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\begin{document}
\title{Mathematics 126:  Test I}
\author{September 22, 1998}
\maketitle
\thispagestyle{empty}
{
\vspace{0.2in}
\noindent\Large Name: $\underline {\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }$
 \vspace{0.1in}
}

\begin{center}
\large 
\begin{itemize}
\item The test runs for one hour.
\item All work is to be your own. 
\item You may use one sheet of paper with formulae and notes,
but should not use any other notes or your
book. 
\item You may use graphing calculators (but not palmtops or laptops).
You may not use
\begin{enumerate}
\item notes in your calculator;
\item  numerical integration facilities on your calculator; or
\item algebra, differentiation, or integration facilities of your calculator. \end{enumerate}
\end{itemize}
\end{center}

\medskip

{\large  The   possible answers to the seven multiple choice questions are either correct or wrong.
{\em Please write the letter of the correct answer in the space provided below.}
The remaining questions will require that you demonstrate 
knowledge of a correct procedure for arriving at the answer.
Please circle the answer you want us to consider.
The correct answer is not enough for full credit: 
you must indicate how you arrived at it.
Please mark out any work on a problem that you do not want us
to consider.

\smallskip

\begin{center}
\begin{tabular}{|c|c|c|c|}\hline
Problem & Answer&Possible Points&Points Lost\\ \hline\hline
1&\ \ \ \ &\ 7 \ &\ \ \ \ \\ \hline
2&\ \ \ \ &\ 7 \ &\ \ \ \ \\ \hline
3&\ \ \ \ &\ 7 \ &\ \ \ \ \\ \hline
4&\ \ \ \ &\ 7 \ &\ \ \ \ \\ \hline
5&\ \ \ \ &\ 7 \ &\ \ \ \ \\ \hline
6&\ \ \ \ &\ 7 \ &\ \ \ \ \\ \hline
7&\ \ \ \ &\ 7 \ &\ \ \ \ \\ \hline
8&{\bf ---------}&\ 12 \ &\ \ \ \ \\ \hline
9&{\bf ---------}&\ 12 \ &\ \ \ \ \\ \hline
10&{\bf ---------}&\ 12 \ &\ \ \ \ \\ \hline
11&{\bf ---------}&\ 15 \ &\ \ \ \ \\ \hline
\end{tabular}
\end{center}
}
\smallskip
\noindent \hfill 
{\Large Total Points Lost $\underline {\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }$\ \ \ \ \ \ \ \ \ \ \ \ \ }



\newpage


\begin{problem}[7 points total] $f(x)=x^5+x+1$ is a differentiable increasing function defined $(-\infty,\infty)$.  Let $g(x)$ denote the inverse of $f(x)$.  Noting that $f(0)=1$, compute $g'(1)$. 
\end{problem}
\begin{tabbing}
abcdefghabcdefg\=abcdefghabcdefg\=abcdefghabcdefg\=abcdefghabcdefg\=abcdefghabcdefg\kill
a)\ \ \ \ \ \ $\displaystyle 3 $
\>b)\ \ \ \ \ \ $\displaystyle 6$
\>c)\ \ \ \ \ \ $\displaystyle\frac{1}{6}$
\>d)\ \ \ \ \ \ $\displaystyle\frac{1}{3}$ 
\>e)\ \ \ \ \ \ $\displaystyle 1$
\end{tabbing}
\vspace{3.3in}

\begin{problem}[7 points total] Find
$\displaystyle  \lim_{x\to \infty}
x^{\frac{1}{\ln(x+2)}}.$
\end{problem}
\begin{tabbing}
abcdefghabcdefg\=abcdefghabcdefg\=abcdefghabcdefg\=abcdefghabcdefg\=abcdefghabcdefg\kill
a)\ \ \ \ \ \ $\displaystyle0$ 
\>b)\ \ \ \ \ \ $\displaystyle e$
\>c)\ \ \ \ \ \ $\displaystyle e^{\frac{1}{2}} $
\>d)\ \ \ \ \ \ $\displaystyle1$
\>e)\ \ \ \ \ \ $\displaystyle \frac{1}{2}$
\end{tabbing}
\newpage


\begin{problem}[7 points total] Charcoal caused by a volcanic eruption  has 25\% of the carbon-14 found in living matter.  About how long ago was the
volcanic eruption.  Assume that the half life of carbon-14 is 5,700 years. You may use
$\ln 2=0.69$ and $\ln 0.25 =-1.38$.
\end{problem}
\begin{tabbing}
abcdefghabcdefg\=abcdefghabcdefg\=abcdefghabcdefg\=abcdefghabcdefg\=abcdefghabcdefg\kill
a)\ \ \ \ \ \ 14,250
\>b)\ \ \ \ \ \ 2,850
\>c)\ \ \ \ \ \ 5,700 
\>d)\ \ \ \ \ \ 8,550
\>e)\ \ \ \ \ \  11,400
\end{tabbing}
\vspace{3.3in}

\begin{problem}[7 points total] For $x$ satisfying $|x|<1$, $\displaystyle \tan(\sin^{-1}(x))=$
\end{problem}
\begin{tabbing}
abcdefghabcdefg\=abcdefghabcdefg\=abcdefghabcdefg\=abcdefghabcdefg\=abcdefghabcdefg\kill
a)\ \ \ \  $\displaystyle\frac{1-x^2}{x}$
\>b)\ \ \ \  $\displaystyle\sqrt{1-x^2}$
\>c)\ \ \ \  $\displaystyle\frac{x}{\sqrt{1-x^2}}$
\>d)\ \ \ \  $\displaystyle\frac{1}{1+x^2}$ 
\>e)\ \ \ \  $\displaystyle\frac{x}{1+x^2}$ 
\end{tabbing}

\newpage
\begin{problem}[7 points total] $\displaystyle \int_0^4\frac{2x\mbox{\rm d}x}{x^2+2}=$
\end{problem}
\begin{tabbing}
abcdefghab\=abcdefghab\=abcdefghabcdefghhhhhhhhccc\=abcdefghabcdefghccchhhhh\=abcdefghab \kill
a)\ \  $\displaystyle\ln 16$
\>b)\ \  $\displaystyle\ln 36$ 
\>c)\ \ $\displaystyle 2(\tan^{-1}(18)-\tan^{-1}(2)) $ 
\>d)\ \ $\displaystyle \sec^{-1}(18)-\sec^{-1}(2)$
\>e)\ \  $\displaystyle 2\ln 3$
\end{tabbing}

\vspace{3.3in}

\begin{problem}[7 points total] Use Euler's method with \mbox{\rm d}x=0.5 to estimate $y(2.5)$ if 
$\displaystyle y'=1-y/x$ and $ y(2)=-1$. 
\end{problem}
\begin{tabbing}
abcdefghabcdefg\=abcdefghabcdefg\=abcdefghabcdefg\=abcdefghabcdefg\=abcdefghabcdefg\kill
a)\ \ $\displaystyle 0.7$
\>b)\ \ \ $\displaystyle 0.75$
\>c)\ \ \ $\displaystyle -0.25$
\>d)\ \ $\displaystyle 0.25$ 
\>e)\ \ \ $\displaystyle 0.5$ 
\end{tabbing}

\newpage
\begin{problem}[7 points total] Which of the following functions satisfies  the initial value problem $\displaystyle x(1+x^2)y'=\tan(y)$ with $y(0)=0$.
\end{problem}
\begin{tabbing}
abcdefghabcdefga\=abcdefghabcdefga\=abcdefghabcdefga\=abcdefghaa\=abcdefghabcdefga\kill
a)\ \ \ \  $\displaystyle\frac{1}{1+x^2}$ 
\>b)\ \ \ \  $\displaystyle \tan^{-1}(x)$
\>c)\ \ \ \  $\displaystyle\tan^{-1}(x^2)$
\>d)\ \ \ \  $\displaystyle x $
\>e)\ \ \ \  $\displaystyle\tan^{-1}(x)+1$
\end{tabbing}

\newpage
\begin{problem}[12 points total]  $\displaystyle \int\sqrt{1+\sinh^2(x)}\mbox{\rm d}x=$  
\end{problem}

\newpage
\begin{problem}[12 points total] $\displaystyle \int\frac{\mbox{\rm d}x}{4x^2+16x+17}=$
\end{problem}

\newpage
\begin{problem}[12 points total] $\displaystyle \int\frac{\mbox{\rm d}x}{\sqrt{4-x^2}}=$
\end{problem}

\newpage
\begin{problem}[15 points total] Solve the differential equation $\displaystyle y'+\sin(x)y=2xe^{\cos(x)}$.
\end{problem}
\newpage
\ \ \ \ \ \ 
\newpage
\ \ \ \ \ \

\end{document}


