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1998 | Book | 2. edition

Tokens from the Gods Read chapter Read first chapter

Calculus: A Liberal Art

Author: William McGowen Priestley

Publisher: Springer New York

Book Series : Undergraduate Texts in Mathematics

Included in: Professional Book Archive

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About this book

reason for delaying its study has to do with the question of mathematical maturity. * No use is made here of trigonometric, logarithmic, or expo­ nential functions except in occasional optional material indicating how such functions can be handled. A perceptive remark made by George P6lya suggests how we can simultaneously learn mathematics and learn "about" mathematics-i.e., about the nature of mathematics and how it is developed: If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference. The reader will find plenty of opportunity here for guessing. The early chapters go at a gentle pace and invite the reader to enter into the spirit of the investigation. Exercises asking the reader to "make a guess" should be taken in this spirit-as simply an invitation to speculate about what is the likely truth in a given situation without feeling any pressure to guess "correctly". Readers will soon realize that a matter about which they are asked to guess will likely be a topic of serious discussion later on.

Table of Contents

Frontmatter
Chapter 1. Tokens from the Gods
Abstract
A calculus is a pebble, or small stone.* Playing with pebbles, or “calculating”, is a primitive form of arithmetic. The calculus, or calculus, refers to some mathematics that was developed principally in the seventeenth century.
William McGowen Priestley
Chapter 2. Rational Thoughts
Abstract
The purpose of this chapter and the next is to remind ourselves of the elements of precalculus mathematics by studying a little history. Some readers may have the urge instead to jump flat-footed into an attack upon the problem that arose in Chapter 1, the problem of how the highest and lowest points of a curve may easily be found. Those readers may scan these short chapters quickly and jump into Chapter 4 if they wish, but they are warned that flat-footed jumps are awkward without a firm foundation from which to leap. Studying history builds foundations.
William McGowen Priestley
Chapter 3. To Measure Is to Know
Abstract
As we have seen in the previous chapter, the word ratio is connected with the idea of rational thought or calculated study. The Greek word for the same notion is logos, which has similarly acquired overlays of meanings stemming from the idea of measurement. We find it as a suffix in many academic words derived from Greek: anthropology (“study of man”), biology (“study of life”), and so on. The word logic, of course, also comes from logos.
William McGowen Priestley
Chapter 4. Sherlock Holmes Meets Pierre de Fermat
Abstract
Given a curve, such as the one below, how can one locate its lowest point? This problem arose naturally in Chapter 1, along with the analogous problem of finding the highest point on a curve. Both problems can be solved by the same method, to which we now turn.
William McGowen Priestley
Chapter 5. Optimistic Steps
Abstract
What is calculus? It is the study of the interplay between a function and its derivative. There are quite a few aspects to this interplay, some of which may be surprising. In this chapter we shall learn more about the use of derivatives in solving optimization problems. To do this efficiently, the major part of the chapter is concentrated upon the development of shortcut rules for finding derivatives.
William McGowen Priestley
Chapter 6. Chains and Change
Abstract
Things change. The world is in flux. How can one understand a world in which change plays so great a role? The seventeenth-century answer given by Leibniz and Newton is simplicity itself:
Study change.
William McGowen Priestley
Chapter 7. The Integrity of Ancient and Modern Mathematics
Abstract
When minds of first order meet, sparks fly, even across the centuries. The fundamental theorem of calculus, to be discussed in this chapter, is the result of such a pyrotechnic fusion of ideas. When Leibniz and Newton met Eudoxus and Archimedes, the calculus was rounded out into a whole. By the end of the seventeenth century it was becoming evident that calculus was not a bag of unrelated tricks but was an entity complete unto itself.
William McGowen Priestley
Chapter 8. Romance in Reason
Abstract
Having reached the fundamental theorem of calculus, we have come to a natural place to pause and take stock of our accomplishments and aspirations. Let us close this volume by discussing the way calculus was viewed in the seventeenth century, looking both backward and forward in time to put this remarkable century’s mathematical thought in true perspective—and perhaps to learn something about the true nature of mathematics itself.
William McGowen Priestley
Backmatter
Metadata
Title
Calculus: A Liberal Art
Author
William McGowen Priestley
Copyright Year
1998
Publisher
Springer New York
Electronic ISBN
978-1-4612-1658-2
Print ISBN
978-1-4612-7233-5
DOI
https://doi.org/10.1007/978-1-4612-1658-2