Polynomials, Power Series and Calculus (University Series in Undergraduate Mathematics)

169 Pages·1968·5.796 MB·English

by  Howard Levi

Polynomials, Power Series,
and Calculus
THE UNIVERSITY SERIES IN
UNDERGRADUATE MATHEMATICS
Editors
John L. Kelley, University of California
Paul R. Halmos, University of Michigan
PATRICK SuPPEs-Introduction to Logic
PAUL R. HALMos-Finite-Dimensional Vector Spaces, 2nd Ed.
J.
EDWARD McSHANE and TRUMAN A. BoTTS-Real Analysis
J.
JoHN G. KEMENY and LAURIE SNELL-Finite Markov Chains
PATRICK SuPPEs-Axiomatic Set Theory
PAUL R. HALMos-Naive Set Theory
JoHN L. KELLEY-Introduction to Modern Algebra
IvAN NIVEN-Calculus: An Introductory Approach, 2nd Ed.
A. SEIDENBERG-Lectures in Projective Geometry
J.
MAYNARD MANSFIELD-Introduction to Topology
FRANK M. STEWART-Introduction to Linear Algebra
LEoN W. CoHEN and GERTRUDE EHRLICH-The Structure of the
Real Number System
ELLIOTT MENDELSON-Introduction to Mathematical Logic
HERMAN MEYER-Precalculus Mathematics
ALBERT G. FADELL-Calculus with Analytic Geometry
JoHN L. KELLEY-Algebra: A Modern Introduction
ANNITA TULLER-A Modern Introduction to Geometries
J.
K. W. GRUENBERG and A. WEIR-Linear Geometry
HoWARD LEvi-Polynomials, Power Series, and Calculus
A series of distinguished texts for undergraduate mathematics.
Additional titles will be listed and announced as published.
Polynomials, Power Series,
and Calculus
by HOWARD LEVI
Professor of Mathematics
Hunter College of the
City Uniuersity of New York
•
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D. VAN NOSTRAND COMPANY, INC.
Princeton, New Jersey
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Copyright© 1968, by D. VAN NOSTRAND COMPANY, INC.
Published simultaneously in Canada by
D. VAN NosTRAND CoMPANY (Canada), LTD.
No reproduction in any form of this book, in whole or
in part (except for brief quotation in critical articles or
reviews), may be made without written authorization
from the publisher.
Library of Congress Catalog Card No. 67-27975
PRINTED IN THE UNITED STATES OF AMERICA
Preface
This book is not intended for use as a text for the calculus course now
generally given in the United States, but rather as a text for a proposed
replacement for that course.
During the author's career as a teacher of college mathematics, he
has seen three courses disappear from the standard college sequence in
mathematics in most colleges and universities-college algebra, trigo
nometry, and analytic geometry. This book is written with the thought
that the traditional calculus course is moving toward the same fate. Each
year many mathematics departments change their calculus textbook
because the old one proved unsatisfactory, only to find that the "new"
one leads to a similar sense of defeat. The whole situation, in the author's
opinion, suggests that the course itself is at fault and that no text can be
satisfactory for it.
The traditional calculus course seeks to impart mastery of the concepts
oflimit, derivative, and integral. In principle, this is a fine program, but
in practice, it seems to be getting out of hand. This could be due to the
fact that, whether he knows it or not, the student of calculus is studying
a class offunctions with some very, very pathological members, and his
methodology has been elaborated in order to cope with them (for in
stance, Riemann introduced his integral in order to investigate highly
discontinuous functions). If these difficulties were the necessary price for
achieving the benefits of calculus, the teacher could simply apologize
and proceed.
The present author believes, on the contrary, that the actual benefit
of the calculus is not the mastery of these concepts but the acquisition
of skill in dealing with relatively well-behaved functions. He offers a
more direct way of acquiring this skill and believes that nothing essential
is lost thereby.
The course for which this text has been written has been laid out so
that, with a little briefing on nomenclature, the student who wants to
go on can reenter the usual sequence where functions of several vari
ables are introduced, with at least as adequate a background as that of
v
VI PREFACE
the students who have followed the sequence all along. The student for
whom this is a terminal course will have learned
how to formulate problems in terms offunctions;
how to find unknown functions by solving differential and other types
of functional equations;
how to find realistic approximations to these functions; and
how to deduce other properties of these functions from their power
series representations.
This new course in analysis covers the calculus of functions of one
variable but has a mathematical base that differs from the traditional
one. Specifically, it builds around the notion of best approximating
polynomials, deemphasizes limits of functions, and omits the Riemann
integral altogether (obtaining all its usual applications in other ways).
A major source of the current difficulty with calculus is the need to
make simultaneous provision for the divergent professional needs of two
broad groups of students: the future scientists (physical, biological and
social) and engineers; and the much smaller group of future mathe
maticians and theoretical physicists. His hope that this book will help
solve the problem is based on the following observations.
( 1) The large group of future users of mathematics needs to be able
to handle only the relatively simple class of piece-wise analytic
functions; they do not need to master subtleties appropriate to
the far more complicated functions that the pure mathematician
will encounter.
(2) The approach to analytic functions presented here is mathema
tically sound, is an appropriate way to deal with these functions,
and is easier to master than the usual approach to the wider class
of functions currently treated.
(3) The future mathematician can profitably begin his study of ana
lysis this way. Even though it differs from the standard approach,
it gives a sound preparation for a modern advanced calculus
course, omitting nothing essential and introducing nothing that
needs to be unlearned.
(4 ) In this development, theory and practice are interconnected so
that they reinforce one another; in particular, proofs do not enter
as irrelevant niceties, but, in many cases, involve computations
that the student will continue to make as long as he uses mathe
matics.
Most of all, the author wishes to state that he has taught the course
set forth here, that he has been pleased with the results and that he
hopes that some of his readers will be pleased, too.
--HOWARD LEVI
Contents
Exercises follow each of the sections listed.
I. BACKGROUND MATERIAL I
Order and Absolute Values I
Functions and Function Notation 3
Sequences 5
Some Polynomial Algebra 8
2. POLYNOMIALS I5
Polynomial Functions I5
Arithmetic Sequences and Polynomial Functions 20
The Algebra of Residues 23
Composition, Inverses, and the Extraction of Roots in Rn[x] 28
3. FUNCTIONS HAVING POLYNOMIAL
APPROXIMATIONS 33
Orders of Magnitude 33
Approximation by Polynomials 37
Continuous Functions 44
Derivatives 49
4. APPLICATIONS OF THE DERIVATIVE AND
ANTIDERIV ATIVE 55
Tangents: Maxima and Minima 55
Some Useful Theorems 60
Applications of the Antiderivative 65
Applications of the Derivative 7I
Higher Derivatives; Taylor's Theorem 75
5. INFINITE SEQUENCES AND INFINITE SERIES 80
Convergence of Infinite Sequences 80
vii
Vlll CONTENTS
Cauchy Sequences 84
Infinite Series 87
Algebra of Power Series 92
6. FUNCTIONS DEFINED BY POWER SERIES 98
Interval of Convergence 98
Analytic Functions 103
The Method of Undetermined Coefficients 109
7. SERIES EXPANSIONS OF THE ELEMENTARY
FUNCTIONS 114
The Exponential Functions 114
Logarithms; the Binomial Theorem 118
Trigonometric Functions 123
Inverse Trigonometric Functions 129
Angle Measure and the Trigonometric Functions of Angles 133
Curvature and the Osculating Circle 138
APPENDICES A. Some Peculiarities oflnfinite Series 143
B. Other Notations for the Derivative and
Differential 145
C. The Riemann Integral 14 7
ANSWERS TO SELECTED EXERCISES 149
INDEX 155