%$Modified: Fri Jun  8 16:23:31 2001 by wgd $

\magnification1200\footline{\ifnum\pageno=1 \hfil\else\hfil\number\pageno\hfil\fi}
\def\whichexam{Math 120, Exam II}
\def\examdate{Thursday, March 22, 2001}
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\answersheet}



%%%%%%%
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%\centerline{\examdate}
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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 {\bf one hour and fifteen minutes}.
\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!}
\vskip4pt
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PLEASE MARK YOUR ANSWERS WITH AN X, not a circle!
\correctanswers
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%
%                  Generate the answer sheet
%
\ifsampletest\relax\else\ifproofreading\relax\else\answersheet\fi\fi
%
\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.
Find the slope of the tangent line to the curve $y=\tan^{-1}(3x^2)$ at
$x=1$.
\ANp
(a) $\frac 35$
(b) $2$
(c) $\frac 23$
(d) $\frac\pi3$
(e) $\frac12$
\Pra31254

\newp 2. 
Calculate $\displaystyle\int_1^2\left(\sqrt x+\frac{1}{\sqrt
x}\right )\sqrt x \,dx$.
\ANp
(a) $\frac 52$
(b) $2^{3/2}-1$
(c) $\frac 12$
(d) $\frac\pi4$
(e) $2$
\Pra42315

\one
\newp 3.
\def\1{\frac{x^2+2x+3}{(x-1)^2(x-2)(x^2+4)}}
Which of the following equations describes the partial fraction decomposition of a
rational function of the form $\1$?
\ANp
(a) $\1=\frac A{x-1}+\frac B{(x-1)^2} + \frac C{x-2} +\frac {Dx+E}{x^2+4}$
(b) $\1=\frac A{x-1}+\frac B{(x-1)^2} + \frac C{x-2} +\frac {D}{x^2+4}$
(c) $\1=\frac A{x-1}+\frac {Bx+C}{(x-1)^2} + \frac D{x-2} +\frac {E}{x^2+4}$
(d) $\1=\frac A{x-1}+\frac B{(x-1)^2} + \frac C{x-2} +\frac {Dx}{x^2+4}$
(e) $\1=\frac A{x-1}+\frac {Bx}{(x-1)^2} + \frac C{x-2} +\frac {Dx+E}{x^2+4}$
\Pra54312

\newp 4.
According to the method of partial fractions, there is an equation of
the form
$$
\frac x{(x-1)(x-2)(x-3)}= \frac A{x-1}+\frac B{x-2}+\frac C{x-3}
$$
for some numbers $A$, $B$, and $C$. What is the number $B$?
\ANp
(a) $-2$
(b) $\frac 12$
(c) $-1$
(d) $3$
(e) $0$
\Pra15243

\newpage
\three
\newp 5.
Evaluate $\displaystyle\int_1^2t^2\ln(t)\,dt$.
\ANp
(a) $\frac83\ln(2)-\frac79$
(b) $4\ln(2)+1$
(c) $\frac92\ln(2)-\frac72$
(d) $3\ln(2)+\frac29$
(e) $4(\ln2)^2$
\Pra54321


\newp 6.  
Suppose that $x=3\tan \theta$. Find an expression for $\cos\theta$ in
terms of $x$. (Hint: draw a triangle.)
\ANp
(a) $\frac3{\sqrt{9+x^2}}$
(b) $\frac{\sqrt{x^2-9}}{x}$
(c) $\frac 3{\sqrt{9-x^2}}$
(d) $\frac{\sqrt{9+x^2}}{3x}$
(e) $\frac{\sqrt{9-x^2}}{\sqrt{9+x^2}}$
\Pra31542

\three
\newp 7.
Calculate $\displaystyle\int_0^{\pi/2}\sin^2x\cos^5x\,dx$.
\ANp
(a) $\frac 13-\frac25+\frac17$
(b) $\frac16$
(c) $\frac15+\frac47-\frac19$
(d) $1-\frac23+\frac15$
(e) $0$
\Pra41325

\two
\newp 8.
Evaluate the indefinite integral $\displaystyle\int x\cos2x\,dx$.
\ANp
(a) $\frac{x\sin2x}{2}+\frac{\cos 2x}{4}+C$
(b) $\frac{x\sin2x}{2}+\frac{\cos4x}{2}+C$
(c) $\frac{x^2\cos2x}{2}- \frac{x^2\sin2x}{4}+C$
(d) $\frac{x^2\sin2x}{4}+C$
(e) $\sin x^2+C$
\Pra53124

\newp 9.
Calculate $\displaystyle\lim_{x\to\infty}\sin(\tan^{-1} x)$.
\ANp
(a) $1$
(b) $0$
(c) $\infty$
(d) $-\infty$
(e) $2$
\Pra32154

\ifcompressedspacing\newpage\fi
\three
\newp 10.
Consider the integral
$\displaystyle\int\frac{x^5}{\sqrt{x^2-9}}\,dx$. Which of the
following trigonometric substitutions would eliminate the square root
from this integral?
\ANp
(a) $x=3\sec\theta$
(b) $x=3\cos\theta$
(c) $x=3\tan\theta$
(d) $x=3\cot\theta$
(e) $x=3\sin\theta$
\Pra12534

\newp 11.
What is $\sin^{-1}(\frac{\sqrt3}{2})$?
\ANp
(a) $\frac\pi3$
(b) $\frac\pi4$
(c) $\frac\pi2$
(d) $\pi$
(e) $\frac12$
\Pra43512

\newp 12.
Calculate $\displaystyle\lim_{x\to0}\frac{\sin^{-1}x}{2x}$
\ANp
(a) $\frac12$
(b) $0$
(c) $\infty$
(d) $-2$
(e) $-\infty$
\Pra31254

\newchoice{Partial Credit}
\newp 13.
Calculate $\displaystyle\int\frac{2x^2+1}{x(x^2+1)}\,dx$.

\newp 14.
Find the length of the curve $y=\ln(\cos x)$ for $0\le x\le\frac\pi4$.


\newp 15. Calculate $\displaystyle\int\frac{dx}{(x^2+1)\sqrt{x^2+1}}$


\five
\newp 16.
Calculate $$\displaystyle\lim_{x\to0}\frac{\cos x -1
+({x^2}/2)}{x^4}$$
(Hint: don't give up too soon!)
\vfill\eject}
\exam{}
%\exam{Byers}
%\exam{Dwyer}
%\exam{Wong}
%\exam{Xavier}
%\exam{Vassiliev}
%\exam{Neusel}
%\marked{}
\end
