sequence does not converge.
  \item If $\int _ { a } ^ { b } f ( x ) d x = 5$ and $\int _ { a } ^ { b } g ( x ) d x = - 1$, which of the following must be true?\\
I. $f ( x ) > g ( x )$ for $a \leq x \leq b$\\
II. $\quad \int _ { a } ^ { b } ( f ( x ) + g ( x ) ) d x = 4$\\
III. $\quad \int _ { a } ^ { b } ( f ( x ) g ( x ) ) d x = - 5$\\
(A) I only\\
(B) II only\\
(C) III only\\
(D) II and III only\\
(E) I, II, and III
  \item Which of the following is equal to $\int _ { 0 } ^ { \pi } \sin x d x$ ?\\
(A) $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \cos x d x$\\
(B) $\quad \int _ { 0 } ^ { \pi } \cos x d x$\\
(C) $\quad \int _ { - \pi } ^ { 0 } \sin x d x$\\
(D) $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \sin x d x$\\
(E) $\int _ { \pi } ^ { 2 \pi } \sin x d x$
\end{enumerate}

\section*{1993 AP Calculus BC: Section I}
\includegraphics[max width=\textwidth, alt={}, center]{19ceef41-3979-486d-8719-654b937becf0-103_386_435_345_848}\\