User:Mikesmith00

Martin's Page

Exercise Template

Factor the polynomial $4x^{2}+4x+1$ completely

Quiz template

1

Solve the inequality $4x^{2}+4x+1<0$ All real numbers No Solution $x<1/4$ $x>1/4$ 2

 Solve the system $\left[{\begin{array}{c c}3&4\\2&1\end{array}}\right]\left[{\begin{array}{c}x\\y\end{array}}\right]=\left[{\begin{array}{c}0\\0\end{array}}\right]$ $x=$ $y=$ My final project

For my final project, I would like to do a more in depth analysis of Gaussian quadrature integration methods. I would do an example using the change of variables to [a, b] from [-1, 1]. I would also do a discussion and example on how we can make our error term as small as we want numerically by increasing the number of zeros used, instead of just two points. So I would add 4 or 5 examples, both of normal polynomials, and functions that are not polynomial in nature, and I would make a quiz on the subject.

Final Report

For my final project, I felt that it was necessary to add information to Wikiversity on numerical integration. This is a huge part of mathematics that was completely neglected previously. I added the entire page Gaussian Quadrature, and there has only been one edit, which I can't determine what they changed. I also felt that if I were to add an entire new section, I shouldn't assume prior knowledge at levels higher than being able to perform simple integrations, so as to make it available to the widest number of possible users, while making it technically advanced enough to be of use to high level students. Before I started on the final project, I had done some homework problems and also done the in class presentation on Gaussian Quadrature, and found that in my opinion, it is a very powerful tool for doing quick integration with very little cost computationally. I did some introductory work earlier in the quarter getting comfortable with posting on Wikipedia by making some changes to the condition number page on Wikipedia. I wanted to add some information the condition number rule of thumb, although my instructor said that it wasn't quite enough material, so I added a little bit more quantifiers related to the rule of thumb. I posted my proposal to the discussion page, and there were no comments, so I went ahead and added my change, and it has not been edited. At this point I moved forward with my final project. My goal for the page was to move from simple problems with a narrow reach of uses but simple application, to simple problems with a wider reach of uses. Then in the end I wanted my page to show the full scope of uses for Gaussian quadrature. I feel like my examples meet this. Then in examples, I wanted first to confirm that the reader understands and can apply the algorithm to their own problems, and finally be able to prove that the algorithm works exactly for any polynomial of degree less than $2n$ . I want the reader to be able to complete this proof, as I feel it gives a deeper understanding of what is happening in the algorithm, and what could cause the algorithm to fail. I didn't understand until I completely understood the proof why continuity was important for approximation. At the end, I wanted to add a quiz that was primarily theoretical. I don't feel like being able to turn the crank on an algorithm imparts much knowledge at all, although it is a useful skill to have. For our purposes though, understanding the theoretical underpinning of the material is far more important than being able to use the algorithm.