[OS X TeX] pdf corrupted, again.

Fri May 2 15:23:20 EDT 2008

Hi,

I reported to this list a while ago about  the problem with the output pdf file (corrupted). I thought it was a package or something like that. But now, I have the problem almost every time, I am using skim and carbon emacs. This last time I decided to open acroread and open the file every time I run latex. It worked. If you have any new news about this please send to the list. It will be very helpful to everyone. 

Regards, Javier

PS. In case you want to see the file I was running here it goes.

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%To appear in \\ {\sc Journal of Algebraic Geometry}
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%%%%%%%%%%%%%%%%%%%%    Top Matter (amsart style)  %%%%%%%%%%%%%%%%%%%%%

%\date{ }
%\title[Chow varietihttp://www.rae.es/es and Euler-Chow series]{Some remarks on Chow
%varieties and Euler-Chow series} 
%\author{E. Javier Elizondo}
%\address{Instituto de Matem\'aticas, UNAM, Mexico}
%\thanks{The first author was supported in part by grants UNAM-DGAPA
%IN119298 and CONACYT 27969E} 
%\email{javier at math.unam.mx}

%\author{V. Srinivas}
%\address{School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha
%Road, Mumbai-400005, India}
%\thanks{ Srinvias, here you have to write if you like to thanks} 
%\email{srinivas at math.tifr.res.in}

\begin{document}
%\maketitle

\begin{flushright}
Name:\underline{\hspace{7cm} }\\
UIN:\underline{\hspace{7cm} }
\end{flushright} %\vspace{1.5cm}
\begin{flushleft}
MATH-172-501
\end{flushleft}
\begin{center}{\Large {\bf Final Exam}}\end{center}
\begin{center}{\large Javier Elizondo}\end{center}\vspace{.3cm}
You should name and write the tests for the convergence of series that
you need in solving a problem. 

\begin{enumerate}
\item[I.] % 7-review- 453-18
Let $\cal R$ be the region in the first quadrant bounded by
  the curves\ $y=x^3$ \ and\ $y=2x-x^2$. Calculate the volume obtained
  by rotating $\cal R$ about the $y$-axis. \vspace{.3cm}\\
Sol. $V= \int_0^1 2\pi x(2x-x^2-x^3)dx = 13\pi/13  $

\item[II.]  % 7-review-453-29
A force of 30 N is required to maintain a spring stretched from its
natural length of 12 cm to a length of 15 cm. How much work is done in
stretching the spring from 12 cm to 20 cm? \vspace{.3cm}\\
Sol. $30N = f(x)= kx =k(0.03m) \Rightarrow k=(30/0.03) (N/m) =1000
(N/m)$. \newline Now, $W = \int_0^{0.08} kx\,dx = 3.2 \,J$

\item[III.] % 8.3-476-27
Compute $$\int e^x \sqrt{9-e^{2x}}\, dx$$ 
Sol.$u = e^x \Rightarrow du = e^x \,dx.\Rightarrow \int e^x
\sqrt{9-e^{2x}} =\int \sqrt{9-u^2} \int (3\cos\theta)3\cos\theta =$
\newline $= 9/2 (\theta + \sin\theta \cos\theta) + C$   

\item[IV.]  % 8.4-485-31
Compute $$\int \frac{5x^2 + 3x -2}{x^3 + 2x^2}\, dx$$
Sol. $$ \int \left(\frac{5x^2 + 3x -2}{x^3 + 2x^2}\right) \, dx =
\int \left(\frac{2}{x}-\frac{1}{x^2}+\frac{3}{x+2}\right) dx =
2\ln|x| + \frac{1}{x} +3\ln|x+2| + C     $$

\item[V.] % 8 review-519-45
Evaluate the integral or prove that it is divergent 
$$ \int_{0}^{3} \frac{dx}{x^2 -x -2}.$$
Sol. $$ \int_{0}^{3} \frac{dx}{x^2 -x -2} = \int_0^2
\frac{dx}{(x+1)(x-2)} + \int_2^3 \frac{dx}{(x+1)(x-2)} $$
$$\text{ But }
  \int_2^3 \frac{dx}{(x+1)(x-2)} = \lim_{t\rightarrow 2^{+}} \left[
    \frac{-1/3}{x+1} + \frac{1/3}{x-2}\right] = \infty  $$

\item[VI.] % 9.1- 534-30
A Tank contains 1000 L of pure water. Brine that contains 0.05 Kg of
salt per liter of water enters the tank at a rate of 5 L/min. Brine
that contains 0.04 kg of salt per liter of water enters the tank at a
rate of 10 L/min. 
The solution
is kept thoroughly mixed and drains from the tank at the same
rate of 15 L/min. How much salt is in the tank after $t$
min?\vspace{.3cm}\\ 
Sol. $$ \frac{dy}{dt} = (0.05)(5) +(0.04)(10) -
\left(\frac{y(t)}{1000}\right) (15) =0.65 -0.015y =
\frac{130-3y}{200}$$
$$\int \frac{dy}{130 - 3y} = \int \frac{dt}{200} \text{ since } y(0)
= 0 \Rightarrow  |130 - 3y| = 130 e^{-3t/200} $$

\item[VII.] % 9.2-540-20
Solve the following differential equation 
$$
x\, \frac{dy}{dx} - \frac{y}{x+1} = x \,\,\,\,\,\,\, y(1)=0, \,\,\,\,\, x>0
$$
Sol. $$ y^\prime - \frac{y}{x(x+1)} = 1 \, \, (x>0) \Rightarrow I(x)
= \int e^{[{1}/{x(x+1)}]dx}= \frac{x+1}{x}  $$
$$ \left( \frac{x+1}{x} \, y\right)^\prime = \frac{x+1}{x} $$
$$ y = \frac{x}{x+1} (x+\ln x +C). \text{ But } y(1) = 0 \Rightarrow C= -1$$

\item[VIII.] % 10.5 617-12
Find the radius and interval of converge of 
$$ \sum_{n=1}^{\infty} \frac{(-1)^n x^{2n-1}}{(2n-1)!}$$
Sol. $$ \lim_{n\rightarrow \infty} \left|
  \frac{a_{n+1}}{a_n}\right|=\lim_{n\rightarrow \infty}
  \frac{x^2}{(2n+1) 2n} = 0 < 1$$
Therefore the series converges for all $x$.  

\item[IX.] % 10.7-633-11
Find the Taylor series for\, $f(x)=\sin x$\ at\ $x=\pi /4$.\newline What is the
radius of convergence?\vspace{.3cm}\\
Sol. $\sin x = f(\pi/4) + f^\prime (\pi/4) (x- (\pi/4)) +
\frac{f^{\prime\prime}(\pi/4)}{2!}(x-\frac{\pi}{4})^2 +
\cdots =$ Therefore
$$ \sin x = \frac{\sqrt{2}}{2}\sum_{n=0}{\infty} \,
\frac{(-1)^{n(n-1)/2} (x-\frac{\pi}{4})^n }{n!} $$
$$\lim_{n\rightarrow \infty} \left| \frac{a_{n+1}}{a_n}\right| =
\lim_{n\rightarrow \infty} \frac{|x-\frac{\pi}{4}|}{n+1} = 0 < 1\vspace{.3cm} $$
Therefore, the series converge for all the real numbers.

\item[X.] % 10.7-633-32
Evaluate the indefinite integral $$\int e^{x^3} dx$$ as an
  infinite series \vspace{.3cm}\\
Sol. 
$$\int e^{x^3} dx = \int \sum_{n=0}^{\infty} \frac{(x^3)^n}{n!} dx =
C + \sum_{n=0}^{\infty} \frac{x^{3n +1}}{(3n+1)n!} \,\, \text{ with }
R = \infty   $$

\end{enumerate}
%\underline{\hspace{16.5cm}}\\
%{\large Some formulas}
%
%$$\int \sec x \, dx = \ln |\sec x + \tan x| + C$$
%$$ \int \csc x \,dx = \ln |\csc x - \cot x | + C$$
\end{document}

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      ____________________________________________________________________________________
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