The above demonstration is on tangent lines. This is a concept learned early in calculus. A tangent line is a linear representation of an equation at some point. The closer to the point of origin, the closer (i.e. the less error) the function is approximated.

In this demo, the red bar can be moved to select the location at which the tangent line will be calculated. The resulting graph will show the calculated tangent line in green. As with previous math demos, the black dots may be moved around to modify the function, and more or less points can be used.

The equation for the tangent line at some point a is g( x ) = m ( x – a ) + b, where m = f '( a ), b = f( a ), and f is the function. This demo creates a polynomial function for f based on the points the function needs to intercept. The resulting polynomial function can be expressed:

f( x ) = c₀ + c₁ x + c₂ x² + c₃ x³ + … + c_n xⁿ

The derivative is then:

f '( x ) = c₁ + 2 c₂ x + 3 c₃ x² + … + n c_n x^n-1

This is quite easy to compute. So the slope (m) of the line is based on the derivative, and the intercept (b) is based on the root function.

The demonstration is (I hope) the first part of a more complected demonstration I plan on doing latter on.

   I've been playing with VirtualBox for sometime, and I learned a new fetcher I like. DrQue.net runs on a Ubuntu-based web server. When developing the PHP code for demos I've made in the past, I've always had the server next to me. However, without instant access to the server, I needed a way to have a LAMP setup running. I decided VirtualBox was the answer. Using “Host-only networking” I was able to use a Ubuntu virtual machine, and setup a LAMP server visible to the local host. I was able to use my local web browsers to develop the PHP and Javascript just as if I had DrQue.net sitting next to me.
   The picture is one in a series of shots taken at Disaster House.

   One in a series of shots taken at Disaster House.

   One in a series of shots taken at Disaster House.

   Amy and Aislinn at Zen and Amy's wedding reception.

   Back in March I had a class in linear algebra which unfortunately I didn't feel was taught all that well.  Since I felt the homework wasn't doing a good job of getting me to learn the techniques we were being taught, I decided to write a C++ library that implemented several of the methods this class covered. 
   For my linear algebra class, I used OpenOffice Calc to do my matrix calculations.  This made it easy to check my steps and eliminate arithmetic mistakes (which I seem to make a lot).  One thing I noticed was that OpenOffice often ended up with tinny rounding errors.  I would get results like 1E-15 (1^-15) that I had to manually replace with 0.  The reality was, all the matrix math I was doing used integer numbers.  Since all the matrix operations I did were just addition, subtraction, multiplication, and division, everything could be done using rational numbers.  So my first task was to write a rational number template class.
   A rational number is a number that can be expressed as a quotient, so in the form a/b or "a over b".  Rational numbers (set Q) are made of two integer number (set Z), a and b.  In C/C++, there are several integer types, so I decided a template class would work the best so the programmer could select the integer type they needed.  In addition, as long as the type supported basic arithmetic (subtraction, multiplication, and division), any class that overloaded these operators could be used.  This allows the use of arbitrary precision libraries such as the GNU Multiple Precision Arithmetic Library. 
   The matrix class is also a template.  The elements of the matrix can be of any type (integer, floating-point, rational, ext.)  I implemented the following functions: addition/subtraction, transpose, row echelon reduction, determinants, dot product, matrix product, inversion, and LU decomposition.
   The libraries are available here, and linked on the projects page.

So in my calculus III class, we're learning how to solve multiple integrals. I like to write about what I'm learning in my math classes, but I don't really have a lot of application for this. So, I decided to expand on my favorite integral. Here is my result:

For those of you who can solve this, go ahead—it's fun. But for those of you who can't, let me walk you through it. We first solve the inside most integral.

Substitute this back...

Now solve the inside integral again...

Substitute this back... (math is sometimes a lot of 'rinse and repeat')

Now, solve the finial integral...

There you have it. Mathematically I've shown that if you use the correct limits of integration, and do so three times (i.e. if you do it right) then can end up being 69. Don't get it? Then try and read the letters of the previous equation. Yep, I've learned so much now that I am attending a university raked in the top 20 in the country, I'm making licentious innuendos using math for lack of anything else to demonstrate.