Aspects of Calculus

525 Pages·1986·12.955 MB·English

by  Gabriel Klambauer (auth.)

Undergraduate Texts in Mathematics
Editors
F. W. Gehring
P. R. Halmos
Advisory Board
C. DePrima
I. Herstein
Undergraduate Texts in Mathematics
Apostol: Introduction to Analytic Halmos: Finite-Dimensional Vector
Number Theory. Spaces. Second edition.
Armstrong: Basic Topology. Halmos: Naive Set Theory.
BaklNewman: Complex Analysis. Iooss/Joseph: Elementary Stability and
Bifurcation Theory.
Banchoff/Wermer: Linear Algebra
Through Geometry. Janich: Topology.
Childs: A Concrete Introduction to Kemeny/Snell: Finite Markov Chains.
Higher Algebra.
Klambauer: Aspects of Calculus.
Chung: Elementary Probability Theory
with Stochastic Processes. Lang: Undergraduate Analysis.
Croom: Basic Concepts of Algebraic Lang: A First Course in Calculus. Fifth
Topology. Edition.
Curtis: Linear Algebra: Lang: Calculus of One Variable. Fifth Edition.
An Introductory Approach.
Lang: Introduction to Linear Algebra. Second
Dixmier: General Topology. Edition.
Driver: Why Math? Lax/Burstein/Lax: Calculus with
Applications and Computing, Volume I.
Ebbinghaus/FJum/Thomas Corrected Second Printing.
Mathematical Logic.
LeCuyer: College Mathematics with APL.
Fischer: Intermediate Real Analysis.
Lidl/Pilz: Applied Abstract Algebra.
Fleming: Functions of Several Variables.
Second edition. Macki/Strauss: Introduction to Optimal
Control Theory.
Foulds: Optimization Techniques: An
Introduction. Malitz: Introduction to Mathematical
Logic.
Foulds: Combination Optimization for
Undergraduates. Marsden/Weinstein: Calculus I, II, III.
Second edition.
Franklin: Methods of Mathematical
Economics.
continued after Index
Gabriel Klambauer
Aspects of Calculus
With 82 Illustrations
Springer-Verlag
New York Berlin Heidelberg Tokyo
Gabriel Klambauer
Department of Mathematics
University of Ottawa
Ottawa KIN 9B4 Ontario
Canada
Editorial Board
F. W. Gehring P. R. Halmos
Department of Mathematics Department of Mathematics
University of Michigan University of Santa Clara
Ann Arbor, MI 48019 Santa Clara, CA 95053
U.S.A. U.S.A.
AMS Subject Classification: 26-01, 26-A06
Library of Congress Cataloging in Publication Data
Klambauer, Gabriel.
Aspects of calculus.
(Undergraduate texts in mathematics)
Bibliography: p.
Includes index.
1. Calculus. I. Title. II. Series.
QA303.K654 1986 515 85-30283
© 1986 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover 1st edition 1986
All rights reserved. No part of this book may be translated or reproduced in any form without
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Typeset by Asco Trade Typesetting Ltd., Hong Kong.
9 8 7 6 5 4 3 2 1
ISBN-13 :978-1-4613-9563-8 e-ISBN-13 :978-1-4613-9561-4
DOl: 10.1007/978-1-4613-9561-4
To Agnes Catherine
Preface
This book is intended for students familiar with a beginner's version of
differential and integral calculus stressing only manipulation offormulas and
who are now looking for a closer study of basic concepts combined with a
more creative use of information. The work is primarily aimed at students in
mathematics, engineering, and science who find themselves in transition from
elementary calculus to rigorous courses in analysis. In addition, this book
may also be of interest to those preparing to teach a course in calculus.
Instead of exposing the reader to an excess of premature abstractions that
so easily can degenerate into pedantry, I felt it more useful to stress instruc
tive and stimulating examples. The book contains numerous worked out
examples and many of the exercises are provided with helpful hints or a
solution in outline. For further exercises the interested reader may want to
consult a problem book by the author entitled Problems and Propositions in
Analysis (New York: Marcel Dekker, 1979). For the history of calculus I
recommend the book by C. B. Boyer, The Concepts of the Calculus (New
York: Dover, 1949).
This book is made up of seven chapters and the Contents gives detailed
information concerning the topics covered. The book begins with a study of
the logarithmic and exponential functions. The treatment of these functions
is geometric rather than arithmetic in nature and quickly leads to the evalua
tion of certain limits that are of crucial importance; the approach, which
depends on a specific relation between hyperbolic segment and logarithmic
function, goes back to A. A. de Sarasa (1618~ 1667). In the Bibliography at
the end of the book the reader will find suitable references for further study.
I thank Professor P. R. Halmos, Indiana University, for the kind interest
he has shown in my work, my son Peter who prepared the illustrations for
this book, and my friends Dr. E. L. Cohen, University of Ottawa, and Dr.
viii Preface
G. K. R. Rao, University of Kenya, for valuable help and steadfast encourage
ment. I also wish to express my gratitude to the Board of Governors of the
University of Ottawa for the benefit of a sabbatical leave during a portion of
the writing of this book, and to the staff of Springer-Verlag for their fine
cooperation.
Gabriel Klambauer
Contents
CHAPTER 1
The Logarithmic and Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . 1
1. An Area Problem .............................................. .
2. The Natural Logarithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3. The Exponential Function ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4. The Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5. Miscellaneous Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Exercises to Chapter 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
CHAPTER 2
Limits and Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1. Limits........................................................ 49
2. Continuity..................................................... 68
3. Monotonic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4. Miscellaneous Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Exercises to Chapter 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
CHAPTER 3
Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
1. Basic Rules of Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
2. Derivatives of Basic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3. Mechanics of Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4. Asymptotes.................................................... 131
5. Tangent to a Conic Section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Exercises to Chapter 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
x Contents
CHAPTER 4
Applications of Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
1. Mean Value Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
2. Taylor's Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
3. Concave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
4. Newton's Method for Approximating Real Roots of Functions. ... ..... 190
5. Arithmetic and Geometric Means. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
6. Miscellaneous Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
Exercises to Chapter 4........................................... . . ... 232
CHAPTER 5
Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
1. Examples of Area Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
2. Area of a Planar Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
3. The Riemann Integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
4. Basic Propositions of Integral Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
5. Numerical Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
Exercises to Chapter 5. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 302
CHAPTER 6
Additional Topics in Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
1. The Indefinite Integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
2. Some Techniques ofIntegration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
3. Integration of Rational Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
4. Integration by Rationalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
5. Some Applications of Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
Exercises to Chapter 6................................................ 370
CHAPTER 7
Infinite Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
1. Numerical Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
2. The Formulas of Wallis and Stirling............................... 406
3. Numerical Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
4. Groupings and Rearrangements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
5. Uniform Convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448
6. Power Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466
7. Some Important Power Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
8. Miscellaneous Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483
Exercises to Chapter 7. . .. . . . . . . . . .. . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . .. . . 497
Bibliography 512
Index 513
CHAPTER 1
The Logarithmic and
Exponential Functions
1. An Area Problem
The curve y = l/x, for x > 0, is of special interest to us; it is located above the
x-axis in the first quadrant of the x, y plane and it is seen to be symmetric
with respect to the line y = x (because the equation xy = 1 remains un
changed when x and yare interchanged). See Figure 1.1 for a display of the
curve under consideration.
°
Definition. If < a < b, let Aa, b denote the area of the region bounded above
by the curve y = l/x, bounded below by the x-axis, bounded °to the left by
the line x = a, and bounded to the right by the line x = b; if < b < a, let
Aa,b = - Ab,a'
It is easily seen that Aa,a = 0, Aa,b = - Ab,a, and
Aa,c = Aa,b + Ab,c (1.1)
for any points a, b, and c on the positive part of the x-axis [see Figure 1.2 and
Figure 1.3 in connection with equation (1.1)]. Perhaps less obvious is the
relation
(1.2)
for any r > 0. To verify equation (1.2) we° pr oceed as follows. Suppose r > 1.
(The case r = 1 needs no proof; the case < r < 1 reduces to the case under
consideration when we interchange the roles of rand l/r.) In Figure 1.4 the
region indicated by vertical cross-hatching has the same area as the region
indicated by horizontal cross-hatching because the rectangle R2 (with ver
tical cross-hatching) and the rectangle Rl (with horizontal cross-hatching)