Sheldon & Company's Text~:Sooks, PROFESSOR OLNEY'S NEW MATHEMATICAL SERIES The success of Prof. Olney's series has been most wonderful. With all their admitted excellencies, both the Author and Pub- lishers have felt that it was possible to retain their many attractive features and yet adapt the books more perfectly to the special school-room wants. To accomplish this most desirable end, Professor Olney has been accumulating very valuable suggestions. He has also, for several years, had associated with him in the preparation of this new series, some of the best practical teachers in the country. The design is to present to the educational public the best and most teachable series of Mathematics ever published. The work is now so far advanced that the Publishers are able to make the above pleasing announcement, which they feel will be of great interest to all who are engaged in teaching. THE NEW SERIES EMBRACES: I. Olney' s First Lessons in Arithfuetic. Just Published. II. Olriey's Practical Arithinetic. This book has been published but a short time, but it has already had the most wonderful success. They are models of beauty and cheapness. For schools of a high grade, Professor Olney has prepared — III. The Science of Arithmetic. lY. The First Frincij^les of Algebra. An Introduction to the Author's Complete and University Algebras. V. Olney' s Complete Algebra, New Edition, in large type. This book is now entirely re-el ectrotyped in larger and more attractive type. Theexplanatory matter is greatly lessened. The attractive features of this book, which have made it the most popular Algebra ever published in this country, are all retained. .051 Sheldon & Company's Text-^Books^ OLNEY'S SERIES OF MATHEMATICS. Olney's First Lessons in Arithinetlc Illus- trated Olney's Practical Arithmetic Olney's Science of Arithmetic, (For High-Schools only.) Olney^s First Principles of Algebra Olney's Complete Algebra Olney's Book of Test Examples in Algebra.., Olney's University Algebra Olney's Elements Geom, 4& Trigonom. (Sch. Ed.) Olney's Elements of Geometry. Separate Olney's Elements of Trigonometry. Separate. . Olney's Elements of Geometry and Trigonom," etry. (Univ. Ed., with Tables of Logarithms.) Olney's Elements of Geometry and Trigonom- etry. (University Edition, without Tables.) Olney's General Geometry and Calculus The universal favor with which these books have been received by educators in all parts of the country, leads the publishers to think that they have supplied a felt want in our educational ap- pliances. There is one feature which characterizes this series, so unique, and yet so eminently practical, that we feel desirous of calling special attention to it. It is The facility with which the books can be used for classes of all grades, and in schools of the widest diversity of purpose. Each volume in the series is so constructed that it may be used with equal ease by the youngest and least disciplined who should fee pursuing its theme, and by those who in more mature years and with more ample preparation enter upon the study. Ij Library ciW.H.Metzleri Class Noi OLNEY'S MATHEMATICAL SERIES. A W. I' GENERAL GEOMETRY AND CALCULUS. INCIitTDING BOOK I. OF THE GENEEAIj GEOMETRY, TREATING OP LOCI IN A PLANE ; AND AN ELEMENTARY COURSE IN THE DIFFER- ENTIAL AND INTEGRAL CALCULUS. BT EDWAED OLNEY, FBOFESSOB OF MATHEMATICS IN THE Xmi^^^^F^^jpOMOeUXj NEW YOEK : SHELDON AND COMPANY. 1881. Entered according to Act of Congress in the year 1871, by SHELDON & COMPANY, In the Office of the Librarian of Congress at Washington. PROF. OLNEY'S MATHEMATICAL COURSE. INTRODUCTION TO ALGEBRA COMPLETE ALGEBRA KEY TO COMPLETE ALGEBRA --,---. UNIVERSITY ALGEBRA KEY TO UNIVERSITY ALGEBRA A VOLUME OF TEST EXAMPLES IN ALGEBRA ELEMENTS OF GEOMETRY AND TRIGONOMETRY ELEMENTS OF GEOMETRY AND TRIGONOMETRY, University Edition - ELEMENTS OF GEOMETRY, separate - ELEMENTS OF TRIGONOMETRY, separate GENERAL GEOMETRY AND CALCULUS - - - - BELLOWS' TRIGONOMETRY PROF. OLNEY'S SERIES OF ARITHMETICS. PRIMARY ARITHMETIC ELEMENTS OP ARITHMETIC PRACTICAL ARITHMETIC SCIENCE OF ARITHMETIC - - - - - - - G. &. C. 150249 PREFACE. -•♦i- ^ This volume presents a course in the General Geometry and the Infinitesimal Calculus, which is thought to be as extended as is pr,acticable for the general student in the regular undergrad- uate course in our American colleges. If we can secure a suffi- ciently high grade of preparation, so that students in the Fresh- man year can complete a respectable course in Elementary Geometry, including Plane and Spherical Trigonometry, and in Algebra, it is thought that during the Sophomore year the 3on- tents of this volume can be readily mastered. Such is the purpose in this University ; and it is already well nigh reahzed. As to the propriety of including the study of both these sub- jects in the regular undergraduate course, there can be but one opinion among those competent to judge. No man can justly claim to have a good general education, who is ignorant of the elements of the processes by which all extended operations in the exact sciences are carried forward, and which are the foun- dation of all the arts based upon mathematical science. The man who is ignorant of the General Geometry and the Calculus, is not only a stranger in one of the subHmest realms of human thought, but knows nothing of the instruments in most familiar use by the engiueer, the astronomer, and the machinist in any of the higher walks of art. In short he is ignorant of the charac- teristic processes of the mathematician of his day. Nor is it impracticable for the majority of students to become intelligent in these subjects. They do not he beyond the reach of good common minds, nor require peculiar mental character- istics for their mastery. The difficulty hitherto has been in the methods of presentation, in the limited and totally inadequate amount of time assigned them, and more than all in the precon- ceived notion of their abstruseness. The mathematician will see in the plan of the first part of IV PREFACE, this volume, as well as in its title ( General Geometry), a recog- nition of the profound views of Comte upon the philosophy of the science. This science is a method of Geometrical reasoning. Its characteristic feature is that it represents form, as well as magnitude, by equations, and hence makes algebra its instru- ment. It is consequently indirect. Its ultimate object is breadth of comprehension, — the discussion of general problems. In accordance with this conception, the first purpose is to exhibit the method of translating geometrical forms into algebraic equations, i. 6. to show how loci are represented by equations. While the prominence is given to the Conic Sections which their importance in physical science demands, the student is not led to think that this is merely a scheme for treating these curves. He is taught to look upon it as a method of investiga- tion — as designed to embrace the discussion of all loci. For this purpose many Higher Plane Curves are treated. After the student has become famihar with the equation as the represent- ative of a locus, and has learned how to ]3roduce the equation of a locus from its definition, he has obtained the instrument. He is now to learn how to apply it for the purposes of Geometrical investigation. In carrying forward this part of his study the Calculus renders invaluable service. Moreover, this preparatory study of the General Geometry gives him exactly the needed means for illustrating the elementary processes of the Calculus. He has, therefore, come to a point where his further progress requires a knowledge of the Differential Calculus, and he has also the requisite preparation for its study. Hence, after having become famihar with the first three chapters of General Geom- etry, he reads the Differential Calculus. By this arrangement the Calculus is seen in its true relations, as an independent abstract science, grand and beautiful in itself, and rendering most efficient service in the more immediately practical science of Geometry, as it is afterwards seen to do in Physics. Having obtained the needed acquaintance with the Differential Calculus, the student returns to pursue his Geomet- rical studies, with equations of loci as his instruments, and the Calculus to aid in the manipulation of them. But the pecuhar features of the treatise are too numerous to be enumerated here, and can be seen in their true light only by a perusal of the work. PBEFACE. V In the treatment of the Calculus I have used the Infinitesi- mal method instead of the method of Limits, on account of its greater simplicity, as well as because it is the only conception which enables us to apply the Calculus to practical problems with any degree of facility. The general use of the method of limits in our text books has done not a little to prevent the com- mon study of this elegant and useful branch of mathematics. This method is not only exceedingly cumbrous, but it has the misfortune that its element, a differential coefficient, is a ratio. The abstract nature of a ratio, and the fact that it is a com- pound concept, pecuharly unfit it for elementary purposes. The beginner will never use it with satisfaction, for it does not give him simple, direct and clearly defined conceptions. But while I have adopted the infinitesimal theory, I have felt free to intro- duce the doctrine of Hmits, and to illustrate and apply it. The metaphysical objections to this method, if not rebutted by equal difficulties of a similar character encountered in the method of Hmits, are immensely overborne by its practical advantages ; for, let it be remembered that no writer adheres to the Newton- ian method throughout, but ghdes into the other in the Integral Calculus, and adopts it exclusively in most geometrical and physical applications. The sources from which the material has been drawn will be readily perceived by the mathematician, and need not be enum- erated here. That the treatise is sufficiently different from others of a similar purpose to justify its existence, the author feels more sure than that these differences will commend them- selves to his fellow laborers in the work of mathematical train- ing. One thing, however, is certain, nothing in matter, arrange- ment, or manner of treatment, has been introduced without careful reference to the capabilities and wants of such students as I have been accustomed to meet in the class room for more than twenty years ; and few things will be found in the volume but what have been put to the test of class room use many times over. A second volume, treating of Loci in Space, and affording a more extended course in the Calculus, will be published as soon as it can be prepared. The present is thought sufficient for all students except such as make mathematics a specialty ; and for the latter the other volume will be designed. VI PREFACE. In conclusion I must do myseK the pleasure to acknowledge mj indebtedness to mj accomplislied colleague and friend, Prof. J. C. Watson, Ph. D., for the original, direct, and simple method of demonstrating the rule for differentiating a logarithm, which is given on page 25, and which banishes from the Calculus the last necessity for resort to series to establish any of its funda- mental operations. I am also indebted to my friend and pupil, J. B. Webb, B. S., for many valuable suggestions, and much care- ful labor in reading both the manuscript and proof. To his quick and accurate eye, and his good taste and logical acumen, I am indebted for the ehmination of not a few defects which might otherwise have disfigured the work. That there is not much of the same sort of pruning yet needed, I have not the vanity to think. But, such as it is, I commend my work to the consideration of teacher and student, with the hope that it may contribute to aid the one in imparting, and the other in acquiring, a knowledge of the elements of two branches of science which, in their fuller developments, exhibit the profoundest and most sagacious workings of the human mind, and reach to the farthest verge of the hitherto explored realms of human thought. EDWAED OLNET. Ann Abbob, Mich., July, 1871. N. B. — A shorter course in tlie General Geometry, ivithout the Calculus, may he taken from this volume hy such as desire it. For this purjoose, the first three chapters are to he read, and then the course completed hy reading the XIV. and XV. Sections of Chap- ter IV. If time and purpose permit, Articles {194:, lOo) might he read loith profit hy such students. This ivill he found to comprise a course on Plane Co-ordinate Geometry somewhat more full than is found in our common text-hooks. CONTENTS. INTMOnUCTION. A BRIEF SURVEY OF THE OBJECTS OF PURE MATHEMATICS AND OP THE SEVERAL BRANCHES. PAGK PuBE Mathematics. — Definition (i) ; Brandies enumerated {2f 3) 1 Quantity. — Definition (4) 1 NuMBEK. — Definition {5) ; Discontinuous and continuous {6, 7f 8) 2, 3 Definition of the Several Branches of Mathematics. — Arithmetic {9 , ; Algebra (JO); Calculus {11); Geometry {12) \ Descriptive Geometry (15) 3, 5 General Geometry divided into Two Books {14:) 5 — -tt^^^ GENERAL GEOMETRY BOOK I. OF PLANE LOCI. CHAPTEE I. THE CABTESIAN METHOD OF CO-OBDINATES. SECTION L DEFINITIONS AND FUNDAMENTAL NOTIONS. Locus.— Definition {1) 6 General Geometry. —Definition {2) 6 Method of Co-ordinates.— What (5) ; Two Systems (4) ; Varieties of Eec- tUinear {5) ^' "^ Definitions.— Axes {6, 7) ; Origin {8) ; Co-ordinates {9, 10, 11) ; Illus- tration « • ' » " Notation.— Of Co-ordinates {12) ; The Four Angles {13) ; Signs of the Co-ordinates {14) ^' ^ OnvN-T-rr..-, —Constant and Variable US' ; Definition of each {16, 17 \ 9 El istration : Caution. Sch. 1 TlU CONTENTS. FAGS Indetebminate Analysis. — ^What, Sch. 2 9 To CoNSTEUCT AN EQUATION. — ^What (^18) 10 SECTION IL CONSTBTJCTING EQUATIONS, OE FINDING THEER LOCI. Deitnitions. — A Continuous Curve {19) ; Branch {20) ; Symmetry {21) ; Independent and Dependent Variables {25) 10-12 To LOCATE A Point {22) 10 To CONSTBUCT AN EQUATION {23) 11 Discussing an Equation. — Wliat ; Intersection ; Limits ; Symmetry {26) 12, 13 "R-yAATPT.-B^S ,^. 11-16 SECTION IIL THE POINT IN A PLANE. Deitnitions. — ^Equations of a Point {27) 17 Equations op a Point. — ^What {28) ; In different angles, In the axes, In the origin, ScKs. 1, 2 ; Points, how designa' ed, Sch. 3 17 Distance between two Points. — General Formulae {29) ; Special cases, Cor. and Sch , 18 Examples. . , , , , 17, 18 SECTION IV. THE EIGHT LINE IN A PLANE. Definit i on. — Equation of a Locus {30) 19 Equations of a Right Line. — Through Two Points {31) ; Through One Prnnt {32 i ; Common Form {33) ; Referred to Oblique Axes {34:) ; Meaning of ^,~^„ Cor.l 19-21 X — X Discussion oiy = ax -\- h, Sch's. 1 and 2 20 ^Methods of Constbuctestg y = ax -\- h 21 Locus of an Equation of the Ftest Degeee (55) 22 EXAfcLPLES. 21-23 SECTION K OF PLANE ANGLES, AND THE INTEESECTION OF LINES. Tangent op a Plane Angle. — Formulae for {36) 23 Equation of a Line making any given Angle with anothee Line. — Com- CONTENTS. IX PAGE mon form, When passing through a given point (57) ; When Parallel to a given Line, Comnion form, Passing through a given point {38) ; When Perpendicular to the given line, Common form. Passing through a given poiat {39) -. 24 Examples 25, 26 To Find the Intersection of Lines {40) 26 ExAMPiiES , 26-28 Distance feom a Point to a Line {41) ; Between Parallels, Cor 28 Examples 28, 29 SECTION VL OF THE CONIC SECTIONS. Boscovich's Definition {4=2) 29 To CONSTBTJCT A CoNIC SECTION {43) 29 Definitions. — Directrix, Focus, Focal Tangents, Transverse Axis, Conjugate Axis, Latus Kectum, Vertices, Focal Distances, Eccentricity {44) 29, 30 Axis of Hyperbola, Transverse, Conjugate, Conjugate Hyperbola, Equi- lateral Hyperbola {47) • 32, 33 Examples 30-34 Boscovich's Ratio = Eccentbicitt {48) 34 Fundamental Relations {45^ 46 , 49) 30, 31, 34, 35 To pass a Conic Section theough Thkee Points {50) 36 Examples 37 Equations of Conic Sections. — General Equation (5 J? ) ; Referred to their Axes, In terms of A and e {52), Common Forms {53y 54, 50, 57 f 59) ; Referred to Axis and Tangent at Vertex {55, 56, 57) \ Of Conjugate Hyperbola {58) 37-40 Comparison op Equations of Ellipse and Hypebbola {60) 41 Locus OF Equation of Second Degree {61) 41 Features op the Equation which chaeacteeize the dippebent Conic Sec- tions {62) ; Species dependent on A, B, C, {63) ; All varieties included in Aij^ + Or^ -{-I>y-\- Ex-j-F= {64) 42, 43 Examples 42, 43 Varieties.— Of Ellipse {65), Hyperbola {66), Parabola {67) ; Eccentricity of Circle {68) 43-45 Examples 46-49 Exercises in producing various forms of the equation of the Conic Sections directly from the definition 49-51 The Origin op the name Conic Section {69) 51 Five Points in the Curve determine a Conic Section {70) 52 Examples 52-54 Exercises in producing the equations of Conic Sections from various defini- tions. 54-57 X CONTENTS. SECTION VIL EQUATIONS OF HIGHER PLANE CURVES. PAGE Definitions. — Function {71)', Classes ofcT'^); Algebraic (75) ; Trigo- nometrical (7^) ; Circular (.75) ; Logarithmic (7^) ; Exponential {77) 57 Loci Ciassified.— Higher and Lower, Algebraic and Transcendental {78, 70) 58 Cisson). — Definition {80) ; Construction {81) ; Origin of name, Sch. 1 ; Mechanical method of Constructing, ISch. 2 ; Equation of {82 j ; Discus- sion of Equation, Sch. 1 ; Duplication of cube by means of, Sch. 2 . . . . 58-60 Conchoid. — Definition {83) ; Construction {84) ; Mechanical Construc- tion, Sch. ; Equation of {8S) ; Discussion of Equation, Sch. 1 ; Be- comes the equation of circle, Sch. 2 ; Trisection of an angle by means of, Sch. 3 60-62 Witch.— Definition {86) ; Construction {87) ; Equation of {88) ; Dis- cussion of Equation, Sch 62 Lemniscate, —Definition [89) ; Construction {90) ', Equation of {91) ; Discussion of Equation, Sdi. 1 ; How related to Equi-lateral Hyperbola, Sch. 2 63 Cycloid. — Definitions, of the Locus, Generatrix, Base, Axis, {92,93) ; To put the Generatrix in position {94:) : Equations of the Cycloid, 1st form {95), 2nd form (96) ; Discussion of Equation, Sch. to {95), and Cor. and Sch. 1 to (96) 64, 65 Equations of some i.oci written dikectly feom the definitions {98) .... 66 NuMBEB OF PLANE CUBVES INFINITE. — A fcw Suggested {99) 66 ■^♦» CHAPTEE II. THE METHOD OF POLAR CO-ORnU^ATES. SECTION L OF THE POINT IN A PLANE. How A Point is designated by Polar Co-ordinates {100) 67 Definitions. — Pole, Prime Radius, Eadius Vector, Variable Angle, Polar Co- ordinates ilOl) 67 Equations of a Point {102) ; Examples 67, 68 Distance between two Points {103) ; Examples 68 SECTION IL OF THE RIGHT LINE. Equations op the Right Line. — 1st form, 2nd form {104) ; Discussion of 1st form, Sch, 1 ; Diseussion of 2nd form, Sch. 2 ; Examples 68-70 CONTENTS. XI SUCTION III. OF THE CIRCLE. PAGE Equation "when the Pole is the Circumference, and the Polar Axis is a diame- ter {105) ; Discussion, Sch 70, 71 General Polab Equation {100) ; Discussion, Sch. ; Geometrical Illus- tration, ^.5 71-73 Examples 72, 73 SECTION IV, OF THE CONIC SECTIONS. Polar Equation op Conic Section {107) ; Of Parabola {108) ; Of El- lipse and Hyperbola {109) ; Discussion of Equation of Parabola, Sch. 1, Of Ellipse, Sch. 2, Of Hyperbola, Sch. 3 73-75 Examples 75, 76 SECTION V. OF HIGHER PLANE CURVES. Polar Equation oe Cissoro {110) ; Discussion, Sch 76 PoLAH Equation of Conchoid {111) ; Discussion, Sch 77 PoLAB Equation of Lemniscate {112) ; Discussion, Sch 77 OF PLANE SPIRALS. Definitions. — Of Spiral, Measuring Circle, Spire {113) 77 Spiral of Aechimedes. — Definition {lid) ; Construction {115) ; Equa- tion of {116) 78 Eecipsocal or Hyperbolic Spibal (J[j?7) ; Equation; Construction 78 The Lituus {118) ; Equation ; Construction 79 LoGAKiTHMic Spieal {119) ', Definition, Equation, Construction {119) 79 -♦-♦"^ CHAPTEE III. TMANSFOMMATION OF CO-ORiyiKATES. SECTION I PASSING FROM ONE SET OF RECTILINEAR AXES TO ANOTHER. Definitions. — Transformation ; Two aspects of the Problem ; Primitive Axes or System ; New Axes or System ; Illustba.tions {120) ; Practi- cal Advan'agec, Sch 80, 81 Xii CONTENTS. PAGE FoBMULaj FOB Passing from onb Eectelixear Set of Axes to Anotheb.— Ganeral Formulae {122)', From aay set to a Parallel set {123); From Eectangular to Oblique {124); From Rectangular to Rectangular {125); From Oblique to Rectangular {126); The foregoing where the origin is unchanged {127); From Oblique to Rectangular, when a and a' sig- nify the angles which the Oblique or Primitive axes make with the Rect- a igular, or New axis of x, Sch.. . 82-84 EXAMPIIES 84-90 SECTION IL PASSING FEOM RECTILINEAR TO POLAR CO-ORDINATES, AND VICE VERSA., Formula fob passing from Rectilineab to Polar {128) 90 Fobmuils; fob passing from Polab to Rectilineae {129) 91 ExAMPiiES « 91, 92 ^-♦-> — CHAPTER lY. TBOPERTIES OF PLANE LOCI INVESTIGATED BY 3IEANS OF THE EQUATIONS OF THOSE LOCI. SECTION I TANGENTS TO PLANE LOCI. (rt) BY RECTILmEAR CO-ORDINATES. Definitions. — Consecutive points {130); Tangent {131); Tangent has the same direction as the Curve, Cor. {132) 93 (111 Geometricai. Signification of -^ {133); A Tangent which makes any dx given angle with the axis of x, \Vhica is parallel, Which is perpendicular yl34:); Signification of ~ when the axes are oblique {135); Examples. 93-96 Equations of Tangenis. — Gensral Equation {136) ; Of the Ellipse, The Hyperbola, The ParaboU, The Circle?, and other Examples ; The Intercepts of the Axes by a tangent {137), Wi h the axis of x in Ellipse, Hyperbola, Parabola ; Other Examples ; To draw a tangent to an Ellipse {138), To an Hyperbola {139), To a Parabola {140) 96-101 SuBTANGExVTS. —Definition {141} ; General value of {142} ; Of an Ellipse, Hyperbola, Parabola, other Examples ; Use in drawing tangents 101, 102 Length of Tangent.— General formula {143) ; Of an Ellipse, Hyperbola, Parabola • 102, 103 Asymptotes (rectilinear).— Definition {144\ Illustrations; To examine a curve for A.sr/mptoies,— General Method (145 , By Inspection {148), By Develop^no: the function 149 ; An Asymptote the limiting position of a Tangent {146 ; Equation of (i^T* ; Examples 103-107 CONTENTS. XIU PAGS (6) TANGENTS TO POLAB CURVES. How Deteemi»ed {150) 107 SuBTANGENT.— Definition {151) ; General Value {152) ; Examples. .. 107-109 Asymptotes. — How Determined {153) ; Examples 109, 110 SECTION 11. NOEMALS TO PLANE LOCI. (a) BY RECTANGULAR CO-ORDINATES. Definition or Nobmal {154:) HO General Equation (155).— Signification of — — {156) ; Normal to El- lipse, Hyperbola, Parabola, and otber Examples 110, 111 Subnormal.— Definition {157) ; General Value {158) ; To Cycloid {159) ; To draw a Tangent to the Cycloid {160) ; To draw a Tangent making a given angle {161 ) ; Examples HI, 112 Length of Noemal {162) ; Examples 112 Pekpendiculaii upon a Tangent {163) ; From the focus of a Parabola {164) ; Examples • 112, 113 (&) NORMALS TO POLAR CURVES. SuBNOEMAii. — ^Definition {165} ; General Value {166) ; Examples ... 113, 114 Length of Normal to Polar Curve {167) H^ Length of Perpendicular from the Pole upon the Tangent op a Polar Curve {168) H^ SECTION IIL DIRECTION OF CURVATURE. (a) BY RECTANGULAR CO-ORDINATES Criteria for determining Direction of Curvature. Sign of p{ {169) ; Sign of ^' {170) ; Sign of yp{ {171) ; Ex- dxr ay- ax^ amples 114r-116 {h) BY POLAR CO-ORDINATES. Definition op Direction of Curvature of Polar Curves {172) 116 Ceitebia for determining {17 3 f 174) ; Examples 116, 117 SECTION IV. SINGULAR POINTS. Definition and Enumeration {175) 117 Maxima and Minima Ordinates.— Definition {176); To determine their XIV ^ CONTENTS. PAOS position and value {177) ', A. negative maximum or minimum {178) ; Examples 118, 119 Points of iNFiiExioN. — Definition {170), Illustration ; How determined by Kectangular Co-ordinates {180) ; By Polar Co-ordinates {181) ; Ex- amples 119-121 Multiple Points. — Definition, Species {182) ; How determined {183) ; Examples 121-123 Cusps. — Definition, Kinds {184) ; How determined {185) ; Examples 124, 125 Conjugate Points. — Definition {186) ; Two Criteria {187 > 188) ; How to examine a curve for Conjugate Points {180) ; Examples. , 125-127 Shooting Points. — Definition {100) ; Examples 127, 128 Stop Points. — Definition {101) ; Examples 128 SECTION K TRACING CURVES. Definition (102) 129 General Method {103, and Sch.) ; Examples. . , 129-132 To tbace a cubve of the Second Okdee. — By direct inspection of its equa- tion {104:) ; Examples ; By Transformation of Co-ordinates {105) ; Examples 132-135 To Tkace a Polab Cubve {106) ; Examples 135-137 SECTION VL RATE OF CURVATURE. Definitions. — Curvature {107), Illustration ; Osculatory Circle {108), Il- lustration ; Radius of Curvature, Centre of Curvature {100) ; Parameter {202) 137-140 Contact. — What, How closeness of Contact is characterized, Orders of Con- tact {200), Geometrical Illustration {201) ; Order of Contact dependent upon Parameters {203) ; Order of Contact of Eight Line {201', Of Circle, Of Parabola, ElHpse, Hyperbola {205, 206) ; Eestriction of these statements {207) ', Contact of a Eight Line at Point of Inflexion {215) ; Contact of Osculatory Circle at points of Maximum and Mini- mum Curvature {216), At the Vertices of the Conic Sections {217) ■ ■ 139-146 Eadius of Cuevatuee. —General Formula in terms of Eectangular Co-ordi- nates {208) ; Signification of the sign {200) ; Eadius of Curvature of the Conic Sections, At the vertices {210, 212), Varies how {211, 213) ; Centre of Curvature in the Normal {214) ; Examples 141-145 When Osculatory Curves intersect and when not (218) ; When the Os- culatory Circle Cuts a Conic Section (210) 146, 147 Eadius op Curvature of Polar Curves {220) ; Involving the Normal (^221) ; Examples 147, 148 CONTENTS, XV PAGX SECTION VIL EVOLUTES AND INVOLUTES. Definition (222), Illustration 148 To FIND THE EVOLUTE {223) ', Examples ; The Evolute of a Cycloid an Equal Cycloid {225) ; Same Geometrically, ^eh 149-151 NoBMAii TO Involute Tangent to Evolute {220) 151 Kadius of Cubvatuee vabies as Aug of Evolute {227) 151 A CURVE desckibed mechanically feom its Evolute {228) 152 a curve has but one evolute, but an evolute has an infinite numbeb of involutes {229) 152 SECTION VIIL ENVELOPES TO PLANE CURVES. Definition {230), Illustration 152, 153 To find the Envelope {231) ; Examples 153-159 Envelope tangent to the inteesecting seeies {232) 154 Caustics. — General Equation {233) ; When the incident rays are parallel to the axis of the reflector {23d), When perpendicular {235) ; Illustra- tion ; Examples 156-159 SECTION IX, RECTIFICATION OF PLANE CURVES. Definition {237) 159 By Eectangulae Co-oedinates. — General Formula {238) ', Examples ; Circumferences of Circles are to each other as the radii {240) ; Value of 7t [24:1 ) ; Arc of Cycloid equals twice the corresponding chord of the gen- eratrix i24:2) 159-162 By Polab Co-obdinates. — General Formula [24:3) ; Examples 163, 164 SECTION X. QUADRATURE OF PLANE SURFACES. Definition {244:) 164 By Kectangulab Co-oedinates.— General Formula {245) ; Examples ; Areas of Circles to each other as squares of radii [246) ; Area of Circle whose radius is 1 {247) ; Area of Circle = ^r X circumference {248) ; Area of Segment of Circle (249) ; Area of Ellipse compared with Circumscribed and Inscribed Circles (250) 164-168 B^ PoLAE Co-ordinates.— General Formula ^251) ; Examples 168, 169 XVl CONTENTS. SECTION XI QUADRATURE OF SURFACES OF REVOLUTION. Definition {252) ; Illustrations l(il) I^ENEEAL FoKMULA {233) ', Examples ; Surface of a Sphere =: 4 great Cir- • cl.-s, or Circumference X Diameter, Cor. 1 ; Area of Zone {254:) ; Sphere / and Circumscribed Cylinder, Sch 169-170 SECTION XII CUBATURE OF VOLUMES OF REVOLUTION. Geneeal Foemula {255) ; Examples ; Volume of a Sphere = the surface X 3 radius {256) ; Volumes of Spheres are to each other as the cubes of their radii {257) ', Volume of a Segment {258) ; Volume of Sphere and Circumscribed Cylinder {259) 171, iTa SECTION XIII EQUATIONS OF CURVES DEDUCED BY THE AID OF THE CALCULUS. Teacteix. — Dehuition {200) ; Equation (261) 172, 173 Locus WHOSE SUBNOEMAL IS CONSTANT [262) 173 Locus WHOSE NOEMAL IS CONSTANT {263) 174 Locus WHOSE SUBTANGENT IS CONSTANT {264) 174 Locus WHOSE SuBNOEMAL VAEIES AS THE SqUAEE OF ITS AbSCISSA {265) 174 Locus WHOSE Area is twice the peoduct of its Co-oedinates {266) 174 Locus WHOSE AeC VAEIES AS THE SQUAEE EOOT OF THE THIED POWEE Ol' I'rS abscissa {267) 174 SECTION XIV. OF TANGENTS AND NORMALS. [WITHOUT THE AID OF THE CALCULUS.] Tangents. — General Method of producing the equation of {268) ; Ex- amples, — Tangent t > Parabola, Ex. 1 ; Elhpse, Ex. 4 ; Hyperbola, Ex. 10 ; Tangent of the angle which a tangent to a Conic Section makes with the axis of X {260), Examples ; To find the point on a curve from which a tangent must be drawn to make a given angle with the axis of x, be paral- lel, be perpendicular {270) ; Examples 175-179 SuBTANGENTs.— Definition {271) ; To find the length of {272) ; Ex- amples, — Subtangent in Parabola, To draw a tangent by means of {273) ; Subtangent of Ellipse, To draw a tangent by means of {274, 275) ; Sub- CONTENTS. XVll PAGE tangent of Hyperbola {270), To draw a tangent by means of {277) ', Half either axis a mean proportional between its intercepts by a tangent and ordinate, Ex. 4 ; Analogy between the equations of the Conic Sections and the equations of their tangents {278) 179-181 Normals. — Definition {279) ; To produce the Equation of Normal {280) ; Tangent of angle which Normal makes with axis of x {281) ; Examples, Normal to Ellipse, Hyperbola, Parabola, Circle ; Expressions for tangent * of the angle which a Normal to a Conic Section makes with the axis of x {282) 181, 182 SuBNOEMAiiS. — Definition {283) ', Examples in the Conic Sections, Is con- stant in the Parabola and = p, To draw a tangent by means of the latter property. Ex's 1 and 2 183 The Peepediculae feom the focus or a Paeabola upon the tangent {284:) ; Cor. {285) ; To find the focus of a Parabola when the curve and its axis are given, Also to draw a tangent {280 f 287 ? 288) 183 SECTION XV, SPECIAL PROPERTIES OF THE CONIC SECTIONS. Eadh Vectoees. — Definition {289) ; Sum of in Ellipse and diflference in Hyperbola {290) ; Length of each {291) ; To construct an EUipse and Hyperbola on this principle {292) ; Eadii Vectores make equal angles with the tangent in Ellipse and Hyperbola {293) ; Corresponding property in Parabola {290) ; Angles included by the Eadii Vectores and Normal, in Ellipse and Hyperbola {294:) ; To draw a tangent upon these principles, 1st, from a point in the curve, 2nd, from a point without {295) ; Same problems in reference to the Parabola {298) 184-187 The eectangle of PEEPENDicuiiAES FEOM FOCI UPON Tangent {299) 187 The Semi-conjugate axis a mean peopoetional between focal distances {300) 187 Supplementaey Choeds and Conjugate Diameters. — Definition of Ordinate {301), Of Supplementary Chords {302), Of Conjugate Diameters {303) ; Fundamental property of Supplementary Chords {304, 305) ; When drawn on the Conjugate Axis {300) ; When drawn from a point in the Conjugate Hyperbola {307) ', This property in the Circle {308) ; Paral- lelism of Sup. Chords to the axes {309) ; The — sign m axi' = j {310) ; Discussion of the Angle included by Sup, Chords {311) ; Sup. Chords parallel to Tangent and Diameter {312 , 313)*; To draw Tangents by this property {314) ; Eelations between Conjugate Diameters and the Axes {318) ; Examples 188-194 Oedinates. — Eelation to each other, in Ellipse {319), in Hyberbola {322), in Parabola {333) ', Corresponding properties of oblique ordinates {325) ; Eelation of an ordinate to the corresponding segments of its diameter {320) ; Latus Eectum a third proportional to the axes {321 ) ; The rela- tion of ordinates in the circle {323) ; Eelation of ordinates to the conju- gate axis of Ellipse {324) ; Parallel chords bisected hj Diameter {320 , 334) ; To find the centre, axes and foci of a Conic Section when the curv- XVIU CONTENTS. PAGB ature is given {327 f 335) ; Ordiuates of different Ellipses on same axis {Ji28); Of Ellipse and Circle on same axis {329) ; The Trammel {330); Ordinates to different ellipses on same Conjugate Axis {331) ; Of Ellipse and Inscribed Circle {332) 195-199 EcjsNTEic Angle. —Definition (336) '■ Sine and cosine of this angle {337); Advantages, Sch. ; Equation of Tangent to Ellipse in terms of this angle {338) ; Eccentric angles of the vertices of the Conjugate Diameter {339} ; To draw a Conjugate Diameter on this principle {340) ; Kect- angle of Kadii Vectors = Square of Conjugate Diameter {341) ; Sum of the Squares of Conjugate Diameters constant {342) ; Examples 199-201 The Intercepts or a Secant between the Htpeebola and its Asymptotes {343) ; To construct an Hyperbola on this principle, 8ch 201, SOS Parajmetee to ai^ Diameter of a Conic Sectiok. — Definition (344) ; Distance from point in Parabola to focus {345) ; Parameter to any dia- meter of Parabola {340); Parameter to any diameter of Parabola a double ordinate through focus {347) ; Chord of Ellipse through focus {348) ; Sch. {347) not applicable to Ellipse {349) 202, 203 Chord of Curvature. — Definition {350) ; In the Parabola chord of cur- vature through focus a parameter to the diameter through potut of con- tact {351} ; Intercept on this diameter by the osculatory circle equals this chord of curvature {351) 203, 204 CONTENTS. XIX THE INFINITESIMAL CALCULUS. *♦* INTnonUCTION. PAGE Definitions. —Quantity {!) ; Number {2) ; Discontinuous and Continuous Number {3, 4:, 5), Illustrations ; An Infinite Quantity (6) ; An Infini- tesimal (7) ; Caution {8). 1, 2 Infinites and Infinitesimals eecipeocals of each other (9, 10) 3 Obders of Infinites and iNFiNiTESiMAiiS.— What {11} ; Belations to eacb other {12) 3 Axioms {IS, 14:, 15, 10, 17, 18) ; Illustrations ; Examples 4, 6 Constants and Variables. — What {19,20); Any expression containing a va- riable is a variable when taken as a whole {21) ; Distinction of Depend- ent and Independent Variables {22, 23, 24), Illustration ; Equicrescenfc Variable {2S) ; Contemporaneous Increments {20} ; Illustration 6, 7 Functions and their Forms. — Definition of Function {27), Illustration ; Exact limitation of the term, Sch. ; Functions classified as Algebraic and Transcendental, and the latter as Trigonometrical, Circular, Logarithmic and Exponential, with Definitions {28, 29, 30, 31, 32, 33) ; Functions Explicit or Implicit {34, 35, 36), Notation {37) ; Functions Increasing or Decreasing {38, 39, 40) 7-9 The Infinitesimal Calculus. — What {41) t Illustration; Two Branches {42) , 9,10 ^»» CHAPTER I. TSE DIFFERENTIAL CALCULUS, SECTION L DIFFEEENTIATION OF ALGEBRAIC FUNCTIONS. Definitions. — The Differential Calculus {43) ; A Differential {44) ; Con- secutive Values {45), Illustrations 11 Notation for a Differential {4G) 11 EULES FOR DIFFERENTIATING ALGEBRAIC FUNCTIONS. EuLE 1.— To Differentiate a Single Variable {47), Geometrical Illustration 12 Bule 2. — Constant Factors {48), Geometrical Illustration 12, 13 BuLE 3. — Constant Terms (49) ; Geometrical Illustration ; An infinite variety of functions may have the same differential {50) 13 BuLE 4. — The Sum of Several Variables (51^. Illustration ; Character of dr, dy, dz. etc , Sch 14 XX CONTENTS. PAGE; Rule 5. — The Product of Two Variables (52), Illustration ; Rate of Change 14 Rtile 6. — The Product of Several Vaiiables (33) 15 RT7ii£ 7. — Of a Fraction with variable numerator and denominator (54) , With constant numerator (55) ; With constant denominator, Sch 15, 16 Rule 8. — Of a Variable with exponent {36) : Square Root (57) ; Other special rules, Sch 16 ExEECiSES in differentiating 16-22 Tt.t.ustbative Examples showing the significance of differentiation 22-25 SECTION 11. DIFFERENTIATION OF LOGARITHMIC AND EXPONENTIAL FUNCTIONS. Definition. — Modulus {S8) 25 To DiEEEBENTiATE A LoGABiTHM, Common, Napierian {SO) 25 To DiFFEBENTiATE EXPONENTIALS. — With Constant base {00), With reference to Napierian logarithms {01) ; When the base of the Exponential is the base of the system of logarithms {62) ; Of exponential with variable base (63) 26 Exercises 27, 28 DrFEEKENTIATING A VARIABLE WITH TmAGINABY EXPONENT {6S) 28 IujUSTBAtive Examples 28-30 SECTION IIL DIFFERENTIATION OF TRIGONOMETRICAL AND CIRCULAR FUNCTIONS. Of Teigonometeical Functions. — Of a sine {66) ; Of a cosine {67), Signifi- cance of the sign {68) ; Of a tangent {69) ; Of a cotangent (70) ; Of / a secant {71) ', Of a cosecant {72) ; Of a versed-sine {73) ; Of a co- ' versed-sine (74) 30-32 Exercises 32 , 33 Illustbative Examples 34, 35 Of Cieculab Functions. — In terms of sine (75), Relation io differentiating trigonometrical functions {76) ; In terms of cosine {77) ', In terms of tangent {70) ; In terms of cotangent {80) ; In terms of secant {81); In terms of cosecant {82) ; In terms of versed-sine {83) ; In terms of co- versed-sine {81:) 35, 36 Exercises ; Geometrical Illustration 37-39 SECTION IV, SUCCESSIVE DIFFERENTIATION AND DIFFERENTIAL COEFFICIENTS. Successive Differentiation. — Definiiions — Of successive differentials {87), IllustraMons ; Of Second, Third, etc., differentials {88} ; To produce cnccossivc diff^reiitial^ (89) 40, 41 CONTENTS. Xxi PAGR EXEECISES 41 , 42 DuTEEENTiAii CoEPFiciENTS. — A first, A second, A third {90) ; Illustration ; Differential coefficients generally variable, Sch 42-44 Exercises .......,,,,,,,,,,,,,,, 43, 44 SECTION V. FUNCTIONS OF SEVERAL VARIABLES, PARTIAL DIFFERENTIATION, AND DIFFERENTIATION OF IMPLICIT AND COMPOUND FUNCTIONS. Functions oi' Independent and of Dependent Vabiables {91) ; lUustra- j tions 44 Definitions.— Partial Differential (92) ; Total Differential (93) ; Illustra- tions ; Partial Differential Coefficient {94) ; Total Differential Coefficient {95) ; Equicrescence of variables, Sch 45 Total Deffseential equals the Sum or the Paetial Diffeeentials {97) ; Illustrations ; Exercises 45-48 Notation of Differential Coefficients {98) 48 Total Diffeeential Coefficient. — Of function of two variables, Formula, (99 ) ; Meaning of -i- in sucb cases, and distinction between I — and — , I ^ dx [dxj dx i Sch. ; Of three variables, Formula {100) ; "When u = f{y, z, w), and J y = q){x), z = cpiix), and w = qj^ {x) {101) ; Exercises 48-50 Implicit Functions. — To differentiate f{x, y) = {102) ; Why — p= 0, and —J or — not, Sch. ; Exercises 51, 52 dx dy Compound Functions. — Definition, and methods of expression {103) ; To differentiate u=f{y), when y = cp{x) {104:); Exercises; To differen- tiate u = cp{z), when z =f{x, y) {103) 52, 53 k SECTION VL SUCCESSIVE DIFFERENTIATION OF FUNCTIONS OF TWO INDEPENDENT VARIABLES, AND OF IMPLICIT FUNCTIONS. Both Vaeiables may be Equiceescent {106) ; Illustration 53 Successive Paetial Diffeeentials. — ^Definition {107) '■> Notation {108) ; Partial Differential Coefficients, Sch. ; Examples ; Order of differentiation ■unimportant (109) ; Examples ; To form successive Partial Differentials of a function of two Independent Variables {110) ', Law of the formula, Sch 54-58 To FOEM Successive Diffeeential Coefficients of an Implicit Function of A Single Vaeiable {111) ; Examples 58-60 Deeived Equations. — What; Orders of; First and Second produced from u = =f{x, y) {112) 60, 61 XXU CONTEMS. SECTION VIL CHANGE OF INDEPENDENT VAEIABLE. PAGB Why necessaet {113) 61 FoBMS OS -^, -~, -— , -vrlien neither variable is equicrescent {114:) ; Ex- (XX CuC CfriC amples . . 62, 63 FoBMUIiaJ FOE CHANGING FROM X 10 p {116) 64 FoEMuxLiE FOE Inteoducing A NEW VAEiABiiE 6 as the equicrescent {117) 64 Examples 65, 66 To Express the Partial Duteeential Coefficients of u =f{x, y), in terms of r, and 6, when x = cp{r, 6), and 2/ =
— -, y has two
numerically equal, real values, affected with opposite signs. .*. The curve is
symmetrical with respect to the axis of x, and has two infinite branches extending
to the right.
Again -- = ± . ' ' " — . which for x= — y, becomes
dx
s/a + hx
dx
db ex, and for ic = 0,
dy
dx
rh s/a- . * . At ( , ) the curve cuts the axis of x perpendicularly, and at
(0, 0) it cuts it in two directions, viz., at tan-i( -f- v^a), and tan-i( — \/a). This
also shows that (0, 0) is a multiple point, a double point.
Examining for direction of curvature, we have -^ = ±: -, which is
^' 4(a + hx)^
db between and — -, and ± between 0, and -\- cc, . • . At the left of the origin,
the curve is concave towards the axis of x, and at the right, convex.
"We have a maximum and a minimum ordinate at a; = — -^^f y
36'
— -v^d«, as
ct 1 ^hx
appears by solving the equation ± — — ^— = 0.
s/a -j- hx
It only remains to examine the infinite branches for
asymptotes.
^ dx ihx^
X =^x — y-r =
^dy
= CO, for re = 00 ; and
a -)- f 6x
_ dy ^ ihv'^ .
Y = y — x-f- = — ' = =P GO, for x = oo.
^^ s/a 4" bx
Therefore there are no asymptotes.
From this investigation the curve is readily conceived to
have the form given in the figure, which is constructed
assuming a = 36.
Fig. 133.
Ex. 2. Trace tlie curve y^ = a^x\
Results. The curve is symmetrical with respect to the axis of x ;
extends only to the right ; is convex to the axis of x ; has two
infinite branches ; has a cusp of the first kind at the origin, with
the axis of x for the common tangent ; and has no asymptote.
X
Ex. 3. Trace the curve 2/ =^ i , ,-
^ 1 -\- x^
Results. The curve cuts the axes at the origin under an angle of \7t ;
has one infinite branch extending to the right above the axis of x.
TRACING CURVES.
131
and another extending to the left below this axis ; has a maxi-
mum ordinate at ^" = + 1, and a minimum at j; = — 1 ; has the
axis of X as an asymptote to both branches ; has points of in-
flexion at (0, 0), and at ^ = \/3, and x = — v 3 ; between the
latter points is concave towards the axis of x, and beyond them
is convex.
Ex. 4. Trace y^ =: a^ — x^.
Ex. 5. Trace {y — x'^)" == x\
Ex. 6. Trace ay^ — x^ + bx'^ = 0.
Results. The curve cuts the axis of x at right
angles at (6, 0) ; has a conjugate point
at the origin ; has points of inflexion at
x= ^h ', is concave to the axis of x from
x=ib iox==^b, and convex beyond ; has
two infinite symmetrical branches with-
out asymptotes.
Ex. 7. Trace ay^ — .r^ + {b — c)x-^ + bcx
= 0.
The form of the curve is given in the
figure. Observe that when c = this locus
becomes identical with the preceding, which
is sometimes called the campanulate (bell
shaped) parabola.
Ex. 8. Trace the FoHum of Des Car-
tes, whose equation is y'^ — Saxy -f
x^ = 0.
Ex. 9. Trace y^ = 2ax'^ — x^
Fig. 134.
Fig. 135.
Ex. 10. Trace ?/ =
x^
X
-. Examine
a
the curve for asymptotes, for maxima
and minima ordinate s, for cusps, for
direction of curvature, and points of
inflexion.
Fig. 136.
132
PROPERTIES OF PLANE LOCI.
•Ex. 11. Trace y' =
x^ + x-^
Examine
X — 1
the curve for asymptotes, for limits, and
for maxima and minima ordinates.
194, J*TOb, — To trace a curve of the
second order, that is, the locus of Ay^ -f-
Bxy + Cx2 + Dy + Ex + E = 0, by direct
inspection of its equation.
Solution. — One method of solving this problem has been given on pages 46 — 49.
The present method is given as a good algebraic exercise, and in illustration of
the remark in the preceding schoHum upon equations which take the form
y = (p[x) =t ipix).
Solving the equation for y we have
1st. K we construct the straight line of which
y = — - — (B.X -[- D) is the equation (let it be rep-
resented by M N in the figure), any value of x
(as AD) which locates a point (as P) in this line,
locates, in general, two points (P', P") in the
curve, on opposite sides of the line and equally
distant from it, this distance being the radical part
■ 4:A G)x^ + 2(i?i> — 2AE)x + {D^ — 4.AF).
of the value of y.
Therefore 2/ = ■ — irii.^^ + -^)»
Fig. 138.
is a diameter of the locus.
2nd. For such value or values of x as render the
radical 0, y has but one value, and at this point, or these points, the locus cuts its
diameter. Hence {B'^ — 4.AG)x-^ + 2(£i) — 1AE)x + (X>^ — 4.AF) = deter-
-(JBx + X»). In general, this
2^^
mines where the locus cuts the diameter y =
gives two values of a*, indicating that the locus cuts its diameter in two points, as
in the ellipse and hyperbola. But if B^ — 4.AC = 0, the equation becomes
2{BD — 2AE)x -f (1)2 — 4.AF) = 0, which gives only one point of intersection,
as in the parabola, a result which agrees with the fact that B^—4AG=0 characterizes
a parabola {62). Locating the point, or points, at which the curve cuts its diam-
eter, we know, if there are two points, and the curve is an ellipse, that it Hes be-
tween these hmits, or, if an hyperbola, beyond. These facts will readily appear
by observing whether intermediate values of x give real or imaginary values to ?/.
Thus the limits of the curve appear,
3rd. If the locus is an ellipse, the values of y midway between the two values of
TRACING CURVES.
133
X which correspond to the extremities of the diameter, make known a diameter
parallel to tangents at the extremities of the former, and hence determine the cir-
cumscribed parallelogram. Thus the situation of the ellipse becomes known.
dth. If the locus is an hyperbola, we can determine a few values of y corres-
ponding to values of x without the limits, and thus locate the curve. It is often
expedient to find the intersections with the axes.
5th. If the locus is a parabola, having determined its diameter and vertex, a few
values of x will make known sufi&cient points to enable us to sketch the curve.
The intersections with the axes may also be of service.
Ex. 1. Trace the curve whose equation is
y2 — 2a;y + 2^7* + 2?/ + a? + 3 = 0.
Sug's. — Since B'^ — 4:AC <^ 0, the locus is an el-
lipse. Solving for y, we have
y = OS — 1 ± \/ — .r2 — Sx — 2 ;
whence y = x — 1 is a diameter, which we construct.
n/ — x- — dx — 2=0, gives x = — 1, and — 2, the
limits of the curve. Between these limits y is real,
and without them it is imaginary. For a; = — li,
2/ := — 2, and — 3. Thus we find the circumscribed
parallelogram.
Fig. 139.
Ex. 2. Trace the curve whose equation is t/2 -|- 2xy — 2x^ — 4?/ —
07 + 10 = 0.
Sug's. — As B^ — iAC >* the locus is an hy-
perbola. ?/ = — ^ + 2drv/3(x2— iP — 2). y = — X
-f- 2 locates N M . From 3(aj'^ — x — 2) = 0, we
find P and P ", at .r =-- 2, and — 1. Between
these values y is imaginary ; hence the locus lies
beyond these points to the right and left. Put-
ting 2/ = 0, we have — 2a;2 — a; -[- 10 = 0, whence
a; = 2, and —21, and the curve cuts the axis of
a; at C and B. For ic = 4, 2/ = 3 • 5 and — 7-5
nearly, and we have 1 and 2. In hke manner
a§ many points as we wish may be found ; but
with the diameter and intersections with the
axis, little or nothing more is necessary in order
to form a pretty definite idea of the situation of
the curve.
Fig. 140.
Ex. 3. Trace the locus y^ — 2xy -\- x^ — 4y -f- a; -f 4 = 0.
134
PEOPEETIES OF PLANE LOCI.
Sug's. — Since JB^ — 4:AC = 0, the locus is a parab-
ola, y = x -\- 2 is the equation of a diameter. For
X =1 0, y Tz=.2. For x negative, y is imaginary. For
.r = 3, 2/ = 8, and 2.
Ex's. 4 to 7. In like manner trace the fol-
lowing : 2/2 _|_ ^xy + 3j^2 — 4^ = ; ?/« —
^xy + 2j72 — 2^ = ; y^ _|_ 4^1/ _|_ 4j;2 — y
4=0; and 2/" — ^^y + 2^2 _j_ 2i/ — 2^ +
3 = 0.
Fig. 141.
lOS, I*VOh, — To trace a locus of the second order hy means of
transformation of co-ordinates.
Solution. — "We will illustrate this method by an example. The method itself is
altogether too tedious for practical purposes, but is highly important as giving a
clear view of a process which we have occasion to use for other purposes. Let us
trace the locus whose equation is y^ -\- 2xy -{- Sx^ — 4:X = 0.
This is an ellipse, since B'^ — 4:A C'<^ 0. We will find its equation when referred
to its own axes. This requires transformation from one rectangular system to
another. The formulce for this transformation are a; = x,^ cos a — yi sin a -\- m,
and y z=z Xi sin a -\- yi cos a -\- n. Substituting these in the equation, we have
=0.
(Eq. A.)
As the required form of the equation is Ay'^ -\- Bx- -\- F= 0, we desire to elimi-
nate the terms containing x^y^, and y^ and ccj. To find the direction of the new
axes, I. e. to determine the value of a, and to find the position of the new origin,
i.e., to determine the values of m and n, which will effect this reduction, we place
the coefficients of the terms to be eliminated each equal to 0, and solve the result-
ing equations. These equations are
(1) 2 sin a cos a — 2 sin2 a -f- 2 cos"^ a. — 6 sin a cos a: = ;
(2) 2n cos a -\- 2m cos a — 2n sin a — 6m sin a -}- 4: sin a = \
cos'^a
?/r 4-2 sin a cos a
Viyi-\- sin^a
.•Ci2-f-27icosa
?/i-}-2nsin a
Xi-\- n^
■2 sin a cos a
— 2 sin -a
-f-2sinacosa
4-27ncosa
-f 2msin ' = 90° -j- (p ; whence sincp' = cos cp, and P' A'^= A- —
e2 • A-cos'2 (p ^= A^ — e-x^. .' . rr' = P'A^. q. e. d.
342* JPvop,—The sum of the squares of any pair of conjugate
diameters is constant and equal to the sum of the squares of the axes.
Dem. — In Fig. 223 we have P' A' =^ x^- -{- y^^ = B-sin^cp' -f- ^2cos2^' ; or
since (p' =90° -|-^ sin q)' = cos cp, and cos cp'^= — sin cp,
■ — -, 2
P'A = A^ sin2 q) -^ B- cos^ qj ; and in like manner,
P A' = ^- cos- q) 4- B- sin2 q).
Adding P A' + P' A" = A"- -{- B^-. Multiplying by 4,
4PA' + 4P'A'=:4^2_|_4^e. Q. E. D.
ScH. — This proposition has been demonstrated before {318 ^ a), but is
inserted here as its demonstration affords an example of the utility of the
eccentric angle.
Ex. 1. What is tlie eccentric angle of the extremity of the trans-
SPECIAL PBOPERTIES OF THE CONIC SECTIONS.
201
verse axis ? What of the extremity of the latus rectum ? What of
the extremity of the conjugate axis ?
Ans., (p = 0°, cp = cos~^ e = sin~^ — cp = 90®.
Ex. 2. In an elHpse whose axes are 8 and 6, what is the eccentric
angle at :r = 1 ? What are the co-ordinates of the point of which
the eccentric angle is 60° ? 45° ? 30° ?
Ex. 3. In an ellipse whose axes are 12 and 8 what is the length of
the diameter from the point whose eccentric angle is 60° ?
SuG.— Calling the semi-diameter A^ we have A2- = A'^ cos^ g) -\- B'^sin'^q)^
36 X {ky -+- 16 X (iv'3)2 = 21, and A^ = v/21.
343, JProp, — The intercepts of a secant between the hyperbola and
its asymptotes are equal.
Dem. — Let DD' be any secant, and P
and P', the points in which it cuts the
curve, be designated respectively as {x, y')
and (x", y"). Since DD' is a line pass-
ing through the two points (x, y'), and
{x", y"), we have for its equation y — y' =
y
y
-{x — x). And since (.r', y') and
X — cc" '
{X", y") are points in the curve x'y' =
x"y" = m. If in the equation of D D '
we make y = 0, x = AD, and x — x' =
CD. Hence we have CD = x — x' =
X y — y X
Fig. 224.
_ x"y" — y'x"
y" — y'
= X" =
y" — y'
C'P'. Now as PCD and P'C'D' are equiangular and have CD = C'P',
the triangles are equal, and PD = P' D'. Q. e. d.
ScH. — This proposition afibrds an ele-
gant and convenient method of construct-
ing the hyperbola. If the axes are given,
put them in position and draw the
asymptotes, which are the diagonals of
the rectangle on the axes. Then, through
the extremities of the transverse axis,
draw a convenient number of radiant
lines, as aa' , 'hh' , cc\ dd\ and make the
intercepts \a' , 2b', 3c', 4c?' respectively
equal to Ba, Bb, Be, Be?. Then are 1, 2,
3, 4, points in the curve.
If the asymptotes are given, or the
Fia. 225.
202
PROPERTIES OF PLANE LOCI.
angle included, and any point in the curve as P, the asymptotes can be
drawn ; and then radiant hues through P will be secants whose intercepts
will make known points in the curve.
PARAMETER TO ANT DIAMETER.
o44. A Parameter to any iPiameter of an Ellipse or
Hyperbola, is a third proportional to that diameter and its conjugate.
In the Parabola it is a third proportional to any abscissa and its cor-
responding ordinate.
34:S, J^rop, — The distance from any point in a Parabola to the
focus is one fourth the parameter to the diameter from that point.
Dem.— Let A2 F =/ ; then is y2^ = ¥^z- From Ex. 12, page 88, we have 2/2 2
2p
= ■ , X2 j and also 2n sin a' — 2r) cos a'
0.
From the latter, n^ sin^o:'
p2sin"a'; whence sin^a'
p2 cos'-'a' = p2 —
p2
n2-f p2 •
2p 2(7i2 4- p2) 2(2pm -f- p2)
Hence
sin-'a' p p
4(to -}- ip), since n^ = 2pm. But m -f- sP =
TF = FA2 = /. Therefore 2/2^= ¥^2,
or iCg : 2/2 • • 2/2 '■ ¥ '} and 4/ is the parameter
to the diameter A. 2^2 Q. e. d.
Fig. 226.
34:0, CoR. 1.— The parameter to any diameter of a Parabola is four
times the distance from the vertex of that diameter to the directrix.
34:7* CoE. 2. — The double ordinate to any diameter of a Parabola,
which {ordinate) passes through the focus, is the parameter to that dia-
meter.
Dem. — Let AgH =X2, and LH = 2/2 1 whence 2/2^= ^f^i- Now A^H =;
Xi= TF = A2F =/. Wherefore ?/2 2 = 4/2 ; and 2/2 = 2/. But IL =
2LH = 2y.2. =^ 4f. . • . I L is the parameter to A2a;2'
34S, JProp, — Any chord ichich passes through the focus of an
Ellipse is a third proportional to the transverse axis and a diameter par-
allel to the chord.
Dem.— Let PF=:r, PFB = a:, and P'F = r'. Then
P _„,-, ... V
-, and r' = _ ,
1 — e cos a 1-j-ecosa:
{107)', whence r
4-r'= PP' =:
86, 5i2 =
2p
1 — e- coti'^a
But from JEx. 10, page
AHl — e2)
A'^ siii'^a -f- B^ cos- a
e- cos- a
P'
Fig. 227.
Ap
1 — e^ cos^a
2
SPECIAL PROPERTIES OF THE CONIC SECTIONS.
; by substituting A^(l — e^) for B^ and reducing. Therefore
203
pp_
, or 2 J. : BjCi : : BiCj : PP'. q. e. d.
349. ScH. — Tlie statement in [347) is not true in case of the ellipse, as
will appear from this proposition.
CHORD OF CURVATURE.
[Note.— The following proposition is designed to be read by those who have taken the
Dlfterential Calculus, and have studied Section VI, Chapter IV, or have some knowledge of the
subject of radius of curvature.]
350. A Chord of Curvature is a chord of the Osculatory
Circle, drawn from the point of contact.
351. I^rop, — In the parabola, the chord of curvature which
passes through the focus is the parameter to the diameter passing
through the point of contact.
Dem. O being the centre of the os-
culatory circle at 3P, in the parabola
whose focus is "F, IE*^wC is the chord of
curvature passing through the focus, and
is the parameter to IP3D, the diameter
through IE*. For, draw IFXj perpendic-
ular to the tangent through P, and we
have from the similar triangles Jr^JbdiiMC
and 1P:E*'Lj,
■F:E<, : lE^IS^C : : IT : FI-., or.
:!E*1^ =
-p-p
2N^
But 1*1^ = — j », I[ being the normal,
and p the semi-latusrectum of the curve {211); !FIj = ^!F*IEj = \]Sf {164:
or 284), and IT = A/^^ + XJF
*' = \/\n^-\-~N'^1
4pi
pi
1 /2>2 4- ^2 7^2
g N'Y ^ / - — {143, Ex. 2), remembering that p^ + y^ = W\ Substituting
these values, we have, IPIMI = — s- x -pr-
2i^3 ]^ 2p 2i\r2
p'
2 W'
= = 4Jr'Jb' and hence is
P
the parameter to IE*ID {340).
204 GENEEAL SCHOLIUM.
352, CoK. — The chord l»s intercepted on the diameter I»ID is
equal to the chord of curvature j^assing through the focus, since
angle CI*ID = TI^OSJ:.
EKD OF PART FIRST.
GENERAL SCHOLIUM.
Book Second, treating of Loci in Space, is reserved for a second volume.
The present volume is deemed sufficient for the use of all students in our
colleges, except such as pursue mathematical studies as a specialty. Yol-
ume n. -^11 contain Loci in Space, and a more extended course in the
Calculus.
THE
INFINITESIMAL CALCULUS.
INTRODUCTION.
[>ToTE. — The four following chapters on the Diflferential Calcixlus are to be read iniinediately
after the first three chapters of the General Geometry, that is, the first 92 pages of this volume.]
1, Qtiatltity is the amount or extent of that which may be
measured ; it comprehends number and magnitude. (See Akt. 4,
General Geometry, and the two SchoHums under it on pages 1
and 2.)
2, NlilfYlbeT is quantity conceived as made up of parts, and
answers to the question, "How many?" (See Art. 5, Illustration,
General Geometry.)
S* Number is of two kinds, DiscontiTlttOUS and CoTltin-
uous.
4, Discofltiflttous JVtcmber is number conceived as made
up of finite parts ; or it is number which passes from one state of
aggregation to another by the successive additions of finite units,
i, e., units of appreciable magnitude.
S» ContiflUOUS JVttmber is number which is conceived as
composed of infinitesimal parts ; or it is number which passes from
one state of value to another by passing through all intermediate
values, or states.
Ill's. — The method of conceiving number with which
the pupil has become familiar in arithmetic and algebra,
characterizes discontinuous number. Thus the number
1 3 is conceived as produced from 5 by the successive ad-
ditions of finite units, either integral or fractional. In
either case we advance by successive steps oi finite length.
If we say 5, 6, 7, etc., tiU we reach 13, we pass by one •^^^- ^^
kind of steps; and, if we say 5.1, 5.2, 5.3, etc., till we reach 13, we pass by
another sort of steps {tenths), but as really hy finite ones. If, however, we call the
hne A B, Fig. 1, x, and C D, x', and conceive AB to slide to the position CD,
increasing in length as it moves so as to keep its extremities in the lines O M and
B D '*
B D
Fig. 3.
2 INFINITESIMAL Cx\LCULUS.
O N , it will pass by infinitesimal elements of growth from the value x, te the value
x' ; or, it will pass from one value to the other by passing through all intermediate
values, and thus becomes an illustration of continuous number.
Again, if the line A B, Fig. 2, be considered as gen- ^
erated by a point moving from A to B, and we call AC B
the portion generated when the point has reached C, • ^^^' 2-
X, and the whole line x', x will pass to x' , by receiving" infinitesimal increments,
or by passing through all states of value between x and x'.
A surface may be considered as generated by the mo-
tion of a Hue, and thus afford another illustration of
continuous number. Thus let the parallelogram AF
be conceived as generated by the right line A B moving
from AB to EF. When AB has reached the po-
sition CD, call the surface traced, namely A BCD,
X, and the entire surface A B E F, x' ; then will x pass to x' by receiving infinites-
imal increments, or by passing through all intermediate values.
Finally, as volumes may be conceived as generated by the motion of planes, all
geometrical magnitudes -afford illustrations of continuous number.
We usually conceive of time as discontinuous number, as when we think of it as
made up of hours, days, weeks, etc. But it is easy to see that such is not ttie
way in which time actually grows: A period of one day does not grow to be a
period of one week by taking on a whole day at a time, or a whole hour, or even
a whole second. It grows by imperceptible increments (additions). These incon-
ceivably small parts of which continuous number is made up are called Infinites-
imals.
Motion and force afford other illustrations of continuous number. In fact, the
conception which regards number as continuous, vsdll be seen to be less artificial —
more true to nature — than the conception of it as discontinuous.
6. Jin Infinite Quantity is a quantity conceived under such
a form, or law, as to be necessarily greater than any assignable quan-
tity.
7. A.n Infinitesimal is a quantity conceived under such a
form, or law, as to be necessarily less than any assignable quantity.
8. ScH. — By an infinite quantity is not meant one larger than any other,
or the largest possible quantity. It simply means a quantity larger than
any assignable quantity ; i. e. , larger than any one which has limits. The
mathematical notion concerns rather the manner of conceiving the quantity,
than its absolute value. Thus, a series of Is, as 1 1 1, etc., repeated with-
out stopping, represents an infinite quantity, because, from the method of
conceiving the quantity, it is necessarily greater than any quantity which
we can assign or mention. If we assign a row of 9s reaching around the
world, though it is an inconceivably great number, it is not as great as a
series of Is extending without limit. Moreover, one infinite may be larger
than another ; for a series of 2s extending without limit, as 2 2 2 2, etc., is
INTEODUCTION. 3
twice as large as a series of Is conceived in the same way. It is never of
any use to try to comprehend the magnitude of an infinite quantity ; we
cannot do it ; although we can compare infinites just as well as finites.
Again, and- what is more to our purpose, an infinitesimal quantity is not
a quantity so small that there can be no smaller. There would be but one
such quantity and hence no comparison of infinitesimals. All that is meant
by the term as used in mathematics is, a quantity which is to be treated
in the argument as less than any assignable quantity. Whether we can
or cannot comprehend its absolute magnitude is of no manner of con-
sequence. Nor is absolute value usually of any importance in pure mathe-
matical reasoning. Thus 2 times 5 is 10 whether 5 be mites or moun-
tains. In order to free himself from needless embarrassment in the use
of infinitesimals, the student needs to keep constantly in mind the fact that,
In pure mathematics, it is the relation of quantities, rather
than their absolute values, with which we are concerned.
9» JPfop, — Any finite quantity divided by an infinite is an infinites-
imal ; and any finite quantity divided by an infinitesimal is an infinite.
Dem. — Let a represent any finite quantity and x any infinite. Then - is an in-
X
finitesimal ; for the value of a fraction depends upon the relative values of its
numerator and denominator, and is less as the ratio of numerator to denominator
is less. Now, in this case, a is infinitely less than x, by the definition of an infinite.
Hence - is an infinitesimal. Again - is infinite if x is infinitesimal, since a is in-
X X .
finitely greater than cc.
10, CoK. — The reciprocal of an infinite is infinitesimal, and the re-
ciprocal of an infinitesimal is infinite.
11, The products of infinites by infinites, and of infinitesimals by
infinitesimals are denominated Ot^CTS : thus, if x and y are in-
finites, x% 7/2, and coy are infinites of the Second Order ; if x, y,
and z are infinites, x^, z^, xyz, x^y, xy^, etc., are infinites of the Third
Order, The corresponding expressions are used with reference to
infinitesimals, the product of two infinitesimals being caUed an infin-
itesimal of the second order, of three, the third, etc.
12, ScH. — An infinite of a lower order sustains a relation to the next
higher similar to that which a finite sustains to an infinite. Thus if x and y
are infinites, x-, xy, and y'^ are infinitely greater than x and y. On the other
hand if x and y are infinitesimals, x'^, xy, and y^ are infinitely less, and sus-
tain a relation to x and y, similar to that which infinitesimals sustain to
finites.
INFINITESIMAL CALCULUS.
AXIOMS.
13* From expressions containing the sum or difference of finites
and infinites, the finites may be dropped without affecting the ratio.
14:, From expressions containing the sum or difference of infin-
ites of different orders, the terms containing the lower orders may be
dropped without affecting the ratio.
15, The order of an infinite is not altered by multiplying or divid-
ing it by a finite.
10» From expressions containing the sum or difference of finites
and infinitesimals, the infinitesimal terms may be dropped without
affecting the ratio.
17* From expressions containing the sum or difference of infini-
tesimals of different orders the terms containing the higher orders
may be dropped without affecting the ratio.
18, The order of an infinitesimal is not changed by multiplying
or dividing it by a finite.
Ill's. — Although the above are conceived to be axioms in the strictest sense,
that is truths to which the mind at once assents as soon as the terms used are
clearly comprehended, the true notion of infinites and intinitesimals is so removed
from common thought that a familiar illustration or two may aid the comprehen-
sion. Suppose, then, that the quantities under consideration were the masses of
matter in the earth and in the sun. If a grain of sand were added to or subtracted
from each or either it would not appreciably affect the ratio of these masses. But
in this instance the grain of sand is by no means infinitesimal with reference to
either mass ; it is & finite, though very small part, of either mass.
Again, let x and y be two infinite quantities, and a and 6 two finite ones. There
can be no difference between — ==^ and - : since to assume such a difierence
y ±zh y
would be to assign some values to a and h, as respects x and y. But by hypoth-
esis, the former have no assignable values in relation to the latter.
^ ..„.,, ^ a±x a
Once more, if a and h are finite quantities, and x and y infinitesimal, = -,
since x and y have no assignable values as compared with a and h. So also, x and
y still being infinitesimal, — = — = -, as x^ and y^ are infinitesimals, (have no
y±y' y
assignable values) with respect to x and y.
INTKODUCTIGN. 5
ETALUATION OF EXPRESSIONS CONTAINING EINITES AND
INFINITESIMALS, AND FINITES AND INFINITES,
Ex. 1. What is the value of the fraction - — — - if x is infinite and
dx -\- b '
a and 6 finite ?
Solution. — Since a and h liave no assignable values in relation to x they must
^x 2
be dropped, and we bave ^. Now dividing both terms by x, we have - as the
value of when re is infinite and a and & finite.
'dx -\- h
Ex. 2. What is the value of the fraction in the last example if x is
infinitesimal and a and h finite ?
Solution. — As x is infinitesimal ^x and 3ic are also infinitesimal, and hence have
no value in relation to a and h, and must be dropped. Hence the value of the
fraction is — -.
Ex. 3. What is the value of -— — when x is infinite ? When
X is infinitesimal ?
Ans., When x is infinite, 6 ; when infinitesimal, 3.
^x
X
Ex. 4. What is the value of y in the equation y = when x
--\-x
X
is infinite ? When x is infinitesimal ?
Atis.j When x is infinite, — 5 ; when infinitesimal, -.
Ex. 5. What is the value of y in the expression y = — - — when x
JL "j~ X
is infinite ? When x is infinitesimal ?
Ans., When x is infinite, ; when infinitesimal, 1.
-^ ^ -^x, . . .-, , . ax^ -\- hx"^ -^ ex -\r d . ••/»..«
Ex. 6. What IS the value of when x is mnmte?
m^3 _|_ ifirjQ'i -\- px 4- q
When X is infinitesimal ?
Ans., WTien x is infinite, — ; when infinitesimal, -.
m q
2x^ — 5m^x
Ex. 7. What is the value of y m the expression y = — •
ox • TTIX
when X is infinite ? When x is infinitesimal ?
Ans., When x is infinite, ; when infinitesimal, 5m,
6 INFINITESIMAL CALCULUS.
• doc'^ A- 2 7*^ 1
Ex. 8. "What is the value of y in the equation y = -
when X is infinite ? "When x is infinitesimal ?
Ans., When x is infinite, y=Qo', when infinitesimal, y = ■ — \.
?)X
Ex. 9. When x and v are infinitesimals what is the value of *-- ?
^/zs., We cannot tell ; as we know nothing about the relation be-
tween X and y.
3x
Ex. 10. What is the value of — when y^ = 9x and x and y are
infinite ? Ani^., oo.
Ex. 11. Same as Ex. 10, only x and y infinitesimal? Ans., 0.
Ex. 12. What is the value of y in the equation y^ = —- , when
X is infinitesimal ? Ans., 0.
CONSTANTS AND TARIABLES.
19. A. CottStatlt quantity is one which maintains the same
value throughout the same discussion, and is represented in the no-
tation by one of the leading letters of the alphabet.
20* VuTiable quantities are such as may assume in the same
discussion any value, within certain limits determined by the nature
of the problem, and are represented by the final letters of the
alphabet.
2JL, CoR. — Any exjjression containing a variable is, when taken as a
whole, a variable. Thus the value of the entire exjjression (4a — 3^^ -[-5)2
varies if x varies ; so that taken as a whole it is a variable.
[Note. — These notions should be already familiar from General Geometry, page 9, and are in-
troduced here only to give completeness, and for review.]
22, Variables are distinguished as Independent and Dependent.
23. An Independent Variable is one to which we assign
arbitrary values, or upon whose law of variation we make some arbi-
trary hypothesis.
24:, A- Dependent Yariahle is one which varies in value in
consequence of the variation of the independent variable or vari-
ables.
IiiL. — ^ThuB, in the equation of the parabola, y'^ = 2px, if we assign arbitrary
INTRODUCTION.
values to x and find tlie corresponding values of y, we make
X the independent variable, and p the dependent variable.
Again, and what is more to our present purpose, if we as-
sume X to vary in some particular way, as by taking on equal
increments, as DD', D'D", D"D"', etc., 2/ will evi-
dently vary ia some other way, but still in a way depending
upon the way in which x varies, and upon the nature of the
curve, or, what is the same thing, upon the form of the
equation of the curve. In this case also, x is the independ-
ent and 7/ the dependent variable.
Fig. 4.
ScH. — This distinction is made simply for convenience, and is not founded
in any difference in the nature of the variables ; either variable may be
treated as the independent variable.
2S, A.n JSquicresceflt variable is one which is assumed to in-
crease or decrease by equal increments or decrements, as x in the last
illustration.
26. Contemporaneous Tncrefnents are
such as are generated at the same time.
iLii. — Thus let^=/(cc) represent the equation of AM in
the figure. Suppose we contemplate the values of x and y
at the point P'. Now if ic takes the increment D'D", y
takes the contemporaneous increment P"E'. So also we
see that DD', P'E, and PP' are contemporaneous incre-
ments of the abscissa, ordinate, and arc, respectively.
Fig. 5.
FUNCTIONS AND THEIR FORMS,
27* JL Function is a quantity, or a mathematical expression,
conceived as depending for its value upon some other quantity or
quantities.
III. — A man's wages for a given time is a function of the amount received per
day ; or, in general, his wages is a function of both the time of service and the
amount received per day. Again, in the expressions y = 2ax-, y = x^ — 26x + 5,
2/ = 2 log ax, y = a'', y is, a function of x ; since, the numbers 2, 5, a and & being
considered constant, the value of y depends upon the value we assign to x. For
a like reason \/a'^ — x"^, and 3aa;2 — 2\/a; may be spoken of as functions of x.
Once more, the ordinate of a curve is a function of the abscissa.
ScH. — There is a sense in which the dependent variable (or function) is a
function of the constants as well as of the variable or variables which enter
into its value. So also it is a function of the form of the expression, that
is, its value depends in part upon the form of the expression as well as
upon the value of the independent variable. Thus if we have y = a\ogx
8 INFINITESIMAL CALCULUS.
-f- h, and y = x^ — ex, though in each case ?/ is a function of x, speaking
according to the definition, nevertheless it is not the same function in both
cases. Its value depends upon the value of x, upon the constants, and
upon the form of the expression involving these quantities. But the con-
ception expressed in the definition is the ordinary one.
28* Functions are classified by their forms as Algebraic and
TraTiscendentalf and the latter are subdivided into Trigono-
metrical and Circular^ Logarithmic and Exponential.
29, An Algebraic Function is one which involves only the
elementary methods of combination, viz., addition, subtraction, mul-
tiplication, division, involution and evolution. Thus in y= ax^ — 3^",
y is an algebraic function of x.
30, A Trigonometrical Function is one which involves
sines, cosines, tangents, cotangents, etc., as variables ; thus ?/=:sinar,
y = sin X tan x, etc.
31, A Circular Function is one in which the concept is a
variable arc (in the trigonometrical the concept is a right line). These
are written thus : y = sin~^^, read " y equals the arc whose sine is a; ";
y == isiii'^x, read " y equals the arc whose tangent is x."
III. — Notice that in the expression 2/ = tan— i .-j;, it is the arc which we are to
think of, while in the expression x = tan y it is the tangent, which is a right line.
Trigonometrical functions are right lines ; circular functions are arcs. These
functions are mutually convertible into each other ; thus y = sin—' x is equivalent
to a; = sin y, the only difference being that in the former we think of the arc, the
sine being given to tell what arc, and in the latter, we think of its sine, the arc
being given to tell what sine.
The circular functions y = sin~^^, y = cosr'^x^ y = sec~^x, etc., are
often called Inverse Trigonometrical Functions.
32, A Logarithmic Function is one which involves loga-
rithms of the variable ; as y = log x, log^ 2/ = 3 log ax, etc.
33, An Fxponential Function is one in which the vari-
able occurs as an exponent ; as y = a"", z = x'-', etc.
34, Functions are further distinguished as Explicit and Im-
plicit,
35, An Explicit^ Function is a variable whose value is ex-
pressed in terms of another variable or other variables and constants.
Thus in y = 2ax^ — 3^"^, y is an explicit function of x.
* From explicitum, unfolded. The function is disentangled from the other quantities.
INTRODUCTION. 9
36, An Implicit^ Fmiction is a variable involved in an
equation which is not solved. Thus in x"^ — ^xy -f 2?/ = 16, ?/ is an
implicit function of x, or x is an implicit function of y. When we
can solve the equation, an implicit function may always be expressed
as exphcit.
37* dotation. When we wish to write that y is an explicit
function of x, and do not care to say precisely what the form of the
function is, we write y =f(x), read "i/ = a function of x." If we
wish to indicate several different forms of dependence in the same
discussion, we use other letters, as 2/=/(^), y=F{x), y= cp{x), etc.,
or use subscripts or accents as y =f(x), y ==zf'(^x), etc. Such
symbols are read "y = the/", large F, cp, f sub-one, f prime, etc.,
function of x" as the case may be.
When we wish to write that x and y are functions of each other, or
that y is an implicit function of x, or x an implicit function of y,
without being more specific, we write F{x, y) = 0, or f{x, y) == 0,
or (p{x, y) r= 0, etc. ; and read "function x and y = 0," the F func-
tion X and 2/ = 0, etc. This form symbolizes any equation between
two variables with all the terms transposed to the first member.
3S, Again, functions are distinguished as Incveasitig and
Decreasing, '
30* A.n Increasing Function is a function that increases
as its variable increases, and decreases as its variable decreases.
4:0, Jl Decreasing Function is a function which decreases
as its variable increases, and increases as its variable decreases.
III,. — In the expressions t/- = 'Ipx, y = log 'X,, y t^ a^, y is an increasing function
of X. In the expressions y = —, y"- -\- x^ = E-, y = log — , y is a decreasing func-
tion of X. For what vahies of cc is 2/ an increasing function of its variable, and
for what a decreasing, in the following : y^ = ax^ — x'^, y = sin a;, y =z cosa; ?
41, OOhe Infinitesimal Calculus treats of Continuous
Number^ and is chiefly occupied in deducing the relations of the con-
temporaneous infinitesimal elements of such number from given re-
lations between finite values, and the converse process, and also in
pointing out the nature of such infinitesimals and the methods of
using them in mathematical investigation.
* From implicituiti, infolded, entangled.
10
INFINITESIMAL CALCULUS.
III. — Let y^ = 8x be the equation of the parabola in the
figure. Here we have the relation between finite values of
y and x expressed. Now suppose x takes an infinitesimal
increment as. D D ' *, what increment does y take ? The cal-
. 4
cuius shows us that the increment which y takes is - times
y
as large as the increment which x takes ; that is, it shows us
the relation between the elements of the variables y and x,
when we know the relation between finite values. This is
the province of the Differential Calculus, The converse of
Fig. 6.
this problem is, What is the equation of the curve whose ordinate varies - times
as fast as its abscissa ? that is, having given the relation between the infinitesimal
elements of y and x, to find the relation between finite values. This is the prov-
ince of The Integral Calculus.
4:2, There are two branches of the Calculus, yiz., XTie Differ-
ential Calculus^ and The Integral Calculus,
* Of course all such attempts to represent infinitesimals to tlie eye, are egregious exaggerations;
nevertheless they are of great service to the roiad.
THE
INFINITESIMAL CALCULUS.
CHAPTER I.
THE JDIFFEBENTIAL CALCULUS.
SECTION L
DifFerentiation of Algebraic Functions.
4:8, The DiffereTitial Calculus is that branch of the Infin-
itesimal Calculus which treats of the methods of deducing the
relations between the contemporaneous infinitesimal elements of vari-
ables, from given relations between finite values.
4:4:, A. Diffevential is the difference between two consecutive
states of a function, or variable. It is the same as an infinitesimal.
45, Consecutive Values of a function or variable are values
which differ from each other by less than any assignable quantity.
Consecutive Points on a line are points nearer to each other than
any assignable distance.
III. — Suppose y = 2x^ — Sx. Now let x be supposed to increase infinitesimally,
2/ will also change infinitesimally. Call the new value of y, y\ Then y' =2x'^ —
3x'. In such a case a; and x' are consecutive values of the variable, and y and y'
are consecutive values of the function. But by this we do not mean that x and x'
{or J and y' ) are so nearly equal thai there can he no intermediate value, for this would
be to make an infinitesimal mean a quantity so small that there can be no smaller,
which is not its meaning as used in mathematics (7). AH that is meant by saying
that y and y' are consecutive values is that they are to he reasoned upon as having
no assignable difference.
So also in speaking of consecutive points on a line, as D and D', or P and P',
Mg. 6, we do not conceive them as actually in juxtaposition ; but we mean simply
that we are to reason upon them as nearer each other than any assignable distance.
46, Wotatiofl, The differential of a variable (one of its infini-
tesimal elements) is represented by writing the letter d before it.
12 THE DIFFEKENTIAL CALCULUS.
Thus, doc, read, " differential x" Of course the letter d is not to be con-
founded with a factor ; it is simply an abbreviation for differential.
[Caution. — The student should be careful and not allow himself to read such
expressions as dy, dx, etc. , by merely naming the letters as he would ay, ax, etc.
The former should always be read " differential y," "differential x," etc.]
RULES FOR DIFFERENTIATING ALGEBRAIC FUNCTIONS.
47. BULE 1. To DIFFERENTIATE A SINGLE YAEIABLE SIMPLY WHITE
THE LETTER d BEFORE IT.
Dem. — Let us take the function y =z x. The consecutive state of the variable
is ic -|- dx. Now representing the change in y which is produced by this change
in X by cZy {dx and dy being the contemporaneous increments of the variable and
the function), we have
1st state of the function, y = x,
2nd, or consecutive state, y -\- dy ^x -\- d^.
Subtracting the 1st from the 2nd, dy = dx, which
being the difference between two consecutive states of the function is its differen-
tial {4:4:). Q. E. D.
ScH. — This rule is evidently only the same thing as the notation requires,
and its formal demonstration would be unnecessary except for the purpose
of uniformity in treating the several cases of differentiation.
III. — Let M N be a line passing through the origin and making an angle of
450 with the axis of x. Its equation is ?/ = cc. Let P be any point in the line,
AD = ic, and P D = ?/. Let a; take the infinitesimal increment D D ' {dx), then
y becomes P' D'. Now the first state of the function is
P D = A D, or 2/ = .T,
The second or consecutive state is PD + P'E = AD+ DP", ""r y^dy = x-\-dx.
Subtracting we have P' E ^ D D', or dy = dx.
Now that the increment of y (or dy) is equal to the in- Y /jyi
crement of x (or dx) in this case is readily seen from
the figure ; for, as P'PE = 45°, P'E = PE, or DD'.
dy = dx, then, means that the contemporaneous incre-
ments of X and y are equal, or that x and y increase at
the same rate.
DDT X
Fig. 7.
48, B ULE 2. — Constant factors or divisors appear in the differ-
ential THE SAME AS IN THE FUNCTION.
Dem. — Let us take the function y = ax, in which a is any constant, integral or
fractional. Let .v take an infinitesimal increment and become x -\- dx ; and let dy
be the contemporaneous increment of 2/, so that when x becomes x -\- dx, y be-
comes y -|- dy. We then have
DIFFERENTIATION OF ALGEBRAIC FUNCTIONS.
13
1st state of tlie function, y ■= ax\
2nd, or consecutive state, y -\- dy = a'x -\- dx)
Subtracting the 1st from the 2nd,
a.x -j- adx:
dy = adx, ■
■which being the difference between two consecutive states of the function is its
differential {44:) . Now the factor a appears in this differential just as it was in
the function, q. e. d.
Iljl. — Let y =z ax he the equation of the line MN.
PD and P'D' representing consecutive ordinates, DD'
represents dx, and P' E represents dy. Here it is evident
that P'E =a X DD'; for from the triangle PP'E we
have P'E = tanP'PE X PE. But tan P'PE =
tan MAX = a. Th'e meaning in this case is, therefore,
that the ordinate increases a times as fast as the abscissa. ^^^ °-
If « :■= 1, or tan 45°, the ordinate and abscissa increase at equal rates ; if a <^ 1,
i. e. , if the angle is less than 45° the ordinate increases more slowly than the
abscissa ; if « ^ 1, the ordinate increases more rapidly than the abscissa.
40, RULE 3. — Constant terms disappear in differentiating: or
THE DIFFERENTIAL OF A CONSTANT IS .
Dem. — Let US take the function y = ax -±1 h, in which a and h are constants.
Let x take an infinitesimal increment and become cc -\- dx ; and let dy be the con-
temporaneous increment of y, so that when x becomes x -\- dx, y becomes y -\- dy.
We then have
1st state of the function, y = ax ±h \
2nd, or consecutive state, y -\- dy = a(x + (^^) rt &,
or y -\- dy = ax -\- adx =fc h.
Subtracting the 1st state from the 2nd, dy =^ adx, which
being the difference between two consecutive states of the function is its differen-
tial {44). Now from this differential the constant term ± 6 has disappeared.
We may also say that as a constant retains the same value there is no difference
between its consecutive states (properly it has no consecutive states). Hence the
differential of a constant may be spoken of (though with some latitude) as 0.
Q. E. D.
III.— Let y==ax-\-hhe the equation of the line M N.
Now the relative rates of increase of the abscissa and or-
dinate, that is the relation of dy to dx, is evidently not
affected by h which is A B ; for, if we were to draw a line
through the origin parallel to M N , the contemporane-
ous increments of its co-ordinates would be the same as
those of M N . Again, we can see that the constant term
does not affect the differential, i. e., the difference between
the consecutive states of y, by observing that these two states are represented by
PD and P' D', each of which contains the constant as a part of it, whence the
difference between them is not affected by it.
SO, Cor. — An infinite variety of functions differing from each other
only in their constant terms still have the same differential.
Fig. 9.
14 THE DIFFEEEKTIAL CALCULUS.
31, RULE 4. — To differentiate the algebraic sum of several va-
riables, DIFFERENTIATE EACH TERM SEPARATELY AND CONNECT THE DIFFEREN-
TIALS WITH THE SAME SIGNS AS THE TERMS.
Dem. — Let u = x -{- y — z, u representing the algebraic sum of the variables
X, y, and — z. Then is the differential of this sum or du=.dx -\- dy — dz. For
let dx, dy, and dz be infinitesimal increments of x, y, and z ; and let du be the in-
crement which u takes in consequence of the infinitesimal changes in x, y, and z.
We then have
1st state of the function, m = £c + 2/ — ^ ■>
2nd, or consecutiye state, u -\- du = x -\- dx -\- y -\- dy — {z -\- dz),
or u -^ du := X -}- dx -{- y -^ dy — z — dz.
Subtracting the 1st state from the 2nd, du = dx -{- dy — dz. q. e. d.
III. — "We may illustrate this by conceiving x and y to be forces acting to raise
a weight, and z a force acting to prevent the raising, u being the aggregate effect of
all, i. e. their algebraic sum (Complete Algebra, 65). Now if x, y, and z each re-
ceive an infinitesimal increment, which we will call respectively dx, dy, and dz, it
is evident that the increment of lifting force is dx -\- dy, and as the increment of
the depressing force is dz, the combined effect of the change is dx -\- dy — dz,
which is the change in u. Moreover, since this quantity dx -\- dy — dz is the ag-
gregate of a finite number of infinitesimals, it must be itself infinitesimal. Hence
the change in u is infinitesimal, or du.
ScH. — It is important to notice that the above reasoning is entirely inde-
pendent of the relative values of the infinitesimals dx, dy, and dz. These
may be conceived as equal, or as sustaining any finite ratio whatever to
each other, only so that they remain infinitesimal.
32, RULE 5. — The differential of the product of two variables
15 the differential of the first into the second, plus the differential
OF THE SECOND INTO THE FIRST.
Dem, — Letw =^ xy be the first state. The consecutive state is w -f- dw =
{x -\- dx){y -\- dy) =xy -\- ydx -\- xdy -\- dxdy. Subtracting the 1st state from the
2nd, or consecutive state, we have dxL = ydx -f- ^^y + <^^' • ^V- Now ydx and xdy
are infinitesimals of the 1st order, and dx • dy, being the product of two infinitesi-
mals, is of the 2nd order and must be dropped (17). Therefore du = ydx -f- xdy.
Q. E. D.
III. — Let u represent the area of the rectangle A BC D, £C = ^ e
C
* A B, and y = *BC. Then u = xy. Let B& represent dx, and p -
Oc",dy. Whence ShCc' = *ydx, DdCc"=*xdy, Cc'cc" ==*
dx ' dy, and du = *BhCc' + DdOc" -\- Cc'cc". Now since
cc' is infinitesimal and c'h is finite, Cc"cc" is infinitesimal with A B
reference to B&Cc', as for a like reason it is with reference to Fig. 10.
DdCc" ; hence it is to be omitted as having no assignable value with reference to
them.
Another view which may be taken of this is to consider that it is the rate at which
* In such cases = signifies " is represented by," and is used for brevity.
DIFFEBENTIATION OP ALGEBRAIC FUNCTIONS. 15
the rectangle is increasimj when a; = A B and 1/ = BC, not the amount of change
in the area after x and y shall have increased more or less : in other words, we seek
for the difference between consecutive values of the area. Now it is easy to see
that the rale at which the rectangle A BCD starts to increase, depends upon the
length of the side BC iy) and the rate at which it starts to move to the right, -j- the
length of DC (x) and the rate at which it starts to move upward. Letting dx
represent the rate at which A B starts to increase (by being the amount which it
would increase in an infinitesimal of time), and dy represent in like manner the
rate at which y starts to increase, we readily see that du = ydx -f- scdy is the rate
at which the area starts to increase. Moreover, we see that this is equally true
whether dy =^ dx, or whether one is any finite multiple of the other ; all that is
necessary being that both be infinitesimals of the same order.
^3, B ULE 6. — The differential of the product of several varia-
bles IS THE SUM OF THE PRODUCTS OF THE DIFFERENTIAL OF EACH INTO THE
PRODUCT OF ALL THE OTHERS.
Dem. — Let u := xyz ; then du = yzdx -f- xzdy -f- xydz.
For the 1st state of function is u = xyz,
2nd, or consecutive state, u-\-duz= {x-\-dx){y-\-dy){z-\-dz),
or u-{-du ■= xyz -\- yzdx -f- xzdy -\- xydz -f- xdydz -f- ydxdz -\- zdxdy -\- dxdydz.
Subtracting and dropping infinitesimals of higher orders than the first we have
du =- yzdx -f- xzdy -\- xydz.
In a similar manner the rule can be demonstrated for any number of variables.
Q. E. D.
S4:, RULE 7. — The differential of a fraction having a variable
NUMERATOR AND DENOMINATOR IS THE DIFFERENTIAL OF THE NUMERATOR
MULTIPLIED BY THE DENOMINATOR, MINUS THE DIFFERENTIAL OF THE DENOMI-
NATOR MULTIPLIED BY THE NUMERATOR, DIVIDED BY THE SQUARE OF THE
DENOMINATOR.
Dem. — Let u = - ; then is du = — ^ — - — -. For clearing of fractions yu = x.
Differentiating this by Kul^ 5, udy -\- ydu = dx. Substituting for u its value, we
have ' — : — f- ydu = dx. Finding the value of du, we have du = - — '■ — -.
y y
Q. E. D.
SS» CoR. — The differential of a fraction hamng a constant numerator
and a variable denominator is the product of the numerator with its sign
changed into the differential of the denominator, divided by the square of '
the denominator.
Dem. — Let u = -. Differentiating this by the rule and calling the differential
of the constant (a), 0, we have du = = ~. o. e. d.
16 THE DIFFERENTIAL CALCULUS.
ScH. — ^If the numerator is variable and the denominator constant it falls
under Rule 2.
S6. RULE 8. — The differential of a vaeiable affected with an
EXPONENT IS THE CONTINUED PEODUCT OF THE EXPONENT, THE VARIABLE WITH
ITS EXPONENT DIMINISHED BY 1, AND THE DIFFERENTIAL OF THE VARIABLE.
Bem. — 1st. When the exponent is a positive integer. — Let y = x"^, m being a pos-
itive integer ; then dy = mx^~^dx. For y z= x^ ::= x • x • x ' xio m factors. Now
differentiating this by Bute 6, we have
dy = {XXX to m — 1 factors) dx -f- {xxx to m — 1 factors) dx + etc., to m terms,
or dy = ic"»-^dx + x'^—^dx -\- x"'—^dx -f- etc., to m terms.
. • . dy =: mx'^—^dx.
- m .
2nd. When the exponent is a positive fraction. — Let y =z xn, — being a positive
TO !^ — 1
fraction ; then dy = — X" dx. For involving both members to the nth power
n
we have 2/" = ^C". Differentiating as just shown, ny"~^dy = mx^—^dx. Now
m mn — m
from y = a «, we have y"—^ = x~i . Substituting this in the last form, we have
mn — m »j. mv — m ™ in
nx n~dy = mx'^—^dx, or dv = — x^ Ti dx = — X"~ dx.
^ n n
3rd. When the exponent is negative. — Let y = x—", n being integral or fractional ;
then dy= — nx-^—^dx. For i/ = x—" = — , which differentiated by jRwte 7, Cor.,
gives dy=: ;; — '— = — nxr-"—^dx. All three of which forms agree with the
enunciation of the rule. q. e. d.
S7* Cor. — The differential of the square root of a variable is the dif-
ferential of the variable divided by twice the square root of the variable.
Dem. — Let y = \/x = x . Differentiating by the rule we have dy =-ix^ dx=.
1 -2 , dx
iX dX = =. Q. E. D.
2v/x
ScH. — Special rules can be readily made for other roots, but it is un-
necessary. The square root is of such frequent occurrence as to make the
special process expedient. Of course the general rule can always be used,
if desired.
EXERCISES.
[Note. — The following examples are designed to give practical skill in applying tlie rules for
differentiating algebraic functions. The student should not advance beyond these, till he has
the rules firmly fixed in memory, and can apply them with facility to all forms of algebraic func-
tions.]
Ex. 1 . Differentiate y = Qx — 4. dy = 6dx.
QuEKT. — What three iiales apply? Be careful to repeat the rules in applying
them to the solution of the examples, and thus render them familiar.
DIFFEBENTIATION OF ALGEBRAIC FUNCTIONS. 17
-Ex. 2. Differentiate y = a'' + da'x-' + Sa'x^ + x^
Solution. — The differential of i/ is d^/- [Bepeat Rule 1.] To differentiate the
second member we notice 1st, that it consists of several terms, and hence proceed
to differentiate each term separately. [Repeat liule 4.] a^ being a constant term,
disappears. [Eepeat Eule 3.] To differentiate Sa^x^, we notice l«t that the con-
stant factor 3a4 will be a factor in the differential. [Repeat Eule 2.] The differ-
ential of a;2 is 2xdx. [Repeat Bule 8.] Hence the differential of Sa'^x'^ is GalTc/a?.
[In like manner proceed with the other terms, giviiig the reason for each step hy
repeating the appropriate rule.^
Ex. 3 . Differentiate u = 2ax — 3^^ _|_ ahx^ — 5.
Result, du = (2a — 6x -{- dabx^)dx.
Ex. 4. Differentiate y = ^x^ — ■ 2x — Bin.
Ex. 5. Differentiate u = ab — ■ 6x^ -{- 2ax.
Ex. 6. Differentiate it = ax-y\
QuEBiES. — What is the most general feature of the function ax^^? What rula
applies first? Rule 5. What other rule apphes ?
Result, du = 2axy^dx -f dax^y^dy.
2
Ex. 7. Differentiate u = 6ax^y^.
1 2.
Result, du = 4,ax~^y^dx -\- ISax^y^dy.
Ex. 8. Differentiate y = 2bz-^ + Sax'^z'^.
6
2 1 'Sax'^dz Abdz
Result, dy = 5ax^z^dx + r- •
^ 2^z ^'
11 -r. 7 xdy -f ydx
Ex. 9. Differentiate u = x'^y'^. Besult, — ■ — —'- — .
2^-^i/^
P
Ex. 10. From y^ = 2px find the value of dy. dy = -dx.
Ex. 11. From A^y- -\- B^x^ = A^B^ find the value of dy.
Ex. 12. From A^y^ — B^x^ = — A^B^ find the value of dy.
dy = -— -a^.
X
Ex. 13. From ^24-2/2 = R2 find the value of dy. dy = dx.
Ex. 14. From 2xy^ — ay^ £= x^ find the value of dy.
^j, = ^dx.
4:Xy — "lay
18 THE DIFFEKENTIAL CALCULUS.
Ex. 15. Differentiate u = r-~. Result, du = ~ -,
32/3 Sy^
1 dx
Ex. 16. Differentiate v = -. dy = .
^ X x^
Ex. 17. Differentiate u = - — %r—. du = ^^ ^
b — 22/2* (6 — 2]/2)2*
Ex. 18. Differentiate y = —zr-- dy = -- x 2xdx = -—dx.
SiTG. — Do not treat this as a fraction under Kule 7.
Ex. 19. Differentiate u == x-y^z.
2.^2 — 3
Ex. 20. Differentiate u =
4j7 4- X'
Opebation. dK = -^(^^^ - 3)(^-^ + »^) - d(ix + x^)(2g_--_3) _
4:Xdx{ 4:X + a;g) — (4da; + 2a;d.-r) (2a;3 — 3) _ {4x(4x-{-x'^) — (4 + 2a;) (2.^^ — 3)} da;
(8a;^ + 6a; + 12)da;
~ (4x + a;2)2 *
Stjg^s. — The first step is the application of the rule for fractions, since the func-
tion is a fraction with a variable numerator and a variable denominator. The
second step is to perform the difierentiation of 2,'r2 — 3, and 4.x -|- s;^. This step
involves the rules for constant factors, variables affected with exponents, constant
terms, and the sum of variables. The remainder of the work is reduction and
addition of terms.
^ r.^ ^n^ .■ , 2a:^ 7 8<22^3 — 4075^
Ex. 21. Differentiate u = . du = — —dx.
(a2 — x^y
^ ^ a,
Ex. 22. Differentiate y = . dy = dx.
/][ 2j7 x'^^dx
Ex. 23. Differentiate y = ^ ' '^ . dy == rr-— r
^ t < -"> ^ (1 _|_ a;2)
Ex. 24. Differentiate y =
Ex. 25. Differentiate v = .
^ 1 — X
Ex. 26. Differentiate y ==: Sx"^ — 4. dy == Smx'^-'dx.
m 29712 ">—"
Ex. 27. Differentiate y = 2mx\ dy == x « dx.
^ n
a-' —
- x^'
a —
X
X
1 +
X
1 +
x^
1 +
372
1
^2*
X
DIFFERENTIATION OP ALGEBRAIC FUNCTIONS. 19
m 1
Ex. 28. Differentiate w = Sno;"!/".
m — n 1 v% 1 — n
du = 1mx~^y^doc + 2x''y~dy.
1 71
Ex. 29. Differentiate i/ = — . dy = zTl^^'
Ex. 30. Differentiate y = ^ x:^ — a^.
Opebation. — By the special rule for the square root (57), we have dy ==
d(iK3 — a:^ 3x-cZx
Ex. 31. Differentiate y = V~ax ■\- Vc^.
ad.v 3c-.r-dr , , i -i 3c -i-, , a^ -f 3ca7
^2/ = — -^ + —7= = {W^ + 17^ )^^> o^ -=-dx.
iVax ■ Wc:^x^ ^ 2va7
/- X
Ex. 32. Differentiate y = av a; — -.
o
Ex. 33. Differentiate y = V ax -\- hx'^ -\- cx^.
Ex. 34. Differentiate y = {ax- — x^y.
Solution. — Kegarding' «x2 — x^ as a variable, it is affected with the exponent 4 ;
hence we have dy = ^{ax'^ — x^)'^ X d^ax- — x^), the operation of differentiating
the variable ax- — x^ being as yet unperformed. Performing this operation and
reducing, we have dy = 4(aa;- — x^Y X (2ax — ^x'^)dx =
8ax^(a — xydx — 12a;8(a — xydx.
Ex. 35. Differentiate y= {a + bx")^. dy==^^-i-{a f hx'^)'^bxdx.
Ex. 36. Differentiate y == (a^ + ^2)3. dy = 6x{a''- + x'^ydx.
a , 6a^ -
Ex. 37. Differentiate y = r- dy = — - — ■ — -r-dx.
•^ {ly^ + x^y^ {b- + ^2)'«
Ex. 38. Differentiate 7/ = (1 + 2j72)(1 -}- 4^;^).
Solution.— Regarding this function as the product of the two variables 1 -|- 2x^
and 1 + 4x^ we have dy = d^l + 2x^) X (1 + 4.x^) + d(l + 4.^3) X (1 + Sx^).
Performing the operation of differentiating 1 + 2.1?^ and 1 + 4.x^, we have dy =
4a;(l 4- 4a;3)(2a; + 12x2(1 -f 2£c2)da; = 4a;(l + Zx + 10x3)dx.
Ex. 39. Differentiate y = {x^ -\- a.){Sx^ + 6).
dy = (1507^ + 35^2 + 6ax)dx.
x^ , 3^2 4- ^3
Ex. 40. Differentiate y = 7-— -. dy = — — -^^.
^ (1 + :r)2 ^ (1 + ^)3
a , 3a^.r
Ex. 41. Differentiate y == -t r-. dy
(a — ar)3 (a — :r)^
20 THE DLFFEBENTIAL CALCULUS.
Ex. 42. Differentiate y = —z -. dy— — j^—- — r — •
^ {ab ^2)3 ^ (a6 — ^r2)4
"^x. 43. Differentiate, without first expanding, y = {1 -^ x)*{l -\- x^y.
dy = 4(1 + xy{l -i- x^){l -}- X + 2x^)dx.
Ex. 44. Differentiate y == x^ — \/i — x^
„ , dx'-dx
av = 2xdx -\ ■
^ ^ 2v/l~^-
/ : ( n ^ j dx
Ex. 45. Differentiate u == v2ax — x"^. du = — - — .
'^^2ax — xi
Ex. 46. Differentiate u = Va^ + ^^ X Vb"^ + j/^.
(^^4- y^)xdx + (a2 + x-^)ydy
. \/a2 _j_ ^-2 X V 62 + ?/2
Ex. 47. Dinerentiate i/ = — - a^/ =
v^a2 — ^2 v/(a2 — x^y
Ex. 48. Differentiate j/
X
\/l + J72
Sug's. 2/=a;(l+x2) I .-. d?/ = dx(l +3:2) ^ -f a;. d(l+a;2) ^==da;(l+a;2) ^ —
£c2(l -j- a;2) *dx = — ^ = — -. Or, we may apply the rules
(1+^2)2 (1+^2)2
„ - ,. ^ i XT, J dxVl 4- £c2 — a;dv/l + ic2
tor a fraction and a square root, thus du = ■ —. — =
xdx
dXs/l -f- iC2 £C
v^l + ^^ dx(l -f- a;2) — x^dx dx
1 _l_ r2 3. Ji
(1 — 3^)cZj;
Ex. 49. Differentiate u= {l-\-x)vl — x. du
2v'l —
X
Ex. 50. Differentiate u = — . du =
\/{i — x^y (1 — ^2)i*
Ex. 51. Differentiate u = — .
2v a2^2 — ^4
— 04(^2 — 2x^)dx
du= ^.
2j;2(a2 — 072)2
Ex. 52. Differentiate u = \^ x -\- \/l + x\
xdx
Sug's. — Squaring u2 = a; -}- s/1 -f- x'^. 2iLdu == dx + — . du =
v/l + a:-'
dx -j-
DIFFERENTIATION OP ALGEBRAIC FUNCTIONS. 21
xdx (a; -f y/l 4- x')dx
a;2
Or, we may differentiate without squaring, thus du
2v- 1 4- a;'
xdx
dx -j y-
2 Va; 4- n/IT^^ 2 v/r+¥' V a; -f v/r+^'^ ^^^ + '^^
x^dx
X
Ex. 63. Differentiate u
V a'^ -\- x-^ — X
Sug's. — As the denominator is more involved in the differential of a fraction
than the numerator, it is expedient to reduce the fraction to a form having as sim-
ple a denominator as possible. Bationalizing this denominator, we have u =
^2 — a2 a-^ aVa;-^ + a^ «^
Ex. 54. Differentiate w = ' .
V x'^ 4-14-^
du = 2 \ 2x g ^l±-L I dx,
VX'' 4- 1
11 s/x
Ex. 55. Differentiate u =
1 + \/:
X
„ , 1 — v/.'T Vl — s/x s/l — X . dx
SUGS. U = K. z — : = — = . du
^ + ^a? \/l4-v/x l + v^a; 2(l4->/a;)ya; — a;2
X \" , nx''~^dx
(X \
) . du =2
1 -^ x/
Ex. 57. Differentiate u =
(i 4- x)"-^''
\/l -i- X — VI — X
1 + v/l — ^^ .
du = ; — dx.
Ex. 58. Differentiate w = >/a; • V ^a; 4- 1
, 7a;^ 4- 4 ,
du = — dx.
22
THE DIFFERENTIAL CALCULUS.
Ex. 59. Differentiate u = N/2:r— 1— V25;~l — \/2a7-^l— -, etc.,
to infinity.
Sug's.— We liave u = v^'^x — 1 — m ; whence m^ = 2x — 1 ~ m, and u
— si iv/8x — 3. . • . di* = ±
v^Sx — 3
Ex. 60. Differentiate u = ,y
du
v/
+ V (c2 — a;2
J?
)']■
36
4^7
— "^
ILLUSTRATIVE EliLOIPLES.
[Note. — The following examples are designed to illustrate more fully the significance of the
process of differentiation.]
Ex. 1. In a parabola whose parameter is 12, which is increasing
the faster at a; = 2, the ordinate or the abscissa, and how much ? At
^ = 3 ? At ^ = 8 ? At ^ = 24 ? How does the relative rate of
change vary as we recede from the vertex? At what point are ordi-
nate anid abscissa varying equally ?
Solution. — The equation of this parabola is y- = 12x.
Differentiating, -we have dy = -dx. Now, as differentiat- ^
ing is the process of finding the difference between
two consecutive states of a function, dx represents one
of the infinitesimal increments of a;, as DD', and dy
the coniemporaneous, infinitesimal increment of y, as
n
P' E. We, therefore, learn from dy = -dx that in gen-
eral dy is - times as great as dx ; or. in other words, that
2/
a —
y changes _ times as fast as x. At P where a; = 2, y ■= \/24. Hence, at this
y 6
point, dy = — ■z=dx = ^v/edx ; that is, y is increasing nearly li times as fast as x.
>/2i
At P" where re = 3, ?/ = 6, and dy = dx ; that is, x and y are increasing equally.
In general, at the focus the ordinate and abscissa of a parabola are increasing
equally, since at this point y =^ p. At P''^ where cc = 8, y is, increasing only about
.G as fast as x. At P^' where cc = 24, y is increasing at the still slower rate of
about .35 as fast as x. Finally, it is evident, fi'om a slight inspection of the figure,
Fig. 11.
DIFFERENTIATION OF ALGEBRAIC FUNCTIONS. 23
that y increases less and less rapidly as x becomes larger, x continuing to increase
at a uniform rate. At cc = co, y ceases to increase, i. e. the branches become par-
allel to the axis of x.-
Ex. 2. Examine the relative rates of change of the ordinate and
abscissa in the eUipse.
JB-x
Solution.— Differentiating A-y^ + B-x"^ = A'^B'^, we find dy = — -jr-dx —
-A- y
p.,
:dx. On this we observe 1st, That the — sign shows that x and y
AVA^ — a;2
are decreasing functions of each other ; that is, that as a; takes an increment y
takes a decrement. This is evident from a consideration of the curve. 2nd, That
Bx
in general terms y diminishes times as fast as x increases. 3rd, At
A\^A^ ~ a;2
X = 0, I e. at the extremity of the conjugate axis, y is not increasing or decreas-
Bx
ing, since here = 0, and dy = Q -dx — O. At the extremity of the
As/A^ — x^
transverse axis dy = — oc • dx, i. e. y is decreasing infinitely faster than x increases.
There are, therefore, all relative rates of change between x and y from to oc.
Moreover as x begins to increase from 0, y commences to decrease (at first slowly,
Bx
as the fraction • — is small when x is small), and then more and more
AVA^ — x'^
rapidly as x increases, till it reaches an infinitely rapid rate of decrease at x = A.
Bx
This, it is easy to see, is the law of change in the fraction — _ as x in-
AvA'^ — x'^
creases. The same law is also rendered probable from an inspection of the curve.
Finally, we may inquire at what point the relative rates of change sustain any
given relation to each other, as, for example, when y decreases twice as fast as x in-
creases, or just as fast, or 10 times as fast. Thus when y decreases twice as fast as
Bx
X increases, we must have dy= — Mx, i. e. ^rr = 2. From this we find
2A^
a; = i . — . - ; hence at these points, y is diminishing twice as fast as x is
v/4^2 -f J52
increasing.
Ex. 3. A boy is running on a horizontal plane directly toward the
foot of a tower 100 feet in height. How much faster is he nearing the
"foot than the top of the tower? How far is he from the foot of the
tower when he is approaching the base twice as fast as he approaches
the top ? How far off must he be to be approaching both base and
top equally ? Where is he when he is not approaching the top at all,
or is making infinitely more progress toward the base than towards
the top? When he is at 200 feet from the base of the tower how
much faster is he approaching the base than the top ?
Sug's. — Let AB represent the tower, and AX the line in the plane of the base
in which the boy is approaching the base. Suppose the boy at any point, as P
24 THE DIITEEENTLIL CALCULUS.
and let AP =^ x, and PB = y. Then
2/2 — .-r^rrr 10000. WhencB dy = -dx. Hence
we see that in general he is only approach-
ing the top an -th part as fast as he is the
base ; i. e., letting PP' represent an infin- ^^'^^ ^^'
itesimal element of the distance to the foot of the tower, P F represents a contem-
poraneous, infinitesimal element of the distance to the top ; and also, that P F is
X
an -th part of PP'. Secondly, when he is approaching the foot of the tower
1 X 1
twice as fast as he is the top ; we have dy = ^dx, or - = - ; whence y = 2cc. But
^ 2 2/2
100
2/- — x^=^ 10000 ; and, substituting, 3a;'- = 10000, or x = — - = uS nearly. Lastly,
\/3
/J. 200 25
when he is at 200 feet from the base y = \/50000 = 224 nearly, and - = -— = —
y 224 2o
25 25
nearly. Hence dy = ^-^^x, or he is approaching the top — as fast as he is the
28 2o
base. [Let the pupil decide the other points himself. ]
Ex. 4. A sliip is sailing northwest at 15 miles an hour. At what
rate is she making north latitude ?
An^., At 10.6054- niiles an hour.
Sitg's. — Let 2/ represent any distance run in the northwest course, and x the cor-
responding northing. Then as the course is northwest there is made in the same
time X westing, and we have y^ = 2x-. From this dy = — dx, and the ship is run-
2x — 2x
3iing -- times as fast as she is making northing. But y = x\/% whence -— =
^x - - -
— ^: = \/2, and dy = \^1dx, or dx = ^V'ldy ; i. e., she is making northing
a:\/'2
, 707-f- as fast as she is running.
Ex. 5. In the function y = 27x -|- 8^72, required the value of x when
y is increasing 45 times as fast as :3t. Result, x = o.
Ex. 6. What is the relative rate of variation of the side and alti-
tude of an equilateral triangle? i. e., if the side takes an infinitesi-
mal increment, what is the contemporaneous infinitesimal increment
of the altitude ? When the side is increasing at the rate of 2 inches
per second how rapidly is the altitude increasing ? Is the relative rate
of increase constant or variable ; that is, does the altitude increase
more or less rapidly in comparison with the side when the side is
small than it does when it is large, or is the relative rate of increase
always the same?
Sug's. —Let y = the altitude and x one cf the sides of the tdangle. Then
LOGARITHMIC AND EXPONENTIAL FUNCTIONS. 25
3x Sx ~
?/2 = |aj2 and di/ = ~-dx = ^-dx = i\/ddx. Hence we see that the infinitesi-
4?/ 2s/ '3x
mal increment of y is always i\/3 times as much as the contemporaneous infini-
tesimal increment of x. When x is increasing at 2 inches per second y is increas-
ing 2>/3 times 2 inches, or \/3 inches per second.
ScH. — The student should now be able to comprehend with considerable
clearness the object of the Differential Calculus; viz., having given the
relation between finite values of variables, to find the relation between the
contemporaneous infinitesimal increments of those variables, or their rela-
tive rate of cl^ange. Thus, in the last example, the relation between the
altitude and one side of an equilateral triangle, 3/2 = lx~, is the relation
between finite values of the variables, from which we find the relation be-
tween the contemporaneous infinitesimal increments d^ and dx, by the Dif-
ferential Calculus.
■4»»-
SUCTIOJSr IL
Differentiation of Logaritlmlio and Exponential Functions.*
S8, The JI£odulus of a system of logarithms is a constant
factor which depends upon the base of the system and characterizes
the system.
SO, I^vop. — The differential of the logarithm of a variable is the
differential of the variable multiplied by the modulus of the system, di-
vided by the variable ; or, in the Napierian system the modulus being 1,
the differential of the logarithm is the differential of the variable divided
by the variable.
Dem. — Let 2/ = a*.", n being constant. Then log y = n log a;. Differentiating
dy
y = x", we have dy = nx^—^dx, orn = fr- = — - = -r-> since X"—^ = --. Again,
^ .^ x'^-Hx y ^ dx X ^
-dx —
X X
whatever the differentials of log y and log x are, we have d{log y) ■= n - dQiOgx),
ox n = - \ . Placing these values of n equal to each other, we obtain
d(logic) ° \
dy
- , ' . = -— . Now let m be the factor by which -- must be multiplied to make
a(logfl;) dx V
X
it equal to ^(log?/), then is (^(logic) = .
* See 3^, 33,
26 THE DIFFERENTIAL CALCULUS.
"We are now to show that m is a constant depending upon the base of the system.
To do this take y ^z""', from which we find as before n = / °^ ^^ = ~. But m
a{log z) dz
z
is the ratio of d{logy) to — ; hence d(logz) = . Thus we see that in any case
y 2
*the same ratio exists between the differential of the log. of a number, and the differ-
ential o! the number divid^-d by the number. Therefore m is a constant factor.
To show that m depends il^)wu tue base of the system we have but to recur to the
definition of a logaritlim to see that the only quantities involved are ihe number, its
logarithm, and the base of the system. Of these the two former are variable, whence,
as the base is the only constant in the scheme, m is a function of the base. *
Finally, as m depends upon the base of the system, the base may be so taken
that m := 1. The system of logarithms founded on this base is called the Napie-
rian system, q. e. d.
00, JPtoj)- — The differential of an exponential function with a
constant base is the function itself, into the logarithm of the hose, into the
differential of the exponent, divided by the modulus.
Dem. — Let 2/ = «'. Taking the logarithms of both members log 2/ = £c log a.
Differentiating — - = log adx, or dy = ^^ , remembering that y = a^, and
that log a is constant, q. e. d.
Ql, Cor. 1. — The differential of an expjonentiol function with a con-
stant base, taken with reference to the Napierian system, is the function
itsef, into the logarithm of the base, into the differential of the exponent.
Thus if J == a"", cly = p/ log adx.
02 » Cor. 2. — If the base of the exponential is the base of the system cf
logarithms in reference to ivhich the differeyitiation is made, we have, in
general, dy = , or in the Najjierian system dy = e'^dx, since the log-
aritlim (f the base of a system, taken in that system, is 1, and in the Na-
pierian system e is used to represent the base and m = 1.
G3, JPvo2>o — The differential of an exponential with a variable base
is best obtained by passing to logarithms, and then differentiating.
III.. — Let w = 2/"". Passing to logarithms, log it = a; log y. Differentiating, we
, . , mdu , , , mxdy , , ulogy dx , u x dy
have m neneral, = lo!? y dx 4- , whence du = + — =
-^ u y m ' y
11^ loc 11 dx ii^xdii
' ^^-^ — '■ 1- — — '-. If the logarithms are taken in the Napierian system, m = 1,
* What tliis relation is, it does not concern us at present to know. It will be determined here-
after.
LOGARITHMIC AND EXPONENTIAL FUNCTIONS. 27
and du = y* log y dx -{- y—^xdy. If in addition y = x, so that u = a;*, du ==
x*(loga; -f- l)dx.
EXERCISES.
[Note. — The following exercises are designed to familiarize the rules for differentiating loga-
rithmic and exponential functions, and give the needed facility in applying them.]
Ex. 1. Differentiate u = x log x.
du = logxdx -\- mdx, or {logx + l)dx.
Ex. 2. Differentiate u == log x^ du = 2m--, or .
° XX
d X dx
Ex. 3. Differentiate u = log^ x. du = 2m log x- — , or 2 log x — .
X X
Ex. 4 Differentiate u = x""'.
du = af'x' ] log x{\og ^ 4- 1) + - r dx/
a}°^ -^ loff a ,
Ex. 5. Differentiate u = a}"^', du == ^-dx.
X
, xdx
Ex. 6. Differentiate u = losf \/l — x^. du = — .
^ 1 — x^
#
6^+1-7
Ex. 7. Differentiate u = log (3x^ + x). du = ■ dx.
^ ^ ^ 6x^ + X
Ex. 8. Differentiate u = log {x -\- \/l + x'^). du =
yi +a;2
2
Ex. 9. What is the differential of w = a"" in the common system
2 2
when a is the base of the system ? du = — a"" xdx.
m
Ex. 10. Differentiate u = e^"^"", in the common system, e being the
. , udx . mudx
base of the Napierian system. du = log e == — — .
SuG. — If the student has studied the subject of logarithms as usually presented
in our higher Algebras, he has learned that the common logarithm of the Napier-
ian base is the modulus of the common system ; L e. , in this example log e = m.
This fact will also appear hereafter.
V X'i -\- I X
Ex. 11. Differentiate u = log •
V X- 4- 1 + ^
SuG.— First rationalize the denominator of the fraction, obtaining u
Mx
log (\/x-2 4- 1 — ic)2 = 2 log( \/x2 -)- 1 — x), and then differentiate, du = —
\/iC2-j-l
* The student will observe for himself whether common or Napierian logarithms axe used.
28 THE DIFFERENTIAL CALCULUS.
64» ScH. — The differentiation of algebraic functions is often performed
with greater faciUty by first passing to logarithms.
Ex. 12. Differentiate u = .
1 — x^
SuG. — Passing to logarithms we have log u = log (1 -f- x^) — log (1 — ^-)- Biffer-
du 2xclx — 2xdx 4:Xdx Axdx
entiatmg, — = — - — = —— — — tj —- .' . clu
X — ^^^- = — ^ — . This example illustrates the method
(1 -\-x-^){l — x-^) 1 — x^ (1 — rC2)2
referred to in the scholium, although the student will find the direct method quite
as expeditious.
Ex. 13. Differentiate u = x{a^ + x^)va^ — x% by first passing to
logarithms. du = — — dx.
va^ — X-
Ex. 14. Differentiate w = (a" + l')^. du = 2a"(a" + 1) logadx.
a"" — 1 - , 2^"" log- adx
Ex. 15. Differentiate u == —- -. du = -.
a'' -^ 1 {a'' + 1)2
66. Cor. — The ordinary rule (SO) for differentiating a variable
affected with an exponent applies when the exponent is imaginary.
Dem — Let u = x^ *'— ^ Passing to logarithms, log u = a V — 1 log x. Differentiat-
du , — -dx , . — -xidx , — - aVZl_i
mg, — = av — 1 — . . • . dit = av — 1 = av — 1 x dx. q. e. d.
ILLUSTRATIYE EXAMPLES.
Ex. 1. Which increases the faster, a number or its logarithm ?
Solution. — Let x represent any number and y its logarithm, so that y = log x.
"We now wish to find the relation between the contemporaneous, infinitesimal in-
crements of X and y ; i. e., if the number {x) changes how does the logarithm (y)
change? Hence we differentiate, and have dy = -dx. Prom this we see that the
increment of the logarithm {dy) is — times the increment of the number {dx).
Therefore when ic << m the logarithm increases faster than the number ; when
X > m t'le logarithm increases more slowly than the number ; and when x = m
they incujabe equally.
[Note. —The student should not fail to see in every such example the real object of the Dif-
ferential Calculus {42). In the last example the relation between finite values of the A-ariables x
and v is y = log. x. The relation between the contemporaneous, infinitesimal elements of these
variables is found by differentiating, this being the object of the Differential Calculus.]
Ex. 2. When the number is 2124 and is conceived as passing on to
larger values by the law of growth of continuous number, i e. by
LOGARITHMIC AND EXPONENTIAL FUNCTIONS. 29
taking on infinitesimal increments, liow much faster is the number
increasing than its common logarithm ? If this relative rate of change
continued uniform (which it does not) while the number passed to
2125, i. e. increased by 1, how much would the logarithm have in-
creased ?
Solution. — Letting x be any number and y its logarithm, we have found that
dy = —dx. But m, the moduhis of the common system = .434:29448. Hencef
X
. 43429448
when X = 2324, we have dy = '- — — — — dx = .000204dc, or the increment of the
logarithm is .000204 part of the increment of the number. The number is, there-
1000000
fore increasing — — — — , or about 4902 times as fast as the logarithm. Secondly,
If this relative rate of change continued the same while the number passed from
2124 to 2125, I. e. increased by 1, the logarithm
would increase once .000204, or .000204.
Hence the logarithm of 2125 would be .000204
larger than the logarithm of 2124.
GEOMETEicAii Illustbation. — Let M N be
the curve whose equation is y = log x. Take
A D = 2124 ; then will P D represent its loga-
rithm. Let D D ' represent dx ; then will p -. o
P'E represent (Z?/.*
Ex. 3. The common logarithm of 327 is 2.514548. What is the log-
arithm of 327.12, on the hypothesis that the relative rate of change
of the number and its logarithm continues uniformly the same from
327 to 327.12 that it is at 327?
43429448
SuG.— At 327 dy = '- — — — dx = .001328d'r. Now as the number 327 increases
.12 to become 327.12 ; according to the hypothesis the logarithm increases .12
times .001328 or .000159. Hence the logarithm of 327.12 is 2.514707.
ScH. — The hypothesis that the relative rate of change of a number and
its logarithm continues constant for comparatively small changes in the num-
ber, is sufficiently accurate for practical purposes, and is the assumption made
in using the tabular difference in the table of logarithms, as explained in The
Complete School Algebka {125), and in the introduction to the table of
logarithms {14:) in the volume on Geometry and Trigonometry.
Ex. 4. What should be the tabular difference in the table of loga-
rithms for numbers between 2688 and 2689 ? Ans., .00016156+.
QuEKT. — How is it that the tabular difference found in the table of logarithms for
* The figure is necessarily out of proportion, as the true relation of y and x requires that A D
be nearly 700 times as long as PD.
30 THE DIFFERENTIAL CALCULUS.
numbers between 2688 and 2689, is 162? Sbow how the method of nsing this
tabular difference makes the result agree substantially with the method of inter-
polating now being presented.
Ex. 5. According to the arrangement of our common tables, show
that the tabular difference corresponding to 7487 is 58.
■#♦»
SUCTION III
Differentiation of Trigonometrical and Circular Functions.
TEIGONOMETKICAL FUNCTIONS.
SG, J^TOp, — The differential of the sine of an arc {or angle) is the
cosine of the same arc into the differential of the arc.
Dem. — Let X be any arc (or angle) and y its sine, i. e. let y = sin.r. If a; takes
an infinitesimal increment (dx), let dy represent the contemporaneous infinitesi-
mal increment of y. Then the consecutive state of the function is
y -\- dy ^ sin {x -\- dx) = sin x cos dx -j- sin dx cos x.
Now cos da; = 1, since as an angle grows less its cosine approaches the radius in
value, and at the limit, is radius. Moreover, as an angle grows less the sine and the
corresponding arc approach equality, and at the limit we have sin dx = dx.* The
consecutive state may therefore be written y -{- dy = sin x -|- cos x dx.
From this subtract y = sin x
and we have dy = cos x dx,
which, being the difference between two consecutive states of the function, is the
differential, q. e. d.
67 » JPvop, — The differential of the cosine of an arc (or angle) has
the opposite algebraic sign from the function, and is numerically equal to
the sine of the same arc into the differential of the arc.
Dem. — Let x be any arc (or angle) and y its cosine, so that y = cos a. Since
cos X = sin (90° — x) we have y = sin (90° — x). Differentiating this by the pre-
ceding proposition, we obtain
dy = cos (90° — cc) X d{90° — x) = cos (90° — x){— dx) = — sin ic dx,
since cZ 90° — x) = — dx, and cos (90° — x) = sin x. q. e. d.
68, ScH. — The opposition in the signs of the differential of the cosine,
and of the corresponding arc, signifies that they are decreasing functions
of each other [40) ; ^. e., if one takes an incremeyit the other suffers p de-
crement.
* The student m&Y be inclined to say that at the limit siu dx = 0. This is time, and no error
would follow from the assumption ; but the statement in the demonstration is equally true, and
we consider sin dx = dx instead of = 0, simply because we do not wish to have dx vanish from
the formula, our object being to find the relation between dy and dx.
TRIGONOMETEICAL FUNCTIONS. 31
69 • JProp* — The differential of the tangent of an arc (or angle) is
the square of the secant of the same arc into the dfferential of the arc ;
or for the square of the secant we may write the reciprocal of the square
of the cosine.
Dem. — Let 11 = tana;. Now tan x = '-, whence y = '-. Differentiatine
■^ cos X cos X
Sin ^
this, observine that '- is a fraction with a variable numerator and denominator,
cos a;
and hence can be differentiated by the rule for fractions {54:), and the two propo-
^^ ^^. -, , cos cede sin a;') — sin.'rd(cosa;) coH-xdx-\-s\u'^xdx
sitions (oo. 07), we have ay = = ■
■^ ^ cos2a; cos- a;
cos2 X 4- sin2 .r , 1 ,
= ■ ax = ax = sec'^xax. q. e. d.
cos2 a; cos^ a;
Another Demonstbation. — The consecutive state of the function y = tan x,
, . . , , , -, V tan =r 4- tan di3 isnax 4- d.v ,, _,„
being y -\- dy = tan (a; + dx) = ; , = -,-> the difference
1 — tan ic tan da; 1 — tan a; aa;
between the two states, i. e. the differential is dy = -^ — ^ tan x =
1 — tanxaa;
^ _j_ tan- X
—dx =(14- tan2a;)di; = sec~xdx. [Let the student give the detailed
1 — tan xdx .
explanation of the process. ]
70. JPvop, — The differential of the cotangent of an arc (or angle)
has the opposite algebraic sign from the function, and is numerically equal
to the square ofjhe cosecant of the same arc into the differential of the
arc : or for the squafe cf the cosecant we may turite the reciprocal of the
square of the sine.
Dem. — Let y = cot x = tan (90° — a;). Differentiating by the last proposition
dy = sec2 (90^ — x) X cZ(90o — a;) = cosec^ x{ — dx) = — cosec^ a;da;, or r—^dx.
Q. E. D.
cos X
[Let the student demonstrate this rule by remembering that y = cot x =
Binx
and also by taking the difference between the consecutive states y = cot x, and
y ~\- dy = cot (x -{- dx), developing and reducing as in the second demonstration
under {69).}
Query. — What is the significance of the opposition in signs?
71» JPvop, — The differential of the secant of an arc (or angle) is
the tangent of the same arc into its secant into the differential of the arc.
Dem. — Let y = sec a; = . Differentiating by {55, 67) "we have dy =
sin a; da; sin a; 1 , ,
= X X dx = tan x sec a;aa;. q. e. d.
cos2 X cos x cos X
82 THE DIFFERENTIAL CALCULUS.
72* JPvop. — The differential of the cosecant of an arc (or ang>) ],as
the opposite algebraic sign from the function, and is numerically equal to
the cotangent of the same arc into its cosecant into the di^erential of the arc.
Dem. — Let y = coseccc = sec (90° — x). Differentiating by the last proposition,
dy =z tan(90° — a;)sec(90° — x)d{'dO° — a;) = cot£ccosecjK( — dx) =^ — cotxcosecadx.
Q. E. D.
[Let the student demonstrate this proposition from the relation y ^ cosec x =
sin x'
Query. — What is the significance of the opposition in signs?
73, IPTOp,— The differential of the versed-sine of an arc (or angle)
is the sine of the same arc into the differential of the arc.
Dem.— Let y = versic =1 — cosic. Differentiating by {S7), dy = smxdx.
Q. E. D,
QuEEY. — Why should the differential of the versed-sine be numerically the same
as the differential of the cosine, but have an opposite sign ? Illustrate geometri-
cally.
74. J^vop, — The differential of the coversed-sine of an arc {or angle)
has the opposite algebraic sign from the function, and is numerically equal
to the cosine of the saine arc into the differential of the arc.
Dem. — (Similar to the preceding.)
QuEBY. — Why should the coversed-sine have the same differential as the sine,
but with an opposite algebraic sign ? Illustrate geometrically.
EXERCISES.
Ex. 1. Differentiate u = sin x cos x.
SuG. — Observe that we have here the product of two variables, viz., since and
cos X. Hence du = cos .-c d(sin x) -f sin x d(cos x) = cos- xdx — sin^ xdx =
(cos^a — sin2 x)dx, or (2 cos2 x — l)dx, or (1 — 2 sin^ x)dx, or cos 2 x dx.
• Ex. 2. Differentiate u = cos^a;.
SuG. — Observe that this is the cube of the variable cos re. Hence apply (56)
and we have dii = 3 cos^ x d{cos x) = — 3 cos^ re sin a dx = 3(sin3 x — sin x)dx.
Ex. 3. Differentiate u = tan 5x.
SuG. du = sec2 5xd{5x) = 5 sec2 5x dx.
Ex. 4. Differentiate u = cot^ x^-. du = — Qx^ cot x^ cosec^ x^ dx.
Ex. 5. Differentiate w == sins x cos x.
du = sin2 x(^ — 4 sin^ x)dx.
TEIGONOMETEICAL FUNCTIONS. 33
Ex. 6. Differentiate w == 3 sin'' x. du = 12 sin^ cc cos x dx.
Ex. 7. Differentiate u = cos mx. du = — m Binmxdx.
Ex. 8. Differentiate u = sin 3^ cos 2^.
du = (3 cos 3^7 cos 2x — 2 sin ^x sin 2x)dx.
Ex. 9. Differentiate u = sec^ ^x. du = 10 sec^ 5a: tan 5a7 c?a:.
Ex. 10. Differentiate u = tan" nx. du = 7i^ tan"~^ no; sec^ no: d'^.
Ex. 11. Differentiate u = log sin x.
Solution. — ^We have here a logarithm to differentiate, {. e. the logarithm of sin a;.
Hence the differential is the differential of sin x, divided by sin x, in the Napier-
ian system, or m times this, in the common system. Therefore du = — -^ • =
sin X
m cos X dx
smx
m cot X dx, or in the Napierian system, cot x dx.
Ex. 12. Differentiate u = log cos x.
du = — m tan x dx, or — tan x dx.
Ex. 13. Differentiate u = log tan x.
- sec2 X _ dx 2dx
du = ■ dx =
tan X sin x cos x sin 2x'
Ex. 14. Differentiate u = log cot j;.
Ex. 15. Differentiate u = log sec x.
Ex. 16. Differentiate u = log cosec x. du = — cot x dx.
Ex. 17. Differentiate u = e^'cos x, e being the Napierian base.
Sug's. du = Cdicosx) + cos x die") = — e'sinxdx + e^cosxdx =
e*(cos X — sin x)dx.
Ex. 18. Differentiate u = cce*'*"'^
Sug's. du ■= e''°'"dx + xe''°^'d{cosx) = e^^'^dx — xe'^^'"' Bin x dx =
grosx^l _ a; sin (c)c?a;.
Ex. 19. Diffsj-entiate w = — ^ TXl
du = e*^ sin x dx.
Ex. 20. Differentiate u = log v sin ^ + log v cos a;.
Sug's. w = 5 log sin a; + ^ log cos as. .•.(!« = J(cot x — tan x)dx = - — —.
tan aX
34
THE DIFFEBENTIAL CALCULUS.
ef^
ILLUSTRATIVE EXAMPLES.
[Note. — The object of these examples is to still farther illustrate the meaning of the process of
differentiation.]
Ex. 1. Whicli changes the faster an arc or its sine ? "What is the
relative rate of change ? When is the disparity greatest and when
least? What is the relative rate of change when the arc is 60°?
When 20° ? When 80° ?
SoiiTTTiON. — From y = sin a;, we have by differen- RrT=SlP
tiating, dy = cosxdx. The meaning of this is, that if
the arc {x, AP) takes an infinitesimal increment
{dx, Pp) the sine {y, P/) takes an infinitesimal incre-
ment {dy, pE) which is cosjc times the increment of
the arc. Now cos x is, in general, less than unity, so
that the increment of the sine is, in general, less than
the contemporaneous increment of the arc. But as x
grows less cos x becomes greater and approaches unity
as X approaches 0. So, also, cos a? approaches as x
approaches 90°. Hence the disparity between the contemporaneous increments
of an arc and its sine is less as the arc is less, disappears when the arc is 0,
and becomes infinite when the arc is 90°. For x = we may, therefore , con-
sider the arc and its sine to be increasing at equal rates. For x = 90°, the arc is
increasing infinitely faster than its sine. When x = 60° cos. r = ?. Hence at 60°
the sine is increasing just i as fast as the arc. In the figure, letting P'p' represent
dx,p'E.' represents dy andp'E' = iP'p'. "When .a; = 20° cos a; = .94 nearly.
Hence at 20° the sine is increasing . 94 as fast as the arc. At 80°, the sine is in-
creasing only about .17 as fast as the arc. These facts are illustrated in the figure.
Ex. 2. Assuming that the relative rate of increase remains con-
stantly the same as at 40°, how much does the sine increase when the
arc increases from 40° to 40° 10' ? What when the arc increases to
41°?
8 14159
StJG. — Since the arc of 10' = — ^ = .0029088 ; we find the increase of the
180 X 6
sine, on the above hypothesis, to be . 002228 , which is slightly in excess of the real
increase, as will be found by examining a table of natural sines in which the de-
cimals are extended to 7 places. The table gives ,0022156.
At the same rate of increase the sine of 41° should be .01^9 above the sine of
40° ; whereas from a table the increase is found to be .0132714.
[The student will observe that the cause of this disagreement is that the rela-
tive rate of increase of the sine as compared with its arc, is greater at 40° than at
any point between 40° and 40° 10', or at any point after 40°,]
Ex. 3. The natural tangent of 27° 20' is .5168755. ' Assuming that
the relative rate of increase of the tangent as compared with its arc
TRIGONOMETRICAL FUNCTIONS. Zo
remains the same as at this point, for the next 25" increase of the arc,
what is the natural tangent of 27° 20' 25" ? Ans., .517029.
Ex. 4. Which increases faster, the arc or its tangent ? "When is this
difference greatest? When least? What is the value of the arc
when the tangent is increasing just twice as fast as the arc?
Ansiver to the last, 45°.
Ex. 5. The natural cosine of 5° 31' is .995368. Assuming that the
relative rate of change of the cosine as compared with the arc re-
mains the same as at 5° 31', while the arc increases to 5° 32'_, what is
the cosine of 5° 32'? Ans., .995340.
Ex. 6. At 36° what is the relative rate of increase of the arc and
the logarithm of its tangent?
SuG. — From ii, = log tan x, we have du=m {teinx-\-cotx)dx. When x = 36° this
becomes dit = .43429 X 2.102925da; ; or the logarithm of the tangent increases
about .91 times as fast as the arc.
Ex. 7. The logarithmic cosine of 67° 30' is 9.582840. Assuming
that the relative rate of change of the logarithmic cosine and the arc
remains the same as at this point while the arc passes to 67° 31',
what is the logarithmic cosine of the latter arc ? Ans. , 9. 582535.
Ex. 8. The log cot 58°21' = 9.789868. On the same assumption .
as above, what is the decrease of this logarithm for 1 second increase
in the arc? Ans., .OOOOO'ill.
Ex. 9. The log cos 42° 14' = 9.869474. What is the corresponding
tabular difference ?
Ex. 10. At what rate relative to its velocity, is a point in the cir-
cumference of a wheel revolving in a vertical plane, ascending, when
it is 60° above the horizontal plane through the centre of motion?
Ans., One half as fast.
CIRCULAR FUNCTIONS.
'^S, I^TOp* — The differential of an arc in terms of its sine is the
differential of the sine difjided by the square root of 1 minus the square
of the sine ; or the differential of the sine divided by the cosine.
Dem. — Let y = sin— ^a;*, whence x = siny. Differentiating and finding the
dx
value of dy, we have dy = . But cos y = \/l — sin^y = s/l — x^. .'. dy i=
dx
, Q. E. D.
\/l — a;2
* This notation is explained in the Trigonometry of this series. It means simply "y = the aro
wh.066 Bine is «, and hence y = 6in""'x is equivalent to x = sin y."
36 THE DIFFERENTIAL CALCULUS.
70' ScH. — The student should not fail to observe the essential identity
of this proposition with [06). Thus, when we differentiate u = sin a;, we
get du = cos xdx, which expresses the differential of the sine (u) in terms
of its arc (x). From this we have dx = = — , which expresses
cosa; ^/l—u^
the differential of the arc (x) in terms of the sine [u). The one conception
is the converse of the other.
77. JPvop* — The differential of an arv in terms of its cosine has the
opposite sign from the function, and is nuvfierically equal to the differential
of the cosine divided by the square root of 1 minus the square of the co-
sine ; or the differential of the cosine divided by the sine.
Dem. — Let y=cos—^ x, whence x = cos y. Differentiating, and finding the value
f]l* cl'X
of dy, we have dy = r^ — = . q. e. d.
^ ^ sm2/ v^l — X-'
7S, ScH — Compare this and the following propositions with their equiv-
alents in Trigonometrical functions, as was done in the case of the preced-
ing proposition.
79. JPvop* — The differential of an arc in term.s of its tangent is the
differential of the tangent divided by 1 plus the square of the tangent.
Dem. — Let y = tan— i cc, whence x = tan?/. Differentiating and finding the
dx dx
value of dy, we have dy = — ~ = q — ■. — -, since sec^w =1-1- tan^?/ = 1 -f x^
^ sec-?/ 1 -[- x^
Q. E. D.
^0. JPvop, — The differential of an arc in terms of its cotangent has
the 02:)posite sign from tlie function, and is numerically equal to the differ-
ential of the cotangent divided by 1 idIus the square of the cotangent.
Dem. — Let y = cot-i cc, whence x = cot y. Differentiating, and finding the value
(It S.PO
of dy, we have dy =^ ^r— = — :; — ; — ;;. Q- e. d,
^' ^ cosec2 y 1 j^ X'
SI, Pvop, — The differential of an arc in terms of its secant is the
differential of the secant divided by the secant into the square root of the
square of the secant minus 1.
Dem. — Let y = sec-i x, whence x = sec y. Differentiating and finding the value
dx dx
'>f dy we have dy = "- — — = ■ since tan y = s/sec^j/ — 1 = \/x--^ — 1.
secy tan 2/ ^cVx^ — l
Q. £. D.
CIRCULAR FUNCTIONS. 37
S2, Prop. — The differential of an arc in terms of its cosecant has
the opposite sign from the function, and is numerically equal to the differ-
ential of the cosecant divided by the cosecant into the square root of the
square of the cosecant, minus 1.
Dem. — Let y = cosec-i a;, whence x = cosec?/. Differentiating, and finding tlie
dx dx
valtie of dy, we have dy = = . since coty = \/cosec^y — 1
cosecycot?/ ^s/x- 1
V X^ 1. Q. E. D.
S3, I^TOp, — The dfferential of an arc in terms of its versed-sme is
the differential of the versed-sine divided by the square root of twice the
versed-sine minus the square of the versed-sine.
Dem. — Let y = Yers—^x, whence x = \ersy. Differentiating and finding tho
(Xx
value of dy we have dy = . ' . But sin y = \/l — coss y = \/l — (1 — vers y)^ =
dx
\/l — (1 — xy^ = \/-Ax — a;2. Therefore, substituting, dy = — — "- . q. e. d.
V2x — a;2
S4:, JPvop. — The diffey^ential of an arc in terms of its coversed-sine
has the opposite sign from the function, and is numerically equal to the
d,ifferential of the coversed-sine divided by the square root of twice the
coversed-sine minus the square of the coversed-sine.
Dem. — Let y = covers—' a;, whence x = covers y. Differentiating, and finding the
, . , , , cZa; dx dx
value of dy, we have ay= — =
cos y ^i _ gij^sy yi _ ^1 _ covers s^)2
dx dx
=zr. Q. E. D.
v/1 — (1 — xy s/'tc — x^
EXERCISES.
Ex. 1. Differentiate y == sin""'^- ; y = cos~^- ; y = tan~^- ; y =
cot~^- ; y = sec~^— ; y = cosec""^- ; y = vers~^- ; y = covers"^-.
StTG. — We have dy = — . -■ ' . by (75). Now since dl -) = — , we have
dx
r dx ^ ,., - ,/ a; . dx
(X
cos— ^ - ) =
38
THE DIFFERENTIAL CALCULUS.
rdx
rdx
.. = .(t.n-.^-) = ^-^^; ., = .(eot-.5) = --^,; ., = .(sec- ?) =
rdx / x\ rdx / x\
— - ; dy = d{ cosec-i - ) = :=:z ; dy = d[ vers-i - ) = _-
zVx^—r^ ^ ^^ xVx'^ — r^ ^ rJ v/9
-. , ,/ £c\ dx
and dy = a( covers—' - I =
\ ^/ ^2ra; — x^
Geometeical IiiiiUSTBATioN. — Let O A = 1, and OA'
= r. Let y = arc C A (to radius 1), and x= C B' the
sine of the same number of degrees as y, but to a radius
r. Now CB =
C B
^, and we have y (or CA) =
C B' V
sin—' C B = sin—' = sin— ' -, the arc {y) being
taken to radius 1 while the sine x is taken to the radius r.
dx
\/2rx — jc2
B A B' AT
Fig. 15.
Ex. 2. Differentiate ^ = sin-^- ; ^ = cos"^- ; ^ = tan"^- ; ^ =
f IT IT y V TV
L-1^ y _i^ y 1^ V 1^ -. V ^^
cot ^- ; - = sec ^-- ; - = cosec - ; - = vers" - : and - = covers -.
f f T T TV TV V
Besults ill order : dy
r^dx
rdx
dy
r- -\- x^
; dy
\/r2 — x^
r^dx
^; dy =
rdx
\/ri
fi x2
-; dy.
r'^dx
rdx
s/^rx — x'-^
; and dy =
XV X- — r^
o^dx
; dy =
r^-dx
XV X-
j-2
r2 + 072 '
; dy =
v2rx — x^
Geometrical L^lustbation, — Li Fig. 15 let OA' = r, C'A' = y, and
V X
C'B' = X. Now if OA = 1, we have CA = -, and CB = -. Hence
r r
dy = d(sin— ' a;)to radius r =
rdx
v/r2 — a;2
r, etc.
85, ScH. — The results in the last example will be seen to correspond with
those in Ex. 1, by noticing that in Ux. 1 y represents CA, whereas in
Ex. 2 it represents C'A'. Now an increment of C'B' (which is x in both
cases) which makes an increment in C A, will make r times as great an incre-
ment in C'A'. Hence we have but to multiply the increments of CA (the
c?y's) as found in Ex. 1, by r to get the corresponding increments in C'A',
which are the c?y's in Ex. 2.
X
Ex. 3. Differentiate u = tan~^-.
y
du =
Ex. 4. Differentiate u = BLvr^{2x\/l — x^). du =
ydx — xdy
'Idx
v/l — ar»
CIECULAR FUNCTIONS. 39
Ex. 5. Differentiate u = cos~\a7\/l — x^).
Sug's. — By the rule the differential of the arc u is negative, and numerically
equal to the differential of its cosine, jc\/l — x^' divided by the square root of 1 minus
the square of its cosine. The differential of ck \/l — x~ is dx\/l — x^ —
and 1 — {x^l — a;2)2 = 1 — x2 ^ a;4 ... ciy,
\/l — x\
(1 — 2x2)dx
V{1 — x2 -f x4)(l — x^)
X dx
Ex. 6. Differentiate u = sin~' — • . du ==
3^.37
Ex. 7. Differentiate u = sin""^(3a: — 4^3). du = --
V 1 ^2
Ex. 8. Differentiate u = vers"^?/ — \/2r2/ — y^, understanding that
vers~^?/ is taken to radius r.
SuG. dw = ^^1 _ rdy—ydy ^ ydy
s/'lry — y- \/2ry — y~ \/2ry — y^
Ex. 9. Differentiate u = tan~^(\/l + x'^ — x).
du =
Ex. 10. Differentiate u = log a^/y— ^- 1- i- tan~^a;.
dx
2(1 + ^=)'
dr
SuG. w = 4 log (1 + as) — 4 log (1 — ic) + ^ tan-» x. du = — £-
fl;4
mdx
Ex. 11. Differentiate v = sin~'ma;^ — - cZv = / — =
^ vl — m2^-5
Ex. 12. Differentiate y = e»'*'^~\ tZy = e*"~'" ^"^
1 + ^■^'
Ex. 13. Differentiate y = tan""^^ ^ — . dy
1 — x^' 1 + x^'
Ex. 14. Differentiate ?/ = x^^''~\
1
dy == x'^'' ^ :j — ^i— ^ ^ — Y dx.
x{l — x^)
GENERAL SCHOLIUM,
86, The preceding sections comprise the fundamental rules of the differ-
ential calculus ; and it only remains to extend and apply them, in order to
complete this portion of our subject.
40
THE DIFFERENTIAL CALCULUS.
SUCTION IV,
Successive Differentiation and Differential Coefficients.
SUCCESSIVE DIFFEKENTIATIOK
87 » Bef. — Successive Differentials are differentials of dif-
ferentials ; or a successive differential is the difference between two
consecutive states of a differential.
III. — Let M N Fig. 16, be a straight line whose
equation is y ^=^ ax -\- b ; whence dy = adx. Now
suppose X to be considered equicrescent, and let
DD', D'D", D"D"',and D"'D''^ represent the
successive equal increments. P'E, P"E', P"'E",
and P'vE'" represent the contemporaneous incre-
ments of y, i. e. the dy's. But in this case the dy's
are all equal. Hence there being no difference be-
tween two successive states of dy, as between P'E
and P"E', there is no successive differential, or the
differential of dy is 0, since dy is constant. This fact appears also from the rela-
tion dy = adx, in which, if we conceive dx to be constant (i. e., x equicrescent),
adx is constant ; whence dy, which equals adx, is constant.
But consider in a similar manner the parabola in Fig. 17,
Fig. 16.
whose equation is y'^
2px ; whence dy =^— .
Still con-
FiG. 17.
sidering dx as constant, i. e. DD' = D'D" = D"D"'
= D"'D'^, etc., it is evident that the dy's, which are
represented by P'E, P"E', P"'E", etc., are not constant.
Now the difference between any two successive values of
dy, as between P'E and P"E', is a successive differ-
ential, i. e. a differential of a differential. The fact that
dy is a variable in this case when dx is constant is also
Tfdx
readily seen from its value dy =- — . In this pdx is constant, but y is variable.
Hence dy varies inversely as y.
88. Def. — A Second Differential is a differential of a first
differential, is represented by d^y, and read " Second differential y."
A. Third Diff'erential is a differential of a second differential,
is represented by d?y, and read " Third differential y." In like man-
ner we have fourth, fifth, etc., differentials.
ScH. — The student should be careful not to confound d^y with dy'^. The
latter is the square of dy. Nor should the superior 2 in d'^y be mistaken
for an exponent : it has no analogy to an exponent. Observe the significa-
SUCCESSIVE DIFFERENTIATION. 41
tion of the several expressions d'^y, dy^ and d{y'^). The latter is equivalent
to lydy.
89, JProp, — Second differentials are formed by differentiating first
differentials, third differentials by differentiating second differentials, etc.,
according to the rules already given.
This proposition is self-evident, since the differentials are expressed as algebraic,
trigonometric, logarithmic, or exponential functions, the rules for differentiating
which are those heretofore given.
Ex. 1. Produce the several successive differentials of y == ax*.
Solution. — Differentiating y = ax\ we have dy = 4:ax^dx. Differentiating this
differential remembering that d{dy), i. e. the differential of dy is written d~y, and
that dx is constant, we have d-y = 12ax'^dx dx, or 12ax^ dx-. In like manner dif-
ferentiating d'^y = VlaxHx"^, we have d'^y = 24«a; dx? . And again d'*y = 2^adx*.
Here the operation terminates, since d'^y being equal to 24«dx-* is constant.
Ex. 2. Produce the several successive differentials oi y = Sx* —
dx^ — 5x.
' dy = (32^3 — 9^i2 — h^dx.
Results, -
dHj = (96^2 — 18^)^^%
dHj = (19207 — l%)dx\
d^y = ld2dx^
Ex. 3. Produce the first six successive differentials oi y = sin x.
r dy = cos x dx, d^y = • — sin x dx^,
Results, \ d^y = — cos x dx^, d^y == sin x dx*,
[ d^y = cos X dx-', d^y = — sin x dx^.
QuEBY. — Does the above process ever terminate ?
Ex. 4. What is the 3rd differential oi y = af ?
d^y = n{n — l)(n — ■1)x''~^dx^.
Ex. 5. Produce the 4:th differential oi y = ax^.
15a dx*
Ex. 6. Produce the first six successive differentials of y = cos^.
Ex. 7. Produce the first four successive differentials oi y = logx,
in the common system.
^ , , mdx ^ m dx^ , 2m xdx^ 2m dx^ _
Results, dy = , d^y = — , d^y = = ^, d*y =
' ^ X ^ x^ ^ X* x^ ^
6??i dx*
X*
42 THE DIFFERENTIAL CALCULUS.
Ex. ^ Produce the first four successive differentials oi y =:
log (1 + ^)> in the connnon systera.
^ , , mdx ^ m dx^ , ImilA-x) dx"^ 2m dx^
Results, dy = , d^y = — — , d^y = — y-—^ — '- = ,
^ l^x ^ {1+xy ^ {1 + xy {l-{-xy
, Qmdx*
Ex. 9. Produce the fourth differential oi y = ef.
d^y = e'dx*.
Ex. 10. Produce the fourth differential oi y = a^, in the common
- log^a ,
system. dni = a'^dx*.
DIFFERENTIAL COEFFICIENTS.
90, Defs. — A First JDifferenticd Coejficient is the ratio of
the differential of a function to the differential of its variable, and is
dy
represented thus, -7^, y being a function of the variable x.
A Second Differential Coefficient is the ratio of the
second differential of a function to the square of the differential of
its variable, and is expressed thus, -r-^.
A Third Differential Coefficient is the ratio of the third
differential of the function to the cube of the differential of its vari-
d^y
able, and is represented thus, -7^. In Hke manner the nth differen-
(XX
d^y
tial coefficient is -7—.
dx"
S/u civ
111. — Having y = ax\ we obtain -- = 5ax^. In strict propriety — is a symbol
representing the general conception of the ratio of an infinitesimal increment of
the function to the contemporaneous infinitesimal increment of its variable ; and
5ax* is, in this case, its value. But it is customary to speak of either as the dif-
ferential coefScient. The appropriateness of the term differential coefficient arises
from the fact that the Sax' is the coefficient by which the differential of the vari-
able has to be multiplied in order to produce the difi'erential of the function.
Strictly, therefore, the differential coefficient is the coefficient of the differential
of the variable ; but it is customary to speak of it as the differential coefficient
of* the function.
* The "of" meaning, perliaps, "derived from," or "appertaining to."
DIFFERElilTIAL COEFFICIENTS. 43
Ex. 1. Given y ■= ax^ — x^, to find the 1st, 2nd, and 3rd differential
coefficients.
Results, — = 3ax^ — 2x, — ^ = 6ax — 2, -— ^ = 6a.
ax ax'^ dx^
1 -\- X
Ex. 2. Given v = z , to find the 5th differential coefficient.
. 1 — X
dni 240
dx'^ (1 — a;)«
Ex. 3. Given y = •'- "-
, logx
logy.=o=-Y-
X
sin'x
2 sin X cos x
2 sin X cos x
-
COSiC
iccosa;
cosx — xsina;
cosx
sin X
sin2 X ?
2 sin X = 0. . • .
y
= 1.
EVALUATION OF INDETERMINATE EXPRESSIONS. 87
, . . . 1 . , . loar sin x , . ^ ...
Also y = (smir)*'"== gives logy = sm a; log sm ^ = —- (differentiating)
COS6C *Cx=
cot a; 1 .
= 0. . • . 2/ = 1.
cosec X cot X cosec x
Ex. 3. Evaluate y = (cot^)^*'^* for x = 0.
SuG. — Put this in tlie form — . Thus logv = sin x log cot a; = — ■ (dif-
00 ^ cosec iCa:=0
cosec2 a;
- . . cot X cosec X sm cc .
lerentiatme) = = = = - = 0.
— cosec X cot X cot^ x cos^ x 1
Ex. 4. Evaluate y = (1 + nxy for x = 0,
„ . log(l 4- nx) ^.^ 1. X- 1 5^
SuG. log ?/ = ^^ ■ = -. Differentiating, log 2/2=0 = - = n.
X x = ^ •*•
, • . y=e\
Ex. 5. Evaluate y = {Qosiax)}"""'"^'^^ for ^ = 0.
SuG. y = {cos(ax)}<'°^«<=^^'=^) = 1", when x = 0. Passing to logarithms
log cos («x)
s (ax) = — ^ ^^ = -, when x = 0. Differentiat-
sin2 (cxj
— a tan (ax) a tan (ax) — a- sec2(ax) a-
mg twice ogT/^^o 2c sin (ex) cos (cic) csin(2ca;) 2c2cos(2ca;) 2e^
« *
88 APPLICATIONS OF THE DIFFERENTIAL CALCULUS.
SUCTION III.
Maxima and Minima of Functions of One Variable,
14:8, Def. — A M^aociniutn value of a function of a single vari-
able is a value which is greater than the immediately preceding and
the immediately succeeding values ; i. e., the value when the variable
takes an infinitesimal decrement, and the value when the variable
takes an infinitesimal increment.
Ill's. — Let y = sin x. When x = —, y \b b, maximum, since it is greater than
the immediately preceding and the immediately succeeding values. If x takes an
increment h, making y' = sinf^ f- ^ )> or a decrement, — h, so that y" =
sinf — — ^^\y is evidently greater than y' and y" , as at 90° the sine is greater
than it is at a little more or a little less than 90°.
Again, constructing the equation y'^ = 603^ — x'^, we
find the right hand branch to be as given in the figure.
Here y =:f{x), and y is & maximum when a; = A D =
4, since for x infinitesimally less or greater than 4, y is
less than for x = 4. The maximum value of y is,
therefore, y = -^6-4^ — ^4^ = 3^ nearly. ^xq 19
Once more, let y = 8x — a;^. If x = 1, y = 7 ', if
x = 2, 2/ = 12 ; if a; = 3, 2/ = 15 ; if x = 4, 2/ = 16 (a maximum) ; if cc = 5, ?/ = 15 ;
if a; = 6, 2/ = 12 ; and if a; = 7, y = 7. Hence it appears that as x increases y in-
creases till it has attained a certain value, when although x is made to continue
its increase, y begins to diminish. The point at which the function ceases to in-
crease and begins to decrease is its maximum. In this case it will be found that
however little x varies from 4, either way, y becomes less than 16. Thus if x ^=
3.9, y = 15.99 ; and if a; = 4.1, y = 15.99.
149, I^EF. — A. ]\Hnil7lU7¥l value of a function of a single vari-
able is a value which is less than the immediately preceding and the
immediately succeeding values ; i. e. , the value when the variable
takes an infinitesimal decrement, and the value when the variable
takes an infinitesimal increment.
tc
Ill's. — Let 2/ = coseca;. As x approaches — y diminishes and approaches 1,
reaching 1 at a; = - . "When x passes — , y begins to increase, so that 2/ = 1> is a
it A
minimum value of the function y = cosec x.
Again, y = x" — Q>x -\- 10, has a minimum value for a; = 3, at which value
MAXIMA AND MINIMA OF FUNCTIONS OF ONE VARIABLE.
89
y = 1. By substituting values of « a little greater than 3,
as 3.01, and a little less as 2.09, y will be found to be greater
than 1 in both cases. The locus of the function is given in
Mg. 20, where P D represents the minimum value of y.
ISO, CoE. — The same function may have several
maxima or several minima values, and these may he
equal or unequal. Moreover, a maximum value may
be equal to or even less than a minimum value of the
same function.
Ill's.— The function y := x^ — 8x3 _^ 22x2 — 24a; -}- 12, has
minima values for x = 1, and x = 3, which values are both
?/ = 3 ; or two equal minima values, as illustrated by the ordi-
nates at P and P" in the figure. For x =^2 y = i,, a maximum
value, as illustrated by P'D'.
Again, let y = /(.t?) be the equation of M N referred to AX
and AY Fig. 22. Then PD, P"D", and P"'D'"^are maxima
values of y ; and P'D', and P"'D"' are minima values.
But the several maxima values are unequal and the minimum
P' D' is greater than the maximum P^vQiv.
Fig. 20.
ADD' D"
Fig. 21.
151, ScH. — It will be observed
that the terms maximum and mini-
mum, as here used, do not mean the
greatest possible and least possible.
Thus, if we ask for the maximum
value of 3/ in 3/ = «;•* — Zax'^ — 5, we
do not inquire, "what is the greatest
possible value which y can have ?
but simply, whether if x vary con-
tinuously through all possible values,
there is any point at which y will at-
tain a greater value than it had immediately preceding that point, and than,
it will have immediately after passing that point ; and, if there be such a
value of y, what it is.
Fig. 22.
1S2, JPfop. — In an explicit function of a single variable, y = f(x),
dy
the first diffey^ential coefficient, —jChanges sign from, -f to — ,for contin-
uously increasing values of the variable, where the function is at a maxi-
mum, and from — to -{- where the function is at a minimum. Hence for
such values the first differential coefficient == or oo.
Dem. — Let y =f^x) be the function. First, For x = x', suppose y becomes y' , a
maximum. Then ?/' = f'^x') is at a maximum. Now the immediately preceding
90
APPLICATIONS OF THE DIFFERENTIAL CALCULUS.
dx') —fix')
flu* "fi %
state of tlie function is fix' — dx"), and we have -4-, = — — ; tt-
''^ " dx {x'—dx)
By
hypothesis /(a' —dx) — f{x') is — *, and as (x' — dx') — x' is evidently — , we have
dv'
— -j— Again, the immediately succeeding state to y' =f{x') isf{x'-\-dx');
hence we have -r—, =
dx
dy' f{x'-\-dx')—f{x')
\x' -j- dx) — x
and as {x -\- dx) — x is evidently -\-, we have
By hypothesis /(a' -\-dx') — /(.'r') is
dy'
dx
Therefore where y' =
dy'
fyX ) is a maximum -— changes sign from
to
dx')-f{x')
dx ) — x
x' is evidentlv —
is — , since by
Again
dv' fix' ■
Second. If y' = fix') is at a minimum -^^ = — — -
dx (X
hypothesis fix — dx') — fix') is -|-, and (cc' — dx')
dy' f(x' + dx') —fix) . ^-u • ^- ' , ^ 'N ^/ 'N • I A
d^' ^ ix + dx) — x' "^' "'^ ^ hypothesis /,a: + dx ) —fix ) is +, and
{x -\- dx) — x is evidently -j--
Finally, since when a varying function changes sign it passes through or go,
dy
we have -^ = or oo for maxima and minima values of the function, o. e. r».
dx
Geometeical IiiLUSTEATioN. — T"S being
tangent to the curve M N at P, P' being a
consecutive point so that P E represents dx,
and P'E dy, we observe that the angle
P'PE = a, the angle which the line makes
with the axis of abscissas. Hence tan a =
tan P'PE =
-=r-r=- = - - ; i. e. the first dif-
PE dx
Fig. 23.
ferential coefficient of the ordinate regarded
as a function of the abscissa, represents the tangent of the angle which a tangent
to a plane curve makes with the axis of abscissas.
Now, observing Fig. 22 we see that as x is increasing, and y approaching a
maximum value as PD, the tangent to the curve makes an acute angle ; hence
dy
approaching P from the left -- is
dx
At P the tangent becomes parallel to the
axis of X ; tan a
d'd^
-- = 0. Immediately upon passing P, a becomes obtuse,
and consequently tan a = -^is — .
dx
So also in approaching a minimum value as P' D' from the left it appears that
a is obtuse, and hence --
ax
passing P', a becomes acute and -- 4-.
dx
; at this point, P', a = 0, and
dy
dy
dx
0; and after
* The hypothesis is that y' =J\x') is a maximum, i. e. is greater than either the immediately
preceding or the immediately succeeding state of the function. But fix' — dx') is the immedi-
ately preceding state, and/(a;' + dx') is the immediately succeeding state. Hence/(a;''— flfa;0<
/(xO, and/(a3' + cfaK) (a:0.
MAXIMA AND MINIMA OF FUNCTIONS OF ONE VARIABLE. 91
rh%t
To illustrate the case in which -;- changes sign ^
by passing through oo, consider y =/(.r) as the
equation of M N , tig. 24. P D is evidently a
maximum ordinate. But in approaching PD
from the left, a is an acute angle, and --, -}-•
D X
At P, a = 90O, and j(= oo. After passing PD, ^^^- 2^'
dv
a is obtuse and --, — . A similar illustration may be given of the case in which
<^V .1 -. , . .
■J- passes through go at a minimum.
ScH. — The student needs to guard against the error of supposing that all
values of the variable which render the first differential coefficient or oo,
necessarily render the function a maximum or minimum. These values of
the variable correspond to the maxima and minima values of the function if
it has any maxima or minima values, since if the first differential coefficient
changes sign, it must pass through or oo ; but a quantity may pass through
or 00 without changing sign, so that the values of the variable which
render the first differential coefficient or oo are simply critical values, i. e.
values to be examined.
153. JPvop, — In an explicit function of a single variable, y = f(x),
the second differential coefficient, —^, if not 0, is — where the function
is at a maximuin, and + where it is at a minimum.
Dem. — Let y =/(x) be the function. We have seen that when the function
dv
passes through a maximum -~ changes sign from -|- to — for continuously in-
dx
dy
creasing values of x, i. e. -- is decreasing ; and when the function passes through
a minimum -- changes sign from — to +, i. e. -- is increasing. jNow -— =
ClX CIX CtX"
d^y
dx df'{x) f (x -\- dx) — fix) ,. , . V xu J. • J
== -^-— ^ — = '^—^ ! '—-—, which IS — when the numerator is — , and
dx dx dx
dy
-\- when the numerator is -f- , since dx is -J- by hypothesis. But at a maximum --
is decreasing for increasing values of x, and f{x -f- dx) — f{oc) is — ; and at a
dv
minimum -- is increasing for increasing values of x, and /'(a; + dx) — f{x) is -\- .
dHi
Therefore -r^ is — at a maximum value of the function and 4- at a minimum,
unless it is 0, a case which is not yet provided for. •
154, ScH. 1. — The ordinary method of examining an explicit function
92 APPLICATIONS OF THE DIFFERENTIAL CALCULUS.
of a single variable for maxima and minima values is to form the first differ-
ential coefficient, put it equal to 0, and solve the resulting equation. Some or
all of the values of the variable thus foulid may correspond to maxima and,
minima values of the function. They are then to be examined separately.
To do this, form the second differential coefficient of the function and sub-
stituting in it the value of the variable to be examined, if it gives a — re-
sult, this value of the variable corresponds to a maximum value of the
function ; but if it gives a -|- result, it corresponds to a minimum vahie of
the function. Thus all the values of the variable arising from equating the
first differential coefficient with 0, are to be examined. Tf, however, any one
of these critical values renders the second differential coefficient 0, it is best
to examine the first differential coefficient for this value and see if it actu-
ally does change sign in jiassing from a value of the variable infinitesimallj
less to a value infinitesimally greater than that being examined.
155, ScH. 2. — The following axiomatic principles often facilitate the ex-
amination of a function for maxima and minima values :
1st. Whatever value of x renders u = f[x) a maximum or minimum, ren-
ders u' = afix) or u" = —-!■ a maximum or minimum. Hence constant fac-
a
tors or divisors may be dropped from the function.
2nd. Whatever value of x renders u =/{x) positive and a maximum or
minimum, renders u = [/(•'c)]" a maximum or minimum, h being a positive
integer; but if u=f{x) is rendered negative for the particular value of x,
u = [f{x)y"' is a minimum when it =f[x) is a maximum, and a maximum
when u =f{x) is a minimum. Hence the function may be involved to any
power.
3rd. Whatever value of x renders u = log [/(a;)] a maximum or minimum
renders u' = f(x) a maximum or minimum. Hence to examine the log-
arithm of a function we have to examine simply the function itself dropping
the symbol log.
Ex. 1. What values of x render y = v 4a-^- — ^ "lax;^ a maximum or
minimum ; and what are the maxima and minima values oi y?
Solution.* — Whatever value of x renders y = s/ia x- — 'Aiw-' a maximum or
minimum rendei"S y^ or y' = -ia'x- — 2ax^ a maximum or minimum {155). And
for a similar reason we maj' drop the constant factor 2a, and examine y"='2ax'^ — x^,
since any value of x which renders the original function a maximum or minimum
dy"
will also render this a maximum or minimum. Differentiating we have -^ =
4ax — 3x2. Now whatever A^alue of x renders the function a maximum or mini-
4a
mum renders 4ax — 3x2 = 0. From this x =: 0, .r = — . If, therefore, there are
o
any maxima or minima values of the function, they are those which correspond to
* This solution may seem needlessly prolix, but the author finds that comparatively few stu-
dents really follow the argument through unless required to give it thus in dt-tail.
MAXIMA AND MINIMA OF FUNCTIONS OF ONE VARIABLE. 93
one or the other or both of these vahies of x. Diiferentiating again, —4 = 4a — 6cc.
For X =0, — ^ = 4^a ; hence a; = corresponds to a minimum value of the func-
dx'^
tion. For x = — , — = 4a — 8a = — 4a ; hence x = -- corresponds to a maX'
3 dx' o
imum value of the function.
Substituting these values of x we find y = v/4a'av^ — 2ax- = 0, a minimum
6i«* 128«' 8a2
value ; and y = >/4a2ic2 — 'lax:^ = —r- ^r^r- = -, a maximum.
\ y ^' 3\/3
Ex. 2. What values of x render y = x-^ — 9^=^ + 24.x — 16 a maxi-
mum or a minimum, and what are the maxima values of y ?
Results, X = 2 corresponds to a maximum, and x = 4: to a mini-
mum. The maximum value is y = 4, and the minimum y = 0.
Ex. 3. Examine y = x^ — ^x^- — 24.x + 85 for maxima and minima.
Results, For ^ = 4, ?/ = 5, a minimum ;
For X = — 2, y = 113, a maximum. ,
Ex. 4. Examine y = 5(x — ^r^) for maxima and minima.
SuG. — Drop the 5. x = i, gives y ==. ^, a maximum.
Ex. 5. Examine y = {2ax — x^)'^ for maxima and minima.
SuG. — Use y' = 2ax — x^. x = a, gives y = a, a maximum, and — a, a min-
imum.
Ex. 6. Examine y = x* — Sx^ -\- 22a;2 — 24^ + 12 for maxima and
minima.
Sug's. ^f = 4x'^ — 24.'k2 4- 44a; — 24 =-. 0, or x^ — Gx2 + IL-r — G = 0. To
ax
find the roots of this equation, observe that the factors of the absolute term with
its sign changed are 1, 2, and 3 (Complete School Algebra, 111). By trial
these are found to be the values oi x, x = l gives ?/ = 3, a minimum ; .^• = 2 gives
2/ = 4, a maximum ; .r = 3 gives ?/ = 3, a minimum (see III. Fig. 21).
Ex. 7. Examine y=x^ — 5x^ + 5x^ + 1 foi' maxima and minima.
Results, The critical values of x are 0, 0, 1, 3. For x ■= 1, ij = 2,
a maximum ; for x = d, y = — 26, a minimum, x = does
not correspond to either a maximum or minimum value of y.
SuG. — That x = does not correspond to either a maximum or a minimum is
determined as follows :
dy
Having -- = 5x4 — 20x3 + 15x2, substitute — h and -\- h for x, and evaluatef^
dx
dv
the expression for k infinitesimal, thus determining whether -- changes sign or not^
cvX
94 APPLICATIONS OF THE DIFFERENTIAL CALCULUS,
in passing through x = 0. Thus JJ = 5(0 — h)" — 20(0 — hy -f 15(0 — A)2 =
5;i4 + 20/13 + 157^2 = 15;i2, when h is infinitesimal. Again -- = 5/i4 — 20A3
ux-
-f- 15^2 ^= 15^2, when h is infinitesimal. Therefore, as -- has like signs on
both sides of a; = 0, and consecutive with it, it does not change sign in passing
through a; = 0. Hence jb = does not correspond to either a maximum or a min-
imum.
Ex. 8. Examine y z=h -\- \x — a)^ for maxima and minima.
dv
Stjg's. — = 3(a; — aY = 0, gives x = a. Hence if there is any maximum or
dv
minimum it must be v = &, as no other value of x than x = a will render -^ = 0.
dx
d^V dv
Again, since this value renders -r— = 0, we examine it by ascertaining whether -rr
dx^ dx
dv dv
changes sign a.tx = a. -^ = 3(a — h — a)2 = 3h' is the value of -^ next preced-
(XX ax
dv
ing X = a; and —■ = 3(a -f- ^ — a)^ = STi^ is the next succeeding value. There-
fore, as -- does not change sign at a; = a, the function has no maximum nor mini-
dx
mum value.
Ex. 9. Examine y = a{x — by -\- c for maxima and minima values.
Sug's. -^ = 4:(x — hy = 0. .-. x = h. % = 4(& — h — hy = — 4.h^, and
dx ^ ' dx
dv' dv'
-J— = 4(6 4" ^ — 6)3 = 47i3, are the values of — — immediately preceding and suc-
(X*o CuC
dy'
ceedmg x = h ; hence, as — — changes sign from — to 4- ^t this point, x = 6 cor-
dx
responds to a minimum. .-. y = a(b — 6)^ -J- c = c is a minimum.
Ex. 10. Examine y = {x — l)^{x -\-2y for maxima and minima.
Sug's. -^ = A{x — iy{x + 2)3 -f- 3[x — iy{x -f 2)2 == {{x — l)3(x + 2)-'}
{4(a;-f2)4-3(a; — 1)} =(« — l)3(a;+2)2(7a;-f5)=0. .-. .r— 1 = 0, a;+2 = 0,
7a; -|- 5 = 0, give x = 1, x := — 2, x = — fas the critical values of x.
d^V
-^=.3(a; — l)2(a;-|-2)2(7x + 5)4-2(x — l)3(x + 2)(7a; + 5)-l-7(a; — l)3(cc + 2)2 =
(123) . 92
for X =^1, and x = — 2, but is — -^ — —- for x =^ — ^. The latter value, there-
' 74 '
12 9'
fore, corresponds to a maximum, and gives y =z ( — f — ly{ — ^ -|- 2)3 =: — — — , a
maximum.
To ascertain whether x = \ corresponds to a maximum or minimum, notice
MAXIMA AND MINIMA OF FUNCTIONS OF ONE VARIABLE. 95
that -^ = n—h~ 1)3(1 — h-^ 2)2(7 — 7h-{-5)= — h^S — /i)2(12 — Ih) is -
ax
and '^^ = a+h- 1)3(1 + /i + 2)2(7 + 7/i + 5) = h\3 + 70^(12 + 7h) is +.
Hence at x = 1, -- changes sign from — to +, and there is a minimum at this value.
dx
dy
dy_
dx
This minimum is 2/ = 0.
Finally, to test a; =— 2,^ = (— 2 — /i — 1)3(— 2 — /i + 1)\— 14 — 7/i + 5) =
(— 3 — K)\-~ h)\— 9 — 7/i), which is -f . Again, ^ ^ (— 2 + /i — 1)^
(_ 2 4- /i + 2)2(— 14 -I- 7/i 4- 5) = (— 3 -f /i;3(+ )i)\— 9 + 7/i) is also +.
Therefore a; = — 2 does not correspond either to a maximum or a minimum.
156. ScH. — It is usually easy to see, without going through with the de-
tails of the substitution, whether -^ changes sign with h in such cases as
dx
dy
the above ; that is, whether \i x= a is the critical value we are testing, -~
will have a different sign when we substitute a -\- h, iox x, from what it will
when we substitute a — % for x.
Ex. 11. Examine v = -^ — J, for maxima and minima.
^ (a; — 3)2
Stjg's. -- = - = 0, gives for the critical values re = — 2, x = 13.
d« (a? — 3)3
dy _ (x + 2)2(c.-13) _ ^ _ 3 _ ^j^^^^^ ^ _ 3_
da; (X — 3)3 . 5 V ^
d^v
In this case it is better not to form -^ as it is complicated, but test the critical
dx"^
dv
values by noticing whether -- changes sign or not for these points, x = — 2
does not correspond to either a maximum or a minimum, x = 13, gives :y = 33|,
a minimum, x = 3, gives i/ = oo, a maximum. ^
ScH.— The first 10 examples give x = co iov -j = oo, and hence give rise
to no critical values, as .r = oo cannot correspond to a maximum or mini-
mum, there being no succeeding value of the function.
{X 1)2
Ex. 12. Examine y == -, =^- for maxima and minima.
{x-\-iy
Sug's.— Putting-5- = 0, gives x = 1, and 5, as the critical values. Putting
— == oc, gives X = — 1. When x = 1, 2/ = 0, a minimum. When x — 5, y = j^y,
dx
a maximum. When x =. — 1, y is neither a maximum nor a minimum.
96 APPLICATIONS OF THE DIFFERENTIA!, CALCULUS.
3
Ex. 13. Examine y = b -{- {x — aj"^ for maxima and minima.
Result, The critical value oi x i^ x = a. But this does not corres-
dy
pond to either a maximum or a minimum, since -^ does not
change sign at this value.
m
SuG. —In this example — ^ = rb ex for a; — a, and hence cannot be used to dis-
criminate between maxima and minima.
4
Ex. 14. Examine y = h -{- {x — a)^ for maxima and minima.
Result, ?/ = 6 is a minimum.
Ex. 15. Examine y = i — {x — aj^ for maxima and minima.
Result, 2/ = 6 is a maximum.
Ex. 16. Show that y = x^ — Sx- + 6^+7 has neither a maximum
nor a minimum value.
X
Ex. 17. Show that y = - is a maximum when x = cos a;.*
1 4- j; tana?
1- ''
dv cos"^ X .__. dy
SuG. — = ————— . When x <" cos a;, -- is + ; but when x >> cos a,
dx (l + a;tanx)2 ^ ' dx ' ^
^is
dx
Ex. 18. Show that y = sin^ x cos x is sl maximum when x = 60°.
sin X
Ex. 19. Show that y = is a maximum when x ^ 4:5°.
^ 1 + tan^
GEOMETRICAL PROBLEMS.
Ex. 1. Eequired the altitude of the maximum cylinder which can
be inscribed in a given right cone with a circular base.
Solution. — Let SO = a be the altitude, and AO = 6 the radius of the base
of the given cone. Let ac = xhe the altitude, and cO = of = y he the radius
of the base of the required cylinder. The function which is to he a maximum is the
volume of the cylinder. Calhng this V, we have V = Tty^x. In this form V is a
function of two variables x and y. But these variables being dependent upon
each other, we can find the value of one in terms of the other. Thus, S/ : S O : :
«/ : AO ; or, in the notation, a — x : a : : y :h ; whence y =i -[a — x). Substi-
* When X = cost, x = 42"* 21' nearlj'.
GEOMETRICAL PEOBLEMS.
97
tutmg this value of y, we have V = — (a — x^x,
which is to be a maximum. Dropping the con-
stant factor — {153, 1st), we have V'= {a — x^x
a^x — 2ax-^-j-x^.
dV'
dx
= a" — 4aa; + 3x2 _ o ;
whence a; = ia ; that is, the axis of the cyHnder is
i of the axis of the cone. From this we readily
find y, the radius of the base of the cylinder =|&.
. • . volume of cylinder = -^^ita}fi. But volume of A
cone =: \Tidb~ ; whence volume of cylinder = f
volume of cone.
Fig. 25.
Ex. 2. To find the axis of the maximum cone which can be inscribed
in a given sphere.
771
Sug's. — Let AmB be the semicircle which, re-
volved upon A B, generates the sphere, and Aa& the
triangle which generates the cone. Let AO=r,
A&=::ic, anda&=:2/- Then V = inr2/2x=i7rx2(2r — a),
—2
since a6 =2/^ = A6 X6B=£c(2r — x). .•.a; = |r,
or the altitude of the cone is f of the diameter of
the sphere. Volume of sphere = ^itr'^, volume of
maximum cone = -/f X a^rr^ ; or the cone :==: -^^ of the sphere.
ScH.— In attempting the solution of such problems, first notice wliat the
function is which is to be a maximum. Thus, in Fx. 1, it is the volume of
a cylinder ; in Ex. 2, it is the volume of a cone. Having obtained the equa-
tion expressing the function in terms of the variable or variables on which
it depends, if there are two dependent variables involved, find from the
conditions of the problem the relation between these variables, and sub-
stitute for one of them its value in terms of the other. Finally, we have
a function of a single variable, which can be examined for maxima and
minima values in the usual way.
Ex. 3. Required the cylinder of greatest convex surface which can
be inscribed in a given right cone with a circular base.
SuG. — The function is the convex surface of a cylinder. Using the same notation
as in Ex. 1, and letting S represent the function, we have S = 27tyx =
(a — x)x. .' . X =-la, and S = -3— ; that is, the altitude of the cylinder is i
that of the cone ; and the convex surface of the cylinder is to the convex surface
of the cone as - ; s/a^ -j- ^^j o^ ^s i the altitude of the cone is to its slant height.
Ex. 4. Required the maximum Cylinder which can be cut from a
98 APPLICATIONS OF THE DIFFERENTIAL CALCULUS.
given fephcre. The axis of the cylinder = f \/3 times radius of sphere.
The cylinder is to the sphere as 1 : y'S.
Ex. 5. Eequired the area of the greatest rectangle which can be
inscribed in a given circle.
The rectangle is a square, and its area = 2r2.
Ex. 6. "What is the altitude of the maxi-
mum rectangle which can be inscribed in a
given parabola ?
Sug's. — Let ac = X, af = y, and AX = «. Let
A be the function, the area of the rectangle. Then
A = 2a'?/. From the equation of the parabola
aj" = 2p X A/, or y^ = 2p{a — x) ; whence A =
2x\/2p^a — X). A' =^ ax^ — x^, and x = |a.
Ex. 7. Bequired the axis of the cone of maximum convex surface
which can be inscribed in a given sphere.
The axis == ^ the radius of the sphere.
Ex. 8. Required the altitude of the maximum cone which can be
inscribed in a given paraboloid, the vertex
of the cone being at the intersection of the
axis of the paraboloid with the base.
Sug's. — Let ABC be the parabola whose revo-
lution about AS as an axis generates the parabo-
loid. Let AS ==^ a the axis of the paraboloid,
oS = X, the altitude of the cone, and ao = y the
radius of the base of the cone. The result is
X = ia.
Ex. 9. Required the maximum para-
bola which can be cut from a right cone
With a circular base, knowing that the
area of a parabola whose limiting co-ordi-
nates are x and y is |-.r?/.
Sug's. — LetSO=a, BO = &, AX=a;,and
f X = t/. The function is A (the area) = ^xy.
But aX = y = v^BX X XC ; and CX :
Fig. 28.
C B : AX : S B, or CX : 26 : : a; : v/a'^ -f 62 ;
2hx
whence CX = — p, letting S = Va'^ + ^^ ^o^
brevity. Then BX
CB — CX
25
26x
Fig. 29.
GEOMETRICAL PROBLEMS.
99
->
x). Finally, A = ^x
m
86
-x{S — x) ■= --\/x\S — X), and A' = Sx^
— X*. The result is a; = |/S^, that is, the axis of the parabola is 4 the slant height
of the cone. The area of the parabola = ibS\/3. Notice that CX = |CB.
Ex. 10. From a given quantity of material a cylindrical vessel with
circular base and open top is to be made, so as to contain the greatest
amount. What must be its proportions ?
Sug's. — Let X = the altitude, y the radius of the base, and V the volume.
dV dy
Then V = Tty'^x is to be a maximum. Hence -y- = S^/x-^ -\- y'^ z= Q, or y =
— 2a;--. But iTtyx -f 7ty^
litydy = ; whence -- = ■
dx
dx dx
s, the surface. Differentiating ^itxdy -f- 27tydx -f-
y
Substituting, y = — -— . . • . y = x, that
x + y_ ^' ^ x + y
is, the altitude = the radius of the base.
The altitude = — •
Ex. 11. Of all right cones of a given convex surface to determine
that whose solidity is the greatest.
The altitude = \/2 into the radius of the base.
Ex. 12. To find the maximum rectangle inscribed in a given ellipse.
Sug's. A = 4xy. A' = xy. — r— r= ?/ + ^'y "^ ^' '^^y^
-f ^2x2 — ^2^2.
dy
dx
dx
B'^x
A^y
y =
dx
dy B^x'i
X— = .
dx A^y
X '.y :: A: B. That is, the sides of the rectangle are to each
other as the axes of the ellipse. The sides of the rectangle
are A\/2, and B\/2.
Fig. 30.
Ex. 13. To find the maximum cylinder which can be inscribed in a
given ellipsoid, generated by the revolution of an ellipse about its
2
The axis of the cylinder = — -=A.
transverse axis.
Ex. 14. A person being in a boat 3 miles from the nearest point of
the beach, wishes to reach in the shortest time a place 6 miles from
that point along the shore ; supposing he can walk 5 miles an hour,
but pull only at the rate of 4 miles an hour,
required the place where he must land.
Sug's. — Let AX =;r, and f = the time required to
reach A by rowing from B to X, and walking from
5 4
is to be a mini-
mum. He must land at X, 1 mile from A.
FlO. 31r
100 APPLICATIONS OF THE DIPFEEENTIAL CALCULUS.
Ex. 15. Divide a into two factors the sum of which shall be a min-
imum. Result, The factors are equal.
Ex. 16. The difference between two numbers is a ; required that
the square of the greater divided by the less shall be a minimum.
Result, The greater = twice the less.
Ex. 17. To find the number of equal parts into which a must be
divided, so that their continued product shall be a maximum.
Sug's, — Tlie function is u=zl-\. logw = a;(loga — \ogx). u ^= .^'log« —
a; logic. — = log a — log a; — 1 ^= 0. x = -. Arithmetically the problem is
possible only when — is integral.
Ex. 18. Eind a number x such that its ^th root shall be a maxi-
mum. X = e.
Ex. 19. A privateer wishes to get to sea unobserved, but has to pass
between two lights, A and B, on opposite headlands, the distance be-
tween Avhich is a. The intensity, at a unit's distance, of A is h, and
of B, c. At what point must the privateer cross the line joining the
lights, so as to be as little in the light as possible ; it being under-
stood that the intensity of a light at any point equals its intensity
at a unit's distance divided by the square of the distance from the
light. .
6 c
SuG. — Letting x = the distance from A, the function is u = — -] —
x-i ' (a — x)2
J.
ao
X = — —
^ 3 , 3
-\- C
Ex. 20. The intensity of illumination from a given light varies as
the sine of the angle under which the light strikes the illuminated
surface, divided by the square of its distance from the surface. Re-
quired the height of a light directly over the centre of a given circle,
so that it shall illuminate the circumference as much as possible.
Sug's. — Let /represent the illumination at P, which is to be a
sin l_PO
maximum ; PO = B ; and LO =: .r. 7 = — ; — . But
LP'
Bin LPO =—-=—-. .•.!=:=—,= -^ ; whence
^^ ^^ LP' ^R.J^x^Y
dl (E->-fx2)- — 3a-2(/?2 + ar2)- f i .
T,. = ^R:^ -I- a:^). ' = «. im-\-x^-f-Zx'^{R^+x^)l=6
Ji2 _^ .,.< _ 3.,;: ^ 0, and X = Bs/\.
GEOMETRICAL PROBLEMS. 101
Ex. 21. To find in a line joining the centres of two spheres, the
point from which the greatest portion of spherical surface is visible.
Sug's. — The function is the sum of the two
zones whose altitudes are, M D and md ;
hence we must obtain an expression for the
areas of these zones Let CO =^ R, co=: r,
Oo == a, PO = X, and Po = x =a — x. ^~^ 1^ ^ ^
From the right angled triangle PCO, R- =z Fig. 33.
]^x 7?2
DO X aJ ; whence MD=-R — DO=^ — , the altitude of the zone seen
X
fV'rv* . rt*3
on this sphere. In like manner md = ; — . Now the area of a zone being to
X
the area of the surface of its sphere as the altitude of the zone is to the diameter
of the sphere, letting Z and z be the zones, Z : 4:7tR'^ : : — — ; : 2E, . • . Z =
„ ^Rx — R^ A 1 • Ti ^ rx' —r^ a — x — r
zTtR . And m like manner z = znr -, = Aitr- .
x X a — a;
X Ji
Hence, letting S represent the function, we have, S = 27tR'^ -f-
^ a — x — r ,„ ^ R3 , r3 dS' R^ r^ . ,
27rr-2 , S' = R^ ^ r^ . -- = -; ^- = ; whence
a — X X a — X dx x- {a — xy^
9 3. a.
x= " '' — ■■"
a. 3.
)2
r (r^ 4- i?^)2-|
; and the entire surface = 27r r^ -f- J^^ •
r' +R'
Since 27tr^ -f- ^jtR^ is the sum of the hemispheres, S is always less than this sujn
except when a = oo.
GENERAL SCHOLIUM.
The student should now resume the study of G-eneral Geometry at Chap-
ter IV.
OHAPTEE IE.
THE INTEGMAL CALCULUS.
SECTION I.
Definitions and Elementary Forms.
157. The Integral Calculus is that branch of the Infinites-
imal Calculus which treats of the methods of deducing the relations
between finite values of variables, from given relations between the
contemporaneous infinitesimal elements of those variables. It is the
inverse of the Differential Calculus.
158. The Tvttegval of a differential function is another func-
tion which being differentiated produces the differential.
150, IlfltegvatiOTl is the process of deducing the integral func-
tion from its differential.
I(y0, The Sign of Integration is J", which is a form derived
from the old, or long s. It is the initial of the word simi, and came
into use from the conception that integration is a process of summing
an infinite series of infinitesimals.
Ill's. — Suppose we have given dy = -—^ — — . This is a differential function,
and we have given in the equation the relation between dy and dx. The Integral
Calculus proposes to find the relation between y and x from such a relation between
their differentials ; or, in other words, to find the function which being differen-
tiated produces the given differential. The function in this case is ?/ = = , as
1 — iC^
will be proved by differentiating. The latter is therefore called the integral of the
// Axidx ^x^
dy== I
^xdx
and read, "the integral of dy equals the integral of .^ ' ~-_^, which equals
(1 — X2)2 1—X^
Ix
(1 — "x2)2'
1—X^
The conception of integration as a process having for
its object the summation of an infinite series of infinites-
imals may be illustrated by considering the area of an
ellipse as composed of an infinite number of infinitesi-
mal segments, as represented in the figure. Let A rep-
resent the area of the ellipse ; whence cZA represents one
of the infinitesimal segments, or elements of the axea.
DEFINITIONS AND ELEMENTABY FOKMS. 103
Now it is found that dA = -{a^ — x^^) dx. By integration it is found that the
entire area is Ttah, h and 6 being the semi-axes. But, as the entire area is the sum
of the infinitesimal segments, the process of integration may be considered as
having for its object the summing, or adding together of all the infinitesimals
which go to make up the entire area.
101. Important General Statement. — Strictly speaking, there is
no such thing as a Process of Integration. Whenever a differential is
proposed for integration, the first question is, Is this a Knoivn Form f
that is. Can we see by inspection what function, being differentiated, pro-
duces this ? If we cannot thus discern the integral by a simple inspec-
tion, the only question remaining is, Can we transform the differential
into an equivalent expi^ession the integral of luhich we can recognize?
Thus, in any case, we pass from the differential to its integral by a
simple inspection ; and the sufficient reason always is, This expression
is the integral of that, because, being differentiated, it produces it.
THREE ELEMEIVTART PROPOSITIONS,
102, JPvop. 1. — Constant factors or divisors appear in the integral
the same as in the differential, and hence may be written before or after
the sign of integration at 2)leasure.
Dem. — This is a direct consequence of the fact that constant factors or divisors
appear in the differential the same as in the function {48).
lOS* ^vop, 2, — To integr'ate the algebraic sum of several differen-
tials, integrate each term separately, and connect the integrals by the same
signs as their differentials were connected.
Dem. — This is a direct consequence of the rule for differentiating the algebraic
sum of several variables (51).
104, JPvop, 3, — An indeterminate constant must always be added
td the integral of a function.
Dem. —Since, in difierentiating, constant terms disappear, in returning from the
diflferential to the integral we have to represent any possible constant terms by an
indeterminate constant.
ScH. — The method of disposing of this constant term, which we usually
represent by C, will be presented hereafter.* The fact that there may be
such a term is all that the student is expected to see at this point. To illus-
trate, suppose y =^ dax^ -\- 12b, dy = ^ax dx. Now, if the latter alone were
given, we might see that y = dax'^ was its integral, since being differentiated
it would produce dy = 6axdx. But so will y = dax^-\~ any constant, as 126,
or, as we represent it, y = 3«.r2 -J- G.
* Section VII., closing illustration.
104 THE INTEGRAL CALCULUS.
TWO ELEMENTARY RULES.
lOS, R ULE 1. — Whenever a differential can be separated or trans-
formed into three factors ; viz., 1st. Its constant factors ; 2nd. A vari-
able factor affected with any exponent except — 1 ; and 3rd. A differen-
tiat factor which is the differential of the 27id factor without its exponent,
its integral is
The product of the second factor with its exponent increased by 1,
INTO the 1st or constant factor divided by the new exponent.*
Dem.— This rule is evident from {162), and the rule for differentiating a variable
affected with an exponent {56.. Thus, if y = m[/(a;)]«, dy = mn[/(a;)]«-i d.f{x),
or mn X [/(''''^)]"~^ X d.f[x) ; whence to pass from the latter to the former, we have
to suppress the differential factor, d.f{x), increase the exponent n — 1 by 1 making
it n, and divide the constant factor mn by this n.
In the exceptional case the exponent by which we would be required to divide
according to the rule would be 1 — 1=0, whence the result would be oo.
Ex. 1. Integrate dy = dax^dx.
Solution, dy = 3a Xx^Xdx; whence y = jSax^dx = — x^ -\-G= ax^ -f" G.
O
Ex. 2. Integrate dy = ax^dx.
/CL
axHx = -x4 _[- C.
Ex. 3. Integrate dy = (a -\- 3x^y6xdx.
Solution, dy = 1 X{ci-\- Sx^y X Qxdx, which corresponds to the requirements
of the rule, since d{a + 3x^) = 6xdx. .'. y = J{a-\- Sx^y6xdx = i{a + 3x2j3 -|- C.
Ex. 4. Integrate dy = (a + ^x^yxdx.
Solution. — The differential of the quantity within the parenthesis being Qxdx,
we write dy = \ {a -\- 3a;2)3 x Qxdx, which conforms to the requirements of the
rule. .'. y — f\{a + 3.r2)36arc?a; = -^{a + ^x'^Y + G.
Ex. 5. Integrate dy = a{ax -\- hx^y^dx -\- '2h{ax -\- bx^)^xdx.
Sug's. y=fla{ax-{-hx^ydx + 2b{ax-\-bx-^yxdx'] = f{{ax-\-lx^)\a-}-2hx)dx']
= /[I X (ax + hx^f X (« + 2hx)dx'] = i{ax -}- bx^^Y -f- G.
100, RULE 2. — "Whenever a differential can be written in, or
transformed into a fraction whose numerator is the exact differen-
tial OF its denominator, the integral is the Napierian logarithm of
the denominator.*
* In giving such rules the constant term of the integral is not mentioned, as its addition is
always implied.
DEFINITIONS AND ELEMENT AllY FOllMS. 105
Dem. — This is a direct consequence of the rule that the differential of the Na-
pierian logarithm of a number is the differential of the number divided by the
number. [This will be seen to be the exceptional case under the preceding rule.]
167. ELEMENTARY FORMS.
1. V = Cx^'dx = -^"+^ + G. Same as Bule 1.
^ -^ • n + 1
/dx
— = log X -\- G. Same as Bule 2.
S. y= fa'dx = -^a^ + G.
^ log a
3i. y = fe'-dx = e' -\- G.
4.. y = fcos X dx = sin x -\- G.
5. y = r — sin X dx = cos x -j- G.
6. V = / > or fsec^ x dx = tan x -{- G.
J cos^^ -^
7. V = / — — > or f — cosec^^ dx = cot^ + G.
J sin2^ -^
8. 2/ = rtan x sec xdx = sec x -{- G.
9. y = f — cot X cosec xdx = cosec x -\- G,
10. ?/ = fsin X dx = vers j: -f (7.
11. y z= C — cos X dx = covers x -\- G.
12. y = I . =: sin~^^ + G.
13. 2/ = / ; = COS-^^ + U.
dx
14. 1/ = fzr^- = tan-^j7 + a
15. y= f— r^- = cot-^o; + G.
^ J 1 -{- x^
16. 2/ = f — 7=^= = sec-^^ 4- G.
J XV X'^ ■ 1
17. y = I — = cosec~^a; + C,
J XV x-^ 1
18. ?/ = / = vers~^^ + G.
^ Vix 072
/dx
. = covers-^o; + G.
V 2,r — J72
Converse of (60).
«
(61),
((
(66).
«
(67),
it
(69),
-a "
(70),
^->
a( --. — ;- ) = -; — T-dx + ; — r-dy,
\dxdy/ dx^dy dxdy^
and cZ( -— ) = dx A- -^—dy. Substituting, we have
Kdy'^J dy^dx ' dy^ ^ ^
d^w , , d^u , ^ , ^ d'^u , ^ , ^ dht , , , d^u , ^ , d% ,
= -v-cZx3 _L -^ — -dx^dy 4- 2- — -dx-dy 4- 2 , , dxdy'^ 4- ——-dy^dx 4- -—dy^
dx^ ^ dxHy '^ ^ dx;^dy "^ ' dxdy'^ "^ ^ dy^dx ^ ^ dy^ ^
^^^ 7 , . O <^^W 7 » 7 , r, ^^^* 7 „ 7 , <^^W 7 ,
= -r;-dx^ 4- 3- — -dx^dy 4- 3- — -dy^dx 4- —-dy\
dx^ ^ dxHy -^ ^ dyHx -^ ^ dy^ ^ .
d^u , „, d-u _ , _ dhi , , „ ^
since -; — --dx-dy = ^— ; — — dxdydx = --^ — dydx% etc.
dx^dy dxdydx ^ dydx^ ^
In like manner we may proceed to differentiate as often as desired.
ScH. — A little observation will enable the student to write out any re-
quired differential of it = /{x, y) by analogy from the above. He only-
needs to notice that every distinct form of the partial differential of the
required order is involved, and making x the leading letter insert the coef-
ficients as in the binomial formula. Thus d^u = — 'dx''> + 5 dx^dy -j-
dx^ dx^dy
^^ d'u , , , „ , ^„ d^u , „ , „ , _ d'm ^ , , dm , .
10- — —dx^dy'^ + 10 dx'^dy^ + 5 dxdy* 4 dy:
dbfldy^ ^ dx'^-dy^ ^ dxdy^ ^ dy'^ ^
111, ^TOh, — To form the successive differential coefficients of an
implicit f miction of a single variable.
Solution. — ^Let u =f{x, y) = 0, in which y is an implicit function. We are to
. d% d^y ^
form -7^, — ^, etc.
dx2' dic3'
du
dii dx- '
First we have -^ = by {102). (1).
dy
The form for differentiating this is
/du\du /du\du
d^y \dx/7ly \dy)dx . , . „ , , • ^ ^.
•T-, = — , „ • — dx, since the second member is a fraction.
\dy)
To perform the operations thus indicated we have to remember that -r- and tt-
dx dy
are functions of x and y.
Hence d( — ) = --dx 4- -r—-dy*
\dx/ (Zx2 ^ dxdv^
SUCCESSIVE DIFFERENTIATION OF FUNCTIONS OF TWO VARIABLES. 59'
and d( ~~) = - — —dx 4- -r-dy. Dividing these values by dx and substituting:,
\dy/ dydx dy- °
du/d-u d"u dy\ du/ d-u d'u dy\
d^j diKdx'^ dx dy dx / dxXdydx dy^dxj
we have -^ = — ^ 7 . > ■
\dy)
dii
Finally, substituting in this the value of — as given in (1) we have
du f d-u d'^u dx \ du I d^u
I
dy \ dx^ dx dy du | dx | dy dx
dx^ /duy
\dy)
/du\^/d"u\ d^u du du d"u du du d^u/du\^
\dy/\dx-/ dxdy dx dy dydx dx 7ly dyAclx)
\d4/)
d'U/duY n ^"^ ^^ ^^ _t d^u/duY
dxAdyJ dxdy dx dy dyAdx/
(-
\d
duy
.dyJ
In like manner the higher coefficients may be produced, but the forms are too
complicated for elementary purposes.
Ex. 1. Form the first and second differential coefficients of y as a
function of x, when y^ — 2axy -f x^ — 62 = 0.
du
dy dx — 2aw 4-2^7 ay — x ^
Solution. -- = — •— = — —- = . For convemence of
dx du 2?/ — 2ax y — ax
dy
notation put -- = = p, whence p is a function of x and y. Hence | - - r=
dx y — ax f ^ ^ LdxJ
dp* dp* dy
dx dy dx'
— (y — ax) 4- aiciy — ^) , dp* /ay — ^\ . ^ ^''iy — ^'^^ — («?/ — ^)
I (99). But \f\ = % f = dri^^^) - dx
X LdxJ dX' dx \y — ax/-
and f = d./'^^IH^) ^ dy = 2<
dw \y — ax/
Y2
(y — ax)^ dy \y — ax/ ' {y — ax)
Reducing, -- = -, and -~ = '-. Substituting these values and
dx {y — axy-i dy {y — ax)'-^
dv
also the value of -- as at first found, we have,
dx
d^y _ (a^ — Vy (a^ — l)a; ay — x ( a^ — l)(y ^ — 2axy -f- x'^)
<^^ {y — ^^y^ (2/ — ^^)^ y — «* iy — «^)^
Ex. 2. Form the first and second differential coefficients of ?/ as a
function of x, when y^ -\- x^ — r^ = 0.
dy X d-^y r^
\' dx y' dx^ T/3*
* Beiuembar that these are partial differeutial coafficidnts.
60 THE DIFFERENTIAL CALCULUS.
SuQ. — Be particular to use the method now being illustrated.
Ex. 3. Given ^3 _}_ Saxy -f- i/3 = 0, to form -j-, and —^, by the method
for differentiating implicit functions.
_ , di/ x'^ -\- ay d y ^a?xy
Results, -^ =
dx V' -\' ^•^' ^'^■^ (Z/' + CLXf
Ex. 4. Given y^ — %jcy -f o^- = 0, to form the first and second differ-
ential coefficients of ?/ as a function of x, by substituting in (1) and
''2) of the preceding demonstration.
^ , du ^ du ^ „ d% d^u d^u dy
d^u/du\^ ^dH( dii du d^u/du\~
2w t^ d'v dxAdyJ "dxdy dx dy di/Adx/
,•/ — ^ g^jj^ _^ __ — :
. (22/ — 2 a;)2 — 2(— 2)(— 22/)(2y — 2x^ + 2(— 2.y)2 ^ _ —mj:y — x)-\-8y'^ ^
(2?/ — 2iCj3 (2?/ ~ 2a;/
y(y — 2a;)
(y — x)3 *
. dy ^ d'^y ...
Ex. 5. Given cos {x + y) = 0, to form -^, and — by substituting
as above.
du . , , du • / , N <^'^^ , I N d^u
— cos (a; + ?/), -^ = — cos (.r + 2/). Substituting, -- = — 1, and ~ =
— cos (■'g + y) sin-^(.r -f y^ 4- 2 cos (a; -{- ;?/) sin^. x + y^ — cos(a; -|- ?/ ■ sin^ ;r + y^ _
' — tsm^^^x 4-2/;
These results are as might have been anticipated, since for cos ^.v -^ y) = 0,
X + 2/ = 90° ; hence as one arc (x) increases, the other {y) decreases at the same
rate. Therefore -/ = — 1, and, consequently, t^ = 0.
dx a*''
Ex. 6. Solve Ex's 1 — 3 inclusive by substituting in the general/orm-
ulce (1) and (2).
DERITED EQUATIONS.
112. From w = =f{x, y), we have
du
^ == _ — (1)
dx du
dy
CHANGE OF THE INDEPENDENT VARIABLE. 61
d'^u dy\ du/ d"u d^u dy\
dxdii dx) dxSdiidx dy- dx^
du/d^u d^u dy\ du/ d-u d^u dy
^ d'^y dy\dx^ dxdy dx) dxKdiidx dy ^^
and— - = ' T , (2),
dx-^ /du^ ^ "
\d^)
-r, ^x -. du dy du ^ /-, s , • ■, . ,, -, rw^^
From (1), we nave j~t^+3~^=0- (J-O' which is called TJie
First Derived JEqiiation, or The Differential Equa-
tion of the First Order,
From (2) we obtain
, du/d^u d^u dy\ du/ d^u d^u dy\
du d^y dy\dx^ dxdy dx/ dxxdydx dy^ dx/
dy dx^ du
dy
du
d'^u d^u dy dx/ d'^u d^u dy\
dx^ dxdy dx du\dydx dy^ dx/
Ty
dHi d-u dy dy/ d^u d^u dy\
dx^ dxdy dx dxXdydx dy^ dx/
dHL d'^u dy d/Hi/dyy
dx^ dxdy dx dy^\dx/ '
Whence, transposing, we have,
du d^y d^u dy dHi./dy\" d'^u
dy dx^ dxdy dx dy'^dx) ' dx^ ' ^ ^^'
which is called The Second Derived Fquation or The
Differential Fquation of the Second Order,
In a similar manner the Third Derived Equation is found to be
du d^y f dm d^u dy ] d^y dH(./dy\^ d^u /dy\^
dy dx^ I dxdy dy^ dx ) dx^ dyAdx/ dxdy^dx/
^ d^u dy dHi
ax"ay dx dx'^
SECTION YIL
Change of the Independent Variable.
113, In considering functions of a single variable, as ?/ = f{x),
the hypothesis which we usually make that x is equicrescent, and
hence that dx is constant, gives to all the differentials and differential
coefficients of the function after the first, a different form from what
they would have had if such hypothesis had not been made. Thus
&2 THE DIFFERENTIAL CALCULUS.
^/dy\ d^ , . . ,■ d^ydx — d'^xdy
d{-r-\-= —-, when x is equicrescent, but —— '-, when neither
•variable is regarded as equicrescent (i. e. when dy and dx are both
treated as variable). In the course of a discussion it sometimes be-
comes important to change the conception and regard y as the equi-
crescent, or independent variable, and x as the function. Or it may
be desirable to introduce a new variable of which ^ is a function, and
make it the equicrescent variable.
Either of these changes can be readily effected by first giving to
the expression under consideration the form which it would have had
if neither variable had been treated as equicrescent. Then, to make
y equicrescent, remember that all its differentials above the first are 0,
and drop out the terms affected by them. To introduce a new inde-
pendent equicrescent variable, as 6, of which j: is a function, simply
substitute in the general form in which neither x nor y is equicres-
cent, the values of x, dx, d-x, etc., in terms of the new equicrescent
variable 0,
dy d'Y d^y
114z. JProh, — To find the forms which ^, — -, —-, etc., take when
-' ^ dx dx^ dx^
neither variable is considered equici^escent.
Dem. — Since the hypothesis of au equicrescent variable has not modified the
form of --, in it a; or v may be considered equicrescent, or neither, at pleasure.
dx "
Again — = — —— . Now differentiating the latter without regarding dx as
dx' dx
/ dy\ d^y dx — d'^x dy