\documentclass[12pt]{article}

 
\setlength{\parindent}{0pt}
\everymath={\displaystyle}
%\pagestyle{empty}
\newcommand{\answerspace}{\vfill}
\newcommand{\problemspace}{\newpage}
\setlength{\textwidth}{6in}
\setlength{\oddsidemargin}{.30in}
\topmargin=-0.5in
\textheight=9in

\begin{document}

\textbf{Math 126: Calculus II}
\hfill\makebox[2.5in][r]{Name:\hrulefill}

\textbf{Exam I}\quad \textsl{September 30, 1999}
\hfill\makebox[2.5in][r]{Section:\hrulefill}

\bigskip

There are 5 problems worth a total of 80 points
 on 6 pages plus a page to be used as a scratch
sheet.
You start with 20 points.  Unless otherwise noted, each part of a
 problem
is worth the same number of points.

\medskip

Please read the questions carefully, especially the {\bf bold} words.

\medskip
You may use a calculator if you wish.

\medskip
If you want partial credit on any problem, be sure to indicate
{\bf clearly} what you are doing.  {\bf Show your work}
and {\bf all} important steps. {\bf No partial credit} will be
given
for any numerical answer that is not explained clearly.  

\bigskip


1. (10 points)  Compute the following limits. 

\bigskip

a) 
\begin{math}
  \lim _{x\to 0} \frac{\sin ^{-1}(3x)}x
\end{math}

\answerspace

b) 
\begin{math}
  \lim _{x\to 0} \frac {5-5\cos x}{e^x-x-1}
\end{math}

\answerspace

\problemspace

\hfill\makebox[2.5in][r]{Name:\hrulefill}
\medskip

2. (20 points) 
In each part of this problem you are presented with two functions 
\begin{math}
  f(x)
\end{math}
and 
\begin{math}
  g(x)
\end{math}
and you are to decide which of the following three possibilities holds
as
\begin{math}
  x\to\infty.
\end{math}
i) 
\begin{math}
  f(x)
\end{math}
grows faster than 
\begin{math}
  g(x),
\end{math}
 ii) 
\begin{math}
  g(x)
\end{math}
grows faster than 
\begin{math}
  f(x),
\end{math}
iii) 
\begin{math}
  f(x)
\end{math}
and
\begin{math}
  g(x)
\end{math}
grow at the same rate.
Write the answer as i), ii), or iii);  if the answer is iii)
{\bf you must also give}
\begin{math}
  \lim _{x\to\infty} \frac{f(x)}{g(x)}.
\end{math}

\bigskip
{\bf THINK!}


\bigskip

a) 
\begin{math}
  f(x)=(\ln x)^2
\end{math}
and 
\begin{math}
  g(x) = 2^{\ln x}.
\end{math}
\hfill\makebox[1.5in][r]{Ans. a) \hrulefill}

\answerspace


b) 
\begin{math}
  f(x)=\frac 1x
\end{math}
and 
\begin{math}
  g(x) = \cos (\tan^{-1}x).
\end{math}
\hfill\makebox[1.5in][r]{Ans. b) \hrulefill}

\answerspace


c) 
\begin{math}
  f(x)= \tan ^{-1}(x^2)
\end{math}
and 
\begin{math}
  g(x) = (\tan^{-1}x)^2.
\end{math}
\hfill\makebox[1.5in][r]{Ans. c) \hrulefill}
\answerspace

d) 
\begin{math}
  f(x)= x
\end{math}
and 
\begin{math}
  g(x) = \cosh (\ln x)
\end{math}
\hfill\makebox[1.5in][r]{Ans. d) \hrulefill}

\answerspace

\problemspace

\hfill\makebox[2.5in][r]{Name:\hrulefill}
\medskip

3. (15 points)  Radioactive radon gas has a half-life of 
\begin{math}
  3.85 
\end{math}
days. A leaky basement of a house has a dangerous 
\begin{math}
  30
\end{math}
picocuries of radioactive radon per liter of air.  After the leaks are
repaired how long will it take for the air in the basement to get down
to a safe 4 picocuries of radioactive radon per liter?  ({\em
  Remember:}
The half-life of a radioactive element is
the time required for half of the radioactive nuclei present in the
sample
to decay.)


\problemspace

\hfill\makebox[2.5in][r]{Name:\hrulefill}
\medskip

4.  (10 points)  Solve the differential equation
$\frac{dy}{dx}=e^{x-y}$.

\problemspace

\hfill\makebox[2.5in][r]{Name:\hrulefill}
\medskip

5.  (25 points, 5 points each for the first three parts, 10 points for
  the last part)
  Compute the following integrals.
 
\bigskip
 
a)
$\int_1^4 \frac{1}{\sqrt{x} + x}\, dx$
 
  \answerspace

b) 
$\int \frac{e^x }{1+4e^{2x}}\, dx$

  \answerspace
 
\problemspace

\hfill\makebox[2.5in][r]{Name:\hrulefill}
\medskip

c) 
 $\int_0^{\pi/2} x^2\sin(x)\,dx$
 
  \answerspace

d)  $\int\frac{2x}{\sqrt{6x-x^2}}\,dx$

\answerspace


\end{document}



