Numerical Integration Please!

If you read my introduction, you probably already knew which lab I would pick. This lab is talking about Integration. It explains different way we can integrate, such as: Riemann Sum (Rectangle), Trapezoid Sum (Trapezoid, and Simpson’s Rule (parabola (dotted)). Integrations help us to calculate the approximate area underneath a given curve.
A question that will come up while working on this lab may be about comparing the few integration techniques listed previously. We will also be working with the difference of left hand, right hand and midpoint integration.
The reason I would like to work on this lab is because I have taken two and half calculus classes working with integrals and I would like to see if this opens my mind more to what exactly we are doing when we integrate.

Carrie and Bioinformatics

“How might disease spread in populated areas in the event of a bioterrorism incident, and how would it be contained?”

It is questions like the one above that motivated me to become an applied mathematics major. The general definition of Bioinformatics is an interdisciplinary field that develops and improves methods for storing, retrieving, organizing and analyzing biological data. The part that I am interested in is the study of disease. One part of bioinformatics involves taking the data collected from studies to form a complete picture for interpretation and analysis.

In the Journal for Cancer Research, “Serum Protein Fingerprinting Coupled with a Pattern-matching Algorithm Distinguishes Prostate Cancer from Benign Prostate Hyperplasia and Healthy Men” (http://cancerres.aacrjournals.org/content/62/13/3609.full#sec-2) is an article that uses a decision tree algorithm. This is a powerful tool for classification and prediction. It is a classifier in a tree form having a

  • Decision node: specifies a test on a single attribute
  • Leaf node: indicates the value of the target attribute
  • Arc/edge: split of one attribute
  • Path: a disjunction of test to make the final decision

Source: www.cse.ust.hk/~twinsen/Decision_Tree.ppt

I think that bioinformatics is a huge and very important tool in mathematics that I hope to be apart of one day.

Samantha Intro For Funsies

I am Samantha (obviously). I am only three and a half semesters away from graduating with my Math Ed degree. Many of my current classes are not “new” to me. We are working on more of the teaching aspect. However, I am in Calc 3, that stuff is new. I love Calculus. I would rather not teach it, but I love working out the problems. Currently we are working with Integrals again. Integrals would be in my list of “Favorite Math Topics”. If you would like to know all about integrals, Wikipedia knows it!….: http://en.wikipedia.org/wiki/Integral
This is what they look like:
$\int_a^bf(x)dx$
In Calc three we are working with multi-variables, so they look like this:
$\int_a^b\int_c^dF(x)dxdy$

Fun, right?!

The Polyhedra

Definition: A polyhedra is a solid figure with many plane faces, typically more than six.

I’ve always learned better with some sort of visual help. In this chapter, it has to do with the polyhedra and its properties. In geometry there are usually a lot of visual aids. The book that I have doesn’t actually have the questions written out (accidentally ordered the teacher’s guide). It seems to be focusing on the number of vertices, edges, and faces of polyhedra and the extension of some simple formulas in plane geometry to polyhedra.

Another important aspect of this chapter that I like is that it does not require a computer. The coding has definitely been difficult for me in this class and it would be nice to get a little break from that.

p-Adic Numbers

From the start, most students are introduced to rational and irrational numbers. We are taught how to construct these numbers by setting axioms and properties. We also learn about prime numbers and the properties associated with it. But at the end before most students graduate, they may not even know that there exists a whole set of different type of numbers known as the p-adic numbers.

The p-acid numbers builds upon the arithmetic of the rational numbers. As for the arithmetic of p-acid numbers, where $p$ is a prime, the process is different than what we have been used to seeing. The difference comes from the alternative definition of absolute values on $Q$.

An interesting question asked is:

Find the $7$-acid expansion of $-1$.

Euclidean Algorithm for Complex Integers

We got into the Euclidean Algorithm at the beginning of this course and one of the questions I had at the end of the lab was how the algorithm applied to other numbers systems beyond the integers.  I was curious how it worked with polynomials and complex numbers.  Imagine my delight to discover this lab.

In this lab, we investigate the $Gaussian$ $Integers$, which take the form $a+bi$ and are named after Karl Friedrich Gauss who was the first person to discover the correct way to generalize the Euclidean Algorithm for these numbers.

Much like we did with the first Euclid lab, we will investigate using complex numbers in the division algorithm.  This lab also defines a new term.  In the complex plane, the distance a point is from the origin is called its $modulus$  and the $norm$ of a complex number $z$ is denoted by $N(z)$, which is the square of its modulus.

The Euclidean Algorithm is very fascinating and I’m excited to see how else it can be applied to other number systems.

Numerical Integration

Numerical Integration is a lab exploring numerical methods for computing integrals. That is, using a computer program or calculator to find an approximation to the integral of some function $f(x)$.

Of course, because we are talking about integration we can’t go very far without the fundamental theorem of calculus: $ F(x) = \int_a^x{f(t)dt} $. Further, in this lab we will talk about a few methods for numerically computing integrals, namely: Rectangle/ Riemann Sum, Trapezoidal Sum, Parabola/ Simpson’s Rule, just to name a few.

In Calculus courses, we are usually given “nice” functions, functions that are “easy” to solve or do not require numerical methods to compute. However, the set of functions that are “nice” is very small. Thus, we must resort to numerical methods. For example, there is no elementary antiderivative to the following integral:

$$ \int{e^{e^x}dx} $$

But we can approximate it using one of the methods that we explore in this lab.

I was initially drawn toward this lab because other courses have introduced numerical integration and I have used other numerical methods by hand and wanted to further explore the topic by automating it and exploring the different methods.

Randomized Response Surveys-Janae

I found the lab about Randomized Response Surveys, from chapter 6, interesting. This lab has the reader explore how to get accurate data from a survey and how to evaluate that data.

An important mathematical term introduced in this lab would be bias, which is the difference between an estimate’s expected value and the true value to be estimated. Expected value would actually be another term defined in this chapter, there is actually a fair amount of vocabulary.

A good example question from this lab is without doing any simulations, guess the general shape of the functional relation between Pr(Heads) for the penny and the SD of the estimate. How do you think the estimator will behave if Pr(Heads) is near 0? near 1? Record your guess in the form of a sketch of a graph of SD (theta) as a function of theta = Pr(Heads).

I think the general nature of statistics, having to adjust for unexpected circumstances and how you apply logic and patterns to those circumstances is what fascinates me about this lab.