![]() Sequences and Series of Numbers 228 Pointwise Convergence of Sequences of Functions Taylor Polynomials 199 The Lagrange Remainder Theorem 203 The Convergence of Taylor Polynomials 209 A Power Series for the Logarithm 212 The Cauchy Integral Remainder Theorem 215 A Nonanalytic, Infinitely Differentiable Function The Weierstrass Approximation Theorem 223ĩ SEQUENCES AND SERIES OF FUNCTIONS 9.1 9.2 Solutions of Differential Equations 175 Integration by Parts and by Substitution 178 The Convergence of Darboux and Riemann Sums The Approximation of Integrals 190Ĩ APPROXIMATION BY TAYLOR POLYNOMIALS 8.1 8.2 8.3 8.4 8.5 8.6 8.7 Solutions of Differential Equations 116 The Natural Logarithm and Exponential Functions The Trigonometric Functions 125 The Inverse Trigonometric Functions 132Ħ INTEGRATION: TWO FUNDAMENTAL THEOREMS 6.1 6.2 6.3 6.4 6.5 6.6ĭarboux Sums Upper and Lower Integrals 135 The Archimedes-Riemann Theorem 142 Additivity, Monotonicity, and Linearity 150 Continuity and Integrability 155 The First Fundamental Theorem: Integrating Derivatives 160 The Second Fundamental Theorem: Differentiating Integrals 165ħ* INTEGRATION: FURTHER TOPICS 7.1 7.2 7.3 7.4 The Algebra of Derivatives 87 Diffe renti ating Inverses and Compositions 96 The Mean Value Theorem and Its Geo metr ic Consequences 101 The Cauchy Mea n Value Theorem and Its Anal ytic Consequences The Nota tion of Leibnitz 113ĥ* ELEMENTARY FUNCTIONS AS SOLUTIONS OF DIFFERENTIAL EQUATIONS 116 5.1 5.2 5.3 5.4 3.2 The Extreme Value Theorem 58 3.3 The Intermediate Value Theorem 62 3.4 Unifo rm Cont inuit y 66 3.5 The E-o Crite rion for Cont inuit y 70 3.6 Images and Inverses Mon oton e Func tions 3.7 Limits 81 In order to put his system into mathematical form at all, New ton had to devise the con cep t of differential quotien ts and pro pou nd the laws of mot ion in the form of differential equ atio ns- per hap s the greatest advance in thou ght that a single indi vidu al was ever privileged to make.Īn essay On the one hundredth anniversary of Maxwell's birth James Clerk Max wel l: A Com mem orat ive Volu meġ TOOLS FOR ANALYSIS 5 1.1 The Completeness Axio m and Some of Its Consequences 5 1.2 The Distr ibuti on of the Integers and the Rational Numbers 12 1.3 Inequalities and Identities 16Ģ CONVERGENT SEQUENCES 23 2.1 The Convergence of Sequences 23 2.2 Sequences and Sets 35 2.3 The Mon oton e Convergence Theo rem 2.4 The Sequential Compactness Theo rem 2.5 Covering Properties of Sets* 47ģ CON TINU OUS FUN CTIO NS 53 3.1 Cont inuit y 53. This boo k is ded icat ed to Benjamin Patrick Evans Any additional questions about permissions can be submitted by email to thomsonrights of Congress Control Number: 2004117805 ISBN 3-7 Printed in the United States of America 1 2 3 4 5 6 7 09 08 07 06 05įor more information about our products, contact us at: Thomson Learning Academic Resource Center 1-80 For permission to use material from this text or product, submit a request online at. No part of this work covered by the copyright hereon may be reproduced or used in any form or by any means-gr aphic, electronic, or mechanical, including photocopying, recording, taping, Web distribution, information storage and retrieval systems, or in any other manner-w ithout the written permission of the publisher. Thomson Higher Education 10 Davis Drive Belmont, CA 94002-3098 USAĪLL RIGHTS RESERVED. Thomson, the Star logo, and Brooks/Cole are trademarks used herein under license. © 2006 Thomson Brooks/Cole, a part of The Thomson Corporation. Print Buyer: Rebecca Cross Permissions Editor: Chelsea Junget Production Service: Matrix Productions Text Designer: John Edeen Copy Editor: Carol Dean Illustrator: Interactive Composition Corporation Cover Designer: Jennifer Mackres Cover Printer: Phoenix Color Corp Compositor: Interactive Composition Corporation Printer: R.R. ![]() Fitzpatrick Publisher: Bob Pirtle Assistant Editor: Stacy Green Editorial Assistant: Katherine Cook Technology Project Manager: Earl Perry Marketing Manager: Tom Ziolkowski Marketing Assistant: Jennifer Velasquez Marketing Communications Manager: Bryan Vann Project Manager, Editorial Production: Cheryl! Linthicum Art Director: Vernon Boes O Mexic o o Singap ore o Spain United Kingdo m Īdvanced Calculus, Second Edition Patrick M. ![]() Fitzpatrick University of Maryland, College Park
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