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Math 300: Preparation for Senior Comprehensives |
Part 1: An Exam on the Five Fundamental Courses of Calculus 1, Calculus 2, Multivariable Calculus, Linear Systems and Discrete Math.
How we prepared: We spent a class day devoted to each of the fundamental courses and then had each student write 2 questions which could appear on the Comps Part 1 Exam. Here is a full listing of the questions:
We also compiled a List of Important Topics for each of the Fundamental Courses.
Part 2: A paper and 30-minute talk about a topic of mathematical significance.
How we prepared:Each student in the class gave an 8 minute talk generally related to the concept of "infinity" on Tuesday April 27th. Here is a listing of the talk titles and abstracts.
| Name | Abstract |
|---|---|
| Nick Biller | This paper will describe and analyze the problems and certain paradoxes related to infinity and the infinitesimal that were encountered in bringing rigor to Integral Calculus. In our introduction, we discuss the relationship between the infinite, infinitesimal and Integral Calculus. Zeno's paradoxes are discussed as is the reason why mathematicians needed to discover a way to incorporate Calculus. It is shown how set theory helped us with the infinite and how an epsilon and delta proof facilitated the use of infinitesimals in calculus. |
| Sara Blaski | A fractal is a geometric shape that is complex and detailed at every level of magnification. In general, fractals consist of simple rules: a starting image called the initiator, and the generator (by which the initiator is replaced). By definition, they are recursive, and therefore go on infinitely by iterating the function. However, they are quite complex and are strictly self-similar meaning that it does not matter which part of the fractal we analyze because it always looks exactly like a scaled-down copy of the whole. Fractals, with respect to mathematics, exist at the limit point of an infinite number if recursions. Each section contains infinite smaller sections. The theoretical result of multiple iterations is the creation of a finite area with an infinite perimeter, meaning the dimension is incomprehensible. This paper will discuss what fractals are in terms of mathematics--more specifically, geometry, infinity--and give examples of famous fractals. |
| Sandra Fuentes | |
| Veronica Juarez | An infinitesimal can be defined as a quantity, which yields 0 after some limiting process. In this paper we will look at two ways to define and grasp the concept of infinitesimal. Followed by a brief history of the development of our understanding of infinitesimal, from it’s origins in ancient Greece to modern times, and the conflicts in comprehending these concepts. Then finally a look at how our understanding of infinitesimals applies to modern mathematics, more specifically calculus. |
| Matt Piazza | This paper deals with the topic of infinity, in particular infinite series. A series is defined to be a summation of terms defined by a specified rule. A series, which consists of an infinite number of sums, is considered an infinite series. An infinite series is usually classified as either converging or diverging. These terms will be defined more thoroughly later in the paper. In order to talk about infinite series in more detail, the background of historical figures responsible for further research in the subject of infinite series will be discussed. Finally, ways that infinite series are used to solve problems will be covered. Therefore, the purpose of this paper is to provide historical background on key figures, provide the reader with an understanding of what infinite series are and what it means to be an infinite series, and finally to describe problems that would use infinite series as solutions |
| Joe Salazar | |
| Marie Smith | Infinity is a familiar concept to every mathematician. It is used in an array of mathematical areas. However, along with the concept of infinity come many logical difficulties. For instance, when evaluating a problem that tends to infinity, it is impossible to define where real numbers end and infinity begins. While infinity is not a number, it logically must take over for natural numbers at some point. Infinity, in essence, is simply based on theories and faith. Since it has no precise definition and can never be physically observed, there is no hard evidence it exists. Despite the illogic ideas associated with infinity, it is a necessary theory in the functioning of our mathematical world. Without the concept of infinity, the world as we know it, would collapse. So, how can something with inherent illogic be so central to mathematical thought today? |
| Tae Youn |
Last Updated January 22, 2007