\section*{50 Minutes-No Calculator}
Note: Unless otherwise specified, the domain of a function $f$ is assumed to be the set of all real numbers $x$ for which $f ( x )$ is a real number.

\begin{enumerate}
  \item $\int _ { 1 } ^ { 2 } \left( 4 x ^ { 3 } - 6 x \right) d x =$\\
(A) 2\\
(B) 4\\
(C) 6\\
(D) 36\\
(E) 42
  \item If $f ( x ) = x \sqrt { 2 x - 3 }$, then $f ^ { \prime } ( x ) =$\\
(A) $\frac { 3 x - 3 } { \sqrt { 2 x - 3 } }$\\
(B) $\frac { x } { \sqrt { 2 x - 3 } }$\\
(C) $\frac { 1 } { \sqrt { 2 x - 3 } }$\\
(D) $\frac { - x + 3 } { \sqrt { 2 x - 3 } }$\\
(E) $\frac { 5 x - 6 } { 2 \sqrt { 2 x - 3 } }$
  \item If $\int _ { a } ^ { b } f ( x ) d x = a + 2 b$, then $\int _ { a } ^ { b } ( f ( x ) + 5 ) d x =$\\
(A) $a + 2 b + 5$\\
(B) $5 b - 5 a$\\
(C) $7 b - 4 a$\\
(D) $7 b - 5 a$\\
(E) $7 b - 6 a$
  \item If $f ( x ) = - x ^ { 3 } + x + \frac { 1 } { x }$, then $f ^ { \prime } ( - 1 ) =$\\
(A) 3\\
(B) 1\\
(C) $\quad - 1$\\
(D) - 3\\
(E) - 5
  \item The graph of $y = 3 x ^ { 4 } - 16 x ^ { 3 } + 24 x ^ { 2 } + 48$ is concave down for\\
(A) $x < 0$\\
(B) $x > 0$\\
(C) $x < - 2$ or $x > - \frac { 2 } { 3 }$\\
(D) $x < \frac { 2 } { 3 }$ or $x > 2$\\
(E) $\frac { 2 } { 3 } < x < 2$
  \item $\frac { 1 } { 2 } \int e ^ { \frac { t } { 2 } } d t =$\\
(A) $e ^ { - t } + C$\\
(B) $e ^ { - \frac { t } { 2 } } + C$\\
(C) $e ^ { \frac { t } { 2 } } + C$\\
(D) $2 e ^ { \frac { t } { 2 } } + C$\\
(E) $e ^ { t } + C$
  \item $\frac { d } { d x } \cos ^ { 2 } \left( x ^ { 3 } \right) =$\\
(A) $\quad 6 x ^ { 2 } \sin \left( x ^ { 3 } \right) \cos \left( x ^ { 3 } \right)$\\
(B) $\quad 6 x ^ { 2 } \cos \left( x ^ { 3 } \right)$\\
(C) $\sin ^ { 2 } \left( x ^ { 3 } \right)$\\
(D) $- 6 x ^ { 2 } \sin \left( x ^ { 3 } \right) \cos \left( x ^ { 3 } \right)$\\
(E) $- 2 \sin \left( x ^ { 3 } \right) \cos \left( x ^ { 3 } \right)$
\end{enumerate}

Questions 8-9 refer to the following situation.\\
\includegraphics[max width=\textwidth, alt={}, center]{19ceef41-3979-486d-8719-654b937becf0-108_478_764_406_627}

A bug begins to crawl up a vertical wire at time $t = 0$. The velocity $v$ of the bug at time $t$, $0 \leq t \leq 8$, is given by the function whose graph is shown above.\\
8. At what value of $t$ does the bug change direction?\\
(A) 2\\
(B) 4\\
(C) 6\\
(D) 7\\
(E) 8\\
9. What is the total distance the bug traveled from $t = 0$ to $t = 8$ ?\\
(A) 14\\
(B) 13\\
(C) 11\\
(D) 8\\
(E) 6\\
10. An equation of the line tangent to the graph of $y = \cos ( 2 x )$ at $x = \frac { \pi } { 4 }$ is\\
(A) $y - 1 = - \left( x - \frac { \pi } { 4 } \right)$\\
(B) $\quad y - 1 = - 2 \left( x - \frac { \pi } { 4 } \right)$\\
(C) $y = 2 \left( x - \frac { \pi } { 4 } \right)$\\
(D) $y = - \left( x - \frac { \pi } { 4 } \right)$\\
(E) $\quad y = - 2 \left( x - \frac { \pi } { 4 } \right)$\\
\includegraphics[max width=\textwidth, alt={}, center]{19ceef41-3979-486d-8719-654b937becf0-109_354_414_371_867}\\
11. The graph of the derivative of $f$ is shown in the figure above. Which of the following could be the graph of $f$ ?\\
(A)\\
\includegraphics[max width=\textwidth, alt={}, center]{19ceef41-3979-486d-8719-654b937becf0-109_300_336_893_485}\\
(B)\\
\includegraphics[max width=\textwidth, alt={}, center]{19ceef41-3979-486d-8719-654b937becf0-109_300_336_880_1067}\\
(C)\\
\includegraphics[max width=\textwidth, alt={}, center]{19ceef41-3979-486d-8719-654b937becf0-109_295_334_1239_485}\\
(D)\\
\includegraphics[max width=\textwidth, alt={}, center]{19ceef41-3979-486d-8719-654b937becf0-109_300_450_1237_1018}\\
(E)\\
\includegraphics[max width=\textwidth, alt={}, center]{19ceef41-3979-486d-8719-654b937becf0-109_300_446_1576_429}\\
12. At what point on the graph of $y = \frac { 1 } { 2 } x ^ { 2 }$ is the tangent line parallel to the line $2 x - 4 y = 3$ ?\\
(A) $\left( \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right)$\\
(B) $\left( \frac { 1 } { 2 } , \frac { 1 } { 8 } \right)$\\
(C) $\left( 1 , - \frac { 1 } { 4 } \right)$\\
(D) $\left( 1 , \frac { 1 } { 2 } \right)$\\
(E) $( 2,2 )$\\
13. Let $f$ be a function defined for all real numbers $x$. If $f ^ { \prime } ( x ) = \frac { \left| 4 - x ^ { 2 } \right| } { x - 2 }$, then $f$ is decreasing on the interval\\
(A) $( - \infty , 2 )$\\
(B) $( - \infty , \infty )$\\
(C) $( - 2,4 )$\\
(D) $( - 2 , \infty )$\\
(E) $( 2 , \infty )$\\
14. Let $f$ be a differentiable function such that $f ( 3 ) = 2$ and $f ^ { \prime } ( 3 ) = 5$. If the tangent line to the graph of $f$ at $x = 3$ is used to find an approximation to a zero of $f$, that approximation is\\
(A) 0.4\\
(B) 0.5\\
(C) 2.6\\
(D) 3.4\\
(E) 5.5\\
\includegraphics[max width=\textwidth, alt={}, center]{19ceef41-3979-486d-8719-654b937becf0-110_530_848_992_644}\\
15. The graph of the function $f$ is shown in the figure above. Which of the following statements about $f$ is true?\\
(A) $\lim _ { x \rightarrow a } f ( x ) = \lim _ { x \rightarrow b } f ( x )$\\
(B) $\lim _ { x \rightarrow a } f ( x ) = 2$\\
(C) $\lim _ { x \rightarrow b } f ( x ) = 2$\\
(D) $\lim _ { x \rightarrow b } f ( x ) = 1$\\
(E) $\lim _ { x \rightarrow a } f ( x )$ does not exist.\\
16. The area of the region enclosed by the graph of $y = x ^ { 2 } + 1$ and the line $y = 5$ is\\
(A) $\frac { 14 } { 3 }$\\
(B) $\frac { 16 } { 3 }$\\
(C) $\frac { 28 } { 3 }$\\
(D) $\frac { 32 } { 3 }$\\
(E) $8 \pi$\\
17. If $x ^ { 2 } + y ^ { 2 } = 25$, what is the value of $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point $( 4,3 )$ ?\\
(A) $- \frac { 25 } { 27 }$\\
(B) $- \frac { 7 } { 27 }$\\
(C) $\frac { 7 } { 27 }$\\
(D) $\frac { 3 } { 4 }$\\
(E) $\frac { 25 } { 27 }$\\
18. $\int _ { 0 } ^ { \frac { \pi } { 4 } } \frac { e ^ { \tan x } } { \cos ^ { 2 } x } d x$ is\\
(A) 0\\
(B) 1\\
(C) $e - 1$\\
(D) $e$\\
(E) $e + 1$\\
19. If $f ( x ) = \ln \left| x ^ { 2 } - 1 \right|$, then $f ^ { \prime } ( x ) =$\\
(A) $\left| \frac { 2 x } { x ^ { 2 } - 1 } \right|$\\
(B) $\frac { 2 x } { \left| x ^ { 2 } - 1 \right| }$\\
(C) $\frac { 2 | x | } { x ^ { 2 } - 1 }$\\
(D) $\frac { 2 x } { x ^ { 2 } - 1 }$\\
(E) $\frac { 1 } { x ^ { 2 } - 1 }$\\
20. The average value of $\cos x$ on the interval $[ - 3,5 ]$ is\\
(A) $\frac { \sin 5 - \sin 3 } { 8 }$\\
(B) $\frac { \sin 5 - \sin 3 } { 2 }$\\
(C) $\frac { \sin 3 - \sin 5 } { 2 }$\\
(D) $\frac { \sin 3 + \sin 5 } { 2 }$\\
(E) $\frac { \sin 3 + \sin 5 } { 8 }$\\
21. $\lim _ { x \rightarrow 1 } \frac { x } { \ln x }$ is\\
(A) 0\\
(B) $\frac { 1 } { e }$\\
(C) 1\\
(D) $e$\\
(E) nonexistent\\
22. What are all values of $x$ for which the function $f$ defined by $f ( x ) = \left( x ^ { 2 } - 3 \right) e ^ { - x }$ is increasing?\\
(A) There are no such values of $x$.\\
(B) $\quad x < - 1$ and $x > 3$\\
(C) $- 3 < x < 1$\\
(D) $- 1 < x < 3$\\
(E) All values of $x$\\
23. If the region enclosed by the $y$-axis, the line $y = 2$, and the curve $y = \sqrt { x }$ is revolved about the $y$-axis, the volume of the solid generated is\\
(A) $\frac { 32 \pi } { 5 }$\\
(B) $\frac { 16 \pi } { 3 }$\\
(C) $\frac { 16 \pi } { 5 }$\\
(D) $\frac { 8 \pi } { 3 }$\\
(E) $\pi$\\