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\begin{document}
{\bf Part I: Multiple Choice Questions} (5 {\em points each})

\mm
{\bf 1.} $\ds{\frac{d}{dx}(x^{3}e^{4x})} = $ \vspace{1em}

\mm
(a) $12x^{2}e^{4x}$ \hfil (b) $3x^{2}e^{4x} + 4x^{4}e^{4x - 1}$ \hfil (c)
$x^{3}e^{4x} + 12x^{2}e^{4x}$

(d) $3x^{2}e^{4x} + 4x^{3}e^{4x}$ \hs (e) $4x^{3}e^{4x - 1}$

\vskip1.5in
{\bf 2.} Suppose $f(x)$ is a function which satisfies $f'(3) = 0$, $f'(5) = 0$,
$f''(3) = -4$, and $f''(5) = 5$.  Which of the following statements is true?

\mm
(a) $f(x)$ has a relative minimum at $x = 3$ and at $x = 5$.\\
(b) $f(x)$ has a relative minimum at $x = 3$ and a relative maximum at $x =
5$.\\
(c) $f(x)$ has a relative maximum at $x = 3$ and at $x = 5$.\\
(d) $f(x)$ has a relative maximum at $x = 3$ and a relative minimum at $x =
5$.\\
(e) None of the above is true.

\vskip1.5in
{\bf 3.} Let $y = \sqrt{\ln x}$.  Then $y' = $

\mm
(a) $\ds{\frac{1}{2x}} \hfill$ (b) $\ds{\frac{1}{2x \sqrt{\ln x}}} \hfill$ (c)
$\ds{\frac{1}{2 \sqrt{x}}} \hfill$ (d) $\ds{\frac{1}{2 \sqrt{\ln x}}}
\hfill$ (e)
$\ds{\frac{x}{2 \sqrt{\ln x}}}$


\newpage
Questions 4 - 5 involve the following table of values\vspace{1em}

\begin{large}
\begin{tabular}{|c||c|c|c|c|} \hline
$x$ & $f(x)$ & $g(x)$ & $f'(x)$ & $g'(x)$ \\ \hline \hline
0 & 1 & 2 & $-1$ & 1/2  \\ \hline
1 & 0 & 3 & 1/2 & 0  \\ \hline
2 & 1/3 & 5/2  & 2  & $-1/3$  \\ \hline
\end{tabular}
\end{large}

\mm
{\bf 4.} Let $h(x) = \ds{\frac{f(x)}{g(x)}}$.  Then $h'(0) = $

\mm
(a) 2 \hfill (b) $-3/8$ \hfill (c) 5/8 \hfill (d) $-2$ \hfill (e) $-5/8$

\vskip1.0in
{\bf 5.} Let $k(x) = f(g(x))$. Then $k'(0) = $

\mm
(a) 1/6 \hfill (b) $-1/2$ \hfill (c) 1 \hfill (d) 4 \hfill (e)$-2/3$

\vskip1.0in
{\bf 6.} Let $f(x)$ be a function with $f'(2) = 0$.  Which of the following
\textbf{must} be true about $f$?

\mm
(a) At $x = 2$, $f$ changes from increasing to decreasing or from decreasing to
increasing.\\
(b) The tangent line to the graph of $f$ at $(2,f(2))$ is the line $x = 2$.\\
(c) $f$ has a local maximum or a local minimum at $(2, f(2))$.\\
(d) $(2, f(2))$ is an inflection point.\\
(e) The tangent line to the graph of $f$ at $(2, f(2))$ is horizontal.

\vskip.5in
{\bf 7.} Suppose that $f(x)$ is increasing and concave up everywhere. Further,
$f(2) = 2$ and the tangent line at the point $(2,2)$ is the line $y = x$. Which
of the following can you conclude?

\mm
(a) $f(5) > 5$ \hfil (b) $f(5) = 2$ \hfil (c) $f(5) < 5$ \hfil (d) $f(5) = 5$
\hfil (e) none of the preceding

\newpage
{\bf 8.} The graph of a function $f(x)$ is sketched below.  Which of the
following
statements is true?

\vskip.8in

(a) $f(x)$ is increasing and $f'(x)$ is decreasing.\\
(b) Both $f(x)$ and $f'(x)$ are increasing.\\
(c) $f(x)$ is decreasing and $f'(x)$ is increasing.\\
(d) Both $f(x)$ and $f'(x)$ are decreasing.\\
(e) None of the above is true.

\bm
{\bf 9.} Suppose that $f(x)$ is a function with $f''(x) > 0$ on the interval
$(2,\infty)$ and $f''(x) < 0$ on the interval $(-\infty,2)$.  Which of the
following could be the graph of $f(x)$? (Use the letter to the upper left
of each graph.)

\mm
(a) \hs\hs\hs\hs\hs\hs\hh (b) \hs\hs\hs\hs\hs\hs\hh (c)

\vskip1.2in

(d) \hs\hs\hs\hs\hs\hs\hh (e)

\vskip1.2in

\bm

\bm
{\bf 10.}  Suppose that $f(x)$ is a function whose {\bf derivative} is $f'(x) =
x\ln x$.  Identify the $x$-value(s) of all inflection point(s) of $f(x)$.

\mm
(a) $e$ \hfil (b) $0, 1$ \hfil (c) $0$, $e$ \hfil (d) $1/e$ \hfil (e)
$f(x)$ has
no inflection points

\newpage
{\bf Part II: Partial Credit Questions}

Show all work and put your final answer in the space provided.  You will
receive
no credit if the answer is not in the space provided and no partial credit
for a
wrong answer if you do not show your work.\\
Answers may be approximated by decimals if desired, but full points will be
also
be awarded for correct answers left in exact form in terms of logarithms or
exponentials.

\mm
{\bf 11.} (7 {\em points}) A pot of boiling water ($100^\circ$ Celsius) is
left to
cool in a room whose temperature is $20^\circ$ Celsius so that after $t$
minutes, the temperature of the water is $H = 20 + Ae^{kt}$ for some constants
$A$ and $k$.  After 30 minutes, the water is $60^\circ$.  At what {\bf
rate} (with
respect to time) is the temperature of the water changing after 90 minutes?

\vskip3.0in
\ans

\bm

{\bf 12.} (5 {\em points}) The half-life of carbon-14 is 5730 years, so carbon
decays according to the equation $y = Ae^{-kt}$, where $k = .000121$.  An
ancient
document written on papyrus is found to have 37.5 $\%$ of the carbon-14 that a
freshly cut sample has.  How old is the document?

\vskip2.3in
\ans

\newpage
{\bf 13.} (6 {\em points}) Find the the equation of the line tangent to $y =
\ds\frac{e^{x^{2} + 2x}}{x^{3} + 1}$ at $(0, 1)$.

\vskip3.2in
\ans

\bm
{\bf 14.} (6 {\em points}) The period $P$ in seconds of a simple pendulum is
completely determined by its length $L$ according to the formula
$\ds P = \frac{2\pi}{\sqrt{10}}\sqrt{L}$, where $\pi \approx 3.14$.
Therefore, if
the length of a pendulum increases, its period will increase.  (That is why
clocks
with metal pendulums tend to run slow in the summertime.)  Find the rate (with
respect to time) at which a simple pendulum's period is increasing when the
pendulum is 4m long and its length is increasing at a rate of .001m/day.

\vskip3.3in
\ans

\newpage
{\bf 15.} (6 {\em points}) The demand $q$ for Widgets is a function
of the price $p$, i.e.\ $q = f(p)$. When the price $p = \$10$, we have $q =
f(10)
= 100$ and $f'(10) = -4$.

\mm (a) The revenue $R$ is determined by $R = pq$.  Find
$\ds{\frac{dR}{dp}}$ when
$p = 10$.

\vskip2.0in
\ans

\bm (b) The cost of producing $q$ Widgets is $C(q) = 100q + 5000$. Find
$\ds{\frac{dC}{dp}}$ when $p = 10$.

\vskip2.0in
\ans

\bm (c) If $P$ represents profit, find $\ds{\frac{dP}{dp}}$ when $p = 10$.

\vskip2.0in
\ans

\newpage
{\bf 16.} (6 {\em points}) Consider the function $f(x) = x - 3x^{\frac23}$.

(a) Identify the intervals on which $f(x)$ is increasing and those on which
$f(x)$ is decreasing.

\vskip2.0in
\ans

\bm (b) Identify the $x$-value(s) of all relative extreme point(s) of
$f(x)$ and
specify whether each is a maximum or a minimum.

\vskip1.0in
\ans

\bm
{\bf 17.} (6 {\em points}) Consider the function $f(x) = x^4 - 54x^2 + 27x
- 17$.

\mm (a) Identify the intervals on which $f(x)$ is concave up and those on which
$f(x)$ is concave down.

\vskip2.0in
\ans

\bm (b) Identify the $x$-value(s) of all inflection point(s) of $f(x)$.

\vskip0.8in
\ans

\newpage
{\bf 18.} (8 {\em points}) Suppose that $f(x)$ is a function which
satisfies the
following list of properties:
\begin{tabbing}
\hs \=$\bullet$\ $f(-3)$ \= $= -1$, $f(0) = -3$, $f(2) = -5$, and $f(4) = 0$.\\
\>$\bullet$\ The only solutions to $f'(x) = 0$ are $x = -3$ and $x = 2$.\\
\>\>$\bullet$\ $f'(x) > 0$ on the intervals $(-\infty,-3)$, $(2,\infty)$.\\
\>\>$\bullet$\ $f'(x) < 0$ on the interval $(-3,2)$.\\
\>$\bullet$\ The only solution to $f''(x) = 0$ is $x = 0$.\\
\>\>$\bullet$\ $f''(x) > 0$ on $(0,\infty)$.\\
\>\>$\bullet$\ $f''(x) < 0$ on $(-\infty,0)$.
\end{tabbing}
\mm
(a) List all the relative extreme points and inflection points of $f(x)$.

\vskip1.5in
(b) Sketch a possible graph of $f(x)$, labeling the points you listed in
part (a).

\end{document}
