CALCULUS MichSapeilv ak CALCULUS Fourth Edition Copyright© 1967, 1980, 1994, 2008 by Michael Spivak ALL rights reserved Library ef Congress Catalog Card Number 80-82517 Publish or Perish, Inc. PMB 377, 1302 Waugh nv , Houston, 't xas 77019 www.mathpop.com Manufactured in the nited State ef America ISBN 978-0-914098-91-l Dedicated to the Memory of Y P PREFACE J hold l]Jery man a debtor to his profession, of from the which as men course doe seeke to receive countenance and profit, so ought they of du~ to endeavour of themselves by amends, W'!)' to be a kelp and ornament thereunto. FRANCISB ACON PREFACE TO THE FIRST EDITION E,·ery aspect of this book was influenced by the desire to present calculus not merely as a prelude to but as the first real encounter with mathematics. Since the foundations of analysis provided the arena in which modern modes of math ematical thinking developed, calculus ought to be the place in which to expect rather than avoid, the strengthening of insight with logic. In addition to devel oping the students' intuition about the beautiful concepts of analysis, it is surely equally important to persuade them that precision and rigor are neither deterrents to intuition, nor ends in themselves, but the natural medium in which to formulate and think about mathematical questions. This goal implies a view of mathematics which, in a sense, the entire book attempts to defend. No matter how well particular topics may be developed, the goals of this book will be realized only if it succeeds as a whole. For this reason, it would be of little value merely to list the topics covered, or to mention pedagogical practices and other innovations. Even the cursory glance customarily bestowed on new calculus texts will probably tell more than any such extended advertisement, and teachers ,vith strong feelings about particular aspects of calculus will know just where to look to sec if this book fulfills their requirements. A few features do require explicit comment, howe,·er. Of the twenty-nine chap ters in the book, two (starred) chapters are optional, and the three chapters com prising Part V have been included only for the benefit of those students who might want to examine on their own a construction of the real numbers. l\loreover, the appendices to Chapters 3 and 11 also contain optional material. The order of the remaining chapters is intentionally quite inflexible, since the purpose of the book is to present calculus as the evolution of one idea, not as a collection of "topics." Since the most exciting concepts of calculus do not appear until Part III, it should be pointed out that Parts I and II will probably require less time than their length suggests~ although the entire book covers a one-year course, the chapters are not meant to be covered at any uniform rate. A rather natural dividing point does occur between Parts II and III, so it is possible to reach differentiation and integration even more quickly by treating Part II ,-cry briefly, perhaps returning later for a more detailed treatment. This arrangement corresponds to the traditional organization of most calculus courses, but I feel that it will only diminish the value of the book for students who ha,·e seen a small amount of calculus pn'viousl); and for bright students with a reasonable backgTound. The problems have been desig11cd with this particular audience in mind. They range from straightforward, but not m·erly simple, exercises which develop basic techniques and test understanding of concepts, to probkms of considerable diffi culty and, I hope, of comparable interest. 'There are about 625 problems in all. Those which emphasize manipulations usually contain many examples, m1ml1ercd \'11