%$Modified: Fri Jun  8 16:21:27 2001 by wgd $

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\def\whichexam{Math 120, Exam I}
\def\examdate{February 13, 2001}
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%%%%%%%
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\itemitem{$\bullet$}
The Honor Code is in effect for this examination.
All work is to be your own.
\itemitem{$\bullet$} The exam lasts for one hour.
\itemitem{$\bullet$} Be sure that your name is on every page in case
pages become detached. 
\itemitem{$\bullet$} Be sure that you have all 9 pages of the
test. Good luck!
%\vskip5pt
%\centerline{Good Luck!}
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PLEASE MARK YOUR ANSWERS WITH AN X, not a circle!
\correctanswers
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%
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%
\multichoicetrue
%
%                   Start the exam
%
\newchoice{Multiple Choice}
\five
%
%
\ifsampletest\par\noindent{\bf This test consists of $12$ 
multiple-choice questions and four partial credit questions.
The correct answer to each multiple choice question is (a).
}\par\smallskip\fi
%
%
\ifproofreading\par\noindent{\bf 
\centerline{PROOFREADING COPY. TOP SECRET!} 
The correct answer to each multiple-choice question is (a). }
\par\smallskip\fi

\newp 1.
The area of the region bounded by the curves $y
= |x|$ and $y = x^2$ is
\ANp
(a) $\frac{1}{3}$
(b) $\frac{1}{7}$
(c) $0$
(d) $1$ 
(e) $2$
\Pra42315

\two
\newp 2. 
Use the disk method (in other words, {\it not} the method of cylindrical
shells) to set up an integral which gives the volume of the solid
obtained by revolving the region bounded by
$y = x$ and $y = \sqrt{x}$ about the line $y=-2$. The resulting
integral is
\ANp
(a) $\DS\pi \int_0^1 [(2 + \sqrt{x})^2 - (2 + x)^2]\,dx$
(b) $\DS\pi \int_0^1 [(2 + y)^2 - (2 + y^2)^2]\,dy$
(c) $\DS\pi \int_0^1 [(2 - x)^2 - (2 - \sqrt{x})^2]\,dx$
(d) $\DS\pi \int_0^1 [(x^2 + 2) - (x-2)]\,dx$
(e) $\DS\pi \int_0^1 [(x^2 + 2) - (x-2)]\,dx$
\Pra31254

\two
\newp 3.
Use the method of cylindrical shells to set up an integral which gives the 
volume of a sphere of
radius $r$ with center at the origin. The resulting integral is
\ANp
(a)$\DS4\pi \int_0^r x \sqrt{r^2 - x^2}\,dx$ 
(b)$\DS4\pi \int_0^r x (r^2 - x^2)\,dx$ 
(c)$\DS4\pi \int_0^r x^2 \sqrt{r^2 - x^2}\,dx$ 
(d)$\DS4\pi \int_0^r (r^2 - x^2)\,dx$ 
(e)$\DS4\pi \int_0^r (r - x) \sqrt{r^2 - x^2}\,dx$ 
\Pra15243

\newp 4.
Let $f(x) = 4 - 3x^2, 0 \le x \le 2$. Find the number $c$ in $[0,2]$
such that $f(c)$ is equal to the average of $f$ on the
interval $[0, 2]$. The number $c$ is
\ANp
(a) $\frac{2}{\sqrt{3}}$
(b) $1$
(c) $2$
(d) $\frac{1}{2}$
(e) $0$
\Pra54321


\newp 5.
Suppose that $g$ is the inverse of a differentiable function
$f$. Assume that $f(-1) = 0$ and that $f'(-1) = -2$. What is the
number $g'(0)$?
\ANp
(a)$- \frac{1}{2}$ 
(b)$2$ 
(c)$0$ 
(d)$-2$ 
(e)$-1$ 
\Pra14352


\five
\newp 6.  
If $f(x) = \sqrt{2+x}$, find a formula for the inverse function
$f^{-1}(x)$.
\ANp
(a)$x^2 - 2$ 
(b)$\frac{x^2}{2}$ 
(c)$\sqrt{x^2 - 2}$ 
(d)$x - 2$ 
(e)$(x - 2)^2$ 
\Pra41325

\newp 7.
If $y = e^{e^x}$ then $\frac{dy}{dx} =
$
\ANp
(a) $e^{e^x} e^x$
(b) $e^{e^x}$
(c) $e^{e^{x-1}}$
(d) $e^{x}e^{e^{x-1}}$
(e) $e^{x} e^{x-1}$
\Pra53124

\newp 8.
If $f(x) = \ln \frac{x}{x-1}$ then $f'(x) = $
\ANp
(a)$\frac{1}{x} - \frac{1}{x-1}$
(b)$\frac{1}{x} + \frac{1}{x-1}$
(c)$\frac{x-1}{x}$ 
(d)$\frac{x}{x-1}$ 
(e)$\frac{1}{x} - 1$ 
\Pra32154

\newp 9.
Simplify the expression $\ln 3 + \frac{1}{3} \ln 8$.
\ANp
(a) $\ln 6$
(b) $\frac{1}{3} \ln 11$
(c) $\frac{1}{3} \ln 26$
(d) $1$
(e)$\ln 2$ 
\Pra12534

\three
\newp 10.
Evaluate the indefinite integral $\int e^x \cos e^x\,\,dx$.
\ANp
(a) $\sin e^x + C$
(b) $e^x \sin e^x + C$
(c) $\cos\, (e^x)^2+C$
(d) $\frac{(e^x)^2 \cos x}{2}+C $
(e) $\DS e^{\sin x}+C$
\Pra31542

\newp 11.
Evalute the limit $\lim_{x \to
\infty} \frac{e^x}{1+e^{2x}}$.
\ANp
(a)$0$ 
(b)$1$ 
(c)$\infty$ 
(d)$2 $ 
(e) $\frac{1}{2}$
\Pra31254

\newp 12.
A large spring is extended one foot beyond its natural length, and
measurements show that a force of $6$ lb is required to hold it in
that position. The spring is then released and it returns to its natural
length.  
How much {\it work} would now be done in stretching the spring $6$
inches? (Remember that there are $12$ inches in a foot.) 
\ANp
(a) $\frac{3}{4}$ ft-lb
(b) $3$ ft-lb
(c) $\frac{9}{2}$ ft-lb
(d) $12$ ft-lb
(e) $35$ ft-lb
\Pra13542

\newchoice{Partial Credit}
\newp 13.
Set up an integral which gives the volume of a right circular cone with
height $5$ and base radius $4$.  

\newp 14.
A cylindrical tank stands upright with its circular base on the
ground. It has a base radius $1$ meter and is $2$ meters high, and
it is filled to the brim with water. 
Find the work done in
pumping half of the water in the tank up to the top.
(The density of water is $1000 \,kg/m^3$.) 

\one
\newp 16. 
 Sketch the region bounded by $y = \frac{1}{x^2}, y =
0, x = 1, x = 3$ and find the volume of the solid
obtained by revolving the region about the line $y = - 1$.

\five
\newp 17.
Find the equation of the line tangent to the graph
of the function $y = \frac{e^x}{x}$ at the point $(1, e)$.

\vfill\eject}
%\exam{Byers}
%\exam{Dwyer}
%\exam{Wong}
%\exam{Xavier}
%\exam{Vassiliev}
%\exam{Neusel}
\exam{}
%\marked{}
\end
