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In most modern Calculus courses, the history behind the useful mathematical results are often left ignored. Though the pragmatic uses for Calculus are numerous, without a fundamental understanding of the origins of its methods, the student is left applying memorized techniques--often lacking an understanding of why those techniques work. It is our intent to explore the historical path, in significant mathematical detail, to the elementary methods of the Calculus.
We have grown accustomed to utilizing calculators and mathematical software to aid in solving Calculus problems. Increases in raw computational power have led to incredible savings in the amount of time required by mathematicians to perform laborious tasks. In addition, computational improvements have made solvable problems that were previously thought unsolvable. Many of the uses for computational systems in mathematics seemed obvious applications of such power. However, an area that struck us as a particularly interesting use for computers in mathematics is Computer Algebra Systems. The symbolic capabilities of programs such as Maple and Mathematica mysteriously simulate human problem-solving techniques. We will explore the methods behind such Computer Algebra Systems, and in order to truly understand the complexity of such methods, write our own system that mimics some of the elementary capabilities of these robust packages.
The project consists, then, of exploring two basic ideas. First, we believe that in order to truly understand any science, it is necessary to study the path that led to its creation. In this case, we have chosen to research the history of the mathematics from the 17th century that led to the basic methods of Calculus. In doing so, we hope to both improve our own understanding of Calculus and provide a comprehensible guide to others who wish to improve their understanding. Second, technology--specifically the computer--has had a large effect on the way mathematics is taught and used. We are taught to rely on Computer Algebra Systems and Graphing Calculators as an aid to solving problems. Rather than briefly describe the many ways that computer have influenced mathematics, we have chosen to explore one area in-depth. We will provide a discussion of the methods behind Computer Algebra Systems, a description of our own approach to writing a Computer Algebra System, and a Web accessible version of our software for those interested in seeing first-hand how the system works. It is our opinion that the best way for us to understand how computers have influenced mathematics is to ourselves develop a breed of software that has significantly influenced our own education in mathematics.
The project consists of two major portions that are intended for those with an introductory background in Calculus. The paper is written in such a way that anyone who has taken the usual introductory Calculus sequence should be able to understand the mathematics involved. In addition to examining the mathematics, we will explore the people that were pivotal in developing the Calculus. Though we believe understanding their mathematics is crucial, it is of equal importance to study the personal characteristics of the revolutionary thinkers that enhanced our understanding of nature.
The intent is to provide a look at Calculus from two directions: examining the technicalities of how the past has led to present methodologies and how present technology has automated the methods from the past.
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