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Consider 1( z) (1) = 1. But by Axiom 4, there exists distinct1 1 such that z(1) = 1, so = 1. Suppose there exists xsuch that 0= 1, but = 0 and 1 are distinct, so zero has no reciprocal. A+ ( ) = 0;0 + 0 = 0. Then a+ ( a) + b+ ( b) = (a+ b) + ( a) + ( b) = 0 (a+ b) = a+ ( b) = a b Exercise 6. A + ( ) = 0;b b, so.

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Apostol CALCULUS VOLUME 1 One-Variable Calculus, with an Introduction to Linear Algebra SECOND EDITIONJohn Wiley & Sons, Inc.New York l Santa Barbara l London l Sydney l TorontoCONSULTING EDITORGeorge Springer, Indiana UniversityXEROX @ is a trademark of Xerox Corporation.Second Edition Copyright 01967 by John WiJey & Sons, Inc.First Edition copyright 0 1961 by Xerox Corporation.Al1 rights reserved. Permission in writing must be obtainedfrom the publisher before any part of this publication maybe reproduced or transmitted in any form or by any means,electronic or mechanical, including photocopy, recording,or any information storage or retrieval system.ISBN 0 471 00005 1Library of Congress Catalog Card Number: 67-Printed in the United States of America. PREFACEExcerpts from the Preface to the First EditionThere seems to be no general agreement as to what should constitute a first course incalculus and analytic geometry. Some people insist that the only way to really understandcalculus is to start off with a thorough treatment of the real-number system and developthe subject step by step in a logical and rigorous fashion. Others argue that calculus isprimarily a tool for engineers and physicists; they believe the course should stress applica-tions of the calculus by appeal to intuition and by extensive drill on problems which developmanipulative skills. There is much that is sound in both these points of view. Calculus isa deductive science and a branch of pure mathematics.

At the same time, it is very impor-tant to remember that calculus has strong roots in physical problems and that it derivesmuch of its power and beauty from the variety of its applications. It is possible to combinea strong theoretical development with sound training in technique; this book representsan attempt to strike a sensible balance between the two. While treating the calculus as adeductive science, the book does not neglect applications to physical problems. Proofs ofa11 the important theorems are presented as an essential part of the growth of mathematicalideas; the proofs are often preceded by a geometric or intuitive discussion to give thestudent some insight into why they take a particular form. Although these intuitive dis-cussions Will satisfy readers who are not interested in detailed proofs, the complete proofsare also included for those who prefer a more rigorous presentation.The approach in this book has been suggested by the historical and philosophical develop-ment of calculus and analytic geometry. For example, integration is treated beforedifferentiation. Although to some this may seem unusual, it is historically correct andpedagogically sound.

Moreover, it is the best way to make meaningful the true connectionbetween the integral and the derivative.The concept of the integral is defined first for step functions. Since the integral of a stepfunction is merely a finite sum, integration theory in this case is extremely simple. As thestudent learns the properties of the integral for step functions, he gains experience in theuse of the summation notation and at the same time becomes familiar with the notationfor integrals. This sets the stage SO that the transition from step functions to more generalfunctions seems easy and natural.vii CONTENTS11.1 1.1 1.1 1.1 1.1 1.12.1 2.12.1 2.1 2.13.1 3.1 3.1 3.1 3.1 3. INTRODUCTIONPart 1.

Historical IntroductionThe two basic concepts of calculusHistorical backgroundThe method of exhaustion for the area of a parabolic segmentExercisesA critical analysis of Archimedes’ methodThe approach to calculus to be used in this bookPart 2. Some Basic Concepts of the Theory of SetsIntroduction to set theoryNotations for designating setsSubsetsUnions, intersections, complementsExercisesPart 3. A Set of Axioms for the Real-Number SystemIntroductionThe field axiomsExercisesThe order axiomsExercisesIntegersand rational numbers719192121ixX Contents1 3.7 Geometric interpretation of real numbers as points on a line1 3.8 Upper bound of a set, maximum element, least upper bound (supremum)1 3.9 The least-Upper-bound axiom (completeness axiom)1 3.10 The Archimedean property of the real-number system1 3.11 Fundamental properties of the supremum and infimum.1 3.12 Exercises.1 3.13 Existence of square roots of nonnegative real numbers.1 3.14 Roots of higher order. Rational powers.1 3.15 Representation of real numbers by decimalsPart 4. Mathematical Induction, Summation Notation,and Related Topics14.1 An example of a proof by mathematical induction1 4.2 The principle of mathematical induction.1 4.3 The well-ordering principle1 4.4 Exercises.14.5 Proof of the well-ordering principle1 4.6 The summation notation1 4.7 Exercises1 4.8 Absolute values and the triangle inequality1 4.9 Exercises.14.10 Miscellaneous exercises involving induction####### 1.

THE CONCEPTS OF INTEGRAL CALCULUS1.1 The basic ideas of Cartesian geometry1.2 Functions. Informa1 description and examples.1.3 Functions. Forma1 definition as a set of ordered pairs1.4 More examples of real functions1.5 Exercises1.6 The concept of area as a set function1.7 Exercises1.8 Intervals and ordinate sets1.9 Partitions and step functions1.10 Sum and product of step functions1.11 Exercises1.12 The definition of the integral for step functions1.13 Properties of the integral of a step function1.14 Other notations for integrals-646669xii Contents3.3 The definition of continuity of a function3.4 The basic limit theorems.