2009
10.31

## The limit of a function f(x)

The limit of $\emph{f}(x)$ as $x$ approaches $a$ is the number $L$, written

$\mathop {\lim }\limits_{x \to a} \emph{f}(x) = L$

provided the $\emph{f}(x)$ is arbitrarily close to $L$ for all $x$ sufficiently close to, but not equal to, $a$. A Limit is independent of the way (from the left or from the right) in which $x$ approaches $a$. Here are some properties of limits:

• $\text{If }\emph{f}(x) = c \text{ is a constant function, then}\mathop {\lim }\limits_{x \to a} \emph{f}(x) = \mathop {\lim }\limits_{x \to a} c = c$
• $\mathop {\lim }\limits_{x \to a} x^{n} = a^{n} \text{ for any positive integer } n$
$\text{If }\mathop {\lim }\limits_{x \to a} \emph{f}(x) = L_{1} \text{ and } \mathop {\lim }\limits_{x \to a} \emph{g}(x) = L_{2} \text{, where } L_{1} \text{ and } L_{2} \text{ are real numbers, then}$

• $\mathop {\lim }\limits_{x \to a} [ \emph{f}(x) \pm \emph{g}(x) ] = \mathop {\lim }\limits_{x \to a} \emph{f}(x) \pm \mathop {\lim }\limits_{x \to a} \emph{g}(x) = L_{1} \pm L_{2}$
• $\mathop {\lim }\limits_{x \to a} [ \emph{f}(x) \cdot \emph{g}(x) ] = \mathop {\lim }\limits_{x \to a} \emph{f}(x) \cdot \mathop {\lim }\limits_{x \to a} \emph{g}(x) = L_{1} \cdot L_{2}$
• $\mathop {\lim }\limits_{x \to a} [ c\emph{f}(x) ] = c \cdot \mathop {\lim }\limits_{x \to a} \emph{f}(x) = cL_{2} \text{, where } c \text{ is a constant}$
• $\text{If }\emph{f} \text{ is a polynomial function, then}\mathop {\lim }\limits_{x \to a} \emph{f}(x) = \mathop {\lim }\limits_{x \to a} \emph{f}(a)$
• $\mathop {\lim }\limits_{x \to a} \frac{ \emph{f}(x) }{ \emph{g}(x) } = \frac{ \mathop {\lim }\limits_{x \to a} \emph{f}(x) }{ \mathop {\lim }\limits_{x \to a} \emph{g}(x) } = \frac{ L_{1} }{ L_{2} } \text{, if } { L_{2} } \neq 0$
• $\mathop {\lim }\limits_{x \to a} \sqrt[n]{\emph{f}(x)} = \sqrt[n]{\mathop {\lim }\limits_{x \to a} \emph{f}(a)} = \sqrt[n]{L_{1}}$ *

* If $n$ is even, $L_{1}$ must be positive

Tip! Wolfram Alpha makes calculating limits really easy. Check this out.

2009
10.30

## Editing- and running R scripts with Notepad++

Many people, including myself, might not like editing .r (source) files in R itself. Therefore I present you a very ingenious way of avoiding this.

“NppToR provides R language syntax highlighting, code folding, and auto-completion to Notepad++. In addition, it provides Rgui style code passing between Notepad++ and the Rgui.”

To use NppToR the first step is to download and install R. Secondly, we dowload and install notepad++. Finally, we download and install NppToR.

Open NppToR by clicking on . In the settings window we can see/edit which hot-keys to use and adjust a few thing.

An .r file in notepadd++ now look like

nppToR Code Snippet (Series Problem)

Passing it to R ( F8 (lines), ^F8 (file) ) results in the following

nppToR code passing

My friend Alexander, pointed out that writing R code is even easier (and more fancy) when editing in Komodo (in combination with SciViews-K). I may come back to that later!

2009
10.29

## How to style br.spacer_ with tinyMCE Advanced in WordPress

When using the “Stop removing the <p> and <br /> tags when saving and show them in the HTML editor” switch with tinyMCE Advanced in WordPress you finally have newlines! The only problem is that they become huge and thereby really ugly. One way, I found, to overcome this problem is to style br.spacer_

To do this, you have to put:

br.spacer_ {
line-height: 0;
}


in your theme’s css file. You can find this css file in wp-content/themes/####/

Note that after upgrading your theme you have to do this trick again…

2009
10.29

The graph of the quadratic function $\emph{y} = \emph{f}(x) = ax^{2} + bx + c$ is parabola.

• If $a > 0$, the parabola opens upwards. If $a < 0$, it opens downwards
• The vertex is $\left ( -\frac{b}{2a} , \: \emph{f}\left ( -\frac{b}{2a} \right ) \right )$
• The y-interacept = $c$
• The x-intercepts are obtained by setting $\emph{y} = 0$ and solving for $x$

This relationship between the value inside the square root (the discriminant), the type of solutions (two different solutions, one repeated solution, or no real solutions), and the number of x-intercepts (on the corresponding graph) of the quadratic is summarized in this table:

 $x^{2} - 2x - 3$ $x^{2} - 6x + 9$ $x^{2} + 3x + 3$ a positive number inside the square root zero inside the square root a negative number inside the square root two real solutions one (repeated) real solution two complex solutions two distinct x-intercepts one (repeated) x-intercept no x-intercepts

References:

2009
10.28

## Types of functions

A function is a rule that assigns to each input number exactly one output number. The set of input numbers to which the rule applies is called the domain of the function. The set of all output numbers is called the range.

A variable that represents input numbers for a function is called an independent variable. A variable that represents output numbers is called a dependent variable.

Output numbers such as $\emph{f}(-4)$ are called function values; they are in the range of $\emph{f}$.

Lets be specific about the domain of a function. Unless otherwise stated, the domain consists of all real numbers for which the equation makes sense and gives function values that are real numbers.

Hint,

• We cannot divide by zero
• $\sqrt{x}$ where $x < 0$ does not give a real number (imaginary number)

# Types of functions

Any function in the form $\emph{f}(x) = c$, where $c$ is a constant, is called a constant function.

A constant function belongs to a broader class of functions, called polynomial functions. in general a function of the form $\emph{f}(x) = c_{n}x^{n} + c_{n-1}x^{n-1} + c_{n}x^{n} + \ldots + c_{1}x + c_{0}$ The number $n$ is called the degree of a function, and $c_{n}$ is the leading coefficient. Note that a nonzero constant function, such as $\emph{f}(x) = 5$ [which can be written as $\emph{f}(x) = 5x^{0}$], is a polynomial function of degree 0.

A function that is the quotient of polynomial functions is called a rational function. $\emph{f}(x) = \frac{x^{2}-6x}{x + 5}$

Sometimes more than one equation is needed to define a function.

$\emph{f}(x) = \left\{ \begin{array}{rcr} 1, & \text{if} & -1 \leq s < 1, \\ 0, & \text{if} & 1 \leq s \leq 2, \\ s - 3, & \text{if} & 2 < s \leq 3. \end{array} \right.$

This is called a compound function.

The function $\emph{f}(x) = \left | x \right |$ is called the absolute function. Recall that that the absolute value, of a real number $x$ is denoted $\left | x \right |$ and is defined by,

$\left | x \right | = \left\{ \begin{array}{rcr} x, & \text{if} & x \geq 0, \\ -x, & \text{if} & x < 0. \end{array} \right.$

# Graphs

The graph of a function $\emph{f}$ is simply the graph of the function $\emph{y} = \emph{f}(x)$. It consists of all points $(x, y)$.

In general, the domain consists of all x-values that are included in the graph, and the range is all y-values that are included.

f(x) = x^2 + 2x + 3

It is clear that the domain of this function is all real numbers and the range is all reals $\geq -4$

2009
10.28

## Exponents and Radicals

The product $x \cdot x \cdot x$ is abbreviated $x^{3} = x^{n}$ where $n = 3$. The letter $n$ is called the exponent and the letter $x$ is called the base.

The principal nth root of $x$ is that the $n$th root of $x$ which is if positive if $x$ is positive, and is negtive if $x$ is negative and $n$ is odd.

Thus,

$\sqrt[n]{x} \text{\: is \:} \left\{\begin{array}{ll} \text{positive \: if \:} x \text{\: is \: positive} \\ \text{negative \: if \:} x \text{\: is \: negative \: and \:} n \text{\: is \: odd} \end{array}\right.$

The symbol $\sqrt[n]{x}$ is called a radical. Here $n$ is the index, $x$ is the radicand and $\sqrt{}$ is the radical sign. With principal square roots we usually omit the index and write $\sqrt{x}$ instead of $\sqrt[n]{x}$.

2009
10.28

## Changing the timezone in Debian (Sarge)

If the timezone is not set or is wrong, you can run tzconfig to configure it after the operating system is installed. tzconfig will ask you a few simple questions and after that the time will be correct (again).

Your default time zone is set to 'Europe/Amsterdam'.
Local time is now: Wed Oct 28 11:41:01 CET 2009.
Universal Time is now: Wed Oct 28 10:41:01 UTC 2009.

2009
10.27

## Installing TeX Live on Debian Etch

After doing some math assignment for my study I’d really like to publish these things (in wordpress). I found a few ways to do so. One of them was wpmathpub, but I’d rather use LaTeX. Therefore I choose WP LaTeX and because I like it the non easy way I used my server’s installation of LaTeX. A requirement for this is that you have a running LaTeX installation. No problem I thought. Unfortunately, apt was giving me all sorts of errors related to this bug? Fortunately, I found a solution!

First we have to add an apt repositoy

nano /etc/apt/sources.list.d/texlive.list


and add the following lines:

deb http://people.debian.org/~preining/TeX/ tl2007/
deb http://people.debian.org/~preining/TeX/ context/
deb http://people.debian.org/~preining/TeX/ lmodern/
deb http://www.backports.org/debian etch-backports main


Then, we update apt and install the needed packages:

apt-get update

apt-get -t etch-backports install \
texlive texlive-base texlive-base-bin \
texlive-context texlive-fonts-recommended texlive-latex-base \
texlive-latex-recommended texlive-metapost \
texlive-pdfetex texlive-math-extra

apt-get install install dvipng


If you want to get rid of the warning of your apt-get/aptitude, import this public key into the apt keyring by calling:

wget -qO - http://www.logic.at/people/preining/dsa.asc | apt-key add -


References:

2009
10.27

## Installing osTicket on CentOS 5.3

First we have to download osTicket and make it a available via http:

cd /var/www
wget http://osticket.com/dl/osticket_1.6.rc5.tar.gz
tar -xzvf osticket_1.6.rc5.tar.gz
ln -s /var/www/osticket_1.6.rc5/upload /var/www/html/osticket


Then we create a database by means of the MySQL client:

mysql>
CREATE USER 'osticket'@'localhost' IDENTIFIED BY '****************';

GRANT USAGE ON * . * TO 'osticket'@'localhost' IDENTIFIED BY '****************'
WITH MAX_QUERIES_PER_HOUR 0 MAX_CONNECTIONS_PER_HOUR 0
MAX_UPDATES_PER_HOUR 0 MAX_USER_CONNECTIONS 0 ;

CREATE DATABASE IF NOT EXISTS osticket ;

GRANT ALL PRIVILEGES ON osticket . * TO 'osticket'@'localhost';

FLUSH PRIVILEGES;


Change the owner of the osticket folder,

cd /var/www
chown -R apache:apache osticket_1.6.rc5


because include/settings.php must be writable for the installer (user: apache).

Access the installer through http://192.168.1.1/osticket and fill in the right data (e.g. MySQL username and password)

Finally, follow the suggestions of the installer:

• Change permission of include/settings.php to remove write access
• Delete or move setup directory
cd /var/www
chown -R root:root osticket_1.6.rc5
cd osticket_1.6.rc5
mv setup _setup


Optional (for password encryption):

yum install php-mcrypt

2009
10.27

## Installing OTRS on CentOS 5.3

wget http://ftp.otrs.org/pub/otrs/RPMS/fedora/4/otrs-2.4.5-01.noarch.rpm


Install it!

rpm -ivh otrs-2.4.5-01.noarch.rpm
error: Failed dependencies:
perl-DBD-MySQL is needed by otrs-2.4.5-01.noarch
mysql-server is needed by otrs-2.4.5-01.noarch


You’ll probably get an error.

Install the dependencies and (re)install the OTRS package:

yum install mysql-server
rpm -ivh otrs-2.4.5-01.noarch.rpm


Restart your webserver (apache):

service httpd restart


Use the MySQL client,

mysql>

to execute the following queries (creating a new user + database, and granting all privileges to that database):

CREATE USER 'otrs'@'localhost' IDENTIFIED BY '****************';

GRANT USAGE ON * . * TO 'otrs'@'localhost' IDENTIFIED BY '****************'
WITH MAX_QUERIES_PER_HOUR 0 MAX_CONNECTIONS_PER_HOUR 0
MAX_UPDATES_PER_HOUR 0 MAX_USER_CONNECTIONS 0 ;

CREATE DATABASE IF NOT EXISTS otrs ;

GRANT ALL PRIVILEGES ON otrs . * TO 'otrs'@'localhost';

FLUSH PRIVILEGES;


Then, follow the installer and you should have a working OTRS instance soon…