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 \date{{\large December 16, 1998}}
                                      \lastversion=0
                                      \input exam.perm4
                                      \title{{\large\bf Math 126, Final}}
\comment{\large \hspace*{-1cm}
\begin{minipage}[l]{5.8in}%
  1. Please cross \fbox{$\times $} the correct answers. \\
  2.  This test will be exactly 120 minutes in length. When you
  are told to begin, but not before, glance through the entire
  test and put your name on each page.  It is YOUR RESPONSIBILITY
  to make sure your test consists of 13 PAGES with 25 PROBLEMS.
  Each problem has an equal point value of 6 points. Use the back
  of the test pages for scratch work.\vspace{-3mm}
\end{minipage}} 
\answersheetfootline{\large\bf\fbox{\rule[-2mm]{0mm}{8mm}Sign
    your name:\hspace{6cm}} \hspace{6cm}}\twocolumn{16}
%\noprofessor 
\nosection \vertmultchoice=1.3in \multiplechoiceskip=0.55cm
\let\ansborder=\boxborder\boxwidth=1cm\boxdepth=0.5cm

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%1
\problem   Let $\displaystyle f(x) = x^3 + 4x - 3$, $x \ge 0$.  Find the
value of $\displaystyle\frac{d\ }{dx}f^{-1}$ at the point 
$\displaystyle x= 2 = f(1)$.

\correct  $\displaystyle \frac{1}{7}$
\wrong    $\displaystyle \frac{1}{16}$
\wrong    $\displaystyle 7$
\wrong    $\displaystyle 16$
\wrong    $\displaystyle \frac{1}{2}$

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%2
\problem $\displaystyle \frac{d\ }{dt}(\sqrt{t})^t=$  

\correct $\displaystyle \frac{(\sqrt{t})^t(1+\ln t)}{2}$
\wrong  $\displaystyle \frac{1+\ln t}{2}$
\wrong $\displaystyle (\sqrt{t})^{(t-1)}$
\wrong $\displaystyle \frac{(\sqrt{t})^{(t-2)}}{2}$
\wrong $\displaystyle t(\sqrt{t})^{(t-1)}$

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%3
\problem  Assume that a colony of bacteria is grown under ideal conditions,
and has its population increase exponentially with time.  At the end of one hour 
there are 10,000 bacteria. At the end of three hours there are 40,000 bacteria.  
How many of bacteria were present initially.

\correct  5,000
\wrong 2,500
\wrong $\displaystyle \frac{10,000}{\ln 3}$
\wrong $\displaystyle \frac{5,000}{\ln 3}$
\wrong $\displaystyle \frac{10,000\ln 3}{\ln 4}$

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%4
\problem  $\displaystyle \lim_{x\to \infty} \left(1+\frac{3}{x}\right)^x=$  

\correct $\displaystyle e^3$
\wrong $\displaystyle 1$ 
\wrong $\displaystyle \infty$
\wrong $\displaystyle 4$
\wrong $\displaystyle 20$

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%5
\problem Which of the functions
$$
\mbox{\rm 1)}\ \ 2^x\ \ \ \ \ \ \ \ \ \ \ \ 
\mbox{\rm 2)}\ \ \frac{x^2}{10000}\ \ \ \ \ \ \ \ \ \ \ \ \ 
\mbox{\rm 3)}\ \ 10000\ln x\ \ \  \ \ \ \ \ \ \ \ \ \ 
\mbox{\rm 4)}\ \ \sqrt{x}\ \ \ \ \ \ \ \ \ \ \ \ \ 
\mbox{\rm 5)}\ \ e^x
$$
 is the slowest growing as $x\to \infty$.

\correct $\displaystyle 10000\ln x$
\wrong  $\displaystyle 2^x$
\wrong $\displaystyle \frac{x^2}{10000}$
\wrong $\displaystyle \sqrt{x}$
\wrong $\displaystyle e^x$


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%6 
\problem  Simplify the expression $\displaystyle \log_{11} 121$

\correct 2
\wrong   $e^2$
\wrong   11
\wrong   1
\wrong   $e^{11}$


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%7
\problem  $\displaystyle \int \frac{e^{\arcsin(x)}}{\sqrt{1-
x^2}}\mbox{\rm d}x=$

\correct $\displaystyle e^{\arcsin(x)}+C$
\wrong   $\displaystyle e^{\sqrt{1-x^2}}+C$
\wrong   $\displaystyle e^{\sqrt{1-x^2}}+C$
\wrong   $\displaystyle \mbox{\rm arcsec}(e^x)+C$
\wrong   $\displaystyle \frac{e^{\arcsin(x)}}{\sqrt{1-x^2}}+C$


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%8
%\problem  $\displaystyle \lim_{x\to 0}\frac{2\arctan(3x)}{5x}=$

%\correct $\displaystyle \frac{6}{5}$
%\wrong   $\displaystyle 0$
%\wrong   $\displaystyle \frac{2}{5}$
%\wrong   $\displaystyle \frac{12}{5}$
%\wrong   $\displaystyle \frac{3}{25}$

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%8 
\problem  Solve $\displaystyle \frac{dy}{dx}=2(x+y^2x)$.

\correct $\displaystyle \tan(x^2+C)$
\wrong   $\displaystyle \frac{x^2}{\arctan(x+C)}$
\wrong   $\displaystyle \sin(x^2+C)$
\wrong   $\displaystyle \sinh(x^2+C)$
\wrong   $\displaystyle e^{x^2+C}\arctan(x)$

% End Sommese
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% Begin Dorabiala
\problem If $a_1 = 2$ and $ a_{n+1} = (-1)^{n+1}\frac{a_n}{2}$
then $a_3 = $ ?

\correct $-\left(\frac{1}{2}\right)$
\wrong   $\frac{1}{4}$
\wrong   $\frac{1}{2}$
\wrong   $-\left(\frac{1}{4}\right)$
\wrong   $\frac{1}{8}$

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\problem  Find $\lim_{n\to \infty} \left(\frac{3}{n} \right)^{\frac{1}{n}} $

\correct 1
\wrong   0
\wrong   3
\wrong   $e^2$
\wrong   $\frac{-1}{3}$


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\problem  Evaluate $ \int\limits_e^{+\infty } \frac{1}{x(\ln (x))^2}dx $

\correct $1$
\wrong   $\frac{\pi }{2}$
\wrong   $\frac{1 }{3}$
\wrong   $\frac{1 }{2}$
\wrong   $e^2$

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\problem  Evaluate $ \int \frac{dx}{x^2 + x}$

\correct $\ln |x| - \ln|x+1| + C$
\wrong   $\ln |x| + \ln|x+1| + C$
\wrong   $\ln |x^2 + x|  + C$
\wrong   $\ln (x^2 ) + \ln|x| + C$
\wrong   $-\left(\frac{1}{x}\right) + \ln|x| + C$


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\problem  Evaluate $ \int \frac{dx}{x^2 + 4}$

\correct $\frac{1}{2} \arctan \left(\frac{x}{2}\right)  + C$
\wrong   $ \arctan \left(\frac{x}{2}\right)  + C$
\wrong   $ -\left(\frac{1}{x}\right)  + 4 + C$
\wrong   $ \arccos\left(\frac{x}{2}\right) + C$
\wrong   $\ln ( 4 + x^2) + C$




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\problem  Evaluate $ \int x^2 \sin(x) dx$

\correct $ -x^2 \cos(x) + 2x \sin(x) + 2\cos(x) + C$
\wrong   $ x^2 \cos(x) - 2x \cos(x) - 2\cos(x) + C$
\wrong   $ -x^2 \cos(x) + 2 \cos(x) + C$
\wrong   $ 2x\sin (x) + x^2 \cos(x) - 2\sin(x) + C$
\wrong   $ -x^2 \sin(x) - 2x\cos(x) + 2\sin(x) + C$




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\problem Find the sum of the series $\sum\limits_{n=2}^{\infty } 5
{\left(\frac{1}{2}\right)}^n$

\correct $\frac{5}{2}$
\wrong   $\frac{5}{4}$
\wrong   5
\wrong   10
\wrong   15




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\problem Which of the following series converge: %
(1)  $\sum\limits_{n=1}^{\infty } \frac{(n+1)!}{3!\ n!\ 3^n}$ , %
(2)  $\sum\limits_{n=0}^{\infty }{\left(\frac{1}{2}\right)}^n$, %
(3)  $\sum\limits_{n=2}^{\infty } \frac{1}{2n - 1}$

\correct (1) and (2)
\wrong   (1) and (3)
\wrong   (2) and (3)
\wrong   (1)
\wrong   (2)



% End Dorabiala
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% Start Rosenthal
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\problem   Find the length of the curve:
$$
(x(t),y(t))=\left(3t^2,3t-t^3\right),\ 
0\leq t\leq 1.
$$

\correct 4
\wrong 5
\wrong 7
\wrong 15
\wrong 6

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\problem   For what values of $x$ does the series
$\displaystyle \sum_{n=0}^\infty (25x^2)^n$ converge absolutely.

\correct $|x|<\frac{1}{5}$
\wrong  $|x|< 1$
\wrong $|x|< 5$
\wrong $|x|<2$
\wrong $|x|<\frac{1}{25}$

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\problem  The coefficient of the degree 6 term of the Maclaurin series for
$x^2\cos(x^2)$ is:

\correct  $-\frac{1}{2}$
\wrong $\frac{1}{2}$
\wrong $0$
\wrong $\frac{1}{6!}$
\wrong $-\frac{1}{6!}$

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\problem  Find the area of the surface generated by revolving the
circle\\
$(x(t),y(t))=(\cos(t),7+\sin(t))$, $0\leq t\leq 2\pi$, around the
$x$-axis.

\correct $28 \pi^2$
\wrong $20 \pi^2$ 
\wrong $7 \pi^2$
\wrong $ \pi^2$
\wrong $49 \pi^2$

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\problem Compute the Eccentricity of the Ellipse
$\frac{x^2}{16}+\frac{y^2}{9}=1$

\correct $\frac{\sqrt{7}}{4}$ 
\wrong  $5$
\wrong $\frac{5}{4}$
\wrong $\frac{\sqrt{7}}{3}$
\wrong $\frac{5}{3}$


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\problem Consider the curve
$(x(t),y(t))=\left(\sqrt{t^2+3},t^3\right)$.  Let $P$ be the
point $(x(1),y(1))$. The tangent line to the curve at $P$ has
then the equation:

\correct $y=6x-11$
\wrong $y=3x-10$
\wrong $y=-3x-8$
\wrong $y=3x-13$
\wrong $y=6x-25$

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\problem Find the second order Taylor polynomial for $f(x)=e^x$
at $a=1$.

\correct $e+e\left (x-1\right )+{\frac {e}{2}}\left (x-1\right )^{2}$
\wrong ${\frac {5}{2}}+2\,\left (x-1\right )+\frac {1}{2}\left (x-1\right )^{2}$
\wrong $\frac {5e}{2}+2\,e\left (x-1\right )+\frac {e}{2}\left (x-1\right )^{2}$
\wrong $\frac {7e}{3}+\frac{5e}{3}\left(x-1\right)+\frac{e}{3}\left(x-1\right)^{2}$
\wrong $e{x}^{2}+ex+1$


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\problem  The polar equation $r=-8\cos(\theta)$, where $r\geq 0$ and
$0\leq \theta\leq 2\pi$, is the same as the cartesian equation:

\correct $x^2+8x+y^2=0$
\wrong $(x-4)^2+y^2=16$
\wrong $(x+4)^2+y^2=8$
\wrong $(x-4)^2+y^2=8$
\wrong $x^2-8x+y^2-8=0$

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%25
\problem Letting $i$ denote $\sqrt{-1}$, $\displaystyle \frac{1}{5+i} =$

\correct $\displaystyle \frac{5-i}{26} $
\wrong $\displaystyle \frac{5+i}{26} $
\wrong $\displaystyle \frac{5-i}{24} $
\wrong $\displaystyle \frac{5-i}{4} $
\wrong $\displaystyle \frac{5+i}{24} $


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\endtest
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\problem  If $x\ge 0$ then $\displaystyle \sin\left(\arctan(\sqrt{x^2+2x})\right)=$

\correct  $\displaystyle \frac{\sqrt{x^2+2x}}{x+1}$
\wrong  $\displaystyle \frac{\sqrt{x^2+2x}}{1-2x-x^2}$
\wrong  $\displaystyle \frac{x+1}{\sqrt{x^2+2x}}$
\wrong  $\displaystyle \frac{x+1}{x^2+2x}$
\wrong  $\displaystyle \frac{x^2+2x}{x+1}$

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 \end{document}