Modern Calculus and Analytic Geometry

1169 Pages·2014·52.449 MB·English

by  Silverman, Richard A

MODERN CALCULUS
AND ANALYTIC GEOMETRY
MODERN CALCULUS
AND ANALYTIC GEOMETRY
Richard A. Silverman
DOVER PUBLICATIONS, INC.
Mineola, New York
Copyright
Copyright © 1969 by Richard A. Silverman
All rights reserved.
Bibliographical Note
This Dover edition, first published in 2002, is a corrected republication of the edition published by The
Macmillan Company, New York, 1969.
Library of Congress Cataloging-in-Publication Data
Silverman, Richard A.
Modern calculus and analytic geometry / Richard A. Silverman
p. cm.
Originally published: New York : Macmillan Co., 1969.
Includes index.
eISBN 13: 978-0-486-79398-6
1. Calculus. 2. Geometry, Analytic. I. Title.
QA303.2 .S55 2002
515'.15—dc21
2002023778
Manufactured in the United States by Courier Corporation
42100702
www.doverpublications.com
To Joan
PREFACE
The philosophy of this book can perhaps be summarized in the following two commandments:
“Motivate all new ideas, both mathematical and physical” and “Prove all theorems.” Concerning
the second precept, it seems to me that the author of a calculus book should avoid recourse to
arguments beginning with phrases like “Considerations beyond the scope of this book show that .
. .” or “In a more advanced course you will learn that . . . ,” even if such arguments are
accompanied by suitable references to other books. In the first place, it is unreasonable to assume
that the average reader of a calculus book (not a mathematics major) will ever take the
appropriate “more advanced course.” But, more to the point, the presence of too many unproved
propositions introduces an atmosphere of expediency which subverts the spirit of a mathematics
book, even at this level. After all, how is a student to learn to think for himself (and this includes
discovering that plausibility alone is never enough in any science) if he is constantly being
exhorted to take things on faith, even things which sound quite reasonable? Fortunately, there is
almost always an elementary method of proof, allowing one to avoid all “handwaving” without
getting too fancy. For example, there exists a simple proof of the equality of double integrals and
iterated integrals of continuous functions of two variables (see Sec. 98), the legitimacy of term-
by-term differentiation and integration of power series can be established without explicitly
invoking the concept of uniform convergence (see Sec. 114), and so on. Some of these proofs
may have to be skipped on a first reading or in an abbreviated course, but at least the student
knows where to find them stated in familiar language, and what’s more, he will not get an
exaggerated idea of their difficulty!
It will be a rare student who has not already encountered some of the material in Chapters 1
to 3, and an even rarer one who is familiar with all of it. (For example, Secs. 17 and 18 contain
some things that the average student is unlikely to know from high school trigonometry.) This
material, which might aptly be called “precalculus,” is intended to make the book as self-
contained as possible and equally accessible to students with diverse backgrounds. If the students
are particularly well-prepared, Chapters 1 to 3 can be skimmed first and consulted later, as the
need arises (to facilitate this, I have tried to make the index unusually complete). In any event,
some time should be spent on Secs. 3 to 5 and 13 to 16. On the other hand, two extended
passages on the use of decimals to construct the real number system R can safely be omitted, as
explained in the “shortcuts” on pp. 44 and 63. The approach sketched in these passages is an
elementary alternative to the construction of R by using the more abstract notions of Dedekind
cuts or equivalence classes of Cauchy sequences. “The decimals,” to quote J. F. Ritt, “have the
advantage of lacking profundity and of not putting the student through a mathematical
revolution.” In particular, the key property of R, namely its completeness, is very easily proved
by using decimals (see Theorem 2.11).
The key concept of the limit is confronted squarely in Chapter 4, in sufficient detail to exhibit
its full generality. In other words, nontrivial calculations of limits are given, as well as examples
illustrating how a function can fail to have a limit at a given point. Continuous functions are
studied in Sec. 27, again with examples showing the various ways in which a function can fail to
be continuous. Having mastered the limit concept, the student is more than halfway home to an
understanding of the concepts of the derivative and integral, presented in Chapters 5 and 7. In my
opinion, the more traditional approach of treating derivatives and integrals first and then picking
up the necessary background information about limits, continuity, etc., in a series of flashbacks is
counterindicated in a modern treatment of calculus, in which the limit itself is recognized as the
central notion. The price for premature exposure of the student to derivatives and integrals is the
necessity for repeatedly shoring up his flagging understanding of what these concepts really
mean, as he encounters functions which are less and less well-behaved. This can only be
prevented by sending him to battle armed with a full grasp of limits and continuity.
Chapter 6 is a unified treatment of material that is scattered in many books under such
headings as “applications of the derivative,” “transcendental functions,” “underlying theory,”
“basic properties of continuous and differentiable functions,” etc. In keeping with the spirit of
modern functional analysis, I have organized these topics around the central idea of “well-
behaved functions.”
As already noted, integrals enter the picture in Chapter 7. Once equipped with integrals, the
student is immediately exposed to differential equations, rather than to the more conventional
applications such as hydrostatic pressure, volume of solids of revolution, centroids, etc. In the
first place, many of these applications are best handled by using multiple integrals, as in Chapter
13. But an even more compelling reason for highlighting differential equations at this point is
that they have always been and remain the most characteristic and important application of
calculus to science and technology. Newton’s second law of motion appears in Sec. 53, and the
program of showing how the second law leads naturally to the key concepts of work, energy,
center of mass, and moments of inertia is begun in Sec. 54 and completed in Secs. 74 and 100. I
do not see how these concepts, so intimately associated with calculus, can be brought in as a
deus ex machina, as is so often done.
Analytic geometry, touched upon in Chapter 3, is pursued in Chapter 8 from a contemporary
standpoint emphasizing such concepts as point and coordinate transformations, invariance, etc.
Conics are treated in detail, with due regard for improper conics. Calculus methods are used
freely in dealing with tangents and areas, both in rectangular and polar coordinates.
By the end of Chapter 8, roughly halfway through the book, most of the key ideas of calculus
have put in an appearance. There is now time for various generalizations, notably the
introduction of the notion of a parametric curve in Chapter 9 and the transition from two-space to
three-space in Chapters 10 and 11. The powerful technique of vector algebra is developed in
Chapter 9, extended in Chapter 10, and applied repeatedly in Chapter 11. A modicum of linear
algebra is introduced in Chapter 10, in anticipation of subsequent needs in connection with
vector algebra and analytic geometry in three-space and the treatment of Jacobians in Problem
Set 104. Thus Secs. 75 and 76 are devoted to determinants and their properties, and Sec. 77 to
systems of linear equations and their solution both by Cramer’s rule (when appropriate) and,
equally important, by elimination.
The differential and integral calculus of functions of several variables is developed in
Chapters 12 and 13. Here the reader will find suitable definitions of the terms “region” and
“surface,” too often left vague and undefined in first courses on calculus. The earlier formal use
of the technique of implicit differentiation is justified in Sec. 92, on the implicit function
theorem. The treatment of multiple integrals in Chapter 13 goes much further than usual in the
direction of proving the main results of the subject, but only elementary tools are used. In both
Chapters 12 and 13 it is made clear that propositions involving functions of one variable often
have counterparts for functions of several variables, which are proved in much the same way. It
is only in Sec. 104, on change of variables in multiple integrals, that it becomes hard to abide by
the precept “Prove all theorems.” Here we inevitably arrive at the boundary between elementary
and advanced calculus, by anybody’s standards.
By Chapter 14 the student has the time to spend on perfecting his computational technique,
first acquiring L’Hospital’s rule and then Taylor’s formula. The technique of integrating rational
functions is treated in more detail than usual in Sec. 108, and then applied in Sec. 109 to the
evaluation of difficult integrals. Sec. 110 is devoted to numerical integration, including proofs of
the all-important error estimates. This material on integration technique is traditionally placed
earlier in most books, at a stage where, in my opinion, the student can more profitably devote his
attention to new concepts rather than special tricks.
The book ends with Chapter 15, on infinite series. As already mentioned, avoiding the topic
of uniform convergence is no obstacle to giving a full treatment of power series and Taylor
series.
There are well over 1600 problems, arranged in 115 problem sets, one at the end of each
section. A problem number unaccompanied by a page number always refers to a problem at the
end of the section where the reference is made. Many problems were drawn from Russian
sources in the public domain. Harder problems are indicated by asterisks, as are problems which
pursue certain topics somewhat beyond the needs of a first course on calculus. For example, such
things as one-sided derivatives, a curve with no length, and the general formula for the radius of
convergence of a power series will be found in starred problems. Thus the book becomes suitable
for an “honors course” if the starred problems are solved and the harder proofs are worked
through.
Answers to all the odd-numbered problems are given at the end of the book. These answers
are often detailed enough to constitute complete solutions, which are meant to be read sooner or
later. This is particularly true of starred problems dealing with points of theory. The results of
problems are sometimes used in the text, but only if they are odd-numbered problems
accompanied by answers complete enough to make the whole treatment self-contained.
The whole manuscript was read in detail by Professor Richard M. Pollack of New York
University. I am grateful to him for making numerous suggestions and critical comments leading
to important improvements in the book.
R. A. S.
New York, N.Y.
CONTENTS
1
CHAPTER SETS AND FUNCTIONS
1. Sets
2. Ordered n-Tuples. Cartesian Products
3. Relations, Functions and Mappings
4. Real Functions
5. Operations on Functions
6. Counting and Induction
7. Binomial Coefficients. The Binomial Theorem
2
CHAPTER NUMBERS AND COORDINATES
8. Rational Numbers
9. Incompleteness of the Rational Number System
10. Decimals and Real Numbers
11. Completeness of the Real Number System
12. The Real Line: Coordinates
13. The Real Line: Intervals
3
CHAPTER GRAPHS
14. Rectangular Coordinates
15. Graphs in General
16. Graphs of Functions
17. Trigonometric Functions: Basic Properties
18. Trigonometric Functions: Graphs and Addition Formulas
19. Straight Lines and Their Equations
20. More About Straight Lines
4
CHAPTER LIMITS
21. The Limit Concept
22. More About Limits
23. One-Sided Limits
24. Infinite Limits. Indeterminate Forms
25. Limits at Infinity. Asymptotes
26. The Limit of a Sequence
27. Continuous Functions
5
CHAPTER DERIVATIVES
28. The Derivative Concept
29. More About Derivatives
30. Curves and Tangents
31. Technique of Differentiation
32. Differentials. Further Notation
33. Implicit Differentiation. Related Rates
34. Higher-Order Derivatives
6
CHAPTER WELL-BEHAVED FUNCTIONS
35. More About Continuous Functions. Absolute Extrema
36. Uniform Continuity
37. Inverse Functions
38. Exponentials and Logarithms
39. More About Exponentials and Logarithms
40. Hyperbolic Functions
41. The Mean Value Theorem. Antiderivatives
42. Relative Extrema
43. Concavity and Inflection Points
44. Applications
7
CHAPTER INTEGRALS
45. Indefinite Integrals