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CALCULUSand Analytic Geometry

CALCULUSWadsworth

John F. RandolphUniversity of Rochesterand Analytic GeometrySecond EditionPublishing Company, Inc.-, Belmont, California

Calculus and Analytic Geometry 2nd EditionJohn F. RandolphFifth printing: July 1969 1961,1965 by. Wadsworth Publishing Company, Inc., Belmont, California.All Rights Reserved. No part of this book may be reproduced in any form,by mimeograph or any other means, without permission in writing from thepublisher.L.C. Cat. Card No. 65-11579.Printed in the United States of America.

PrefaceThis Second Edition incorporates improvements in exposition, severalnew and clearer proofs, expansion of some problem sets, a review of AnalyticTrigonometry, a few remarkable airbrush figures, and bonuses on pagespreviously only partially used. A quick evaluation of the alterations may bemade by comparing old and new pages 3-4, 54-56, 76-77, 118, 179-186,323-329, 352 (figure), 474-475, but all changes are too numerous to list.The First Edition anticipated the trend in secondary and college mathematics so well that an extensive survey of users revealed little need fordisturbing the content or order of the book, In fact, it was found that mostof the suggested improvements could be made within the present format andwithout altering the compact size of the book. Thus a casual examinationmay not reveal the extent of the revision, but, as a matter of interest, themanuscript for it outweighed the book itself.I deeply appreciate the help of Professors Billy J. Attebery and MarianBrashears of the University of Arkansas, Irving Drooyan, Los AngelesPierce College, M. L. Madison, Colorado State University, Gary Mouck,Santa Barbara City College, Karl Stromberg, University of Oregon, andothers who took the trouble to volunteer criticism or responded to requestsfor it. I hold no delusions of perfection and continue to seek advice fromboth students and faculty.The preface of the First Edition is reproduced below, except for theapologetic justification of set notation and terminology at this level. In thesix years since the First Edition went to press, elementary set theory hasbecome so common that its omission would be untenable.There is clear evidence that some students whose records qualify themfor a first course in calculus are not ready for a rigorous treatment of thesubject. It is gratifying, however, that an increasing number of students areimpatient with vague ideas and half-truths, and that a generation of collegeinstructors has emerged with sufficient training and judgment to presentv

viPrefacematerial within reach of most students, but also dedicated to elevating thepotential of more gifted students. Democratic education must be continuedto provide many technicians who have absorbed enough intuitive backgroundfor routine applications, but in the present international scientific marathonit is imperative that extra effort be extended toward providing earlier andbetter training for those students who have possibilities of making originalcontributions in their fields.Since most colleges cannot section their students according to ability,this book was written in such a way that various types of students may betaught in the same class. On page 55, for example, the inequalities1-x2 --- l for 0 Ixl 2replace the usual ones with little extra effort. An instructor may now pointout that "as x gets small, then " and in most classes it might be well to doso, but there is the intriguing possibility that an additional moment or two devoted to showing "for e 0, then 6 Ve . " might pay dividends.Many students resent an instructor's insistence upon a standard abovethe texts, but readily accept a less accurate one if they are not frustrated by apatchwork of required and omitted portions. For this reason much of therigorous development is placed in appendices with full confidence that astudent capable of appreciating such work is also capable of interlacing itproperly. An instructor may thus relax to fairly intuitive presentations inclass, if this seems desirable, but can help his better students to higher goalsby encouraging them to read critically and to discuss fundamental conceptswith him. All students will have the opportunity of seeing the utility ofpowerful underlying principles, whereas the averse ones need not be subjectedlong, if at all, to ideas beyond their comprehension.Even should the happy situation arise of a whole class of serious students,it might be well to omit the pertinent appendices until after Chapter 6. Thisprocedure would fulfill the early service obligation to a companion physicscourse and give an overall picture of the main features of calculus. With thisrough picture as a guide, the student would be in a better position to appreciate the value and significance of Appendices Al-A4.The definite integral is undoubtedly the most erudite concept of calculus,but even so there seems to be an obsession to move it ever nearer the beginningof the course. It is argued that this shift is dictated by the needs of a concurrent physics course. Experience has shown, however, that reasonablephysicists make no such demands, whereas others would not be satisfiedunless definite integrals (even curvilinear integrals) were taken in high school.Consequently, in this book students are given time to polish their algebraicmanipulations and equation-graph concepts, and to have transcendental

Prefaceviifunctions in hand before tackling definite integrals late in the first or early inthe second semester. If forced against his better judgment, an experiencedinstructor may work in an early exposure to definite integrals by postponingportions of Chapters 1, 2, and 4 and all of Chapters 3 and 5 until after selectedproblems of Chapter 6 have been worked.Vector analysis is a handy tool which has either been sadly neglected orglorified beyond its due by making analytic geometry and calculus its slaves.With the plethora of new ideas that cannot be avoided in the early portion ofthe course, it seemed prudent to allow the students the comfort of theirfamiliarity with plane rectangular coordinates for any new work on standardgraphs and to. delay introducing vectors until Chapter 8. Even here thethird dimension is postponed by using vectors in the plane for curvilinearmotion, parametric equations, rotation of axes, and polar coordinates.Now with vector ideas and notation entrenched, their power and economyas an aid to space considerations is capitalized upon in succeeding chapters.This book was written with the numerous potential engineers and scientistsmore in mind than the small group of mathematics majors. The rigor andset theory included here are far surpassed in an introduction to analysiscourse and thus serves the mathematics major as a preview and only slightacceleration. Most undergraduate engineering programs, on the other hand,are so crowded that mathematics courses are confined to the first two yearsand hence students in these programs are forever handicapped unless theysee some careful reasoning in their calculus course. Vector analysis ispresented from the directed line segment point of view as used and understood by engineers rather than the linear space approach more satisfying tomathematicians. Also, the present tendency to subdue the traditional firstcourse in differential equations is followed by including a short chapter toprovide further manipulative skill in integration, introduce some of thestandard techniques in finding so called solutions, and give a glimpse ofapproximations so important in modern high speed computing. It is hopedthat students who continue their mathematical studies will not be spoiled forthe solid and practical work in store for them when they are ready for anhonorable differential equations course.It is inevitable that the first half of the book is richer in theory than thesecond half. Once a fairly firm foundation has been laid, much of the super-structure follows without ostensive honor to the underlying theory. If astudent has developed reasonable thought patterns, then he will be able tofill in many details for himself (or at least see there are gaps), but if he has not,there seems little point in plaguing him further. To be sure, Jacobians couldhave been developed for the sake of less intuitive discussions of pdpd6, upperand lower limits for some proofs or neater proofs,* functions of bounded* In particular for L'Hospital's Rules, see A. E. Taylor, American MathematicalMonthly, Vol. 59 (1952), pp. 20-24.

viiiPrefacevariation for arc length, Stieltjes integrals for work and other concepts, or arigorous Fubini theorem, but the effort would seem wasted on all studentsexcept those who will continue anyway.Finally, this book steers a path between the terse theorem-proof listing ofbare essentials and the bulky down-to-the-student-level books that try tousurp the role of the instructor by a wordy and chatty style. It is hoped thatmost students can, and will, read the book, but it is also assumed that eachcollege course has an instructor who will augment the text according to theneeds of his current class and motivate the work in a more effective andspontaneous manner than an impersonal author could hope for. If the bookwere used for self-study, then Chapter 1 would seem to start rather abruptly,as would the notion of the limit of a function. Lives there an instructor,however, who does not set the stage for his course by a ten or fifteen minutetalk the first class period, or who does not generate limits and discontinuitiesbefore the very eyes of his students?My former co-author and publisher did me the great favor of granting'free use of material from Analytic Geometry and Calculus, The MacmillanCompany, 1946, by John F. Randolph and Mark Kac. Professors MelvinHenriksen and Warren Stenberg went far beyond their assignment of acritical reading of the manuscript and offered many constructive criticisms,but I asked them to forgive my not following all of their suggestions. ProfessorHewitt Kenyon helped greatly in version after version of the appendices.I thank (and exonerate) each of the following for criticism of at least onechapter and an independent set of its answers: Theodore A. Bick, GordonBranche, Mrs. Martha Burton, David Burton, Michael Lodato, David F.Neu, Patrick S. O'Neill, William Pitt, Vemuri Sarma, Earl R. Willard, andRonald Winkleman.John F. Randolph

ContentsCHAPTER 1Rectangular Coordinates11. Inequalities and Absolute Values. 2. Linear Coordinate System.3. Intervals, Half-lines, and Linear Motion. 4. Sets of Numbersand Sets of Points. 5. Plane Rectangular Coordinates. 6. Slope andEquations of a Line. 7. Sets and Ordered Pairs. 8. Functions.9. Some Special Functions. 10. Distance Formula, Circles. 11.Properties in the Large. 12. Translation of Coordinates. 13. ConicSections. 14. Parabola. 15. Ellipse and Hyperbola.CHAPTER 2 Limits and Derivatives4816. Limit of a Function. 17. Limit Theorem. 18. Limits of Trigonometric Functions. 19. Composition Functions. 20. ContinuousFunctions. 21. Tangents. 22. Velocity. 23. Derived Function.24. Derivative Theorems. 25. Power Formulas. 26. The Chain Rule.27. Second Derivatives.CHAPTER 3 Applications of Derivatives8428. Equations of Tangents. 29. Solutions of Equations. 30. Newton'sMethod. 31. Maxima and Minima. 32. A Mean Value Theorem.33. Points of Inflection. 34. Simple Econometrics. 35. Rates. 36.Related Rates. 37. Linear Acceleration. 38. Simple Harmonic Motion.CHAPTER 4 Additional Concepts11939. Derived Functions Equal.40. Derivative Systems. 41. Differentials. 42. Differential Systems. 43. Increments. 44. Approximations by Differentials.CHAPTER 5 Elementary Transcendental Functions55. Trigonometric Functions. 46. Inverse Trigonometric Functions.47. Exponents and Logarithms. 48. Log Scales. 49. Semi-LogCoordinates. 50. Log-Log Coordinates. 51. The Number e. 52.Derivatives of Log Functions. 53. Exponential Functions. 54.Variable Bases and Powers. 55. Hyperbolic Functions.ix134

xContentsCHAPTER 6 Definite Integrals16756. Sigma Notation. 57. Definite Integrals. 58. Area and Work.59. The Fundamental Theorem of Calculus. 60. Algebra of Integrals.61. Area Between Curves. 62. Pump Problems. 63. HydrostaticForce. 64. Integration by Parts. 65. First Moments and Centroids.66. Second Moments and Kinetic Energy. 67. Solids of Revolution.68. Improper Integrals.CHAPTER 7 Indefinite Integration21169: Four Basic Formulas. 70. Trigonometric Integrals. 71. AlgebraicTranscendental Integrals. 72. Exponential Integrals. 73. Trigonometric Substitutions. 74. Integral of a Product. 75. IntegralTables. 7V. Partial Fractions. 77. Resubstitution Avoided.CHAPTER 8 Vectors23878. Definitions. 79. Scalar Product. 80. Scalar and Vector Quantities.81. Vectors and Coordinates. 82. Parametric Equations. 83. Vectorsand Lines. 84. Vector Functions. 85. Curvature. 86. RectifiableCurves. 87. Parametric Derivatives. 88. Rotation of Axes. 89.Polar Coordinates. 90. Polar Analytic Geometry. 91. Polar Calculus.CHAPTER 9 Solid Geometry29892. Preliminaries. 93. Coordinates. 94. Direction Cosines andNumbers. 95. Parametric Equations of Lines. 96. Planes. 97.Determinants. 98. Cross Products. 99. Triple Products. 100. SpaceCurves. 101. Surfaces and Solids. 102. Functions of Two Variables.103. Cylindrical and Spherical Coordinates.CHAPTER 10 Multiple Integrals347104. Double and Iterated Integrals. 105. Volumes of Solids. 106.Mass, Moments, Centroids. 107. Polar Coordinates. 108. ReversingOrder Transformations. 109. Triple Integrals. 110. Attraction.CHAPTER 11Partial Derivatives111. Definitions. 112. Normals and Tangents to a Surface. 113. TheSchwarz Paradox. 114. Area of a Surface. 115. Partial DerivativeSystems. 116. Differentiable Functions. 117. Exact Differentials.118. Implicit Functions. 119. Families. 120. Functions of ThreeVariables. 121. Change of Variables. 122. Second Partials. 123.Directional Derivatives. 124. Vectors and D

glorified beyond its due by making analytic geometry and calculus its slaves. With the plethora of new ideas that cannot be avoided in the early portion of the course, it seemed prudent to allow the students the comfort of their familiarity with plane rectangular coordinates for any new work on standard graphs and to. delay introducing vectors until Chapter 8. Even here the third dimension is .

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