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THE GIFT OF ALEXANDER Ziw DIFFERENTIAL AND INTEGRAL CALCULUS AS DIFFERENTIAL AND 1 PRAL CALCULUS BY CLYDE ovs, PH.D. ASSISTANT PROFESSOR OP MATHEMATICS IN THE UNIVERSITY OF MICHIGAN Neto Bork THE MACMILLAN COMPANY 1916 Al right reserned. Corraisnr, 1918, Br THE MAOMILLAN COMPANY. Set up and electroryped. Published September, 1916. Nortoagd Press. 3. 8, Oushing Co. — Berwiok d Bmith Oo. Norwood, Mass, D.B,A, vi PREFACE its meaning and deriyation, are of little value. In the present text the non-geometric applications are taken sys- tematically from one subject, mechanics, and the theory is developed as fully as in the calculus proper. A feature of the book is its insistence on the importance of checking the results of exercises, either directly or by solving in more than oue way. The latter method is largely used in the integral calculus, on account of the variety of elementary transformations possible with defi- nite integrals. The answers to many of the exercises are given, but seldom where a knowledge of the answer would help in the solution, or where a simple means of checking the answer exists. Ñ Topics of minor importance are presented in such a way that they may be omitted if it is desired to give a short Course, The chapter on curve tracing is introduced as early as possible, so that the results are available for use through- out the course. Some instructors will wish to begin the use of integral tables immediately after the chapters on formal integra- tion. This of course can easily be done. In spite of obvious difficulties, a chapter embodying a first treatment of centroids and moments of inertia is introduced before multiple integrals have been defined. By this arrangement the student.is brought to realize the fact that in most cases of practical importance mass- moments of the first aud second orders can be found by simple integration, whereas from the usual treatment he gets exactly the opposite idea. In the chapters on differential equations, emphasis 19 laid on those types most likely to be met by the student of engineering or the mathematical sciences. In the last chapter the average student will doubtless require con- PREFACE vii siderable help from the instructor, but it is hoped that, if properly presented, the chapter may give the student some facility in writing and solving the simpler differen- tial equations of mechanics and in interpreting the results. To Professor Alexander Ziwet, who has read the entire manuscript, the author makes grateful acknowledgment, not only for valuable advice and criticism, but for his unfailing encouragement and support. Thanks are also due to Professor T. H. Hildebrandt, who has kindly assisted in reading the proofs, and has made a uumber of useful suggestions. CLYDE E. LOVE. ANN ARBOR, August, 1916. ar. 1 2 4 8. 7 10. 11. 13. 14 15. 16. 17. 18. 20. CONTENTS CHAPTER I FUNCTIONS. LIMITS. CONTINUITY Functions + o Geometric represemtatlon 20 . Independent variable... Kinds of junetiona . . . . . :. One=valued and many-valued fanetions Raté of change; slope . 0... Limite . + a |. Theorems on limite e . Limit ofa function. . . . . Infnitesimale. . . oo. Limit of the ratio of two infultesimels Combi Infinity. . o Funotion with infinite argument 2 CHAPTER II THE DERIVATIVE The derivative . . . . . . Higher derivatives a CHAPTER III DIFFERENTIATION OF ALGEBRAIC Introduction . . Derivativo of a constant . Derivative of a sum; a product; a quolient. Derivative of a function of a function . ix FUNCIIONS 14 ART. 21. 22. 23. 24, 25. 26. 2. 287 29. 31 32. 34. CONTENTS Derivative of 2%, n a positive integer +». . +. Derivative of x*, n fractional . . . » . The general power formula Implicit functions... e Differentiation of implicit Janetions Inverse functions . - - . . . . . CHAPTER IV GEOMETRIC APPLICATIONS Tangents and normals to curves + Length of tangent, enbtangent, normal, and subnormal Tnoreasing and decreasing functions... + Maxima and minima... Coma vit Points of inflection —. . : Summary of toste for maxima and minima, eto. +. Applications of maxima and mínima +. + 4 . Deriva Curves CHAPTER Y PAGE 24 24 26 Prd DIFFERENTIATION OF TRANSCENDENTAL FUNCTIONS L TRIGONOMETRIC AND INVERSE TRIGONOMETRIC FUNCTIONS 30. 37. 38. 30. 40. 4l 42. 43. 45. 46. 47. 48. Trigonometric functions... Differentiation of sine. . e Limit of sin 6/00 as a approaches O 2 Diflerentiation of cos z, tan x, ete. Inverse trigonometrio functions... +. + Restriction to a single braneh . Diferentiation of tito inverse trigonometric functions - IT. EXPONENTIAL AND LoGARtrmMIc FUuNoTIONS Exponentials and logarithms +. + Properties of logaritluns ++ The derivative of the logarithn . 2. The limite. . oo. Differentiation of the exponential function +; + Hyperbolic functions + + + 74. 76. 76. 77. 78. 79. g sl 82, 85, e. 88, 89. 90. eL CONTENTS II. TranscENDENTAL CURVES Tracing of transcendental curves Curve tracing by composition of ordinates . . . .. Graphic solution of equations +. +. + The oyo The epiegeloid The bypocgoloid CHAPTER X CURVE TRACING IN POLAR COORDINATES Slope of a curve in polar codrdinates . . . +. Maxima and minima . . e Curve tracing. . . . +. o. o. CHAPTER XI THE INDEFINITE INTEGRAL Integration. . e Integration an indirect process 20 Constant of integration. —. o Functions having the same derivative Geometric interpretation of an integral oo. Variable of integration . —. o» Change of the variable of integration - 2 2 + Integration by substitation . . o. + CHAPTER XU STANDARD FORMULAS OF INTEGRATION Standard formulas. . . . +. . Formulas (1)-(8) +. - Formula (4): Powers. . . Formulas (6)-(8): Logarithme and exponentials Formulas (7)-(9): Trigonometric functions. . Formulas (10)-(11): Inverse trigonometric fanctions Formula (12): Integration by parts Integration by substitution 2 ne 114 116 118 119 120 122 122 123 1268 127 197 129 181 132 132 134 arm 93. 94, 95. 96. 97. 93. 29. 100. 101. 102, 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. CONTENTS CHAPTER XII INTEGRATION OF RATIONAL FRACTIONS Preliminary step... Partial fractions . . - , - Distinot linear factora . . . . . Repented linear factors... +. Quadratic factors... 2... CHAPTER XIV THE DEFINITE INTEGRAL The definite integral . . . Geometrio interprotation of a definito integral Interchanging limits Change of limits corresponding to a change of variable CHAPTER XV THE DEFINITE INTEGRAL AS THE LIMIT A SUM Aren under a curve Evaluation of the limit. The fundamental theorem . +. Plane areas in cartesian cosrdinates . + Plane areas in polar cosrdinates . Volumes of revolution . + o. Volumos of revolution; second method A theorem on infinitesimals Other volumes. + Line integrale + Geomotrio interpretation of ths line integral Fundamental theorem for line integrals —. Evaluation of line integrals. +. + Length of a curvilinear aro...» Surfaces of revolution +. +. + Cylindrical surfaces... +. + OF Paox 137 137 187 139 140 143 144 145 145 148 150 150 151 154 156 157 158 161 163 164 165 165 167 168 170 asr. 139. 140. 141. 142, 143, 144, 145. 146. 147. 148, 149. 150. 151. 159. 160. 161. CONTENTS The indeterminate forms o Po o The indeterminate forms Ú . «0, oo — 00 - . . General remarks on evaluation of limits. . , CHAPTER XX INFINITE SERIES. TAYLOR'S THEOREM L Serik8 or CONSTANT TERMS Series of n terms . . . . . . . Infinite series . . . . . . . Sum of an infinite series. . . . . . Convergence and divergenes Tests for convergence . a Cauchy's integral test... Comparison test... .. . . +. Rabo deste Alternadimg series Absolute CONVErgenCe . 2 TI. Power Series Power series . . . . . “ . - . . Maclaurinta 8erien. . Taylors series . Taylor's theorem . - o . Approximate computation by sexies 277 Operations with power gerlen 2 0 . Computation ol logaritlms +. 0 CHAPTER XXI FUNCTIONS OF SEVERAL VARIABLES L Parrian DirreRENTIATION Functions of several variables... + + Limits; combimaity Partial derivabives 2 0 racer 202 204 206 220 222 223 226 223 280 234 236 230 237 ART, 163, 164. 165. 173. 174 175. 176. CONTENTS . Geometric interpretation of partial derivatives . . + Higher derivatives —. . + Total differential. +. Differentiation of implicit functions. U. APPLICATIONS TO SOLID ANALYTIC GEOMETRY . Tangent plane to a surface... . Normal line to a surface. + . . Angle between two surfaces; between a line and a surfaco Space curves. o .. Tengent line and normal plano to a spaoo curve Direction cosines of the cagon - e Length of a space curve — . o CHAPTER XXII ENVELOPES. EVOLUTES Envelope of a family of plane curves . a Determination of the envelope +... + Envelope of tangente. Theo evolute CHAPTER XXIIT MULTIPLE INTEGRALS Volume under a surface +. As ; “Volume under a surface: second melhod 22 . Interpretation of the given function +... + The double integral . The double integral in polar cosrdinates oo. . Transformation of double integrales +... + . ATOR OÍ a BUrÍaCO . Triple integrales. . Heterogeneous masses . Centroids and moments of inertia: the general case PaOR 238 238 240 241 252 252 254 256 258 262 263 264 265 286 268 270 273 276 
