My Latex is not working
I have to apologize to those of you that use my script to render latex symbols in their blog. My website was down and then the hosting company of my site move me to another server. I didn’t remember to put everything back as it was. So the location of the script was changing. Now if you still want to use the mathtex you can use the code below. To adjust with your template change the color from black into whatever color that suitable for you (you may need to find some latex documentation to see what colors are well defined).
The weakness of the mathtex script is that there is no reliable public server for mathtex. John Forkosh doing a great job for giving this public server but sometimes if you produce a lot of math symbols from time to time there is a warning that you need to install mathtex on your own.
I will discuss how to use another method (for example mathjax that I use it myself on my question and answer forum: http://www.ask.watchmath.com)
If you still have problem with the script below please let me know.
<script src="http://www.watchmath.com/main/cgi-bin/mathtex3.js?" type="text/javascript"></script> <script type="text/javascript">window.mathPreamble = '\\usepackage[usenames]{color}\\color{black} \\gammacorrection{1.3}\\png \\normal ';replaceMath( document.body );</script> <center> <a href="http://www.watchmath.com"> <img alt="" src="http://www.watchmath.com/main/images/formula.png" width="100"></a> <br> <a href="http://watchmath.com/vlog/?p=1461"> Install LaTeX?</a></center>
As usual here is the demo site:
http://www.watchmath.blogspot.com
Fundamental Theorem of Calculus
If you want to know for example of how to compute

First we need the statement of the fundamental theorem of calculus
Let
be a continuous function on
. Then the function defined on the interval
by

is differentiable and

Ok, so what the statement really says?
1) If
is a continuous function, then
has an antiderivative (on
)
2) Roughly the statement says that taking integral and taking derivative cancel and give you back your original function.
Well it doesn’t sound so important! Then don’t worry about that
, you can just move on with your life
.
Practice Problems:
Find the derivative of
1)
2)
3)
4)
Please try to do the above problems. You may write your answer on the comment section below.
The set of polynomials with integer coefficients is not a PID
Let
(i.e., the set polynomials with even constant term). Show that
is an ideal and show that
for any
.
Answer
If
then
. Hence
. If
then
. Hence
. Therefore
is an ideal.
Suppose on the contrary
for some
. Since
, then
for some
. By the degree consideration
must be a constant polynomial, and
divides
. It follows that
or
. If
, then
which is not true since
. If
, then
. Then
for some
which is not true since
(contradiction!). Therefore
for any
.
