INTRODUCTORY
ANALYSIS
A Deeper View of Calculus
Richard J. Bagby
Department of Mathematical Sciences
New Mexico State University
Las Cruces, New Mexico
San Diego San Francisco New York Boston London Toronto Sydney Tokyo
,Sponsoring Editor Barbara Holland
Production Editor Julie Bolduc
Editorial Coordinator Karen Frost
Marketing Manager Marianne Rutter
Cover Design Richard Hannus, Hannus Design Associates
Copyeditor Amy Mayfield
Composition TeXnology, Inc./MacroTEX
Printer Maple-Vail Book Manufacturing Group
This book is printed on acid-free paper.
∞
Copyright
c 2001 by Academic Press
All rights reserved. No part of this publication may be reproduced or
transmitted in any form or by any means, electronic or mechanical, includ-
ing photocopy, recording, or any information storage and retrieval system,
without permission in writing from the publisher.
Requests for permission to make copies of any part of the work should
be mailed to: Permissions Department, Harcourt, Inc., 6277 Sea Harbor
Drive, Orlando, Florida, 32887-6777.
ACADEMIC PRESS
A Harcourt Science and Technology Company
525 B Street, Suite 1900, San Diego, CA 92101-4495, USA
http://www.academicpress.com
Academic Press
Harcourt Place, 32 Jamestown Road, London NW1 7BY, UK
Harcourt/Academic Press
200 Wheeler Road, Burlington, MA 01803
http://www.harcourt-ap.com
Library of Congress Catalog Card Number: 00-103265
International Standard Book Number: 0-12-072550-9
Printed in the United States of America
00 01 02 03 04 MB 9 8 7 6 5 4 3 2 1
, CONTENTS
ACKNOWLEDGMENTS ix
PREFACE xi
I
THE REAL NUMBER SYSTEM
1. Familiar Number Systems 1
2. Intervals 6
3. Suprema and Infima 11
4. Exact Arithmetic in R 17
5. Topics for Further Study 22
II
CONTINUOUS FUNCTIONS
1. Functions in Mathematics 23
2. Continuity of Numerical Functions 28
v
, vi CONTENTS
3. The Intermediate Value Theorem 33
4. More Ways to Form Continuous Functions 36
5. Extreme Values 40
III
LIMITS
1. Sequences and Limits 46
2. Limits and Removing Discontinuities 49
3. Limits Involving ∞ 53
IV
THE DERIVATIVE
1. Differentiability 57
2. Combining Differentiable Functions 62
3. Mean Values 66
4. Second Derivatives and Approximations 72
5. Higher Derivatives 75
6. Inverse Functions 79
7. Implicit Functions and Implicit Differentiation 84
V
THE RIEMANN INTEGRAL
1. Areas and Riemann Sums 93
2. Simplifying the Conditions for Integrability 98
3. Recognizing Integrability 102
4. Functions Defined by Integrals 107
5. The Fundamental Theorem of Calculus 112
6. Topics for Further Study 115
VI
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
1. Exponents and Logarithms 116
2. Algebraic Laws as Definitions 119
ANALYSIS
A Deeper View of Calculus
Richard J. Bagby
Department of Mathematical Sciences
New Mexico State University
Las Cruces, New Mexico
San Diego San Francisco New York Boston London Toronto Sydney Tokyo
,Sponsoring Editor Barbara Holland
Production Editor Julie Bolduc
Editorial Coordinator Karen Frost
Marketing Manager Marianne Rutter
Cover Design Richard Hannus, Hannus Design Associates
Copyeditor Amy Mayfield
Composition TeXnology, Inc./MacroTEX
Printer Maple-Vail Book Manufacturing Group
This book is printed on acid-free paper.
∞
Copyright
c 2001 by Academic Press
All rights reserved. No part of this publication may be reproduced or
transmitted in any form or by any means, electronic or mechanical, includ-
ing photocopy, recording, or any information storage and retrieval system,
without permission in writing from the publisher.
Requests for permission to make copies of any part of the work should
be mailed to: Permissions Department, Harcourt, Inc., 6277 Sea Harbor
Drive, Orlando, Florida, 32887-6777.
ACADEMIC PRESS
A Harcourt Science and Technology Company
525 B Street, Suite 1900, San Diego, CA 92101-4495, USA
http://www.academicpress.com
Academic Press
Harcourt Place, 32 Jamestown Road, London NW1 7BY, UK
Harcourt/Academic Press
200 Wheeler Road, Burlington, MA 01803
http://www.harcourt-ap.com
Library of Congress Catalog Card Number: 00-103265
International Standard Book Number: 0-12-072550-9
Printed in the United States of America
00 01 02 03 04 MB 9 8 7 6 5 4 3 2 1
, CONTENTS
ACKNOWLEDGMENTS ix
PREFACE xi
I
THE REAL NUMBER SYSTEM
1. Familiar Number Systems 1
2. Intervals 6
3. Suprema and Infima 11
4. Exact Arithmetic in R 17
5. Topics for Further Study 22
II
CONTINUOUS FUNCTIONS
1. Functions in Mathematics 23
2. Continuity of Numerical Functions 28
v
, vi CONTENTS
3. The Intermediate Value Theorem 33
4. More Ways to Form Continuous Functions 36
5. Extreme Values 40
III
LIMITS
1. Sequences and Limits 46
2. Limits and Removing Discontinuities 49
3. Limits Involving ∞ 53
IV
THE DERIVATIVE
1. Differentiability 57
2. Combining Differentiable Functions 62
3. Mean Values 66
4. Second Derivatives and Approximations 72
5. Higher Derivatives 75
6. Inverse Functions 79
7. Implicit Functions and Implicit Differentiation 84
V
THE RIEMANN INTEGRAL
1. Areas and Riemann Sums 93
2. Simplifying the Conditions for Integrability 98
3. Recognizing Integrability 102
4. Functions Defined by Integrals 107
5. The Fundamental Theorem of Calculus 112
6. Topics for Further Study 115
VI
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
1. Exponents and Logarithms 116
2. Algebraic Laws as Definitions 119
