^1 IN MEMORIAM FLORIAN CAJORl (^^ £t4Ji /^^«.. f^^^. By D* A* MQRRAY, Ph.D., Formerly Instructor in Matiikmatics in Cornell University ; Professor of Mathematics in Daliiousie College, Halifax, N.9. INTRODUCTORY COURSE IN DIFFERENTIAL EQUATIONS, FOR Students in Classical and Engineering Colleges. Pp. xvi + 2o6. PLANE TRIGONOMETRY, for Colleges and Secondary Schools. With a Protractor. I'p. xiii + 206. SPHERICAL TRIGONOMETRY, for Colleges AND Secondary Schools. Pp. xiv + 114. PLANE AND SPHERICAL 'J RIGONOMETRY. In One Volume. With a Protractor. LOGARITHMIC AND TRIGONOMETRIC TABLES. Five-place and Four-place. Pp. 99. PLANE TRIGONOMETRY AND TABLES. In One Volume. With a Protractor. Pp. 318. A FIRST COURSE IN INFINITESIMAL CAL- CULUS. Pp. xvii + 439. NEW YORK: LONGMANS, GREEN, & CO. A FIRST COURSE INFINITESIMAL CALCULUS BY DANIEL A. MURRAY, Ph.D. (Johns Hopkins) r PROFESSOR OF MATHEMATICS IN DALHOUSIE COLLEGE, HALIFAX, N.S. LONGMANS, GREEN, AND CO. 91 AND 93 FIFTH AVENUE, NEW YORK LONDON AND BOMBAY 1903 Copyright, 1903, by LONGMANS, GREEN, AND CO. All rights reserved. J. S. Cashing & Co. — Berwick ^ Smith Co. Norwood, Mass., U.S.A. PREFACE. This book has been written for beginners in calculus. Its purpose is to provide an introductory course for those who are entering upon that study either to prepare themselves for ele- mentary work in applied science or to gratify and develop their interest in mathematics. This purpose has determined the choice and the arrangement of the topics and the mode of presentation. Little more has been discussed than what may be regarded as the essentials of a primary course in calculus. An attempt is made to describe and emphasise the fundamental principles of the subject in such a way that, as much as may reasonably be ex- pected, they may be clearly understood, firmly grasped, and intelligently applied by young students. There has also been kept in view the development in them of the ability to read mathematics and to prosecute its study by themselves. Excepting in a few instances, only real functions of real variables are considered. Simple, practical applications of the more elementary notions are introduced as early as possible; and, subject to the requirements of a logically counected develop- ment of the study, the more difficult and abstract discussions appear later. In accordance with this plan, the time-honoured division into differential calculus and integral calculus has not been made, and, to mention one instance in particular, following the example set by Professors Lamb, Gibson, and others, the development of functions in series is taken up in the latter, instead of in the earlier, part of the course. The book, however, can be divided easily into differential and integral sections, and thus can be adapted, in this respect at least, for use in cases in which such a division is deemed necessary. With regard to simplicity and clearness in the exposition of the subject, it may be said that the aim has been to write a book vi PREFACE. that will be found helpful by those who begin the study of calculus without the guidance and aid of a teacher. For these students more especially, throughout the work suggestions and remarks are made concerning the order in which the various topics may be studied, the relative importance of the various topics in a first study of calculus, the articles that must be thoroughly mastered, and the articles that may advantageously be omitted or lightly passed over at the first reading, and so on. The notion of anti-differentiation is presented simultaneously with the notion of differentiation, and exercises thereon appear early in the text; but in the chapter in which integration is formally taken up the idea of integration as a process of summa- tion is considered before the idea of integration as a process which is the inverse of differentiation. In this matter I have followed the order adopted in my Integral Calculus, although there is considerable difference of opinion as to the propriety or the advantage of this order. The decision to follow it here has been made mainly for the reason that students appear — at least so it seems to me, but other teachers may have a different experience — to understand more clearly and vividly the relation of integration to many practical problems when the summation idea is put in the forefront. In teaching the 'one order can be taken as readily as the other. In several technical schools the time assigned to calculus is not sufficient for a fair study of Taylor's theorem. What may be regarded as the irreducible requisite for a slight working acquaintance with Taylor's and Maclaurin's series is indicated at the beginning of Chapter XIX., and may be taken at an early stage in the course. The evaluation of indeterminate forms, which affords interest- ing exercises in the application of differentiation, is far from being as important as many other applications of the calculus; and in the few cases in which this evaluation is required it can be effected by other means. Useful exercises in applying inte- gration can be given to students who have a knowledge of mechanics. In many cases, however, these students make but a small fraction of the class, and, besides, in a large number of technical schools the curriculum provides that mechanics shall PREFACE. Vll follow calculus. Accordingly, it seemed better not to treat inde- terminate forms and mechanics in the body of the text, but to deal briefly with them in the appendix. An explanation of hyperbolic functions can be made more naturally and more fully, perhaps, in a course in calculus than in any other course in elementary mathematics. For this reason, and also because students will meet them in their later work and reading, a note on these functions appears in the latter part of the book. Owing to the pressure of other subjects the time allotted to mathematics in quite a number of technical schools is rather brief. Where this is the case, and where there is a lack of maturity in the students, it is better not to try to cover too much ground, but to lay stress on fundamental principles, to drill in the elementary processes, and to train in making simple applications. Thus this book, small as it may be regarded even for a short course, contains more matter than can be thoroughly studied in the few months allotted to calculus in colleges and technical schools where such conditions exist. Several topics, however (for example, the investigation of series), which in some cases are not studied by technical students owing to lack of time, are very important, particularly for those who take a first course in the calculus as an introduction to a more extended study of the subject and as j^art of the preparation necessary for more advanced work in mathematics. For the sake of these students more especially, but not exclusively on their account, many definite references for collateral reading or inspection are given throughout the text. It is hoped that these references will add to the helpfulness of the book. With but very few exceptions those are chosen which are easily accessible to all college students. Some of the references will aid the learner by presenting an idea of the text in the words of another ; but the larger number of them are intended to direct students to places where they will either receive fuller information or be impressed with some of the important modern ideas of mathematics. Turning up such references as these will increase the mathematical interest of the student and widen his outlook. It will also help to train viii PREFACE. the pupils in the use of mathematical literature, and, by arous- ing and exercising their critical faculties, will greatly benefit those who may intend to teach mathematics in the secondary schools. Of course the lists of references are not exhaustive, and, while care has been taken in making them, it is to be expected that several other equally serviceable lists can be arranged. It is intended that these lists shall be revised and supplemented by those who may use the book. For learners who can afford but a minimum of time for this study the essential articles of a short course are indicated after the table of contents. The exposition given here is, in the main, a result of my experience in teaching the calculus to a large number of pupils. Accordingly, it is my duty to acknowledge my indebtedness to many students whose difficulties and original opinions have interested and stimulated me. In preparing the text many works and articles have been consulted. I feel myself to be especially indebted to the writers to whom references are made in various places in the book. Not many examples involving a technical knowledge of engi- neering, physics, or chemistry have been inserted. Few young students understand examples of this kind without considerable explanation, and thus it seems better to refer the pupils to the more specialised text-books dealing with calculus (for instance, those of Perry, Young and Linebarger, and Mellor), which contain many examples of a technical character. I take this opportunity of thanking A. T. Bruegel, M.M.E., Professor of Mechanical Engineering in the Drexel Institute, Philadelphia, for advice and suggestions concerning the draw- ings, and Louis C. Loewenstein, Ph.D., Instructor in Mechanical Engineering in Lehigh University, for the interest and care taken by him in making the figures. I also wish to thank Miss A. A. Stewart, B.Sc, and my colleague. Professor H. Murray for kindly help in the revision of the proof-sheets. Miss Stewart also gave valuable assistance in verifying many of the examples. D. A. MURRAY. Dalhousie College, Halifax, N.S. August 15, 1903. CONTENTS. CHAPTER I. Introductory Problems. ART. PAGE 2. Speed of a moving train 3. To determine the speed of a falling body 4. To determine the slope of a tangent 5. To determine the area of a plane figure . 6. To find a function when its rate of change is known To find the equation of a curve when its slope is known 7. Elementary notions used in infinitesimal calculus . 2 2 6 10 11 11 11 CHAPTER XL Algebraic Notions which are frequently used in the Calculus. 8. Variables 13 9. Functions , . .15 10. Constants 16 11. Classification of functions 17 12. Notation 17 13. Geometrical representation of functions of one variable ... 19 14. Limits 20 15. Notation 24 16. Continuous functions. Discontinuous functions . . . . ■ 24 CHAPTER III. Infinitesimals, Derivatives, Differentials, Anti-derivatives, and Anti-differentials. 18. Infinitesimals, infinite numbers, finite numbers 30 19. Orders of magnitude. Orders of infinitesimals. Orders of infinites 31 20. Theorems on limits and infinitesimals 35 21. Fundamental theorems of the calculus 36 ix X CONTENTS. ART. PAGE 22. The derivative of a function of one variable , .... 38 23. Notation 41 24. The geometrical meaning and representation of the derivative of a function 43 25. The physical meaning of the derivative of a function ... 45 26. General meaning of the derivative : the derivative is a rate . . 46 27. Differentials 47 27a. Anti-derivatives and anti-differentials 50 CHAPTER IV. Differentiation of the Ordinary Functions. General Besults in Differentiation. 29. The derivative of the sum of a function and a constant, say 0(x) + c 51 30. The derivative of the product of a constant and a function, say C(/)(x) 53 31. The derivative of the sura of a finite number of functions . . 54 32. The derivative of the product of two or more functions ... 55 33. The derivative of the quotient of two functions .... 57 34. The derivative of a function of a function 59 35. The derivative of one variable with respect to another when both are functions of a third variable 60 36. Differentiation of inverse functions 61 Differentiation of Particular Functions. A. Algebraic Functions. 37. Differentiation of m» 61 B. Lof/arithmic and Exponential Functions. 38. N'oTE. Tofind.lim^=oo( 1 + --)'" 66 39-41. Differentiation of log„ w, a", w" 67-71 C. Trigonometric Functions. 42-48. Differentiation of sin u., cos m, tan i«, cot m, sec w, esc m, vers u 71-76 D. Inverse Trigonometric Functions. 49-55. Differentiation of sin"^ u, cos~i u^ tan"^ i«, cot~i ?/, sec~^ w, csc-i w, vers-i u 76-80 56. Differention of implicit functions : two variables .... 80 CONTENTS. XI CHAPTER V. Some Geometrical, Physical, and Analytical Applications. Geometric Derivatives and Differentials. ART. PAGE 58. Slope of a curve at any point : rectangular coordinates ... 84 69. Lengths of tangent, subtangent, normal, and subnormal : rectangu- lar coordinates 84 60. Slope of a curve at any point : polar coordinates .... 88 61. Lengths of tangent, subtangent, normal, and subnormal: polar coordinates 89 62. Applications involving rates 91 63. Rolle's theorem 93 64. Theorem of mean value 94 65. Small errors and corrections ; relative eiTor 96 66. Applications to algebra 98 67. Geometric derivatives and differentials 99-106 CHAPTER VI. Successive Differentiation. 68. Successive derivatives 107 69. The 7ith derivative of some particular functions . . •, .111 70. Successive differentials 112 71. Successive derivatives of y with respect to x when both are func- tions of a third variable 112 72. Leibnitz's theorem . 113 73. Application of differentiation to elimination 114 CHAPTER VIL Further Analytical and Geometrical Applications. 74. Increasing and decreasing functions 116 75. Maximum and minimum values of a function. Critical points on the graph, and critical values of the variable . . . .117 76. Inspection of the critical values of the variable for maximum or minimum values of the function 120 77. Practical problems in maxima and minima ..... 123 78. Points of inflexion : rectangular coordinates . , . . . 127 xu CONTENTS. CHAPTER VIIL variables Differentiation of Functions of Several Variables. ART. 79. Partial derivatives. Notation .... 80. Successive partial derivatives .... 81. Total rate of variation of a function of two or more 82. Total differential 83. Approximate value of small errors . 84. Differentiation of implicit functions ; two variables 85. Order of partial differentiations commutative . 86. Condition that an expression of the form Pdx + differential 87. Euler's theorem on homogeneous functions 88. Successive total derivatives .... Qdy be a total PAGE 130 133 134 136 138 139 140 141 142 143 CHAPTER IX. Change of Variable. 80. Change of variable 144 90. Interchange of the dependent and independent variables . . 144 91. Change of the dependent variable 145 92. Change of the independent variable 145 93. Dependent and independent variables both expressed in terms of a single variable 146 CHAPTER X. Integration. 94. Integration and integral defined. Notation . 95. Examples of the summation of infinitesimals 96. Integration as summation. The definite integral . 97- Integration as the inverse of differentiation. The indefinite in Constant of integration. Particular integrals 98. Geometric or graphical representation of definite integrals Properties of definite integrals 99. Geometric or graphical representation of indefinite integrals Geometric meaning of the constant of integration . 100. Integral curves 101. Summary tegi-al 148 150 154 160 162 163 164 166 167 168 169 CONTENTS. XUl CHAPTER XI. Elementary Integrals. ART. PAOK 103. Elementary integrals 172 104. General theorems in integration 173 105. Integration aided by substitution 175 106. Integration by parts . 177 107. Further elementary integrals 180 108. Integration of f{x)dx when f{x) is a rational fraction . . 184 109. Integration of a total differential 188 CHAPTER XII. Simple Geo3ietrical Applications of Integration. 111. Areas of curves : Cartesian coordinates 192 112. Volumes of solids of revolution 199 113. Derivation of the equations of curves 203 CHAPTER XIII. Integration of Irrational and Trigonometric Functions. Integration of Irrational Functions. 115. The reciprocal substitution 206 116. Differential expressions involving y/a + hx 207 117. A. Expressions of form F{x^ Vx^ + ax + b)dx. B. Expressions of form F(x, V-x2 + ax + h)dx 208 118. Tofind fx"»(a + 6x")Pda: 211 Integration of Trigdnometric Functions. 119. Algebraic transformations 215 120. Integrals reducible to | F(ii)du, in which u is one of the trigo- nometric ratios 216 121. Integration aided by multiple angles 217 122. Reduction formulas 218 xiv CONTENTS. CHAPTER XIV. Approximate Integration. Mechanical Integration. ART. PAGE 123. Approximate integration of definite integrals .... 223 124. Trapezoidal rule for measuring areas and evaluating definite inte- grals 223 125. Parabolic rule for measuring areas and evaluating definite integrals 225 126. Integration in series 227 127. Mechanical devices for integration 228 CHAPTER XV. Successive Integration. Multiple Integrals. • Applications. 129. Successive integration : one variable. Applications . . . 230 180. Successive integration : several variables 232 131. Finding areas : rectangular coordinates 234 132. Finding volumes : rectangular coordinates 235 133. Finding volumes : polar coordinates 238 CHAPTER XVI. Further Geometrical Applications of Integration. 135. Volumes of solids of known cross-section 240 136. Areas : polar coordinates 24? 137. Lengths of curves : rectangular coordinates 245 138. Lengths of curves : polar coordinates ...... 248 139. Areas of surfaces of revolution 249 140. Areas of surfaces z = /(ic, y) 253 141. Mean values 255 CHAPTER XVII. CoNCAViTr and Convexity. Contact and Curvature. Evolutes AND Involutes. 142. Concavity and convexity : rectangular coordinates . . . 260 143. Order of contact 261 144. Osculating circle 264 145. The notion of curvature 265 146. Total cuiTature. Average curvature. Curvature at a point . 266 CONTENTS. XV ART. PAOB 147. The curvature of a circle 267 148. To find the curvature at any point of a curve : rectangular coordi- nates 267 149. The circle of curvature at any point of a curve .... 268 150. The radius of curvature : polar coordinates . . • . .271 151. Evolute of a curve 272 152. Properties of the evolute . . . . . . . . 273 153. Involutes of a curve 276 CHAPTER XVIII. Special Topics relatixg to Curves. 154. Family of curves. Envelope of a family of curves . . . 277 155. Locus of ultimate intersections of the curves of a family . . 278 156. Theorem 280 157. To find the envelope of a family of curves having one parameter . 281 158. Envelope of a family of curves having two parameters . . . 284 159. Rectilinear asymptotes 286 160. Asymptotes parallel to the axes 288 161. Oblique asymptotes 290 162. Rectilinear asymptotes : polar coordinates 292 163. Singular points 293 164. Multiple points 293 165. To find multiple points, cusps, and isolated points . . . 296 166. Curve tracing 298 CHAPTER XIX. Infinite Series. 167. Infinite series : definitions, notation 300 168. Questions concerning infinite series 301 169. Study of infinite series 303 170. Definitions. Algebraic properties of infinite series . . . 304 171. Tests for convergence 307 172. Integration of infinite series 310 173. Differentiation of infinite series 312 174. Applications of the integration and differentiation of series . . 313 CHAPTER XX. Taylor's Theorem. 176. Derivation of Taylor's theorem 318 177. Another form of Taylor's theorem 323 XVI CONTENTS. AKT. PAGK 178. Maclaurin's theorem and series 324 179. Relations between the circular functions and exponential functions 327 180. Another method of deriving Taylor's and Maclaurin's series . 329 181. Application of Taylor's theorem to the determination of condi- tions for maxima and minima 331 182. Application of Taylor's theorem to the deduction of a theorem on the contact of curves 332 183. Applications of Taylor's theorem in elementary algebra . . 333 CHAPTER XXI. Differential Equations. 184. Definitions. Classifications. Solutions 384 185. Constants of integration. General solutions. Particular solutions 335 Equations of the First Order. 186. Equations of the form /(a;) (Zx + i^(2/)^?/ = 335 187. Homogeneous equations 336 188. Exact differential equations. Integrating factors . . . 336 189. The linear equation 337 190. Equations not of the first degree in the derivative : The form x = f(y, p) ; the form y = /(x, p) ; Clairaut's equation 338 191. Singular solutions ." . . 340 192. Orthogonal trajectories . . 341 Equations of the Second and Higher Orders. 193. Linear equations with constant coefficients. Homogeneous linear equations 346 194. Special equations of the second order : ;^^=-^w >/(S-l-)=«- 128.8 + 16.1 xfe It is evident that the less the increase given to the 4 seconds, the more nearly does the average speed during this additional time approach to 128.8 feet per second. The last line of the table shows that, no matter how short a time h may be, the average speed during this time has a definite value, namely (128.8 + 16.1 x h) feet per second. The number in brackets becomes more and more nearly equal to 128.8 when h is made smaller and smaller ; the difference between it and 128.8 can be made as small as one pleases, merely by decreasing 7i, and will become still less when h is further diminished. Since the number (128.8 + 16.1 x h) behaves in this way, the speed of the falling body at the end of the fourth second is manifestly 128.8 feet per second. 4 INFINITESIMAL CALCULUS. [Ch. I. (6) To find the speed after the body has been falling for ti seconds. Let si denote the distance in feet through which the body has fallen in the ti seconds. It is known that ^ _ i ^^ 2 (i\ Let A^i (read " delta ii ") denote any increment given to tu and Asi denote the corresponding increment of si. Note 1. Here A^i does not mean A x ^1. The symbol A is used with a quantity to denote any difference, change, or increment, positive or negative (i.e. any increase or decrease), in the quantity. Thus Ax and Ay denote " increment of x,"" " increment of y," " difference in x," " difference in y." Then si + Asi = i g(ti + A^i)'^. (2) Hence, by (1) and (2), Asi = gti • A«i + ^ 9'(A«i)2. .'.^=gt,-]-ig-Ati. (3) Ati Here =^ is the average speed for the time Ati and the corresponding A«i Asi distance Asi. Now the sihaller Ati is taken, the more nearly will ^ approximate to the actual speed which the falling body has at the end of the ^ith second. But when A^i is taken smaller and smaller (in other words, when Ati approaches nearer and nearer to zero), the second member of equa- tion (3) approaches nearer and nearer to gti. Equation (3) also shows that =^ can be made to differ as little as one pleases from gti, merely by taking Ati Ati small enough. Hence it is reasonable to conclude that at the end of the ^ith second the speed of the falling body = gti feet per second. (4) Here ti may be any value of t. So it is usual to express conclusion (4) thus : the speed of a body that has been falling for t seconds is gt feet per second. This result (speed = gt feet per second) is a general one, and can be applied to special cases. Thus at the end of the fourth second the speed is g^ X 4 or 128.8 feet per second, as found in (a) ; at the end of 10 seconds the speed is 10 g or 322 feet per second. The two principal points to be noted in this illustration are : (1) No matter what the value of A^i may be, or how small A^i may be, the quantity — ^ has a definite value, namely, gti + \g - A^j ; (2) When A^i is taken smaller and smaller, — ^ gets nearer and nearer to gt^ ; and the difference between them can be made as small as one pleases by giving Afi a definite small value; this difference remains less than the assigned value when A^j further decreases. 4.] INTBODUCTORT PROBLEMS. 6 Note 2, The definite small value referred to in (2) can be easily found. For example, suppose that — ^ is to differ from gt\ by not more than k say {k being any small quantity, as a millionth, or a million-millionth). Then ^' - gh ^ k. But ~-9h = ig' Ah by (3). ^ — —2k . •. ^- g • Ati^k; accordingly A^i < — • Note 3. It should be observed, as shown by equation (3), that the value of -^ depends upon the values of both ti and A^i. On the other hand, the Ah Asi value to which — tends to become equal as Ah decreases, depends (see (4)) upon h alone. The quantity A^i is any increment whatever of h, but it does not depend upon the value of h- 4. To determine the slope of the tangent to the parabola y = x- : (a) at the point whose abscissa is 2 ; (6) at the point whose abscissa is x^. v\ (a) Let VOQ, Fig. 1, be the parabola y = x^, and P be the point whose abscissa is 2. Draw the secant PQ. If PQ turns about P until Q coin- cides with P, then PQ will take the position PT and be- come the tangent at P. The angle QPE will then become the angle TPB. Note 1. This conception of a tangent to a curve has probably been already employed by the student in finding the equations of tangents to circles, parabolas, ellipses, and hyperbolas. The process generally followed in the analytic treatment of the conic sections is as follows : The equation of the secant PQ is found subject to the condition that P and Q are on the curve ; then Q is supposed to move alo7ig the curve until it reaches P. The resulting form of the equation of the secant is the equation of the tangent at P. The calculus method (now to be shown) of finding tangents to curves is preferred by some teachers of analytic geometry; e.g. see A. L. Candy, Analytic Geometry, Chap. V. Draw the ordinates PL and QM-, draw PM parallel to OX. Let PR be denoted by Ace, and EQ hy Ay. Then the slope of the secant FQ is ^- f For tan EPQ = ^-^ Ax \ PE J Q. ^ /^ y AX R /G L M Fig. 1. INFINITESIMAL CALCULUS. [Ch. L The following table shows the value of -^ for various values of Aic. Ax Value of X. Corresponding value of y. (Increase over a;). Ay (Increase over y). Corresponding value of ^y. Xv. 2. 4. _ 2.1 4.41 .1 .41 4.1 2.01 4.0401 .01 .0401 4.01 2.001 4.004001 .001 .004001 4.001 2.0001 4.00040001 .0001 .00040001 4.0001 2-\-h A + ^h + h^ h 4 /i + /i2 4 + /^ It is apparent from this table that the less Ax is, the more nearly does —2 approach the value 4. The last line shows that, no matter how small Ax Ax ^ (or K) may be, -^ has a definite value, namely A-\-h. This number becomes Ax more and more nearly equal to 4 when h is made less and less ; the difference between it and 4 can be made as small as one pleases, merely by decreasing h to a certain definite value, and will continue to be as small or smaller when h is further diminished. Because the number \ -\- h behaves in this way, it is evident that — ^ will reach the value 4 when Ax decreases to zero. Ax Accordingly the slope of the tangent PT is 4 ; and hence angle TPR or PWL is 75° 57' 49". (h) To determine the slope of the tangent at the point whose abscissa is x^. Let (Fig. 1) P be the point (xi, yi). Draw the secant PQ, and the ordinates PL and Q3I ; draw PB parallel to OX. Let PB, the difference between the abscissas of P and Q, be denoted by Axi, and let BQ, the difference between the ordinates of P and Q, be denoted by Ayi. Then tangent QPB=^ = ^^-1. ^ PB Axi If Q be moved along the curve toward P, the secant PQ will approach the position of PT, the tangent at P ; at last, when Q reaches P, the secant PQ becomes the tangent PT. As Q approaches P, Axi becomes less and less, and when Q reaches P, Axi becomes zero. Conversely, as Axi decreases, PQ approaches the position PT. Accordingly, the slope of the tangent PT can be determined by finding what the slope of the secant PQ, namely -^- approaches when Axi approaches zero. Axi 4.] INTRODUCTORY PROBLEMS, 7 yi + Ayi{= MQ) = (xi -^ Axiy. Hence, on subtraction, Ayi = 2 xi • Axi + (Axi)2. (1) ... :^ = 2 a:i + Axi. (2) Axi This equation shows that -^ approaches nearer to 2 xi when Axi decreases. It also shows that ^^ can be made to differ as little as one pleases from 2 xi, Axi merely by taking Axi small enough, and that this difference will become smaller when Axi is further diminished. (For instance, if it is desired that ^ — 2 xi be less than any positive small quantity, say e, it is only necessary Axi to take Axi less than e.) Accordingly. the slope of PT (the tangent at P) = 2 xi. (3) The two principal points to be noted in tliis illustration are : (1) No matter what the value of Ax^ may be, or how small Aa?i may be, the quantity -^* has a definite value, namely 2 x^ H- A4. ^^ A?/ (2) When Ax^ decreases, the quantity -— approaches the Av, ^1 value 2 x^ : the difference between — ^ and 2 Xi can be made as ^ '. AXi small as any number that may be assigned, by giving Ax-^ a definite small value ; this difference remains less than the assigned value when Ax^ further decreases. Avi Note 1. The value of -^, as shown by Equation (2), depends upon the A values of both xi and Axi. On the other hand, the value to which -~^ tends to become equal as Axi decreases, depends (Equation (3)) upon xi alone. The value of Axi does not depend upon the value of Xi ; for Q (Fig. 1) may be taken anywhere on the curve. Note 2. The method used in getting result (3) does not depend upon the particular value of Xi. The result is perfectly general, and may be expressed thus : '''■the slope of the curve y = x? is 2 x." This general result can be used for finding the slope at particular points on the curve. For instance, if Xi = 2, the slope is 4, as found in {a) ; if Xi =— 1, the slope is — 2, and accordingly, the angle made by the tangent with the x-axis is 116° 34'. (It is advisable to make a figure showing this.) Note 3. In the infinitesimal calculus, as well as in other branches of mathematics, it is very important for the student always to have a clear 8 INFINITESIMAL CALCULUS. [Ch. I. understanding of the meaning of the operations which he performs with numbers^ and to interpret rightly the numerical results obtained by these oper- ations. Thus, if it is stated tliat 6 men work 5 days at 2 dollars per day each, the numbers 6, 5, and 2 are treated by the operation called multiplication, and the number 60 is obtained. The calculator then applies, or interprets, this numerical result as meaning, not 60 men, or 60 days, but that the men have earned 60 dollars. In the curve above, y — y?. This does not mean that at any point on the curve the ordinate is equal to the square on the abscissa, i.e. a length is equal to an area. By y = x^ it is meant that the number of units of length in any ordinate is equal to the square of the num- ber of units of length in the corresponding abscissa. Again, the result in Equation (3) does not mean that the slope of FT is twice OL. The result means that the number which is the value of the trigonometric tangent of the angle TFB is twice the number of units of length in OL. Many persons who can perform operations of the calculus easily and accurately, cannot correctly or confidently interpret the results of these operations in concrete practical problems in geometry, physics, and engi- neering. Thus, some engineers who have had a fairly extended course in calculus discard it when possible, and solve practical problems by much longer and more laborious methods. Such a misfortune will not happen to those who early get into the habit of giving careful thought to finding out the real meaning of the operations and results of the calculus. They will not only "understand the theory," but they can use the calculus as a tool with ease and skill. Note 4. In Fig. 1 let a point Qi be taken on the curve to the left of P, and draw the secant Q\F. (The drawing for this note is left to the student.) It is obvious from the figure that the same tangent PT is obtained, whether the secant QiP revolves until Q-^ reaches P, or QP revolves until Q reaches P. This may also be deduced algebraically. Let the coordinates of ^i be Xi — Acci, y\ — A^i. [Here the ^Xx and Ayi are not necessarily the same in amount as the Axi and A«/i in (6).] Draw the ordinate QiM\. Then yx{=LP)=x^% yi - Ayi (= MiQi) = (xi - Axi)2. Whence, it follows that — ^ = 2 cci — Axi. AXi Accordingly, when AiCi approaches zero, -^ approaches the value 2xi. Axi Note 5. Thoughtful beginners in calculus are frequently, and not un- naturally, troubled by the consideration that when A^i (Art. 3 b) is diminished to zero, ~ has the form - ; and likewise, when Axi (Art. 4 b) becomes A^i zero, -^ becomes ^. It is true that ^ is indeterminate in form : and, if Axj V 4.] INTRODUCTORY PROBLEMS. 9 it is presented without ariy information being given concerning the whence and the wherefore of its appearance, then its value cannot be determined. In the cases in Arts. 3, 4, however, there is given information which makes it possible to tell the meaning of the quantity ^ that appears at the final stage of each of these problems. In these cases one knows how the quantities -— and -r^ are behaving when Mi and Axi respectively are approaching zero ; and by means of this knowledge he can confidently and accurately state what these ratios will become when A^i and Aa^i actually reach zero.* Note 6. Moreover, it should be carefully noted that at the final stages in the solution of the problems in Arts. 3 and 4, ~ is not regarded as a A 61 fraction composed of tico quantities, Asi and A«i, but as a single quantity, namely the speed after ti seconds ; likewise, that ~~ is then not regarded as a fraction at all, but as a single quantity, namely the slope of the tangent at P. Note 7. The student should not be satisfied until he clearly perceives, and understands, that the method employed in solving the problems in Arts. 3 and 4 is not a tentative one, but is general and sure, and that the results obtained are not indefinite or approximate, but are certain and exact. EXAMPLES. 1. Assuming the result in Art. 4 (6), namely, that the slope of the tangent at a point (xi, y{) on the curve y = x^ is 2a:i, find the slope and the angle made with the ic-axis by the tangent at each of the points whose abscissas are .5, 0, 1, 1.5, 2, 2.5, 3, 4, -2, -3, - f, - 1, - f . 2. In the curve in Ex. 1 find the coordinates of the points the tangents at which make angles of 20°, 30°, 45°, 60°, 85°, 115°, 145°, 160°, 170°, respec- tively, with the a;-axis. 3. Draw figures of the following curves. Find the value of — at any Ay point (x, y) in the case of each curve ; then find what -- is approaching when Ax approaches zero : (a) x2 + 2/2 = 16; (b) y = x^ -[- x -\- 1; (c) y = x^ ; (d) y^ = Sxi (e) 9x2 + 16y2 = 144 ; (/) 9 .x2 _ 16 1/2 = 144 ; (9) y^ = ^px] (h) b'^x'^ + a:^y^ = a'^b'^ ; (i) b^x^ - a^y'^ = a'^b^. * The mathematical phraseology and notation employed to express these ideas is given in Chapter II. 10 IN FINITE SIM A L CALC UL US. [Ch. I. TSuGGESTiON. In (a), (x + Ax)2 + (2/ + A2/)2 = 16. It can then be de- *- Ay 2x4- Ax 1 ducedthat^ = -2^^^.J Compare the results found in (g), (h), and (i), with those found in analytic geometry. 4. Using the results obtained in Ex. 3, find the slopes and the angles made with the X-axis by the tangents in the following cases : (a) The curve in Ex. 3 (a), at the points whose abscissas are 4, 2, 1, 0, - 1.5, - 3.5. (&) The curve in Ex. 3 (c), at the points whose abscissas are -3, -2,-1, 0, 1.5, 2.5. (c) The curve in Ex. 3 (d), at the points whose abscissas are 0, 1, 2, 3, 6, 8. (d) The curve in Ex. 3 (e), at the points whose abscissas are 0, 1, 2, 4, - .5, - 1.5. (e) The curve in Ex;. 3 (/), at the points whose abscissas are 4, 8, 10, -5, -7. 5. Using the results obtained in Ex. 3, find the points on the curve in Ex. 3 (a) the tangents at which make angles 40"^ and 136° with the x-axis. 6. Do as in Ex. 5 for the curves whose equations are given in Ex. 3 (c), (d), (e), and (/). 7. Do some of the examples in Art. 59. Make careful drawings in each 5. To determine the area of a plane figure. A plane area, say ABCD, may be supposed to be divided into an infinitely great number of infinitely small rec- tangles. It will be seen later that the limit of the sum of these rectangles when they are taken smaller and smaller, is the area. The calculus furnishes a way to find this limit. Even at this stage in the study of the calculus the student can get some useful ideas concerning this problem by making a brief inspection of Art. 95, Exs. (a), (h), (c). [Art. 14 discusses the term "limit."] 5-7.] INTBODUCTOBY PBOBLEMS, 11 6. (a) To find a function when its rate of change at any (every) moment is known, or, in more general terms, when its laic of change is known. In Art. 3 (5) a particular example has been given of this general problem, viz. to determine the rate of change of a func- tion at any moment. The calculus not only provides a method of solving this general problem, but also provides a method of solving the inverse problem which is stated above. (6) To find the equation of a curve when its slope at any (every) point is known. In Art. 4 (6) a particular example has been given of this general problem, viz. to determine the slope of a curve at any point on it. The calculus not only provides a method of solving this problem, but it also provides a method of solving the inverse problem which has just been stated. Problem (6) is a special case of problem (a), for the slope at a point on a curve really shows " the law of change " existing between the ordinate and the abscissa of the point (see Art. 2Q). A brief inspection of Arts. 24-26, 97, 99, at this time, will repay the beginner. Note. Diflferential calcnlns and integral calculus. The subject of infinitesimal calculus is frequently divided into two parts ; namely, differential calculus and integral calculus. This division is merely a formal division ; though oftentimes convenient, it is by no means necessary. Examples of the kind given in Arts. 2-4 formally belong to "the differential calculus," and those described in Arts. 5, 0, to "the hitegral calculus." 7. Elementary notions used in infinitesimal calculus. The prob- lems used in Arts. 2-4 put in evidence some notions and methods, the consideration and development of which constitute an impor- tant part of infinitesimal calculus. These notions are : (1) The notion of varying quantities which may approach as near to zero as one pleases, such as M^ and AiCi in the last stages of the solution of the problems in Arts. 3 and 4. (2) The notion of a varying quantity, such as — ^ in Art. 3 or — ^ in Art. 4 j, which approaches a fixed number when Afj (or Aa^i) becomes more nearly equal to zero, and approaches in such a way that the difference between the varying quantity and the fixed number can be made to become, and remain, as small as one pleases, merely by decreasing A^i (or Aa^i). 12 INFINITESIMAL CALCULUS. [Ch. I. The infinitesimal calculus gives mathematical definiteness and exactness to these notions, and a convenient notation has been invented for dealing with them. From these notions, with the help of this notation, it has developed methods and obtained results which are of great service in such widely separated fields of study as geometry, astronomy, physics, mechanics, geology, chemistry, and political economy. A review of certain notions of algebra is not only highly advan- tageous but absolutely necessary for a satisfactory understanding of the calculus and for good progress in its study. Accordingly, Chapter II. is devoted to the consideration of the notions of a variable, a function, a limit, and continuity. Note. Reference for collateral reading. Perry, Calculus for Engi- neers, Preface, and Arts. 1-18. CHAPTER II. ALGEBRAIC NOTIONS ^ATHICH ARE FREQUENTLY USED IN THE CALCULUS. 8. Variables. When in the course of an investigation a quan- tity can take different values, the quantity is called a variable quantity, or, briefly, a variable. For instance, in the example in Art. 3, the distance through which the body falls and its speed both vary from moment to moment, and, accordingly, are said to be variables. Again, if the x in the expression or + 3 be allowed to take various values, then x is said to be a variable, and ar -|- 3 is likewise a variable. If a steamer is going from New York to Liverpool, its distance from either port is a variable. Note 1. Numbers and their graphical representation. The measures of quantities are indicated by means of numbers. For instance, if a distance is 30 feet, its measure (when a foot is taken as the unit of measurement) is 30 ; and its measure is 360 when an inch is taken as the unit. When a quantity^ varies, the number which indicates its measure varies. Numbers which involve V— 1 are called imaginary numbers ; other numbers are said to be real numbers.* The (so-called) real numbers can be represented graphically on a straight line L' OL extending to an infinite distance in both directions from O. Let unity be represented by some arbitrarily chosen * Real numbers are divided into two classes, algebraic numbers and tran- scendental numbers. Every (real) root of an algebraic equation, ax" + bx^-^ + •'• + bx + m = 0, with integral coefficients is called an algebraic (real) number. These numbers include integers, irrational numbers such as V2 and v^3, and fractional numbers formed from integers and irrational numbers, A real number which cannot be a root of an algebraic equation of the form described is called a transcendental number. A well-known number of this kind is tt, the ratio of the circumference of a circle to its diameter. Transcendental numbers are irrational.- There are far more transcendental numbers than algebraic. For an interesting brief element- ary discussion on transcendental numbers see Klein, Famous Problems in Elementary Geometry (Beman and Smith's translation, Ginn & Co.), in particular, pages 51—54. 13 14 INFINITESIMAL CALCULUS. [Ch. II. length, say OM. Let the distances of the points on tlie line he measured from 0, and, according to the usual convention^ let the distances of points on the right of be regarded as positive (and be given a plus sign), and the distances of points on the left of be regarded as negative (and be given a minus sign). To each point P on OL there corresponds a definite number, — I ^ \ 1 ^ L! M PL Fig. 3. viz. the ratio OP : OM, the number w say ; and to each number, for instance n, there corresponds a definite point P such that 0P= n- OM. Positive numbers are represented by the points on the right of 0, and negative numbers by the points on the left of 0. When a point moves along the line from to X, it passes over every point from to i in succession, and represents successively each number from zero to the ratio OL : OM. Some of these numbers are integral, such as 1, 3, 12 ; some are fractional, such as \, f, -i^ ; and some are incommensurable, such as V^, V8, v'7, tt. The value of an incommensurable number can be expressed by fractional numbers to as close an approximation to exactness as one pleases ; and the correspond- ing point on L'OL can be located as nearly to absolute correctness as one pleases. For instance, V2 = 1.4142 ••• ; accordingly, the point corresponding to \/2 lies betv^een the points corresponding to 1.4 and 1.5; betw^een the points corresponding to 1.41 and 1.42 ; between the points corresponding to 1.414 and 1.415 ; and so on. As to the graphical representation of imaginary numbers see Chrystal, Algebra (ed. 1886), Part I., Chap. XII., § 2. In this course, with the exception of a few instances, only real numbers are met. The value of a number without regard to sign is called its absolute value. Thus the absolute values of the numbers 1, — 2, ^, — i are 1,2, i, ^. The absolute value of a number x. is denoted by the symbol j x | . Note 2. Infinite numbers. The student has a general idea of the set of numbers ordinarily called Jinite numbers. There is also a set of numbers each of whose (absolute) values is "greater than any number that can be named " or is "beyond all bounds. " These numbers are said to be infinitely great numbers or infinite numbers. Finite numbers have each one distinct symbol at least, as 2, \/2, ], |, or .4, etc. ; but infinite numbers have each the same symbol, namely oc, which is called "infinity." Instead, however, of reading x = cc, "x is equal to infinity," it is better to say " xis infinitely great,"*^ or " x is infinite,^'' or " x is beyond all bounds.'*'' The phrase "is equal to infinity " may give the impression that oo denotes a single, definite, immense quantity ; an impression which is erroneous. For instance, consider a number and its logarithm to base 10. Log 10 = 1, log 100 = 2, log 1000 = 3, log 1,000,000 = 0, log 1,000,000,000,000 = 12, and 9.] FUNCTIONS. 15 so on. It is evident that when the logarithm is infinitely great, the corre- sponding number is also infinitely great. Now these infinitely gr.at numbers are very different from each other ; for when the logarithm bt comes infinite, the corresponding number is much further along (so to say) in the set of infinite numbers. But both these numbers (the logarithm and the anti- logarithm) are then denoted by the same symbol, viz. go.* Note 3. References for collateral reading on numbers. Echols, Calculus, Arts. 1-9 ; Harkness and Morley, I)itroductio?i to the Theoi-y of Analytic Functions, Chaps. I., II. ; Whittaker, Modern Analysis, ('hap. I. 9. Functions. The area of a circle varies when the radius varies, and when the radius has a definite value, the area has a corresponding definite value. The volume of a cube varies with its length, and when the length has a definite value, the volume has a corresponding definite value. To the sine of an angle there correspond certain definite values of the angle. The number of deaths per year in a city depends, in some measure, upon the number of people in it, and in each city thei-e is a definite number of deaths per year. These facts illustrate the following definition : When two variables are so related that to a definite value of one of them there corresponds a definite value {or values) of the other, the second is said to be a function of the first. The first is sometimes called the argument of the function. The following definition of a function may also be used : When \\\o variables are so related that the value of one of them depends upon the value of the other, the first is said to be the dependent variable or to be a function of the second, and the second is said to be the independent variable or to be the argument of the func- tion. The exact relation between the variables may, or may not, be capable of definite statement. If ?/ = 2 x2 + 3 a: - 7, (1) the value of y varies when x varies, and the value of y depends upon the value of x; here y is the dependent variable (or the function), and x is the independent variable (or the argument). On the other hand, the value of X varies when y varies, and the value of x depends upon the value of y. If * The two infinitely great numbers here referred to are compared in Appendix, Note C (Art. 3, Ex. 1). 16 INFINITESIMAL CALCULUS. [Ch. II. y be allowed to vary, x must vary in a manner to suit ; in such a case y is the independent variable, and x is the dependent variable (or the function). Precisely the same remarks may be made if x and y are connected by the relation x^y + y% -f a:2 _ 3 y + 7 = 0. (2) When a relation connecting two variables is given, it does not matter, except in so far as convenience is concerned, which vari- able is regarded as independent. When one variable is chosen as the independent variable, the other must be considered the function. The value of a variable may depend upon the values of two or more variables, or it may have a definite value (or several definite values) when two or more other variables have definite values. In such a case the first variable is said to be a function of the other two, or the first variable is said to be a dependent variable, and the other two are said to be independent variables. Thus the distance which a vessel sails from a port depends both upon the time since departure and upon the speed ; in other words, the distance sailed is a function of the time and the speed, ^gain, if y2 j^Zz^-\- x" + 11 = 0, 2; is a function of x and y. Note. On the term "function," read Gibson, Calculus, § 11. 10. Constants. When a quantity remains unchanged during the course of an investigation, the quantity is said to be a constant. Thus in the case of a steamer going from New York to Liverpool the distance of the steamer from either port is variable, and the distance between the ports is constant. If a quantity has the same value in every investigation, it is said to be an absolute constant; if it has a particular value in one investigation, and another value in a second investigation, and so on, it is said to be an arbitrary constant. Thus g (Art. 3), the ratio tt, 2, | are absolute constants. In each of the triangles (Fig. 4) having a common vertical angle a, let x and y denote the lengths of the sides containing this angle, and let A denote the area of the triangle. Then, by trigo- nometry, A = \xy sin a. Here A^ x, and y are yariables, \ is an absolute constant, and a is an arbitrary constant. 10-12.] CLASSIFICATION OF FUNCTIONS. 17 11. Classification of Functions. A. Explicit and implicit functions. When a function is expressed directly in terms of the dependent variable, like y in equation (1), Art. 9, the function is said to be an explicit function. When the function is not so expressed, as in equation (2), Art. 9, it is said to be an implicit function. If relation (2), Art. 9, were solved for y, then y would be expressed as an explicit function of x. B. Algebraic and transcendental functions. Functions may also be classified according to the operations involved in the relation connecting a function and its dependent variable (or variables). When the relation involves only a finite number of terms, and the variables are affected only by the operations of addition, sub- traction, multiplication, division, raising of powers, and extraction of roots, the function is said to be algebraic; in all other cases it is said to be transcendental. Thus 2 x- -{-3 x — 7, -y/x -f -, are X algebraic functions of x ; sin a', tan (x + a), cos~^ x, l"", e^, log x, log 3 X, are transcendental functions of x. The elementary tran- scendental functions are the trigonometric, anti-trigonometnc, expo- nential, and logarithmic. Examples of these have just been given. C. Continuous and discontinuous functions. A discussion on this exceedingly important classification of functions is contained in Art. 16. 12. Notation. In general discussions variables are usually denoted by the last letters of the alphabet, x, y, z, u, v, •••, and constants by the first letters, a, b, c, •••. The mere fact that a quantity is a function of a single variable, X, say, is indicated by writing the function in one of the forms /(a?), F(x), (x), •",fi{x),f{x), •••. If one of these occurs alone, it is read " a function of x " or " some function of x " ; if several are together, they are read " the /function of a;," " the i^-f unction of a;," "the phi-function of a;," •••. The letter y is often used to denote a function of a;. The fact that a quantity is a function of several variables, X, y, z, •••, say, is indicated by denoting the quantity by means of some one of the symbols, f(x, y), (x, y), F(x, y, z), if/(x, y,z,u),'". These are read "the /-function of x and ?/," "the phi-function of X and 2/," " the i^-f unction of x, y, and z,^' etc. 18 INFINITESIMAL CALCULUS, [Ch. II. Sometimes the exact relation between the function and the dependent variable (or variables) is stated ; as, for example, f(x) = aj-+ 3x — 7,ovy = x^-\- 3 a? — 7 ; F(x, y)=2 e^+ 7 e«-{-xy - 1. In such cases the /-function of any other number is obtained by substituting this number for x in f(x), and the i^-function of any two numbers is obtained by substituting them for x and y respvic- tively in F(x, y). Thus /(^) = ^2 + 3 ^ - 7, /(4) = 42 + 3 . 4 - 7 = 2 1 ; F(t, z)=2e'+7e'-i-tz-l, F(2, 3) = 2 e^ + 7 e^ + 5. EXAMPLES. 1. Calculate /(2) and /(.I) when f{x)= 3\/^+- + 7 a;2 + 2. Write /(^), /(w),/(sinx). ^ 2. Calculate /(2, 3), /(-2, 1), and /(-I, -1) when f{x, y) = 3 x2 + 4 xi/ + 7 1/2 - 13 X + 2 ^ - 11. Write /(w, v), /(sin x, 2). 2 -I- 3 a; 3. Calculate 2: as a function of x when y =f(x) = —^ and z = f(y). 4 — 7 X 4. Given that /(x)= x^ + 2 and F(x) =4 + Vx, calculate /[i^(x)] and 5. If /(x, y^ = ax2 + 6x|/ + cy^ write /(i/, x), /(x, x), and /(?/, y). 6. If 2/ = /(x) = ^^Jl^, show that x =/(«/). ex — a 2 X — 1 7. If y-=(/)Cx) = , show that x = 0(^), and that x=zl. e^ 4- 1 Note 7, More formal (and more rigorous) definitions of continuous functions are the following : A. A function f{x) is said to be continuous throughout the interval from x = a to X = b, when (1) /(x) does not become infinite for any value of x between a and b ; and (2) at any point in this interval, as Xi, it is always possible to find a value of h for which J /(xi + h) —f{x{) \ is less than any number as small as one pleases, say e, that may be assigned. B. A function f(x) is said to be continuous when x = a^\i (1) /(«) is not infinite and has a definite value (or definite values). (2) limit,i„/(x)=/(a). An inspection of the definition of a limit, Note 3, Art. 14, and a comparison of definitions A and JS, show that conditions (2) are practically identical. * "Latent heat of fusion." (See text-books on Physics.) t " Latent heat of vaporisation. " 16.] CONTINUITY. 29 Note 8. The following important proposition can be deduced from the definitions of continuity in Note 7, viz. : If a function f(x) is everywhere continuous in the interval from x = a to x = b, and xi^ x-y are two points in this interval, then as x goes through the range of values from x\ to X2 the function assumes, once at least, each value which lies between /(x'l) and f(x-2). In other words, the continuous function does not overleap any values intermediate between two values which it assumes. The meaning of this proposition will be made clearer by a reference to Figs. 5, 20 a, &, c. In Fig. 5, when x varies from 2 to 3, y takes all values from 7 to 9. Note 9. References for collateral reading. On Limits 2iXi^ Continuo^is and Discontinuous Functions : Chrystal, Algebra (Ed. 1889), Part I., Chap. XV., Part II., Chap. XXV. (in particular §§1-13, 24, 26) ; Harkness and Morley, Introduction to the Theory of Analytic Functions, Chap. VI. (VII.); Lamb, Calculus, Chap. I.; Gibson, Calculus, Chaps. IV., V. ; Harnack, Calculus (Cathcart's translation), §§ 9-19 ; Echols, Calculus, Arts. 12-25. Also see references for collateral reading, Art. 21. AVith refer- ence to the topic in Note 8, see Whittaker, Modern Analysis, Art. 30. CHAPTER III. INFINITESIMALS, DERIVATIVES, DIFFERENTIALS, ANTI-DERIVATIVES, AND ANTI-DIFFERENTIALS. 17. In this chapter some of the principal terms used in the calculus are defined and discussed, and one of the main problems of the calculus is described. In the first study of the calculus it is better, perhaps, not to read all this chapter very closely, but after a cursory reading of it to proceed to Chapter IV., and, while working the examples in that chapter, to re-read carefully the articles of this chapter. These articles can also be reviewed most profitably when the special problems to which they are applied are taken up. Articles 22, 23, however, should be care- fully studied before Chapter IV. is begun. 18. Infinitesimals, infinite numbers, finite numbers. An injini- tesimal is a variable which has zero for its limit. (See definition of a limit. Art. 14.) That is, if a denote an infinitesimal, a = 0, or limit a =r 0. For instance, in Ex. (a), Art. 4, when PR is approaching zero it is an infinitesimal. So also, at the same time, are angle QPT and the triangle PQR. Again, when angle is an infinitesimal sin 6 and tan 6 are infinitesimal ; cos 6 is an infinitesimal when is approaching ^ ; when n is increasing beyond all bounds 1-5-2" is an infinitesimal. Note. The infinitesimal of the calculus Is not the same as the infinitesimal of ordinary speech. The latter is popularly defined as " an exceedingly small quantity," and is usually understood to have a fixed value. The infinitesimal of the calculus, on the other hand, is a variable which approaches zero in a particular way. 17-19.] INFINITESIMALS. 31 The following statements are in accordance with, or follow directly from, the definitions of a limit and an infinitesimal. (1) The difference between a variable and its limit is an infinitesimal. That is, on denoting the variable by x and the limit by a, if limit X =^a, i.e. if x = a, then x = a-\- aj in which a = 0. (2) If the difference between a constant and a variable is an infinitesimal, then the constant is the limit of the variable. In "J"^"^^"? ..-.. x=^a-\ -a, in which « = 0, then x = a, i.e. limit x = a. This principle has been employed in the exercises in Arts. 3, 4. It is evident that the reciprocal of an infinitesimal approaches a number which is greater than any number that can be named, namely, an infinite number. Accordingly, an infinite number may be defined as the reciprocal of an infinitesimal. Numbers which are neither infinitesimal nor infinite are called finite numbers. 19. Orders of magnitude. Orders of infinitesimals. Orders of infinites. Let m and n each denote a number which may be finite, infinite, or infinitesimal. When the limiting value of the ratio — is a finite number, m and n are said to be finite with n respect to each other and to be of the same order of magnitude; when the limit of the ratio — is either zero or infinity, m and n n are said to be of different orders of magnitude. For instance, 1,897,000,000 and .000001 are of the same order of magni- tude. Tan 90° and tan 45° are of different orders of magnitude. Log x and X are of different orders of magnitude when x is an infinite number. (See Appendix, Note C, Art. 3, Ex. 1.) That infinitesimals may be of different orders of magnitude is shown by the following illustration. 32 INFINITESIMAL CAL CUL US. [Ch. III. Suppose that the edge BL of the cube in Fig. 8 is divided into any number of parts, and that each part, as Bb, becomes infinitesimal. Through each point of division, as 6, let planes be passed at right angles to BL. The cube is thereby divided into an infinite number of infinitesimal slices like Bd. Now suppose that the edge BA is divided like BL into parts like Bf which become infinitesimal, and let a plane be passed through each point of division / at right angles to BA. The slice Bd is thereby divided into an infinite num- ber of infinitesimal parallelopipeds like Ck. Finally suppose that the edge BC is divided into parts which become infinitesimal like Bg, and that through each point of division, as g, a plane is passed at right angles to BC. Then Ck is thereby divided into an infinite number of infinitesimal parallelopipeds like ^, ^\ ^, is infinite, Bd Ck kg kg. Since the limiting value of each of the ratios the parallelopipeds DL, Bd, Ck, kg, are all of different orders of magnitude. Deflnition. If a is an infinitesimal, and ^ is such that the limit of the ratio — is a finite number, then (3 is said to be an a infinitesimal of the same order of magnitude as a, and ^ is said to be finite with respect to a. If ^ is such that the limit of the ratio — , in which n is a positive integer, is finite, then fi is said a" to be an infinitesimal of the nth order with respect to a. In order to determine the orders of infinitesimals, it is neces- sary to take some one infinitesimal as a standard infinitesimal, and this standard infinitesimal is said to be of the first order. If the standard infinitesimal be denoted by a, then «-, a?, •••, a", are said to be infinitesimals of the second, third, •••, nth orders, respectively. Infinite numbers, being reciprocals of infinitesimals, also have different orders of magnitude. With reference to the standard infinitesimal «, - is an infinitely great number. The numbers 1111^ -, — , — , •••, — (i.e. a~^, a~^, •••, <«"**), are said to be infinites of If a number ^ a a" 0^' a the first, second, •••, nth orders, respectively be such that the limiting value of the ratio yS-r-- (i.e. I3a~^) is a a 10.] THEOREMS ON INFINITESIMALS. 33 finite number, then ^ is said to be an infinite of the first order, and /3 and a~'^ are said to be finite with respect to each other. If the limit oi ^ -. — (i.e. ^a'"") is finite, then /3 is said to be an infinite of order n. Theorems on inflnitesimals. (a) The product of an infinitesimal a, and any finite number k, namely Ka, is an infinitesimal of the same order as a. This follows at once from the definition above.* CoR. 1. The sum of a finite number of infinitesimals of the same order is an infinitesimal of that order. CoR. 2. The algebraic sum of a finite number of infinitesimals is an infinitesimal, t (b) The product of two infinitesimals, ^ and y say, of orders m and n respectively, is an infinitesimal of order m 4- n. For, if a denote the standard infinitesimal, ft = kiu"*, y = Ksa", and hence fty = Ki/f2a'"+'*, which is an infinitesimal of order m + n. (Here K, and K2 are finite numbers.) (c) The quotient ft -i- y (see (6)) is an infinitesimal of order m — n. N.B. These theorems are true for numbers of any magnitude, for finite and infinite numbers as well as for infinitesimals. The student can make the proofs. EXAMPLES. 1. Let (Fig. 8) AB = 1, and let Bg, Bf, fk, be infinitesimals of the first order, (i) Show that the volumes of Bd, Ck, and kg are infinitesimals of the first, second, and third orders respectively, with respect to the volume of DL. (ii) Show that, with respect to Ck, DL and Bd are infinites of the second and first orders respectively, and kg is an infinitesimal of the first order, (iii) Show that, with respect to kg, the volumes of Ck, Bd, and DL are infinites of the first, second, and third orders respectively. *The product of an infinitesimal and an infinite number may be infini- tesimal, finite, or infinite, according to circumstances. Particular instances are given in Appendix, Note C. t The limiting value of the sum of an infinite number of infinitesimals may be infinitesimal, finite, or infinite, according to circumstances. For simple illustrations see McMahon and Snyder, Diff. Cal, page 12. Many instances in which this limiting value is finite will be found later in this book. 34 INFINITESIMAL CALCULUS. [Ch. III. 2. Show that, if Aic in Fig. 1 be an infinitesimal of the first order, then A«/ is an infinitesimal of the first order, and the area of triangle PQB is an infinitesimal of the second order. 3. Show that if angle d be an infinitesimal, sin d and tan d are infini- tesimals of the same order as d. (See Ex. 13, Art. 14, and Plane Trigo- nometry, Art. 83.) This is a very important case in infinitesimals. 4. Let d denote one of the angles of a right-angled triangle, x the adjacent side, y the opposite side, and r the hypothenuse. Show that if 6 is an infinitesimal of the first order, r and x are both finite, or both infini- tesimals, or infinites of the same order ; and show that if r is also an infinitesi- mal, y is an infinitesimal of an order one higher ; and if r is an infinite, y is an infinite of an order one less ; and if r is finite, y is an infinitesimal of the first order. 5. In the triangle in Ex. 4, in which 6 is an infinitesimal of the first order, show that if r be an infinitesimal of order n, r — x is of order w + 2. »-2sin2^ n fSu GGESTiON : r^ — x"^ — y"^ = r^sin^ d ; whence r — x r -\- X 6. In a circle of finite radius the difference between the length of an infinitesimal arc of the first order and its chord is an infinitesimal of at least the third order. Let AB be an arc of a circle of finite radius r and centre 0. Draw the chord AB and the tangents at A and B. Thesa tangents meet at T; OT bisects arc AB, the chord AB, and the angle AOB. Let angle AOC= 6', take 6 for the standard infinitesimal. Then arc AC =rd (trigonometry), an infinitesimal of first order, and AM = r sin 6, an infinitesimal of first order (Ex. 3) ; also AT = rt3ind, an infinitesimal of first order. Now angle MA T = d. Hence, by Ex. 5, AT - AM is an infinitesimal of the third order. But (SiTC AC- AM)<(AT- AM). Hence,2(arc^C-^itf), i.e. arc AB — chord AB, is an infinitesimal of at least the third order. Note. The theorem stated in Ex. 6 holds for any curve of finite curvature. (See McMahon and Snyder, Diff. Cal., Th. 4, page 27 ; also see Byerly, Diff. Cal., Art. 165.) 20.] THEOEEMS ON LIMITS. 35 20. Theorems on limits and infinitesimals, (a) If two variables j X and y, be always equal, and if one of them, say x, approaches a limit a, then the other approaches the same limit; that is, if ^ = 2/) and x = a, then y = a. For x = a -{-a, in which a = ; hence ?/ = a + a ; that is y = a. Ex. 1. The two members of Equation (3), Art. 3 (ft), are always equal, and the second member approaches a limit gh when A^i approaches zero ; hence the first member approaches the same limit. Ex. 2. The two members of Equation (2), Art. 4 (6), are always equal, and the second member approaches a limit 2 xi when AiCi approaches zero ; hence the first member approaches the same limit. (6) The limit of the sum of a finite number of variables, x, y, z, •••, is equal to the sum of their limits. For, if x = a, y = 6, •••, then x = a-\-a, y = b-\-p, •••, in which a = 0, /3 = •••. Hence x + y -\- -" = a + b + -' + (a + P + — )• But a + /3 + ••• i 0. [Art. 19, Theorem (a), Cor, 2 ] Hence lim (a; + y + •••) = a + 6 + ••• = lim X + lim y + •■•. Ex. lim .333 ... = lim (^'^ + j^^ + ...) = ^, lim .141414 ... = ^f, lim (.333 ... + .141414 ...) = lim (.474747 -..) — 9? — I + i?- (c) The limit of the product of a finite number of variables is the product of their limits. For, if a; = a, !/ = 6, « = c, then X = a -\- a, y = b -\- p, z = c + y, in which a = 0, /3 = 0, 7 i^. 0. Hence xyz = abc + bca + ca/3 + aby + apy -\- bya + ca/3 + apy. .'. lim xyz = abc [Art. 19, Theorem (a), Cor. 2, Art. 20, Theorem (a)] = lim x • lim y • lim z. H^.B. Theorems (a), (b), (c), are true if one or more of the variables be replaced by constants. Ex. lim (.333 ... x .141414 ..-)= ^ • i| = ^S. 36 INFINITESIMAL CALCULUS. [Ch. III. (d) The limit of the quotient of two variables, x and y, whose limits are finite, is the quotient of their limits. Since x =- > y, lim x = lim ( - ) • lim y by Theorem (c). y \yJ x\ _ lima; y) lim 2/' Ex. lim •^3^- '^■^•" -- - .1414 ... lim. 1414... " ^" '* (e) The order of an infinitesimal is not altered by adding or subtracting another infinitesimal of higher order. Let a be the standard infinitesimal, and ^ and y be infinitesimals of orders m and n respectively, and n be greater than m. Now ^ + 7 ^ ^ I 7 . hence lim t±y. :^ lim A + lim -^. [Theorem (6)] But lim -^ == 0, a™ since 7 -=- a*" is an infinitesimal of order n — w, and n is greater than m. ...lim^^^^^lim-^. Note. The order of an infinitesimal is not altered by adding an infini- tesimal of the same order, but it may be altered by subtracting one of the same order. E.g. if jS = 2 a^ + 3 a*, 7 = 2 a^ - 3 a^, 5 = a^ - a^, then ^ + 7 = 4 a3 + 3 a* - 3 a^, which is of the third order ; /3 - 7 = 3 a* + 3 a^, which is of the fourth order ; /3 — 5 = a^ + 3.a* + a^, which is of the third order. Ex. Show that if x and y are two variables, and limit (x -^ y) = 1, then X — y is infinitesimal with respect to both x and y. 21. Fundamental theorems of the calculus. A. The limit of the quotient of any two variables, x and y, is not altered by adding to them any two numbers, say a and p, which are infinitesimal to x and y respectively; that is, lim ^^ = lim ^, when ^ = and ? = 0. ^ y + ^ y ^ y l + « For X -\- a _x X 21.] FUNDAMENTAL THEOllEMS. ' 37 T X + a V X ■,- X T X .'. lim — ■ — = lim - . lim = lim -• . y+^ y i+i? y y [Art. 20, Theorems (a), (c), (cZ).] Note 1. This is sometimes called the fundamental theorem of the differential calculus, as it is frequently used in that branch of the infini- tesimal calculus. See Art. 22, Notes 1, 2. Ex. 1. Lim,.,2aM:J^e ^ 2^3 ^ Ex. 2. Lim^^^o, A^ol^T^ = -* t^ee Ex. 3 {a) , Art. 4. ] 2 ?/ + Ay y B. If the limit of the sum of any number of infinitesimals of the same sign be finite, this limit is not altered when any infinitesimal is replaced by another the limit of whose ratio to the first infini- tesimal is unity. Let there be a set of any number of infinitesimals, aj, as, •••, a,,, whose sum approaches a finite limit as n becomes infinitely great. Let /?!, /Sgj •••> Pnj ^6 another set of infinitesimals, such that liin- = l, lira^' = l, ..., lim — = 1. (1) According to (1), in which ^\ = ^, ^9 = 0, •••, e„ = 0. Then ^^=za^-\- Cjaj, ^^ = <^2 + f-^^-i, '", /?« = «« + c«a„ ; and ^1 + A H h /8„ = aj + ao H h a„ +(ciai + Cgao H h c„a„). ••• (A + )S2 + ••• + A.) - (ai + a, 4- ••• + «„) = £,«! + c^ao + ••• + c„a„. .-. lim (/?! + ^2 4- ... + ^,,,) _ lim (a^ + ag + ••• + «,) = lim (citti + €2^2 H h e„a„). But lim (Citti H- £202 + • • • + Cnttn) = 0- Por let r) be the numerically greatest of the e's, then (ejaj -f egog -f- t,. -f- €„a^) <; (^aj -f (Xg + ' " + a„)?y. 38 INFINITESIMAL CALCULUS. [Ch. III. Now lim (ttj + a2 + • • • + a„) is finite, by hypothesis, and lim^ = 0; hence ihn (a, + -.+ -+ a^r; = 0, and accordingly, lim (cjai + caas + h e„a„) = 0. Hence lim (^i + i^, + - + /8„) = lim (a^ + a^ + - + a„). Note 2. This is sometimes called the fundamental theorem of the integral calculus, as it is often used in that branch of infinitesimal calculus. Note 3. A simple proof of B, depending on a theorem on fractions, is given in Gibson's Calculus, page 198. Note 4. References for collateral reading on infinitesimals. McMahon and Snyder, Differential Calculus (American Book Co.), Arts. 1-15; J. J. Hardy, Infinitesimals and Limits (Chemical Publishing Co., Easton, Pa.), pamphlet 22 pages ; Gibson, Elementary Treatise on the Calculus, Arts. 86, 87 ; Byerly, Diff. Cal., Chap. X. 22. The derivative of a function of one variable. Suppose that the function .-/ ") denotes a continuous function of x. Let x receive an increment Aa; ; then the function becomes f{x -h ^x). (a) Hence the corresponding increment of the function is f{x-^^x)-f{x). (b) This may be written A [/(o^)]. The ratio of this- increment of the function to the increment of the variable is f(x + Ax)-f(x) , .^ Ar/(.T)1 .. Ax ' Ax ^ ^ TJie limit of this ratio when Ax approaches zero, i.e. T f(x-\-Ax) — f(x) 1- A fix) ... Ax Ax is called the derived fnnction of /(«) with respect to x ; or the derivative (or the derivate) of f{x) with respect to x-, or the a;-derivative of /(a?). It is also called the differential coefficient of f(x), a name which is explained in Art. 27. 22.] DIFFERENTIATION. 39 If y also be used to denote the function, that is, if then if x receive an increment Aa:, y will receive a corresponding increment (positive or negative), which may be denoted by Ay, i.e. y + Ay =f{x + Ax). Hence Ay =f(x + Ax) — /(x) ; and ' Ay^ /(^ + Ax)-/W . , Aa; Aa; ^ ^ Ax Ax The process of finding the derivative of a function is called differentiation. This process is a perfectly general one, as indi- cated in steps (a), (6), (c), and (d). It may be described in words, thus : (1) Give the independent variable an increment; (2) Find the corresponding increment of the function ; (3) AYrite the ratio of the increment of the function to the increment of the variable. (4) Find the limit of this ratio as the increment of the variable approaches zero. For a slightly different description of the process of differentiation, see Note 4. Note 1. To differentiate a function (i.e. to find its derivative) is one of the three main problems of the infinitesimal calculus, and is the main problem of that branch which is called " the differential calculus.''^ Note 2. The other two main problems of the infinitesimal calculus (see Arts. 27 rt, 94) are the main problems of that branch called " the integral calculus.^'' It may be said here that while the differential calculus solves the proUem, " when the function is given, to find the derivative," on the other hand the integral calculus solves as one of its two main problems the inverse problem, namely, "when the derivative is given, to find the function." EXAMPLES. 1. Find the derivative of x^ with respect to x. Here f(x) = x^ (See Fig., p.4^2.) Let X receive an increment Ax ; then /(x + Ax) = (x + Ax)^ = x-"^ + 3 x^Ax + 3 x(Ax)2 + (Ax)3. 40 INFINITESIMAL CALCULUS. [Ch. III. .-. fix + ^x) - fix) = 3 x^Ax + 3 xiAxy + (Axy. Ax .-. 11111^.^0- ^^^ + ^ -^^—-^^^^- = 3 x2. Ax If y be used to denote the function, thus y = x^, then the first members of these equations will be successively, y, y + Ay, Ay, — ^, lim^^io— • Ax Ax Note 3, It should be observed that the expression (c) depends both on the value of x and the value of Ax, and, in general, contains terms that vanish with Ax, as exemplified in Ex. 1. (This is shown clearly in Art. 176.) On the other hand, the value of the derivative depends on the value which X has when it receives the increment, and on that alone. For this reason, the derivative of a function is often called the derived function. For instance, in Ex. 1, if X = 2, the value of the derivative is 12 ; if x := 6, the value of the derivative is 108. Compare Exs. in Arts. 3, 4. (It is probably now apparent to the beginner that the process used in the problems in Arts. 3, 4, was nothing more or less than differentiation.) Note 4. Sometimes Ax is called the difference of the variable x, (&) is called the corresponding difference of the function, and (c) is called the difference-quotient of the function. The process of differentiation may then be described, thus : (1) Make a difference in the independent variable ; (2) Calculate the corresponding difference made in the function ; (3) Write the ratio of the difference in the function to the difference in the variable ; (4) Determine the limiting value of this ratio when the difference in the variable approaches zero as a limit. 2. Find the derivatives, with respect to x, of x, 2 x, 3 x, ax, x^, 7 x^, 11 x^, 6x2, yfi^ 5 yfi^ 13 gj3^ and cx^. Ans. 1, 2, 3, a, 2x, 14 x, 22 x, 2 &x, 3x2, 15x2, 39x2, ^cx\ 3. Calculate the values of these functions and the values of their derivatives, when x=:l, x = 2, x = 3. 4. Find the derivatives, with respect to x, of : («) x2 + 2, x2 — 7, x2 + ^• ; (6) x^ + 7, x3 - 9, x3 + c. 1 2 5. Differentiate x*, x2 + 4 x — 5, -, - — 3 x + 2 x2, with respect to x. XX o 6. Find the derivatives, with respect to t, of 3 «2, 4 «3 _ g « + -• 3 7 ^ 7. Differentiate %^, - y^ — S y — -, with respect to ?/. 4 y 8. Show that, if n is a positive integer, the derivative of a?»» with respect to X, is nic»»-i. Note 5. The result in M, 8, ^^ wiU be seen l^^er, iq ^r^e for qU con- stant values of n, , 23.] NOTATION. 41 9. Assuming the result in Ex. 8, apply it to solve Exs. 4-7. Note 6. In order that a function may be differentiable (i.e. have a deriva- tive), it must be continuous ; all continuous functions, however, are not differentiable. For remarks on this topic, see Echols, Calculus, Art. 30. For an example of a continuous function which has nowhere a determinate derivative, see Echols, Calculus, Appendix, Note 1, or Harkness and Morley, Theory of Functions, § 65. 23. Notation. There are various ways of indicating the deriva- tive of a function of a single variable. (In what follows, the independent variable is denoted by x. In the case of other variables the symbols are similar to those now to be described for functions of x.) (a) This symbol is often used to denote (d) Art. 22, viz. /'(^). A Thus the derivatives (or derived functions) of F{x), (y), f{t), fi(z), with respect to x, y,' t, and z, respectively, are denoted by F'(x)j '(y), f'(t), fi(z). These are sometimes read " the F-prime function of .t," etc. (b) If y is used to denote the function of x (see Art. 22), the derivative of y with respect to x is frequently indicated by the symbol ^, ^ This is often read " i/-prime " ; but it is better to say " deriva- tive of 2/." (c) The a^derivative of f(x) is also indicated by the symbol The brackets in D are usually omitted, and the symbol is written df(x) doc E Symbols C, D, and E should be read "the ic-derivative of /(a;)." {d) When y denotes the function, the derivative (see Equation (/) Art. 22) is sometimes denoted by 42 INFINITESIMAL CALCULUS. [Ch. III. The brackets in F and G are usually omitted, and the symbol for the derivative is written % H dx This should be read for a while at least by beginners, "the derivative of y with respect to ic," or more briefly " the x-derivative ofy" (Other phrases, e.g. " dy by dx/' are common, but, unfortu- nately, are misleading.) (e) In case (d) the operation of differentiation, and also its result, namely, the derivative, are alike indicated by the symbol (/) Sometimes the independent variable x is shown in the symbol, thus 2) ?/. r Note 1. Mathematics deals with various notions, and it discusses these notions in a language of its own. In the study of any branch of mathe- matics, the student has Jirst to clearly understand its fundamental notions, and then to learn the peculiar .shorthand language, made up of signs and symbols and phrases, which has been in part invented, and in part adapted, by mathematicians. A striking instance of the great importance of mere notation is seen in arithmetic. To-day a young pupil can easily perform, arithmetical operations which would have taxed the powers of the great Greek mathematicians. The one enjoys the advantage of the convenient Arabic notation* for numerals, the other was hampered by the clumsy notation of the Greeks. Note 2. Symbols A and B, and also /and J, have this important quality, namely, they tend to make manifest the fact that the derivative is a single quantity. It is not the ratio of two things, but is the limiting value of a variable ratio. Symbols C and F have the quality that they indicate, in a way, the process (Art. 22) by which the derivative is obtained. The symbol — dx before a function indicates that the operation of differentiation with respect to X is to be performed on the function ; it also serves to indicate the result of the operation. The symbols D and Dx,t in /and J, are simply abbrevia- tions for the symbol — dx * This should really be called the Hindoo notation ; for the Arabs obtained it from the Hindoos. See Cajori, Histortj of Mathematics. t The symbol />^y is due to Louis Arbogaste (1750-1803), professor of mathematics at Strasburg. The symbol -^ was devised by Leibnitz, and the symbol /', by Lagrange (1736-1813), ^* 24.] BEPRt:SENTATION OF THE DERIVATIVE. 48 Note 3. Beginners in the calculus are liable to be misled by the symbols 2), E, G, and H, especially by H. The symbol -^ does not denote a fraction ; dx it does not mean "the ratio of a quantity dy to a quantity dx.''"' Such quan- tities are not in existence at the stage when -^ is obtained. It should be dx thoroughly realized, and never forgotten, that ^ is short for —(y), and dx dx that both these symbols are merely abbreviations for lim_^j.^ —^ /gee EaJf) Art. 22). Some one has remarked that the dy and az in ^ are merely " the dx ghosts of departed quantities ■' ; but perhaps this is claiming too much for them. 24. The geometrical meaning and representation of the derivative of a function. Let f{x) denote a function, and let the geometrical representation of the function, namely the curve be drawn. y=f(^)y (1) Let P(x^, 2/i) and Q(.Ti + Aa^i, 2/1 + ^?/i) be two points on the curve. Draw the secant LPQ. Then tanPiX = AXi Now let secant LQ revolve about P until Q reaches P. Then the secant LP takes the position of the tangent TP, and the angle PLX becomes PTX ; then, also, Aa\ reaches zero. Hence tanPrX = lim Axi=0 (2) 44 INFINITESIMAL CALCULUS. [Ch. 111. Now P(xi,yi) is any point on the curve; hence, on letting (x, y), according to the usual custom, denote ariy point on the curve, and ^ denote the angle made with the ic-axis by the tangent at (x, y), tan<^ = lim^,^o^- (3) The first member of (3) is the slope of the tangent at any point {x, y) on the curve y=f(x), and the second member is the derivative of either member of (1). Hence -^, i.e. /'(x), is the slope of the tangent at any point (a?, y) on the curve y =/(x). This principle has already been applied in the exercises in Art. 4. Curve of slopes. If the graph of f'(x) be drawn, that is, the curve y=f'(x), it is called the curve of sloj^es of the curve y=if{x). It is also called the derived curve, and sometimes the differential curve of y =f{x). For instance, the curve of slopes of the curve y = x^ is the line y = 2x. The curve of slopes is the geometrical representative of the derivative of the function ; the measure of any of its ordinates is the same as the slope of y = f(x) for the same value of x. Ex. Sketch the graphs of the functions in Exs., Art. 22. "Write the equations of these graphs. Give the equations of their curves of slopes, and sketch these curves. (Use the same axes for a curve and its curve of slopes.) Note 1. Produce RQ (Fig. 10) to meet TP in 8, produce PR to R , and draw R'Q'S' parallel to RQ to meet the curve in Q' and TP in S'. Then dx / ^ ^ PR PR' Now, if Ax, = PR,^ = ^; and if Axi = PR', ^ = ^' Also, ' Aa^i PR ' Axi PR' T BQ dy limp^^Op^ = ^, , ,., . ,. R'Q' dy and likewise, hmp^j-^o -p^ - j- • Note 2. Hereafter, in general investigations like the above, the symbol x will be used instead of Xi to denote any particular value of x ; and similarly in the case of other variables. 25.] MEANING OF THE DERIVATIVE. 45 25. The physical meaning of the derivative of a function. Sup- pose that the value of a function, say s, depends upon time ; i.e. suppose s=f(t). After an interval of time A^, the function receives an incre- ment As; and , . ^,^ , .^. As =f{t + M) ~f(t). (1) . A8^ /(^ + A0-/(0 " M At lim^^^ fi.e.^)=f'(t). (2) At V dt^ As Since As is the change in the function during the time At, — At is the average rate of change of the function during that time. As At decreases, the average rate of change becomes more nearly equal to the rate of change at the time t, and can be made to differ from it by as little as one pleases, merely by decreasing A^. Hence the second member of (2) is the actual rate of change at the time t. In words : The derivative of a function with respect to the time is the rate of change of the function. If s denotes a varying distance along a straight line, then — denotes the rate of change of this distance, i.e. a velocity. (For discussions on speed and velocity see text-books on Kine- matics and Dynamics, and Mechanics.) Ex. Show that if s = Igt^, then — = gt. (See Art. 3 b.) dt Note. Newton called the calculus the Method of Fluxions. Variable quantities were called by him fluents or flowing quantities, and the rate of flow, i.e. the rate of increase of a variable, he called the fluxion of the fluent. Thus, if s and x are variable, — and — are their fluxions. Newton dt dt indicated these fluxions thus : 5, x. This notation was adopted in England and held complete sway there until early in the last century, and the other notation, that of Leibnitz, prevailed on the continent. At last the continental notation was accepted in England. "The British began to deplore the very small progress that science was making in England as compared with its racing progress on the continent. In 1813 the 'Analytical Society' was formed at Cambridge. This was a small club established by George Peacock, 46 INFINITESIMAL CALCULUS. [Ch. III. John Herschel, Charles Babbage, and a few other Cambridge students, to promote, as it was humorously expressed, the principles of pure ' D-ism,' that is, the Leibnitzian notation in the calculus against tliose of 'dot-age,' or of the Newtonian notation. The struggle ended in the introduction into Cambridge of the notation ^, to the exclusion of the fluxional notation y. dx This was a great step in advance, not on account of any great superiority of the Leibnitzian over the Newtonian notation, but because the adoption of the former opened up to English students the vast storehouses of continental discoveries. Sir William Thomson, Tait, and some other modern writers find it frequently convenient to use both notations." — Cajori, History of Mathematics, page 283. 26. General meaning of the derivative : the derivative is a rate. When a variable changes, a function of the variable also changes. A comparison .of the change in the function with the causal change in the variable will determine the rate of change of the function tvith respect to the variable. The limit of the result of this com- parison, as the change in the variable approaches zero, evidently gives this rate. But this limit has been defined as the derivative of the function with respect to the variable. Accordingly (see Art. 22, Note 1), the main object of the differential calculus may be said to be the determination of the rate of change of the function with respect to its argument. Note 1. The rate of change of the function with respect to the variable may also be shown in a manner that explicitly involves the notion of time. In the case of the function y, when y =/(x), let it be supposed that x receives a change Ax in a certain finite time A^. Accordingly y will receive a change Ay in the same time At. Then, from the equation preceding (e), Art. 22, Ay ^ f(x + Ax) - fix) ^ fix 4- Ax) - f(x) Ax At 'At Ax 'At' When At approaches zero. Ax also approaches zero. On letting At approach zero, this equation becomes (Art. 20, Th. c). dy dy ^,, ^dx . dir dy dx ,^, „^, dii dt -±=fi(x)—; i.e. ^ = -:^ • —• (1) Whence, /=^- (2) dt '' ^ ''dt' dt dx dt ^ ^ ' dx dx ^ ^ dt Result (1) Can also, by a theorem on limits. Art. 20 (d), be derived from Ay _Ay Ax 'Ax~~Ki^~Ai' 26, 27.] DIFFERENTIALS, 47 Thus the derivative of a function with respect to a variable may be regarded as the ratio of the rate of change of the function to the rate of change of the variable. Note 2. References for collateral reading. McMahon and Snyder, Diff. C'a?., Arts. 88, 89; Lamb, Calculus, Art. 33; Gibson, Calculus, Arts. 31^7, 51. EXAMPLES. 1. A square plate of metal is expanding under the action of heat, and its side is increasing at a uniform rate of .01 inch per hour; what is the rate of increase of the area of the plate at the moment when the side is 16 inches long ? At what rate is the area increasing 10 hours later ? Let X denote the side of the square and A denote its area. Then A = x^. Now M = M . Ax ^^ dA^dA^ dx^ . ^^ 2a; X .01 sq. inches At Ax At dt dx dt dt per hour = .02 x sq. inches per hour. Accordingly, at the moment when the side is 16 inches, the area of the plate is increasing at the rate of .32 sq. inches per hour. Ten hours later the side is 16. 1 inches ; the area of the plate is then increasing at the rate of .322 sq. inches per hour. The area of the square is increasing in square inches 2 x times as fast as the side is increasing in linear inches. 2. In the case of a circular plate expanding under the action of heat, the area is increasing at any instant how many times as fast as the radius ? If when the radius is 8 inches it is increasing .03 inches per second, at what rate is the area increasing ? At what rate is the area increasing when the radius is 15 inches long ? 3. The area of an equilateral triangle is expanding how many times as fast as each of its sides ? At what rate is the area increasing when each side is 15 inches long and increasing at the rate of 2 inches a second ? At what rate is the area increasing when each side is 30 inches long and increas- ing at the rate of 2 inches a second ? 4. The volume of a spherical soap bubble is increasing how many times as fast as its radius ? At what rate (cubic inches per second) is the volume in- creasing when the radius is half an inch and increasing at the rate of 3 inches per second ? At what rate is the volume increasing when the radius is an inch ? 5. A man 5 ft. 10 in. high walks directly away from an electric light 16 feet high at the rate of 3| miles per hour. How fast does the end of his shadow move along the pavement ? 27. Differentials. If y=f(x), (1) then, in accordance with notations A and Hj in Art. 23, |=/(x). (2) 48 INFINITESIMAL CALCULUS. [Ch. III. Suppose that an arbitrary difference {i.e. change or increment) h may be made in the independent variable x, and let the product f'ix) • h be denoted by k ; that is, let lc=f{x)-h. (3) (For instance, in Fig. 10, RS =f'(x) • PE-, here h = PR, and k = RS. Also, R'S' =f{x) . PR' ; here li = PR', and k = R'S'). Now, let h be written in the form dx, and the corresponding value of k be written dy. Then (3) is written dp = f'{x)dx, (4) This, by (2), may be written dy = ^if.dx. (5) dx < As used in Equation (4), dx is called the differeyitial of x, dy is called the differential of y, and f{x)dx is called the differential of fix). Since f\x) is the coefficient of dx in the diiferential of f{x), fix) is frequently called the differential coefficient of f{x). (See Art. 22.) The defining Equations (4) and (5) may be ex- pressed in words : Tlie differential of a function y of an indepen- dent variable x is equal to the derivative of the function multiplied by the differential of the variable, the latter differential being merely an arbitrary increment (or difference), usually small, made in the variable. The letter d is used as the symbol for a differential ; for example, the differential of /(ic) is written df(x) ; thus df(x)=f'(x)dx. Note 1. It is highly important to notice that in Equations (2) and (4), dy and dx are used in altogether different ways.* In (2) and (5), - ^ is used as a symbol for liniAxio— ; and it denotes the definite limiting value of an Ax "indeterminate" form — In (4) and in (5) on the extreme right dx is not zero (although it may happen to be, and usually is, a small quantity),! and the dy is such that the ratio dy : dx is equal to /'(x). For instance, in Fig. 10, * In one respect this double use of dx and dy is unfortunate ; for it tends to confuse beginners in calculus. Other notation is also used. t Later on many examples will be found in which this dx is an infinitesimal. 27.] EXAMPLES. 49 ^^ of Equation (2) is tan SPR. As to Equations (4), (5), if dx = PR, then dy = US', and if dx= PB', then dy = B'S'. This shows that dy, in (4), is the increment of the ordinate of the tangent corresponding to an increment dx of the abscissa. The corresjjonding increment of the ordinate of the curve y — f{x) [i.e. the hicrement of the function /(x)] in some cases can be found exactly by means of the equation of the curve, and in some cases can be found, in general only approximately, by means of a very important theorem in the calculus, namely, Taylor'' s Theorem (see Chap. XX.). Instances of the former are given below ; instances of the latter are given in Art. 176. Note 2. It should be clearly understood that, according to the preceding remarks, cancellation of the dx^s in (5) is impossible. iV.B. For geometric illustrations of derivatives and differentials see Art. 67. EXAMPLES. 1. In the case of a falling body s = I gt^ (see Art. 3) ; on denoting, as usual, the differential of the time by dt, ds, the corresponding differential of the distance is [Ex., Art. 3 (6)] gtdt ; i.e. ds = gldt. The actual change in s corresponding to the change dt in the time is [see Eq. (2), Art. 3 (6)] gtdt + \g(idty\ 2. In the curve y = x"^, dy = 2 x dx. The actual change in y corresponding to the change dx in x is2xdx -\- (dxy. (See Eq. (1), Art. 4. ) Thus if a; = 10 and dx = .001, d^ = 2 x 10 x .001 = .02. The actual change in the ordinate of the curve from x = 10 to a; = 10 + .001 is (10.001)2 ~ 102, j g^ .020001. This change may also be calculated as stated above, viz. 2 x 10 x .001 + (.001)2. The dy = .02 is the change in the ordinate of the tangent at ic = 10 from x = 10 to x = 10.001 (see Note 1). (The student should use a figure with this example.) 3. Write the differentials of the functions in the Exs. in Art. 22. 4. Given that y = x^ — 4: x^, find dy when x = 4 and dx = .1. Then find the change made in y when x changes from 4 to 4.1. 5. Given that y = 2x^ + 7x^ — 9x + 5, find dy when x = 6 and dx = .2. Then find the change made in y when x changes from 5 to 5.2. Note 3. It is evident from these examples that the differential of a function is an approximation to the change in the function caused by a differential change in the variable ; and that the smaller the differential of the variable, the closer is the approximation. When the differential varies and approaches zero it becomes an infinitesimal. Ex. Calculate the differentials of the areas in Ex. 2, Art. 26, when the differential of the radius is .1 inch. Ex. Calculate the differentials of the areas of the triangles in Ex. 3, Art. 26, when the differential of the side is .1 inch. 50 INFINITESIMAL CALCULUS. [Ch. III. Note 4. It may be remarked here that in problems involving the use of the differential calculus derivatives more frequently occur, and in prob- lems in integral calculus differentials (viz. infinitesimal differentials) are more in evidence. Note 5. References for collateral reading. Gibson, Calculus, § 60 ; Lamb, Calculus, Arts. 57, 58. 27 a. Anti-derivatives and anti-differentials. In Arts. 22 and 27 the derivative and the differential of a function have been defined, and a general method of deducing them from the function has been described. With respect to the derivative and the differen- tial the function is called an anti-derivative and an anti-differential respectively. Thus, if the function is x^, the x-derivative and the i»-dift'erential are 2 x and 2 xdx respectively ; on the other hand, Q(? is said to be an anti-derivative of 2 a? and an anti-differential of 2 xdx. To find the anti-derivatives and the anti-differentials of a given expression is one of the two main problems of the integral calculus. (See Art. 22, Notes 1, 2, and Arts. 94, 96, 97.) Note. Reference for collateral reading. Perry, Calculus for Engi- neers, Arts. 12-24, 28, 30. CHAPTER IV. DIFFERENTIATION OF THE ORDINARY FUNCTIONS. 28. In this chapter the derivatives of the ordinary functions of elementary mathematics are obtained by the fundamental and general method described in Art. 22. Since these derivatives are frequently employed, a ready knowledge of them will prevent stumbling and thus make the subsequent work in calcnlus much simpler and easier; just as a ready command of the sums and products of a few numbers facilitates arithmetical work. Accord- ingly these derivatives should be tabulated by the student and memorized. N.B, The beginner is earnestly recommended to try to derive these results for himself. For a synopsis of the chapter see Table of Contents. GENERAL RESULTS IN DIFFERENTIATION. 29. The derivative of the sum of a function and a constant, namely, Put y = {x-\-^x) + c- \_{x) 4- c] = <^ (a; + ^x) — (x -{- Ax) — (x) ' ' Ax Ax 51 52 INFINITESIMAL CALCULUS. Let Ax approach zero as a limit ; then [Ch. IV. lim^^^o Ay Ax (x-\- Ax) — (x) ^ Ax ' I.e. I.e. I=*'(^)^ '^l(.x) + c-] = 4:X^(x)l (1) dx dx Hence, if constant terms appear in a function, they may he neg- lected when the function is differentiated. If u be used to denote <^ (a?), result (1) can be expressed : doc doc (2) CoR. 1. It follows from (1) that the deriyative of a constant is zero. This may also be derived thus : If y = c sl constant, then y-\-Ay = c', and, accordingly, Ay = 0. Hence, — ^ = for all d d ^^ values of Ax; hence, -^, i.e. —(c), is zero. dx dx CoR. 2. If two functions differ by a constant, they have the same derivative. From (2) and Art. 27, d(u + c) = du. Note 1, In geometry y = c is the equation of a straight line parallel to the axis of X and at a distance c from it. The slope of this line is zero ; this is in accord with Cor. 1. Note 2. The curves y = (x) and F{x) denote the functions, and put y = {x) = F(x^ + c. Ex. Show this geometrically. Note 5. If ^ = 'Mi then y = (a5). Put y = c(x). Let X receive an increment Ax; consequently y receives an increment, Ay say. That is, y -{- Ay = c{x -[- Ax). .'.Ay = cl(x-^Ax)-(x) l " Ax [_ Ax J ii,n ^.'^-lim J (x-^Ax)- iim^j^o-T— — ^1"1ax=o ^ 7 Ax [_ Ax dx M]; i.e. ^[c^{x)-\ = e4>\x). (1) 54 INFINITESIMAL CALCULUS. [Ch. IV. That is, the derivative of the product of a constant and a function is the iwoduct of the constant and the derivative of the function. If <^ (x) be denoted by u, then (1) is written In particular, \i u = x, — (ex) = c. ax From the above and the definition in Art. 27, d[c(x)^ = cd[<^(aj)], d(cu) = cdu, d(cx) = cdx. Ex. See Exs., Art. 22. 31. The derivative of the sum of a finite number of functions, say Put y=^(^x)-^F(x) + ->: Then, on giving x an increment Ax (as in Arts. 29, 30), y -\- Ay = ct>(x -\- Ax) + F(x + Ax) + -. -. .-. Ay = (t>(x + Ax) — ct)(x) + F(x + Ax) — F(x) -\ . . Ay ^ cf>(x 4- Ax) - (x) F(x + Ax) - F(x) ^_^ Ax Ax Ax Hence, on letting Ax approach zero, dx dx dx i.e. £l^(x) + F(x) + ...:\=.l>'(x)+F'(x) + .... (2) That is, the derivative of a sum of a finite number of functions is the sum of their derivatives. If the functions be denoted by u, v, iv, •••, i.e. if y = u-^v-{-tv-\ , the result (1) may be expressed thus : dp _ du dv . dw ■ dx dx dx dx 31,32.] DIFFERENTIATION OF FUNCTIONS. 55 rrom this and Art. 27, dy = du + dv + dtv + ••• . Note 1. The differentiation of the sum of an infinite number of functions is discussed in Art. 173. In working the following exercise the result of Ex. 8, xVrt. 22, may be used. Ex. Find the derivatives of 2 x3 + 7 x2 - 10 a; + 11, x^ - 17 x + 10, _ a;2 + 21 x - 5. 32. The derivative of the product of two functions, say ^{x)F{x). Put y = (x)F(x). Then, on giving x an increment Ax, y + Ay = {x -f Ax)F{x + A:^). .-. Ay = 4>(x 4- Ax)F(x -\- Ax) - (x)F(x). . Ay ^ ct>(x + Ax)F(x + Ax) — (x)F(x) .^. " Ax'' Ax ' ^ ^ On letting A.r = 0, the second member takes the form -• In order to evaluate this form, introduce (f>(x -h Ax)F(x) — {x)F'(x) + F(x),l>Xx). (2) That is : The derivative of the product of two functions is equal to the product of the first by the derivative of the second plus the product of the second by the derivative of the first. * Equally icell, ^(x) F{x + Ax) — {x) F(x + Ax) may be thus introduced. The student should do this as an exercise. 5Q INFINITESIMAL CALCULUS. [Ch. IV. If the functions be denoted by u and v, that is, if y = uv, then (2) may be expressed ^ = u^ + v^. (3) di€ doc doc ^ The derivative of the product of any finite number of functions can be obtained by an extension of (3). For example, if y z= nvw, then, on regarding vw as a single function, -^ = (vw) \- u — (vw) dx dx dx du , vw f- dx ( dv , div\ U[ lU—-{- V ) y dx dxj du , dv . dw /.. = vw h i^u h uv — • (4) dx dx dx Similarly, \i y = uvwz, dy du , dv , dw , dz /e-x -^ = vwz 1- uwz h uvz h uvw — (5) dx dx dx dx dx In general : In order to find the derivative of a product of several functions, multiply the derivative of each function in turn by all the other functions, and add the results. Note. Another way of obtaining (5) is given in Art. 39 (a). The differential of the product of two functions. If y = uv, then, from (3) and the definition in Art. 27, it follows that dy = u'^dx-^v—dx, (6) dx dx But, by Art. 27, —dx = dv, and —dx = du. dx dx Hence, (6) may be written d(uv) = udv + vdu, (7) 33.] DIFFERENTIATION OF FUNCTIONS, 57 Similarly, if y = uvw, it follows from (4) that dy = vwdu -f wudv -j- uvdw. On division by uvw, this takes the form d{uvw) du . dv , dw ,q. = 1 1 {O) uvtv uvw Ex. 1. Write dy in forms (7) and (8), when y = uvwz. Ex. 2. Differentiate (x^ + l){x^ — 2 a: + 7) by the above method ; then expand this product and differentiate, and show that the results are the same. Ex. 3. Treat the following functions as indicated in Ex. 2 : x\x -1)(7^ + 4), (aa;2 -\-bx + c){lx + m). Ex. 4. Write the differentials of the functions in Exs. 2, 3. 33. The derivative of the quotient of two functions, say <|>(a?) -^ Fix), Put y=ii^. Then, on proceeding as in Arts. 29-32, " ^ F{x-^Ax) F(x) ^ <^(a; 4- Ax)F(x) - {x + Ax)F(x) - (x)F(x + Ax) ^. * * Ax F{x)F{x + Aa:)Aic ' ^ On letting Ax = 0, the second member takes the form -• In order to evaluate this form, introduce F(x)(x) - F(x)(x) in the numerator of this member. Then, on combining and arranging terms, (1) becomes ^(^T ^(.. + Ax) - Xx) - (x)F'(x) ,.v dx lF(x)y ^ ^ That is : If one function be divided by another, then the derivative of the fraction thus for^ned is equal to the product of the denomi- nator by the derivative of the numerator minus the product of the numerator by the derivative of the denominator, all divided by the square of the denominator. If the functions be denoted by u and v ; that is, if * then (1) has the form u dy _ doc ddc (2) dic~ v2 The differential of the quotient of two functions. If 2/ = -> t^^^i from (2) and the definition in Art. 27, V V — dx — u — dx /Q\ dx dx yy) ^y = ^2- — But, by Art. 27, —dx = du and ^^dx = dv. Hence (3) may , ... dx dx be written ay^ vdu-udv ^ (4) Note. The derivative (1), or (2), can also be obtained by means of Art. 32. For if y = -, then vi/ = u. Whence v^^ + y~ = —- From this V dx dx dx dy ^ldu_ydv^ which reduces to the form in (2) on substituting - for y. dx V dx V dx V Ex. 1. Find the derivatives and the differentials of a;^ 3;2 + 7 x -U 3a:2-7a; + 2' jc'^ + 8' 2x;^-9x + S Ex. 2. Calculate the differentials of the functions in Ex. 1 when x = 2 and dx = .1. 34.] DIFFERENTIATION OF FUNCTIONS. 69 34. The derivative cf a function of a function. Suppose that y = (ti), and that u = F{x), and that the derivative of y with respect to x is required. (Here <^(m) and F(x) are continuous functions.) The method which naturally comes first to mind, is to substitute F(x) for u in the first equation, thus getting y = (u + Au). .'. Ay = (f>(;u + Au) - (u). Ay _(u H- All) — (u) ' ' Ax~ Ax (u -\- Au) — (u) Aw Au Ax Kow Ax, Au, Ay reach the limit zero together. Hence (Art. 20, Th. c) on letting Ax approach zero, dx du^^^ '-^ dx i,e.^ = ^.^. (1) dor. du dx ^ ^ Note. It should be clearly understood that the first member of (1) does not come, and cannot come, from the second member by cancellation of the dw's. Cancellation is not involved at all. Result (1), which may be expressed more emphatically (Art. 23), is an important one and has frequent applications. It may be thus stated : the derivative of a function icith respect to a variable is equal to the product of the derivative of the function with respect to a second function and the derivative of the second function with respect to the first named variable. (Here all the functions concerned are supposed to be continuous.) 60 INFINITESIMAL CALCULUS. [Ch. IV. From (1) and (2) it results that A (2,) ^ due dx Relations (1) and (2), Note 1, Art. 26, are special applications of (1) [or (2) and (3)]. The showing of this is left as an exercise for the student. Ex. 1. Explain why the dw's in (1) may not be cancelled. Ex. 2. Find ^, given that y = u^ and ii = x^ -i- 1. dx Here ^^ = Sn% —=2x. . •. ^ = 6 u^x = 6 x^x"^ + 1)2. du dx dx Ex. 3. Find ^ when y = S iC^ and u = x^-ox-{-l. . Verify the result dx by the substitution method referred to at the beginning of the article. Ex. 4. Find — when z = 2v'^ -Sv + 1 and v = (jt^-\- 1. A^erify the result by the substitution method. Ex. 5. Show that a function of a function is represented by a curve in space. (See Echols, Calculus, Appendix, Note 2.) 35. The derivative of one variable with respect to another when both are functions of a third variable. Let X = Fit) and y = (t). Now — ^ = -^ -; Now A^, Ax, and Ay reach the limit zero Ax At M together. Hence, Art. 20, Th. d, on letting At approach zero, dy dy^dt^ (1) dx dx dt This result may also be derived as a special case of result (3), Art. 34. This is left as ah exercise for the student. Ex. 1. Find -^ when y z=St'^ - 7 t -\- 1, 2ind x = 2t^ - ISt^ + lit. Here^ = 6«-7, ^ = 6«2_26«+ll. .-. ^ = ^-^^ dt dt . dx 6f-^-26«+ll Ex. 2. Find ^ when x = 2t^ + \lt-\ and j/ = 3 «* - 8 «2 _(- 9. dx Ex. 3. Find — when m = 7x* - 3 and v = 3x2 + 14x - 4. dv 35-37.] DIFFERENTIATION OF FUNCTIONS. 61 36. Differentiation of inverse functions. If y is a function of x, then X is a function of y; the second function is said to be the inverse function of the first. This is expressed by the following notation: If y=f(x), then x=f~^{y). Examples of inverse notation have been met in trigonometry. The equation _J^ . — ^ zzz 1 is always true. Accordingly (Art. Ax Ay 20, Th. c), ^ .^' = 1. ' ^' dx dy Hence, dx doc dy DIFFERENTIATION OF PARTICULAR FUNCTIONS. In the following articles u denotes a continnous fanction of or, and differentiation is made with respect to x. The letters a, n, •••, may denote constants. N.B. It is advisable for the student to try to obtain the derivatives before having recourse to the book for help. # A. Algebraic Functions. 37. Differentiation of ii^, (a) For 71, a positive integer. Put 2/ = ^*" ; i.e. y = uuu ••• to w factors. ... ^ = u^-^ — + u^-^ ^ -f- . . . to n terms (Art. 32) du dx dx „^idu dx In particular, — (x) = 1, and — (x") = na;**~^. dx dx Ex. 1. Give the derivatives with respect to x of w2, 3 w*, 7 i<9, x8, 3 xS 7 a;i2, 9 x^ - 17 x'^ + 10 x + 40. 62 INFINITESIMAL CALCULUS. [Ch. IV. Ex. 2. Find the a;-derivative of (2x + ly^. On denoting this function by !/, and putting ii iov 'Ix + 1, y = u^^. Hence dx dx Now ^' = 2 ; hence ^ = 36 w" = 36 (2 x + 7)i7. dx dx The substitution w for 2 a; + 7 need not be explicitly made. For, if y=(2x+7)i8, then ^ = 18 (2 x + 7)i' ^^ (2x4-7) (Art. 34) dx dx =r 36 (2 X + 7)1". Ex. 3. Differentiate (5 x2 - 10)2-1, (-3 ^4 4. 2)1^ (4 x2 + 5)^(3 x^ - 2 x + 7)-\ (6) i^or n, a negative integer. Let n = — m, and put ?/ = u"^. Then ; — (Art. 33) y = u -.=1. dii ir •t(1)- dx dx u- — u^- ■ = nu =(-»)«<— I Ex. 4. Differentiate with respect to x, w-2, w-7, M-11, x-7, 3x-5, 17a:-l^ (^2 - 3)-*, (3x* + 7)-5, 3x5_7x3 + 2-UV^3' X X'' 9x"* (c) Fo7' n, a rational fraction. Let n = --, in which p and q are integers. Put y=n^; then ?/« = ?/,^. On differentiating, qy'^-^-^ ^^pu^-^— • dy _ 1) u^~^ da _ p u^-'^ du _ p g~^du _ n-idu " dx q y^'^ dx q ^(g-i) dx q dx dx 37.] DIFFERENTIATION OF FUNCTIONS. 63 Ex. 5. Find the ^-derivatives of Vw (i.e. zt2), u~^, M^, Vx, x^, Vx^, VSx'^ — d, i/2x^ + 7 X - 3, V2 a; + 7, (3x-7)~5, 3:^2- 7x2 + A + A _ ..2_. x^ a:^ 7x^ (d) For n, an incommensurable number. In this case it is also true that — (w«) = nu''-'—. This is proved in Art. 39 (6). dx dx ^ Hence, for all constant values of n, £(u") = nun-.2. (1) In particular, if u = x, — (a.-") = wa;'*"^ ax Ex. 6. Find the x-derivatives of M^2, X^^ 5 X^^ (2 X + 5)^5, (3 x2 + 7 X - 4)^3. Ex. 7. Write three functions which have x^ for a derivative. Ex. 8. Do as in Ex. 7 for the functions x5, 1, VS, ^/x% ^x,6x^--- i-. ' x2 ' ' ' x-^ Vx Ex. 9. Show that the general form lohich includes all the functions that have x'^ for the derivative^ is 1- c, in lohich c is an arbitrary constant. 71 + 1 Note 1. The result (1) and the general results, Arts. 29-36, suffice for the differentiation of any algebraic function. Note 2. Case (a) can also be treated as follows : Put y = m", and let x receive an increment Ax ; then u and y receive increments An and Ay respectively. Then y -^ Ay = (u + Aw)'*- On expanding the second member by the binomial theorem, then calculating Ay and then -^, and finally letting Ax approach zero, the result will be obtained. Note 3. It is well to remember that — (x) = 1 and — (Vx) =— — dx dx 2 Vx Ex. 10. Do the operations indicated in Note 2. Ex. 11. Differentiate "^ • Find the value of the derivative when ic = 2. Vx"-^ + 2 Put y^x(x^ + 7)^. (x2 + 2)^ 64 INFINITESIMAL CALCULUS. [Ch. IV. (a;2 + 2)^-^[x(a;2 + 7)^] - a;(x2 + 7)^-- (x^ + 2)^ rj^hen ^ = <^^ ^^ On performing the differentiations indicated in the second member, and reducing, it is found that dy_ 4 a;* + 19 g2 -f. 42 ^^ 3 (a;2 + 7)*(a;2 4- 2)^ Hence, when a; = 2, -^=:1.68, approximately. dx Ex. 12. Differentiate the following functions with respect to x : (2x-5)(x2+llx-3), ax'«+-^, ^-^t£!, ^_z:^, VIT^, -^+5^-7x5, ic" 1 — ^2 a 4- ic x* ^^ + ''' , — =^=, — ^, a/t"^^' (1 + ^^'^)"' (« + &^^)S x-(l - x)«, (a + a:) Va — cc. Ex. 13. Find ^ when x^^^^ + 2a; + 3y = 5. Here y is a?i implicit function dx of X. On differentiation of both members with respect to x, a^2# (y^) + y^^ (X^) + 2 4- 3^ ::=: ; c?x dx dx i.e. 3 X V ^ ^_ 2 xy3 + 2 + 3 ^ = 0. dx dx dy 2(l + xi/3) From this - ^ — — ^-^• dx 3 (1 + x^y'^) dy Ex. 14. (a) Find ^ when x and y are connected by the following rela- dx tions : ?/3 + x3 - 3 ax«/ = ; X* + 2 ax'^y -ay^ = 0; 7 x'^y^ + 2 x?/3 _ 3 x^y + 4 x^ -82/2 = 5; (a + yy^b'^ - y^) + (x + a)2?/2 = ; x2 + 2/2 = a2 ; a2?/2 + fe'^a^^ ^ a2&2. In the last case also obtain -^ directly in terms of x. dx (6) In the ellipse 3 x2 + 4 y2 = 7, find the slope at the points (1, 1), (1, -1), (-1,1), (-1, -1). N.B. The following examples should all be worked by the beginner. They will serve to test and strengthen his grasp of the fundamental prin- ciples of the subject, and will give him exercise in making practical applica- tions of his knowledge. For those who may not succeed in solving them 37.] DIFFERENTIATION OF FUNCTIONS. 65 after a good endeavour, two examples are worked in the note at the end of the set. Ex. 15. A ladder 24 feet long is leaning against a vertical wall. The foot of the ladder is moved away from the wall, along the horizontal surface of the ground and in a direction at right angles to the wall, at a uniform rate of 1 foot per second. Find the rate at which the top of the ladder is descend- ing on the wall when the foot is 12 feet from the wall. Ex. 16. Show that when the top of the ladder is 1 foot from the ground, the top is moving 575 times as fast as when the foot of the ladder is 1 foot from the wall. Ex. 17. Find a curve whose slope at any point (x, y) is 2x. Find a general equation that will include the equations of all such curves. Find the particular curve which passes through the point (1, 2). Ex. 18. A man standing on a wharf is drawing in the painter of a boat at the rate of 4 feet a second. If his hands are 6 feet above the bow of the boat, how fast is the boat moving when it is 8 feet from the wharf ? Ex. 19. A man 6 feet high walks away at the rate of 4 miles an hour from a lamp post 10 feet high. At what rate is the end of his shadow increasing its distance from the post ? At what rate is his shadow lengthening ? Ex. 20. A tangent to the parabola y'^ = 16 x intersects the x-axis at 45°. Find the point of contact. Ex, 21. A ship is 75 miles due east of a second ship. The first sails west at the rate of 9 miles an hour, the second south at the rate of 12 miles an hour. How long will they continue to approach each other ? What is the nearest distance they can get to each other ? Ex. 22. A vessel is anchored in 10 fathoms of water, and the cable passes over a sheave in the bowsprit which is 12 feet above the water. If the cable is hauled in at the rate of a foot a second, how fast is the vessel moving through the water when there are 20 fathoms of cable out ? Ex. 23. Sketch the curves y^ = 4:X and x^ = 4 1/, and find the angles at which they intersect. (If 6 denotes the angle between lines whose slopes are m and n, tan^ =(m — w)-^(l + mn) ; see analytic geometry and plane trigonometry.) Ex. 24, Sketch the curves y^ = Sx and x^ = 8 y, and find the angles at which they intersect. Note. Examples worked. Ex. 15. Let FT be the ladder in one of the positions which it takes during the motion, and let FH be the horizontal projection of FT. Let FH=x, and HT=y. Then x2 -h 2/2 = 576. (1) Fig. 13. QQ INFINITESIMAL CALCULUS. [Ch. IV. Now X and y are varying with the time ; the time-rate — is given, and dv ^^ the time-rate -^ is required. Differentiation of both members of (1) with respect to the time give dt dt whence dy^_xdx^ ^ ,^. dt y dt In this case, — =c 1 foot per second, x= \2 feet, and, accordingly, Cit y = V242 - 12-^ feet = 12 V3 feet. .*. -^ = • 1 foot per second = — .577 feet per second. d^ 12 V3 The negative sign indicates that y decreases as x increases. It should be noticed that the result (2) is general, and that all particular solutions can be derived from it by substituting in it the particular val^ies of x, y, and — • dt Ex. 17. Find a curve whose slope at any point (x, y) is 2x. Find a general equation that will include the equations of all such curves ; and find the particular curve which passes through the point (1, 2). Here ^=2x. dx Hence y = x^ + c, ** (1) in which c denotes any arbitrary constant. This is the general equation of all the curves having the slope 2 x. .-. y = x^ + T is one of the curves ; y = x'^ — b is another. If the point (1, 2) is on one of the curves (1), then 2 = 1+0; whence c = 1, and, accordingly, y =x^ -{- 1 is the particular curve passing through (1, 2). As in Ex. 15 it is easier to find first the general solu- tion of the problem in question, and therefrom to obtain any particular solution that may be required. Figure 12 shows some of these curves. B. Logarithmic and Exponential Functions. 38. Note. To find lim/w=ao f 1 + — J . This limit is required in what follows. ^ '"^ (a) For m, a positive integer. By the binomial theorem, \ mj m 1.2 m^ 1.2.3 w»» ^ ^ This can be put in tlie form l(l_n ifi_lVi_2\ fl+l\"'=l4.1 + _A «l/ + J WV ral ^2) \ ml 2 1 3 1 ^ ^ 38, 39.] DIFFERENTIATION OF FUNCTIONS. 67 On letting m approach infinity, and taking the limits, this becomes * lim„^« (l +!)'"= 1 + 1 +± + ±+ ... \ mj 2 ! 3 ! = 2.718281829.... (3) This constant number is always denoted by the symbol e. (&) The result (3) is true for all infinitely great numbers, positive and negative, integral, fractional, and incommensurable. For the proof of (3) for all kinds of numbers, see Chrystal, Algebra (ed. 1889), Part II., Chap. XXV., § 13, Chap. XX VIII., §§ 1-3 ; McMahon and Snyder, Diff. Gal., Art. 30, and Appendix, Note B ; Gibson, Calculus^ § 48. Note on e. The transcendental number e frequently presents itself in investigations in algebra (for instance, as the base of the natural logarithms, and in the theory of probability), in geometry, and in mechanics. The num- bers e and it are perhaps the two most important numbers in mathematics. They are closely allied, being connected by the very remarkable relation gtn- — — l,t which was discovered by Euler. See references above, and Klein, Famous Problems (referred to in footnote. Art. 8), pages 55-67. 39. Differentiation of \oga u. Put y = log.w, and let x receive an increment ^x ; then u and y consequently receive increments A?* and A^/ respectively. Then y -\- ^y = log„ (ii -\- Au). ,'. A?/ = log„ (u -\- Au) - log« u = log.(^)=log.(l + ^). Ax \ u J Ax 1 u On introducinor Au in the second member. u Alt u Ay _ 1 71 ■, /-. A? A A?fc _ 1 1 ^ A Ai*\ A^ Au Ax a Alt " v u I Ax it *" V u I Ax * This conclusion is properly reached only after a more rigorous investiga- tion than is here attempted. (See Arts. 167-171.) t See Art. 179. 68 INFINITESIMAL CALCULUS. [Ch. IV. From this, on letting Aic approach zero and remembering that Aw and A?/ approach zero with I^x, it follows by Arts. 22, 23, 38, that dx u " dx^ liu = x, then -^ (loga a?) = — . loga e. Ifa = e, then ^(log«^) = ^^. If 7^ = ic, and a = e, then -^(log a?) - — . Note. When e is the base it is usual not to indicate it in writing the logarithm. Ex. 1. Find the derivatives of log„ (3 x^ + 4 x - 7), log (3 a;^ + 4 x - 7), logio (3 ^2 + 4 X — 7). Find the values of these derivatives when x = 3. Ex. 2. Find the values of the derivatives of log Vx^ + 10, logio vx^ + 10, when X = 2. Ex. 3. Differe ntiate the following: log^-^l^, logJi-i-^, log^-t^, log (x 4- y/yi^ + «■'), log (log x), X log X. ^ "^ ^ 1-x i_Vx Ex. 4. Find anti-derivatives of ^^^ '"^ — , '^ ^^ ~ '^ , — . x2 + 3x + 5 x3-7x-l 2x (a) Logarithmic differentiation. If ?/ = uvw, (1) then log y = log ?« + log v + log w. On differentiation, l^ = l*f + l*' + l*^, 2/dic i^da; vdx wdx whence ^= „^„ri*' + l*'+ l*^']. da; \_udx vdx wdxj (2) This result can easily be reduced to the form obtained in Art. 32. The same method can be used in the case of any finite number of factors. This method of obtaining result (2) is called 40.] DIFFERENTIATION OF FUNCTIONS, 69 the method of logarithmic differentiation. It is frequently more expeditious than that given in Arts. 32, 33, especially when several factors are involved. Ex. 5. Find ^ when y = ^i(Ei±I)_. (See Ex. 11, Art. 37.) Here, log y = log x + | log (x^ + 7) - ^ log (x2 + 2). On differentiation, 1^ = 1 + ^ 2x ydx X x2 + 7 3 (x2 + 2) From this, on transposing, combining, and reducing, dy ^ 4 x^ + 19 a;2 4- 42 ^ 3 (x2 + 7)^(x2 + 2)^ Ex. 6. Differentiate, with respect to x, the following functions : («) t^+2)! ; (6) {x-l){x-2) V2x + 5^7x -5, (6) Differentiation of an incommensurable (constant) power of a function. This paragraph is supplementary to Art. 37 (d). Let y = u^^ in which n is any constant, commensurable or incommensurable. Then logy = n log ?«. From this Idy^ ydx ndu^ ' udx' and hence doc _ ny du _ u dx -nu^- ^du^ dx 40. Differentiation of a«. Put y = = a". Then log 2/ = = u log i %. On differentiation, Idy^ ydx :loga . du dx . dy _ "dx :2/loga.^^ dx le. ax ax 70 INFINITESIMAL CALCULUS. [Ch. IV. If u=:iX, then — {a^) = a^ • log a, due If a = e, then -^ (et*) = e**— • doc dx If i< = aj, and a = e, then that is, the derivative of e"" is itself e^. Note 1. On the derivation of results in Arts. 39, 40. The derivative of loga u was deduced by the general and fundamental method, and has been used in finding the derivative of a". The latter derivative can be found, however, by the fundamental method, independently of the deriva- tive of loga u. Moreover, the derivative of loga u can be obtained by means of the derivative of a"-. These various methods of finding the derivative of a'^ and log^ u are all employed by writers on the calculus. For examples see Todhunter, Diff. Gal., Arts. 49, 50; Gibson, Calculus, §65, where both these derivatives are obtained independently of each other ; Williamson, Diff. Cal., Arts. 29, 30; McMahon and Snyder, Diff. Cal, Arts. 30, 31, where the derivative of the logarithmic function is first obtained and the derivative of the exponential function is deduced therefrom ; and Lamb, Calculus, Arts. 35 (Ex. 5), 42, where the derivative of the exponential function is obtained first and the derivative of the logarithmic function is deduced therefrom. (See also Echols, Calculus, Art. 33 and foot-note.) Note 2. On the expansion of e^ in a series see Hall and Knight, Higher Algebra, Art. 220; Chrystal, Algebra, Vol. II., Chap. XXVIII., §§4, 5; and other texts. (This expansion is derived by the calculus in Art. 178, Ex. 7.) Ex. Assuming the expansion for e^, show that the derivative of e^ is itself e^. Note 3. The compound interest law. The function e' "is the only [mathematical] function known to us whose rate of increase is proportional to itself ; but there are a great many phenomena in nature which have this property. Lord Kelvin's way of putting it is that ' they follow the compound interest law.' " (See Hall and Knight, Higher Algebra, Art. 234, and, in particular, Perry, Calculus, Art. 97 and Art. 98, Exs. 4, 2.) Ex. 1. Differentiate, with respect to x, e^"", \(f, 10'^''', e^*. Ex. 2. Find the ^-derivatives of e^, lO'', e''+^ 10^+^ Ex.3. Find the a;-derivatives of the following: C'x^, a*", — ^— , a;e-^, £llL.£l!, £!!. e* — 1 e"" -\- e-* x Ex. 4. Find anti-derivatives of e^^, a;e*^ 2 e^+\ 41, 42.] DIFFERENTIATION OF FUNCTIONS. 71 41. Differentiation of u^, in which u and v are both functions of X, Put y = u\ (1) Then log y = v log u. On differentiation, 1 ^ = ^ ?^ + log « . *!. ?/ da; u dx dx (ia; \it da; cix + '°°"-£" Note 1. It is better not to memorize result (2), but merely to note the fact that the function in (1) is easily treated by the method of logarithmic differentiation. Note 2. The beginner needs to guard against confusing the derivatives of the functions m", a'*, and m«. Ex. 1. Find -^ when y = oC'. clx Here log y = x log x. 1 dy X On differentiation, - -^ = - -i- log x ; y dx X " ' whence -^ = a;* (1 + log x) . Ex. 2. Find the ^-derivatives of (3x + 7r^ (3x+7)^ {(3x + 7)*}2, ^x, x-\ e'', f^^^ log^. \x/ a* (7. Trigonometric Functions. 42. Differentiation of sinu. Put 2/ = sin u. Then y + Ay = sin (w + Aw). .-. Ay = sin (w + Au) — sin w = 2 cos [ 1* H- -— ) sin — — (Trigonometry) 72 INFINITESIMAL CALCULUS. [Ch. IV. .*. -^- = 2 COS 1^ + — — sm — - • — Aa; V 2 / 2 Aa; cos . Aw sm— - 2 ^u Au Ax 2 Let Aa; = ; then also Au = 0, and sm Aw Hm^^^ -^ = lim^^^o cos ( i^ -|- -~] • lim^„^o — r • Hm^^^o t^* ; Ax V 2 / Aw Aa; I.e. -^ = cos u ' 1 * — : dx dx 2 i.e. ^(sin«e) = cos«*^. (1) dx dx ^ In particular, if w = a;, ^(siii£c) = cosa;. (2) dx ^ That is, the rate of change of the sine of an angle with respect to the angle is equal to the cosine of the angle. Note 1. Result (2) can also be obtained by geometry. (Ex. Show this.) See Williamson, Dif. Cal, Art. 28, and other texts. Note 2. Result (2) shows that as the angle x increases from to — the rate of increase of the sine is positive, since cos x is then positive. As x increases from ^ to tt the rate is negative (i.e. the sine decreases), since 2 fj^ cos X is then negative. The rate is negative when x increases from w to '-|— , q 2 and the rate is positive when x increases from — to 2 t. This agrees with what is shown in elementary trigonometry, and it is also apparent on a glance at the curve y = sin x. Note 3. Result (2) also shows that if the angle increases at a uniform rate, the sine increases the faster the nearer the angle is to zero, and increases more slowly as the angle approaches 90°. This is also apparent from an inspection of a table of natural sines, or from a glance at the curve y = sin X. Note 4. The derivative of sin u has been found by the general and fundamental method of differentiation. It is not necessary to use this Ex, 4. Find the x-derivatives of — — -, x sin 2 x. 43.] DIFFERENTIATION OF FUNCTIONS. 73 method in finding the derivatives of the remaining trigonometric and anti- trigonometric f unctions, . for these derivatives can be deduced from that of the sine. Ex. 1. Find the x-derivative^of sin 2 u, sin 3 u, sin i u, sin | w, sin y u. Ex. 2. Find the cc-derivatives of sin2x, sin3aj, sin^a;, sin 3x2, sin2 3x, sin4x^ sin5 4x. Ex. 3. Find the derivatives with respect to t of sin 5 1, sin 1 1^. x2sinfx + -V sin3x \ 4/ Ex. 5. At what angles does the curve y = sinx cross the x-axis ? Ex. 6. At what points on the curve y = sinx is the tangent inclined 30° to the X-axis. Ex. 7. Draw the curve ?/ = sin 2 x. At what angles does it cross the X-axis ? Ex. 8. Draw the curve y = sinx -\- cos x. Where does it cross the x-axis ? At what angles does it cross the x-axis ? Where is it parallel to the x-axis? Ex.9. Find the x-derivatives of the following: sin nx, sinx", sin»»x, sin(l 4- ic2), sin(wx + a), sin(a+6x''), sin=^4x, ^^^, sin(logx), log(sinx), sin(e^) • logx. ^ Ex. 10. (a) Find anti-derivatives of cosx; cos3x, cos(2x-K5), xcos(x2 — 1). (5) Find anti-differentials of cos2xdx, cos(3x — 7)dx, x-cosx^^x. Ex. 11. Calculate dCsinx) when x = 46° and dx = 20', and compare the result with sin 46° 20' - sin 46°. (Radian measure must be used in the computation.) Ex. 12. Compare d{sin x) when x = 20° and dx = 30', with sin 20° 30' - sin 20°. 43. Differentiation of cosu. Put y =z cos u. Then 2/ = sm(| l=^^/2 (/s, s = \ gt^, find — in two ways ; (iv) 2c = tan-i(xy), y = e', find — • 12. Compute the angle at which the following curves intersect, and sketch the curves : (i) x^ - ?/2 _ 9 and xy = 4 ; (ii) x^ -\- y^ = 25 and Ay^ = 9x; (iii) ?/2 = 8 (x + 2) and y'^ + 4(x - I) = ; (iv) 1/ = 3 x2 - 1 and ?/ = 2 x2 + 3 ; (v) x2 + y2 ^ 9 and (x - 4)2 + ?/2 _ 2 y = 15. 13. A point P is moving with uniform speed along a circle of radius a and centre O ; AB is any diameter, and Q is the foot of the perpendicular from P on AB. Show that the speed of Q is variable, that at A and B it is zero, and at O it is equal to the speed of P. (The motion of Q is called simple harmonic motion.) rScGGESTiON : Denote angle AOP by 6, and OQ by x. Then x = « cos^ ; hence ^ = - a sin 6^-1 dt dt J 14. Suppose, in Ex. 13, the radius is 18 inches, and P is making 4 revolu- tions per second : what is the speed of Q when AOP is 15°, 30"^, 45°, 60°, 75°, 90°, 120°, 150°, respectively ? CHAPTER V. SOME GEOMETRICAL, PHYSICAL, AND ANALYTICAL APPLICATIONS. GEOMETRIC DERIVATIVES AND DIFFERENTIALS. N.B. The variation of functions, the sketching of graphs, and the determination of maxima aud minima, which are discussed in Chapter VI L, can be studied before entering upon this chapter. For some reasons it may be preferable to do this. 57. This chapter gives some practical applications of the pre- ceding principles of the calculus. The applications in Arts. 58 and 59 are already familiar or obvious. Rolle's theorem and the theorem of mean value, in Arts. 63, 64, are two of the most im- portant theorems in the calculus. The study of the geometric derivatives and differentials, in Art. 67, is of no immediate press- ing importance, but will be found of particular interest when Chapters XIL and XVI. are taken up. A glance over this article, however, will serve to make clearer and stronger the notions of a derivative and a differential. 58. Slope of a curve at any point : rectangular coordinates. It ]ias been shown in Art. 24 that at a point (.Tj, y^) on a curve whose equation is (1) y =f(x), or (2) {x, y) = 0, the slope of the tangent is -^- [Here -^ denotes the result of substituting (x^, y{) for (X3/J ax^ {x, y) in -^ derived from (1) or (2).] Examples have been given ax in the preceding articles. 59. Lengths of tangent, subtangent, normal, and subnormal, for any point on a curve : rectangular coordinates. Let P be a point (xi, 2/i) on the curve y =f(x) [or, cf>(x, y) = 0]. At P let the tangent PT be drawn ; likewise the normal PN and the ordinate PM. The length of the line J*r, namely, that 84 67-59.] LENGTHS OF TANGENT, ETC. 85 part of the tangent which is intercepted between P and the a>axis, is here termed the length of the tan- gent. The projection of TP on the a>axis, namely TM, is called the suhtangent. The length of the line PN, the part of the normal which is intercepted between P and the a;-axis, is termed the length of the normal. The projection of PN on the ic-axis, namely MN, is called the subnormal. Note 1. The subtangent is measured from the intersection of the tangent with the X-axis to the foot of the ordinate ; the subnormal is measured from the foot of the ordinate to the intersection of the normal with the x-axis. Let angle XTP be denoted by a ; then tan a = In the triangle TPM: MP^y^] TM= y, cot a = y^^^^] TP = y^c^ca dyi -MW' or, TP= -y^MP' + TM" = 2/i \ 1 + f — In the triangle PMN: angle MPN = a ; MN= y^ tan MPN= y^^] dxi PN = 2/1 sec 3fPN =y^yjl-{- /"^Y; fov, PN = yJMP^ + J/JV' = 2/r It being understood that y and -^ denote the ordinate and the dx slope of the tangent at any point on the curve, these results may be written : snbtangent = y^; dy subnormal = y ^5 CTOJ length of tangent = 2/ \'l + ( ^^V ; length of normal = 2/Vl + (|^)* For, since tan PTX-^^^ 86 INFINITESIMAL CALCULUS. [Ch. V. Note 2. These results are true, no matter what the figure may be. The student is advised to draw various figures. Note 3. These results may also be derived by means of analytic geometry. dx\ the equation of the tangent at P is y — yi = -^ (x — x^) ; (1) and the equation of the normal at P is (y — y{) -^ -{-(x Xi) = 0. (2) dxi Hence, from (1), the intercept OT = xi — ?/i — ; dyx and from (2) the intercept ON =^ xi + ?/i^- dxi The subtangent TM=OM- 0T = yi^- dyi The subnormal 3IN=: ON- OMr^ yi^- dx\ Then TP and FN can be found from MP, TM, and MN. EXAMPLES. N.B, Sketch all the curves and draw all the lines involved in the follow- ing examples. 1. In each of the following curves write the equations of the tangent and the normal, and find the lengths of the subnormal, subtangent, tangent, and normal, at any point (xi, ?/i), or at the point more particularly described: (1) Circle x^ + y'^ = 25 where x = - 3 ; (2) parabola ?/2 = 8 x at x = 2 ; (3) ellipse b^x^ + a^y^ = a^b'^ ; (4) sinusoid y = smx; (5) exponential curve y = e'. 2. Where is the curve y(x — 2) (x — 3) =-x — 7 parallel to the x-axis ? 3. What must a^ be in order that the curves 16 x^ + 25 y'^ = 400 and 49 x^ + ct^y'^ = 441 intersect at right angles ? X 4. In the exponential curve y = &e« show that the subtangent is constant and that the subnormal is — • a 5. In the semi-cubical parabola Sy^ = (x -\- ly show that the subnormal varies as the square of the subtangent. 6. In the hypocycloid of four cusps, x^ + y^ = a^ : (1) Write the equa- tion of the tangent at (xi, yi) ; (2) show that the part of the tangent intercepted between the axes is of constant length a ; (3) show ttjat the length of the perpendicular from the origin on the tangent at (x, y) is \/axy ; (4) if p, pi be the lengths of the perpendiculars from the origin to the tangent and normal at any point on the curve, 4p"2 -j- pi^ = a^. 59.] EXAMPLES. 87 7. In the parabola x^ -\- y^ — a^, write the equation of the tangent at any point (xi, j/i), and show that the sum of the intercepts made on the axes by this tangent is constant. Show that this curve touches the axes at (a, 0) and (0, a). 8. In the cycloid x = a{d — sin ^), y = a{l — cos 6): (1) Calculate the lengths of the subnormal, subtangent, normal, and tangent at any point (x, y) ; (2) show that the tangent at any point crosses the y-axis at the angle -; (8) show that the part of the tangent intercepted between the axes is ad cosec — 2 9. In the hyperbola xy = d^ : (1) Show that for any point (a;, y) on the curve the subnormal is — ^ and the subtangent is — a; ; (2) find the X- and ?/-intercepts of the tangent at any point (aci, j/i), and thence deduce a method of drawing the tangent and normal to the curve at any point on it. Show that the product of these intercepts is 4 c'^. 10. In the semi-cubical parabola ay'^ = x^, show that the length of the subtangent for any point (x, y) is f x ; thence deduce a way of drawing the tangent and the normal to the curve at any point on it. Q 11. Show that the parabola ic^ = 4 ?/ intersects the witch y = at an angle tan"! 3 ; i.e. ir 33' 54". ^^ + ^ 12. Find at what angles the parabola y'^ — 2ax cuts the folium of Descartes x^ -^ if = 3 axy. 13. In the curve x"*?/" = «"*+" show: (1) That the subtangent for any point varies as the abscissa of the point ; (2) that the portion of the tangent intercepted between the axes is divided at its point of contact into segments which are to each other in the constant ratio m -.n', (3) thence, deduce a method of drawing the tangent and the normal at any point on the curve. (The curves x'"v/" = «"»+", obtained by giving various values to m and n, are called adiahatic curves. Instances of these curves are given in Exs. 9, 10, and in the parabolas in Exs. 11, 12.) 14. Show that all the curves obtained by giving different values to n in (-) +(-) =2? touch one another at the point (a, 6). Draw the curves in which (a, 6) is (4, 7), n = 1, n = 2. 15. Show that the tangents at the points where the parabola ay = x^ meets the folium of Descartes ji^ + y^ = S axy are parallel to the oj-axis, and that the tangents at the points where the parabola y^ = ax meets the folium are parallel to the ?/-axis. Make figures for the curves in which a = 1 and a = 4. 88 INFINITESIMAL CALCULUS. {Cn.Y. 60. Slope of a curve at any point : polar coordinates. Let CM be a curve whose equation is r=f(e), [or <^(r, ^)=0], and P be any point on it having coordinates y\, 6i, with reference to the pole and the initial line OL. Draw OP; then OP=r^, and angle LOP =6^. Through P and Q (a neigh- bouring point on the curve), draw the chord TPQ, and draw OQ. From P draw PR at right angles to OQ. Let angle POQ = A(9i, and OQ = ri + A?'i ; then PE = ?\ sin A^i, and i?Q = ?'i -|- A?'i — r^ cos A^j. The angle between the radius vector drawn to any point P and the tangent at P is usually denoted by i/^. Since x}/ = lim^g^..o angle BQP, then, using the general coordinates r, 6, instead of r^, ^j. MP tani/. = lim^g^ — = lini sin A^ AS^ r + Ar — r cos A^ On replacing cos A^ by its equal, 1 — 2 sin^ ^ A^, and dividing numerator and denominator by A^, this becomes sin A^ r tan i/f = lim^^^ Ar , . . ^ , sm "I A^ dr That is, tantl/^?-^. (1) The angle between the initial line and the tangent at P is usually denoted by <^. LENGTHS OF TANGENT, ETC, ),61.] It is apparent from Fig. 17 that «|) = t|f + 0. 89 (2) Note. Results (1) and (2) are true for all polar curves, whatever the figure may be. The student is advised to draw various figures. 61. Lengths of the tangent, normal, subtangent, and subnormal, for any point on a curve : polar coordinates. In Fig. 18 is the pole and OL is the initial line. At P any point (?*!, Oi), on the curve CR, whose equation is r=f(0), [or cf>(r, ^) = 0], let the tangent PT and the normal PN be drawn. Produce them to intersect NT, which is drawn through at right angles to the radius vector OP. The length of the line PTis termed the length of the tangent at P; the projection of PT on NT, namely OT, is called the polar subtangent for P; the length of PN is termed the length of the normal at P; the projec- tion of PN on NT, namely ON, is called the polar subnormal for P. Note. In Art. 59 the line used with the tangent and the normal is the X-axis. Here the line so used is not the initial line, but the line drawn through the pole at right angles to the radius vector of the point. In the triangle OPT: 0T= OP tan OPT: Fig. 18. 90 INFINITESIMAL CALCULUS. [Ch. V. i.e. (on removing the subscripts from the letters) polar subtan^ent = r tan if/ = r^—; also, TP=OPseGOPT', i.e. polar tangent length = r sec i/^ = r ^'1 + r2 ( — In the triangle OPN : angle iV^PO = 90-1/^; ON=OPta,nNPO', i.e. polar subnormal = r cot j/^ = — ^ ; also, iV7^= OP sec N^PO; i.e. polar normal length = r cosec i/^ =-y/i'2 + l-~\ . ["or : ^P = VOP' + OiV^' = yj r^ + (— Y-1 Note. In Fig. 18 r increases as 6 increases; accordingly — is positive, dd ^*' and hence the subtangent is positive. Thus when — is positive, the sub- dr tangent is measured to the right from an observer at looking toward P. When r decreases as 6 increases, and thus — is negative, the subtangent is dr measured to the left of the observer looking toward P from O. The student is advised to construct figures for the various cases. EXAMPLES. N.B. In the following examples make figures, putting a = 4, say. Apply the general results found in these examples to particular concrete cases, e.g. a = 6 and e = -, a = 2 and d = — , etc. The angle 6, as used in the equa- tions of the curves, is expressed in radians. 62.] APPLICATIONS INVOLVING RATES. 91 1. In the following curves calculate the lengths of the subnormal, sub- tangent, normal, and tangent, at any point (r, ^) : (1) 7"he spiral of Archimedes r = ad ; (2) the parabolic spiral or lituiis r^ = a^d {i.e. r = ad^) ; (3) the hyperbolic spiral (or the reciprocal spiral) rd = a; (4) the general spiral r = ad'K (The preceding spirals are special cases of this spiral.) 2. From the results in Ex. 1 deduce simple geometrical methods of drawing tangents and normals to the spirals in (1), (2), (3), 3. Do as in Exs. 1, 2, for the logarithmic spiral r = e«^. In this curve each of the lengths specified varies as the radius vector. 4. (a) In the spiral of Archimedes r = ad, show that tan \}/ = 6. Find t// and in degrees when angle TOP (Fig. 17) = 40°, and when TOP = 70°. (6) In the curve r = 4:6, find yp and when r = 2. 5. («) In the logarithmic spiral r = ce«^, show that \}/ is constant. This spiral accordingly crosses the radii vectores at a constant angle, and hence is also called the equiangular spiral, (b) Show that the circle is a special case of the logarithmic spiral, and give the values of ^ and a for this case. 6. In the parabola r = asec2-, show that + 1/' = tt. Make a prac- tical application of this fact to drawing tangents and normals of this curve. 7. In the cardioid r = a(l — cos 6), show that =— , V' = -, sub- 6 6 2 2 tangent = 2 « tan - sin^ -. Apply one of these facts to drawing the tangent and normal at a point on the curve. 62. Applications involving rates. Applications of this kind have already been made in Arts. 26, 37. Rates and differentials have been discussed in Arts. 25-27. It has been seen, Art. 26, Eq. (1), that if y =f(x), then ' dt -^ ^^ Ut dx dt In words, the rate of change of a function of a variable is equal to the product of the derivative of the function with respect to the variable and the rate of change of the variable. The following principles, which are proved in mechanics, will be useful in some of the examples : (a) If a point is moving at a particular moment in such a way that its abscissa x is changing at the rate — , and dt 92 INFINITESIMAL CALCULUS. [Ch, V. its ordinate y is changing at the rate ~^', and. if — denote its rate (It dt of motion along its path at that moment, then 'dsY fdx\^ dtj \dt ' (ij (6) If a point is moving in a certain direction with a velocity V, the component of this velocity in a direction inclined at an angle a to the first direction, is v cos a. For instance, if a point is moving so that its abscissa is increasing at tlie rate 2 feet per second and its ordinate is decreasing at tlie rate 8 feet per second, it is moving at the rate V2'^ + 3'^, i.e. Vl8 feet per second. Again, if a point is moving at the rate of 6 feet per second in a direction inclined 60° to the X-axis, the component of its speed in a direction parallel to the aj-axis is 6 cos 60°, i. e. 3 feet per second, and the component parallel to the y-axis is 6 cos 30°, i.e. 5.196 feet per second. EXAMPLES. N.B. Make figures. 1. If a particle is moving along a parabola y^ =Sx at a miiform speed of 4 feet per second, at what rates are its abscissa and its ordinate respectively increasing as it is passing through the point (cc, y) and x has successively the values 0, 2, 8, 16 ? 2. A particle is moving along a parabola y^ = ix, and, w^hen x = i, its ordinate is increasing at the rate of 10 feet per second : find at what rate its abscissa is then changing, and calculate the speed along the curve at that time. 3. A particle is moving along the hyperbola xy = 25 with a uniform speed 10 feet per second : calculate the rates at which its distances from the axes are changing when it is distant 1 unit and 10 units respectively from the y-axis. 4. A vertical wheel of radius 3 feet is making 25 revolutions per second about an axis through its centre : calculate the vertical and the horizontal components of the velocity, (1) of a point 20° above the level of the axis; (2) of a point 65° above the level of the axis. 5. A point is moving along a cubical parabola y = x^ : find (1) at what points the ordinate is increasing 12 times as fast as the abscissa ; (2) at what points the abscissa is increasing 12 times as fast as the ordinate ; (3) how many times as fast as the abscissa is the ordinate growing when oj = 10 ? 63.] BOLLE'S THEOREM. 93 63. RoUe's Theorem. Note 1, Progressive and regressive derivative. In Art. 22 the derivative of /(x) v\ras defined as ,^^^jj^±^^,fM, (1) The process of evaluating (1) is equivalent to the geometrical process of revolving the chord PQ of the curve y =f{x) about P until Q coincides with P, and thus PQ becomes the tangent PT. If in this curve a chord PB be drawn, and BP be revolved about P until B coincides with P, then BP will finally take the position PT. The slope of the tangent ob- Fig. 19. tained by thus revolving BP is evidently /(x)-/(x-Ax) . /rx-Ax)-/(x) "iiiAa;^ ^ , i.e. iim^j.^ ITAx ^ '^ It is customary to call (1) the progressive derivative, and (2) the regressive derivative. In general these derivatives are equal ; that is, in general the tangent on the representative curve is the same, whether the secant which is revolved until it assumes a tangential position be drawn forward or backward from the point under consideration. In some cases, however, these derivatives are not equal ; such a case is represented at P on Fig. 21 c, where the two revolving secants give two different tangents. In such a case the derivative is discontinuous at P, for its value suddenly changes from the slope of TP to the slope of LP. Theorem. If a function f(x) and its derivative f'(x) are continu- ous for all values of x between a and b, and if f{a) =f(b), then f'(x) = for at lea^t one value of x between a and b. Following is a geometrical proof* and representation of this theorem. Let the curve MN (Figs. 20 a, b, c) represent the function f(x). At M and ^ let x=a and x = b respectively. Since the ordi- nates AM and BN are equal, it is evident that there must be at least one point between M and N where the function ceases to increase and begins to decrease, or ceases to decrease and begins * An analytical discussion will be found in the collateral reading suggested in Note 2, Art. 64. 94 IN FINITE SIM A L CALC UL US. [Ch. V. to increase. There may be several such points, as in Fig. 20 c. But at such a point, for instance F, or F^, or F^, or 1\, where x=Xi say, /'(a^i) = 0. (If f(x) is constant, then f'(x) = at every point.) Fig. 20 a. Fig. 20 6. Fig. 20 c. A special case of this theorem is that in which /(a) = and f(b) = 0. The student may construct the figure for himself by merely moving OX to the position MN. For an application to the theory of equations and for the corresponding algebraic statement of the theorem, see Art. 66 B. Note 2. The necessity of the condition relating to continuity is evident from Figs. 21 a, 6, c, d. Fig. 21 a. Fig. 21 6. Fig. 21 c. Fig. 21 d. For a value of x between x = a and x = h : in Fig. 21 a, f(x) is infinite ; in Fig. 21 6, /(x) is discontinuous ; in Fig. 21 c, f{x) is discontinuous ; in Fig. 21 d, f{x) is infinite. 64. Theorem of mean value. If a function f(x) and. its derivative f'(x) are continuous for all values of x from x = a to x = b, then there is at least one value of x, say x^, between a and b such that J{b)-f(a)_ 64.] THEOREM OF MEAN VALUE. 95 Following is a geometrical proof* and explanation of this theorem. Let the curve MN (Fig. 22 a or Fig. 22 b) represent the func- tion f(x). Draw the ordinates^lP and BQ at A and B, where Fig. 22 a. Fig. 22 6. x = a 2iud x=b respectively. Draw PQ and draw PR parallel to OX. Then AP=f(a), BQ=f(b). Hence and BQ=f(b)-f(a), ^ PR b-a Now the chord PQ and the tangent ST drawn at some point V (or Vi and F2) between P and Q evidently must be parallel. At Flet x=Xi, Xi thus being between a and b ; then tan RPQ=f'{x^. Hence b-a (1) Since x^ is between a and b, Xi = a + 6(b — a), in which denotes some number between and 1 (i.e. < ^ < 1). Accord- ingly, theorem (1) may be expressed /(6)=/(a) + (6-a)/Ta + e(6-a)]. (2) If b — a = h, then b = a -\- h, and (2) is written f(a -^h) = f{a) + hf (a -f- efe) . (3) * For an analytical deduction of the theorem of mean value from Rolle' theorem, see Art. 176. 96 INFINITESIMAL CALCULUS. [Ch. V. Eesult (3) has important applications. It is very useful for finding an approximate value of /(a + h) when f{x), a, and li, are given. A closer approximation to the value of f{a -f h) can be found by Taylor's formula, Art. 176. Note 1. The necessity for the condition relating to continuity can be made evident by figures similar to Figs. 21 a, i, c, d. Note 2. References for collateral reading on Rollers theorem and the theorem of mean value: McMahon and Snyder, Diff. Cal., Arts. 59, 66; Lamb, Calculus, Arts. 48, 49, 56 ; Gibson, Calculus, §§ 72, 73 ; Harnack, Calculus, Art. 22 ; Echols, Calculus, Chap. V. The last mentioned text has a particularly full and valuable account of these theorems. EXAMPLES. 1. Find by relation (3) an approximate value of sin 32° 20' taking a=32° : (1) putting 6 = 0, (2) putting 6 = 1; and compare the calculated results with that given in the tables. 2. If /(x) = 2x^ — x + 5, find what 6 must be in order that relation (3) be satisfied : (1) when a = 3 and h = 1 ; (2) when a = 10 and h = 2. 3. Show that for any quadratic function, say /(a?) = Ix^ + mx + n, /(a 4- h) will be obtained by putting ^ = i in relation (3) . What geometrical property of the parabola corresponds to this ? (Deduce the value of 6.) 4. If f(x) = x^, find what 6 must be in order that relation (3) be satisfied when a = S and h = 1. What problem in connection with the cubical parabola y = x^ is the correlative of this ? 65. Small errors and corrections : relative error. If 2/=/(^), (1) then by Art. 27, dy = f'(x) • dx, (2) in which dx is an assigned change in x. It has been seen (Note 3, Art. 27) that d.y is approximately the change in y due to dx. An important practical application may be made of this principle. For it follows that if dx be regarded as a small error in the assigned or measured value of x, then dy is an approximate value of the consequent error in y. The ratio ^ ov -^ • dx (3) is, approximately, the relative error or the proportional error^ i.e. the ratio of the error in the value to the value itself. 65.] SMALL ERBORS. 97 The approximate values of the correction and relative error may also be deduced from the theorem of mean value. For, if y =f{x), and Ax be an error in x, then /(x + Ax) - /(x) is the error in y, i.e. the correction that must be applied to y. Now by (3) Art. 64, on putting a = x and h = Ax, /(x + Ax) - /(x) = /' (x + . Ax) . Ax. Hence, on denoting the error in y by Ay, Ay =/'(x) • Ax approximately. Ay f'(x^ From this the relative error is, approximately, — ^ =''—^-^- Ax. (4) EXAMPLES. 1. The side a of a square is measured, but there is a possible error Aa : find approximately the error in the calculated value of the area. Let A denote the area. Then A = a- ; whence A^ = 2 a • Aa approximately. 2. If the measured length of the side is 100 inches and this be correct to within a tenth of an inch, find an approximate value of the possible error in the computed area, and an approximate value of the relative error. In this case, approximately, Aa = 2 x 100 x .1 = 20 square inches. The relative error is, approximately, -^ or — ; that is, 20 square inches in 10,000 square inches, or 1 square inch in 500 square inches. 3. A cylinder has a height h and a radius r inches ; there is a possible error Ar inches in r : find by the calculus an approximate value of the possible error in the computed volume. If A = 10 inches and the radius is 8 ± .05 inches calculate approximately the possible error in the computed volume and the relative error made on taking r = 8 inches. 4. Find approximately the error made in the volume of a sphere by making an error Ar in the radius r. The radius of a sphere is said to be 20 inches : give approximate values of the errors made in the computed surface and volume, if there be an error of .1 inch in the length assigned to the radius. Also calculate the relative errors in the radius, the surface, and the volume, and compare these relative errors. 5. Two sides of a triangle are 20 inches and 35 inches. Their included angle is measured and found to be 48° 30'. It is discovered later that there is an error of 20' in this measurement. Find, by the calculus, approximately the error in the computed value of the area of the triangle. Compare the relative errors in the angle and in the area. 6. The exact values of the errors in the computed values in Exs. 1-4 happen to be easily found. Calculate these exact values, and compare with the approximate values already obtained. 98 INFINITESIMAL CALCULUS. [Ch. V. 7. (1) Two sides, a, &, of a triangle are measured, and also the included angle C : show that the approximate amount of the error in the computed length of the third side c due to a small error A C made in measuring C, is ah sin C Va2 + 62 - 2 a& AC. (2) Calculate the approximate error in the computed value of the third side in Ex. 5. 66. Applications to algebra. A. If f(x) is a rational integral function* of x, and {x — ay is a factor of f(x), then (x — ay~^ is a factor of f (x). Let f{x) = (x — ay (x) -\- (x - ay ' (x) = {x — a)'-i [rcl>{x) + (x-- a)cf>'{x)^. It follows from this theorem that if f{x) is a rational integral function of x, and a is an r-tuple (or rfold) root of the equation f(x) =0, then a is an (r— Vytuple root of the equation f'(x) = 0. This theorem may be employed in finding the multiple roots of an equation. Ex. 1. Solve aj3 — 2 aj2 — 15 X 4- 36 = by trying for equal roots. The derived equation is 3 a;^ — 4 aj — 15 = 0, i.e. (3 a; + 5) (x - 3) = 0. Trial will show that (x - 3)2 is a factor of x^ - 2 x2 - 15 x + 36, and the first equation is (x — 3)2 (x + 4) = 0. The roots are thus : 3, 3, — 4. Note. The multiple roots of /(x) = loill be revealed on finding the highest common factor of f{x) and /'(x). Ex. 2. Solve the following "equations : (1) 3 x3 + 4 x2 - X - 2 = (2) 4 x3 + 16 x2 + 21 X + 9 = (3) X* - 11 x3 + 44 x2 - 76 X + 48 = (4) 8 x* 4- 4 x3 - 62 x2 - 61 X - 15 = (5) x6 + x4 - 13 x» - x2 + 48 X - 36 = 0. Ex. 3. Find the condition that x" — px^ + r = may have equal roots. * A rational integral function of x is a function in which x has only posi- tive integral exponents and does not appear in the denominator of a fraction ; e.g. x2 — ^ X + 2, ax» + &x"-i + ••• + mx + p, if n is a positive integer. 66, 67.] GEOMETRIC DERIVATIVES, 99 B. An important application of Boilers Theorem may be made to the theory of equations. According to the theorem, geometrically, the slope of a curve y=f(x) is zero once at least, between the Fig. 23 a. Fig. 23 &. points where the curve crosses the a^axis. Hence, at least one real root of the equation f'(oc) =0 lies between any two real roots of the equation fioc) = 0. (In the theory of equations this is called Rolle's Theorem.*) Note. According to this principle r real roots of an equation f(x) = have at least (r — 1) roots oi f'{x) = between them. Now, if the r roots coalesce and thus make an r-tuple root, the (r — 1) roots must also coalesce and thus make an (r — l)-tuple root oif'(x) = 0. (Compare A above.) Ex. Verify Rolle's Theorem in each of the following equations /(x) = ; also sketch the curve y = f(x) : (1) x^ + x-Q = (2) x3 + 2 x2 - 5 a; - 6 = 0. 67. Geometric derivatives and differentials. (a) Derivative and diflferential of an area : rectangular coordinates. Let PQ be an arc of the curve y = f{x) . Take any point on P§, F(x, y) say, and take T(x + Ax, y + Ay) . Construct the rec- tangles VN and TM as shown in Fig. 24. Draw the ordinate BP^ and let the area of BPVM be denoted by A\ then the area of JfFTiV may be denoted by A^. Now, Fig. 24. rectangle VN doc ^ \dx From (2), (3), and Art. 27, and ds = ^ll+{^^y.dv. (5) Ex. 8. Show that for a given dx and the actual derivative - - at M, the second member of (4) gives the length of the intercept of the tangent, namely, MT. Show that for a given dx, and using dy to denote the exact corresponding change in the ordinate, the second members in (4) and (5) give the length of the chord of the arc, namely, the line MN. Note. It is shown in Art. 137 how to find the length of the arc MN corresponding to an increment dx in x. The smaller dx is, the more nearly will MT, arc MN, and chord 3IN, become equal to one another. See Ex. 6, Art. 19. Ex. 9. (1) Calculate the x-derivative and the ?/-derivative of the arc of the parabola ?/2 = 4 ax. (2) Find the x-derivative of the hypocycloid x^ + y^ — a^. Ex. 10. In the cubical parabola y = x^ calculate the differential of the arc at the point (2, 8) when : (1) dx = .2 \ (2) dy = A. Show these differen- tials in a figure. (The actual increments of the arcs can be computed by Art. 137.) 67.] GEOMETRIC DERIVATIVES. 103 (d) Derivative and diflFerential of tlie lengtli of a curve: polar coordinates. Let FQ be an arc of the curve /(r, 0) = 0, On PQ take any point V(r, 6), and take ir(r + Ar, e + Ad). Denote the length of PF by s ; then the length of VW may be denoted by As. Draw the chord VW. Now, as in (c), chord VW lim Ae=o' a rc VW Ad li.e. ^) V ddl imA^^o Ad (1) Fig. 27. About describe a circular arc VM intersecting OW in M, and draw VT at right angles to W. Then angle VO W = Ad, and M W = Ar. .-. TW = OW - OT = r -h Ar - r cos Ad, and VT = r sin A^. /. chord VW = V( VT)'^ + ( rir)^ = V(r sin A^)^ + [r(l - cos A^) + Ary. chord Ffr A^ A^ y ■ |_ ^A^ chord VW r 2 — . sill k Ad -\ , ^ A^J Arn2^ (2) A^ V-^CD' since, if A^ - 0, 5H^ - 1, ^J^^^ = 1, and sin i A^ = 0. ' Ai/ ^A^ ^ Hence, by (1), ds de V-^f)' (3) A^ On multiplying each member of (2) by — , and then letting A^, and con- Ar sequently Ar, approach zero, it will be found that dr ' \drj From (3), (4), and definition Art. 27, and (4) (5) (6) Ex. 11. Find the derivative of the arc of the spiral of Archimedes r = ad : (1) with respect to the angle ; (2) with respect to the radius vector. Ex, 12. Calculate the differential of the arc of the Archimedean spiral r = 2d when d = 2 radians and dd = 1°. Make a figure, (The actual incre- ment of the arc can be computed by Art. 138.) 104 INFINITESIMA L CALCUL tlS. [Ch. V. (e) Derivative and diiferential of the volume of a surface of revolu- tion. Let FQ be an arc of the curve y =f{x). On PQ take any point L{x^ y), and take the point M(x -\- Ax, y + Ay). On letting V denote the volume obtained by revolving arc PL about OX, the volume obtained by revolving arc LM may be denoted by A F. Through L and M draw the lines shown in the figure. The volume obtained by revolving arc LM about the x-axis is greater than the volume obtained by revolving LG^ and is less than the volume obtained by revolving KM. That is, ir.lJL^ .LG< AF<7r . Vm'^ . KM ; Try^ ' Ax <: AV <: w • (if + AyY • Ax. Ax (1) On letting Ax approach zero, the three numbers in (1) become 7r?/2, -— , 7r!/2, respectively. dx Hence, dV doc = iry'K (2) From (2) and Art. 27 dV= ir?/2 . dx. If PQ had been revolved about the y-axis, then dV dy = irx^f and dV = irx^'dy. (3) (4) Note. According to (3), for a given differential dx the corresponding differential of the volume is the volume of a cylinder of radius y and height dx. The smaller dx is, the more nearly does this volume become equal to the actual increment, due to dx, in the volume of the solid of revolution. Ex. 13. Derive the results in (4). Ex. 14. (1) Find the x-derivative of the volume generated by the revolu- tion of the parabola y = x^ about the x-axis. (2) Find the ^/-derivative of the volume generated by the revolution of this curve about the y-axis. Ex. 15. (1) Calculate the differential of the volume in Ex. 14 (1), taking dx = .l at the point where x = 2. (2) Thus also in Ex. 14 (2), taking dy = .2 at the point where x = 4. (The actual increment in the volume of the solid due to changes dx and dy can be computed by Art. 112.) 67.] GEOMETRIC DERIVATIVES. 105 Y O (/) Derivative and differential of the area of a surface of revolu- tion. Let FQ be an arc of the curve y =f{x). On PQ take any point, say Z(x, y), and take the point M{x + Ax, y + Ay). Let S denote the area of the surface generated by revolving arc PL about OX ; then the area generated by revolving arc LM about OX may be de- noted by A.S. There is evidently a straight line whose length is equal to the length of the arc LM. Through L and 31 draw the lines LM' and ML' parallel to OX and equal in length to the arc LM. {LM may be supposed to be a piece of wire, LM' the same piece of wire when it is stretched out in a horizontal straight line from Z, and ML' the same piece of wire when it is stretched out in a horizontal line from M. ) The surface obtained by revolving the arc LM about OX is greater than the surface obtained by revolving LM' ; for, with the exception of the point Z, each point on LM has a greater ordinate than the corre- sponding point in the line LM^ and consequently a greater radius of swing. Similarly, the surface obtained by revolving LM \& less than the surface obtained by revolving ML'. That is, 2 Try . LM' < surface generated by L3K.2 ir (y + Ay) • L'M; i.e. 2 Try • arc Z3/< A/S" < 2 tt (y + Ay) • arc LM. (1) (2) Fig. 29. ...2.y^^^lD(Dy)], ••• ; these are respectively denoted by Dy,Dhj,I)'y,^.;D-y. Sometimes there is an indication of the variable with respect to which differentiation is performed ; thus D^y,DJy,D^%^.',Dj^y. Note. Here n is not an exponent ; D"?/ does not mean {Dyy\ {E.g. see Ex., p. 107.) i)"y is called the derivative of the nth order. (c) Instead of the symbols shown in (a) and (6), for the succes- sive derivatives of y, the following are sometimes used, namely, (d) If the function be denoted by '{x), "{x), "(cc), respectively. Note 1. In this book notation (a) is most frequently used. The symbol Z> is very convenient, and is especially useful in certain investigations. See Byerly's Diff. CaL, Lamb's Calculus, Gibson's Calculus (in particular § 67). For an exposition of simple elementary properties of the symbol D also see Murray's Differential Equations (edition 1898), Note K, page 208. Note 2. Instead of the accent notation in (c), the 'dot '-age notation, y, y\ 'y\ ••• is sometimes used, particularly in physics and mechanics. Note 3. Geometrical meaning of ^^« It has been seen in Arts. 25, 26, du d dx^ that -^, i.e. — (?/), denotes the rate of change of y, the ordinate of the curve, dx dx compared with the rate of change of the abscissa x ; this may be simply denoted as the a;-rate of change of the ordinate. Similarly — ^, i.e. — I — ). dx^ dx \dxj^ is the rate of change of the slope -^ of a curve compared with the rate of dx change of the abscissa x, or, simply, the aerate of change of the slope. 68.] SUCCESSIVE DIFFERENTIATION. 109 On a straight line, for instance, the slope is constant, and hence the x rate of change of the slope is zero. This is also apparent analytically. For, if y = mx + c is the equation of the line, then -^ = m, and hence -^ = 0. dx dx^ Note 4. Physical meaning of ^^» In Art. 25 it has been seen that ds d if s denotes a varying distance along a straight line, — , i.e. — (s), denotes Clt civ the rate of change of this distance, i.e. a velocity. Similarly —, i.e. — ( — V dt^ dt \dt ) denotes the rate of change of this velocity. Rate of change of velocity is called acceleration. For instance, if a train is going at the rate of 30 miles an hour, and half an hour later is going at the rate of 40 miles an hour, its velocity has increased by ' 10 miles an hour' in half an hour, i.e. as usually expressed, its acceleration is 10 miles per hour per half an hour. Again, it is known that if s is the distance through which a body falls from rest in t seconds, s = | gt^. Hence —= gt; accordingly, -^= g. That is, the dt dt" acceleration of a falling body is '^ feet per second' per second. (See text-books on Kinematics, Dynamics, and Mechanics, for a discussion on acceleration.) EXAMPLES. 1. Find the second a:-derivative of: (i) xtan-ix; (ii) 'ix^ — 9x + o _ Vx ; (iii) tan x + sec x ; (iv) x'. X 2. Find D/y, when : (i) y= {x? + a^) tan"! - ; (ii) y = log (sin x). a 3. Find — ^, when : (i) y = sin-i x \ (ii) y = dx^ 1 + x^ 5. Find ^, when a!?/2 + 3 a: + 5 y = 0. By Art. 56, ^ = - ^^ + ^ ■ (1) 4. Find D^y, when : (i) y = a;* log x ; (ii) y = e^ cos x cPy dx^ dx 2xy + b On differentiation, ^^ = ^ ^ ^• dx2 (2 xy + 5)2 On substituting the value of -^, and reducing, dx d^ ^ 2(y2 + 3) (3 xy'^ + lOy-Sx) .gv dx^ (2 xy + 5)3 ^ ^ 6. (i) In the ellipse a^y^ + hH^ = a^lP' calculate B^y. (ii) Given y2 + y = x2, find B^. 110 INFINITESIMA£ CALCULUS. [Ch. VI. 7. Show that the point (J, ^) is on the curve log (x + y) = x — y. Show that at this point -^ = 0, and ^'-^- = i. (k: dx'^ 8. What are the values of -^ and --^^ : (i) at the point (2, 1) on the ellipse 7 ic^ + 10 ?/2 = g8 ; (ii) at the point (3, 5) on the parabola y2 — 4x -\- 13. 9. Calculate ^ for the cycloid in Art. 43, Ex. 6. Compute it when 3' 10. Verify the following : (i) if y = as[nx-{- b cos x, ^ -{- y = ; (ii) if u = (sin-i xy, (1 - a:2) i^ - a: ^^ = 2 ; (iii) \i y = a cos (log x) + h sin (log X) , a;2 ^^ + x ^-'^ + ?/ = 0. 11. Show tlmt if 11 y(21og2/ + l)|^. 2/2 log 2/, and 2/=/(x), l?^=(21og?/ + 3)f^V 12. Find ^ in the following cases : y — A:X^ ^1x-Z, y = 4iK3 + 4 x + 2, y = 4 x3 + 5 a; — 4, 2/ = 4 x^ + ex + ^•. 13. Given that _ -^ = 3 x + 2, find the most general expression for ^ OjX -^ ; then find the most general expression for y. ax 14. A curve passes through the point (2, 3) and Its slope there is 1 ; at any point on this curve -^ = 2 x : find its equation and sketch the curve. d'y? 15. At any point on a certain curve -^ = 8 ; the curve passes through dx^ the origin and touches the line y = x there ; find its equation and sketch the curve. 16. (1) In the case of simple harmonic motion, Ex. 13 (p. 83), show that the speed of ^ is changing at a rate which varies as the distance of ^ from the centre of the circle. (2) What is the acceleration of the velocity of the boat in Ex. 18, Art. 37 ? 17. In Ex. 14 (p. 83), calculate the rate at which Q_ is olianging its speed when ^ is : (i) at an extremity of the diameter ; (ii) 12 inches from the centre ; (iii) 6 inches from the centre ; (iv) at the centre. 18. A body moving vertically has an acceleration or a retardation of g feet per second due to gravitation, g being a number whose approximate value is 32.2 : find the most general expression for the distance of tli« moving point from a fixed point in its line of motion, after t seconds. Explain the physical meaning of the constants that are introduced in the course of integration. 69.] SUCCESSIVE DIFFERENTIATION. Ill 19. A body is projected vertically upwards with an initial velocity of 500 feet per second : liiid how long it will continue to rise, and what height it will reach, if the resistance of the air be not taken into account. 20. A rifle ball is fired through a three-inch plank, the resistance of which causes an unknown constant retardation of its velocity. Its velocity on entering the plank is 1000 feet a second, and on leaving the plank is 500 feet a second. How long does it take the ball to traverse the plank ? (Byerlj'^, Problems in Differential Calculus.) 69. The /7th derivative of some particular functions. In a few cases the nt\\ derivative of a function can be found. This is done by differentiating the function a few times in succession, and thereby being led to see a connection between the successive derivatives. EXAMPLES. 1. Let y = cc'". Then Dy = rx''-^ ; j)2y = r(^r - l)a;'-2 ; j)3y ^ r(r - l)(r - 2)x'-3. From this it is evident that D»y = r(r - 1) (r - 2) ••• (r -n + l)x'-«. Show that i>»a;»» = w ! 2. Find the nth derivative of the following functions : (a) e*; (6) a^ ; (c) e«*; (d) aK 3. Show that the wth derivative of sinx is sin (x + — V Suggestion: coss = sin f 2r + ^ V 4. Find the wth derivatives of (a) cos x ; (&) sin ax ; (c) cos ax. 5. Find the ?ith derivatives of log x, log (x — 2)2. 6. Find the wth derivatives of -, ^ ^ ^ X I -\-x 3-x (6 + ra)'" 7. Find the nth derivatives of — — , -^ 1 - a:^ 1 - a [Suggestion : Take the partial fractions.] 112 INFINITESIMAL CALCULUS. [Ch. VI. 70. Successive differentials. In Art. 27 it has been shown that if y=f(x), (1) then dy=f'(x)dx. (2) The differential in (2) is, in general, also a function of x ; and its differ- ential may be required. In obtaining successive differentials it is usual to give a constant differential increment dx to x. Then (Art. 27), on taking the differentials of the members in (2), didy) = d lff(x)dx^ = [f"(x)dx'] dx. (3) On taking the differentials of the members of (3), d{didy)} = d{[f"(x)dx']dx}=f>"(x)dx • dx - dx. (4) It is customary to denote results (o) and (4) thus : d^y=f"ix)dx-^ and d^y =f"'(x)dxK In this notation the nth diflPereatial is written in which /"(x) denotes the nth derivative of f(x), and dx» denotes (dx)». 71. The successive derivatives of / with respect to x when both are functions of a third variable, / say. By Art. 26, Note 1, ^ = ~ dt ' ^ — A. f^^ — ^ f^l\ . ^ C^y ^^® principle in Art. 34, dx^~~ dx\dx) ~ dt\dxj ' dx Eq. (2)] dx ^ _dy ^ ^ dx d^y dy d^x dt ' df dt ' df dt dt ' df dt ' df fdxV dx fdx^ ^ dt The method of obtaining the higher derivatives is similar. Thus, d^^d_/^\^df^\ ^ d£^d/(Py\^dx^ dx^ dx\dxy dt\dx'J ' dx dt\dxy ' dt' 70-72. J SUCCESSIVE DIFFEEENTIATION. 113 And, in general, dx" ~ dx [dx""-^) ~ dt \dx''-^) ' dx~ dt \dx''-y dx ~dt Ex. 1. See Ex. 9, Art. 68. Ex. 2. Find D^y and D^y when x = a — b cos 6 and y = ad + bsin 0. 72. Leibnitz's theorem. This theorem gives a formula for the nth deriva- tive of the product of two variables. Suppose that u and v are functions of X, and put y = uv. Then, on performing successive differentiations, Dy = u • Dv -\- V ' Du ; D^y = u . DH + 2 Du- Dv -\-v • DHi ; Dhf = u-DH^^ Du . D^v + 3 DHi • Dv -^ v - Dhi ; D^y = u ' Dh -{- 4 Du ■ DH -\- 6 Dhi ■ D'v + 4 Dhi - Dv + v - D^u. Thus far the numerical coefficients in these derivatives are the same as the numerical coefficients in the expansions (m + v), (m + i?)^, (u + v)^, and (u + vy respectively, and the orders of the derivatives of u and v are the same as the exponents of u and v in those binomial expansions. Now sup- pose that these laws (for the numerical coefficients and the orders) hold in the case of the ?ith derivative of uv ; that is, suppose that D'^{nv) = u ' D^v + nDu • D'^-^v + ^^^ ~ ^^ DH • D^-^v + ... n(n-l).-(n-r + 2) j^,_^^ _ ^„_,^i^ 1.2...(r-l) + ^'^ - ^>-<'' -''^^^ Dru. D-ry +... + !;. Pr^u. (1) 1 . 2 ••• r Then these laws for the coefficients and the orders hold in the case of the (w + l)th derivative of uv. For differentiation of both members of (1) gives Z)"+i(wv) = u . D^+iv + (w + V)Du • D^v + i'^'^^)''^ Dhi • D'^-^v + ••• 1 • ^ + {n + l)n(n-l).--(n-r + 2) ^,^ ^ j^n-r+i^ + ... + v D-+^u. l-2...(r-l)r Hence, if formula (1) is true for the nth derivative of uv, a similar formula holds for the (w + l)th derivative. But, as shown above, formula (1) is true for the first, second, third, and fourth derivatives of uv ; hence it is true for the fifth, and for each succeeding derivative. 114 INFINITESIMAL CALCULUS. [Ch. VI. Ex. 1. Find 2)/'«/ when ij = aj^e^. = e*[x2 + 2 nx + ?i(« - 1)]. Ex. 2. Calculate the fourth :*;-derivative of x^ sin x by Leibnitz's theorem. Ex. 3. Find Dx'^y when : (i) y = xt"" \ (ii) y — a'e-^. Note. Reference for collateral reading on successive differentiation. Echols, Calculus, Chap. IV., especially Art. 56. 73. Application of differentiation to elimination. It is shown in algebra that one quantity can be eliminated between two inde- pendent equations, two quantities between three equations, and that n quantities can be eliminated between n-|-l independent equations. The process of differentiation can be applied for the elimination of arbitrary constants from a relation involving vari- ables and the constants. For by differentiation a sufficient num- ber of equations can be obtained between which and the original equation the constants can be eliminated. EXAMPLES. 1. Given that y = Acosx -{- B sin x, (1) in which A and B are arbitrary constants, eliminate A and B. In order to render possible the elimination of these two constants, two more equations are required. These equations can be obtained by differen- tiation. Thus, ^ = -Aiimx + Bcosx, (2) dx -r-^ = — Acosx — B sin x. i^) dx^ On eliminating A and B between (1), (2), (3), there is obtained the relation Note 1. Equation (4) is called a differential equation, as it involves a derivative. It is the differential equation corresponding to, or expressing the same relation as, the "integral" equation (1). The process of deducing the integral equations (or solutions, as they are then called) of differential equations is discussed, but for a very few cases only, in Chapter XXI. 73.] SUCCESSIVE DIFFERENTIATION. 115 2. Eliminate the arbitraiy constants m and b from the equation y = mx + b. Ans. -^ = 0. In this case the given equation represents all lines, m and b being arbi- trary. Accordingly the resulting equation is the differential equation of all lines. For the geometrical point of view see Art. 68, Note 3. 3. Eliminate the arbitrary constants a and b from each of the follovr- ing equations : (1) ?/ = wx^ -f 6. (2) y = ax^ + bx. (3) (j/ — 6)2 = 4 ax. (4) y-^ - 2 a?/ + x2 = a^. (5) y^ = b{aP- - x^). 4. Find the differential equations which have the following equations for solutions, Ci and Cg being arbitrary constants : (l)y = ci. {2)y = cix. (3) ?/ = cix + C2. (4) y = Cie* + 026"=^. (5) y = cie'^ + C2e~™'=. (6) ?/ = Ci cos mx + C2 sin wx. ( 7 ) ?/ = ri cos (mx + C2) . 5. Obtain the differential equations of all circles of radius r: (1) which have their centres on the x-axis ; (2) which have their centres on the y-axis ; (3) which have their centres anywhere in the xy-plane. 6. Show that the elimination of n arbitrary constants Ci, C2, •••, c„, from an equation f(x, ?/, Ci, C2, •••, c„) = gives rise to a differential equation involving the nth derivative of y with respect to x. Note 2. For geometrical explanations relating to differential equations the student is referred to Murray, Differential Equations., Chap. I., wliich may easily be read now. The reading will widen his mathematical outlook at this CHAPTER VII. FURTHER ANALYTICAL AND GEOMETRICAL APPLICATIONS. VARIATION OF FUNCTIONS. SKETCHING OF GRAPHS. MAXIMA AND MINIMA. POINTS OF INFLEXION. N.B. This chapter may be studied before Chap. V. is entered upon. 74. Increasing and decreasing functions. When x changes con- tinuously from one value to another, any continuous function of x, say (x), in general also changes. The function may either be increasing or decreasing, or alternately increasing and decreas- ing. By means of the calculus it is easy to find out how the function behaves when x passes through any value on its way from — 00 to -f- GO. Let Ax be a positive increment of x, and A(^x) be the corre- sponding increment of cf)(x). If (f>(x) continually increases when x is changing from a; to x -|- Ax, then A(x) is positive ; and accord- ingly, ^^ ^ is positive. Moreover, this is positive for all positive iJkX values of Ax, however small ; hence lim^^^o is positive, i.e. Ax <^'(x) is positive. In a similar way it can be shown that if cf>{.v) continually decreases when x is changing from x to x -f Ax, then <^'(x) is negative. These facts may be stated thus : <^'(x) is positive when <^(x) is increasing, and | j <^'(x) is negative when <^(x) is decreasing ; and conversely. J These facts will also be apparent on an inspection of the accom- panying graphs. Let <^(x) be graphically represented by the curve ABCDE, whose equation is ./ . 116 74, 75.] MAXIMUM AND MINIMUM. 117 At any point on this curve, dx ^\x). By Art. 24, the slope of the curve represents the derivative of the function. Now at A, D, and E, the slope is negative, and the ordinate y (the function) is evidently decreasing as x is passing in the positive direction through the values of x at A, D, and E. On the other hand, at B, C, and F, the slope is positive, and the ordinate y is evidently increasing as x is passing in the positive \ r1 Y » ti i i^ h I. S r ^ L \ r Li 3fi N, — n — » Y J K A « k ' X Fig. 30 6. Fig. 30 c. Fig. 30 a. direction through the values of x at B, C, and F. In Fig. 30 6 when X is increasing from 07>i to OM^, the ordinate ?/ is decreas- ing from LiL to M^M and the slope at points on LM is negative ; when X is increasing from OMi to OiYi, the ordinate is increasing from M^M to N^N and the slope at points on MN is positive. Fig. 30 c also exemplifies principles marked A on page 116. 75. Maximum and minimum values of a function. Critical points on the graph, and critical values of the variable. The values of the function at points such as Pj, P^, P3, M, and K (Art. 74), where the function stops increasing and begins to decrease, or vice versa, may be called turning values of the function. When a function ceases to increase and begins to decrease, as at P2, P4, and K, it is said to have a maximum value ; when a function ceases to decrease and begins to increase, as at Pj, P3, and M, it is said to have a minimum value. Therefore, at a point (on the graph) where the function has a maximum value the slope changes from positive to negative; at a point where the function has a minimum value the slope changes from negative to positive. (Examine Fig. 30.) 118 INFINITESIMA L CALCUL US. [Ch. VII. Accordingly, at each of these points the slope (i.e. the derivative of the function) is generally (see j^ote 3) either zero or infinitely great. It should be observed that, although the derivative of a function may be either zero or infinitely great for values of the variable for which the function has a maximum or a minimum value, yet the converse is not always the case. The function may not have a maximum or minimum value when its derivative is zero or infinity. o Fig. 31 b. This is exemplified by the functions whose graphs are given in Figs. 31 a, b. Thus at P the slope is zero and the function is increasing on each side of P; at Q the slope is zero and the function is decreasing on each side of Q; at i? the slope is infi- nitely great, and. the function is increasing on each side of R; at S the slope is infinitely great and the function is decreasing on each side of S. Accordingly, a point where the slope of the graph of a function is zero or infinitely great is, for the purpose of this chapter, called a critical point. Such a jjoint must be further criticised, or ex- amined, in order to determine whether the ordinate has either a maximum or a minimum value there. In other words, that value of the variable for which the derivative of a function is zero or infinitely great is called a critical value ; further examination is necessary in order to determine whether the function is a maxi- mum or a minimum for that value of the variable. Note 1. The points ^, P, i?, 8 (Figs. 31 a, 6), are examples of what are called pom^s of inflexion (see Art. 78). Note 2. By saying that a function {x) has a minimum value, for x = a say, it is not meant that 0(a) is the least possible value the function can have. It is meant that the value of the function for a; = a is less than the values of the function for values of x which are on opposite sides of a, 76.] MAXIMUM AND MINIMUM. 119 and as close as one pleases to a ; i.e. h being taken as small as one pleases, (a) < 0(« - h) and («) < ^(a + h). (See Pi in Fig. 30 a.) Likewise, if ^(x) is a maximum for x = b, this means merely that 0(6) > 0(6 — h) and 0(&) >0(& + ^)» in which ^ is as small as one pleases. (See P2 in Fig. 30 a.) EXAMPLES. 1. Examine sin x for critical values of the variable. Here 0(x) = sinx. The graph of this function is on page 409. In order to find the critical 0'(a;) = cos a; = 0. points solve the equation Accordingly, the critical values of x are — , — , — , • •• . &j, 222' 2. Examine (x — iy(x + 3) for critical values of the variable. Here 0(x) = (x - l)2(x + 3). The solution of 0' (x) = (x - 1) (3 x + 5) = 0, gives the critical values of x, viz. 1, — f. 3. Examine (x —1)^ + 2 for critical values of the variable. Here 0(x) = (x - 1)^ + 2. On solving 0'(x) = 3(x - 1)^ = 0, the critical value of x is obtained, viz. x = 1. 2 4. Examine (x - 2) '^ + 3 for critical values of x. 2 Here Fi Y n c. J / /^ i lO -\-* X On solving 0(x) = (x-2)3 + 3. ^'(x) = ^ = c 3(x-2)3 the critical value x == 2 is obtained. 6. Examine (x — 2)^ + 3 for critical values of x. Here and 0(x) = (x-2)3+3, 0'(x) = = Fig. .si d. Y H Fig. 31 e. Y ) ^2- X 3(x - 2)^ gives the critical value x = 2. p^^ g^ , Note 3. A function may have a maximum or minimum value when its derivative changes abruptly ; see Art. 164, Note 3, and Fig. 21 (c). 120 INFINITESIMAL CALCULUS. [Ch. VIL 76. Inspection of the critical values of the variable (or critical points of the graph) for maximum or minimum values of the function. Let the function be (.^')• The equation of its graph is 2/ = <^(^)? and the slope is -^ or ^'{x). The solutions of the equations (f>'(x) = and '{x) = cc, give the critical values of the variable. Suppose that ABODE (Fig. 30 a) is the graph, and that the critical values are x = a and x = b. There are three ways of testing whether the critical values of the variable will give maxi- mum or minimum values of the function, viz. : (a) By examining the function itself at, and on each side of, the critical value ; (b) By examining the first derivative on each side of the critical value ; (c) By examining the second derivative (see Art. 68) at the critical value. Note 1. It follows from the definition of maximiini and minimum values, and Note 2, Art. 75, that if (a), v^0(a), 0^(a), •••, are maximum (or minimum) values of 0(cc) + w, c4>(x), V(a — h) and cf>(a-i-h), in which h is as small as one pleases, are both compared with cf>{a). (This is the obvious and natural method .of testing the critical values.) If (ci) is greater than both {a — li) and {a-\-h), then {a) is less than both {a — h) and (a-\-h), then {a) is a minimum ; if (a — h) and <^(a4-/i), then (a) is neither a maximum nor a minimum. Ex. 1. In Ex. 1, Art. 75, examine the function at the critical value J of x. .. ) < sin -, and sin ( - + /i J < sin - . Accordingly, a^ = ^ gives a maximum value of sin x. 76.] MAXIMUM AND MINIMUM. 121 Ex. 2. (a) In Ex. 2, Art. 75, examine the function at the critical value x = \. Here 0(1) = O, 0(1), and 0(1 -\- h)> 0(1). Thus 0(1) is a minimum value of 0(x). (6) Inspect this function at the critical value a; = — |. Ex. 3. In Ex. 3, Art. 75, examine the function at the critical value x = 1. Here 0(1) = 2, (t>(\ - h) = - h^ ■\-2, and 0(1 + /i) = ^^ _|_ 2. Accordingly, 0(1 — ^) < 0(1) < 0(1 + /i), and thus 0(1) is not a turning value of the function. Ex. 4. Examine the functions in Exs. 4, 5, Art. 75, at the critical values of x. (6) Examination of the first derivative of tlie function. When a function is increasing, its derivative is positive and the slope of its graph is positive ; when a function is decreasing, its derivative is negative and the slope of its graph is negative (Art. 74). Hence, h being taken as small as one pleases, if '(a — h) is positive and (f>'{a + Ji) is negative, then (a) is a maximum value of (x). On the other hand, if (f>'(a — h) is negative and '{a -f- h) is positive, then <^(.r) is decreasing when x is approaching a, and (x) is increasing when x is leaving a, and accordingly ^(a) is a mini- mum value of ^(x). Note 2. Test (b) is generally easier to apply than test (a). For test (a) the functions 0(a — h) and 0(a + h) must be computed ; for test (&) merely the algebraic signs of 0'(a — h) and 0'(«t + h) are required, Ex.5, (a) InEx.l, Art. 75. 0'f- — ^ J is positive and 0'f^ + /ij is nega- tive. Accordingly, ( - ] , i-e. sin ^ or 1, is a maximum value of sin x. (b) Apply this test at the other critical values in Ex. 1, Art. 75. Ex. 6. (a) In Ex. 2, Art. 75^ 0'(1 — ^) is negative and 0'(1 -}- h) is posi- tive. Accordingly 0(1), i.e. 0, is a minimum value of (x — iy{x -f 3). (b) Apply this test at the other critical value in Ex. 2, Art. 75. Ex. 7. In Ex. 3, Art. 75, 0'(1 — h) is positive and 0'(1 -|- h) is positive. Accordingly, 0(1), or 2, is neither a maximum nor a minimum. Ex. 8. Apply test (6) at the critical values of the functions in Exs. 4, 5, Art. 75. (c) Examination of tlie second 'derivative of tlie function. It has been seen that the sign of the derivative of a function <^ (a*) changes from 'positive to negative when the function is passing through a 122 INFINITESIMAL CALCULUS. [Ch. VII. maximum value. If the derivative <^'(.t) passes from a positive value to zero, and then becomes negative, the derivative is contin- ually decreasing, and hence its derivative, namely (f>"{x), must be negative for the critical value of x. On the other hand, when the function passes through a minimum value, the derivative changes sign from negative to positive. If then the derivative cf>'(;x) passes through zero, it is continually increasing, and hence its derivative, namely <^"(.^), must be positive for the critical value of X. Therefore, if <^'(a) is zero and "(a) is negative, (f>(a) is a maximum value of ^{x); if (ji'(a) is zero and cj)"(a) is positive, (a) is a minimum value of (^(x). Note 3. When the second derivative can be obtained readily, test (c) is the easiest of the three tests to apply. Note 4. Sometimes (p"(a) is also zero. A procedure to be adopted in this case is discussed in Art, 181. One of the other tests, however, may be used. Note 5. Historical. Kepler (1571-1630), the great astronomer, "was the first to observe that the increment of a variable — the ordinate of a curve, for example — is evanescent for values infinitely near a maximum or minimum value of the variable." Pierre de Fermat (1601-1665), a celebrated French mathematician, in 1629 found tlie values of the variable that make an expre.s- sion a maximum or a minimum by a method which was practically the calculus method (Art. 75). Note 6. Many problems in maxima and minima may be solved by ele- mentary algebra and trigonometry. For the algebraic treatment see (among other works) Chrystal. Ali/fhra, Part II., Chap. XXIV.; Williamson, Diff. Cal., Arts. 133-137 ; Gibson, Calculus, § 76 ; Lamb, Calculus, Art. 52. Note 7. Maxima and ininiina of functions of two or more inde- pendent variables. For discussions of this topic see McMahon and Snyder, Diff. Cal., Chap. X., pages 183-197; Lamb, Calculus, pages 135, 596-598; Gibson, Calculus, §§ 159, 160 ; Echols, Calculus, Chap. XXX. ; and the treatises of Todhunter and Williamson. EXAMPLES. 9. (a) In Ex. 1, Art. 75, ' du is called the total differential of u. Relation (2) is also written du = d^u -\- dyU -f dgU. Definitions (1) and (2) may be extended to functions of any number of variables. 138 INFINITESIMAL CALCULUS. [Ch.VIII. 6. Given u = x^ -\- y"^ -\- 2 z, find du when x = 2, ?/ = 3, 5; = 4, dx = .1, dy — .4, dz —— .2). Also find the actual change in u. 7. The numbers r«, x, ?/, and 2; being as in Ex. 6, dx — .01, dy = .04, and dz = — .03, calculate the difference between du and the actual change in u. 8. Find du when w = x^". 83. Approximate value of small errors. A practical application of relations (1) and (2)^ Art. 82, may be made to the calculation of approximate values of small errors. The ideas set forth in the first part of Art. 65 may be applied to any number of variables. If u = f(x,ij,z,---), and dx, dy, dz, •••, be regarded as errors in the assigned or measured values of x, y, z, •••, then du = —- dx -\- —- dy -\ — - dz-\ ox dy ' dz is, approximately, the value of the consequent error in the com- puted value of u. Illustrations can be obtained by adapting Exs. 2, 3, 5, 6, 7, Art. 82. In applying the calculus to the com- putation of approximate values of errors it is usual to denote the errors (or differences) in u,x,y, •••, by Au, /^x, A?/^ '•• rather than by du, dx, dy, •••. Other notations are also used ; e.g. Su, 8x, Sy, •••. EXAMPLES. 1. In the cylinder in Ex. 3, Art. 65, give an approximate value of the error in the computed volume due to errors A^ in the height and Ar in the radius. Let F denote the volume. Then V= irr^h. .-. AF = 2 irrh ■ Ar + tv^ • Ah. The relative error is ^ = ?-^ + M . V r h 2. Do as in Ex. 1 for a few concrete cases, and compare the above approximate value of the error with the actual error. What is the difference between the actual error in the volume in Ex. 1 and its approximate value obtained by the method above ? 3. In the triangle in Ex. 7, Art. 65, let Aa, A6, AC, be small errors made in the measurement of a, b, C: show that the approximate relative error for the computed area ^ is — + — -|- cot (7 • AC. a b 83, 84.] IMPLICIT FUNCTIONS, 139 Find, by the calculus, an approximate value of A J, given that a = 20 inches, 6 = 35 inches, C = 48° 30', Aa = .2 inch, Ab = .1 inch, AC = 20'. How can the actual error in the computed area be obtained ? 4. Show that for the area A of an ellipse when small errors are made in the semiaxes a and 6, approximately — = — H A a b In this general case, and in several concrete cases, compare the approxi- mate error in the computed area with the actual error. 5. In the case described in Ex. 3 show that if Ac denote the consequent error in the computed value of c, then, approximately, Ac = cos B ■ Aa -\- cos A • Ab + a sin B - AC. N.B. For remarks and examples on this topic see Lamb, Calculus, pp. 138-142, Gibson, Calculus, pp. 258-260. 84. Differentiation of implicit functions, two variables. This topic has been taken up in one way in Art. 56. Let the relation connecting two variables x and y be in the implicit form /(^, y) = c, ■ (1) in which c denotes any constant, including zero. Let u denote the function f(x, y) ; then (1) may be written u = c. (2) Since u remains constant when x and y change, — = ; i.e. (Art. 81, Eq. 3, and Note 1) dudx du dy _ ^ /o\ dx dt dy dt dy du Q^ ^^^^ (^)' I = - 1' "^"^^^ ^^''- ''' ^^- ^^)]' ^ = - i' ^^) dt dy dy Ex. 1. Express relation (4) in words. Note. It should not be forgotten that the relation between the function and the variable should be expressed in form (1) before (4) is applied. Ex. 2. Do Exs. 13, 14, Art. 37, and exercises, Art. 56, by the method of this article. Compare the methods of Arts. 37, 56, and 84. 140 INFINITESIMAL CALCULUS. [Ch. Vlll. 85. Order of partial differentiations commutative. The theorem stated in Art. 80, Note 3, and illustrated by the exercises there, will now be proved. (See Gibson, Calculus^ § 93, especially pages 221, 222.) Suppose that u=f{x,y) (1) and that u and its first and second partial derivatives are continuous over a finite range of the variables ; then, as will now be shown, dy doc doc dy Let X receive an increment h^ and y remain constant ; then by the theorem of mean value (Art. 64, Eq. 3) f{x + /I, y) - f(x, y) = h ^-f(x + dih, y), in which < ^i < 1. (3) Now let y receive an increment k, and x remain constant ; then on applying the theorem of mean value to the second member of (3), [fix + h,y^ k) -fix, ?/ + ^•)] - [fix + h, y) -fix, ?/)] = k^- [h4-f(x -f dih, y + ^2^0l = hk-^ l-f /(x + M, y + e2k)\ (4) . dyl dx J dyldx J in which < ^2 < 1- On giving the increments in the reverse order, [/(x -^h,y-\-k)- fix -\-h,y)^- [fix, y + A:) - f(x, y)l = hk-^ rf fix + dsh, y + d,k)\ (5) dxldy J in which 63 and 0^ lie between and 1. Hence, on equating the values of fix + fe, y + k) -fix + h, y) -fix, y + k) -\-fix, y) hk derived from equations (4) and (5), V- -f /(^ + ^1^' y + ^2^0 = ^ ^ fix + Bsh, y + ^4^), (6) dy dx dx dy provided that x -\- h and y -\- k are within the range referred to above. Now let h and k approach zero. Then, since the first and second partial deriva- tives are continuous, equation (6) becomes d^fjx, y) ^ d^fix, y) . .^ d^u ^ d^u dy dx dxdy ' * dy dx dxdy The proof of the commutation theorem can be extended to derivatives of higher orders and to functions of more than two variables. 85,80.] CONDITION FOR TOTAL DIFFERENTIAL. 141 Note 1. Keferences for collateral reading on partial differentiation, total differentials, and the commutative property; Todhunter, Diff. Cat.; McMahon and Snyder, Diff. Cal, Arts. 91-102 ; Lamb, Calculus (ed. 1897), Arts. 45, 46, 60-62, 209, 210; Edwards, Treatise on Diff. Cal, Chap. VI. Especially full and clear treatment of differentiation of functions of more than one variable, with various illustrations and geometrical interpretation, is given in Gibson's Calculus, Chap. XI. (see in particular pp. 204-225 and Ex. 1, p. 222), and in Echols' Calculus, Chaps. XXV. -XXIX., pp. 282-313. Note 2. Applications of partial differentiation : («) To the determi- nation of the maximum and minimum values of functions of tw^o or more variables (see references in Art. 76, Note 7); (6) To the study of surfaces, and curves in space (see references, Art. 166, Note 2). 86. Condition that an expression of the form Pdx -\- Qdy be a total differential. This article may be regarded- as supplementary to Art. 82. Suppose that /^{x, y) and f^ix, y) are two arbitrarily chosen functions : does a function exist which has J\(x, y) for its partial a>derivative and /2(a;, y) for its partial .y-deri vative ? A little thought leads to the conclusion that in general such a function does not exist. The condition that must be satisfied in order that there may be such a function will now be found. Suppose that there is such a function, and let it be denoted by u. Then, according to the hypothesis, ^=/i(a?, 2/) and -~=f.lx,y). (1) By Art. 85, _^=-^. ' (2) oy ox axoy Hence, from (1) and (2), Kesult (3) is directly applicable to the differential expression Pdx + Qdy on substituting P for f^(x, y) and Q for f2(x, y). Otherwise : If Pdx -f- Qdy is a total differential, du say, then f' = P and fi = Q. (4) OX dy Hence, from (2) and (4), ^=^Q. (5) 142 INFINITESIMAL CALCULUS. [Ch.VIII. When condition (5) is satisfied, Pdx -f- Qdy is also called an exact differential. Note 1. That this condition is not only necessary (as shown above), but also sufficient^ is shown in works on Differential Equations. {E.g. see Professor McMahon's proof in Murray, Diff. Eqs., Note E.) Note 2. For the condition that an expression of the form Pdx + Qdy + Bdz (see Art, 82, Eq. 2) be a total differential, see works on Differential Equations; e.g. Murray, Diff. Eqs., Art. 102 and Art. 103, Note. Ex. 1. Apply test (5) in the following cases : (a) w = 3 x^ + 2 ?/2 ; y (6) u — tan - ; (c) x dy -\- y dx ; (d) xdy — y dx. Ex. 2. Illustrate by examples the phrase, '■'• in general such a function does not exist," which occurs in this article. 87. Euler's theorem on homogeneous functions. Let u be a homogeneous function of x and y of degree n ; i.e. let u = Ax"" H h BxPy'' + Cxy -\ \- My", in which j9-[-g = 7'-f-s= ••• = n. Then ^ = uAx""-^ -\ f- pBx^-y + rCx'-^y' -\ ; ax -^ = ••• 4- qBx^r^ 4- sCxY~^ H h nMy'^'K dy From this, on multiplication and simplification, doc dy ^ ^ This result can be extended to homogeneous functions of any number of variables ; thus, ^|^+2/^+«^ + ... = nt«. (2) due dy dz Result (2) is called Euler's theorem.* (See Williamson, Diff. Cal., Arts. 102-104, 123 ; McMahon and Snyder, Diff. Cal, Art. 100 ; Gibson, Calculus, page 412.) * From Leonhard Euler (1707-1783), an eminent Swiss mathematician, who worked at Berlin and St. Petersburg. He greatly advanced the subjects of algebra, trigonometry, and the calculus. 87,88.] SUCCESSIVE TOTAL DERIVATIVES. 143 Ex. 1. Prove theorem (2) when u is a homogeneous function of oj, y, z. Ex. 2. Illustrate (1) and (2) by examples in which n is an integer. Ex. 3. Verify Euler's thieorem in the following cases : (i) w = (x^ + ?/3) (x* + y^) ; (ii) u = {x^ + vh ^ iS^ + vh \ (iii) if = sin-i ^^ ~ ^^ (Here ?i = 0.) . Vx + Vy Ex. 4. Verify Euler's theorem when u—f\^-\\ and apply to tan ^, sin~i ^, log^, in particular. (In this /-function n = 0.) X X 88. Successive total derivatives. An example will be given in order to show the procedure. If w=/(x, 2/), (1) then (Art. 81, Eq. 3) du^dndx dudy^ ,2) dt dx dt dy dt ^ ^ On differentiation with respect to t again, dt^ ~ dt\dtj~ dt\dxl ' dt dx dt^ dt\dy ) ' dt dy dt^' ^' ^ Now, in general, ^ and ^ are functions of x and y ; hence, on applying dx dy the principle enunciated in Art. 81, djdu\d_(di(\ ,dx_^d_(du\ , dy dt\dx) dx\dx) ' dt dy\dx) dt' dt\dy) dx\dy) dt dy\dy) dt On substituting these values in (3), using the notation of Art. 80, and remembering that o ^ — ^ ^^ , Equation (3) becomes dxdy dydx ^^d^fdxy ^_^dxdy d^(dyy du^, /4>) dt^ dx^dtl dxdy dt dt dyAdt) dx dt^ dy dt^ N.B. Questions and exercises suitable for practice and review on the subject-matter of this chapter will be found at page 386 CHAPTER IX. CHANGE OF VARIABLE. N.B. If it is thought desirable, the study of this chapter may be post- poned until some of the following chapters are read. 89. Change of variable. It is sometimes advisable to change either, or both, of the variables in a derivative. If the relation between the old and the new variables is known, the given derivative can be expressed in terms of derivatives involving the new variable, or variables. Arts. 91-93 are concerned with showing how this may be done. In Art. 90 an expression for the given derivative is found when the dependent and independent variables are interchanged ; in Art. 91, when the dependent variable is changed ; in Art. 92, when the independent variable is changed ; and in Art. 93, when both the dependent and the independent variables are expressed in terms of a single new variable. In Note 1, Art. 93, an example is worked in which the dependent and the independent variables are both expressed in terms of two new variables. N.B. Principle (2) of Art. 34 is repeatedly employed in Arts. 90-93. 90. Interchange of the dependent and independent variables. Let y be the dependent and x the independent variable. This article shows how to express the successive derivatives of y with respect to X in terms of the derivatives of x with respect to y. From the fact that ^ . ^ = 1, and Art. 20 (c), it follows that Ay~ ■1, ai dy dx 1 ~d.x dy 144 89-92.] CHANGE OF VARIABLE. Again, d'y_ d (dy\_ d fdy\ dy ... o^x dor' - dx[dx) - dy[dx) ' dx ^^'^^ ^> d r 1 1 d^x dx dy^ '^dy dx dy. ' dy~ fdxV \dyj Ex. Express the third aj-derivative of y in terms of ^-derivatives of x. 145 91. Change of the dependent variable. Let the dependent and independent variables be denoted by y and x respectively. It is required to express the successive derivatives of y with respect to x, in terms of the derivatives of z with respect to x when y = F(z). dy^dyd^^^dz^^ dx dz dx ^ ^ dx' ^^d^/dy dx^ dx \dx = lh)|] ^Ux' dx dx ^^ ^^d^ dx dz^ ^^-' dx Ex. Given that y = F(z) , show that dx^ ^ ^ dx^ ^ ^ dx^dx ^ ^ \dx) 92. Change of the independent variable. Let the dependent and independent variables be denoted by y and x respectively. It is required to express the successive derivatives of y with respect to ic, in terms of the derivatives of y with respect to z when x=f(z). ad hence, dy _dy dz __ 1 dy dx~ dz dx~ f'(z) dz !=/(.), and hence, 1 = ^ 146 INFINITESIMAL CAL CULUS. [Ch. IX. d^y _ d^ (dy\ _ d fdy\ dz _d_/' 1 dy\ dz_ dx^ dx \dxj dz \dxj dx dz \f'(z) dzj dx 1 d'y f"(z) dy f{z)y\z) dz' \_r(z)J dz] d^y Ex. Find -r-| when x—f{z). 93. Dependent and independent variables both expressed in terms of a single variable. Let yz=z(fi(t) and x=f(t). Then dy_dy ^ dx /oxn _ <^'(0 d^y _ d fdy\ _ d fdy \ dt _d \^(^(dy\ J dt \dxj dx- dx \dxj dt \dxj dx dt _ f{t)^"{t)-xt)r(t) [.f'it)7 1 fit) (Compare Art. 71.) EXAMPLES. 1. In the above case find — |- 2. Given that x = a(d — sin 6) and y = a(l — cos 6), calculate b + {l^yf^%- (See EX. 9, An. 68.) 3. Given that x = a cos 6 and y = asm d, calculate the same function as in Ex. 2. What curve is denoted by these equations ? 4. Given that x = acosd and y = bsm d, calculate the same function as in Ex. 2. What curve is denoted by these equations ? Note 1. Both dependent and independent variables expressed in terms of two new variables. Following is an example of this. Ex. Given the transformation from rectangular to polar coordinates, viz. aj=:rcos^, ?/=:rsin^, (1) express ^ and — ^ in terms of r, ^, and the derivatives of r with respect to 6. 93.] CHANGE OF VARIABLE. 147 From (1), — = cose~-rsme, ^ = sin^— + r cos^. ^ ^ dd dd dd dd •••i=(i4:'---.-^(^)) dr sm ^ ~ 4- r cos d dr cosd — — r sin 8 dd d^_ d fdy\_d fdy\ dd _ \dd J dff^ d^-^W-d^Uy'^~/cos^|-rsin^V' Note 2. For more complex cases of change of the variables in a derivative, see other text-books. Note 3. References for collateral reading. Williamson, Diff. Cal., Chap. XXII. ; McMahon and Snyder, Diff. CaL, Chap. XI.; Edwards, Treatise on Diff. Cal, Chap. XIX. ; Gibson, Calculus, §§ 98, 99. EXAMPLES. N.B. In working these examples it is much better not to use the results or formulas derived in Arts. 90-93, but to employ the method by which these results have been obtained. 1. Change the independent variable from x to y in : (i) — ^ +2y[-^j =0; (ii) s(m-^-iy^-fyl^y=o. ^""^ ^^""^ ^ ^ [djc^l dxdx^ dxAdxj 2. In ^ = 1 + ^^^ + y^ f ^ Y, change the dependent variable from y to dx- 1 + y^ \dx I 0, given that y = tan z. Z^ Change the independent variable under the following conditions : niWf^+x^ + u = 0,y = \ogx;iu)(l-x'^)'^-xf-- + y=0,x=:cost-, ^^-^ dx^ dx dx^ dx {m)(l-x^)^-x^ = 0,x = cost; {iv) x'^fy + 2x^ -{-^y = 0, xz = 1; dx^ dx vv A ^ ^ *- ^ J, dx^ dx x^ ^Sx^ + y = \ogx,x = e'. dx 4. Find ^ and ^ when : (i)x = a (cos t -^tsint), y = a (sin t-tcost); dx dx^ (ii) X = cot t, y = sin-^ t. 5. If r.^-?f^Y+^ = 0, andx = ye^showthat2/^ + ^ = 0. dx^ y\dx) rfx ' ^ ' dy^ dy CHAPTER X. INTEGRATION. N.B. If thought desirable, Art. 97 may be studied before Arts. 95, 96. (Remarks relating to the order of study are in the preface.) 94. Integration and integral defined. Notation. In Chapter III. a fundamental process of the calculus, namely, differentiation^ was explained. In this chapter two other fundamental processes of the calculus, each called integration, are discussed. The process of differentiation is used for finding derivatives and differentials of functions ; that is, for obtaining from a function, say F{x), its derivative F'(x), and its differential F'{x)dx. On the other hand the process of integration is used : (a) For finding the limit of the sum of an infinite number of infinitesimals which are in the differential formf(x)dx (see Art. 96) ; (h) For finding functions whose derivatives or differentials are given ; that is, for finding anti-derivatives and anti-differentials (see Arts. 27 a, 97). Briefly, integration may be either (a) a process of summation, or (h) a jy^'ocess which is the inverse of differentiation, and which, accordingly, may be called ariti-differentiation. Integration, as a process of summation, was invented before differentiation. It arose out of the endeavor to calculate plane areas bounded by curves. An area was (supposed to be) divided into infinitesimal strips, and the limit of the sum of these was found. The result was the ivhole (area) ; accordingly it received the name integral, and the process of finding it was called integration. In many practical applications integration is used for purposes of sum- mation. In many other practical applications it is not a sum but an anti-differential that is required. It will be seen in Art. 96 that a knowledge of anti-differentiation is exceedingly useful in the process of summation. Exercises on anti-differentiation have appeared in preceding articles. 148 94. j INTEGRATION. 149 Note. The part of the calculus which deals with differentiation and its im- mediate applications is usually called The Differential Calculus^ and the part of the calculus which deals with integration is called The Integral Calculus. With Leibnitz (1646-1716), the differential calculus originated in the problem of constructing the tangent at any point of a curve whose equation is given. This problem and its inverse, namely, the problem of determining a curve when the slope of its tangent at any point is known, and also the problem of determining the areas of curves, are discussed by Leibnitz in manuscripts written in 1673 and subsequent years. He first published the principles of the calculus, using the notation still employed, in the periodical. Acta Eruditorum, at Leipzig in 1684, in a paper entitled Nova methodus pro maximis et minimis, itemque tangentibus, quae nee fractas nee irrationales quantitates moratur, et singulare pro illis calculi genus. Isaac Newton (1642-1727) was led to the invention of the same calculus by the study of problems in mechanics and in the areas of curves. He gives some description of his method in his correspondence from 1669 to 1672. His treatise, Jlethodus fluxionum et serierum infinitarum, cum ejusdem applicatione ad curvarum geometriam, was written in 1671, but was not published until 1736. The principles of his calculus were first published in 1687 in his Principia {Fhilosophiae Naturalis Principia Mathematical. It is now generally agreed that Newton and Leibnitz invented the calculus independently of each other. For an account of the invention of the calculus by Newton and Leibnitz, see Cajori, History of Mathematics, pp. 199-236, and Cantor, Geschichte der Mathematik, Vol. 3, pp. 150-172. " There are certain focal points in history toward which the lines of past progress converge, and from which radiate the advances of the future. Such was the age of Newton and Leibnitz in the history of mathematics. During fifty years preceding this era several of the brightest and acutest mathe- maticians bent the force of their genius in a direction which finally led to the discovery of the infinitesimal calculus by Newton and Leibnitz. Cavalieri, Roberval, Fermat, Descartes, Wallis, and others, had each contributed to the new geometry. So great was the advance made, and so near was their approach toward the invention of the infinitesimal analysis, that both Lagrange and Laplace pronounced their countryman, Fermat, to be the true inventor of it. The differential calculus, therefore, was not so much an individual discovery as the grand result of a succession of discoveries by different minds." (Cajori, History of Mathematics, p. 200.) Also see the "Historical Introduction" in the article, Infinitesimal Cal- culus {Ency. Brit., 9th edition), and, at the end of that article, the list of works bearing on the infinitesimal method before the invention of the calculus. Notation. In differentiation d and D are used as symbols ; thus, df{x) is read " the differential of /(»)," and Df(x) is read " the 150 INFINITESIMA L CALCUL US. [Ch. X. derivative of /(a?)." In integration, whether the object be sum- mation or anti-differentiation, the sign | is most generally used as the symbol ; thus, | f(x) dx is read " the integral off(x) dx^ * Other symbols, viz. d'~^f{x)dx and D~'^f(x), are used occasionally (see Art. 97, Xote 2). The quantity f(x) which appears "under the integration sign," as the mathematical phrase goes, is called the integrand. 95. Examples of the summation of infinitesimals. These examples are given in order to help the student to understand clearly what the phrase " to find the limit of the sum of a set of infinitesimals of the ioTiR f(x)dx {i.e. a set of infinitesimal differentials)" means. (a) Find the area between the line y = mx, the x-axis, and the ordinates drawn, to the line at X = a and x = b. Let PQ be the line whose equation is y = mx, OA = a, and OB = b. Draw the ordinates ^Pand BQ ; it is required to find the area APQB. Suppose that AB is divided into n equal parts each equal to Ax, X so that n • Ax = b — a. r Pj P ■P«-l ? ^ X-^ p, y G s -S ^ L Jl h^ M. Mn -1^ 3 3 Fig. 39. Draw the ordinates at each point of division, M^ M2, •••, il/n-i ; complete the inner rectangles PMi, Pi, ^¥2, •■•, Pn-iB ; and complete the outer rectan- gles PiA, P2M1, •••, QMn-i. The area APQB is evidently greater than the sum of the inner rectangles and less than the sum of the outer rectangles ; i.e. sum of inner rectangles < APQB < sum of outer rectangles. * The word integral appeared first in a solution of James Bernoulli (1654- 1705), which was first published in the Acta Eruditornm in 1690. Leibnitz had called the integral calculus calculus .ncmmatorius, but in 1696 the term calculus integralis was agreed upon by Leibnitz and John Bernoulli (1667- 1748). The sign \ was first used in 1675, and is due to Leibnitz. It is merely the long S which is the initial letter of summa, and was used by earlier writers to denote " the sum of." 96.] * INTEGRATION. 151 The difference between the sum of the inner and the sum of the outer rectangles is the sum of the rectangles PPi, P1P2, •••, P"~^Q- The latter sum is evidently equal to the rectangle QS, i.e. to CQ ■ Ax. This approaches zero when Ax approaches zero. Therefore APQB is the limit of the sum of either set of rectangles when Ax approaches zero. The limit of the sum of the inner rectajigles will now be found. At^, X = a^ and hence, AP = ma ; atilfi, x = a-\- Ax, and hence, MiPi = m{a + Ax) ; at Jf2, x = a + 2Ax, and hence. MiPi - m(a + 2 Ax) ; at Mn-i, x = a + n — 1 Ax, and hence, ilf„_iP„_i = 7)i(a + n — 1 • Ax), .'. sum of inner rectangles = ma • Ax + m(a + Ax) • Ax + m(a + 2 Ax) • Ax + ••• + m(a + n — I ' Ax) • Ax. /. area APQB = lim^^^ [ma Ax + m(a-]-Ax)Ax-{--"-\-m(a-\-n-l • Ax) Ax] = limAx=o»i[a+(rt + Ax)+(a+2 Ax)H [-{a + n — 1 • Ax)]Ax. Hence, on summation of the arithmetic series in brackets, mn Ax , area APQB = limAx^ [ {2 a -^ n - I ■ Ax}. On giving n Ax its value h — a, this becomes area APQB = limAx-o ^"^^~^^ (6 + a - Ax) -"{1-1} Note 1. In this example the element of area, as it is called, is a rectangle of height y and width Ax when Ax is made infinitesimal, i.e. the element of area is y dx or mx dx in which dx = 0. (See Art. 27, Notes 3, 4, and Art. 67rt.) Note 2. It may be observed in passing that on taking the anti-differential of mxdx, namely ^^, substituting h and a in turn for x therein, and taking the difference between the results, the required area is obtained. Ex. Find the limit of the sum of the outer rectangles when Ax approaches zero. (&) Find the area between the parabola y = ofi, the x-axis, and the ordinates atx = a and x = b. 152 INFINITESIMA L CALC UL US. [Ch. X. Let LOQ be the parabola, OA = a, OB = h ; draw the ordinates AP and BQ; the area APQB is required. As in the preceding problem, divide AB into n parts each equal to Aoj, so that draw ordinates at the points of division, and construct the set of inner rectangles and the set of outer rectangles. As in (a), it can be seen that sum of inner rectangles < area APQB < sum of outer rectangles ; and also that (sum of outer rectangles) — (sum of inner rectangles) = CQ • Asc, which approaches zero when Ax approaches zero. Hence the area APQB is the limit of the sum of either set of rectangles when Ax approaches zero. The limit of the sum of the inner rectangles will now be found. At^, X = a, and hence. AP=a'^', at ilfi, x = a-h Ax, and hence. i¥iPi = (a + Ax)2; atiJfa, X = a + 2 Ax, and hence, M2P2 = (a + 2 Ax)'^ ; at Mn-i, x = a + n — 1 ■ Ax, and hence, Mn-iPn-i = (a + n — 1 - Ax)2. .-. sum of inner rectangles = a'^Ax + (« + Ax)2Ax + (a + 2 Ax)2Ax + ••• + (a + w - 1 • Ax)2Ax. area APQB = \im^^^{a^ + (« + Ax)2 -f (« + 2 Ax)2 + + (a + w - 1 • Ax)2}Ax = liraAx^o{na2 + 2 a Ax(l + 2 + 3 H h n - 1) + (Ax)2(12 + 22 + 32 + ... + w - r)}Ax. Now and 1 + 2 + 3 + .-. + w-l = i n(n - 1) ; 12 4. 22 + 32 + ... + n - r = i (n - l)n(2 n - 1).* area APQB = limAx:^ n Ax {a^ + aw Ax — a Ax + | (n Ax)2 -Jw(Ax)2+KAx)2}. * It is shown in algebra that the sum of the squares of the first n natural numbers, viz. I2, 22, 32, ..., n2, is ^ n(w + 1) (2 n + 1). 95.] INTEGRATION. 153 But w Aic = 6 — a, no matter what n and Ax may be. .-. area APQB = limAxio {h - a){a^ -\r a{h - a) - a ^x ^- \(h - a)^ 3 3" Note 1. In this example the element of area is a rectangle of height y and width Aoj, when Ax becomes infinitesimal, i.e. the element of area is y dx^ I. e. x^ dx, in which dx = 0. Note 2, It may be observed in passing that the result (1) can be ob- tained by taking the anti-differential of x'^ dx, namely — , substituting b and a in turn for x therein, and calculating the difference ^• 3 o Ex. Find the limit of the sum of outer rectangles. (c) Find the distance through which a body falls from rest in ti seconds, it being known that the speed acquired in falling for t seconds is gt feet per second. [Here g represents a number whose approximate value is 32.2.] Note 1. If the speed of a body is v feet per second and the speed remains uniform, the distance passed over in t seconds is vt feet. Let the time ti seconds be divided into n intervals each equal to At, so that nAt = ti. The speed of the falling body at the beginning of each of these successive intervals of time is 0, g ' At, 2 g ' At, •••, (n - l)g • At, respectively ; the speed of the falling body at the end of each successive interval of time is g • At, 2g • At, S g • At, •••, 7ig • At, respectively. For any interval of time the speed of the falling body at the beginning is less, and the speed at the end is greater, than the speed at any other moment of the interval. Now let the distance be computed which would be passed over by the body if it successively had the speeds at the beginnings of the intervals ; and then let the distance be computed which would be passed over by the body if it successively had the speeds at the ends of the intervals. The first distance = + g^Aty^ + 2 g^Aty -I- ••• + (n - 1) — Ao;, and hence, iJf„_iP„_i =/(^ — Ax). .: area APQB = lini_^^^o lf(a)Ax-{-f(a-{-Ax)Ax-{-f{a-^2Ax)Ax+"' -\-f(b-Ax)Axl. (2) The second member, which is the sum of the values, infinite in number, that f(x)Ax takes when x increases from a to b by equal infinitesimal increments Ax, may be written (i.e. denoted by) \im:,,^Q^f(x)Ax.* It is the custom, however, to denote the second member of (2) by putting the old-fashioned long *S' before f{x)clx and writing at the bottom and top of the *S' respectively the values of x at which the summation begins and ends ; thus f(3c)dic; or, more briefly, I f{x)dQC, (3) This symbol is read "the integral of f(;x)clx between the limits a and b" or " the integral of f(x)clx from a; = a to a; = 6." Note 1. The numbers a and b are usually called the Iqicer and upper limits of X. It would be better, perhaps, not to use the word limit in this connection, but to say "the initial and final values of x," or simply, "the end-values of a:." f Note 2. The infinitesimal differential f(x)clx is called "ff/i element of the integral. It is the area of an infinitesimal rectangle of altitude f{x) and infinitesimal base dx. * The latter part of this symbol denotes, and is to be read, "the sum of all quantities of the type" [or "form"] "/(x)A:t, from x = a to x = 6" [or " between x = a and x — &"]. t Joseph Fourier (1768-1830) first devised the way shown in (3) of indi- cating the end-values of a:. 156 INFINITESIMAL CALCULUS. [Ch. X. Note 3. It is not necessary that the infinitesimal bases, i.e. the increments Ax of X, toe all equal ; but for purposes of elementary explanation it is some- what simpler to take them as all equal. (See Lamb, Calculus^ Arts. 86, 87, and the references in Art, 97, Note o ; also Snyder and Hutchinson, Calculus, Art. 150.) Note 4. For the calculation of \ eHlx and i s'mxdx by the process shown in Art. 95, see Echols, Calculus, Art. 125. The sum in brackets in (2) will now be calculated, and then its limit, which is indicated by the symbol (3), will be found. Let the anti-differential (A rt. 27 a) of f(x) dx * be denoted by <^(x);thatis,let f(,)ax^a.i>{x). Then, by the elementary principle of differentiation (see Art. 22, Note 3) for all values of x from a to 6, i(^±^|^iiM =/(») + «, (4) in which e denotes a function whose value varies with the value of X, and which approaches zero when Aa; approaches zero. On clearing of fractions and transposing, (4) becomes f{x) ^x = {x-{- A.t) — <^ (.t) — e • Ax. (5) On substituting a, a -f Aa.', a -i-2 Ax, •••, h — Ax in turn for x in (5), and denoting the corresponding values of e by e^, e^, e.^, ••♦, e„, respectively, there is obtained : f(a) /\x = (a -|- Ax) — e.2 • Ax, f(a + 2 Ax) Ax = ^ (a + 3 Ax) — <^ (a + 2 Ax) — e^ • Ax, f(b-Ax)Ax=(b) -<^(6-Ax) -e„.Ax. * If f(x) is a continuous function of oj, /(x) dx has an anti-differential. For proof see Picard, Traite cf Analyse, t. I. No. 4 ; also see Echols, Calculus, Appendix, Note 9. 96.] INTEGRATION. 157 Addition gives /(a) Ax -\-f(ci 4- Ax) Ax +/(a + 2 Ax) Ax H \-f{h - Ax) = 4>(h)- <\>(a) - (ei + e, + ^3 + ... + e„) Ax. (6) Oil taking the limit of each member of (6) when Ax approaches zero, jj{x) dx = cf>(b)-cf> (a) - lim^^o (e^ 4- e, -f- . . . 4- e,) Ax. (7) Let ej be one of the e's which has an absolute value E not less than any of the others ; then evidently (ei -h ^2 H + e„) Ax < iiE^x ; i.e. by (1), (e^ + ^^ + • • • + e„) Ax < (b - a) E. Hence, lim^^^ (^i + ^9 H- • • • + e„) Ax = 0, since E approaches zero when Ax approaches zero ; and therefore, JV(a?) dx = <|>(6) - c|>(a). (8) That is, expressing (8) in words : The integral I f(x) dx, which • /a is the limit of the sum of all the values, infinite in number, that f(x) dx takes as x varies by infinitesimal increments from a to b, is obtained by finding the anti-differential, cf>(x), off(x)dx, and then calculating {a). Note 5. Many practical problems, such as finding areas, lengths of curves, volumes and surfaces of solids, and so on, can be reduced to finding the limit of the sum of an infinite number of infinitesimals of the form f(x) dx. (See Arts. Ill, 112, 135-140.) As has been seen above, the anti-differential of /(x) dx is of great service in determining this limit ; accordingly, con- siderable attention must be given to mastering methods for finding anti- differentials. Note 6. The process of finding the anti-differential of f(x) dx is nearly always more difficult than the direct process of differentiation, and frequently the deduction of an anti-differential is impossible. Wlien the anti-differential of /(x) dx cannot be found in a finite form in terms of ordinary functions, approximate values of the definite integral can be found by methods dis- cussed in Chapter XIV. The impossibility of evaluating the first member of (8) in terms of the ordinary functions has sometimes furnished an occasion for defining a new function, whose properties are investigated in higlier mathematics. (On this point see Snyder and Hutchinson, Calculus, Art. 123, 158 INFINITESIMAL CALCUL US. [Ch. X. foot-note.) For instance, the subject of Elliptic Functions arose out of the study of what are called the elliptic integrals (see Art. 137, Ex. 4, Art. 174, Note 4, Art. 122, Note 4). (The ordinary elementary functions can be defined by means of the calculus, and their properties thence developed.) Note 7. At the beginning of this article the principle was enunciated that the area bounded by a smooth curve PQ (Fig. 41), the ic-axis, and a pair of ordinates, is the limit of the sum of certain inner, or outer, rectangles constructed between the ordinates. The student can easily show that this principle holds for the smooth curves in Figs. 42 a, b, c. A B X Fig. 42 a. B X N Fig. 42 c. Note 8. This article shows that a definite integral may be represented geometrically as an area. For a general analytical exposition of integration as a summation, see Snyder and Hutchinson, Calculus, Art. 148. Their exposition depends on Taylor's theorem (Art. 176). Also see the references mentioned in Art. 97, Note 5. Ex. Show that the calculus method of computing the area in Fig. 42 c bounded by PMNRQ, AB, AP, and BQ really gives area^PJf + area J? ^B — area MNR. [As a point moves along the curve from P to Q, dx is always positive. In APM y is positive, in MNB negative, in BQB positive. Accordingly, the elements of area, /(x) dx or y dx, are positive in APMdi.nd.RQBy and negative in MNB.I EXAMPLES. N.B. The knowledge already obtained in Chapter IV. about anti-differen- tials is sufficient for the solution of the following examples. It is advisable to make drawings of the curves and the figures whose areas are required. 1. Find the area between the cubical parabola y X-axis, and the ordinates for which a; = 1, x = 3. x8 (Fig., p. 412), the 96.] INTEGRATION. 159 According to (3) and (8), the area required = i x^dx = 20 sq. units of area. 2. Find the area between the curve in Ex. 1, the x-axis, and the ordi- nates for which x = — 2, x = S. Ans. 16i sq. units. 3. Explain the apparent contradiction between the results in Exs. 1, 2. 4. Find the actual number of square units in the figure whose boundaries are given in Ex. 2. Ans. 24i sq. units. 5. Find the area between the parabola 2y = 7 x^, the ic-axis, and the ordinates for which : (1) x = 2, a; = 4 ; (2) a: = — 3, x = 5. Ans. (1) 651 sq. units ; (2) 177^ sq. units. N.B. A table of square roots will save time and trouble. 6. Find the area between the parabola y^ = Sx, the cc-axis, and the ordinates for which : (1) x = 0, x = S ; (2) x = 2, x = 7. Ans. (1) 9.798 sq. units ; (2) 29.59 sq. units. 7. Find the area of the figure bounded by the parabola y'^ = 6x and the chord perpendicular to the x-axis at x = 4. Ans. 26.128 sq. units. 8. Find, by the calculus, the area bounded by the line y = 8 x, the X-axis, and the ordinate for which x = 4. Ans. 24 sq. units. 9. (1) Find, by the calculus, the area of the figure bounded by the line y = Sx, the x-axis, and the ordinates for which x = 4, x = — 4. (2) How many sq. units of gold leaf are required to cover this figure ? Ans. (1) ; (2) 48 sq. units. 10. (1) Find the area between a semi-undulation of the curve y = sin x and the x-axis. (2) Find the area of the figure bounded by a complete undulation of this curve and the x-axis. (3) How many sq. units of gold- leaf are required to cover this figure. Ans. (1) 2 ; (2) ; (3) 4. 11. Compute the area enclosed by the parabola y^ = 4 x and the lines X = 2, X = 5. Ans. 22.27 sq. units. 12. Compute the area enclosed by the parabola y = x'^ and the lines y = 1, y = 4. Ans. 9^ sq. units. 13. Find the area between the parabolas x^ = y and y^ = Sx. Ahs. 2| sq. units. 14. Find the area between the curves : (1) y'^ — x and y^ — x^\ (2) x^ = y and ?/2 — x^. (Make figures.) Ans. (1) y*j sq. units ; (2) J- sq. units. 15. Find the area bounded by the curves in Ex. 14 (2) and the lines X = 2, X = 4. Ans. 8.129 sq. units. ]V.B. Art. Ill may be taken up now. 160 INFINITESIMAL CALCULUS. [Ch. X. 97. Integration as the inverse of differentiation. The indefinite integral. Constant of integration. Particular integrals. In many cases there is required, not the limit of the sum of an infinite number of infinitesimals of the form f(x)dx, but the function whose derivative or differential is given. The following is an instance from geometry. When a curve's equation, y=f(x), is known, differentiation gives the slope at any point on the curve in terms of the abscissa x, namely, -^=f'(x) (Art. 24). On the other hand, if this slope is given, integration affords a means of finding the equation of the curve (or curves) satisfying the given condition as to slope. Again, an instance from mechanics : if a quantity changes with time in an assigned way, differentiation determines the rate of change for any instant (Art. 25). On the other hand, if this rate of change is known, integration provides a means for determining the quantity in terms of the time. (See Art. 22, Notes 1, 2, and Art. 27 a.) EXAMPLES. Ex. 1. The slope at any point (x, y) of the cubical parabola y = x^ is Sx^ ; that is, at all points on this curve, -^ = Sx^ and dy = 3x^ dx. dx Now suppose it is known that a curve satisfies the following condition^ namely, that its slope at any point (oj, ?/) is 3 a;^ ; i.e. that for this curve, ^ = 3 x2, (whence, dy=^Zx'^ dx). dx Then, evidently, y = x^ + c, in which c is a constant which can take any arbitrarily assigned value. This number c is called a constant of integration; its geometrical meaning is explained in Art. 99. Since c denotes any constant, there is evidently an infinite number of curves (cubical parabolas, ?/ = x^ + 2, y = x^ — 10, ?/ = x^ + 7, etc., etc.) which satisfy the given condition. If a second condition is imposed, the constant c will have a definite and particular value. For instance, let the curve be required to pass through the point (2, 1). Then, 1 = 23 -f- c ; whence c = — 7, and the equation of the curve satisfying both the conditions above is y = x^ — 1. (Also see Ex. 17, Art. 37.) 2. Suppose that a body is moving in a straight line in such a way that (the number of units in) its distance from a fixed point on the line is always 97.] INTEGRATION. 161 (the number of units in) the logarithm of the number of seconds, t say, since the motion began ; i.e. so that s = log t. Then, the speed, ^ = 1 and ^ = ^ • Now suppose it is known that at any time after the beginning of its motion, after t seconds say, the speed of a moving body is -; i.e. that = -, [whence, ds=-) Then, evidently, s = log « + c, in which c is an arbitrary constant. If a second condition is imposed, the constant c will take a definite value. For instance, let the body be 4 units from the starting-point at the end of 2 seconds, i.e. let s = 4 when t = 2. '^^^^ 4 = log 2 + c ; whence c = 4 - log 2, and s = log ^ + 4 — log 2. 3. In Ex. 1 determine c so that the cubical parabola shall go through (a) the point (0, 0); (6) the point (7, -4); (c) the point (-8, 2); {d) the point (A, k). Draw the curves for (a), (6), (c). 4. Find the curves for which the slope at any point is 4. Determine the particular curves which pass through the points (0, 0), (2, 3), (—7, 1), respectively. Draw these curves. 5. Find the curves for which (the number of units in) the slope at any point is 8 times (the number of units in) the abscissa of the point. Determine the particular curves which pass through the points (0, 0), (1, 2), (2, 3), ( — 4, 2), respectively. Draw these curves. 6. How are the curves in Exs. 4, 1, 3, 5, respectively, affected when the constants of integration are changed? 7. If at any moment the velocity in feet per second at which a body is falling is 32 times the number of seconds elapsed since it began to fall from rest, what is the general formula for its distance, at any instant, from a point on the line of fall ? In this instance, — = 32 «, (whence, ds = d!it dt). dt Hence s = iet^ + c. 8. In Ex. 7, at the end of t seconds what is the distance measured from the starting-point ? What is the distance at the end of 2 seconds ? of 4 seconds ? of 5 seconds ? What are the distances, in these respective dis- tances, measured from a point 10 feet above the starting-point ? If at the time of the beginning of fall, the body is 20 feet below the point from which 162 INFINITESIMAL CALCULUS. [Ch. X. distance is measured, what is its distance below this point at the end of t seconds ? Explain the meaning of the constant of integration in the general formula derived in Ex. 7 ? Derive the results in Ex. 8 from this general formula. Suppose that cl(x)=f(x)dXy (1) then also (Art. 29), d \ cf>(x) + c | = f(x)dx, (2) in which c is any constant. Hence, if (;x) is an anti-differential of f(x)dx, (x) H- c is also an anti-differential of f{x)dx. That is, if d (x), and draw the ordinates for which x — l and x = m respectively. The figure bounded by the curve, the ic-axis, and these ordinates, is the graphical representative of the integral, and (Art. 96) the number of units in the area of this figure is the same as the number of units in the integral. * See Art. 12, Note, 164 INFINITESIMAL CALCUL US, [Ch. X. TTie foUoicing properties of definite integrals are important. Prop- erties (h) and (c) are eg^^ily deduced by using the graphical representatives of the integrals. {a) If dct>(x)=f(x)dx, then (Art. 96) f{x) dx = (f>{b) — cfi (a) and I f(x) dx = {a) — (x) =f(x) dx, then (Art. 97) Cf(x) dx = {x)', (2) let AB be the curve. Give c the particular values 2 and 10, and draw the curves, y = <^{x) + 2 (3) and y = (x) -f- 10. (4) *For clear proof that this is the mean value, see Art. 141, where the topic of mean values is more fully discussed, and Echols, Calcidus, Art. 150 (and Arts. 151, 152). 99.] INTEGRATION. 167 Let CD and EF be these curves. In the case of each one of the curves obtained by giving particular values to c, dx f{^); Fig. 45. and hence, at points having the same abscissa the tan- gents to these curves have the same slope, and, accord- ingly, are parallel. For in- stance, on each curve, at the point whose abscissa is m the slope of the tangent is f{m). Moreover, the distance between any two curves obtained by giving c particular values, measured along any ordinate, is always the same. For, draw the ordinates KR and ST at x = m and X = n, respectively, as in the figure. Then, by Equations (3) and (4), MK= (n) + 2 ; and ME = ct>{m) + 10 ; NT= ^(n) -\- 10. Hence KE = 8, and ST=S. Accordingly, the graphical representation of the indefinite integral, I f(x) clx, consists of the family of curves, infinite in number, whose equations are of the form y = (x) + c, and which are severally obtained by giving c particular values ; and the effect of changing c is to move the curve in a direction parallel to the ?/-axis. (Also see Art. 29, Note 2.) Ex. 1. How many different values can be assigned to c ? How many particular integrals are included in the general integral ? How many different curves can represent the indefinite integral ? Ex. 2. Write the equations of several curves representing each of the following integrals, viz.: ixdx, ix'^dx, \Sxdx, \Sdx, f(2x + 5)dx. Draw the curves. 168 INFINITESIMAL CALCULUS, [Ch. X. (1) 100. Integral curves. If d <\>{x) =f(x) dx, then (Art. 96) f /(a^) d^ = <{^(^) - (0). The curve whose equation is y = 4>(x) - {0), i.e. y= \ /(«) dx, which is one of the particular curves representing y = (0) (Art. 96), the number of units of length in the ordinate at the point of abscissa x on the curve (1), is the same as the number of units of area in this figure. Accordingly, if the first integral curve of a given curve be drawn, the area bounded by the given curve, the axes, and the ordinate at any point on the avaxis, can be obtained merely by measuring the length of the ordinate drawn from the same point to the integral curve. Consequently, it may be said that this ordinate graphically represents the area, and thus, the integral. ^ Note 1 . The original curve y = f(x) is the derived or differential curve of curve (1). Ex. For instance, for the line y = ^x + S; (2) since P (^ x + 3) dx = |- x^ -f 3 a;, the first integral curve of curve (2) is the parabola y = lx^ -}-Sx. (3) These two curves are shown here. If M be any point on the X-axis, and OM=m units of length, and the ordinate 3ILG be drawn, (the number of units of length in MG) = (the number of units of area in OKLM). For, length MG, by (3), is J m^ + 3m; and area OKLM 100, 101.] INTEGRATION. 169 Just as a given curve — it may be called the original or the fundamental curve — has a first integral curve, this first integral curve also has an integral curve. The latter curve is called the second integral curve of the fundamental curve. Again, the second integral curve has an integral curve ; this is said to be the third integral curve of the fundamental curve. On proceeding in this way a system of any number of successive integral curves may be constructed belonging to a given fundamental curve. Note 2. The integral curve can be drawn mechanically from its funda- mental by means of an instrument called the integraph, invented by a Russian engineer, Abdank-Abakanowicz. Note 3. Integral curves are of great assistance in obtaining graphical solutions of practical problems in mechanics and physics. For further in- formation about integral curves and their uses and the theory of the integraph, and for other references, see Gibson, Calculus, §§ 83, 84 ; Murray, Integral Calculus, Art. 15, Chap. XII., pp. 190-200 (integral curves), Appendix, Note G (on integral curves), pp. 240-245 ; M. Abdank-Abakanowicz, Les Integraphes : la courhe integrate et ses applications (Paris, Gauthier-Villars), or BitterlVs German translation of the same, with additional notes (Leipzig, Teubner). Also see catalogues of dealers in mathematical and drawing instruments. EXAMPLES. 1. Show that, for the same abscissa, the number of units of length in the ordinate of the fundamental curve is the same as the number of units in the slope of its first integral curve. 2. Does the first integral curve belong to the family of curves referred to in Art. 99 ? 3. Show how the members of the family of curves in Art. 99 may be easily drawn when an integraph is available. 4. Write the equations of the first, second, and third integral curves of the following curves : (a) y = x ; (b) y = 2x + b; (c) y = smx; (d) y = e*. Draw all these fundamental and integral curves. Can the curve x^y = 1 be treated in a similar manner ? 5. Find and draw the curve of slopes for each of the curves (a), (6), (c), (d), Ex. 4. Then find and draw the first, second, and third integral curves of each of these curves of slope. 101. Summary. The two processes of the infinitesimal calculus, namely, differentiation and integration, have now been briefly described. 170 INFINITESIMAL CALCULUS. [Ch. X. The process of differentiation is used in solving this problem, among others : the function of a variable being given, find the limiting value of the ratio of the increment of the function to the increment of the variable when the increment of the variable approaches zero (Art. 22). This problem is equivalent to finding the ratio of the rate of increase of the function to the rate of increase of the variable (Art. 2Q). If the function be represented by a curve, the problem is equivalent to finding the slope of the curve at any point (Art. 24). The process of integration may be regarded as either : (a) a process of summation ; or (b) a process which is the inverse of differentiation. Integration is used in solving both of the following problems, viz. : (1) To find the limit of the sum of infinitesimals of the form f(x) dx, X being given definite values at which the summation begins and ends (Arts. 94-96) ; (2) To find the anti-differential of a given differential f{x) dx (Art. 97). Problem (1) is equivalent to finding a certain area; problem (2) is equivalent to finding a curve when its slope at every point is known. In solving problem (1) the anti-differential oif{x) dx is required (Art. 96). Hence, in both problems (1) and (2) it is necessary to find the anti-differentials of various functions of the form/(ic) dx. Chapters XI. and XIII. are devoted to showing how anti-differ- entials may be found in the case of several of the comparatively small number of functions for which this is possible. It may be stated here that, in general, integration is more difficult than the direct process of differentiation. CHAPTER XI. ELEMENTARY INTEGRALS. 102. In this chapter the elementary or fundamental integrals (anti-differentials) are obtained, and some general theorems and particular methods which are useful in the process of anti-differ- entiation are described. There is one general fundamental process (Art. 22) by which the differential of a function can be obtained. On the other hand, there is no general process by which the anti- differential of a function can be found.* The simplest integrals, which are given in Art. 103, are discovered by means of results made known in differentiation. In Art. 104 certain general theorems in integration are deduced. Two particular processes, or methods, of integration which are very serviceable and frequently used, are described in Arts. 105, 106. A further set of fundamental integrals is derived in Art. 107. When f{x) is a rational fraction in x, the anti-differential of f(x)dx may be found by means of the results in Arts. 103, 107; for this reason examples involving rational fractions are given in Art. 108. The integration of a total differential is considered in Art. 109. So far as finding anti-differentials is concerned, this is the most important chapter in the book. The student is strongly recom- mended to make himself thoroughly familiar with the chapter and to work a large number of examples, so that he can apply its results readily and accurately. T7ie list of formulas, I. to XX VL (Arts. 103, 107), should be memorized. Every function, f(x)dx, whose integral can be expressed in finite form in terms of the functions in elementary mathematics, is reducible to one or more of the forms in this list. It is often necessary to make reductions of this kind. A ready knowledge of these forms is not only useful * There is a general process by which the value of a definite integral can be found approximately, as described in Art. 123. 171 172 INFINITESIMAL CALCULUS. [Ch. XI. for integrating them immediately when presented, but is also a great aid in indicating the form at which to aim, when it is neces- sary to reduce a complicated expression. 103. Elementary integrals. The following formulas in integra- tion come directly from the results in Arts. 37-55, and can be verified by differentiation. Here w denotes a function of any variable, and c, Co, c^ denote arbitrary constants. I. (u^^du = — h c, in which 7i is a constant. J n + 1 ' Note 1. This result is applicable in the case of all constant values of w, excepting n =— I. The latter case is given in II. II. f ^ = logu + co = log «* + log c = log cu. J u Note 2. The various ways in which the constant of integration can appear in this integral, should be noted. NoTK 3. Formula II. can also be derived by means of I. (See Murray, Integral Calculus, p. 37, foot-note.) ' ni. (a^du = -^— + e. J log a IV. ie^'du = e^' + c. V. ( sin u du = - cos «* + c. VI. ( cos udu = ^mu + c, VII. ( sec^ udu = tan u -\- c, VIII. j csc^ u du = - coin -\- c, IX. isecutSLnudu = secu-i-€, X. \ CSC u cot udu= — CSC u-\- c. XI. f_^_=sm-i«* + c = -cos-i«*-i-Ci. [Remark. By trigonometry sin-i w = — cos-i m + 2 wtt -}- -• See Art. 97, Ex. 10 and Note 1.] ^ ' du -i&n -^u + c. du sec-^M + c. uVu^- 1 du vers^M + c. 103,104.] ELEMENTARY INTEGRALS. 173 XU. XIII. f XIV. f Note 4, Integrals XII., XIII., XIV., may also be written — cot'^u + c, — csc-i w + c, — covers"! m + c, respectively. 104. General theorems in integration. A. Let f(x), F(x), (^x), •••, denote functions of x, finite in number. By Arts. 29, 31, 97, the differentials of j"[/(ic) + 1^(«;+ <|>(a?) + -.Ociic + Co and (fix)dijc -{-(F(x)dac + r(ic)/a2 - a;2 -^ Va^ - a^ sin^ ^ *^ « This integral may be found by another substitution. For, put x = az', then dx :^adz, and f J^= = f ,ad^_ ^ f-^g- = sin-i z + c = sin-i - + c. a 5. fVrt2-x2dx. Put X = a sin ^. Then dx = a cos ^ d^ ; and f Va2_x2(?a;= C V a^ - a^ sin^ ^ . a cos ^ d^ = a^ fcos2^tZ^=-^ f(l+cos20)de ^«^(^4-«y^)+c = «-(^ + sin^cos^) + c = ^fsin-1^ + ^ J^"^^:^") + c = i(«^ sin-i^+XA/«2_a;-^)+c. 2 \ a a ^ a^ ) a This important integral may also be obtained in other ways ; see Ex. 4, Art. 118, and Ex. 5, Art. 106. C^^JiH — (Put u = az.) Ans. Itan-i -^* + c. J cfi + w2 a a 6 + u 7. r ^^ (Put u = az.) Ans. - sec-i - + c. 8. f ^^ (Putw = a0.) ^9is. vers-i-+c. •^ V2 ait - m2 « Put V^TT=2;. Thenx + l=^^ dx=2 0(?^,and f ^^'^^ = CifsLDl?^ 2 ({z^- l)dz = ^ z(z^ - 3) + c = |(x - 2) Vx + 1 + c. 106. J ELEMENTARY INTEGRALS. 177 10. f^^dx. Put sin x = t. Then cos x (^ = (^^, cos^ xdx = cos^ x • cos a; (Za; = ( 1 - r-^) dt. = I «3(4 _ ^2) _[. c ^ 3 sin^a;(4 - sin2a;). 11. I sin^ X cos x dx, \ tan^ x sec* x dx, i sec- (4 — 7 x) dx, f g-^^ cZx. ^2- irXT^' J-^^-^cZx, f-^c^., fx(x-2)^... 13. r v(x+a)^dx, rc/o«+/ix)3dx, f_— ^_-, f — ^y *^ -^ -^ >/3 - 7 X *^ i/(4 + 6 2/)8 14. re'«+-cZx, f45-3xfZa;, T ^ , pin (log x)^^ -' -^ ^ (1 + x-)tan-ix J X 15. ^t(t-l)^dt, ( (a -{- by)^ dy, ( (m -\- z)^ dz, fcosfxcZx. 16. i cos^ X rfx, I sec* x cZx, i sin^ x (?x, i sec^ (-\d0. -- f sin X dx r cos x cZx r sec^ x (^x T sec^xdx J3 + 7cosx' J9-2sinx' J V4 - 3 tan x •' VlO - 3 sed^' 18. f ^^"^ , f(a2_x2)%(?x, fVc^H:^).^.^^^ f ^^^ . •^Va-^ + x-^ -^ -^ •^(a2-x-^)t 106. Integration by parts. Let u and v denote functions of a variable, say x ; then [Art. 32 (7)] d (uv) = udv -{- V dnj whence u dv = d (uv) — v du. Hence, on integration of both members, iudv = uv - \v du. (1) If an expression f(x)dx is not readily integrable, it may be divided into two factors, ic and dv say. The application of formula (1) will lead to the integral | v du, and it may happen that this integral can easily be found. Note 1, The method of integi-ating by the application of formula (1) is called integration by parts. This is one of the most important of the par- ticidar methods of integration. 178 INFINITESIMAL CALCULUS. [Ch. XL EXAMPLES. 1. Find \ xe* dx. Put u = X', then dv = e^ dx, du = dx, and v = e'. .'. \ xe^ dx = xe^ — I e^ dx = ice^ — e^ + c. 2. Find i sin~i x dx. Put t< = sin~i X ; then (?y = dx, du = — ^ ) and v = x. Vl -x2 X dx .-. i sin-i xdx = X sin-^ ^ ~ I " Vl - x-^ = X sin-i X + Vl -x2 + c. (See Ex. 18, Art. 105.) 3. Find I x cos x dx. Put M = cos X ; then dv = x dx, du = — sin X dx, and v = ^x^. .'. J X cos X dx =: I x2 COS X + ^ ^J^ siu X dx. Here the integral in the second member is not as simple a form, from the point of view of integration, as the given form in the first member. Accord- ingly, it is necessary to try another choice of the factors u and dv. Put u=x; then dv = cos x dx, du = dx, and v = sin x. .'. I X cos X dx = X sin X — j sin x dx = x sin x + cos x + c. 4. Find i x^ cos x dx. Put u = x^ ; then dv = cos x dx, du = 3 x2 dx, and v = sin x. .'. j x^cosxdx = x^ sinx — 3 I x^sinx dx. (1) It is now necessary to find i x^ sin x dx. Put ti = x^ ; then dv = sin x dx, du = 2x dx, and v =— cos x. /. I x2 sin X dx = — x^ cos X + 2 j X cos x dx. (2) 106.] ELEMENTARY INTEGRALS, 179 It is now necessary to find \ x cos x dx. By Ex. 3, ix cos xdx = x sin x + cos a: -f c. Substitution of this result in (2), and then substitution of result (2) in (1), gives j x^ cos X dx = x^ sin x + 3 x- cos x — 6xsmx — 6 cos x + Ci. When the operation of integrating by parts has to be performed several times in succession, neatness, in arranging work is a great aid in preventing mistakes. The work above may be arranged much more neatly; thus: \ x^ cos X c^x = x^ sin x — 3 I x^ sin x dx = x^ sin X — 3 — x2 cos X 4- 2 I X cos x dx = x^sinx — 3[— x2cosx+ 2(xsinx + cosx + c)] = x^ sin X + 3 x^ cos x — 6 x sin x — 6 cos x -\- C = x(x2 - 6) sin X + 3(x2 - 2) cosx + C. The subsidiary work may be kept in another place. 5. Find f Va^ - x'^ dx. (See Ex. 5, Art. 105.) Put u = Va^ — x2 ; then dv = dx, du = ^^^ , and v = x. Va2 - x2 ... C Va2 _ x^ dx = xVa'^-x^ + ( ^^^^ ■ (1) •^ *^ Va^ — X- Kow va^ — x^ = Va'^ — x2 Va^ — x"-^ Va^ — x^ hence ' — ^^r= = — -=. — Va^ — x^. Va-^ - x2 Va2 _ a;2 Substitution in (1) gives f Va2 _ x^ dx = xVd' - x2 + r_^!^__ f VfT^^fix. (2) Hence, on transposition of the last integral in (2) to the first member, division by 2, and Ex. 4, Art. 105, f Va2 - x2 dx = - (x \/a2 - x^ f a2 gin-i ^V 180 INFINITESIMAL CALCULUS. [Ch. XI. 6. I e^ cos xdx = ^ e^ (sin x + cos x). (Integrate, putting u = e"", then integrate, putting u = cosaj. Take half the sum of the two results.) 7. xxe'^^'dx. 11. ixlogxdx. 16. jx^siniccte. 8. ixe-='dx. 12. \x^\ogxdx. 16. ie='x'>'dx. 9. ix^f^dx. 13. (tan-ixdx. . 17. ( a; sin x cos x die. 10. flogxdx. 14. fxtan-ixfZx. 18. f -- -^^^ dx. J J *^ V i - x2 19. Derive \ e^ sinx dx = | e^ (sin x — cosx). (See Ex. 6.) 107. Further elementary integrals. A further list of elementary integrals is given here. They can be verified by differentiation. Some of the ways in which they may be derived are indicated in the latter part of the article. XV. I tan u du - log sec u-\- c, XVI. ( cot u du - log sin «* + c. XVII. ( sec u du = log (sec u + tan u) + c, = logtan(|4-|) + c. XVIII. 1 cosec u du = log tan ^ + ^J* XIX. C du _,in-ii^ + c. ^ Va2 _ ^2 a XX. XXI. r ^^ -^sec-i^ + c. J u Vw2 - a2 a a XXII. r du _ 1 ?* 1 — vcis -r */• .B. See Note 1. 107.] ELEMENTARY INTEGRALS. 181 XXIV. ( — — — = log (u + Vu^ + a^) + c. XXV. r ^^^ = log («* + ^V^ - a^) + c, ^«*2 _ (ji XXVI. y ^^a^ -u^du=^(u Va2 _ ^2 + «2 gi^-i ^\ + c. Integral XXII. is also reducible to form XIX. For 2 au — i*^ = a- — (?^ — a)^, and du = d{u — a); •^ V2 «i* - t*2 J Va^ - (u - ay "* Ex. Show that this result and that in XXII. are equivalent. Remarks on integrals XV. to XXVI. Formulas XV., XVI. For derivation, see Exs. 6, 7, Art. 104. Formulas XVII., XVIII. cosec u — cot u Since cosec u = cosec u cosec u — cot u C J C - cosec ?/ cot 11 4- cosec2 u , I cosec u du = \ — clu J J cosec u — cot u ^ rd (cosec u- cot tQ ^ j^g ^^^gg^ ^^ _ ^Q^. ^^>j J cosec u — cot w 1-^Q^^ = log ?— = log tan ^. Substitution of m + - for u in the last two lines gives f cosec (u + -^ d?< = log tan [^^ + J^ , i.e. f sec m fZit = log tan (^ + j ) 5 = log I cosec ( «+ -) -cot /"?<+ I'j I = log (sec ?<+tan u). There are various methods of deriving XVIL and XVIII. 182 INFINITESIMAL CALCULUS. [Ch. XI. Formulas XIX., XX., XXI., XXII., XXIII. For derivation, see Exs. 4, 6, 7, 8, Art. 105, and the following suggestion : Suggestion : — = — ( | ; — = — ( 1 ) . u^ — a^ 2a\u — a u-\-aj a^ — u^ 2a\a+u a — nj Formula XXIV. Put w2 + a2 ^ ^2 . then udu = z dz, whence ^— = — • z u Hence, __Jm_ ^dji^dz^ ■ v'm2 + a^ ^ u ^ ... du du -\- dz dCii + z) On composition, — :::33:^^ = ■ = — ^^ — — — ^• Vm2 ^ a^ u-Y z u + z .-. r ^^ = f^^-^^^^t^ = log(M + 0) + C = log (it + Vlt^ -h «2) + c. •^ Vm2 4 a2 J If + 2; The last result may be written log (u + Vm2 + a^) - log a + c', le. log ^^ + ^^^^ + ^^ + c', a a form which is convenient for some purposes. See Note 3. Formula XXV. can be derived in the same way as XXIV. Formula XXVI. For derivation, see Ex. 5, Art. 105, and Ex. 5, Art. 106. Note 1. Integrals XIX., XX., XXL, XXII., may be respectively written - cos-i - + c', -- cot-i - + c', - - csc-i - + c', - covers-i - + c'. a a a a a a Ex. Show this. :Note 2. Integrals XXIII., XXIV., XXV. , may be written thus : f ^,^ = - hy tan-' ~ + c'(ii^ < a2), f-T^ = --hycot ''' + «'(^->«2), r^^^ = ±hycos-i^ + c'. ^ Vt*2 _ «2 a 107.] ELEMENTARY INTEGRALS. 183 The functions whose symbols are here indicated are the inverse hyperbolic tangent of -, the inverse hyperbolic sine of -, and the inverse hyperbolic a a cosine of — For a note on hj'perbolic functions see Appendix, Note A. a The close similarity between XX. and these forms of XXIII. may be remarked ; so also, between the forms of XIX. and these forms of XXIV. and XXV. Note 3. The same integral may be obtained by various substitutions, and may be expressed in a variety of forms. Instances of this have already been given ; another example is the following : Integral XXIV. can also be derived by changing the variable from w to 2; by means of the substitution y/u"^ + a'^ = z — u: this leads to the form J: ^" log (u + Vm2 + a2) + c. Vu^ + a^ The first member can also be integrated by changing the integral from u to z by means of the substitution Vu^ + a'^ = zu ; this leads to the form y/u^ 4- d^ + u\h ^ Vu^ + a'2 *- Vu'^ + a- — M ^ It is left as an exercise for the student, to employ* these substitutions in the integration of XXIV., and, the arbitrary constants of integration being excepted, to show the identity of the various forms obtained for the integral. EXAMPLES. 1. fi±^dx=f(-A_ + _L^y:. = 2tan-i^ + Ilog(4 + a:2)+c. J 4 + a;"-^ J V4 + x2 4 + X- y 2 2 2. (±tl^dx= ff ^ +-I^Vzx=4sin-ig-7(4-xM+c. 3 ( (^^ =( ^(^ + 2) ^1 tan-i ^- + ^ + c Jx2 + 4x + 20 J (x + 2)2 + 16 4 4 4«. C ^^ _ ^ r — g( x + 2) ^ioc,(-a;+2+\/x2+4x+20)+c. •^ Vx2+4a:+20 -^ V(x + 2)2 + 16 4?,. r ^x ^r d(x + 2) ^si^-i^±j + ,. *^ V12 - x-! - 4 X *^ Vie - (X + 2)2 4 Notice should be taken of the aid afforded {e.g. in Exs. 3, 4 a, 4 6) by completing a square involving the terms in x. ¥-■ 184 INFINITESIMAL CALCULUS. [Ch. XL X = dx Put X = Wm-x^ 1 Then dx = dt, and dx V16 --I tdt 1 ..n.o .X* , . (16-X^)^ 16 10. 11. 12. 13. 14. r ^ J »;2 + 5 a a;2-f 6X+17 7 — 6 X — x^ ' dx Vie f2 - 1 dx VlT + 6x — X' ^J(16«2_i)-J^(16«2_i) 10 X ^ + c. i^ («)]■ dx Vx2 + 6x + 10 dx Vx2 - 5 X + 7 r dx J 4 x2 - 5 X -I 6 (2)1 (2) f-=^i==; (3) f: •^ V 7 — O X — x2 •^ (2) r <^ ; (3) f ^^ J x^ + 5 X - y J \/4x2-8x + (2) r ^^ — - ; (3) r ^ -'a/9-5x-4x2 J7-5x 4x2 \/8x X fZx dx 5xV9x2-25 f ^ ^^ ; (2) f V9=:^(7x; (3) ('V2o-x^dx. •^ (x-l)Vx'-2x-3 ^ *^ -^0 I VSQ ~ ix^dx ; (2) J sec 3 x dx ; (3) Tcosec (4 x — «) dx. f tan (3 X + a) dx ; (2) Toot (4 x2 + a2)x dx ; (3) ("sec 2 xdx. 15. Derive integrals 62 a, b, 63 a, &, p. 406. 16. •^ ^* *^ (-4 4. x2)t •^xV'12x 108. Integration of f(x)dx when /"(jr) is a rational fraction. In order to find lf(x)dx when f(x) is a rational fraction, the procedure is as follows : Resolve f(x) into component fractions, ayid integrate the differ- entials involving the component fractions. Note. It is here taken for granted that in his course in algebra the student has been made familiar with the decomposition of a rational fraction into component fractions, or, as it is usually termed, the resolution of a. rational fraction into partial fractions. Reference mky be made to works on algebra, e.g. Chrystal, Algebra, Part I., Chap. VIII. ; also to texts on calculus, e.g. Snyder and Hutchinson, Calcithis, Arts. 132-137. 108.] ELEMENTARY INTEGRALS. 185 Examples 1, 2, 4 will serve to recall to mind the practical points that are necessary for present purposes. EXAMPLES. J x2 + ic - 6 x^-\- x — H x2 -\- X — 6 The fraction in the second member is a proper fraction, and is in its lowest terms. Accordingly, the work of resolving it into fractions having denominators of lower degree than the second, may be proceeded with. Since its denominator, x^ + a; - tj, i.e. (x — 2)(x + 3), is the common denom- inator of the component fractions, one of the latter evidently must have a denominator x — 2, and the other a denominator x + 3, Since these frac- tions must be proper fractions, their numerators must be of lower degrees than the denominators, and, accordingly, must be constants. Accordingly, put 14 X - 10 / 14 X - 10 \ A , B Q_ ^ / 14X-10 \ ^ A 6 \ (x - 2) (X + 3) / X - 2 x2 + X - 6 \ (x - 2) (X + 3) / X - 2 ■ X -f- 3 ' ^ ^ Here A and B are to be determined so that the two members of (1) shall be identically equal. On clearing of fractions, 14x- 10 =r ^(x-h 3)-h J?(x-2). (2) Since the members of (2) are to be identically equal, the coefficients of like powers of x must be equal. That is, ^ 4- ^ = 14, 3 ^ - 2 J5 = - 10. On solving these equations, A = V-, B = Y- r ,3_3.-2 + 4x+14 ^^^r/ _ 18 52 \^^ J y^i + x-Q J \ 5(x - 2) 5(x -f 3) / ::, ^ a-2 - 4 X + ^^ log (x - 2) -h ^V log (x -|- 3) -}- c. Another way of finding A and B in (2) is the following : The two members of (2) are to be identically equal, and accordingly equal for all values of x. Now, put X = — 3 ; then — 5 5 = — 52 ; whence, B = ^J^. Put x = 2; then 5^1 = 18; whence, yl = V". Note 1. Any other values, e.g. 3 and 7, may be assigned to x ; in this case, however, the values 2 and — 3 give the most convenient equations for determining A and B. Note 2. For a more rapid way of finding A and B in such cases as (1), see Murray, Integral Calculus, Appendix, Note A. 186 INFINITESIMAL CALCULUS. [Ch. XL X- + 21:>: - 10 J x^ + x^ — 5 X + 3 The fraction in the integrand is a proper fraction, and is in its lowest terms. Accordingly, the work of decomposing it into fractions having de- nominators of degrees lower than the third may be proceeded with. Since the denominator x^ + x^ — 5 x + 3, i.e. (x — 1)^ (x + 3) is the common denominator of the component fractions, one of the latter evidently must have a denominator x + 3, and another must have a denominator (x — l)^. It is also possible that there may be a component fraction having the denom- inator X — 1 ; for, if there is such a fraction, it does not affect the given common denominator. Accordingly, put x2 + 21 X - 10 „ ^ , B , C ,oN + 77 — :r:Ti + - — V KP) (x - l)-^(x + 3) X -h 3 (X - 1)2 in which A., B, C are constants to be determined. On clearing of fractions, equating like powers of x (for reasons indicated in Ex. 1), and solving for A, i?, O, it is found that A=-i, B = 'S, C=b. r X3 + 21X-10 ^^^r/^-£ 3 ^\ J x3 + x'^ - 5 X -i- 3 J \ X 4- 3 (x - 1)2 X - 1 / dx = 51og(x-l)-41og(x + 3)--^+c=:log<^i-:4T5-^+c. X — 1 (x + 3)4 X — 1 Note 3. It may be asked why the numerator assigned to the quadratic denominator (x — 1)2 in the second member of (3) is not an expression of the first degree in x, say Bx + 7>, instead of a constant. The reason is, that if such a numerator were assigned, the fraction would immediately reduce to the forms in (3). For Bx + D ^ jB(x- 1)+ Z)+ i? ^ B D + B (X-l)2 (X-l)2 X-1 (X-l)2' forms which appear in (3). Note 4. If a factor of the form (x — a)'" appears among the factors of the denominator of the fraction to be resolved, there evidently must be a com- ponent fraction having (x — ay for its denominator. There may also possi- bly be fractions having as denominators (x — a) of various powers less than r, e.g. (x — a)''"^, (x — a)''-2, ..., x — a. Accordingly, in such a case it is necessary to allow also for the possibility of the existence of fractions of the M F L (x-ay-^ (x-ay-^ in which M, F, •••, L, are constants. 108.] ELEMENTARY INTEGRALS. 187 J2 a;2 — 8 a; — 10 dx. (Compare denominators in Exs. 2, 3.) 4. (- 5x^ + 3. + 17 ^^ J x3 - x=^ + 4 X - 4 The fraction in the integrand is a proper fraction and is in its lowest terms. If it were not so, division as in Ex. 1 and reduction would be necessary. Since the denominator x^ — x- + 4 x — 4, i.e. (x^ + 4)(x — 1), is the com- mon denominator of the component fractions, one of the latter must have a denominator x^ + 4, and the other a denominator x — 1. Accordingly, put 6x2 4- 3 X + 17 _Ax-]- B . C 1 r » . (x-^ + 4)(x-l) x2 + 4 in which A, B, C, are constants to be determined. On clearing of fractions, equating coefficients of like powers of x, and solving for A, jB, C, it is found that A = 0, B = S, C = 5. ._ r 5x^ + 3x4-17 ^^^r/__3_ + -^Ux Jx3-x2 + 4x-4 JVx2 + 4 x-iy = |tan-if+51og(x-l)+c. Note 5. The expression x^ + 4 has factors x + 2 i, x — 2 i (i — V— 1) ; if these be taken, component fractions imaginary in form, are obtained. It is usual, however, not to carry the decomposition of a fraction as far as the stage in which component fractions imaginary in form may appear. Note 6. The numerator Ax + J5 is assigned above ; for the numerator over a quadratic denominator whose factors are imaginary, may have the form of the most general expression of the first degree in x. Note 7. When a quadratic expression x^ -\- px -\- q has imaginary factors and is repeated r times in the denominator of a fraction, in the process of decomposition of this fraction allowance must be made for fractions of the forms, ^^ + ^ , __Cx + D .. Mx + N , {x^ + px + qy. (x:' + px + q)'-' x'' + px + q 5. (1) rn x^-4x + 28^ r .s^2_i3x-5 ^^ (Compare the ^^Jx3-x-^ + 4x-4 '^^Ja;3-x2 + 4x-4 ^ denominators in Exs. 4, 6.) 188 INFINITESIMAL CALCULUS. [Ch. XI. Find the anti-derivatives of the following fractions : 12. a; + 37 xi _ ;j a; _ 28 8x+ 1 2 x-^ - 9 X — 35 X3 _ 2 X2 - 1 X-^-1 x'^ - a:2 -H 1 x^-x a:2_i0x-5 a;(2 .x^ + 3 a; - 5) x2+pg x(x-p)(x + (/) Ilx3-llx2-74x + 84 6. •^"^''' 17. x-2 _ 3 a; _ 28 _ 8X+1 2 x^ - 9 X — 35 8. ^^-^^^-^ 19. x^ - 1 9. ?^^1 + 1. 20. X^ — X 10. a:2--10x-5 ^ 2^ 11. . 2 x2 x2 - 3 X + 3 X* - 13 x2 + ^3- /"^.\./ 24. (X - 1)^ 14. ^^ + ^ . 25. (4x+5)2 15. 5a;2 + x-10^ 26. x(x2 + 3) 12 - X - x2 (3x-2)(x2 + 5) (X+1)2 x^ + x X3-1 x3 + 3x 2 x2 + 3 X + 6 x3 + 3 X 7x2 + 9 x« + 3x 2 x3 - x2 + 8 X + 12 x-^(x2 + 4) 2 + 3 X - x2 (x-l)(x2-2x+5) 1 + 7 X + x" + x3 16. x2(2 X + 5) 30x2 + 43x-8 (x + 4)(3x + 2)2 Ex. 27. Show that any expression of the form f 0^^^ + n)dx ^ .^ which w, n, a, 6, and c are constants, is integrable. ^ ^ 109. Integration of a total differential. In Art. 86 it has been shown that the necessary condition for the existence of a function ^'^""^ Pd^^qay (1) for its differential, is that -r- = ^» (2) ' dy doc ^ ^ It has also been stated (Art. 86, Note 1) that condition (2) is sufficient for the existence of such a function. In other words, if the expression (1) has an anti-differential (or integral), relation (2) must be satisfied ; conversely, if relation (2) is satisfied, the expression (1) has an integral. Accordingly, relation (2) is called the criterion of integrability for the expression (1). If this criterion 109.] ELEMENTARY INTEGRALS, 189 is satisfied, the expression (1) is said to be a coynplete differential^ a total differential, and also an exact differential. If test (2) is satisfied, the integral of (1) can easily be found. This integral's partial ^-differential, Pdx, can only come from terms containing x (Art. 79). Hence, the integral of Pdx with respect to x, namely, " /* jPdx + c, (3) must yield all the terms of the required integral that contain x. Also, Qdy can only come from terms containing y. Hence the integral of ^ dy with respect to y, namely. / Qdy + c, (4) must yield all the terms of the required integral that contain y. Some of these terms mg,y contain x ; if so, they have already been obtained in (3), and need not be taken this second time. Hence, if the integral of a differential of the form Pdx+Qdy is required, apply the test for integrability, namely, dP^dQ^ dy dx ' if this test is satisfied, integrate Pdx tvith respect to x ; then integrate Qdy with respect to y, neglecting terms already obtained in | Pdx ; add the results and the arbitrary constant of integration. EXAMPLES. 1. Integrate (2xy -^2 + ?>}/- + 12 x) dx + (a:2 + 6 x?/ + 4 y^) dy. Here P =2xy + 2 -\- Zy'^ -^ \2x, and Q = x^ + Qxy + ^y^. .'. ^=2x-\-6y, and ^ = 2 x-\-6y. dy dx Thus the criterion of integrab'ility is satisfied. Also (pdx = x'^y -\- 2 X + S xy^ -{■ Q x"^ ] and I Qdy = x^y -\- S xy^ + y*, in which y* has not been already obtained in ( Pdx. Hence the integral is a:2y 4- 2 a: + 3 y%j- 6x'^ + y^-\-c. 190 INFINITESIMAL CALCULUS. [Ch. XI. 2. Verify the result in Ex. 1 by differentiation. 3. Find \(xdy — ydx). Here ^ = 1, and ^ = — 1 ; hence the test for integrability is not satis- dx dy fied, and there is not an anti-differential. 4. (1) ( e^ (cos ydx- sin ydy). (2) C [(Sx'^+8xy-\-i)dx-\-iix^-Q)dy;\. 5. Integrate : (1) cos x sec^ y dy — (sin x tan y + cos a;) dx. (2) (xey -2x)dy -\- (ey -2y -\-2x) dx. (3) (S - ix - y)dx - (x -{■ y) dy. N.B. An accurate and ready memory of the fundamental inte- grals (Arts. 103, 107), resourcefulness in making substitutions (Art. 105), and quickness in integrating by parts (Art. 106), are three very important things to cultivate in order to insure com- fortable progress in the study of the calculus. EXAMPLES. 1. ( ln^x'^+"' dx, f (a + ft)a:2(«+6)-idfx, ( (r + s)z^+*+'^dz, ( rh^y'-'^ dy, rt^^^ei±l r.3 + 8.'^-9 r^^^^2x^ + 7x-l C 1 dt ^ Jo t + 2 J t?2 + 3 J x?--2 J 9 ^-^ 4-20 r^^, i\^^y^ydy, f^^^, r^^^, f ^'-^ dz. Jz^-\2 Ji ^ ^ ^ ^ ^' J V9^::^6 J Vx6"~9 J (2 ^ - 1)2 TT 2. Ttan (ma; + w) (7x, f (sec 3 a; + 2)2dx, f tan2^(Z^, psin/- + ^"J (Z^. 3. icos~ia;da;, isec-ix(?a;, xcoAr^xdx, j(logx)2c?x, I »;% «cZa;, j x^e-^ dx, i sin x log cos x, j x"* log x. 4. f^Lziclx, fJ^dx, f'-*^, fjiil' 2 sill - r dx r dd c 2 J e2x _ g-2x' J cos2 26- siiV^ 2 ^' J . a: / sin^^ 4 [(1 — sin a; oos y) dx — (cos xsmy ■\-2y) c??/], j [(1 — sinajsini/) dx + (cos a; cosy — 1) dy']. 7. Derive the following integrals : (1) fx(x2±a2rda: = (^i±-«^l^. (2) f-^^^ = V^^T^ *^ 2(H+1) 'J Vx--^±a--^ (3) f a:(a2 _ x^ydx = - (<^' - ^'y'^\ (4) C_^dx_ ^ _ Va^^^. ^^ 2(?i + l) 'J Va2"^^^ 8. Derive the following integrals : (1) f-^ = ^log(a + 6x). (2) r(a4-6x)»dx = -^^i^^, whenn J a + bx b J b{n + 1) is different from - 1. (3) C ^^^ =lra-\-bx-a log (a + 6x)]. J a -{- bx b" (^) r-^!^ = l[i(a + &:^)^-2a(« + 6x) + a-21og(a + 6a:)]. (5) f /^ J a + bx b^ J x{a + bx) = --Mog^ + K (6) f ^^ = -i-+Aiog^L+^. (7) r ^^^ = ^Jlog(a + 6a;)+— ^]. 62 L a + 6a; J 9. Derive the following integrals : (1) r_i??_=A_iog^±-^. (2) f ^^ ^-l^tan-ia^-J^when ^'Ja2-6%2 2a6 ° a - 6a; ^ ^ J a + 6x2 ^^ \a' a>0and6>0. (3) f ^^^ ^ J- log f a;^ + " V i.^) f-^i^^?- ^ ^ Ja + 6a;2 2 6 ''V 6/ ^ ^ J a + 6a;2 6 gf ^^ . (5) f ^ ^J-log ^' . (6) f ^ = -1- bJ a^bx'^ ^ ^ J x{a + 6a;2) 2 a '' a + 6a;2 ^ ' J x'\a + 6x2) ^x _b C dx ,^. C xdx 1 aJ a + bx^' ^^J(a+6x2)"~ 2 6(n -l)(a + 6x2)'»-i' 10. Derive the following integrals : (1) Jx V^TF^ dx ^ _ 2(2 g - 3 6x) A/(a + hx)\ ^2) ^x^V^^Tbidx = 2(8 a2-12 g 6X+15 62x2) \/(a + 6x)3 r x(Zx ^ 2(2 q-6x) ^/— -^ 105 63 ■ ^ ^ J V^+6i -^ ^' (4) r_^i^ = 2(8«2-4«6x + 362x2) ^^^^ ^.^ (' ^^ = •^ Va + 6x 15 63 J X Va + 6x 1 Jqo. Va + 6x-\/a Va Va ij;-^ , for g>0; _A^ tan'i J^i + ^, for a<0. + 6x + Va V- a ^ -a CHAPTER XII. SIMPLE GEOMETRICAL APPLICATIONS OF INTEGRATION. 110. This chapter treats of some simple geometrical applica- tioDS of integration. Examples of some of these applications have already appeared in Arts. 96, 97. In Art. Ill integration is used in measuring plane areas, in Art. 112 in measuring the volumes of solids of revolution. In Art. 113 the equations of curves are deduced from given properties whose expression involves derivatives or differentials. N.B. The student is strongly recommended to draw the figure for each example. In the case of examples which are solved in the text he will find it extremely beneficial to solve, or try to solve, the examples independently of the book. 111. Areas of curves : Cartesian coordinates. A, Rectangular axes. In Art. 96 it has been shown that for a figure bounded by the curve the a7-axis, and the two ordinates for which x = a and x = h respec- tively, the axes being rectangular, area of figure = limit of sum of quantities y \x (or f(x) Ax) when Ax approaches zero and x varies continuously from a to b. This limit is denoted by I ydx or f{x) dx', it is obtained by finding the anti-differential of /(a;) dx, substituting b and a in turn for x in this anti-differential, and taking the difference between the results of the substitutions. In fewer words : the 7iumber of units in the area is the same as the number of units in a certain definite integral; namely, area of figure = f ydie=\ f{x) dx, (X) Ja Ja The infinitesimal differential y dx is called an element of area. 192 110,111.] AREAS OF CURVES. 193 N.B. It will be found that in many problems it is necessary : (1) To find a differential expression for an infinitesimal element of area, or volume, or length, etc., as the case may be. (2) To reduce this expression to another involving only a single variable. (3) To integrate the second expression between limits (end-values of the variable), which are either assigned or determinable. B, Oblique axes. Suppose that the axes are inclined at an angle w, and that the area of the figure bounded by the curve whose equation is y=f(x), the x-axis, and the ordiuates AP and BQ (for which x = a and x= b respectively), is required. Let BM be a parallelo- gram inscribed between A and B, as rectangles were inscribed in the figures in Arts. 95, 96. Area of RM= yAx • sin w. Area APQB = limit of sum of all the parallelograms like RM, infinite in number, that can be inscribed between AP and BQ ; that is, Xx=b /*b y sin (o • cZa; = sin « I y dx. area Unless otherwise specified, the axes used in the examples in this chapter are rectangular. EXAMPLES. 1. Find the area between the line 2^/— 5a; — 7 = 0, the cc-axis, and the ordinates for which x = 2 and a; = 5. The rectangle PM represents an element of area, y dx. The area required is the limit of the sum of these element- ary rectangles, infinite in number, Jrora AB to DC. That is, area = T^^ dx = \ C {bx ^1)dx = \ 1"—+ 7 xT = 36| square units. the unit of length used in drawing the figure one inch, the figure would contain 36f square Fig. 48. 194 INFINITE^lMA L CAL C UL US. [Ch. XII. 2. Solve Ex. 1 without the calcuhis, and thus verify the result obtained by the calculus. 3. (a) Find the area of the circle a;2 + 2/2 = 9 ; (6) find the area of the figure bounded by this circle, and the chords for which X = 1 and x ~ 2. Let APB be the circle whose equation is x2 + y2 — 9^ Take a rectangle PM, sup- posed to be infinitesimal, with a width dx, for the element of area. Its area is ydx. The area of the quadrant AOB is the limit of the sum of all these elements of area, infinite in number, between and A. Hence, OAB = (''~^ydx= CVd^'^dx = i fxVO - x^ + 9sin-i-1^= Itt sq. units. Jx=o Jo L 3 Jo .-. area circle = 4 • OAB = 9ir square units. (6) Draw the ordinates TE and JVL at the points T and N where x = 1 and X = 2 respectively. The area of TBLN is equal to the limit of the sum of all the elements of area, PM, that lie between TB and NL. That is. area TBLN=C \jdx = f Vo - x^dx = i fxVO - x^ + 9 sin-i^]^ = i{ (2 V5 + 9 sin-i |) - ( VS + 9 sin-i i) } = V5- V2 + |(sin-i| sin-ii). Here the radian measures of the angles are to be employed. Now V2 = 1.414 ; sin-i| = (41° 40.8') = .727 radians ; sin-i .340 radians. .-. area required = 2 . TBLN = 5.126 square units. Note 1. Other end-values of x may be used in finding the area of this circle. Thus area circle = 2,AiBA = 2^ ydx = 2C VO-x'^dx = Tx V9^^ + 9 sin-i|l 9sin-il -9sin-i(-l) = 9- (-i)- IT square units. Note 2. These problems may be stated thus : Find by the calculus (a) the area of a circle of radius 3, (&) the area of a segment between two parallel chords, distant 1 and 2 units, respectively, from the centre. In this case it is necessary to choose axes (as conveniently as possible) , to find the equation of the circle, and then to proceed as above. 111.] AREAS OF CURVES. 196 4. Find the area between the curve y = 2x^, the y-axis, and the lines y = 2 and y = i. The area is represented by ABLE. At any point P(x, y) on the arc BL take for the element of area an infinitesimal rectangle 3IF. Its area is x dy. xdy =—^ \ y^dy = § . i_ . 2^ (2^ - 1) = ? ( ^/16 - 1) = 2.2797. 4 2^ ^ Fig. 50. Note 3. The definite integral which gives the area may also be expressed in terms of x. For, since y = 2x^, dy = 6x^dx ; also, when y = 2, x = l, and when y = 4, x — v^. .-. area ABLE = ("^\ dy = (^^^6 x^dx=U v^ - 1) = 2.2797. 6. (a) Find the area of the figure bounded by the parabola y^ = i ax, the X-axis, and the ordinate for which x = xi. Show that this area is equal to two-thirds of the rectangle circumscribing the figure. (6) Find the area bounded by the parabola y^ = 9x, and the chords for which x = 4 and x = 9. 6. Find the area between the curve y^ = 4:X^ the axis of y, and the line whose equation is y = 6. 7. Find the area included between the parabolas whose equations are 1/2 = 8x and x'^ = Sy. ^ The parabolas are OML and OBL ; the area of OBLMO is required. To find the points of inter- y^* section of the curve, solve these equations sinml- taneously. This gives (0, 0) the point O, which is otherwise apparent, and (8, 8) the point L. Area OBLMO = area OBLN - area OMLN = V8j x^dx-\\^x'^dx = H^~ — ¥ = 21| square units. Fig. 51. 8. Find the area included between the parabolas whose equations are Sy^ = 2b3> and 5 x.^ = 9 y. 196 INFINITESIMAL CALCULUS. [Ch. XII. 9. Find the area included between the parabola (y — x — 3)2 = x, the axes of coordinates, and the line x = 9. Figure 52 shows that this problem is ambiguous, for OTGML and OTKNL are each bounded as described. On solving the equation of the curve for y, y -^x ± Vx + 3. Thus if Oq^x, qG = x + y/x-^ 3, and QK = x — Vx + 3. .-. area OTGML \' {x -\- Vx -f 3) dx = 85| square units and Sivea. OTKNL \ (x — Vx + 3) dx = 49| square units. Also, the area MTN (the figure bounded by the curve and the chord for which x = 9) = area, OTGML — a,rea, OTKNL = 36 square units. The area of MTN can also be found as follows : Area MTN — limit of sum of infinite number of infinitesimal strips, like KG^ lying between T and MN. Now strip KG = {QG - QK) dx = 2 Vx dx. area MTN = C^^xdx 36. 10. Apply the second method used in finding area MTN in Ex. 9 to find- ing the areas in Exs. 7 and 8. 11. Find in two ways the area between the parabola (y — x — 5)^ = x and the chord for which x = 5. 12. Find the area between the parabola y = x^ — Sx -\- 12, the X-axis, and the ordinates at x = 1 and x = 9. Area = P~%7 dx = C{x'^-Sx-^ 12) dx = 18| square units. (1) The parabola crosses the x-axis at B and C where X = 2 and x = 6. Area APB = P~% dx = 24 ; area BEC =Cydx = - lOf ; ^re^CQD=j^ydx = 27. ^^^ .^ 111.] ABEAS OF CURVES. 197 Area required = area APB + area BEC + area CQD = 2i - lOf- + 27 = 18f, as in (1). The sign of the area BEC comes out negative, because the element of area, y dx, is negative as x increases from OB to OC ; for dx is then positive and y is negative. On the other hand as x proceeds from Ato B and from C to Z), y dx is positive. The actual area shaded in the figure is 2^ + 10| -f 27, i.e. 40 square units. N.B. It should be carefully observed, as illustrated in this example, that in the calculus method of finding areas bounded by a curve, the a;-axis, and a pair of ordinates, areas above the x-axis come out with a positive, and areas belov^r the .K-axis come out with a negative sign. Accordingly, the calculus gives the algebraic sum of these areas ; and this is really the difference between the areas above the x-axis and the areas below it. 13. (a) Find the area bounded by the x-axis and a semi-undulation of the sine curve y = sin 2x. (6) Find the area bounded by the x-axis and a complete undulation of the same curve, (c) Explain the result zei'o which the calculus gives for (6). (d) What is the number of square units bounded as in (&) ? 14. Construct the figure, and show that, according to the calculus method of computing areas, the area between the curve whose equation is 12 y=(x — l) (x — 3) (x — 5), the X-axis, and the ordinates for which x = — 2 and x = 7, is — fl square units; but that the actual number of square units in the figure thus bounded is 12||. 15. Find the area between the line 2?/ — 5x — 7 = 0, the x-axis, and the ordinates for which x = 2 and X = 5, the axes being inclined at an angle 60°. Area APQB = ('^y sin 60° • dx = sin 60° p(5x-l-7)dx = 63.65 square units. Note 4. In the light of the preceding examples attention may be again directed to the N.B. above. These examples also show: (1) the element of area may be chosen in various ways (compare Exs. 1, 4, 7, 9, 11) ; (2) the end values used in a problem may be chosen in different ways (see Ex. 3, Note 1); (3) the calculas method of computing areas should not be employed in a rule of thumb way, but with understanding and discretiMi (see Exs, 12, 13, 14). 198 INFINITESIMAL CALCULUS. [Ch. XII. Note 5. Precautions to be taken in finding areas and computing integrals. Suppose that the area bounded by the curve y=f(x), the x- axis, and the ordinates at A and B for which x = a and x = h respectively, is required. If the curve has an infinite ordinate between A and B, or if the ordinate is infinite at A or B, or at both A and B, or if either or both the end values a and b are infinite, the area may be finite or it may be infinite. It all depends on the curve ; in one curve the area may be finite, in another curve it may be infinite. When infinite ordinates occur, either within or bounding the area whose measure is required, and also when the end-values are infinite, special care is necessary in applying the calculus to compute the area. The calculus method for finding areas and evaluating definite integrals can be used immediately with full confidence, only when the end values a and h are finite a.nd when there is no infinite ordinate for any value of x from a to 6 inclusive. For illustrations showing the necessity for caution and special investigation in other cases see Murray's Integral Calculus, Art. 28, Exs. 3, 4, 5, 6, Art. 29 ; Gibson, Calculus, § 126 ; Snyder and Hutchinson, Calculus, Arts. 152, 155. Note 6. For the determination of the areas of curves whose equations are given in polar coordinates, see Art. 136. The beginner is able to proceed to Art. 136 now. EXAMPLES. 16. Calculate the actual increases in area described in the Note and in Exs. 2, 4, Art. 67. 17. Find the areas of the figures which have the following boundaries : (1) The curve y = x^ and the line 4y = x. (2) The parabola y^ -\- Sx and the line x + y = 0. (3) The semi-cubical parabola y^ = x^ and the line y = 2x. (4) The curves y^ = x^ and x^ = 4 y. (5) The axes and the parab- ola Vx -\- y/y = Va. (6) The curve x"^ + 6y = and the line y -f- 3 = 0. (7) The curve (y -\- 4)2 + (x -|- 3)2 = and the line ic -f- 6 = 0. (8) The hyperbola xy = 1 and the ordinates : (a) at x = 1, x = 7 ; (h) at x — 1, X = 15 ; (c) at X = 1 and x = n. (d) The hyperbola xy = k^ and the ordi- nates at X = a and x = 6. (And the x-axis in each case.) 18. Find the area of the loop of the curve Sy^ = x4(3 4- x). 19. Show that the area of the figure bounded by an arc of a parabola and its chord is two-thirds the area of a parallelogram, two of whose opposite sides are the chord and a segment of a tangent to the parabola. [Suggestion : First take a parallelogram whose other sides are parallel to the axis of the parabola.] Ex. 20. Prove that the area of a closed curve is represented by iJ(^|-2/f)f?«[oriJ(x#-2,cZx)] taken round the curve. (See Williamson, Integral Calculus, Art, 139 ; Gibson, Calculus, § 128.) 112.] VOLUMES OF REVOLUTION, 199 Fig. 55 112. Volumes of solids of revolution. Suppose that the arc PQ of the curve revolves about the a;-axis. It is required to find the volume enclosed by the surface generated by PQ in its revolution and the circular ends generated by the ordinates AP and BQ. (This is put briefly : the volume generated by PQ.) Let OA = a and OB = b. Suppose that AB is divided into any number of parts, say n, each equal to Ax. On any one of these parts, say LR, construct an "inner" and an '^ outer" rectangle, as shown in Fig. 55. Let G be the point (x, y), and K be the point {x -f- A.t, y -f- Ay). When PQ revolves about the a^axis, the inner rectangle GR describes a cylinder of radius GL {i.e. y), and thickness Ax. At the same time the outer rectangle KL describes a cylinder of radius KR (i.e. y-{-Ay), and thickness Ax. It is evident that the volume PQST is greater than the sum of the cylinders described by the inner rectangles, and is less than the sum of the cylinders described by the outer rectangles. That is, sum of outer cylinders > vol. PQST > sum of inner cylinders. The difference between the volume of the outer cylinders and the volume of the inner cylinders approaches zero when Ax approaches zero. Hence, vol. PQST= liniAx-o J sum of inner (or outer) cylinders J. That is, voL J*Q-S»T= limAxio [sum of cylinders like that generated by GR when x increases from a to b\ limAx-o / (ttLG^ ' Ax) = IT I y^dx, *=« (See Art. 96.) 200 INFINITESIMAL CALCULUS. [Ch. XII. The infinitesimal differential iri/dx, which is the volume of an infinitesimal cylinder of radius y and infinitesimal thick- ness dx, is called an element of voluyne. When PQ revolves about the ?/-axis the element of volume is evidently irx^dy. If the ordinates of P and Q are c and d respec- tively, the volume generated, vol. PQTV= Tc (^^^oc^dy. Note 1. It is almost self-evident that the volume of the inner cylinders and the volume of the outer cylinders (Fig. 55), approach equality -wtien their thickness Ax approaches zero. Note 2. See Art. 67 (e). EXAMPLES. 1. Find the volume generated by the revolution, about the x-axis, of the part of the line 3 x + 10 j/ = 30 intercepted between the axes. The given line is AB. The element of volume is -KXp- dx. At B, X = ; at ^, x = 10. Accord- ingly, the end-values of x are and 10. Hence, rx=io r: vol. cone ABC = tr \ y^dx = Tr \ Jx=0 Jo iV30-3x\2 Fig. 57. V 10 = 94.248 cubic inches. 2. Verify the result in Ex. 1 by finding the volume of the cone in the ordinary way. 3. Derive by the calculus the ordinary formula for finding the volume of a right circular cone having height h and base of radius a. (See Ex. 8.) 4. (a) Find the volume generated by the revolution of the ellipse 9 x2 -f 16 ?/2 = 144 about the x-axis. (b) Find the volume bounded by a zone of the surface and the planes for which x = 2 and x = 3. The element of volume is -rry'^ dx. (a) Vol. ellipsoid = 2\oLABB' = 2Tr ('^^^dx Jx=0 - 2z r\l44 - 9 x2)dx = 48 TT = 150.8 cubic units. 112.] EXAMPLES. 201 Or, vol. ellipsoid = ir \ y^dx = 150.8 cubic units. (6) Vol. segment PQQ'F' = ir ("^y^dx = ^ ir = 17.08 cubic units. 5. Find the volume generated hy revolving the arc of the curve y = x^ between the points (0, 0) and (2, 8), about the y-axis. The arc is OA. The element of volume, taking any point F(x, y) on OA, is ttx- dy. Hence, vol. OAB = IT P"%2 dy = Tr Cy^ dy = -%^- ir Jj/=0 Jo = 60.32 cubic units. The integral may also be expressed in terms of x. Thus, ^^^2 vol. OAB = ir\ ^x^dy. Since y = x^, dy = ^ x^ dx. .'. vol. OAB = 3 IT Cx^ dx = 9/ TT = 60.32, as above. 6. Find the volume generated by revolving about the t/-axis the arc of the catenary y = ^(ea + e a) between the lines x = a and x = — a. AC A' is the catenary ; A and A' are the points whose abscissas are a and — a respec- tively. The volume generated by revolving AC A' about OY is evidently the same as the volume gener- ated by revolving CA. The element of volume is Tx2 dy. * ^^^ .-. vol. ACAM= 7 + BP' = 1 -\- \/25 - y'\ .'. vol. generated by P'G = 28 7r\/25 — y^ • dy. vol. of ring = 2 ^"^28 TrV2b-y^dy=S50 ir^ cubic units. [Or, as in Ex. 4 (a).] vol. of ring = \ . 28 v V25 — y'^ dy = 350 tr^ cubic units, Jy=-5 8. Find the volume of a cone in which the base is any plane figure of area A, and the perpendicular from the vertex to the base is h. 9. Find the volume generated by revolving the arc BEC (Fig. 53) about the x-axis. 10. Find the volume generated by the revolution of MTKN (Fig. 52) about the x-axis. 11. Find the volume generated by the revolution of ORLM (Fig. 51) about the y-axis. 12. Find the volume generated by the revolution of ABLB (Fig. 50) : (a) about the ?/-axis ; (&) about the x-axis. 13. Find the volume generated by revolving the loop in Ex. 18, Art. Ill, about the x-axis. 113.] EQUATIONS OF CURVES. 203 14. Find, by the calculus, the volume generated by the revolution about the X-axis, of the part of each of the following lines that is intercepted bertween the axes, and verify the results by the ordinary rule for finding the volume of a cone : (1) 3x + iy = 2; (3) 7a: + 32/ + 20 = 0; (2)2x-5y = 7', (^i) Sx - 4y + 10 = 0. . 15. Find the volume generated by the revolution about the y-axis, of each of the intercepts in Ex. 14, and verify the result by the usual method of computation. 16. Find the volume generated when each of the figures described in Ex. 17, (l)-(9), Art. Ill, revolves about the ar-axis. 17. Find the volume generated when each of the figures in Ex. 16 revolves about the y-axis. 18. The figures bounded by a quadrant of an ellipse of semi-axes 9 and 5 inches and the tangents at its extremities revolves about each tangent in turn : find the volumes of each of the solids thus generated. 19. Find the volume of a sphere of radius a, considering the sphere as generated by the revolution of a circle about one of its diameters. Note 3. The volume of a sphere may also be obtained by considering the sphere as made up of concentric spherical shells of infinitesimal thickness. The volume of a shell whose inner radius is r and whose thickness is an infini- tesimal dr is (to within an infinitesimal of lower order) 4 wr'^ dr. Accordingly, volume of sphere = I 4 wr'^ dr = ^ ira^. 20. Find the volume generated by the revolution of the hypocycloid x^+ y's = as about the x-axis. {Ans. ^^^ ira^.) 113. Derivation of the equations of curves. The equation of a curve or family of curves can be found when a geometrical prop- erty of a curve is known. Exercises of this kind constitute an important part of analytic geometry. For instance, the equation of a circle can be derived from the property that the points on the circle are at a given common distance from a fixed point. The statement of a geometrical property possessed by a curve may involve derivatives or differentials. To derive the equation of the curve from this statement is, quite frequently, a difficult problem. There are a few simple cases, however, in which it is possible to find the equation of the curve by means of a knowl- edge of the preceding articles. A few very simple examples have been given in Art. 97. 204 INFINITESIMAL CALCULUS. [Ch. XII. Note 1 . It may be worth while merely to glance at more difficult problems of this kind and at the text relating thereto, in Chapter XXI. and in Murray's Introductory Course in Differential Equations^ Chaps. V. and X. Also see Cajori, History of Mathematics^ ^^^.201 -20'^^ " Much greater than . . . integral of it." Note 2. It has been shown in Arts. 58, 59, that for the curve whose equation is f(x, y) = 0, rectangular coordinates, if (x, y) denotes any point on the curve and m is the slope of the tangent at (x, ?/), then m = -^ ; subtangent = y—; subnormal = y ^• dx dy dx Note 3. It has been shown in Arts. 60, 61, that for the curve whose equation is /(r, 6) = 0, if (r, 6) denotes any point on the curve, xp the angle between the radius vector and the tangent at this point, and

, «, &, derive some of the integrals 21-30, 37-46, 53-61, pages 403-406. Note 5. On the integration of irrational expressions also see Snyder and Hutchinson, Calculus, Arts. 129-131, 139, 140. These articles convey valu- able additional information, and, in particular. Art. 139 gives an interesting geometrical interpretation concerning the rationalisation of the square root of a quadratic expression. Also see the references given in Art. 122, Note 2. INTEGRATION OF TRIGONOMETRIC FUNCTIONS. N.B. On account of the numerous relations between the trigonometric ratios, the indefinite integral of a trigonometric differential can take many forms. 119. Algebraic transformations. A differential expression in- volving only trigonometric ratios can be transformed into an algebraic differential by substituting a variable, t say, for one of the trigonometric ratios. The algebraic differential thus obtained may possibly be integrated by some method shown in the preced- ing articles. Knowledge as to what substitution wiK be the most 216 INFINITESIMAL CALCULUS, [Ch. XIII. convenient one to make in a given case can best be acquired by trial and experience. Illustrations of this article have already been met in Art. 105, Exs. 10, 11, 16, 17. Ex. 1. See exercises just referred to. Ex. 2. Do Exs, 1-5, 7-9, Art. 120, making algebraic transformations. 120. Integrals reducible to | F(u) du, in which u is one of the trigonometric ratios. (a) I sin** a? doc and | cos** oo doc are thus reducible when n is an odd positive integer. For I sin" xdx= I sin**"^ x • sin xdx — — ( (1 — cos^ x)~^ d (cos x). The latter form can be expanded in a finite number of terms, ~" being an integer, and then integrated term by term. | cos" x dx can be treated similarly. EXAMPLES. 1. I cos^x^^x = j cos^x • cosicdcc = j (1 — sin2x)2d(sina;) = 1(1—2 sin2 X + sin* x) d (sin x)— sin x — | sin^ x -{■ \ sin^ x -\- c. 2. I sin^ X dx, i cos-^ x dx, \ sin^ x dx. (6) j sin"ircos*^a5c?a? is thus reducible when either n or m is a positive odd integer. 3. I sin^ xcos^xdx = \ sin2 x cos 2 x sin x dx = — I (1 — cos2 x) cos2 X d (cos aj) = — I (cos^ x — cos^ x) d (cos x) 1 n = — f C0S2 X 4- ^Y COS ^ X + c. ^ 4. (1) f^il^^dx, (2) (cos'^xsm^xdz, (3) f52££^, y v^cosx "^ -^ Vsinx (4) j cos^ X sin^ x t^x. Note. Case (a) is a special case of (6). 120, 121.] TRIGONOMETRIC FUNCTIONS. 217 (c) I sec** a? dx and I cosec" x dx are thus reducible when n is a positive even integer. 5. Tcosee^ xdx = \ cosec* x • cosec^ xdx = — i (1 + cot^ x)^ d (cot x) = - cot X (1 + I COt2 X + ^ cot* X). 6. Show the truth of statement (c). 7. (1) I sec* X dx, (2) j cosec* x dx, (3) | sec^ x dx. (d) I taii»»* a? sec»* ac dx and i cot"* x cosec»» x dx are thus reduci- ble when n is a positive even integer^ or when m is a positive odd integer. 8. Show the truth of statement {d). 9. (1) i tan2 X sec* X (?x, (2) i sec^ x Vtan x (^x, (3) | tan^ x sec* x dx, (4) jtan^ X sec^ X (?x, (5) j cot^ x Vcosec x sin- u = \(l — cos 2 u), cos^ u = i (1 + cos 2 u). Accordingly, if n and m are positive even integers, sin** x, cos" x, and sin** X cos"* .T can be transformed into expressions which are rational trigonometric functions of 2 x. Differential expressions involving the latter are, in general, more easily integrable than the original differential expressions in x. Ex. 1. rcos*xdx= ({\{\-\-co&2x)fdx = jf (1 + 2 cos2 x + cos2 2x) dx. Now I 2 cos 2 xdx = sin 2 X, and j cos2 2xdx = ^ | (1 + cos4x) dx = \{x + ^ sin 4 x). .•. \ cos* x dx = | x 4- 5 sin 2 x + ^^ sin 4 x + c. Ex. 2. \ sin2 X cos^ x dx = ^ | sin^ 2 x dx = |^ j (1 — cos 4 x) dx = \x — j^^ sin 4 X 4- c. Ex.3. (1) (sin* xdx, (2) jcos^xdx, (3) j sin* x cos*^ x dx, (4) j sin^ X cos^ X dx, (5) | sin* x cos* x dx. 218 INFINITESIMAL CALCULUS. [Ch. XIII. 122. Reduction formulas. There are several formulas which are useful in integrating trigonometric differentials. A few of them are deduced here ; the deduction of the others is left as an exercise for the student. (a) To find A: \ sin^'xdx, and B: | cos'^xdx, when n is any integer. A. Integrate by parts, putting dv = sin X dx ; then u = sin"~^ x, V = — cos X, du = (?i — 1) sin''~^a.' cos x dx. .: I sin''xdx = — sin"~^ic cos x -\- (n — 1) | sin''~-cc cos^;c dx = — sin'^-^a? cos x -\- (n — 1) j sin''~-ic (1 — sin- a.') dx = — sin"~^cc cos X + (n — 1) j sin"'~^xdx — (n — 1) I sin^'xdx. From this, on transposition and division by n, /' n ^ si n"^^ X COS a; , n — 1 f ■ ,,o 7 /in sm" xdx = 1 — I sm" - x dx. (1) This is a useful formula of reduction when n is a positive integer. From it can be deduced a formula which is useful when the index is a negative integer. For, on transposition and division by ~ " ; formula (1) becomes /. ,,_o -, sin"~^ X cos X , n C - n ^ sin" -xdx = 1 I sm" x dx. 71—1 71 — IJ This result is true for all values of n, and, accordingly, for n = JV+ 2. On putting N+2 for 71, this becomes /. ^ , sin^+^ X cos X , N-\-2 C . ,^,0 7 /o\ sin^ic dx = -^— - I sm^+-xdx. (2) If ^is a negative integer, say — m, (2) may be written / dx _ _ 1 cos X m — 2 r dx ^r^. sin'^a; m — 1 sin"*~'a; m — lJ sin"'"^^ 122.] TRIGONOMETRIC FUNCTIONS. 219 In the above way calculate the following integrals : Ex.1. (1) isin^xdx, (2) iam^xdx, (3) isin^x^x, (4) isin^xdx. Ex. 2. (1) f-:^, (2) f .^^-, (3) f^^. ^ ^ Jsin-^a; ^ Jsin^x ^ Jsin^x Ex. 3. Compare the results in Exs. 1, 2, with those obtained for these hitegrals by methods of the preceding articles. B. Similarly to A there can be deduced results 69, 71, page 407, for B. Formula 69 is useful for positive indices, and 71 for negative indices. Ex. 4. Deduce formulas 69 and 71. Ex. 5. (1) (cos^xdx, (2) fcos^x^x, (3) (*-^^, (4) f-^. J J Jcos*x J cos^x Compare results with those obtained by methods of preceding articles. (b) To find I sec'^xdx ivJien n is a positive integer greater than 1. Put sec"^ xdx = dv; then sec"~^a: = w, tan X = V, (n — 2) sec"~^ x tan x dx = du. .'. I sec'* X dx = sec"~^ x tan x — (n — 2) I sec""^ x tan^x dx. From this, on substituting sec'a; — 1 for tan-ic, and solving for I sec" a; da;, /„ 7 sec^^^x'tano; , n — 2 C „_<, , sec" X dx = — H I sec" - x dx. n ~1 n — lJ Similarly, result 73 for | cosec" x dx can be obtained. Ex. 6. (1) fsec^xdx, (2) Tsec^xdx, (3) sec^xc^x. Ex. 7. (1) fcsc^xf^x, (2) fcsc*x(?x, (3) csc^x^Zx. Ex. 8. Derive formula 73. Ex.9. From formulas 72 and 73 derive formulas for jsec«X(?x and 1 cosec*' X dx which are applicable when n is a negative integer. [Suggestion : Use method employed in deducing formulas 70 and 71.] 220 INFINITESIMAL CALCULUS. [Ch. XIII. (c) To find I taiV^xdx, in ivhich n is a positive integer greater than 1. I tan** xdx = I taii"~^ x tan^ xdx— | tan"~^ x (sec- x — 1) dx = Aaii'^-^ X d (tan x) — ftan'-^ x dx ^tanr^_ r^^^n-.^ax. n-1 J Similarly can be shown result 75 for | cot'' xdx. When n is negative, say — m, then j tan" xdx= I cot"* x dx, and I tan"* a; da; can be expressed in cotangents by formula 75. Formulas applicable to cases in which n is negative, can be deduced from formulas 74 and 75, by the method used in deducing formulas 70 and 71. Ex. 10. Deduce Formula 75, and formulas applicable to ltaii«xd!x and ( cot^xdx when n Is negative. Ex. 11. (1) (tsin^xdx, (2) cot* x dfx, (3) ftan*xc?x, (4) (tdni^xdx. {d) j sm*"a?cos**i»e?a?. When m and n are integers, reduction formulas can be derived for this integral in a manner similar to that used in Art. 118 ; that is, by (i) Connecting it with each of the four integrals in turn, viz. : j sin"'~^xcos'*a;da;, j sin"*a;cos"~^a;rZa;, j sin™+^ a; cos" a? cZo;, j sin"*a7C0S"+^a;dx; (ii) Forming a new function by giving sin x and cos x each an index one greater than the lesser of its indices in the required integral and the integral with which it is connected, and taking the product ; (iii) Giving the connected integral and this newly formed function each an arbitrary coefficient, and equating their sum to the required integral j 122.] TRIGONOMETRIC FUNCTIONS. 221 (iv) Determining the value of these coefficients by proceeding as in Art. 118. The derivation of these reduction formulas is left as an exercise for the student ; they are given in the set of integrals, Nos. 76-79.* Ex. 12. Deduce formulas Nos. 7G-79 by the methods outlined above. Ex. 13. Deduce the formulas in Ex. 12 by integrating by parts. Ex. 14. Apply these formulas to finding the following mtegrals : (1) rsin2a-.cos2a:(Zx; (2) f cos* x sin2 a; ; (3) C^^^dx. J J J sin2 X • Ex. 15. Deduce the integrals in Ex. 14 by the method outlined in {d). Note 1. When w + w is a negative even integer, the above integral can be expressed in the form i /(tan a:)cZ(tan x). Ex. 16. r^Ei? dx = r?llli^ . -J— . dx = ftan^ x sec* x dx J COS^ X J COS^ X COS* X J = \ tan^ X (1 4- tan2 x) c? tan x = ^^(6 + 4 tan2 x) tan* x. Ex.17. (1) f^^^cZx, (2) (^^^dx, (3) f^-^^dx. J sin^ X J cos^ X J sin*5 x Note 2. Special forms. Integrals 80-87 are occasionally required. For their deduction see Murray, Integral Calculus, Arts. 54-57, or other texts. It will be a good exercise for the student to try to deduce these integrals him- self. For a fuller discussion of the integration of irrational and trigonometric functions see the article Infinitesimal Calculus (Ency. Brit., 9th edition), §§ 124 on ; also see Echols, Calculus, Chap. XVIII. Note 3. On integration by infinite series. See Art. 126. Note 4. Elliptic integrals. Elliptic functions. The algebraic inte- grands considered in this book give rise only to the ordinary algebraic, circular, and hyperbolic t functions. (The two last named are singly periodic functions.) Certain irrational integrands give rise to a class of functions treated in hi!?) — X >a + 6 (3) rJ^ — ^c?a; = V(a-x)(6 + x) + (a + 6)sin-iJ?-+|. (4) (* J^dii? ax = V(a + x)(6 + x) + (a - 6) log ( VoT^ + v 6Tx) . ^ '6 + X (5) f ^^ - ..2sin-iJ^HJ. -^ V(x-a){b-x) ^b-a 5. Show that, if /(i<, v) is a rational function of 7i and v, and m and w are m integers, then /{x^, (a + bx'^)>^}xdx can be rationalised by means of the sub- stitution a + &x2 = ;s". (Ex. 14, or Note 3, Art. 117, is a particular case of this theorem.) ^ 6. Show that (1) r sin2- dx = 1 • ^ • 5 ... (2m- 1) . ^ Jo 2. 4. 6... 2 m 2 (^) r 2. 4. 6... 2 m 2 sin2"»+ixdx = — — (m being an integer). 3.5.7... (2w + l) CHAPTER XIV. APPROXIMATE INTEGRATION. INTEGRATION. MECHANICAL 123. Approximate integration of definite integrals. It has been shown in Arts. 95, 96, 98, that : (a) the definite integral I f{x)dx may be evahiated by finding the anti-differential of f(x)dx, (x) say, and calculating cfj(b) — <^(a) ; (b) this last number is also the measure of the area of the figure bounded by the curve y =f(x), the a>axis, and the two ordinates for which x = a and x = b. In only a few cases, however, can the anti-differential of f{x)dx be found ; in other cases an approximate value of the definite inte- gral can be obtained by making use of fact {b). Thus, on the one hand the evaluation of a definite integral serves to give the measurement of an area; on the other hand the accurate measure- ment of a certain area will give the exact value of a definite integral, and an approximate determination of this area will give an approximate value of the integral. The area described above may be found approximately by one of several methods ; two of these methods are explained in Arts. 124 and 125. 124. Trapezoidal rule for measuring areas (and evaluating definite integrals). Let the value of the definite integral I f(x)dx be required. Plot the curve y=f{x) from ic = a to x=b. Let OA = a, OB = b, and draw the ordinates AP Siud BQ. By Art. 96, the measure of the area APQB is the value of the required integral. An approxi- mate value of the area APQB can be found in the following Fig. 63. 223 B X 224 INFINITESIMAL CALCULUS. [Ch. XIV. way. Divide the base AB into n intervals each equal to Ax, and at the points of division A^, ^2? ^3> ••*? erect ordinates A^P^^ A2P2, A^Ps, •••. Draw the chords PI\, P^P.,, P2A, •', thus forming the trapezoids AP^, AiPo, A.2P2, •••. The sum of the areas of these trapezoids will give an approximate value of the area of APQB. ATe3iAP, = ^(AP-^AiP,)Ax, area A1P2 = ^ (^lA + A2P2) ^x, area AoP^ = -J- (AP2 + A^Ps) Ax, area A,_,Q = 1 (^„_iP,_i + BQ) \x. .'. area of trapezoids = (i AP + A^P^^ + A2P2 H -f- A-i^n-i + iBQ)Ax. This result may be indicated thus : . area trapezoids =(| + l + l + ... + l+|) Aac, in which the numbers in the brackets are to be taken with the successive ordinates beginning with AP and ending with BQ. Note. It is evident that the greater the number of intervals into which 6 — a is divided, the more nearly will the total area of the trapezoids come to the actual area between the curve and the x-axis, and, accordingly, the more nearly to the value of the integral. See Exs. 1, 2. EXAMPLES. 1. Find \ x^dx, dividing 12 — 1 into 11 equal intervals. Here each interval, Ax, is 1. Hence, approximate value = (^ . 12 + 22 + 32 + 42 + 52 + 6-2 + 72 + 82 -f 92 + 102 + IP + 1 . 122) = 577|. /•I2 r^3 -| 12 The value of I x^dx= \—+ c\ = 575 1. The error in the result ob- tained by the trapezoidal method is thus, in this instance, less than one- third of one per cent. 2. Show that if 22 equal intervals be taken in the above integral, the approximate value found is 576.125. ri') 8. Show that on using the trapezoidal rule for evaluating I x'^dx^ if 10 Intervals be taken, the result is If units more than the true value, and if 20 intervals be taken, the result is»j\ of a unit more than the true value. 125.] PARABOLIC BULE. 225 4. Explain why the approximate values found for the integrals in Exs. 1, 2, 3, are greater than the true values. 5. Evaluate I cos x dx by the trapezoidal rule, taking 10' intervals. ^^° {Ans. .0148. The calculus method gives .0149.) /•320 6. Evaluate \ sin x dx, taking 30' intervals. •^^^° {Ans. .0506. Calculus gives .0508.) /•380 7. Evaluate i cos x dx, taking 1° intervals. •^^^° (Ans. .1509. Calculus giv6s. 1510.) 125. Parabolic rule* for measuring areas and evaluating definite integrals. Let the area and the integral be as specified in Art. 124. For the application of the parabolic rule, the interval AB is divided into an even y^ number of equal intervals each equal to Aa;, say. The ordinates are drawn at the points of division. Through each successive set of three points (P, P„ P,), (A, Ps, P4), •••, are drawn arcs of parabolas whose axes are parallel to the ordinates. The area between these parabolic arcs and the ic-axis will be approximately equal to the area between the given curve and the x-axis. The area bounded by one 6f these parabolic arcs and the a>axis, and a pair of ordinates, say the area of the parabolic strip APP1P2A2, will now be found. Parabolic strip APP1P2A2 = trapezoid APP2A2 + parabolic segment PP1P2. (1) Now the parabolic segment PP1P2 = two-thirds of its circumscribing parallelogram PPPaP.t (2) Ax A-i A-i, A^ Fig. G4. * This rule, which is much used by engineers for measuring areas, is also known as Simpson's one-third rule, from its inventor, Thomas Simpson (1710-1761), Professor of Mathematics at Woolwich. t See Art. Ill, Ex. 19. 226 INFINITESIMAL CALCULUS. [Ch. XIV. Area trapezoid APP^A^ = \ AA2 (AP -f- A2P2) ; area PPP2P2 = area J.P'P'2^2 — area APP2A2 = 2'^AA2'A,F^-iAA2(AP + A2P2). (3) Hence, by (1), (2), and (3), area parabolic strip APP1P2A2 = (.4P+4AA + ^2/^2)y- Similarly, area of next parabolic strip ^^^2^^4-44 = (A2P2-\-4:A,P, + A,P,)^^-', and so on. Addition of the successive areas gives total area of parabolic strip =(AP^^AP, + 2 A2P2 + 4 A.P, + 2A^,+ -4-^Q)y- This result may be indicated thus : Total parabolic area = (1+4+2 + 4 + . .. + 2 + 4 + 1)^^, (4D o in which the numbers in the brackets are understood to be taken with the successive ordinates beginning with AP and ending with BQ. EXAMPLES. /•lO 1. Find i x^ (Za;, taking 10 equal intervals. Here, each interval = 1. Hence, the result by (4) = (1 . 03 + 4 • 13 + 2 • 23 + 4 . 33 + 2 . 43 + 4 . 53 + 2 . 63 + 4 . 73 + 2 . 83 + 4 . 93 + 1 . 103) X 1 = 2500. t/y.4 nio — + c = 2500. 2. Calculate the above integral, using the trapezoidal rule and taking 10 equal intervals. rn 3. Evaluate i x'^ dx, both by the trapezoidal and the parabolic rules, taking 10 equal intervals. 4. Evaluate Ex. 1, Art. 124, by the parabolic rule. Why is the result the true value of the integral ? 5. Show that there is only an error of 14 in 20,000 made in evaluating \ x'^dx by the parabolic method, when 10 intervals are taken. 126.] INTEGBATION IN SEBIES. 227 6. Find the error in the evakiation of the integral in Ex. 5 by the trapezoidal method, when 10 intervals are taken. 7. Evaluate the integrals in Exs. 6, 7, Art. 124, by the parabolic rule. Note. For a comparison between the trapezoidal and parabolic rules, for a statement of Durand's rule, which is an empirical deduction from these two rules, for a statement of other rules for approximate integration, and for a note on the outside limits of error in the case of the trapezoidal and parabolic rules, see Murray, Integral Calculus, Arts. 86, 87, Appendix, Note P], and foot-note, page 186. 126. Integration in series. The methods described or referred to in Arts. 124 and 125 for evaluating a definite integral | f(x)d^, give a numerical result only, and do not convey any information as to the anti-differential of f(x)dx. Some information, however, about the anti-differential of f{x)dx can be obtained in certain cases by expanding f(x) in a series in ascending or descending powers in x and then integrating f(x)dx term by term. The expanded series can represent f(x) only for the values of x in a certain definite range, namely, the range of values for which the series is convergent. The series obtained by integration is con- vergent for the same range of values of x, and /or values of x in this range represents the anti-differential. See Chapter XIX. (in particular, Arts. 172, 174), where the question of integration in series is more fully discussed. The following examples and note are given here mainly for the purpose of drawing attention to, and arousing interest in, questions relating to series. EXAMPLES. 1. Given that e» = l + a:-f- — + ?-+ •••, show that \ e"" dx — e'' -\- c, in which c is a constant. ■ ^ ' 2. Given that cos x = \ — — -f^ ..., and that sin a: = x — — -|- — 2!4! 3!5! — •••, show that \ cos x dx = sin x + c, and that j sin x (?x = — cos x + c. 3. Do Ex. 2, Art. 174. 4. Find an approximate value of the area of the four-cusped hypocycloid inscribed in a circle of radius 8 inches. (This area can also be found exactly ; see Art. 137, Note 5, Ex.) 228 INFINITESIMAL CALCULUS. [Ch. XIV. Note. Expansion of functions in series: (a) by differentiation; (6) by integration. For remarks on this topic and for the warrant for the operations in Exs. 5, 6, 7, see Chapter XIX., in particular, Arts. 168(e), 172, 173, 174. 5. Suppose it is known that sin y, = X — — + ^ •••, (1) 3 ! 6 ! and thut — (sin x) = cos x. Differentiation of the members in (1) gives cosx = l-^ + ^-.... 2 ! 4 ! 6. By the binomial theorem, (1 - x2)^ = 1 _ I x2 - 1 a;4 - ^i^ccfi . (1) Differentiation of each member of (1) and division by % gives (1 - a:2)~^ = i-f ^ x2 + f a:* + t\ »^^ + •••• (2) Result (2) may be verified by expanding (1 — x^)"^^ by the binomial theorem. 7. Given that d(tan-ix)= — ^ (Art. 51)=l-x2 + x* , find tan-ix. -^^^^^ i r^ x^ On mtegration, tan-i x-\- c — x V- — ••-. 3 5 From this, on putting x = 0, tan-i + c = 0. .-. c = — tan-i = ± nv^ in which n is any integer. .-. tan-i X = WTT + X - — + — . 8. Do Exs. 3-5, 8, 9, Art. 174. 127. Mechanical devices for integration. The value of a definite integral may be determined by various instruments. Accordingly, they may be called mechanical integrators. Of these there are three classes, viz. planimeters, integrators, and integraphs. These instruments are a great aid to civil, mechanical, and marine engineers. The area of any plane figure can be easily and accu- rately calculated by each of these mechanisms. Their right to be termed mechanical integrators depends on the facts emphasised in Arts. 96, 98, 123-125; the facts, namely, that a definite integral can be represented by a plane area such that the number of square units in the area is the same as the number of units in the inte- gral, and hence that one way of calculating a definite integral is to make a proper areal representation of the integral and then measure this area. 127.] PLANIMETERS, INTEGRAPHS. 229 Planimeters, which are of two kinds, viz. polar planimeters and rolling planimeters, are designed for finding the area of any plane surface represented by a figure drawn to any scale. The first planimeter was devised in 1814 by J. M. Hermann, a Bavarian engineer. A jjolar planimeter, which is a development of the planimeter invented by Jacob Amsler at Konigsberg in 1854, is the one most extensively used. By it the area of any figure is obtained by going around the boundary line of the figure with a tracing point and noting the numbers that are indicated on a measuring wheel when the operation of tracing begins and ends. Integrators and integraphs also serve for the measurement of areas ; they are adapted, moreover, for making far greater compu- tations and solving more complicated problems, such as the calcu- lation of moments of inertia, centres of gravity, etc. The integraph (see xirt. 100, Notes 2, 3) is the superior instrument, for it directly and automatically draws the successive integral curves. These give a graphic representation of the integration, and are of great service, especially to naval architects. The measure of an ordi- nate of the first integral curve, when multiplied by a constant belonging to the instrument, gives a certain area associated with that ordinate (see Art. 100). Note 1. A bicycle with a cyclometer attached may be regarded as a mechanical integrator of a certain kind ; for by means of a self-recording apparatus it gives the length of the path passed over by the bicycle. Note 2. Planimeters and integrators are simple, and it is easy to learn to use them. Note 3. A brief account of the planimeter^ references to the literature on the subject, and a note on the fundamental theory, will be found in Murray, Integral Calculus^ Art. 88, and Appendix, Note F. Also see Lamb, Calculus^ Art. 102 ; Gibson, Calculus, § 130. For a fuller account see Henrici, Report on Planimeters (Report of Brit. Assoc, for Advancement of Scienfee, 1894, pages 496-523) ; Hele Shaw, Mechanical Integrators (Proc. Institution of Civil Engineers, Vol. 82, 1885, pages 75-143). For references concerniDg the integraph see Art. 100, Note 3. N.B. Interesting information concerning planimeters, integrators, and the integraph, with good cuts and descriptions, are given in the catalogues of dealers in drawing materials and surveying instruments. CHAPTER XV. SUCCESSIVE INTEGRATION. MULTIPLE INTEGRALS. APPLICATIONS. 128. In Chapter VI. (see Arts. 68, 69, 70), successive deriva- tives and differentials of functions of a single variable were obtained. In Chapter VIII. (see Arts. 79, 80, 82), successive par- tial derivatives and partial differentials of functions of several variables were discussed. In this chapter processes which are the reverse of the above are performed and are employed in practical applications. 129. Successive integration : One variable. Applications. Suppose that 1 /(i«)c?^ =/i(^)? (1) fA(x)dx=f,(x), ' (2) jMx)dx=Mx). (3) Then, by (3) and (2), fs(x)=j\^ff,{x)dx~\dx', (4) By (4) and (1), /.(x) =f\f(^ff(^)^^) ^^] ^^- (^) This k written f^(x) = f f ff{x) {dxf, or, more usually, /3(^) = 1 I \ f{x)dx^. (6) The second member of (6) is called a triple integral. Similarly, the second member in (4) is usually written I | f^(x)dx^, and is called a double integral. In general, I I I "• j f(3o)dac^ denotes the lesult obtained by 230 128, 129.] SUCCESSIVE INTEGRATION. 231 integrating f(x)dx n times in succession. This integral is indefi- nite unless end values of the variable be assigned for each of the successive integrations. This integral and the integrals in (4) and (5) are called multiple integrals. Note. It should be observed that here dx" denotes dxdxdx-"io n factors, i.e. {dxy, and not d • x» {i.e. nx'^-^dx). [Compare Art. 70.] EXAMPLES. 1. Find r r (x'^dx^. = ^ 4- kix" + cax + f3 ; for, since Ci is an arbitrary constant, ^ may be denoted by an arbitrary con- 2 stant k\. 195 ■ - ri>''^-r[j>-]-=j;[f+^]>=fr-='-i 3. Determine the curves for every point of which -^-^ = 0. Which of these curves goes through the points (1, 2), (0, 3) ? Which of these curves has the slope 2 at the point (3, 5) ? TT ^^V On integrating, -^ = Ci. On integrating again, y = cix + C2, which represents all straight lines. For the line going through (1, 2) and (0, 3), 2 = Ci + C2 and 3 = + C2 ; whence ci = — 1, C2 = 3. Hence the line isx + y = S. For the line having the slope 2 at (3, 5), Ci = 2 and 5 = 3 Ci + C2, whence C2 = — 1. Hence the line is y = 2x— 1. 4. Determine the curves for every point of which the second derivative of the ordinate with respect to the abscissa is 6. Which of these curves goes through the points (1, 2), (- 3, 4) ? Which of them has the slope 3 at the point ( - 2, 4) ? 232 INFINITESIMAL CALCULUS. [Ch. XV. N.B. The student is recommended to write sets of data like those in Exs. 3-7, and determine the particular curves that satisfy them. He is also recommended to draw the curves appearing in these examples. 5. Determine the curves for every point of which the second deriva- tive of the ordinate with respect to the abscissa is 6 times the number of units in the abscissa. Which of these curves goes through the points (0, 0) (1, 2) ? Which of them has the slope 2 at (1, 4) ? 6. Determine the curves in which the second derivatives -v4 from point to point vary as the abscissas. Find the equation of that one of these curves which passes through (0, 0), (1, 2), (2, 5). Find the equation of that one of these curves which passes through (1, 1), and has the slope 2 at the point (2, 4). 7. Determine the curves in which the second derivative of the abscissa with respect to the ordinate varies as the ordinate. Which of these curves passes through (0, 1), (2, 0), (3, 5) ? Which of them has the slope ^ at (1, 2), and passes through (— 1, 3) ? 8. A body is projected vertically upward with an initial velocity of 1000 feet per second. Neglecting the resistance of the air and taking the accelera- tion due to gravitation as 32.2 feet per second, calculate the height to which the body will rise, and the time until it again reaches the ground. 9. Do Ex. 20, Art. 68. ^ 10. When the brakes are put on a train, its velocity suffers a constant retardation. It is found that when a certain train is running 30 miles an hour the brakes will bring it to a dead stop in 2 minutes. If the train is to stop at a station, at what distance from the station should the engineer whistle "down brakes" ? (Byerly, Problems in Differential Calculus.) 130. Successive integration : several variables. Suppose that J/ {^, y, ^) ^2; = Ux, y, 2), (1) ffi(^, y, ^)dy =,f2(^, y, ^), ' (2) jf2(x, y, z) dx =fi(x, y, z). (3) The integration indicated in (1) is performed as if y and x were constant; the integration in (2) as if x and z were constant; the integration in (3) as if z and y were constant. (Compare Arts, 79, 130.] SUCCESSIVE INTEGRATION. 233 From (3) and (2), fs(x, y, z) =J\ j^f^(x, y, z)dy \ dx; (4) from (4) and (1), =J | flf-^^'"' ^' ^^ ^^1"^^ \ '^'^' ^^^ The second member in (4) is often written Jjfi(^,y,z)(il/d^; (6) the second member in (5) is often written ffffi^, yy 2) dz dy dx. (7) The integral in (6) is called a double integral, and the integral in (7) a triple integral. Note 1. It should be observed that according to (2), (3), and (4), inte- gral (6) is obtained by first integrating /i(x, y, z) with respect to ?/, and then integrating the result with respect to x ; in (7), according to (1), (2), (3), and (5), the first integration is to be made with respect to z, the second with respect to y, and the third with respect to x. That is, XhQ first integration sign on the right is taken icith the first differential on the left, the second integra- tion sign from the right with the second differential from the left, and so on. When end-values are assigned to the variables, careful attention must be paid to the order in which the successive integrations are performed. Note 2. The notation used above for indicating the order of the variables with respect to which the successive integrations are to be performed, is not universally adopted. Oftentimes, as may be seen by examining various texts on calculus and works which contain applications of the calculus, integrals (6) and (7) are written \ ) /i(^^ y^ ^) ^^ ^y^ Ml •^^^' ^' ^^ ^^^ ^^y ^^ respectively. In this notation the first integration sign on the right belongs to the first differential on the right, the second integration sign from the right to the second differential from the right, and so on ; and the integrations are to be made, first with respect to z, then with respect to ?/, and then with respect to x. In particular instances, the context will show what notation is employed. EXAMPLES. 1. f f f ^ V^ ^^ (iy ^^^ = f f^^^ (f + ^i) ^y ^^ 234 INFINITESIMAL CALCULUS. [Ch. XV. 2. CCC v^yz^ dz dy dx (i.e. P"^* f^T' f -T^^^^^ ^^ ^^^ ^^^ 4 J2 L2 Ji 2 4 J2 3. ^;-j:'^Y-do.^=l'^-{£^'^Vdrj}.l. =f x3[|+ c];^<& iJVo x6-iK3)dx = 28,Vo- 4. Evaluate the following integrals : (1) i i' \ xy'^z dz dy dx. (2) i Y {Zw-'lv^dw dv. (3) j^ J ^st - t^ ds dt. 'TT ^a{l—coa0) r^ cos ddr dd. jr /»2 /•2/r /•2a cos (6) J j \ r^ sin d dr d(p de. n J' 2 /•2a cos X '■*"''■ ■'(£) rd^ (^r. V^ rdr dd. ♦131. Application of successive integration to finding areas: rec- tangular coordinates. EXAMPLES. 1. Find the area between the curve y"^ = Sx, the x-axis, and the ordinate for which x = S. At P, any point within the figure OWM whose area is required, suppose that a rectan- gle PQ having infinitesimal sides dx and dy parallel to the axis is constructed. The area OTFJf is the limit of the sum of all rectangles such as PQ which can be constructed side by side in OWM. Let one of the vertical sides of the rectangle be produced both ways until it meets the curve and the aj-axis in T and S; complete the rectangle TV as in the figure. First, find the area of the rectangular strip TV by finding tlie limit of the sum of the rectangles PQ inscribed in it from >S' to T; then find the limit of the sum of the strips like TV which can be inserted between OF and 31 W. Fig. 65. 131,132.] SUCCESSIVE INTEGBATION. 235 Area TV = lim ^ (rectangles PQ) = f dy dx = y/^x dx. (1) y&tS Area 0.¥Jr = lira 2^ (strips TF) = J^ J J ^ <^1/ J ^^ (2) X atO = 2\/2 I x^ dx = 4VQ square units. The last expression in (2) is usually written ( i ^dydx. The area of O WM may also be found by finding the limit of the sum of the rectangles PQ which may be inserted between R and U, and then finding the limit of the sum of the strips like PL which may be inserted between 03/ and W. Thus, area PL = ^^^'^ dx dy = £^ dx dy = (^S-^^dy; (3) area OMP = J;;;(3 - 1') = angle POZ, <^ = angle XOM, 0^ being the projection of OP on XOY. In other words, ^ = the angle between the plane XOZ and the vertical plane in which OP lies.] Pro- duce OP an infinitesimal dis- tance dr to Pi, and revolve OPPi through an infinitesimal le dO in the plane ZOP to the position OQ. Now revolve OPPiQ about OZ through an infinitesimal angle dcfi, keeping $ constant. The solid PP^QR is thus generated. Its edges PPi, PQ, PR are respectively dr, rdd, r sin deft; its volume (to within an infinitesimal of an order lower than the third) is 7^ sin 6 dr d<^ dd. On determining the proper limits for r, <^, 0, and integrating, the volume required is obtained. Ex. 1. Find the volume of a sphere of radius a, using polar coordinates and taking on the surface of the sphere and OZ on the diameter through 0. (It will be found that the volume is given by the integral in Art. 130, Ex. 4, (6). See Murray, Integral Calculus, Art. 63, Ex. 1.) B. The element of volume can be chosen in another way, which sometimes leads to simpler integrations than are otherwise obtain- able. An instance is given in Ex. 2 below. Fig. 67 EXAMPLES. 2. Another way of doing Ex. 7, Art. 132. In the figure, 0-ABC is one-eighth the sphere, and the solid bounded by the plane faces ALBO, AKO, the spherical face ALBVA, and the cylindrical 133.] SUCCESSIVE INTEGRATION. 239 face AVBOKA is one-fourth of the part of the cylinder intercepted by the sphere. In AOK take any point P. Let 0F = 7\ and angle AOP=d. Produce OP an infinitesimal dis- tance dr to Pi, and revolve OP Pi through an infinitesimal angle dd. Then PPi generates a figure, two of whose sides are dr and rdd. Its area (to within an infinitesimal of an order lower than the second) is r dr dd. (See Art. 136, Note 3, Ex. 8.) On this infinitesimal area as a base, erect a vertical column to meet the sphere in M. Then 'PM — y/a^ — r^, and the volume of the column is Va^ — r^ • rdrdd. This is taken as the element of volume ; the limit of the sum of these columns standing on AOK is the vol- ume required. Keeping 6 constant, first find the limit of the sum of the columns standing on the sector extending from O to K whose angle is dd. /• r=a cos . Since 0K= acosd, this limit is \ Va^ — r'^ • rdrdd. This gives the Jr=Ji volume of a wedge-shaped slice whose thin edge is OB. One-fourth of the volume required is the limit of the sum of all the wedge-shaped slices of this kind that can be inserted between AOB and COB ; tliat is, from ^ = to Fig. 68. 2' vol. required = 4 Jd=0 Jr 7-=a cos 9 \/a2 r^ -rdrdd = ^ira^-^aK [See Art. 130, Ex. 4 (9).] In this instance this is a very much shorter way of deriving the volume than by starting with the element dxdydz, as in Art. 132. 3. Eind the volume of a sphere of radius a, taking O at the centre : (1) choosing the element of volume as in ^ ; (2) choosing it as in B. 4. The axis of a right circular cylinder of radius b passes through the centre of a sphere of radius a (a> b). Find the volume of that portion of the sphere which is external to the cylinder. CHAPTER XVI. FURTHER GEOMETRICAL APPLICATIONS OF INTEGRATION. 134. In this chapter the calculus is used for finding volumes in a particular case, for finding areas of curves whose equations are given in polar coordinates, for finding the lengths of curves whose equations are given either in rectangular or in polar coordi- nates, for finding the areas of surfaces in two special cases, and for finding mean values of variable quantities. N.B. Many of the problems in this chapter are presented in a general form. In such cases the student is recommended, when he obtains the general result, to make immediate application of it to particular concrete cases. 135. Volumes of solids the areas of whose cross-sections can be expressed in terms of one variable. In Art. 112 the volumes of solids of revolution were found by making cross-sections of the solid at right angles to the axis of revolution, taking these cross- sections an infinitesimal distance apart, and finding the limit of the sum of the infinitesimal slices into which the solid is thus divided. This method of finding the volume of a solid can some- times be easily applied in the case of solids which are not solids of revolution. The general method is : (a) to take a cross-section in some convenient way; (b) to express the area of this cross- section in terms of some variable ; (c) to take a parallel cross-sec- tion at an infinitesimal distance from the first cross-section ; (d) to express the volume of the infinitesimal slice thus formed, in terms of the variable used in (b) ; (e) to find the limit of the sum of the infinite number of like parallel slices into which the solid can thus be divided. There is often occasion for the exercise of judg- ment in taking the cross-sections conveniently. 240 134, 135.] VOLUMES. 241 EXAMPLES. 1. Find the volume of a right conoid with a circular base of radius a and an altitude h. Note 1. A conoid is a surface which may be generated by a straight line which moves in such a manner as to intersect a given straight line and a given curve and always be parallel to a . given plane. In the conoid in this example the given plane is at right angles to the given straight line, and the perpendicular erected at the centre of the circle to the plane of the base intersects the given straight line. Let LM be the fixed line and ABB the fixed circle having its centre at C. Take a cross-section PQB at right angles to LM, and, accordingly, at right angles to a diameter AB. Let it intersect AB in D, and denote CD by X. Area PQB =:^PDQB = PD' QD. Now PD — h, and, by elementary geometry Fig. 69. QD = VAD'DB= Via -x)(a + x) = Va^ - x^. .-. area PQB = hVa^ - x^. Now take a cross-section parallel to PQB at an infinitesimal distance from it. Since CD has been denoted by x, this infinitesimal distance may be denoted by dx. Vol. LM-BQABB = 2 vol. LG-TSAT x&tA = 2 lim (sum of slices PQB) = 2 A p Va2 - x-^ dx = l TQ^h. That is, the volume of the conoid is one-half the volume of a cylinder of radius a and height h. (See Echols, Calculus, Ex, 3, p. 266.) Note 2. As already observed, finding the volumes of solids of revolution is a special case under this article. Note 3. Two general methods of finding volumes have now been shown, namely, the method shown in Arts. 132, 133, and the method shown in this article. 242 INFINITESIMAL CALCULUS. [Ch. XVI. 2. Do Ex. 1, denoting AD by x. 3. Do Ex. 8, Art. 112 and Ex. 1, Art. 132 by method of this article. 4. Find the volume of a right conoid of height 8 which has an elliptic base having semi-axes 6 and 4, and in which the fixed line is parallel to the major axis. Find the volume in the general case in which the height is /i, the semi-major axis a, and the semi-minor axis h. 5. A rectangle moves from a fixed point, one side varying as the dis- tance from the point, and the other side as the square of this distance. At the distance of 3 feet the rectangle is a square whose side is 5 feet. What is the volume generated when the rectangle moves from the distance 2 feet to the distance 4 feet ? 6. On the double ordinates of the ellipse h'^x^ + cfiy^ = a^b^, and in planes perpendicular to that of the ellipse, isosceles triangles having vertical angles 2 a are erected. Find the volume of the surface thus generated. 7. A circle of radius a moves with its centre on the circumference of an equal circle, and keeps parallel to a given plane which is perpendicular to the plane of the given circle : find the volume of the solid thus generated. 8. Two cylinders of equal altitude h have a circle of radius a for their common upper base. Their lower bases are tangent to each other. Find the volume common to the two cylinders. 136. Areas: polar coordinates. Suppose there is required the area of the figure bounded by the curve whose equation is /(r,^) = 0, and the radii vectores drawn to two assigned points on this curve. Q{r2,dii) Let LG he the curve f(r, 6) = 0, and F and Q the points (r^, 6i) and (n, O2) respectively; it is required to find the area POQ. Sup- pose that the angle POQ is divided into n equal angles each equal to AO, and let VOW be one of these angles. Denote Fas the point (r, 0). Through V, about as a centre, draw a circular arc intersecting TF in M. 136.] AREAS: POLAR COORDINATES. 243 Through W, about as a centre, draw a circular arc intersecting OV in W. Denote MW hy A?-. Then, area OF3/=i?-A<9 (PL Trig., p. 175), and area ONW = i(r + Ar)-A^. Let "inner " and "outer" circular sectors, like VOM and NOW in the case of VW, be formed for each of the arcs like FTT which are subtended by angles equal to A^ and lie between P and Q. It is evident that total area of inner sectors -(ir-- (2) 138,180.] AREAS OF SURFACES. 249 It can also be shown [see the derivation of result (6), Art. 67 d], that s = j'7V^2^f,)' + l • ^^- (3) Note 1. The quantities under the integration sign in (2) and (3) are the infinitesimal elements of length in polar coordinates. The differential of the arc also has the same forms, Art. 67 d ; see Note 1, Art. 137. Note 2. In (2) the integrand must be expressed in terms of ^ ; in (3), in terms of r. Note 3. The intriusic equation of a curve. See Appendix, Note B. EXAMPLES. 1. Find the length of the card io Id r = a(l — cos d). . = 2£7^V.(|)^.. The substitution of the value of r and -- in the integrand and simplifica- ^. . d9 tion, give = 2aV2 f^Vl -coHdde = 4a ('' siu^ dd = Sa. Jo Jo 2 2. Find the lengths of the following : (1) The circle r = a. (2) The circle r r= 2 a sin ^. (3) The curve a 7' = asm^~- (4) The arc of the equiangular spiral r = «e^«^ta, (a) from o ^ = to ^ = 2 TT ; {b) from ^ = 2 tt to ^ = 4 tt. (5) The arc of the spiral of Archimedes r = ad from (ri, ^i) to (r2, 62) • (6) The arc of the parabola r = a sec2 -, (a) from 6* = to ^ = ^1; (6) from e = --tod = + -• 139. Areas of surfaces of revolution. Note 1. Geometrical Theorem. Let KL and BS (Fig. 74 a) be in the same plane. In elementary solid geometry it is shown that if a finite straight Ihie KL makes a complete revolution about RS, the surface thus generated by KL is equal to 2wT3I • KL, in which TM is the length of the perpendicular lat fall on RS from T, the middle point of KL. Suppose that an arc PQ of a curve y =f(x) revolves about the a'-axis, and that the area of the surface thus generated is required. 250 INFINITESIMAL CALCULUS. [Ch. XVI. Let P and Q be the points (xi, ?/i) and (xg, 2/2) respectively. Sup- pose that PQ is divided into small arcs such as KL, and denote K and L as the points (x, y) and {x -f- Ao?, ?/ + ^V) respectively. QkXo.y^ M Fig. 74 a. SO M Fig. 74 6. Draw the chord KL, and from T, the middle point of this chord, draw TM at right angles to the ic-axis. Then the area generated by the chord KL when the arc PQ revolves about the x-axis = 27rTM-KL = 2 TT (2/ + i Ay)yjl + ff) • Ax. (Note 1.) \Ax The smaller the chords KL are taken, the more nearly will the surfaces generated by them approach coincidence with the surface generated by the arc PQ, and the difference between area of the latter surface and the sum of the areas of the former surfaces can be made as small as one pleases by decreasing Ax. Accord- ingly, the area of the surface generated by the arc PQ is the limiting value of the sum of the areas of the surfaces generated by the chords KL (from P to Q) when these chords become infinitesimal. That is, area of surface generated by JPQ = liin^,^„52 ^(y + | ^y)^l + (|^)'a» (1) « r*2 L , fdv\2^ (Definitions of derivative .t.. 2-lj H^ + [^^) ^^' and integral.) (2) 139.] AREAS OF SURFACES. 251 If the length of the chord KL be denoted by Jl + f—\^y, this integral takes the form \^^/ surface =2.fyyji+(^Ji)W dy (3) Note 2. Each of the expressions to be integrated in (2) and (3) may be denoted by 2 iry ds [Art. 67 /(9)], and is called an element of the surface of revolution. If PQ is revolved about the y-axis, the element of surface is 2 ira? ds ; and the surface dy dy (4) The questions, whether to use form (2) or (3), and which of (4) to employ, are decided by convenience and ease of working. (See Art. 136, Note 1, and Art. 67/0 Note 3. In a similar manner it can be shown that the area of the surface generated by the revolution of an arc of a curve about any straight line in the plane of the arc, is ^^ 2 7ri \ds, (5) in which ds denotes an infinitesimal arc of the curve, I the distance of this infinitesimal arc from the straight line, and e\ and e^ are coordinates of some kind that denote the ends of the revolving arc. An illustration is given in Ex. 4. EXAMPLES. 1. Find the surface generated by the revolution of the hypocycloid a;3 + ?/3 = ^3 about the a;-axis. Surface = 2i'^''2Tr'PN'ds Jx=0 Jo ^ ^ . (See Art. 137, Ex. 1.) 6 Tr«3 p(al _ x3)^d(a^ - x^) 2. X = \^ra^ Fig. 75. 252 INFINITESIMAL CA L C UL ITS. [Ch. XVI. V- In this case an easier integral is obtained by expressing the surface in terms of y and dy^ as in form (3) . Thus, Surface = 2-2 tt P"^"?/^! -i-l^y^Ydy = 4 iraH^jKly = V -jra^, 2. Calculate the surface of the hypocycloid in Ex. 1, using the equations X = a cos^ d, y = a sin^ 6. 3. Derive formula (5). 4. The cardioid r = «(1 — cos^) revolves about the initial line : find the area of the surface generated. Surface = 2 tt i PN • ds. Je=o _ Now P^= rsin^ = a(l — cos^)sin ^, and ds = aV2Vl — cos d dd (see Ex. 1, Art. 138). (9=7r 6>-=0 .*. surface = 2-N/2 7ra2 \^ {I Fig. 76. cos^)"2 sin Odd rV2 L 5 7r«2(l 5"!' 5. Find the area of the spherical surface generated by the revolution of a circle of radius a about a diameter. 6. A quadrant of a circle of radius a revolves about the tangent at one extremity. What is the area of the curved surface generated ? 7. Calculate the area of the surface of the prolate spheroid generated by the revolution of the ellipse b'^x^ 4- a^y^ — a^b"^ about the a:-axis. 8. In the case of an arch of the cycloid x = a(d — sm6), y = a(l—cos6), compute : (1) the area between the cycloid and the cc-axis ; (2) the volume and the surface generated by its revolution about the x-axis ; (3) the volume and the surface generated by its revolution about the tangent at the vertex. 9. Find the volume and the surface generated by revolving the circle x^ j-{y — by = a'^ (& > a), about the x-axis. 140.] AREAS OF SURFACES. 253 10. Find the area of the surface generated by the revolution of the arc of the catenary in Ex. 6, Art. 112. 11. Tlie arc of the curve 7- = a sin 2 6, from ^ = to d =- (<.e. the 4 first half of the loop in the first quadrant), revolves about the initial line : find the area of the surface generated. What is the area of the surface generated by the revolution of the second half of the same loop about the same line ? 12. A circle is circumscribed about a square whose side is a. The smaller segment between the circle and one side of the square is revolved about the opposite side of the square. Find the volume and the surface of the solid ring thus generated. 140. Areas of surfaces whose equations have the form z =/(«, y) or F{pc, y, z) =0. It is shown in solid geometry that: (a) The cosine of the angle between the ac«/-plane and the tangent plane at any point (.r, ?/, z) on such a surface, supposed to be continuous, is {-(l)^-(l)f- (&) The area of the projection of a segment of a plane upon a second plane is obtained by multiplying the area of the segment by the cosine of the angle between the planes. It follows from (a) and (6) that : (c) If there be an area on the a;?/-plane equal to A, then A is the area that would be projected on the a;2/-plane by an area on the tangent plane at (a;, ?/, z) which is equal to (See C. Smith, Solid Geometry, Arts. 206, 20, 31 ; Murray, Integral Calcu- lus, Art. 75.) Let z =f(x, y) be the equation of a surface BFCRAGB [Fig. 66] whose area is required. Let P(x, y, z) be any point on this surface, and Pi the point (cc, y, 0) vertically below P. Let P\Q\ be a rectangle in the x2/-plane having its sides equal to Ax and Ay respectively, and parallel to the x- and ?/-axes. Through the sides of this rectangle pass planes perpendicular to the x^-plane, and let these planes make with the surface the section PQ, and with the tangent plane at P the section PQ^- {Q\Q produced is supposed to meet in Q2 the tangent plane at P.) Then, area Pi ^1 = Ax • Ay. Hence, by (2), area PQ., = a^I + (—Y-^ ifY'^^ ' ^' 254 INFINITESIMAL CALCULUS. [Ch. XVI. Now the smaller Ax and Ay become, the more nearly will the section PQ2 on the tangent plane at P coincide with the section PQ on the surface. Accordingly, the more nearly will the sum of the areas of sections like PQ2 on the tangent planes at points taken close together on the surface, become equal to the area of the surface ; moreover, the difference between this sum and the area of the surface can be made as small as one pleases. Con- sequently, the area of the surface is the limit of the sum of the areas of these sections on the tangent planes when these sections become infinitesimal. That is, ave^BFCRAGB=(''-''^ P^"^ Jl + f/^V+ fi^^. dt/c?^. J.e=0 J!/=0 ^ \dOCJ \dy) Note. The integral f y~^^^]l + (— y+ (^Ydyldx gives the area rx=OA of the strip or zone POL, and the integral \ BGLdx gives the sum of these zones from BOG to A. EXAMPLES. 1. Find the area of the portion of the surface of the sphere in Ex. 7, Art. 132, that is intercepted by the cylinder. The area required = 4 area AVBLA (Fig. 68). In this figure, the equation of the sphere is x^ + y^ -\- z^ = a^, and the equation of the cylinder is x"^ + y^ = ax. The area of a strip L V, two of whose sides are parallel to the zy-plane, will first be found ; then the sum of all such strips in the spherical surface AVBLA will be determined. Since the required surface is on the sphere, the partial derivatives must be derived from the equation of the sphere. Accordingly, ^ = -?, ^ = -??; hence, 1 + f^V+ (^^V= 1 +^ + ^ = ^^^ \dx) \dy) Z' Z'^ Z^ a2 _ a;2 _ y2 Also, BK = Vax .'. area AVBLA = \ \ " - dy dx ]Vax-xi Va2 - X2- a \ sin-i — y dx •r V=i dx. # 141.] MEAN VALUES. 255 This integral can be evaluated by integrating by parts. The integration can be simplified by means of the substitution sin z =\, — - — It will be found that area required = 4 area AVE LA = 2 (tt - 2)d^ = 2.2832 a^. 2. Find the area of the surface of the cylinder intercepted by the sphere in Ex. 7, Art. 132. 3. By the method of this article, find the surface of the sphere x- + y'^ + Z'^ — «2. 4. A square hole is cut through a sphere of radius a, the axis of the hole coinciding with a diameter of the sphere : find the volume removed and the area of the surface cut out, the side of a cross-section of the hole being 2 h. 5. Find the area of that portion of the surface of the sphere inter- cepted by the cylinder in Ex. 4, Art. 133. 141. Mean values. In Art. 98 it has been stated that if the curve y=zf{x) be drawn (Fig. 44), and if 0^1 = a and OB = h, then, of all the ordinates from A to B, the mean value = '^''^'^ f^^^ = jji^^ AB b-a ^ ^ Result (1) can be derived in the following way which has also the advantage of being adapted for leading up to a more general notion of mean value. The mean value of a set of quan- tities is defined as the sum of the values of the quantities the number of the quantities For instance, if a variable quantity takes the values 2, 5, 7, 9, the mean of these values is ^^- or 54. 4 Now take any variable, say x, and suppose that f(x) is a con- tinuous function, and let the interval from x = a to x = b be divided into n parts each equal to Ax, so that n Ax = b — a. Let the mean of the values of the function for the n successive values ' a, a-{- Ax, a + 2 Ax, • ••, a-\-n — l Ax, be required. The corresponding n successive values of the func- lon are ^^^^^ ^^^^ _^ ^^^^ ^^^ _^ ^ ^^^^^ ^ ^^^ _^ —-j ^ ^_^^^ 256 INFINITESIMAL CALCULUS. [Ch. XVI. Hence, mean value of function ^ /(^) +/(^ + Ao;) +/(a + 2 Aa^) + ♦ ■ . +/(« + ii - 1 • Aa;) (2) Now n ^x = h — a, whence n = Substitution in (2) gives mean value Ax ^ f(a)Ax -\-f(a + Aa;) Ax +/(a + 2 Ax) A a? H \-f(a + n—l Aa;^ A g ^ 6 -a ' ^3* Finally, let the mean of all the values that f(x) takes as x varies from a to 6 be required. In this case n becomes infinitely great and Ax becomes infinitesimal ; accordingly [Art. 96 (2), (3)], (3) becomes mean value fV(x) Ja dx h — a (4) as already represented geometrically in Art. 98. Note 1. Reference for collateral reading. Ecliols, Calculus, Arts. 150-152. EXAMPLES. 1. Find the mean length of the ordinates of a semicircle (radius a). the ordinates being erected at equidistant intervals on the diameter. Choose the axes as in Fig. 77. Then the equation of the circle is x-^ + 2/^ = «'^. Let PN denote any of the ordi- nates drawn as directed. Mean value ■ P" PN . dx (" Va'- - y^^ dx (-«) 2a ira^ 2.2a ,7854 a. 2. Find the mean length of the ordinates of a semicircle (radius a), the ordinates being drawn at equidistant intervals on the arc. Let PN be any of the ordinates drawn at equi- distant intervals on the arc, that is, at equal incre- ments of the angle 6. Mean value = '^^^ i e=w PN'de y a sin Odd = ^JL = .Q^ma. 141.] MEAN VALUES. 257 Note 2. A slight inspection will show that it is reasonable to expect the results in Exs. 1,2, to differ from each other. Suggestion : Draw a number of ordiiiates, say 4 or 6 or 8, as specified in Ex, 1, and compare them with the ordinates of equal number drawn as specified in Ex. 2. 3. Find the average value of the following functions: (1) 1 x--\-^x — ^ as X varies continuously from 2 to 6 ; (2) x-^ — 3 x^ -f 4 a; + 11 as x varies from — 2 to 3. Draw graphs of these functions. 4. Find the average length of the ordinates to the parabola y'^ = ^x erected at equidistant intervals from the vertex to the line x = 6. 5. (1) In Fig. 51 find the mean length of the ordinates drawn from OiV to the arc OML, and the mean length of the ordinates drawn from OiVto the arc ORL. (2) In Fig. 50 find the mean length of the abscissas drawn from OY, (a) to the arc OR; (b) to the arc RL; (c) to the arc ORL. (3) In Fig. 52 find the mean ordinate from OL, (a) to the arc TKN; (&) to the arc TGM. 6. (1) In the ellipse whose semiaxes are 6 and 10, chords parallel to the minor axis are drawn at equidistant intervals : find their mean length. (2) In the ellipse in (1) find the mean length of the equidistant chords that are parallel to the major axis. (3) Do as in (1) and (2) for the general case in which the major and minor axes are respectively 2 a and 2 6. 7. On the ellipse in Ex. 6, (3), successive points are taken whose eccen- tric angles differ by equal amounts : find the mean length of the perpen- diculars from these points, (1) to the major axis ; (2) to the minor axis. 8. In the case of a body falling vertically from rest, show that (1) the mean of the velocities at the ends of successive equal intervals of time, is one- half the final velocity ; (2) the mean of the velocities at the ends of succes- sive intervals of space, is two-thirds the final velocity. (The velocity at the end of t seconds is ^ gt feet per second ; the velocity after falling a distance s feet is V2 gs feet per second.) 9. A number n is divided at random into two parts : find the mean value of their product. 10. Find the mean distance of the points on a circle of radius a from a fixed point on the circle. The interval & — a in (1) and (4) through which the variable x passes is called the range of the variable, and dx is an infini- tesimal element of the range. In (1) and Ex. 1 the range is a particular interval on the aj-axis. In Ex. 2 the range is a certain angle, namely tt ; in Ex. 8 (2) the range is a vertical distance j in 258 INFINITESIMAL CALCULUS. [Ch. XVI. Ex. 8 (1) the range is an interval of time. There are various other ranges at (or for) whose component parts a function may take different values. For instance, a curved line as in Ex. 10, a plane area as in Exs. 11, 13 ; a curved surface as in Ex. 15 (1) ; a solid as in Exs. 16, 17. The definition of mean value [or result (4)] may be extended to include such cases, thus : lim 2 {(value of function at each infini- tesimal element of the range) x (this the mean value of a fiinc- ) _ infinitesimal element)} ^ tion over a certain range > the range 11. Find the mean square of the distance of a point within a square (side = a) from a corner of the square. In tliis case "the range" extends over a square. Choose the axes as shown in Fig. 79. Take any point P (x, y) in tlie range, and let its distance from O be d. At P let an infinitesimal element of the range be taken, viz. an element in the shape of a rectangle whose area is dy dx. Now d^ = x"^ -{- y^. .*. mean value of c?2 for all points in Fig. 79. (" P(a;- + 2/-)f/2/(^a^ OACB I B c ' ' a« Ay i / 3 A X a area of square 12. Find (1) the mean distance, and (2) the mean square of the distance, of a fixed point on the circumference of a circle of radius a from all points within the circle. (Suggestion : use polar coordinates.) 13. Find (1) the mean distance, and (2) the mean square of the distance, of all the points within a circle of radius a from the centre. 14. Find the mean latitude of all places north of the equator. 15. For a closed hemispherical shell of radius a calculate (1) the mean distance of the points on the curved surface from the plane surface ; (2) the mean distance of the points on the plane surface from the curved surface, distances being measured along lines perpendicular to the plane surface. 16. Calculate (1) the mean distance, and (2) the mean square of the dis- tance, of all points within a sphere of radius a, from a fixed point on the surface. 17. Calculate (1) the mean distance, and (2) the mean square of the dis- tance, of all points within a sphere of radius a, from the centre. 141.] MEAN VALUES. 259 18. Find (1) the mean distance, and (2) the mean square of the distance, of all points on the surface of a sphere of radius a, from a fixed point on the surface. 19. Find (1) the mean distance, and (2) the mean square of the distance, of all points on a semi-undulation of the sine curve y = asin x, from the X-axis. Note 3. The square root of the mean square in Ex. 19 (2) (viz. .7071 a) is of. special importance in the measurement of alternating currents ; for the heating and dynamometer effects of any current depend directly upon this square root. The latter is generally called " the mean square value of the ordinate of the sine curve" to distinguish it from "the average value" of this ordinate as found in Ex. 19 (1). CHAPTER XVII. CONCAVITY AND CONVEXITY. CONTACT AND CURVA- TURE. EVOLUTES AND INVOLUTES. 142. Concavity and convexity of curves : rectangular coordinates. Definition. At a point on a curve the curve is said to be con- cave to a line {or to a point off the curve) when an infinitesimal arc containing the point lies between the tangent at the point and the given line (or point off the curve). If the tangent lies between the line (or point) and the infinitesimal arc, the arc there is said to be convex to the line (or point). Thus, in Fig. 20 a, at P the curve J/iVis concave to the line OX, and con- cave to the point A ; in Fig. 20 &, at Pi the curve MN is convex to the line OX, and convex to the point A. The arc on one side of a point of inflexion is concave to a given line (or point), and the arc on the other side of the point of inflexion is convex to this line (or point) (see Figs. 36 a, b). The curves passing through P and R have the concavity towards the .T-axis, and the curves passing through Q and S are convex to the a^axis. At P ?/ is positive; and -4 is negative, for -^ decreases dx- dx as a point moves along the curve towards the right through P. At E y is negative ; and -J- is positive, 1 dxr ^ for — increases as a point moves dx along the curve towards the right through R. Hence, at points where a curve is concave to the x-axis y ^=-^ is negative. A similar examination of the curves passing through Q and 8 shows that at points where a curve is convex to the x-axis y y— is positive. 260 142, 143.] CONTACT. 261 Ex. 1. Prove the theorem last stated. Ex. 2. Test or verify the above theorems and Note 1 in the case of a num- ber of the curves in the preceding chapters. Note 1. The curves passing through P and ^S' are concave doimiwards, and here y^ is negative. The curves passing through B and Q are concave upwards, and liere —^ is positive. Note 2. A point where a curve stops bending in one direction and begins to bend in the opposite direction as at L, A, Z>, if, (r, P, Figs. 36 a, 6, 37, is called a point of inflexion. Note 3. A curve /(r, ^) = is concave or convex to the pole at the point , e) according as « + ^ is positive or dff^ McMahon and Snyder, Diff. Cal, Art. 144.) (r, 0) according as « + ^^ is positive or negative, u denoting -. (See 143. Order of contact. If two curves, y = {a) =f(a), <^'(a) =/'(a), and "(a) =f"{ci), but <^"'(a) =^f"(ci), then the curves are said to y-f(x) y-f (X) Fig. 81 a have contact of the second order, as in Fig. 81 c. And, in general, if <^(a) =/(«.) and the respective successive derivatives of <^(x) and f(x) up to and including the nth, but not including the (n -\- l)th, are equal for x = a, then the curves are said to have con- tact of the nth order. Hence, in order to find the order of contact of two curves compare the respective successive derivatives of y for the two curves at the points through which both curves pass. 262 INFINITEmMAL CALCULUS. [Ch. XVII. Note 1. Another way of regarding contact is the following. In analytic geometry the tangent at P (Fig. 82 a) is defined as the limiting position which the secant PQ takes when PQ revolves about P until the point of intersection Q coincides with P. Tlie line then has contact of the first order with the curve. This notion of points of intersection of a line and a curve becoming coincident will now be extended to curves in general. Two curves, C, Fig. 82 a. Fig. 82 h. C\ and C2 (Fig. 82 5), are said to intersect when they have a point, as P, in common. They are said to have contact of the first order at P when the curves (see Fig. 82 c) have been modified in such a way that a second point of intersection Q moves into coincidence with P. (The value of -j^ at P is then the same for both curves, according to the definition of a tangent as given above.) The curves are said to have contact of the second order at P when the curves have been further modified in such a way that a third point of intersection B moves into coincidence with P and Q (see Fig. 82 d). (The dx \dx)'' "■"" dx^ general, the curves are said to have contact of the nth order at a point P when n + 1 of their points of intersection have moved into coincidence with P. (At P the respective derivatives of y up to the nth are then the same for both curves.) See Echols, Calculus^ Art. 98. Note 2. In general a straight line cannot have contact of an order higher than the first with a curve. For in order that a line have contact of the first order with a curve at a given point, the ordinates of the line and the curve must be equal there, and likewise their slopes ; thus two equations must be satisfied. These equations suffice to determine the two arbitrary constants appearing in the equation of a straight line. For example, if the line y = mx + h has contact of the first order with the curve y = f(x) at the point for which x = a, the following two equations are satisfied, viz. : f(a) = ma + b, f'(a) = m ; from these equations w and h can be found. Tliis line and curve have contact of the second order in the particular (and exceptional) case in which f"{a) =0j consequently (Art. 78), if there is a 143.] CONTACT. 263 point of inflexion on the curve y — /(x) where x = a, the tangent there has contact of the second order. The theorem at the beginning of this note is also evident from geometrical considerations. Since, in general, a line can be passed through only two arbitrarily chosen points of a curve, it is to be expected from Note 1 that in general a line and a curve can have contact of the first order only. Note 3. In general, a circle cannot have contact of an order higher than the second with a curve. For in order that a circle have contact of the second order with a curve at a given point, three equations must be satisfied, and these equations just suffice to determine the three arbitrary constants that appear in the general equation of a circle [see Eq. (2), Art. 144]. This theorem is also evident from Note 1 and the fact that, in general, a circle can be passed through only three arbitrarily chosen points of a curve. (In a few very special instances a circle has contact of the third order with a curve. See Ex. 4, Art. 149.) Note 4. It is shown in Art. 182 that inhen two curves have contact of an odd order, they do not cross each other at the point of contact ; but lohen they have contact of an even order, they do cross there. Illustrations : the tangent at an ordinary point on a curve, as shown in Figs. 15, 17 ; the tangent at a point of inflexion, as in Figs. 31 a, h, 36, 37 ; an ellipse and circles having contact of second order therewith (see Ex. 4, Art. 149). This theorem may also be deduced from geometry and the definitions given in Note 1. N.B. As far as possible make good figures showing the curves, lines, and points mentioned in the exercises in this chapter. EXAMPLES. 1. Find the place and order of contact of (1) the curves y = t^ and y = 6 x2 - 9 X + 4 ; (2) the curves y - x^ and ?/ = 6 x^ - 12 x -f 8. 2. Determine the parabola which has its axis parallel to the ?/-axis, passes through the point (0, 3), and has contact of the first order with the parab- ola 2/ = 2 x^ at the point (1, 2). 3. What must be the value of a in order that the parabola y = x + 1 + a(x— 1)2 may have contact of the second order with the hyperbola xy = 3 X - 1 ? 4. Find the parabola whose axis is parallel to the y-axis, and which has contact of the second order with the cubical parabola y = x^ at the point (1, 1). 5. Determine the parabola which has its axis parallel to the y-axis and has contact of the second order with the hyperbola xy = 1 at the point (1, 1). 264 INFINITESIMAL CALCULUS. [Ch. XVIL 144. Osculating circle. It was pointed out in Art. 143, Note 3, that contact of the second order is, in general, the closest contact that a circle can have with a curve. A circle having contact of the second order with a curve at a point is called the osculating circle at that point. In Fig. 83 PT is tangent to the curve C at P. Every circle which passes through P and has its cen- tre in the normal AOTf touches C at P. One of these circles has contact of the second order with (7 at P; let this circle be denoted by K. All the other circles, infinite in number, in general have contact of the first order only. Osculating circle: rectangular coordinates. The radius and the centre of the osculating circle at any point P{x, y) on the curve Fig. 83. y=f(^) (1) will now be obtained. Denote the centre and radius by (a, b) and r. Then the equation of the osculating circle at the point ^^'^)'^ (X-ay+(Y-by = i^. (2) For the moment, for the sake of distinction, x and y are used to denote the coordinates of a point on tlie curve, and X and Y are used to denote the coordinates of a point on the circle. Then at the point where the circle and the curve have contact of the second order , _^ . ., ,_ ., dY _ dy d-Y _ fn/^ dX^dx dX^~d?' X y, (3) From (2), on differentiating twice in succession, X „H.(r-6)||=o, \dXj ^ 'dX' (4) (5) 144, 145.] CURVATURE. 265 and .-. F-6=- X-a = dYyndY ^ cV dx) \dX ' d: Y ' dX^' Accordingly, from (3), (2), (6), (7), [^ -(!)?. and from (3), (6), (7), \dx] dy , d-iy dx ' ^ d3^ dxi (6) (7) (8) (9) Note. For the osculating circle, polar coordinates being used, see Art. 150, Note 2. Ex. 1. Determine the radius and the centre of the osculating circle for each of the curves in Ex. 1 (1), Art. 143, at their point of contact. Ex. 2. Do as in Ex. 1 for the curves Ex. 1 (2), Art. 143. 145. The notion of curvature. Let the curves A, B, C, D have a common tangent FT at P. At the point P the curve A, to use the popular phrase, bends or curves more than the curves B, C, and D ; and D bends or curves less than the curves A, B, and C. These four curves evidently differ in the rate at which they bend, or turn away from the straight line PT, at P. These ideas are sometimes ex- pressed by saying that these curves differ in curvature at P, and that there A has the greatest and D the least curvature. In the case of two circles, say one with a radius of an inch and the other Avith a radius of a million miles, it is cus- tomary to say that the secoad circle has a small curvature, and that the firsfehas a large curvature in comparison with the second. An inspection of a figure consisting of a circle and some of its tangents gives the impression that what is popu- larly called the curvature is the same at all points of that circle. Fig. 84. 266 INFINITESIMAL CALCULUS. [Ch. XVIL On the other hand, an inspection of an elongated ellipse gives the impression that the curvature is not the same at all points of that ellipse, although at two particular points, or at four particular points, it may be the same. Curvature will now be given a precise mathematical definition and its measurement will be explained. Ex. 1. Draw an ellipse, and find by inspection the points where the curva- ture is greatest and where it is least. Show how to obtain sets of four points on the ellipse which have the same curvature. Ex. 2. Discuss a parabola and an hyperbola in the manner of Ex. 1. 146. Total curvature. Average curvature. Curvature at a point. At Ai the curve C has the direction A^T^, which makes the angle <^i with the a;-axis ; at A2 the curve has the direction A2T.2, which makes an angle ^2 with the aj-axis. The difference between these directions represents the angle by which the curve has changed its direction from the direction of the line A^T^ in the interval of arc from A^ to Ao. This difference, namely, T1RT2 or cf)2 — i, is called the total curvature of the arc A1A2. The average curvature for this arc is (<^2 — <^i) ^ length of arc ^1^2- (Here the angle is measured in radians.) Accordingly, if (Fig. 86) A<^ is the angle between the tangents at A and B, then A<^ is the total curva- y ture of the arc AB-, if As is the length of the arc AB, then — ^ is the average As curvature of that arc. Now let B approach A. The arc As and the angle A<^ then become infinitesimal ; and, finally, when B reaches A, — ^ has the ^0 As Fig. 86. 146-148.] CURVATURE. 267 limiting value -^. The limit. ,^o— at any point on a curve, i.e. ds As ^ there, is called the curvature of the curve at that 2»oint. (The phrase " curvature of a curve " means the curvature of the curve at a particular point.) In all curves, with the exception of straight lines and circles, the curvature, in general, varies from point to point. 147. The curvature of a circle. Let A and B be two points on a circle having its centre at 0. In Fig. 87 the angle between the direc- tions of the tangents AT^ and BT^ is A<^, say. Let As denote the length of the arc ^5. ThQXi AOB=T^ET,=:^<^. Hence, by trigonometry. As = rA<^. From this, ^^1. whence ^-l. (\\ ^s r' ^^^"^"^^ ds~r ^ ^ Fig. 87. That is, the curvature of a circle is constant and is the reciprocal of {the measure of) the radius. Note. When the radius increases beyond all bounds, the curvature approaches zero, and the circle approaches a straight line as its limiting position. When the radius decreases, the curvature increases ; as the radius approaches zero and the circle thus shrinks towards a point, the curvature approaches an infinitely great value. It is shown in Ex. 6, Art. 194, that all curves of constant curvature are circles. Ex. Compare the curvatures of circles of radii 2 inches, 2 feet, 5 yards, 2 miles, 10 miles, 100 miles, and 1,000,000 miles. 148. To find the curvature at any point of a curve : rectangular coordinates. Let the curve in Fig. ^Q> be y=f(x), and let its curvature at any point A(x, y) be required. Let k denote the curvature at A, and <^ denote the angle which the tangent at A makes with the a;-axis. Take an arc AB and denote its length by As, and denote the angle between the tangents at A and B by A<^. Then, by the definition in Art. 146, k = ^^tA. as 268 INFINITESIMAL CALCULUS. [Ch. XVII. Now (Art. 58), tan 6 = ^- .-. = tan"^ ^. ^ ^ dx ^ dx d^ ds dsV dx/ da;V d.17 ffo" i , /*y V ' dx [Art. 67 c(2)], fc = ^ (1) i^-am This, by (1) Art. 147 and (8) Art. 144, is the same as the curva- ture of the osculating circle. In order to find the curvature at a definite point (xi, y{) it is only necessary to substitute the coordinates Xi, y^, in the general result (1), Ex. 1. Compute and compare the curvatures of the two curves in Ex. 1 (1), Art. 143, at their point of contact. Ex. 2. Find the curvature of the curve ?/ = ic*^ — 2x'' + 7 ic at the origin. Determine the radius and centre of its osculating circle at that point. 149. The circle of curvature at any point on a curve : rectangular coordinates. The circle of curvature at a point on a curve is the circle which passes through the point and has the same tangent and the same curvature as the curve has there. The radius of this circle is called the radius of curvature at the point, and the centre of the circle is called the centre of curvature for the point. The radius of curvature. Let It denote the radius of curvature and (a, p) denote the centre of curvature for any point (x, y) on the curve y =/(x). Then it follows from Art. 147, and Art. 148, Eq. 1, that ^ (That is, R is the value of this expression at that point.) Note 1. There is an infinite number of circles that can pass through a given point on a curve and have the same tangent as the curve has there but not the same curvature, and there is an infinite number of circles that can 149.] CIRCLE OF CVkVATVRE. 269 pass through this point and have the same curvature but not the same tangent as the curve has there ; but tliere is only one circle passing through the point that has there both the same tangent and the same curvature as the curve. Ex. 1. Illustrate Note 1 by figures. The centre of curvature. Since at any point on a curve the circle of curvature and the curve have the same tangent and curvature, it follows that — and ^. are respectively the same for the circle and the curve at that point. Accordingly (Art. 143, Note 3) the circle of curvature has, in general,* contact of the second order with the curve, and thus (Art. 144) coincides with the osculating circle passing through the point. Accordingly (Art. 144, Eq. 9) dJl. doc' P = 2/ + d^y dx^ (2) Note 2. The coordinates of the centre of curvature may also be obtained in the following manner. Let C be the centre of the circle of cur- vature of the curve TL at P, and let the tangent FT make the angle ^ with the X-axis. Draw the ordinates TM and CN, and draw PB parallel to OX. Let J? denote the radius of curvature. Then NCF = 0, and tan = _dy^ dx In Fig. 88 a= O.V = OM- BP=x- Rs'm(f> b-m' cly dx dJC2 [-(l)t N/T M Fig. 88. d^y dx dx^ 14- Also, /3 = NC = MP + BC = y-\- B cos

, y = b tan . (8) The general spiral r — aaxis. The individual members of the family are obtained by letting c change its values from — 00 to -f 00. A number such as c, whose different values serve to distinguish the individual members of a family of curves, is called the parameter of the family. Thus, to take another example, the equation y = 2x-\-b represents the family of straight lines having the slope 2 ; and y = 2 x-\-o, y = 2x — l, are particu- lar lines of the family. (Let a figure be constructed.) In this case the parameter h can take all values from — 00 to + oo. 277 278 INFINITESIMAL CALCULUS. [Ch. XVIIl. To generalize : f(x, y, a) = (2) is the equation of a family of curves whose parameter is a. The individual members or curves of the family are obtained by giving particular values to a. These curves are all of the same kind, but differ in various ways ; for instance, in position, shape, or enclosed area. A family of curves may have two or more param- eters. Thus, y = mx -\- b, in which m and b may take any values, has two parameters m and b, and represents all lines. The equa- tion {x — hy -\- (y — ky = 25, in which h and k may take any values, represents all circles of radius 5. The equation (x — lif -\-{y — lif = r"^, in which li, k, and r may each take any value, represents all circles. Envelope. The envelope of a family of curves is the curve, or consists of the set of curves, which touches every member of the family and which, at each point, is touched by some member of the family. For example, the envelope of the family of circles in Fig. 92 evidently consists of the two lines y — 2=0 and y-^2 = 0. On the other hand, the family of parallel straight lines y=2x-\-b does not have an envelope ; and, obviously, a family of concentric circles cannot have an envelope. EXAMPLES. 1. Say what family of curves is represented by each of the following equations, and in each instance make a sketch showing several members of the family : (a) x^ + ?/2 = r^, parameter r. (6) y = mx + 4, parameter m. (c) y^ = 4|)x, parameter p. (d) y^ = i a(x 4- a), parameter a. (e) — I- 2_ = 1, parameter a. ( f) — '■ 1 -'- — = 1, parameter k. (g) y = mx H — , parameter m. (h) y = mx + V25 m' + 10, parameter m. m 2. Express opinions as to which of the families in Ex. 1 have envelopes, and as to what these envelopes may be. 155. Locus of the ultimate intersections of the curves of a family. In Eq. (2), Art. 154, the equation of a family of curves, let a be given the particular value a^ ; then there is obtained the equation of a particular member of that family, viz. f{x,y,a,) = 0. (1) 155.] ENVELOPES. 279 Also, f{x, y, a-i -i-h) = is the equation of another member of the family. Let I. and II. be these curves. The smaller h becomes, the more nearly does curve II. come into coincidence with curve I. Moreover, as h be- comes smaller and approaches zero, A, the point of intersection of these curves, approaches a definit-e limiting position. For ^u.V;^ example, if (Fig. 92) the centre L approaches nearer to C, then K, the point of intersection of the circles whose centres are at C and L, moves nearer to j^ Fig. 93. P; and finally, when L reaches C, ^arrives at the definite position P. The locus of the limiting position of the point (or points) of intersection of two curves of a family which are approaching coincidence is called the locus of ultimate intersections of the curves of the family. For instance, in the case of the family of circles in Fig. 92, this locus evidently consists of the lines y — 2 = and y -\-2 = 0. Note. The last-mentioned locus may also be derived analytically. Let (a; _ ci)2 + ?/2 = 4 (1) and (x-cx-hy-vy'^ = ^ (2) be two of the circles. On solving these equations simultaneously in order to find the point of intersection, there is obtained (x - ci)2 _ (x - ci - hY = ; whence A(2 a: - 2 Ci - /i) = 0, and, accordingly, x — Cx-\ — An ultimate point of intersection is obtained by letting h approach zero. If ^ = 0, then X = C\, and by (1) ?/ = ± 2. Thus y =z±2 at the ultimate points of intersection, and therefore the locus of these points is the pair of lines y = ±2. N.B. In the following articles "the locus of ultimate intersections" is denoted by I. u. i. 280 INFINITESIMAL CALCULUS. [Ch. XVIII. 156. Theorem. In general, the locus of the ultimate intersections touches each member of the family. Let I., II., III. be any three members of the family, and let I. and II. intersect at F, and 11. and III. at Q. When the curve I. approaches coincidence with II., the point F approaches a definite position on l. u. i. of the curves of the family. When the curve III. approaches coincidence with II., Q approaches a definite position on I. u. i. When I. and III. both approach coincidence with II., F and Q approach each other along II., and at the same time approach I. u. i. When F and Q finally reach each other on II., they are also on I. u. i. More- over, when F and Q come together, the tangent to II. at F and the tangent to II. at Q come into coincidence as a line which is at the same time a tangent to curve II. and a tangent to t u. i. at the point where F and Q meet. Thus the curve II. and I. u. i. have a com- mon tangent at their common point. Similarly it can be shown that I. u. i. touches every other curve of the family. Since, in gen- eral, each point of l. u. i. may be approached in the manner indicated in this article, the above theorem may be thus supplemented: In general, l.u.i. is touched at each' of its points by some member of the family. Note 1. The family of circles, Fig. 92, will serve to illustrate this theorem. Note 2. An analytical proof oi the theorem is given in Art. 157, Note 3. Note 3. It is necessary to use the qualifying phrase in general in the enunciation of the theorem, for there are some families of curves (viz. curves having double points and cusps, see Arts. 163, 164), in which a part of I. n. i. may not touch any member of the family. It is beyond the scope of this book to go into these cases in detail, (See Edwards, Treatise on the Biff. Cnl., Art. 365 ; Murray, Differential Equations, Chap. IV.) Illustrations may be obtained by sketching some curves of the families {y -f- c)"^ = x^ and (ij 4- c)2 = x{x - 3)2. 156, 157.] ENVELOPES. 281 157. To find the envelope of a family of curves having one pa- rameter. It is in accordance with the definitions and theorem in Arts. 154-156 to say that the envelope of a family of curves fix, y, a) = 0, if there he an envelope, is, in general, the locus of the limiting position of the intersection of any one of the curves of the /am%, say the curve f(x, y, a) = (1) with another curve of the family, viz. /(a^,2/,« + Aa) = (2) when the second cwve approaches coincidence with the first; that is, when Aa approaches zero. From (1) and (2), f{x, y,a + ^a)-f(x, y,a) = 0; hence f(x,y,a + ^a)-f(x,y, a) ^^^ Aa ^ ^ Now Equations (1) and (3) may be used, instead of (1) and (2), to find the points of intersection of curves (1) and (2). If Aa = 0, the point of intersection approaches an ultimate point of inter- section. When (Arts. 22, 79) Aa = 0, Equation (3) becomes £f{x,y,a)=0. (4) Thus the coordinates x and y of the point of ultimate inter- section of curves (1) and (2) satisfy Equations (1) and (4); and, accordingly, satisfy the relation which is deduced from (1) and (4) by the elimination of a. Hence, in order to find the equation of I. u. i. of the family of curves f{x, y, a) = eliminate a betiveen the equations f(x, y,a)=0 and ^ f(x, y, a) = 0. (5) The result obtained is, in general, also the equation of the envelope. Note 1. A slightly different way of making the above deduction is as follows. Let the equations of two curves of the family be f(x, y,a) = (6), and f(x, y,a-\-h) = 0. (7) 282 INFINITESIMAL CALCULUS. [Ch. XVIII. By Art. 64, Eq. (3), Equation (7) may be written /(ic, y, a) + h^ f{x, y, a -j- eh) = 0, in which | ^ |< 1. (8) •J By virtue of (6) this becomes -^f(x, 2/? « + ^h) = 0. (9) Accordingly, the coordinates of the intersection of curves (6) and (7) satisfy (6) and (9). When h becomes zero, the point of intersection becomes an ultimate point of intersection. Hence the ultimate points of intersection satisfy equations /(x, y, a) = and ^/(x, y, a) = 0, and, accordingly, the a-eliminant of these equations.* Note 2. For an interesting and useful derivation of result (5) for cases in which /(x, y, a) is a rational integral function of a, see Lamb's Calculus, Art. 157. Note 3. To show that, in general, the a-eliminant of Equations (5) touches any curve of the family. Let the second of Equations (5) on being solved for a give a = (f)(x, y). Then the equation of the I. u. i. of the family of curves /(x, y, a) = is /(x, y, «) = 9 in which a = 0(x, y). (10) civ The slope ~ of any one of the family of curves /(x, y, a) = is given (see Art. 56), by the equation ' ;)f nf //,, ^ + ^'^ = 0. (11) dx dy dx ^ ' The slope -^ of the I. u. i. is obtained from Equations (10). On taking the total x-derivative in the first of these equations, But by the second of (5), df.dfdydf da dx'^ dy dx^ da dx~ ' (12) >), -;;r- = 0, and accordingly, (12) reduces to df.dfdy dx '^ dy dx ' (13) Thus the slope of the I. u. i. and the slope of any member of the family are both given by the same equation. Hence, at a point common to any curve and the I. u. i., the slopes of both are the same, and accordingly, the curve and the I. u. i. touch at that point. Sometimes the value of ~ obtained from (11) is indeterminate in form, dx ^ ^ and the slopes of the curve and I. u. i. may not be the same. See Arts. 165, 156 (Note 3), and Lamb, Calculus, Art. 158. * This method of finding envelopes appears to be due to Leibnitz. 157.] ENVELOPES. 283 EXAMPLES. 1. Find the envelope of the family of circles (see Art. 154) {X - cY + y^ = 4. (1) Here, on differentiation with respect to the parameter c, 2 (X - c) = 0. (2) The elimination of c between these equations gives y = 4, which represents the two straight lines y = 2, y =—2. 2. Find the envelope of the family of lines y = mx — 2 pm — pm^, (1) in which m is the parameter. (This is the equation of the general normal of the parabola y'^ = 4ipx ; see works on analytic geometry.) On differentiation with respect to the parameter m, = x-2p -Spm^. (2) The m-eliminant of (1) and (2) is the equation of the envelope. On taking the value of m in (2) and substituting it in (1), and simplifying and removing the radicals, there is obtained 27 py2=4(x-2 py. (3) Note 4. In Art. 152 it is shown that the normals to a curve touch its evolute. It also appears from Art. 152 that each tangent to an evolute is normal to the original curve. Accordingly, it may be said that the evolute of a curve is the envelope of its normals, and likewise that the evolute of a curve is the I. u. i. of its {family of) normals. (See Art. 152, Note 2, and Art. 149, Note 5.) Note 5. Compare Ex. 1, Art. 151, Ex. 2 above, and Ex. 1, Art. 158. 3. If A, B, C are functions of the coordinates of a point and m a variable parameter, show that the envelope of Ain^ + Bin + C — is ^2-4^0 = 0. Note 6. The result in Ex. 3 is the same in form as the condition that the roots of the quadratic equation in m he equal. This result is immediately applicable in many instances. It is very easily deduced on taking the point of view explained in the article mentioned in Note 2. 4. Deduce the result in Ex. 3 without reference to the calculus. Apply this result to Ex. 1. 284 INFINITESIMAL CALCULUS. [Ch. XVIII. IV. B. Make figures for the following examples. 5. Find the curves whose tangents have the following general equations, in which m is the variable parameter : (1) y = mx -\- aVl -\- m^. (2) y = mix + y/a^hrir- -\- b'^. (3) y = mx± y/arii^ -\- bm + c. (4) y — mx + a Vm. (5) mH — my-\-a. (Q) y — b — m{x — a) + rVl + w'-^. 6. Find the envelopes of the following lines : (1) x sin ^ — y cos ^ + a = 0, parameter 6. (2) x -^ ysvnd = a cos 0, parameter d. (3) ax sec a — by cosec cj = a^ _ 52^ parameter «. 7. Find the envelopes of (1) the parabolas ?/2 = 4 a{x — a), parameter a ; (2) the parabolas cy'^ = a^(x — a), parameter a. 8. Show that if A, B, C are functions of the coordinates of a point, and a a variable parameter, the envelope of A cos ct + ^ sin a = C is ^2 _|_ ^2 _ (j%^ 9. Find the evolute of the ellipse x = a cos (p, y == b sin and y = c. Note 1. If [see Eq. (2)] x = — also satisfies cx^ -\- dx + e = 0, then a three values of y in F(x, y) = are infinitely great for this value of x. The line X = is then an inflexional tangent (see Art. 78, Note 1) at infinity. Note 2. This method of finding asymptotes parallel to the axes can be applied to curves whose equations are not of the kind considered above. Instances are given in Exs. 7, 8 (6), (9) that follow. EXAMPLES. 4. Find the asymptotes of the curves in Ex. 1. 5. Determine the finite points (if they exist) in which each asymptote in Ex. 4 meets the curve to which it belongs. 6. Show that the line x = a is an asymptote of the curve y = ^1±L when 0(a) and '(x) is discontinuous at the salient points. (See Exs. 5, 6, below.) A salient point such as D may be considered to be the limiting condition of a double point like D (Fig. 96 c), when the loop Dli dwindles to zero but the two tangents at D do not become coincident. (Compare "A Note 2.) There are also stop points, as A, Fig. 99, where the Fig. 99. curve stops and has but one branch. See Ex. 7. 1 5. In the curve y(l -{- e') = x show that when x approaches the origin from the positive side, the slope is zero ; if from the negative side, the slope is 1. The origin is thus a salient point. Suggestion : The slope at the origiH may be taken as linix^o -• | Find the angle between the branches at the origin. ^ -^ 6. In the curve y = x- show that when x approaches the origin e^+1 from the positive side the slope is -|- 1, and if from the negative side, the slope is — 1. The origin thus is a salient point : find the angle between thq branches there. 7. Show that the origin is » gtop point in the curve ?/ ^ aj log x. 296 INFINITESIMAL CALCULUS. [Ch. XVIIL 165. To find multiple points, cusps, and isolated points. From Art. 164 it is evident that in order to determine the character of a point on a curve, it is first of all necessary to examine the tan- gent (or tangents) there. Let the equation of the curve be /(^,2/) = 0, ■ (1) and let f{x, y) be a rational integral function of x and y. Then Sf | = -|. [Art. 84, (4).] (2) dy Kow at a multiple point or a cusp -^ has not a single definite ux value, and, accordingly, at such points — in (2) must have an indefinite form, viz. the form —* Hence, at a multiple point of curve (1) ^ ^=0 and ^ = 0. (3) ax dy The solutions of Equations (3) will indicate the points which it is necessary to examine, t At these points dx 0' (4) the indefinite form in the second member can be evaluated by the method explained in the Appendix, Note C, and applied in Note below. :|: Suppose that the second member of (4) has been evaluated and the resulting equation solved for — • Then: If — has two (iX C(X real and different values at the point under consideration, the point is a double point or a salient point; if — has three real and different values there, it is a triple point ; and so on. If -^ cix * This is frequently called an '■'■ indeterminate,'''' form. The evaluation of (so-called) " indeterminate forms ^' is discussed in the Appendix, Note C. t The values of x and y that satisfy Equations (3), may give points that are not on the curve. Of course these points need not be examinecl further. J Or by othef metl^od^ referred to in Appendix, I^ote C, 165.] SINGULAR POINTS. 297 has two real and equal values at the point which is being examined the point is a cusp, ^f ^ is an isolated point If ^^ has imaginary values at the point, it ctx If the point is a cusp, the kind of cusp can be found by further examina- tion of the curve in the neighborhood of the point. For example, if (xi, yi) is known to be a cusp and it is found that for X = Xi — h (h being infinitesimal), y is imaginary, then the curve does not extend tlirough (a;i, yi) to the left, and thus the cusp is not a double cusp. If for x=xi + h, the value of the ordinate of the tangent at (a^i, yi) is less than the ordinates of both branches of the curve, the cusp is as in Fig. 100. In a similar way tests may be devised and applied in special cases as they arise. Note. The evaluation of the second member of Equation (2) gives, by Appendix, Note C, and Art. 81, (5) dy dx d^f , ay dy dx'^ dy dx dx dV dVdy dx dy dy^ dx (5) If the second member of (5) is not indefinite in form, this equation, on clearing of fractions and combining, becomes dV(dy\ dy^ \dxl dy dydx dx dx^ (6) a quadratic equation in ^. By the theory of quadratic equations, the two dv values of ^ are real and different, real and equal, or imaginary, according as / d'^f V dH dH ( ^-^ j is respectively greater than, equal to, or less than V^ • V^- the point is a double point, a cusp, or a conjugate point, according as Hence, / d'\f \dydx >, or < 5^ dV dy' dx''' If the second member of (5) also is indefinite in form, proceed as required by Note C, remembering that ~ here is constant. The resulting equation will be of the third degree in dy dx 298 INFINITESIMAL CALCULUS. [Ch. XVIII. EXAMPLES. 1. Examine the curve a;^ - ?/2 _ 7 x- + 4 ?/ + 15 x — 13 = for singular points. Here d|/^ _ Sx^ - 14x + 15^ dx -22/4-4 ^ ^ On giving each member the indefinite form -, and solving the equations 3 x2 - 14 X + 15 = 0, -2?/ + 4 = 0, it results that x = 3 or |, and y = 2. Substitution in the equation of the curve shows that x = f , ?/ = 2, do not satisfy the equation, and that x = 3, y = 2 do. Accordingly, the point (3, 2) is the point to be further examined. On evaluating, by the method shown in the Appendix, the second member of (1) for the values a; = 3, y = 2, it is found that dy 6x — 14 , I dy\^ -, , dy dx _2^ dx ■ whence [~\ =2, and -}- = ± V2. \dxl dx Thus the curve has a double point at (3, 2), and the slopes of the tangent there are 4- V2 and — \/2. [The curve consists of an oval between the points (1, 2), and (3, 2), and two branches extending to infinity to the right of (3, 2).] 2. Sketch the curve in Ex. 1. 3. Examine the following curves for singular points : (1) a^y^ = x-2(a2 _ x2). (2) x^ + 9 ^2 - ^/'-^ + 27 x + 2 y + 26 = 0. (3) ?/3 - x2 - 3 ?/2 + 3 y + 4 X - 5 = 0. (4) The curve in Ex. "5 (5), Art. 161. (5) x3 + 2/3 + 3 x;hj + 3 X2/2 - 10 2/^ - 16 xy - 10 x2 + 25 x + 29 2/ - 28 = 0. (6) x3 - 2/2 - 10 x2 + 33 X - 36 = 0. 166. Curve tracing. Some of the matters involved in curve tracing have been discussed in Arts. 75-78, 159-165. To do more than this is beyond the scope of a primary text-book on the calculus. The topic is mentioned here merely for the purpose of giving a few exercises whose solutions require the simultaneous application of methods for finding points of maximum and mini- mum, asymptotes, and singular points. 166.] SINGULAR POINTS. ^99 Note 1. For a fuller elementary treatment of singular points and curve tracing, see McMahon and Snyder, Diff. Cal., Chaps. XVII., XVIII., pp. 275-306; F. G. Taylor, Calculus, Chaps. XVII., XVIII., pp. 250-278; Edwards, Treatise on Diff. Cal., Chaps. IX., XII., XIII.; Echols, Calculus, Chaps. XV., XXXI., pp. 147-164, 329-346. The classic English work on the subject is Frost's Curve Tracing (Macmillan & Co.), a treatise which is highly praised both from the theoretical and the practical point of view.* Note 2. For the application of the calculus to the study of surfaces (their tangent lines and planes, curvature, envelopes, etc.) and curves in space, see Echols, Calculus, Chaps. XXXII. -XXXV., pp. 347-390, and the treatises of W, S. Aldis and C. Smith on Solid Geometry. EXAMPLES. 1. Trace the curves in Ex. 8, Art. 160; in Ex. 5, Art. 161; in Ex. 2, Art. 162; in Ex. 3, Art. 165. . 2. Trace the following curves : (I) y^=: x4(l - x2). (2) y^ = a-.'2(l - x). (3) x^ - 4 x^y - 2 xy'^ -\- 4 y^ = 0. (4) 'iy'^ = 4xy - x^. (5) r = a cos 4 d. * A recent important work on curves is Loria's Special Plane Curves, a German translation of which (xxi. -f 744 pp.) is published by B. G. Teubner, Leipzig. CHAPTER XIX. INFINITE SERIES. EXPANSION OF FUNCTIONS IN INFINITE SERIES. INTEGRATION AND DIFFERENTIATION OF INFINITE SERIES. EXPANSIONS OBTAINED BY INTEGRATION AND DIFFERENTIATION. N.B. There are some students whose time is limited and who require to obtain as speedily as may be a working knowledge of Taylor's and Maclaurin's expansions. These students had better proceed at once to Arts. 175, 180, work the examples in Arts. 176 and 178, and then take up Art. 174. It is, perhaps, advisable in any case to do this before reading this chapter and the other articles in Chapter XX. Those who are studying the calculus as a "culture" subject should become acquainted with the ideas and principles described, or referred to, in Chapters XIX., XX. A thorough understanding of these ideas and principles is absolutely essential for any one who intends to enter upon the study of higher mathematics. 167. Infinite series : definitions, notation. An infinite series consists of a set of quantities, infinite in number, which are con- nected by the signs of addition and subtraction, and which suc- ceed one another according to some law. A few infinite series of a simple kind occur in elementary arithmetic and algebra. For instance, the geometrical series the geometrical series 1 -\- X + x^ -\- 1- x"-i + x« + a;«+i + ..., (2) which may also be obtained by performing the division indicated in the geometrical series 1 -X 1 - a; + ic2 + ... + (- l)nxn-i + ..., (8) which may also be obtained by performing the division indicated in ; the geometrical series "^ ^ a -\- ar -\- ar"^ 4- ••• + «r"-i -f ar" + ar^+^ + ••• ; (4) the series l-f.J--)-A^ f-i_a..... (5) IP 2p Sp nP 300 167,168.] INFINITE SERIES. 301 The successive quantities in an infinite series, beginning with the first quantity, are usually denoted by Uq, Ui, U2y •••, w„_i, ic„, Un+i, •••; or, in order to show a variable, x say, by iio(x), Ui(x), ii.2(x), ..., w„_i(:c), u^(x), n^+i(x), •-. Then the series is Uq -f- U^ + W2 H + ««-l H- «n + ^«,.+l H • (6) The value of the series is often denoted by s ; and the symbol s„ is generally used to denote the sum or value of the series obtained by taking the first n terms of the infinite series ; thus, Sn = Uo -\-Ui-{-U2-\ h ^in-l- The valtie of the infinite series (6) is the limit of the sum of the quantities in the series; i.e. the value of the series is the limit of the sum of n terms of the series when 7i increases beyond all bounds.* This is expressed in mathematical symbols s = lim„ix Su. (7) (This limit s is frequently, but not quite correctly, called " the sum of the series" or "the sum of the series to infinity^") Thus, in (1), — M— i^-(-i^> and hence s = \imn=ooSn = 2; (7) in (2), s„ = 1 + X + x2 + ... + a;"-i = ?^^^, x-1 and hence s = lim„i„ Sn = cc when x-^1 and x "^ — 1, (8) = - — - when - 1 < x< 1. (9) 168. Questions concerning infinite series. The subject of infinite series is highly important in mathematics. Such questions as the following arise and require to be answered : (a) Under what conditions may infinite series be employed in mathematical investigation and used in practical work ? * Thus s is not the sura of an infinite number of terms of the series, but is the limiting value of that sum. 302 INFINITESIMAL CALCULUS. [Ch. XIX. {h) Under what conditions may an infinite series be used to define a function or employed to represent a function? ThuSv in Art. 167, result (8) shows that series (2) does not represent the function when x is greater than 1 or less than —1 or equal to 1 or —1. 1 — X This is obvious on a glance at the series ; in fact, the greater the number of terms of (2) that are taken, the greater is the error comjnitted in taking the series to represent the function. (For instance, put x = 2 ; then the func- tion is — 1 and the series is + go.) On the other hand, the infinite series (2) does represent the function when x lies between — 1 and + 1 ; the \ —X greater the number of terms that are taken, the more nearly will the sum of these terms come to the value of the function. The limit of the sum of these terms when the number of them is infinite is the function. (c) May two infinite series be added like two finite series ? In other words, if U = Ui^-\- III -f- ^^2 + • • • and V — Vq -\-V1-i-V2-] , is u -\-v = no + VQ-{- n^-^Vi+ •" (1) a true equation; and under what conditions is (1) a true equation? (d) May two infinite series be multiplied together like two finite series ? In other words, u and v being as in (c), is a true equation; and under what conditions is (2) a true equation ? (e) May the principles of Art. 31 and Art. 104 A, namely, that the derivative and the integral of the sum of a Jinite number of terms are respectively equal to the sum of the derivatives and the sum of the integrals of these terms (to a constant), be extended to infinite series ? That is, Kq, Ui, Uo, •••, being functions of x^ if Jsdx= I v^/Jx -\- I n^dx -\- | u^dx -\- -", (3) are and A {s) = |- («o) + f (ti,) + f {u.^ + • • •, (4) dx dx dx dx ItjU.] INFINITE SERIES. 303 true equations; and what are the conditions which must be satisfied in order that these equations be true ? Equations (3) and (4) may be expressed : r lim„-ao s,,{x) \(lx = lim„^3, J sXx)dx L |[li.n„_^ «„(.•)] = lim_[|«„(.)]- The above questions then may be stated thus : Is the integral of the limit of the sum of an infinite number of quantities equal to the limit of the sum of the integrals of the quantities ; and is it likewise in the case of the differentials ? For instance, given that = 1 + x + ^^ + a^"' + •••, I — X dx\l-xll (l-x)2j and is f ' -^^ [i.e. log --1-] = x + f + ^' + - ? Jo 1 _ XL 1 — xj 2 8 169. Study of infinite series. Knowledge, elementary knowledge at least, of the theory of infinite series, and practice in their use are necessary in applied mathematics. Infinite series frequently present themselves in the theory and applications of the calculus, and accordingly the subject should be studied, to some extent at least, in an introductory course in calculus. The better text-books on algebra, for instance, among others, Chrystal's Algebra (Vol. II., Ed. 1889, Chap. XXVI., etc.), Hall and Knight's ^j^/ier Algebra (Chap. XXI.), contain discussions on infinite series and examples for practice.* Osgood's pamphlet, Introduction to Infinite Series (71 pages, Harvard University Publications), gives a simple, elementary, and excellent account of infinite series. "This pamphlet is designed to form a supplemen- tary chapter on Infinite Series to accompany the text-book used in the course in calculus." Becent text-books on the calculus, in particular those of McMahon and Snyder, Lamb, and Gibson, contain definitions and theorems on infinite series ; they will especially well repay consultation. More elaborate expositions of the properties of infinite series, which form parts of introductory courses in modern higher analysis, are given in Harkness and Morley, Introduction to the Theory of Analytic Functions, in particular * Also see Hobson, A Treatise on Plane Trigonometry, Chap. XIV., and following chapters. 304 INFINITESIMAL CALCULUS. [Ch. XIX. Chaps. VIII. -XI., and in Whittaker, Modern Anali/sis, in particular Chaps. II. -VIII. These discussions can be read, in large part, by one who possesses a knowledge of merely elementary mathematics. A statement of a few of the principal definitions and theorems which are necessary for an elementary use of infinite series is given in Arts. 170-173. 170. Definitions. Algebraic properties of infinite series. An infinite series has been defined in Art. 1()7. If (see Art. 167) lim„^^ ,s„ is a definite finite quantity, U say, tlie series is called a convergent series, and is said to converge to the value U. If s„ does not approach a definite finite value when n approaches infinity, the series is called a divergent series. In a divergent series, when n approaches infinity, s^ may either approach infinity, or remain finite but approach no definite value. Thus, in Art. i:;7, series (1) is convergent; series (2) is convergent for values of x between — 1 and + 1, for then s = — — ; series (4) is convergent 1—x when r lies between — 1 and + 1, for then s = — ^ . Series (5) is con- \ — r vergent for j) > 1, and divergent for jj = 1 and for j9 < 1. (Hall and Knight, Algebra^ p. 235.) [Note 1. The harmonic series. When j? = 1, series (5) is 2 3 4 5 w n + 1 This series is called the harmonic series.'] The series 1 + 2 + 3 + h « + ••• is divergent. The series 1 — 1 + 1 — 1 + • •• + (— 1)*^"^ + •••, obtained by putting x = 1 in series (3), is divergent ; for its limit is or 1 according as n is even or odd. (A series that behaves like this is said to oscillate. Some writers do not include oscillatory series among the divergent series.) In general only convergent series are regarded as of service in applied mathematics. (For the necessity of the qualifying phrase " in general," see Note 2.) A series may be employed to represent a function, or, what comes to the same thing, a function may be defined by a series, if the series is convergent. Thus series (2), Art. 167, may be used to represent or to define , if x lies between — 1 and + 1. [See questions (a) and (6), Art. 168.*] * Carl Friedrich Gauss (1777-1855), the great mathematician and astrono- mer of Gottingen, and A ugus tin-Louis Cauchy (1789-1857), professor at the 170.] INFIISITE SERIES, 805 Note 2. Oil divergent series. Those who apply mathematics, astroVio- mers in particular, have frequently obtained sufficiently good approximations to true results by means of divergent series. Such series, however, " cannot, except in special cases, and under special precautions, be employed in mathe- matical reasoning" (Chrystal, Algebra, Vol. 11. , p. 102). At the present time considerable attention is being paid by mathematicians to divergent series and to investigations of the fundamental operations of algebra and the calculus upon them. A work on the subject has recently appeared, viz. Leqons sur les series divergentes, par Emile Borel (Paris, Gauthier-Villars, 1901, pp. vi + 182). "It is safe to say that no previous book upon diver- gent series has ever been written." Interesting and instructive information concerning divergent series will be found in reviews on this book, by G. B. Mathews {Nature, Nov. 7, 1901), and E. B. Van Vleck {Science, March 28, 1902). Absolutely convergent series. A series the absolute values (see Art. 8, Note 1) of whose terms make a convergent series is said to be absolutely or unconditionally convergent; other convergent series are said to be conditionally convergent. Ex. 1. Series (1), Art. 167, is an absolutely convergent series. Ex. 2. The series 1 - 2 + i - 4 + i («) may be written (1 _ i) + (^ _ ^) + (| - |)+ ..., i.e. ^ +^1^ + ^1^+ .... Series (a) may also be written Thus the value of the series (a), the terms being taken in the order indi- cated, is less than 1 and greater than \. It can also be shown that this series converges to a definite value. On the other hand (see Note 1, and the state- ment just preceding Note 1), the series is divergent. Thus series (a) is a conditionally convergent series. Theorems. (1) If a series is absolutely convergent, it is obvious that any series formed from it by changing the signs of any of the terms is also convergent. Polytechnic School at Paris, who did much to make mathematics more rigor- ous than it had been during its rapid development in tlie eighteenth century, may be regarded as the founders of the modern theory of convergent series. James Gregory, professor of mathematics at Edinburgh, introduced the terms convergent and divergent in connection with infinite series in 1668. 306 INFINITESIMAL CALCULUS. [Ch. XIX. (2) In a conditionally convergent series it is possible to rearrange the terms so that the new series will converge toward an arbitrary preassigned value. (3) In an absolutely convergent series the terms can be rearranged at pleasure without altering the value of the series. (4) If (see Art. 168) u and v are any two convergent series, they can be added term by term ; that is, Equation (1), Art. 168, is true. (5) If u and v are any two absolutely convergent series, they can be multiplied together like sums of a finite number of quanti- ties ; that is. Equation (2), Art. 168., is true. For proofs and examples of these theorems see Osgood, Intro- duction to Inji)dte Series, Arts. 34, 35; Chrystal, Algebra, Vol. II., Chap. XXVI., §§ 12-14. In a convergent series as n increases, s„ may either: (a) con- tinually increase toward the limiting value of the series ; or (&) decrease toward this limit; or (c) be alternately greater than and less than its limit. Thus iu series (1), Art. 167, Sn continually increases toward its limit (2); in the series 1— -+ — 1- •••, Sn is alternately greater than and less than its limit |. ^ ^"^ ^^ Remainder after n terms. The symbol Vn or Mn is often used to denote the series (and also to denote the value of the series) formed by taking the terms after the nt\\, thus ^"« = U,, -f U^^^ -f Un+2 H • This is usually called the remainder after n terms. Let a func- tion be represented by a convergent series; i.e. let the value of the function be equivalent to the value of this convergent series. Then since ^i j? x.- ^^ the lunction = lim„^ s„, it follows that lim^^^ i\, = 0. Interval of convergence. In general a convergent series, in a variable, x say, is convergent only for values of x in a certain interval, say from x = a to x = h. The series is then said to con- verge within the interval (a, h), and this interval is called the iyiterval of convergence. 171.] INFINITE SERIES. 307 Thus in series (2), Art. 1(57, the interval of convergence extends from x = — ltoaj = + l. In this case, as in many others, the series is not conver- gent for the values of x (in this case — 1 and + 1) at the extremes of the interval. In some cases series are convergent for the values of the variable at the extremes of the interval of convergence as well as for the values between ; in other cases a series may be convergent for the value of the variable at one extreme of the interval but not for the value at the other. Power series. Series of the type aQ + ciioc -T aox'^ + ••• + a^cc'* —, in which the terms are arranged in ascending integral powers of x and the coefficients are independent of Xj are called power series in X. A power series may converge for all values of x, but in general it will converge for some values of x and diverge for others. Theorem. In the latter case the interval of convergence ex- tends from some value x = — r to the value x = -{- v] i.e. the value ic = is midway between the values of x at the extremes of the Divergent Convergent Divergent -r o +^' Fig. 101. interval of convergence. Thus in the power series (2), Art. 167, the interval of convergence extends from —1 to +1. This theo- rem may be graphically represented, or illustrated, by Fig. 101. (For proof of the theorem see Osgood, Lifinite Series, Art. 18.) 171. Tests for convergence. Two simple tests for convergence will now be shown. For nearly all the infinite series occurring in elementary mathematics these tests will suffice to determine whether a series is convergent or divergent. These two tests are : (A) the comparison test and (B) the test-ratio test. A, The comparison test. Let there be two infinite series, Wo+"l + '^2H h^n-l + ?*„+•••, (1) and Vq -f Vi + Vo H 1- ^'„-l + ^'« H • (2) If series (1) is convergent, and if each term of series (2) is not greater than the corresponding term of series (1) {i.e. if v,^ ^ ?/„ for each value of ?i), then series (2) is convergent. If series (1) 308 INFINITESIMAL CALCULUS. [Ch. XIX. is divergent, and if each term of series (2) is greater than the corresponding term of series (1), then series (2) is divergent. Two series which are very useful for purposes of comparison are : (a) The geometric series a-{- ar -\- wi^ -\- ••♦, which is convergent whein | ?• | < 1, divergent when | ?- 1 > 1. (6) The series 1 +;^ + ^ 4-:^ + •••, which is convergent when p > 1, divergent when 2)^1 (see Art. 170). Ex. 1. The series i + i + ^^ + ^Jg + ... is convergent, for it is term by term not greater tlian the geometric con- vergent series -. , , , B, The test-ratio test. In series (6), Art. 167, the ratio (3) is comlnonly called the test-ratio. If when n increases beyond all bounds this ratio approaches a definite limit which is less than 1, then the series is convergent. For, suppose that ratio (3) is finite for all values of n, and suppose that after a certain finite number of terms, say m terms, it is less than a fixed number E which is less than 1. Now The sum of the first m terms is finite. Since it follows that the series beginning with u^ is less than the geometric series /i , n . r>2 . \ and, accordingly, is less than 1 171. J INFINITE SERIES. 309 Hence s{x) ; i.e. let {x) = u,'x-^u,'(x)-\-'-. (.S) It will now be shown that (x) =f'(x). By Theorem (a), Art. 172, (x) is continuous, and the conditions, Art. 172, for the term by term integration of an infinite series are satisfied. Accordingly, J(x) dx= i Uq (x) dx -\- I n^' (x) dx -] a «y a »./a = [m„ (x) - ?<„ («) + K, (a;) - Ml (a)] + • • • 173, 174.] INFINITE SERIES. 313 On differentiation, <^(x) =f'(x). Hence, Equation (2) is true. By the aid of this theorem it can be proved that : A power series can be differentiated term by term for any value of x imthin, but not necessarily for a value at, the extremities of the interval of conver- gence. (For proof see Osgood, Infinite Series, p. 62.) Note 1. For instances of functions defined by convergent series which cannot be differentiated term by term, see Professor Osgood's article men- tioned in Note 1, Art. 172. Note 2. Articles 172, 173, have been taken in large part from Osgood's Introduction to Infinite Series, § V., pp. 52-63. Also see Lamb, Calculus, Arts. 193-198 ; Gibson, Calculus, §§ 147-151, 155 ; the article mentioned in Note 1, Art. 172, §§ 5-9. 174. Applications of the integration and differentiation of series. A, Expansions obtained by integration of known series. Three important examples of the development of functions into infinite series by the aid of integration, will now be given. EXAMPLES. Ex. 1. For - l<.x. (a) On expanding the radical by the binomial theorem and taking the term by term integral of the resulting convergent series it will be found that '--[-Grf-(Hri-(i^rf--]- <^) 7. Apply result (6) of Ex. 6 to find the length of the ellipse whose semi- axes are 5 and 4. (To three places of decimals.) 174.] INFINITE SEEIES, 317 8. The time of a complete oscillation of a simple pendulum of length I, oscillating through an angle «(< tt) on each side of the vertical, is ii: Show that this time "^ , in which k = sin ^ a. (c) -H['-(^?'-{^r^-{hur---} (d) Note 4. Integrals (c) and (a) in Exs. 8 and 6 are known respectively as "elliptic integrals of the first and the second kind." The symbols F{k, 0), E (e, 0) are usually employed to denote these integrals (the upper end value here being 0). Knowledge of these integrals was specially advanced by Adrien Marie Legendre (1752-1838). See Art. 122, Note 4. 9. Show that : ..s r^_ix_ 1,1 1 1.3 1.1.3.5 1 , ^0 ^/nr^ 2 5 2.5 92.4.6 13 ' Vl - x^ ^5 2.59 2.4.6 13 ^^ =l_i.2_^1.2jL5.1__l. 2.5.8 ^ 1. y/(mfiy 4 3 7 1-2 3-^ 10 1-2.3 3^ dx ^.^ 1 1 1 1.4 1 1 1.4.7 1 ^/TZr^b 6*3 iri.2*32 16'l.2.3'33'^ CHAPTER XX. TAYLOR'S THEOREM. (See N.B. at beginning of Chapter XIX,) 175. Taylor's theorem is one of the most important theorems in the calculus. It has a wide application, and several important series, for example, the binomial series (see Ex. 6, Art. 176) can be derived by means of it. Let fi^x) be a function of x which is continuous throughout the interval from x = a to .v = h, and which also has all its derivatives continuous in this interval. Now let X receive an increment h. Taylor\s theorem is a theorem which gives the development of the function f{x + h) in a power series in h. The power series itself is called Taylor^s series. (See Note 2, Art. 178.) N.B. In reading this chapter it is better to take up Art. 180 first. 176. Derivation of Taylor's theorem. Analytic proof of the theorem of mean value. Let f{x) and its first derivative be con- tinuous in the interval from x = a to x = h. Find i^i so that /('>)-/(«) = (6 -a)Bi. (1) Substitute x for 6, and put F(x) =f(x) -/(a) -(x~ a)R,. (2) Then F(b)=0 by (1); also F(a) = identically. Hence, by Eolle's theorem (Art. 63), F'(x,) = 0, in which Xi lies between a and h. But by (2), on differentiation, F\x,)=f\x,)-R,. 318 175,176.] TAYLOR'S THEOREM. 319 Accordingly, E^ =f'{xi), as already shown geometrically in Art. 64. Hence, from (1), on substituting X for b, ^^^^ ^^.^^^ ^ ^^ _ ^^^,^^^^^ ^3^ in which a^x^b, and a < Xi < a;. Result (3) may be written fix) =f(a) + (x- - a)f[a + ^(o; - «)], (4) in which < ^ < 1. Thus the theorem of mean value is deduced analytically from Rolle's theorem. (See Arts. 63, 64.) Taylor's theorem. Taylor's theorem can be derived in various ways. The method adopted in this article is merely an extension of that used in deriving result (4). Let /(if) and its first ?i-derivatives be continuous in the interval from x=a to x = b. Find i?„ so that + ^(t,l"i)";V '"""(") ) = (* - «)"'«»• (5) Substitute x for a, and let F(x) =/(6) -fix) - (6 - x}f'(x) - i (6 - xy-f"{x) - ... ~ V-l)!' -^'"'"^"'^ " ^* ~ ''^"^'" ^^^ Then i^(a;) is continuous in the interval from x = a to x = b, since f{x) and its first 7i-derivatives are continuous there. By (5), F{a) = ; also F(b) = identically. Hence, by Rolle's theorem, in which a < iCj < 6. But, on differentiation and reduction, in (6), F'ix,) = -^^=^^r(x,)+n(b-xy-^R„. (7) in which < ^ < 1. 320 INFINITESIMAL CALCULUS. [Ch. XX. On substituting this value of R^ in (5), and writing x for a and x-\-h for b, there is obtained + |^|/("Kaj + e^). (9) This is Taylor's theorem with the remainder, the last term of the second member being denoted as the remainder. In formula (9) X and x-\-1i must both be in the interval of continuity; in any particular application of this formula, x has a fixed value and h varies. Theorem [or formula] (9) is true for all functions which, with their first ?i-derivatives, are continuous in the assigned inter- val of continuity. If all the derivatives of f{x) are continuous in the interval, and if \im,J'^f^-\x-\-eh) = 0, n ! then /(a? + h) =f(x) + hf'Coc) + ^f"(oo) + ^^/'"(jc) + .... (10) For (by Art. 170) the infinite series in the second member converges to the value of f{x + h) and, accordingly, represents the function f{x->th). Formula (10) is called Taylor's theorem, and the series is called Taylor's series. In (9) and (10) h may be positive or negative, so long as x and x-\-h are in the interval of con- tinuity. " Tlie remainder,''^ the last term in (9), represents the limit of the sum of all the terms after the wth term of the infinite series in (10) ; it is the amount of the error that is made when the sum of the first «-terms of the series is taken as the value of the function. Note. The above method of proving the theorem of mean value was first given by Joseph Alfred Serret (1819-1885), professor of the Sorbonne in Paris, in his Cours de calcul differentiel et integral, 2e ^d., t. I., page 17 seq. The above proof of Taylor's theorem appears in Harnack's Calculus (Cath- cart's translation, Williams and Norgate), pages 65, 66, and in Gibson's Calculus, pages 390-393. The proof in Echols's Calc%ilus (p. 82) is likewise based on the theorem of mean value. Taylor's theorem and series are important in the tlieory of functions of a complex variable, and are more fully investigated iu that subject. 176.] TAYLOR'S THEOREM. 321 EXAMPLES. 1. Express log (x + h) by an infinite series in ascending powers of h. Here /(x + A) = log {x^-h). .-./(x) =loga:, X /'"(x)=4, etc. x^ ,.\og(x + h)=logx + 'f-^ + ^ J>^ + .... X 2x2 3x^ 4x* Here x must not be 0, for then f(x) =- oo, and thus is discontinuous for X = 0. The series is evidently more rapidly convergent the smaller is h and the larger is x. On putting x = 1 and h = 1, this result gives l0g2=:l-l+^-J+..., as found in Ex. 3, Art. 174. K the finite series in (9) is used, then log (X + 70 = logx + ^ + ^-^ + ... + (- l)"-i-T^-''-""T7T7^ < ^< 1. X 2x2 ul(x + eh)» Here, if x = /i = 1, log2 = l -i + i-i + . ..+ (-!)«- w(i + ey On interchanging h and x in formula (10), if that can be done in the interval of continuity, there is obtained the following form of Taylor's theorem : f(x + h) = f{h) + xf'ih) + 1^ f'\h) + 1^ f«'{h) + ..., (11) a form which is often useful. Similarly in the case of formula (9). 2. Express log (x + K) by an infinite series in ascending powers of x. Here/(x + /t) = log(x + /i). •■ f{h) = \ogh, f<{h)=\, f<\h)^-l, etc. ...l0g(X+.)=l0g. + -^-^-f^3-.... lth = l, log(l + x)=x-| + |-|4-..., as already obtained in Ex. 3, Art. 174. 322 INFINITESIMAL CALCULUS. [Ch. XX. 3. Represent sin {x + h) by an infinite series in ascending powers in h. Here f{x + h) = sin (x + h). .-. f{x) = sin x, f'(x) = cos x, /"(x) = - sin a:, etc. Hence, on using formula (10), 7,2 7i3 Jji sin (x + h) = sin x + /* cos x sin x cos x + — sin x + •-. 2! 3! 4! Let X = J, and /i = ji^ of a radian (i.e. W 22".65). o Then f -L V Z3 sin '^ + J- cos ^ 1 sin ^ ^ cos "^ + . . .. ,3 lOOy 3 100 3 (100)^2! 3 (100)33! 3 This is a rapidly convergent series. Now sin J = .86603, cos - = .50000. On making the computations, it will be found that, to Jive places of decimals, sin 60^' 34' 22". 65 = .87099. Note. The last exercise is an example of one of the most useful practical applications of Taylor's theorem. Namely, if a value of a function is known for a particular value of the variable, then the value of the function for a slightly different value of the variable can he computed from the known value by Taylor^ s formula. (See Art. 27, Notes 1,3; Art. 82, Note 3.) 4. Expand sin (x + ^) in a series in ascending powers of x. In this case form (11) is to be used. Here /(x + /i) = sin (x + ^). .V f{h) z= sin h, f'{h) = cos h, f"{h) = - sin h, f"{h) = - cos h, etc. .-. sin (x + h) = sin h + x cos /i — — sin h — — cos h + •••. ^ ^ 2! 3! On letting ^ = 0, the following important series is obtained : sinx = x- — + — . 3! 5! 5. Expand cos (x + h) in series, (a) in ascending powers of 7i, (6) in ascending powers of x. From the latter form deduce the series qf.2 ^4 cosx = l-— + ^^ . 2 ! 4 ! 6. Expand (x + /i)"* by Taylor's formula in a power series in h, and thus obtain the Binomial Expansion (X + h)^ = x^ + mx^-^h + ?^^^^^^ x^-2/12 + .... 1 ■ ^ (This series is convergent for ^ < 1, divergent for ^ > 1. The case in which h=±l requires special investigation.) 177.] TAYLOR'S THEOREM. 323 7. Given that f(x) =ix^ - Sx^+I x + 5, develop f(x + 2) and f(x - 3) by Taylor's expansion. Then find /(x + 2) and /{x — S) by the usual algebraic method, and thus verify the results. 8. (1) Assuming sin 42°, compute sin 44° and sin 47° by Taylor's expansion. (2) Assuming cos 32°, compute cos 34° and cos 37° by Taylor's expansion. (3) Do further exercises like (1) and (2). 9. Derive \ogl. x 2x^ Sx^ 10. Show that log sin (x-\- a)= log sin x + a cot x csc^ x + — .— — h •-••. 177. Another form of Taylor's theorem. On substituting the value of B„ [Eq. (8), Art. 176] in (5) and writing x for b, there is obtained ^ (a.-ar ^:n)^-^^e(;r-a)]. (1) nl If all the derivatives of f(x) are continuous in the assigned interval, and lim^ (^:zJ^/(n)[^ + e(x - a)] = 0, n ! then (Art. 170) the infinite series/(a) + (a;-tt)/'(a) + i(a;-a)y"(a) + ••• represents the f unction /(x) * ; i.e. + ^^^^/"\a)+.... (2) n I Forms (1) and (2) for Taylor's theorem and series, are fre- quently useful. The last term in the finite series (1) is Lagrange^ s form of the remainder in Taylor^ s series. (See Note 4, Art. 178.) Except in ^om§ rare cases. 324 INFINITESIMAL CALCULUS, [Ch. XX. EXAMPLES. 1. Express 5 x^ 4- 7 x + 3 in powers of x — 2. Here f(x) = 5 x^ + 7 x + 3, .-. /(2) = 37, /'(x) = 10x + 7, /'(2)=27, /"(x) = 10, /"(2)=10, f"'(x)=0, /'"(2)=0. Now by (2), /(x) =/(2) + (x - 2)/' (2) + (^ - 2)^ /^2) ^ .... .♦. 5 x2 + 7 X +3 = 37 + 27(x - 2) + 5(x - 2)2. 2. Express 4 x^ — 17 x^ -pll x + 2 in powers of x + 3, in powers of X — 5, and in powers of x — 4, and verify the results. 3. Express by* + 6 y^ — 11 y"^ -{- ISy — 20 in powers of y — 4 and in powers of ?/ 4- 4, and verify the results. Note. Exs. 1-S can be solved, perhaps more rapidly, by Horner'' s process. (See text-books on algebra, e.g. Hall and Knight's Algebra^ § 549, 4th edition, 1889.) 4. Develop e* in powers of x — 1. 5. Show that 1= l_ -l-(x- a) + i(x - a)2- l(x - a)34- -., when x X a a^ a^ a* varies from x = to x = 2 a. 6. Show that log x = (x - 1) - ^(x - 1)2 + 1 (x - 1)3 - ... is true for values of x between and 2. 178. Maclaurin's theorem and series. This is a theorem for expanding a function in a power series in x. As will be seen presently, it is really a special case of Taylor's theorem. Let f{x) and its lirst n derivatives be finite for x = and be continuous for values of x in the neighborhood of a; = 0. In form (9), Art. 176, put ic = ; then fih) =/(0)+7»/'(0) + 1!/"(0)+ ... + ^-^/.-"(O) + ^/'\6h). On writing x for h, this becomes f{x) =X0) +x/'(0)+ 1^,/"(0)+ - + (-£^)-,/'"'"(0) + J/'-Xto). (1) 178.] TAYLOR'S THEOREM. 325 lff(x) and all its derivatives are finite for x = 0, and if lim,^^/"^%) = 0, then Ax) =/(0) + acf'iO) + f^/"(0) + ... + ^/.nKO) + .... (2) z I ft I This is known as Maclanrin's theorem, and the series is called Maclanrin's series. The last term in (1) is called the remmnder in Maclaiirin^s series. It is the limit of the sum of the terms of the series after the ;ith term. EXAMPLES. 1. Show that formula (2) comes from form (11), Art. 176, on putting ^ = ; show that this has practically been done in the derivation above. Show that formula (2) comes from form (2), Art. 177, on putting a = 0. 2. Develop sin x in a power series in x. Here f(x) = sin x. .'. /(O) = 0, :.f>ix)=.cosx, /'(0) = 1, /"(x)=-sinx, /"(0) = 0, /"'(x)=-cosx, /'"(0)=-l, /i^(a:) = sinx, /iv(0) = 0, etc. etc. (Compare Ex. 2 above and Ex. 4, Art. 176.) On applying the method of Art. 171 it will be found that the interval of convergence is from — co to + oo. 3. Calculate sin (^ radian), i.e. sin 5° 43' 46".5. By A, sin (.1 radian) = .1 - ^^ + ^^ = .09983. 4. Calculate sin (.5'') and sin (.2'') to 5 places of decimals. (For Tesults, see Trigonometric Tables.) 5. Show that cos « = 1 -—+ — - — +••• , (B) 2 ! 4 ! 6 ! and show that the interval of convergence is from — go to + oo. 6. To 4 places of decimals calculate the following: sin(.3''), cos (.2)'', sin (.4''), cos (.4'). (See values in Trigonometric Tables.) 326 INFINITESIMAL CALCULUS. [Cii. XX. 7. Show that e* = 1 + X + f- + ^ + .-, (C) and show that this series is convergent for every finite value of x. 8. Substitute 1 for x in C, and thus deduce 2.71828 as an approximate vahie of e. 9. Assuming A and B deduce that the sine of the angle of magnitude zero, is zero, and that the cosine of this angle is unity. Note 1. Expansions A and B were first given by Newton in 1669. He also first established series C. These expansions can also be obtained by the ordinary methods of algebra, without the aid of the calculus. For this derivation see Chrystal, Algebra, Part II., Chap. XXIX., § 14, Chap. XXVIII., § 5, and the texts of Colenso, Hobson, Locke, Loney, and others, on what is frequently termed Analytical Trigonometry, or Higher Trigo- nometry. [This subject is rather to be regarded as a part of algebra (Chrystal, Algebra, Part II., p. vii).] Also see article "Trigonometry" {Ency. Brit., 9th ed.). 10. Develop the following functions in ascending powers in x : (1) sec x ; (2) log sec x ; (3) log (1 + x). Compare the latter with Ex. 3, Art. 174. 11. Show that tan X = X -I- \x^ -\- ^^ x^ + g^ x"^ + .... By this series compute tan (.5''), tan 15°, tan 25°. 12. Find: (1) fe^cosxffe; (2) C^dx; (3) f%-^\?x. J Ja X Jo Note 1 a. The integral in Ex. 12 (3) is important in the theory of probabili- ties. If the end-value x is oo, the value of the integral is ^Vw. (Williamson, Integral Calculus, Ex. 4, Art. 116.) 13. Assuming the series for sin x, prove Huyhen's rule for calculating approximately the length of a circular arc, viz. : From eight times the chord of half the arc subtract the chord of the whole arc, and divide the result by three. 14. State Maclaurin's theorem, and from the expansion for tanx find the value of tan x to three places of decimals when x = 10°. 15. Show that cos« x = 1 - -^ x2 + '^^^ ^ ~ ^) x* • 2! 4! Note 2. Historical. Taylor''s theorem, or formula, was discovered by Dr. Brook Taylor (1685-1731), an English jurist, and published in his Metho- dus Incrementorum in 1715. It was given as a corollary from a theorem in Finite Differences, and appeared without qualifications, there being no refer- ence to a remainder. The formula remained almost unnoticed until Lagrnnue (1736-1813) discovered its great value, investigated it, and found for the 179.] TAYLOR'S THEOREM. 327 remainder the expression called by his name. His investigation was pub- lished in the 3Iemoires de V Academic de Sciences a Berlin in 1772. "Since then it has been regarded as the most important formula in the calculus." Maclaurin'^s formula was named after Colin Maclaurin (1698-1746), pro- fessor of mathematics at Aberdeen 1718 ?-1725, and at Edinburgh, 1725-174-5, who published it in his Treatise on Fluxions in 1742. It should rather be called Stirling''s theorem, after James Stirling (1690-1772), who first an- nounced it in 1717 and published it in his 3Iethodus Differentialis in 17o0. Maclaurin recognized it as a special case of Taylor's theorem, and stated that it was known to Stirling ; Stirling also credits it to Taylor. Note 3. Taylor's and Maclaurin's theorems are virtually identical. It has been shown in Art. 178 that Maclaurin's formula can be deduced from Taylor's. On the other hand, Taylor's formula can be deduced from Mac- laurin's ; e.g. see Lamb's Calculus^ page 567, and Edwards's Treatise on Differential Calculus^ page 81. Note 4. Forms of the remainder for Taylor's series (2), Art. (177). Lagrange's form of the remainder has already been noticed in Art. 177. Another form, viz. • ^x-aY(i ^)'--^ ^(.)[-^ ^ Q^^_ ^)-|^ 0<^<1, was found by Cauchy (1789-1857), and first published in his Leqons sur le Calcitl infinitesimal in 1826. A more general form of the remainder is the Schlomilch- Roche form, devised subsequently, viz. (:, ^).(1 ,)n-. ^^^^^^ + ^(o: - a)], < ^ < 1. (n -\)\ p This includes the forms of Lagrange and Cauchy ; for these forms are ob- tained on substituting n and 1 respectively for p. (The 0's in these forms are not the same, but are alike in being numbers between and 1.) In par- ticular expansions some one of these forms may be better than the others for investigating the series after the first n terms. Note 5. Extension of Taylor's theorem to functions of two or more variables. For discussions on this topic see McMahon and Snyder's Calcu- lus^ Art. 103 ; Lamb's Calculus, Art. 211 ; Gibson's Calculus, § 157. Note 6. References for collateral reading on Taylor''s theorem. Lamb, Calculus, Chap. XIV. ; McMahon and Snyder, Diff. Cal., Chap. IV. ; Gibson, Calculus, Chaps. XVIIL, XIX. ; Echols, Calculus, Chap. VI. 179. Relations between trigonometric (or circular) functions and expo- nential functions. The following important relations, which are extremely useful and frequently applied, can be deduced from the expansions for sinx, cos X, and e^ in Art. 178. 328 INFINITESIMAL CALCULUS. [Ch. XX. The substitution of ix for x in C gives e**^ 1-^ + ^ ^i(x-^-\-^ ^ = COS a? + i sin ic. (1) V 3 ! 5 ! / 2 ! 4 ! The substitution of — ix for x in C gives 2 ! 4 ! V 3 ! 5 ! From (1) and (2), on addition and subtraction. ■ J = COS a? - i sin x, (2) cos£c = ^^2^ (3), sina?=^-^| (4) On putting tt for x in (1), there is obtained the striking relation e**^ = - 1. (See Art. 38, Note on e.) Note 1. The remarkable relations (l)-(4), by which the sine and cosine of an angle can be expressed in terms of certain exponential functions of the angle (measured in radians), and conversely, were first given by Euler (1707-1783). (In connection with the expansions in Arts. 178, 179, see the historical sketch in Murray's Plane Trigonometry, Appendix, Note A ; in particular pp. 168, 169. ) Note 2. Results (l)-(4) can also be deduced by the methods of ordinary algebra; see Note 1, Art. 178, the references therein, and Chrystal's Algebra, Part II., Chap. XXIX., § 23. EXAMPLES. 1. From (3) and (4) deduce that cos^ x + sin^x = 1. . 2. Show that tan x = 3. Express cot x, sec x, cosec x, in terms of exponential functions of x. Note 3. Since, by (1), e'* = cos + i sin 0, and e'*** = cos n 4- i sin 4>) ** = cos n^ + i sin n^, for all values of n, positive or negative, integral or fractional. This very important theorem is called De Moivre''s theorem, after its dis- coverer Abraham de Moivre (1667-1754), a French mathematician who settled in England. It first appeared in his Miscellanea Analijtica (London, 1730), a work in which "he created 'imaginary trigonometry.'" [On De Moivre' s theorem, and results (l)-(4), see Murray, Plane Trigonometry, Art. 98, and Appendix, Note D ; and other text-books on Trigonometry.] N.B. The article on Hyperbolic Functions, Appendix, Note A, may be conveniently read at this time. 180.] TAYLOR'S THEOREM, 329 180. Another method of deriving Taylor's and Maclaurin's series. Following is a method which is more generally employed than that in Arts. 176 and 178 for finding the forms of the series of Taylor and Maclaurin. A, Maclaurin's series. Let f{x) and its derivatives be con- tinuous in the neighbourhood of a; = 0, say from x = — a to x = a. Suppose that f{x) can be expressed in a power series in x conver- gent in the interval — a to + a. That is, assume that (for — aS-A,-i-''-+n(n-l){n-2)A^x--'-\--, /(«)(a;) = ?i.?i-l .?i-2 2.1 A+--, On putting x = in each of these identities it is found that A=/(0), A,=r(0), A, = ^, ^3 = -^, -, A = -^^, -. Hence, on substitution in (1), /(x)=/(0)+ x/'(0) + |:/"(0) + g/"'(0)+ ... +5!/x..(0)+ ..., (2) which is Maclaurin's series (Art. 178). B, Taylor's series. Let f(x) and its derivatives be continuous in the neighbourhood of x = a, say from x = a — h to x = a-\- h. Suppose that f(x) can be expressed in a power series in cc — a 330 INFINITESIMAL CALCULUS. [Ch. XX. which is convergent in the neighbourhood oi x = a. In other words, suppose that there is an identically true equation of the form fix) = A, -^A,(x-a) + A, (x - af + ^3 (^ - a)^ + • • • + A(«^-«r + --- (3) Then, as in case A, the following equations, which are obtained by successive differentiation, also are identically true for values of X near x = a, viz. : f'{x)=A,-\-2A,(x-a)-{-SA.,(x-ay-+"'-\-nA,(x-ay-^ + "', f"(x) = 2 A, + 2 ■ 3 Az(x-a) -^"- -^n- n-1 • A^{x- a)'^-2+ ..., f"{x) =2-3- A^-[-'"^n- n -1-71-2 ■ A^{x- ay-'^ + ••., f'{x) =71-71-1 '71-2 - ...2.1. A+--, On putting .t = a in each of these identities it is found that 71 Hence, on substitution in (3), \ fix) =f{a) +{x- a)f(a) + (^^f"(a) + • •• + ^^^^^/'"'(a) + -, (4) 711 which is series (2), Art. 177. If in (4) X is changed into x-^a, then f(x + a) =f(a) + xf'(a) + ^f"(a) + • • • + ^ f^^Ka) + • . ., (5) which is series (11), Art. 176, with a written for h. On inter- changing a and x in (5), form (10), Art. 176, is obtained. Note. On the proof of Taylor's theorem. The above merely shows the derivation of the fonn of Taylor's series. It is still necessary to examine into the convergency or divergency of the series and to determine the remainder 181.] TAYLOR'S THEOREM. 331 after any number of terms. The investigation of tlie validity of the series is a very important matter in the calculus. For this investigation see, among other works, Todhunter, Diff. Cal., Chap. VI. ; Williamson, Dif. Cal., Arts. 73-77 ; Edwards, Treatise on Diff. Cal., Arts. 130-142 ; McMahon and Snyder, Diff. Cal., Chap. IV. ; Lamb, Calculus, Arts. 203, 204; article, "Infinitesimal Calculus" (Ency. Brit., 9th ed., §§ 46-52). 181. Application of Taylor's theorem to the determination of con- ditions for maxima and minima. This article is supplementary to Art. 76. Let f(x) be a function of x such that f{a-[-li) and f(a — h) can be developed in Taylor's series; and let it be required to determine whether /(a) is a maximum or minimum value of f{x). On developing /(a — h) and f{a -\- h) by formula (9), Art. 176, f(a - h) =f(a) - hf'(a) + |^/"(a) - 1^ /'"(«) + - nl f{a + h) =/(a) -f-;i/'(a) -h|^/"(a) +|^/'"(a) + - + A!/(«)(a + ^^,), (2) in which 6^ and $2 lie between and 1. Suppose that the first n — 1 derivatives of f{x) are zero when x=a, and that the nth derivative does not vanish for x=a. Then /(a-A)-/(a) = t-^/(»)(a_e,70, (3) n ! /(a -h h) -f{a) = ^ /">(a + O.Ji). (4) It follows from the hypothesis concerning /(a;) that the signs of /(">(a — eji) and/("X« + Ooh), for infinitesimal values of h, are the same as the sign of /^"^(a). From (3), (4), and the definitions of maxima and minima, it is obvious that : (a) If n is odd, the first members of (3) and (4) have opposite signs, and consequently, f(a) is neither a maximum nor a minimum value off(x) ; (6) Ifn is even and p'\a) is positive, the first mem- bers of (3) and (4) are both positive, and consequently, /(a) is a 332 INFINITESIMAL CALCULUS. [Ch. XX. miniimim vahie of f{x) ; (c) Ifn is even and p''\a) is negative, the first members of (3) and (4) are both negative, and consequently, /(a) is a maximum value of f{x). The condition for maxima and minima that was deduced in Art. 76, (c), is a special case of this, viz. the case in which n = 2. 182. Application of Taylor's theorem to the deduction of a theorem on contact of curves. This article is supplementary to Art. 14o. (See Art. 143, Note 4.) Theorem. If two curves have contact of an even order, they cross each other at the jooint of contact; if tivo curves have contact of an odd order, they do not cross each other at the point of contact. Let the two curves y = cf)(x) and y = if/(x) (1) have contact of the nth. order at a? = a. Then (a) = ^(a), '(a) = ^'(a), "(a) = ^"(a), ..., 4>^^%a) = ^^^%a). (2) Now compare the ordinates of these curves at x = a — h, i.e. com- pare cf>(a — h) and if/(a — h); also compare the ordinates at x = a + h, i.e. compare cf){a + h) and i/^(a -f h). Let it be further premised that cfi(a ± h) and i/'(a ± h) can be expanded in Taylor's series. On using Taylor's theorem (form 9, Art. 176), and remembering hypothesis (2), it will be found that ^{a - 70 - ^(a - h) = t|^ [<^'"+" (a - 0,K) - f "«>(a - BX)\, (3) ^a + 70 - ^(a + A) = j^-^ [<^<"«>(a - 6Ji) - ^"•+"(« - «.''•)]. (*) in which the four ^'s all lie between and 1. Let h approach zero; then, by the premise above, the signs Of the expressions in brackets are the same as the signs of [<^("+^^(a) - i/^('*+i)(a)]. Hence, ifn is odd, the first members of (3) and (4) have the same sign, and, accordingly, the curves do not cross; if n is even, these first members have opposite signs, and, accordingly, the curves do cross. Ex. Accompany the proof of this theorem with illustrative figures. 182, 183.] Taylor's theorem. 338 183. Applications of Taylor's theorem in elementary algebra. Let f(x) be a rational integral function of x, of the nth degree say. Then p''^'^\x) and the following derivatives are all zero. Hence, Taylor's series for f(x + h) in ascending powers of either h or x [see forms (10) and (11), Art. 176] is finite. That is, /(* + /0=/(x) + ft/'(a.) + g/"(^)+-+^/<">(a;), (1) f(x + h)=f(l,) + xfQi) + ^/'(h)+... + J/<"'('0- (2) A rational integral function f{x) of the ?ith degree can also be e:^pressed in a finite series in ascending powers of x — a [see form (2), Art. 177]. That is, /(x)=/(a) + (a;-a)/'(a) + (-^^V"(«)+ - +^^^>")(a). (3) Exercise. See Ex. 7, Art. 176, and Exs. 1, 2, 3, Art. 177. Note 1. Let /(x) be as specified above. In general the calculation of f{x + h) and the expression of f{x) in terms of x — a, can be more speedily- effected by Horner'' a process* This process is shown in various texts on algebra; e.g. Hall and Knight's Algebra (4th edition), Arts. 549, 572. Note 2. For an applicatiou of Taylor's theorem to interpolation, see McMahon and Snyder, Calculus, Note, pp. 325, 326. Note 3. In expansion (10), Art. 176, if h is a differential dx of x, then A, Ti^, h^^ ..., are respectively differentials of x of the first, second, third, •••, orders; and hf(x), h^f"(x), Ay"(x), •••, are respectively differentials of /(.f) of the first, second, third, •••, orders. If h (or dx) is an infinitesimal, these differentials are also infinitesimals of the respective orders mentioned. * William George Horner (1786-1837), an English mathematician, who discovered a very important method of finding approximate solutions of numerical equations of any degree. CHAPTER XXI. DIFFERENTIAL EQUATIONS. N.B. The references made in this chapter are to Murray, Differential Equations. 184. Definitions. Classifications. Solutions. This chapter is concerned with showing how to obtain solutions of a few differe^i- tial equations which the student is likely to meet in elementary work in mechanics and physics. Differential equations are equations that involve derivatives or differentials. Such equations have often appeared in the preced- ing part of this book. ■ : ' .? ■-> - ^ Thus, in Art. 37, Exs. 2, 11, 13, differential equations appear ; Equations (1), Art. 60, (2)-(5), Art. 67 (a), (2)-(5), Art. 67 (c), (3)-(6), Art. 67 (d), are differential equations; so also, in Art. 68, are (I) a^nd (2), Ex. 5 ; equa- tions in Exs. 13, 14, and some of the equations in Exs. 10, 11 ; several equa- tions in Ex. 1, Art. 69 ; Equations (2)-(4), Ex. 1, Art. 73 ; the answers to Exs. 2-4, Art. 73; in Ex. 4, Art. 79; in Exs. 5-8, Art. 80; Equation (8), Art. 144 ; etc., etc. Differential equations are classified in the following ways, A A. Differential equations are classified as ordinary differential equations and partial differential equations, according as one, or more than one, independent variable is involved. Thus, the equa- tions in Ex. 4, Art. 79, and in Exs. 5-8, Art. 80, are partial differen- tial equations ; the other equations mentioned above are ordinary differential equations. (Only ordinary differential equations are discussed in this chapter.) ^B. Differential equations are classified as to the order of the highest derivative appearing in an equation. Thus, of the exam- ples cited above. Equations (2)-(5), Art. 67 (a), are equations of the first order; Equations (2), Ex. 5, Art. 68, and (8), Art. 144, are , r ,13^4.., >•. 184-186.] DIFFERENTIAL EQUATIONS. 335 equations of the second order; the last equation but one in Ex. 1, Art. 69, is an equation of the nth order. A solatioii (or integral) of a differential equation is a relation between the variables which satisfies the equation. Thus, in p Art. 73, Ex. 1, relation (1) satisfies Equation .j^4), and, accordingly, is a solution of (4). ^ " - 4-^-^ ^ ^ ^JT^^ ^^' ^>^ -cA^^-hfi- Ex, 1. Show that relation (1) satisfies Equation (4) in Art. rSf Ex. 1. Ex. 2. See Ex. 4, Art. 79, and Exs. 5-8, Art. 80. In these examples the equations in the ordinary functions are solutions of the differential equations associated with them. Ex. 3. Show that the relations in Exs. 2-5, Art. 73, are solutions of the differential equations obtained in these respective exercises. 185. Constants of integration. General solution. Particular solu- tions. It has been seen in Art. 73, Ex. 6, that the elimination of n arbitrary constants from a relation between two variables gives rise to a differential equation of the nth order. This suggests the inference that the most general solution of a differential equation of the nth order must contain n arbitrary constants. For a proof of this, see Diff\ Eq., Art. 3, and Appendix, Note C. Simple instances of this principle have appeared in Art. 73, Exs. 1-5. A general solution of an ordinary differential equation is a solu- tion involving n arbitrary constants. These n constants are called constants of integration. Particular solutions are obtained from the general solution by giving the arbitrary constants of integration particular values. The solutions of only a few forms of differential equations, even of equations of the first order, can be obtained. N.B. For a fuller treatment of the topics in Arts. 184, 185, see Diff. Eq., Chap. I. EQUATIONS OF THE FIRST ORDER. 186. Equations of the form f{x) dx -{-F{y)dy = 0. Sometimes equations present themselves in this simple form, or are readily transformable into it; that is, to use the expression commonly used, " the variables are separable." The solution is evidently ^f{x)dx+JF{y)dy^c. 336 INFINITESIMAL CALCULUS, [Ch. XXI. Ex. 1. Solve ydx-\-xdy = 0. (1) On separating the variables, — + ^ zz: 0, i X y and integrating, log x + log y = log c ; whence ' xy = c. (2) Solution (2) can be obtained directly from (1) on noting that ydx + xdy is d(xy). Ex. 2. VI - x-^ dy -i-Vl - y^ dx = 0. Ex. 3. n{x + a) dy + m(y -\- b) dx = 0. 187. Homogeneous equations. These are equations of the form Pclx ^ Qcly = 0, in which P and Q are liomogeneous functions of the same degree in x and y. The substitution of voc for y leads to an equation in v and x in which the variables are easily separable. Ex. 1. (2/2 - a;2) dy + 2xydx = 0. Ex. 3. i/ dx + (xy + x"^) dy = 0. Ex. 2. (a;2 + ?/2) dx + xydy = 0. Ex. 4. {y^ - 2 xy) dx = {x^ - 2 a:y) dy. 188. Exact differential equations. These are equations of the form Fdx+Qdy = 0, (1) in which the first member is an exact differential (see Art. 109). If P and Q satisfy test (2), Art. 109, then (1) is an exact differ- ential equation, and its solution is C(Pdx-\-Qdy) = c. Ex. 1. xdy + ydx = 0. (See Ex. 1, Art. 186.) Ex. 2. (2 xy + S)dx-\- (x^ + 4y)dy = 0. Ex. 3. (e== sin 2/ + 2 x) dx + e^ cos ydy = 0. Ex. 4. (ax - ?/2) dy = (x^ - ay) dx. Integrating factors. Equations that are not exact can be made exact by means of what are termed integrating factors. In some cases these factors are easily discoverable. 187-189.] DIFFERENTIAL EQUATIONS, - 337 EXAMPLES. 6. Solve xdy — ydx = 0. (1) The first member does not satisfy the test in Art. 109 ; thus (1) is not an exact differential equation. Multiplication by 1 ^ xy gives dy_dx_Q, y X whence log y — log x = log c, and, accordingly, y = ex. Multiplication by 1 -^ x'^ gives xdy -y d x _ ^ . whence - = c, i.e. y = ex. X Similarly, multiplication by 1 -=- y'^ makes (1) integrable. The multipliers used above are called integrating factors. In the follow- ing examples these factors can be obtained by inspection. 6. Solve (lx, p, -J- ) say. (6) Possibly, (5) may be solvable and give a relation, say, F{p, X, c) = 0. (6) The p-eliminant between (4) and (6) is the required solution. If this eliminant is not easily obtainable, Equations (4) and (6), taken together, may be regarded as the solution, since they suffice for the determination of x and y by assigning values to a param- eter p. Ex. 4. Ay = x^ + p2. Ex. 5. 2y-\-p'i = 2 x^ C. Clairaut's equation, viz. y =poc -\- f{p). (7) In this case y = cx -\-f(c) (8) is obviously a solution. This solution can be obtained on treating (7) like (4), of which it is a special case. Thus, on taking the 2c-derivatives in (7), dp p=p-]-[x+f'(p)-]^^ dp From this, x+f'(p) = (9), or ^^^' ^^^^ Equation (10) gives p = c. Substitution of this'in (7) gives (8). As to the part played by (9) see Diff. Eq.^ Art. 34. 340 INFINITESIMAL CALCULUS. [Ch. XXI. EXAMPLES. a 6. y=px-\--' T. y =px + aVl -h p'^. 8. x'^(y — px) = yp'^. [Suggestion : Put x^ = u, y^ = v.'] Note 1. Sometimes the first member of an equation f(x, y, p) =0 is resolvable into factors. In such a case equate each factor to zero, and solve the equation thus made. (This is analogous to the method pursued in solv- ing rational algebraic equations involving one unknown.) 9. Solve p^ - p'^(x + y -\- 2) -{- p(xy -\- 2 x + 2 y) - 2 xy = a. On factoring, (p — x) = 0, p — y = 0, p — 2 = 0. On solving, 2y = x^ -{■ c, y = ce*, y = 2 x -\- c. These solutions may be combined together, (2y -x^ -c)(y - ce^) (y -2x-c)=0. Note 2, On Equations of the first order which are not of the first degree see Diff. Eq. , Chap. III. 191. Singular solutions. Let a differential equation f(x, y, p)=0 have a solution f{x, y, c) = 0. The latter is geometrically repre- sented by a family of curves. The equation of the envelope of this family (Art. 154) is termed the singular solution of the differ- ential equation. That the equation of the envelope is a solution is evident from the definition of an envelope (see Art. 154) and this fact, viz. that at any point on any one of the curves of the family the coordinates of the point and the slope of the curve satisfy the differential equation. The singular sohition is obviously distinct from the general solution and from any particular solution. For example, the general solution [(8), Art. 190] of Clairaut's equation is, geometrically, a family of straight lines. The envelope of this family of lines is the singular solution of (7). The envelope of (8) may be obtained by the method shown in Art. 157. Differentiation of the members of (8) with respect to c gives — x A- f'Cc') The envelope is the c-eliminant between this equation and (8). EXAMPLES. 1. Show that the singular solution of Ex. 6, Art. 190, is y"^ = 4 ax. 2. Find the singular solutions of the equations in Exs. 7, 8, Art. 190. 191, 192.] DIFFERENTIAL EQ UA TIONS. Ul 3. Find the general solution and the singular solution of: (1) y=px+p'^. (2) p'^x = y. (3) % a(\ -\- p)^ = 21{x + y)(\ - p^. Note 1. The singular solution can a-lso be derived directly from the dif- ferential equation, without finding the general solution ; see reference below. Note 2. On Singular Solutions see Diff. Eq., Chap. IV., pages 40-49. 192. Orthogonal Trajectories. Associated with a family of curves (Art. 154), there may be another family whose members intersect the members of the first family at right angles. An instance is given in Ex. 1. The members of the one family are said to be orthogonal trajectories of the other family. For example, the orthogonal trajectories of a family of concentric circles are the straight lines passing through the common centre of the circles. A, To find the orthogonal trajectories of the family (1) in which a is the arbitrary parameter. Let the differential equation of this family, which is obtained by the elimination of a (see Art. 73), be , ,, ^ *{<^,y,^) = o. (2) Fig. 102. Fig. 103. Let P be any point, through which pass a curve of the family and an orthogonal trajectory of the family, as shown in Fig. 102. For the moment, for the sake of distinction, let (x, y) denote the coordinates of P regarded as a point on the given curve, and let 342 INFINITESIMAL CALCULUS. [Ch. XXI. (X, Y) denote the coordinates of P regarded as a point on the trajectory. At P the slope of the tangent to the curve and the slope of the tangent to the trajectory are respectively — and — ;:• Since these tangents are at right angles to each other, dy _ dX dx dY Also x= X, and y = Y Substitution in (2) gives But P(X, Y) is any point on any trajectory. Accordingly, (3) or, what is the same equation, is the differential equation of the orthogonal trajectories of the curves (1) or (2). Hence: To find the differential equation of the family of orthog- dx dv onal trajectories of a given family of curves substitute for -^ in the differential equation of the given family. ^ EXAMPLES. 1. Find the orthogonal trajectories of the family of circles which pass through the origin and have their centres on the x-axis. The equation of these circles is x2 + ?/ = 2 ax, (4) in which a is the arbitrary parameter. On differentiation and the elimination of a (Art. 73), there is obtained the differential equation of the family, viz. y2_x'^_2xy^ = 0. (5) dx The substitution of - — for ^ gives the differential equation of the orthogonal curves, viz. •=' y2_r^2^2XV~=0. (6) 192.] DIFFERENTIAL EQ UA TIONS. T 343 Fig. 104. Integration of (6) [see Art. 188, Ex. 6] gives a;2 _|. 2,2 — cy, (7) the ortliogonal family, viz. a family of circles passing through the origin and having their centres on the y-axis. (See Fig. 104.) 2. Obtain the orthogonal trajectories of the circles (7), viz. the circles (4). 3. Derive the equation of the orthogonal trajectories of the family of lines y = mx. 4. Derive the equation of the family of concentric circles whose centre is at the origin. B. To find the orthogonal trajectories of the family (8) in which c is the arbitrary parameter. Let the differential equa- tion of this family, which is obtained by the elimination of c, be l^^r, 0, dr d9 0. (9) 344 INFINITESIMAL CALCULUS. [Ch. XXI. Let P be any point through which pass a curve of the given family and an orthogonal trajectory of the family, as shown in Fig. 103. For the moment, for the sake of distinction, let (r, $) denote the coordinates of P regarded as a point on the given curve, and let {R, ©) denote the coordinates of P regarded as a point on the trajectory. At P (see Art. 60) the tangent to the given curve and the tangent to the trajectory make with the radius vector angles whose tansrents are respectively r— and R^- "" ^ ^ dr dR Since these tangent lines are at right angles to each other, dO 1 , dr „ d® „9 d® r — = 7— : whence — = — rR — = — R- -— - dr j^d®' dd dR dR dR Accordingly (9) may be written But P{R, 0) is any point on any trajectory. Accordingly (10), or the same expression in the usual symbols r and 6, Jf (»•>»> -'•'f:) = o, (10') is the differential equation of the orthogonal trajectories of the curves (8) or (9). Hence : To find the differential equation of the family of orthogo- nal trajectories of a given family of curves, substitute —i^— for — in the differential equation of the given family. EXAMPLES. 5. Find the orthogonal trajectories of the set of circles r = a cos ^, a being the parameter. Differentiation and the elimination of a gives the differential equation of these circles, viz. .i^ — + r tan (9 = 0. dd On substituting, as directed above, there is obtained r^ = tan dr the differential equation of the orthogonal trajectories. Integration gives another family of circles r = c sin ^. (11) 192.] DIFFERENTIAL EQUATIONS. 345 6. Sketch the families of circles in Ex, 5, and show that the problem and result in Ex. 5 are practically the same as the problem and result in Ex. 1. 7. Find the orthogonal trajectories of circles (11), viz. the circles in Ex.5. N.B. Various geometrical problems requiring differential equations are given in the following examples. Note 1. On applications of differential equations of the first order, see Diff. Eq., Chap. V. 8. Find the curves respectively orthogonal to each of the following families of curves (sketch the curves and their trajectories) : (1) the parabolas 1/2 = 4 ax; (2) the hyperbolas xy = ^•2 ; (3) the curves a"-iy = a;«; interpret the cases n = 0, 1, - 1, 2, - 2, ± i, ± |, respectively ; (4) the hypocycloids 2 1^ x^ -{- y^ = a^ ; (5) the parabolas y = ax'^; (6) the cardioids r = a(l-cosd); (7) the curves r" sin nd = a"*; (8) the curves r^ = a" cos n6 ; (9) the lemnis- cates r2 = a2 cos 2 ^ ; (10) the confocal and coaxial parabolas r = — ; 1-1- cos (11) the circles x^ + y^ + 2my = a^, in which m is the parameter. ^ 9. (a) Show that the differential equation of the confocal parabolas y"^ = 4 a(x -\- a) is the same as the differential equation of the orthogonal curves, and interpret the result. (&) Show that the differential equation of the confocal conies — 1 ^^ = 1 is the same as the differential equation a2 + z 6-2 _|. 2 of the orthogonal curves, and interpret the result. 10. Find the curve such that the product of the lengths of the perpen- diculars drawn from two fixed points to any tangent is constant. 11. Find the curve such that the product of the lengths of the perpen- diculars drawn from two fixed points to any normal is constant. 12. Find the curve such that the tangent intercepts on the perpendiculars to the axis of x at the points (a, 0), (-a, 0), lengths whose product is b^. 13. Find the curve such that the product of the lengths of the intercepts made by any tangent on the coordinate axes, is equal to a constant a^. 14. Find the curve such that the sum of the intercepts made by any tangent on the coordinate axes is equal to a constant a. EQUATIONS OF THE SECOND AND HIGHER ORDERS. Only a very few classes of these equations will be solved here ; namely, simple forms of linear equations with constant coefficients and homogeneous linear equations. Three special equations of the second order will also be briefly discussed. 346 INFINITESIMAL CALCULUS. [Ch. XXI. 193. Linear Equations. Linear equations are those which are' of the first degree in the dependent variable and its derivatives. The general type of these equations is in which Pj, P,, ••-, P„, X, do not involve y or its derivatives. (For some general properties of these equations see Murray, Integral Calculus, Art. 113, Diff. Eq., Art. 49.) A, The linear equation ^^^+ Pi ^^ ^+P2^^^ ^^/^.....^p. 2,^0 (1) die" dx''-^ dx'"^-'^ in which the coefficients Pi,P2, •••, P„, are constants. The substitution of e^"" for y in the first member, gives (m" + P-^m''-^ + P.:{m''-'- + ••• + P„)e'"^ This expression is zero for all values of m that satisfy the equation ^. ^ p^^^^^-x ^ p,^^n-2 _^ . . . _^ 7.^ _ ; (2) and, accordingly, for each of these values of m, y = e"^"" is a solu- tion of (1). Equation (2) is called the auxiliary equation. Let mi, mg, •••, m„, be its roots. Substitution will show that y = qe'"!'', y = C2e™2*, ...,?/ = c,,e'^n'=, and also y = de"*!^ + C2e™2^ + • • • + c„e"'"'', (3) in which the c's are arbitrary constants, are solutions of (1). Solution (3) contains n arbitrary constants and, accordingly, is the general solution. Note 1. If two roots of^^ (2) are imaginary, say a + i/3 and a — i/S, z denoting V— 1, the correspon^Jing sohition is , According to Art. 179 this may be put in the form y = ea^(cie''^^ + 026^'^*) = ea=^{ci(cos ^x 4- i sin j8a:) + C2(cos /3x — i sin jSa;)}, = e^^^lCci 4- C2) cos /3x + i(ci — C2) sin jSa;}, = ea=^(^ cos j8x + -S sin /3a:) , in which A and 5 are arbitrary constants, since ci and C2 are arbitrary constants. 193.] DIFFERENTIAL EQUATIONS. 347 ^ Note 2. If ttoo roots of (2) are equal, say nit and m^ each equal to a, the corresponding solution , viz. becomes ?/ 3= (ci + C2)e«*, i.e. y = ce"*, which does not involve two arbitrary constants. Put mo = a -h h ; then the solution takes the form , ^^, ^ y = Cie"^ + C2e(''+*)=', = e«'(ci + 626*=^). On expanding e'"' in the exponential series (Art. 178, Ex. 7), this equation becomes y = e«'^(^l -|- J5x + ^ C2h^x!^ + terms in ascending powers of h), (4) in which A = ci + Co and B — c^h. On letting /i approach zero in (4), the latter becomes \ , ^ ^ x \ y = e'"{A-^ Bx). (The numbers Ci and co can always be chosen so that ci + C2 and C2^ are finite.) If a root a of (2) is repeated r times, the corresponding solution is 2/ = (ci + C2X + csx^ -f ... + Cra;''"^)e"'. Note 3. On Equation (1), see Diff. Eq., Arts. 50-55. EXAMPLES. 1. Solve ^^-3^^ + 2y = 0. dx^ dx The auxiliary equation is m^ — 3 w + 2 = ; its roots are — 2, 1, 1. Accordingly, the solution is y = Cie~'^ + (C2 + Czx)^'. 2. Solve ^+a2y zzO. The auxiliary equation is iii^ + a^ = O; its roots are ai, — ai. Accordingly, its solution is y = cie"'^ + C2e"«** = ^ cos «x + -S sin ax. (See Ex. 1, Art. 73.) 3. Solve the following differential equations : (1) D^y - 4 Dy-^ 13 y = 0. (2) D^y -1 Dy + Qy = 0. (3) ^_12^i^-162/ = 0. (4) ^-10^ + 62^-160«^ + 136y = 0. 348 INFINITESIMAL CALCULUS, [Ch. XXI. B, The "homogeneous" linear equation i m ^t'/w*c/i pi, P2, '", Pn, cire constants. First method of solution. If the independent variable x be changed to z by means of the relation z = log X, i.e. X = e', the equation will be transformed into an equation with constant coefficients. (For examples, see Art. 92 and Exs. 3 (i), (v), (vi), page 147^ 4. Show the truth of the statement last made. 5. Solve Exs. 7 below by this method. Second metliod of solution. The substitution of x"" for y in the first member of equation (5) gives [iftiim — V)'"(jn~-n-\-V) -{-p^m{m — 1) • • • (m — w -f 2) H 1-Ph]-^"*- This is zero for all values of m that satisfy the equation m(m — l)«--(m— n + l)H-j9im(m— 1) '••{m, — n-\-2)-\ \-p^=z{). (6) Let the roots of (6) be m^ m^, •••, m„; then it can be shown, as in the case of solution (3) and equation (1), that y = CiX"^' -f- C2iC'"2 _| _j_ c^Qfh, is the general solution of equation (5). The forms of this solution, when the auxiliary equation (6) has repeated roots or imaginary roots, will become apparent on solving equation (5) by the first method. EXAMPLES. 6. Show that the solution of (5) corresponding to an r-tuple root m of (6), is ?/ = a;'»[ci + C2loga; + C3(logx)2-|- ... + CrClogx)*— 1] ; and show that the solution of (5) corresponding to two imaginary roots a + 1/3, a — i^, of (6) , is y = x'^[ci cos (j8 log x) + C2 sin (/S log a;)]. 194.] DIFFERENTIAL EQUATIONS. 349 7. Solve the following equations : (1) x^D^y - xDy -h2y = 0. (2) x'^D'^y - xDy + y = 0. (3) x'^D-y -3xDy-\-^y = 0. (4) x^D^y + 2 x^^D^y -\- 2 y = 0. Note 3. Equations of the form are reducible to the homogeneous linear form, by putting a -{- bx = z. 8. Show the truth of the last statement. 9. Solve (5 + 2 a;)2^- 0(5 + 2 a;)^+ 8 2/ = 0. Note 4. On Equation (5), see Diff. Eq., Arts. 65, 66, 71. 194. Special equations of the second order. A, Equations of the form -j-^^ = f(y)» dy For these equations 2 y- is an integrating factor. EXAMPLES. 1. ^ + a^i/ = 0. (See Ex. 2, Art. 193.) On integrating, [y j = - aV + ^ = a2(c2 — y-)^ on putting aV for k. On separating the variables, ' = adx, y/c^ — y'^ V and integrating, sin-i- = ax ■}- a. This result may be written y = c sin (ax + a) , or y = Asinax + B cos ax. 2. Show the equivalence of the last two forms. Express A and jB.in terms of c and a, and express c and a in terms of A and J5. 3; Show that 2 y is an integrating factor in case A. C Solve the following equations : (3) If ^2 ~ ~2~' ^'^^ ^' S^^^^ ^^^^t 1~~^ ^^^ X = a, when t = 0. (tt 8 , (It 350 INFINITESIMAL CALCULUS, [Ch. XXI. B, Equations of the form /(|^, il' ^) " ^- (1) On letting j> denote — , this may be written /( -^, p, x\= 0. (2) dx \dx J Integration of (2) may give '- (12) 2/-2 = a2(i+^2). {IZ) {px-y){py -\-x)=h'^p. {\A) pH^ -^ x'^py = \. {\b)x = 2y-^p\ (16) p^ 4- 2py cotx = 2/2, (17) yy/i Jf- p-i ~ a; also find the singular solution. (18) y — px = Vb'^ + a^p^^ ; also find the singular solution. (19) xp^ =(x — a)'^, d^ii d^v and also find the singular solution. (20) -jf^ — a*y = 0. (21) —^ + 4 ?/ = 0- (.24) x^g + 3.^g + .|f + . = 0. (25) (. + a)^g-4(x+a)2+6,=0. ^^=- ^^m=h ^^^^- <-)(S)=«(i)- APPENDIX. NOTE A. ON HYPERBOLIC FUNCTIONS. 1. This note gives a short account of hyperbolic functions and their properties. The student will probably meet these functions in his reading ; for many results in pure and applied mathematics can be expressed in terms of them, and their values are tabulated for certain ranges of numbers.* There are close analogies between the hyperbolic functions and the circular (or trigonometric) func- tions (a) in their algebraic definitions, (b) in their connection with certain integrals, (c) in their respective relations to the rectangular hyperbola and the circle. 2. Names, symbols, and algebraic definitions of the hyperbolic functions. The hyperbolic functions of a number x are its hyper- bolic sine, hyperbolic cosine, hyperbolic tangent, •••, hyperbolic cosecant, and the corresponding six inverse functions. These func- tions have been respectively denoted by the symbols sink x, cosh x, tanlix, cotlix, sechx, cosechx, sinh'^x, etc. These are the symbols in common use. As to symbols for the hyperbolic functions, the following suggestion has been made by Professor George M. Minchin in Nature, Vol. 65 (April 10, 1902), page 531: "If the prefix 7iy were put to each of the trigonometrical functions, all the names would be pronounceable and not too long. Thus, Jiysin x, hytanx, etc., would at once be pronounceable and indicate the * See tables of the hyperbolic functions of numbers in Peirce, Short Table of Integrals (revised edition, 1902), pages 120-123 ; Lamb, Calculus, Table E, page 611 ; Merriman and Woodward, Higher Mathematics, pages 162-168. 353 354 INFINITESIMAL CALCULUS. hyperbolic nature of the functions." This notation will be adopted in this note.* The direct hyperbolic functions are algebraically defined as follows : hysin oc = - — -^ — > hycos dc = ^ — 5 hytan^=^y^^JL^^ ^"-^"" , hycot^ ^ |^^^?^^ ^ ^" + ^"% (1) hycos a? e^ -\- e-i>^ hysma? e^ - e-^ ^^ hysec a? = > hycosec a? hycos a? hysin a? There is evidently a close analogy between these definitions and the definitions and properties of the circular functions. [See the exponential expressions (or definitions) for sin x and cos x in Art. 179.] From the definitions for hysin x and hycos x can be deduced, by means of the expansions for e"' and 6""= (see Art. 178, Ex. 7), the following series, which are analogous to the series for sino? and cos x (Art. 178, Exs. 2, 5) : hysin a3 = a? + 1^ + 1^ + .... hycos a. = l+|^ + |* + ...5 (2) The second members in equations (2) may be regarded as defi- nitions of hysin x and hycos x. EXAMPLES. 1. Derive the following relations, both from the exponential defini- tions of sin aj, cos a;, hysin x, hycos x, and from the expansions of these func- tions in series : (1) cos x = hycos (ix) ; (2) i sinx = hysin {ix) ; (3) cos {ix) = hycos x ; (4) sin (ix) = i hysin x. 2. (a) Show that e^ = hycos x + hysin x, e-^ = hycos x — hysin x. [Compare Art. 179 (1), (2).] (&) Show that hysinO = 0, hycosO = l, hy tan = 0, hysin qo = 00, hycos qo = qo, hytan 00 = 1, hysin (—x) = — hysin x, hycos ( — x) = hycos x, hytan ( — x) = — hytan x. * The symbols used in W. B. Smith's Infinitesimal Analysis are hs, he, ht, hct, hsf, hcsc. APPENDIX, 355 3. Show that the following relations exist between the hyperbolic functions : (1) hycos^x — hysin2x = 1 ; (2) hysec^x + hytan^x = 1 ; (3) hysin {x ± y)= hysin x • hycos y ± hycos x ■ hysin y ; (4) hycos (x ±y)— hycos x • hycos y ± hysin x • hysin y ; (5) hytan (x ± y) = (hytan x ± hytan «/) -^ (1 ± hytan x • hytan y) ; (6) hysin 2 a; = 2 hysin x • hycos x ; (7) hycos 2x = hycos^ x + hysin^ x = 2 hycos^ x — 1 = 1 + 2 hysin^ x ; (8) hytan 2 x = 2 hytan x -^(1 + hytan^ x). Compare these relations with the corresponding relations between the circular functions. 4. Show the following: (1) ^^^^^^^5^ = hycos x ; (2) ^(hycosx)^ dx dx hysin x; (3) ^^(M^^^hysec^x ; (4) ^^^y^^^^)^ -hycsc^x; (5) ^jhysecx) dx dx dx = — hysec x • hytan x ; (6) C ycsca;; _ _ j^y^^g^ ^ , hycot x ; (7) i hysin x dx r ^ r = hycos x; (8) | hycos x dx = hysin x ; (9) | hytan x rfx = log (hycos x) ; (10) \ hycot xdx = log (hysin x) ; (H) l hysec XfZx = 2 tan-^e^ ; (12) i hy esc xfZx = log (hytan -j. Compare these relations with the cor- responding relations between the circular functions. 5. Make graphs of the functions hysin x, hycos x, hytan x. (See Lamb, Calculus^ pp. 42, 43.) y X X 6. Show that the slope of the catenary - = hycos- is hysin-- Sketch ^, . "" a a a this curve. Inrerse hyperbolic functions. The statement "the hyperbolic sine of y is ic" is equivalent to the statement "?/ is a number whose hyperbolic sine is ic." These statements are expressed in mathematical shorthand, hysin y = oc, y = hysin-i ac. (3) The last symbol is read " the inverse hyperbolic sine of a;," or "the anti-hyperbolic sine of x^ The other inverse hyperbolic functions are defined and symbolised in a similar manner. The inverse hyperbolic functions can also be expressed in terms of logarithmic functions, and thus they may be given logarithmic definitions. (This might have been expected, for the direct hyper- bolic functions are defined in terms of exponential functions, and the logarithm is the inverse of the exponential.) 5Q INFINITESIMAL CALCULUS. Let hysin?/ = ic; then a; = ^(e^ — e"*'). This equation reduces to e-^ — 2 ice^ — 1 = 0. On solving for e^, ^ = x -\- Vjtf + 1. (4) (For real values of y, e^ being positive, the positive sign of the radical must be taken.) From (4) y = hysin-i oc = log(x + va?2 + i), (5) N.B. The base of the logarithms in this note is e. In a similar manner, on putting X = hycos ?/ = 1 (e^ -f e-^)j and solving for e^, e^ = x±V^- 1. (6) For real values of y, x is greater than 1 and both signs of the radical can be taken. From (6) and the fact that (x + V^ — l)(x— ^xr — 1) = 1, and thus log (x — Va^ — 1) = — log (x -f -\/otf — 1), it follows that y = hycos-i x = ± \og(ac + V^a - l) . (7) In a similar manner it can be shown that hytaB-ia, = |logl±|, (8) where a^ < 1 for real values of hytan~^ x ; and that hycot-ix=|log|^, (9) where a:^ > 1 for real values of hycot"^ x. EXAMPLES. 7. Derive the relations (7), (8), (9). 8. Solve equations (5), (7), (8), (9), for x in terms of y, and thus obtain the definitions of the direct hyperbolic functions. 9. Show that the differentials of hysin-i x, hycos~i x, hytan-i x, hycot-^ x, are respectively ^^ , ± ^^ . -^- for x2 < 1, ^- for x2 > 1. Vx^ + 1 Vx^ - 1 1 - a;2 x2 - 1 Compare these with the differentials of sin~i x, cos'i x, tan-i x, cot-i x. APPENDIX, 357 10. Following the method by which relations (5) -(9) have been derived, show that : fZf hysin-i ^\ = ^^ ; d[ hycos-i ^ ^ = ± hysin-i ^ = log ^ + ^^^^ + ^^ hycos-i ^ = ± i ^ + V^2 - a'2 a a a a hytan-l - = ^ log ^^^t^ for a:^ < a' ; hycot-i - = | log ^^^^ for x^ > a^. 11. Assuming the relations in Ex. 10, show that the x-differentials are : x\ _ , dx . fZfhytan-i^^ = ^^^ fora:2«2. V a/ a^-x^ \ a) x^ — d^ Compare these differentials with the differentials of sin-i -, cos-i -, tan-^ -, 12. Assuming the relations in Ex. 10 as definitions of the inverse hyper- bolic functions, derive the definitions of the corresponding direct hyperbolic functions. (Suggestion. Follow the plan outlined in Ex. 8.) 3. Inverse hyperbolic functions defined as integrals. It follows from Ex. 11. Art. 2, that r_d^^ = hysin-i- + c ; f ^^ = ± hycos-^ - -f c ; r--^, = ihytan-i-+c,(a)2a2. 358 INFINITESIMAL CALCULUS. These relations between definite integrals and inverse hyperbolic functions may be taken as definitions of the functions. The inverse circular functions can be defined by integrals which are very similar to the integrals appearing in the definitions of the hyperbolic functions. Thus : dx . 1 u C^ d,x -iU _ cin-l I =— cos~ -. Va^ — x^ ^ "^ " Va^ f '^^ ^ sin-''^. ( Jo y„2 _ 3,2 a J X ""^ =itan-i».' r^^^. = -ieot-'« \ o? -\- 7? a a Joo a'- + x^ a EXAMPLES. 1. Find the area of the sector AOP of the hyperbola x^—y^ = a^ (Fig. 106), P being the point for which x = u. Thence show, from tlie definition above, that hycos-i - is the ratio of twice the sector AOP to the square whose side is a. 2. Find the area of the sector BOP' bounded by the ?/-axis, the arc BF of the hyperbola y^ — x"^ = cfi (the conjugate of the hyperbola in Ex. 1), and the line OP' drawn from the origin to the point P , P' being the point for which X = u. Then show, from the definition above, that hysiir^ - is the ratio of twice the sector BOP* to the square whose side is a. ^ 3. Sketch the curve y(a^—x'^)=a^. Calculate the area between this curve, the axes, and the ordinate for which x = u(u'^<::_a^) . Show that hytanT^ - is the ratio of this area to the area of the square ivhose side is a. ^ 4. Sketch the curve y(x'^ - a"^) = a^. Calculate the area bounded by this curve, the x-axis, and the ordinate at x = u(ifi'>a'^). Show that hycot~^ - is the ratio of this area to the area of the square whose side is a. ^ 4. Geometrical relations and definitions of the hyperbolic functions. In Eig. 105 P is any point (x, y) on a circle x^ + ?/- = al Let the area of the sector AOP be denoted by u and the angle AOP by 6. Then, by plane trigonometry, 2^ = ia2^; whence, ^ = ^- (1) In Fig. 106 P is any point on a rectangular hyperbola x^—y^—a^. (The a of the hyperbola bears no relation whatever to the a of APPENDIX. 359 the circle.) Let the area of the sector AOP be denoted by u. Then u = area 0PM — area AP3I =^xy — i Va^ — or dx j whence, « = ?- log ^ + ^^'-»' = ^ log £±l.t 2 a 2 a (2) O From (2), log^tJ = ^5 whence, ^±^ = e< X Fig. 105. Fig. 106. From equations (3), on addition and subtraction, 2m " 2m 2u 2u _2m -^ 2m 6*2 _^ g a2 (4) * That is, M = I a2 hycos-i - ; whence, hycos-i ^ = ?^. a a a^ t If a = 1, log (x + 2/) = 2 M = twice area AOP. On account of the relation between natural logarithms {i.e. logarithms to base e) and the areas of hyper- bolic sectors, natural logarithms came to be called hyperbolic logarithms. The connection between these logarithms and sectors was discovered by Gregory St. Vincent (1584-1667) in 1647- 360 INFINITESIMAL CALCULUS. Relations (4) lead to geometrical definitions of the hyperbolic func- tions. These definitions are given in the following scheme. This scheme, supplemented by relation (1), also shows the close geo- metrical analogies existing between the hyperbolic and the circular functions. N.B. In Figs. 105, 106 the a and u of the circle are not related in any way to the a and u of the hyperbola. In a hyperbola or — y^ — a- (Fig. 106), if P is any point {x, y) and u = area sector AOP, then ^ = hysin^, In a circle x^-^y^ = a'^ (Fig. 105), if P is any point {x, y) and u = area sector AOP, then ^ = sin -^, a a^ X 2u - = cos , a a'' • whence, • ^r = sin-^ = cos-^ = a- a a = tan-i^. X --hycos^, a a2 J^ = hytau2f5 ac a2 ' whence. ^ = hysin-i^ = hycos-i^ = hytan-i K • a? These results may be expressed in words : The circular functions may be defined by means of the relations connecting a point (x, y) on the circle x^ -\- y"- = a? and a certain cor- responding circular sector; and the hyperbolic functions may be defined by means of the relations connecting a point (x, y) on the rectangular hyperbola, x^ — y^ = a^ and a certain corresponding hyper- bolic sector. Each of the inverse circular functions may be expressed as the ratio of tivice the a,rea of a certain sector of a circle of radius a to the square described on the radius of the circle, and each of the inverse liyperbolic functions may be expressed as the ratio of twice the area of a certain sector of a rectangular hyperbola of semi-axis a to the square described on this semi-axis. (For a more general notion see Ex. 3 following.) The term hyperbolic arose out of the connection of these func- tions with the hyperbola. APPENDIX. 361 EXAMPLES. 1. Show that hysin-i | = hycos-^ f = hytan"i i. Represent each of these functions geometrically. Compute hysin-i |. {_A7is. 1.099.] 2. Show that hysin-i | = hycos-^ f = hytan-i |. Represent each of these functions geometrically. Compute hysin~i |. \_Ans. .693.] 3. Show that, if AP (Fig. 105) is an arc of an ellipse b'^x^ + a^y^ = a^b^, and u denote the area of the elliptic sector AOP, it is possible to write ? = cos^, y=:sin^. a ah b ab Also show that, if ^P (Fig. 106) is an arc of a hyperbola ^ - |^ = 1> and u denote the area of the hyperbolic sector AOP^ then and thence show that ^^:=hycos2j^, M = hysin2i£. a ab b ab (Williamson, Integral Calculus^ Arts. 130, 130 a.) X^ y.i 4. Show that a point P(x, y) on the ellipse -3 + b = 1 ^^ ^^- ^ ^^7 ^® represented as (acos^, 6sin^), and show that ^(=: eccentric angle of P) = (2 area sector ^ OP -4- a6). x^ V Show that a point P{x, y) on the hyperbola -^ — ^ = 1 in Ex. 3 may be represented as (a hycos v, 6 hysin v), and show that v =(2 area sector AOP---ab). 5. The Gudermannian. Suppose that sec (f> 4- tan <^ = hycos v + h}' sin v. (1) From (1) and the identities sec- — tan^ = 1, hycos^?; — hysin^u = l, it follows that sec <^ = hycos v, (2) tan <^ = hysin v. (3) Since [see Art. 2, Ex. 2 («)] log (hycos v + hysin v) = v, relation (1) may be written 1 / i , * xn /a\ ^ ^ -^ V = log (sec 4> + tan ^) ; (4) that is, by trigonometry, V = log tan (I + 1) = 2.302585 log^o tan (^ + |J. (5) 362 INFINITESIMAL CALCULUS. When any one of the relations (l)-(5) holds between two numbers V and cfi, (f> is said to be the Gudermannian of v.* This is expressed by this notation : . , z^s ^ «j> z= gd V, (6) In accordance with the usual style of inverse notation each of the relations (4), (6), (6) is expressed v = gd-^^, (7) The second members of (4) and (5) are more frequently denoted by the symbol X(<1>), which is read " lambda (^," than by gd'^ cfj. Geometrical representation of X (<^) or gdr^ <^. If at F (x, y) in Fig. 106, x = a sec ^, then y = a tan <^, since x^ — y'^ = a^. On mak- ing this substitution for x and y, it can be deduced that area sector AOP =^a^ log (sec <^ + tan <^). (8) From this, log (sec + tan <^), i.e. X (cf>) (or gd-^ <^) = ^ • sec^Qr ^OP ^^^ a From (4), (6), (8), ^ = gd ( ^ " ''''^ ^^^P) - (10) If the area of sector AOP be denoted by u, relations (9) and (10) may be expressed 9d-' = —, cf> = gd — - a^ a^ To construct ari angle whose radian measure is . In Fig. 106, about as a centre with a radius a describe a circle. From M draw a tangent to this circle, and let the point of contact be at P* in the first quadrant ; and draw OP'. Now 0M= OP sec MOP ; i.e. x = a sec MOP. But, according to the hypothesis in the last paragraph, x = a sec <^. Hence, an^Ie MOP' = <|>. If a point P(x, y) on the hyperbola x^ — y^ = a?- (see Ex. 4, Art. 4) be denoted as (a sec 0, atan0), is the angle which has just now been con- structed. * This name was given by the great English mathematician Arthur Cayley (1821-1895) "in honour of the German mathematician Gudermann (1798- 1852), to whom the introduction of the hyperbolic functions into modern analytical practice is largely due." (Chrystal, Algebra, Vol. II., page 288.) APPENDIX, 863 EXAMPLES. 1. Derive result (8). 2. (a) Show that, and v being as in equations (l)-(7), gdv = sec"i (hycos v) = tan-i (hysin v) = cos"i (hysec v) = sin-i (hytan v) = cot~i (hycosec v) = cosec-^ (hycot v) ; hytan ^^ = tan x <|>. (6) Show that gd'^ = hycos-^ (sec 0) =hysin-i (tan (p) ; gd x=2ta.n-^ e==— ^- 3. (a) Show that tlie derivative of \(^(p)(i.e. gd-^)=0 say, is called the intrinsic equation of the curve. The term intrinsic is used because the coordinates s and =Mx,y). (3) The elimination of x and y between equations (1), (2), (3), will give the required equation between s and <;^. Fig. 108. APPENDIX. 365 Similarly, let the polar coordinates of P be r and dj and let the polar equation of the curve be F{r,6) = 0. (4) Express s in terms of r, 6 ; suppose that s = F,{r,6). (5) Also express <^ in terms of r, d\ suppose that 4> = F,{:r,e). (6) The elimination of r and 6 between equations (4), (5), (6), will give the required equation between s and <^. Note. A tangent parallel to the x-axis is usually chosen for the tangent of reference. EXAMPLES. 1. Derive the intrinsic equation of the hypocycloid x3 + 2/3 = ai (1) Take the cusp on the positive part of the x-axis for the fixed point, and the tangent there for the tangent of reference. Then at any point P(x, y) on the arc in the first quadrant tan0 = -(i/^-x^), (2) and s = I a^ («^ - x^). (3) From (1) and (2), sec2 = tan2 ^ + i = a^ -- x^. . Substitution for x^ in (3) gives 2s = Za sin^ 0, 2. If in Ex. 1 the chosen fixed point O be at a distance h along the curve from the cusp and the chosen fixed tangent (not necessarily at 0) make an angle a with the tangent at the cusp, show that the intrinsic equation of the hypocycloid is 2 (s+ &) =3asin2(0 + a). 3. Find the intrinsic equation of the cardioid r = a(l — cos 6). Let the polar origin be chosen for the fixed point, and the tangent there be chosen for the tangent of reference. Let P(x, y) be any point on the cardioid. Then s = (Jyj r^ -{- (^Yde = iai I - cos -Y (1) 366 INFINITESIMAL CALCULUS. Also, (Art. 60), = ^ + tan-i ^i^=d + tan-^ f tan '^] = ^e. (2) dr V 2 ,' On substituting in (1) the value of 6 from (2), s = 4 af 1 — cos^V 4. If in Ex. 3 the chosen fixed point be at a distance b from the polar origin and the chosen tangent of reference make an angle a with the tan- gent at the polar origin, show that the intrinsic equation of the cardioid is 5. Derive the intrinsic equation of each of the following curves, the fixed point and the fixed tangent being as indicated : (1) tlie catenary X X ?/ = - (e« + e «), the vertex and tangent thereat ; (2) the parabola ?/2 = 4 ax, 6 the vertex and tangent thereat ; (3) the parabola r = a sec^ -, as in (2) ; (4) the cycloid x = rt(^ — sin^), ?/ = a(l — cos ^), with reference to (a) the origin and tangent thereat, (h) the vertex and tangent thereat ; (5) the logarithmic spiral r = ce"^ ; (6) the semi-cubical parabola 3 ay'^ = 2 x^, the origin and tangent thereat ; (7) the curve (8) the semi-cubical parabola y^ = ax^ ; (9) the tractrix x = Vc- — y'^ 4- clog^-i — C" — y ^ ^jjg point (0, c). (For an account of the tractrix and y for various problems which reveal its properties, see the text-books of Williamson, Byerly, Lamb, and F. G. Taylor, on the calculus.) [Answers : Ex. 5. (1) s = a tan 0, (2) s = a tan

'{a + d2h) ' (The 0's are each less than 1.) Hence, by (1), the value of Z^ = ZW . ^ ^ ^ 0(a) 0'(a) If /'(a) and 0'(a) are both zero, then 0(« + /^) 0(a)+;i0'(a)+|->"(a-fM) ^"(« + ^^'^) Hence, by (1), the value of J^-^ = ^^^^f^ 0(a) 0"(a) On proceeding in this way it can be shown, by means of Taylor's theorem, that, if, for x = a, /(x) and 0(x) and all their derivatives up to and including (2) APPENDIX. their nth derivatives, are zero, while /("+^)(a) and ^("^^^(a), are not both "'•"•""'" the value «f|M = /»;"(«) . Result (2) may also be expressed thus : >"+i)(ac) EXAMPLES. 1. Evaluate ^^-::ii when x = 2. (See Art. 1.) x-2 ^ ^ Valuex-2 ^^^ = value^i2 ^(^^'*) = valuex-2 — ' = 4. x-2 D{x-2) 1 2. Evaluate {x — sin x) -=- x^ when x = 0. In this case, ,., x-sinx ,.^ 1-cosx* i;^ sinx* ,. • cosx 1 linia^ -. = limx-=o — r--; — = Iiuix-^ = Jinii^— — - = - . x^ 3x2 6x b b Xote. The labour of evaluating /(a) -h ^(a) may be lightened in the following cases : (a) If, in the course of the reduction a factor, say ypix), appears in both the numerator and the denominator, this common factor may be cancelled. [Compare Art. 1, Note 2 (a).] (h) If at any stage during the process of evaluation a factor, say ^(x), appears only in the numerator or only in the denominator, and ^(a) is not zero, the value of \}/{a) may be substituted immediately for i/'(x). This will lessen the labour in the succeeding differentiations. 3. Evaluate the following: (1) ^'~^' , when x = 0; (2) !i^::i?, X X whenx^O; (3) ?!lz:^ when x=a ; (4) ^'~^"' , whenx=0; (5) kz^^?^, when;^ = 0. ^-« ^^^^ 4. Find the following : (1) lim^ {x~^y^\nx ^2) lim^.^ (^ " 5)^ ^og (3 - x) . X sin (x — 2) /-QN 1,-rv, e== + e-* + 2C0SX — 4 ,.. ,.^ tan x — sinx (3) lim^^y) — !^ ^^— ; (4) lim^^ : ; X* X - sm X (5)lim^ l-^««^ cos X sin2 X {Answers : Exs. 3. log -, 1, wa«-i, 2, ^ ; Exs. 4. 25, - 9, |, 3, ^.] 6 * Which is in the form -r- 0. B70 INFINITESIMAL CALCULUS. 3. Evaluation of expressions taking the form ~ . Suppose that /(a) = 00 and 0(a) = 00 , and let it be required to determine the value of ^-^ • 0(a) _1_ • Now ^-^ ^t^. The latter is in the form -• Application of result i2, mg, ••♦, and whose distances from any plane are, respectively, dj, c?2, cfg, •••. Let a number D be calculated such that J) ^ midiH-mg^^a + mgC^g"- . ^. ^ ^^^ ^ ^ S md Wi -|- m.2 + mg + • • • ' 2»i For any given plane, D evidently has a definite value. * A real definition of mass, one that is strictly logical and fully satisfac- tory, is explained in good text-books on dynamics and mechanics. (For example, see MacGregor, Kinematics and Mechanics, 2d ed.. Art. 289.) APPENDIX. 373 If (oTi, ?/i, Zy), (x.2, 2/2, 2^2), (a^3, Vs, ^s), ••, respectively, be the coordi- nates of these particles with respect to three coordinate planes at right angles to one anothei", then the point (x, y, z), such that » = -^— J y=^r^^ 2;=——, (1) 2m 2m 2im is called the centre of mass of the set of particles. If the matter " be distributed continuously " (as along a line, straight or curved, or over a surface, or throughout a volume), and if Am denote the element of mass about any point (a;, y, z), then, on taking all the points into consideration, equations (1) may be written : X = l™A>»^S.t--Am ^ and similarly for y and z. (2) hm^,^oSAm From (2), by the definition of an integral, ( oc din \ y dm \ z dm ^ = ^ ,V = ^ , S = =L (3) I dm I dm I dm. If p denote density of an infinitesimal dv about a point, then dTU = pdv (4) ; and, on integration, m=\p dv, (5) Ex. Write formulas (3), putting pdt^ for rim. Suppose that the body is not homogeneous; that is, suppose that the density of the body varies from point to point. Let p denote the density at any point (x, y, z), let dv denote an infini- tesimal volume about that point, and let p denote the average or mean density of the body. Then mass of body _)P^^ ^ ~ vol. of body ~ C^^ ' Note. The term centre of mass is used also in cases in which matter is supposed to be concentrated along a line or curve, or on a surface. In these cases the terms line-density and surface-density are used. 374 INFINITESIMAL CALCUL US. EXAMPLES. 1. In a quadrant of a thin elliptical plate whose semi-axes are a and 6, the density varies from point to point as the product of the distances of each point from the axes. Find the mass, the mean density, and the position of the centre of mass, of the quadrant. Choose rectangular axes as in the figure. At any point P(x, y), let p denote the density and dm denote the mass of a rectangular bit of the plate, say, dx • dy. Let M denote the mass, p the mean density, and (x, y) the centre of mass, of the quadrant. Now dm = p dx dy. Fig. 109. But pccxy ; i. e. p = kxy, in which k denotes some constant. Accordingly, M=\dm=\ \ « kxy dydx — \k a^h"^ _ _ mass of quadr; volume of quadi mass of quadrant _ \k a^lP- volume of quadrant \ irab kab 27r ' •V^^372 dy dx Here Similarly, y ^^kam 1 ka^h-^ ha. dv 31 Hence, centre of mass is (y^^ a, j^ b). 2. Find the centre of mass of a solid hemisphere, radius a, in which the density varies as the distance from the diametral plane. Also find the mean density. Symmetry shows that the centre of mass is in OY. Take a section parallel to the diametral plane and at a distance y from it. The area of this section = TT . CP^ = 7r(a2 _ J/2). For this section, p coy, i.e. p = ky, say. :r r^" ^ /^^ ' -;/ \ / / ■y^~-.\ ^^^^ Fig. 110. Then Also Jo' w(a^-y^)dy kir \ y'i(a^ - y^)dy r X' p7r(a2 _ y2)dy ■|;Ka2 y^)dy vol. I 7ra3 ^ This is the density at a distance | a from the diametral plane. APPENDIX. 375 3. The quadrant of a circle of radius a revolves about the tangent at one extremity. Find the position of the mass-centre of the surface thus generated. In this case let the "surface-density" be denoted by p. Symmetry shows that the mass-centre is in the line PL. Let y denote the distance of the mass-centre from OX. In PL take any point iV, at a dis- tance ?/, say, from OX. Through N pass a plane at right angles to PZ, and pass another parallel plane at an infinitesimal distance dy from the first plane. These planes intercept an infini- tesimal zone, of breadth ds say, on the surface generated. Area of this zone = 2 tt • CiY- c?s = 2 7r(J/iY— MC)ds. Now, at C (x, y) x^ + y^ = a^. Accordingly, ds = Jl ^ I^^Y ■ dy = -^ Hence, area zone = 2ir(a — Va'^ — y'^) -dy. Va^- — dy = 2Tra( ■ 2/2 VV^ .:y = {"^py • (2 TT . CN-ds) 2 TT ap^^y V «2 - y2 -\yy. -l\ly p - area zone 2.ap^J -2 876 a. iVfy 4, In the following lines, curves, surfaces, and solids, find the posi- tion of the centre of mass ; and, in cases in which the matter is not dis- tributed homogeneously, also find the total mass and the mean density ("line-density," "surface-density," or "density," as the case may be). (The density is unitorm, unless otherwise specified.) (1) A straight line of length I in which the line-density varies as (is k times), (a) the distance from one end ; (6) the square of this distance ; (c) the square root of this distance. (2) An arc of a circle, radius r, subtending an angle 2 a at the centre. (3) A fine uniform wire forming three sides of a square of side a. (4) A quadrantal arc of the four-cusped hypocycloid. (5) A plane quadrant of an ellipse, semi-axes a and b. 376 INFINITESIMAL CALCULUS. (6) The area bounded by a semicircle of radius r and its diameter, (a) when the surface density is uniform ; (&) when the surface density at any point varies as (is k times) its distance from the diameter. (7) The area bounded by the parabola \/x-{- Vy = Va and the axes. (8) The cardioid r = 2 a(l - cosd). (9) A circular sector having radius r and angle 2 a. (10) The segment bounded by the arc of the sector in Ex. (9) and its chord. (11) The crescent or lune bounded by two circles which touch each other internally, their diameters being d and ^d, respectively. (12) The curved surface of a right circular cone of height h^ (a) when the surface density at a point varies as its distance from a plane which passes through the vertex and is at right angles to the axis of the cone ; (b) when the surface density is uniform. (13) A thin hemispherical shell of radius a, in which the surface density varies as the distance from the plane of the rim. (14) A right circular cone of height h in which, (a) the density of each infinitely thin cross-section varies as its distance from the vertex; (&) the density is uniform. (15) Show that the mass-centre of a solid paraboloid generated by revolving a parabola about its axis, is on the axis of revolution at a point two-thirds the distance of the base from the vertex. (16) A solid hemisphere of radius r, (a) when the density is uniform ; (&) when the density varies as the distance from the centre. (17) Show that the mass-centre of the solid generated by the revolution of the cardioid in Ex. (8) about its axis, is on this axis at a distance | a from the cusp. (18) If the density /o at a distance r from the centre of the earth is given by the formula p = po 5ilLJ. ^ in which po and k are constants, show that the earth's mean density is „: 7 n ? td 7 r> •^ Q sm kE — kR cos kR ^ k^R' in which R denotes the earth's radius. (Lamb's Calculus.) [Answers : (1) | ? from that end, M= I kl^, ~p = \kl; (h)ll, M-^ kl\ p = ^kl'^ ; (c) ^ I, M = I kl'2, p =^kli. (2) On radius bisecting the arc at dis- tance r from centre. (3) At a distance 4 a from the centre of the a ^ ^ " . square. (4) Point distant | a from each axis. (5) Point distant — from 46 ^^ 4a axis 2 a, — from axis 2 h. (6) (a) On middle radius, at point distant — 3 TT 3 TT from the diameter ; (h) On middle radius, at point .580 a from the diameter, mean density = .4244 maximum density. (7) Point distant I a from each axis. (8) The point (tt, | a). (9) On middle radius of sector, at distance f r from the centre. (10) On the bisector of the chord, at distance APPENDIX. ' 377 ^^" ^^ from the centre. (11) On the diameter through the point a — sin a cos of contact and distant |^ d from that point. (12) (a) On the axis, at distance I h from the vertex ; (&) on axis, at distance | h from vertex. (13) On the radius perpendicular to the plane of the rim, at a distance | a from the centre. (14) («) On the axis, ^h from the vertex; the mean density is the same as the density at the cross-section distant f h from the vertex ; {h) on the axis, at a distance f h from the vertex. (16) (a) On a radius perpendicular to the base, at a distance .375 r from it; (6) on radius as in (a), at distance Ar from the base.] 2. Moment of inertia. Radius of gyration. These quantities are of immense importance in mechanics and its practical applications. Moment of inertia. Let there be a set of particles whose masses are, respectively, m^, m^, ?%, •••, and whose distances from a chosen fixed line are, respectively, )\, r^, rg, •••. The quantity mi/'i^ + ^»'2^'2" + »i3>*3' + •••) *-6. 5 ^rir^ (1) is -called the moment of inertia of the set of particles with respect to the fixed line, or axis, as it is often called. It is evident that for any chosen line and system of particles the moment of inertia has a definite value. In what follows, the moment of inertia will be denoted by J. It can be shown, by the same reasoning as in Art. (1), that definition (1) can be extended to the case of any continuous dis- tribution of matter (whether along a line or curve, or over a sur- face, or throughout a solid) and any chosen axis; thus. = \ r2 dm, in which r denotes the distance of any point from the axis, and dm an infinitesimal element of mass about that point. Radius of gyration. In the case of any distribution of matter and a fixed line, or axis, the number k, which is such that .. the n^oment of inertia .f>--^'» ^ the mass f^.,^ ' is called the (length of the) radius of gyration about that axis. 378 • INFINITESIMAL CALCULUS. EXAMPLES. 1. Find the radius of gyration about its line of symmetry of an isosceles triangle of base 2 a and altitude h. The density per unit of area will be denoted by p. Fig. 112. Let P be any point in the triangle, and make the construction shown in the figure. Denote NO by y. Then A;^ =: ^ 2 PN^ -p-dx dy over AOC 2p ■ dx dy over ABC ry=n p=z.v^2 dxdy p ah Now LN _ AO CN .^ LN CO' ' ' a h — V whence LN= -(h — y). h /. k^ i^ = la2. whence /fcz:.^. ah 6 y/Q In this example, the moment of inertia is ^ a%. 2. Show that the moment of inertia of a homogeneous thin circular plate about an axis through its centre and perpendicular to its plane is ^ pir a*, in which p denotes the surface density, and that its radius of gyration is | aV2. I On using polar coordinates, I = i r^ . (^^,^ — i r- • p • dA = p\ I r^-rdrdd. \ 3. Find the moment of inertia of a solid homogeneous sphere of radius a about a diameter, m being the mass per unit of volume. Suppose that the sphere is gener- ated by the revolution of the semicircle APB about the diameter AB. Let rectangular axes be chosen as in the figure. At any point P(x, y) on the semicircle take a thin rectangular strip PN at right angles to AB APPENDIX. 379 and having a width Aa;. This strip, on the revolution, generates a thin circu- lar plate. It follows from Ex. 2, since m is the mass per unit of volume, that / of this plate about AB = - tt • PN^ • ^x. .'. I of sphere = 2 -tt • PN^Ax from A to B 2 = 2 . ^ P (a2 - x'^ydx = j% mira^ 2 Jq Here, on denotin^the mass of the sphere by M, M = ^ mira^ ; hence, ^ = f ^«^ > accordingly, A:^ = | a2 . and thus, k = .632 a. 4. Find the moment of inertia and the square of the radius of gyration in each of the following cases : (Unless otherwise specified, the density in each case is uniform. The mass per unit of length, surface, or volume is denoted by nij and the total mass by M.) (1) A thin straight rod of length ?, about an axis perpendicular to its length : (a) through one end point, (&) through its middle point. (2) A fine circular wire of radius a, about a diameter. (3) A rectangle whose sides are 2 a, 2 6: (a) about the side 2 6, (&) about a line bisecting the rectangle and parallel to the side 2 b. (4) A circular disc of radius a : (a) about a diameter, (6) about an axis through a point on the circumference, perpendicular to the plane of the disc, (c) about a tangent. (5) An ellipse whose semi-axes are a and b : (a) about the major axis, (&) about the minor axis, (c) about the line through the centre at right angles to the plane of the ellipse. (6) A semicircular area of radius a, about the diameter, the density varying as the distance from the diameter. (7) A semicircular area of radius a, about an axis through its centre of mass, perpendicular to its plane. (8) A rectangular parallelopiped, sides 2 a, 2 6, 2 c, about an edge 2 c. (9) A right circular cone (height = h, radius of base =: 6), about its axis. (10) A thin spherical shell of radius a, about a diameter. (11) A sphere of radius a, about a tangent line. (12) A right circular cylinder (length = I, radius = R) : (a) about its axis, (6) about a diameter of one end. 380 INFINITESIMAL CALCULUS. (13) A circular arc of radius a and angle 2 a : (a) about the middle radius, (&) about an axis through the centre of mass, perpendicular to the plane of the arc, (c) about an axis through the middle point of the arc, perpendicular to the plane of the arc [Lamb's Calculus, Exs., ;j»i;XXIX.]. [Answers: (1) (a) lml\ ^l'; (b) i^ml^, ^^P. (2) \Ma'^, -i a^. (3) (a) r- = ^,ar-, (h) Tc^ = \a'^. (4) (a) k^ = \a^- (b) k^ = ^a^; (c) k^ = fa2. (o) (a) \Mb-^; (b) \Ma'-, (c) \ M{a:^ -V If-) . (6) | Jifa^, |a-. (7) k^ = [\- ^\ a\ (8) A:2 = K«' + &'-^) • (9) ^V ^'^'t ft^/i, ^\ b'^ ( 10) ^'^ = |a2. (11) A.-^^laS. (12) («) I=\MB^; (b) /= iV/(i i?^ + i Z2). (13) (a) ^•2 = i«-2(l-«"y-^^); (/,) /.■3 = «2(i_«mia^. (e) A:^ = 2a2fl-^'"-^ ' J QUESTIONS AND EXERCISES FOR PRACTICE AND REVIEW. 3jcos-1a/^^^ — -; (2) i>sin-i^^ 6x + I)sin-.^(''-^-»')(l-^')=0. a + &x 13. If x^z/S 4- cos X — sin X tan ?/ — sin ?/ = 0, show that dy _ ( — 2 x?/^ + sin x) cos^ y + cos x sin ?/ cos y dx 3 x^?/^ cos"-^ y — sin x — cos^ y 14. Differentiate: (1) ^^1^^ + log VT^^^' ; (2) tan-^ ^^'^ ~ ^"'^^'"^ ; ^Jl —x^ • a + h cos X (3) cos-i ^ + ^^Q«^ ; (4) sin-i^JlA^ilL^; (5) tan-i ^^' " ^' ^^" -^; a + 6 cos X « + 6 sin X 6 + a cos x (6) v'wsin--^x + ncos-^x; (7) (2a^ + x*)'^a^ + x* ; (8) l^iBJ^^; (cosmx)" (9) (cosx)«in^; (10) tan-i ^ ^ + ^'' + V 1 - x'^ ^ Vl + X=^ - Vl - QC^ [Answers to Ex. 14: (1) ^^""^^ ; (2) _v^MZ«i ; (3) ^«^ " ^% L ri-x2^^ 6 + acosx a + 6cosx (^ V^EH; (5) ^^^' , (&-,\(m-n^ giL ^^ a + & sin X a + 6 cos x 2 Vm sin2 x + n cos2 x x«. Wa + SVx . ,j.v WW (sin nx)"»-i cos (mx - nx) , .q. rr.n - e^x?-), - Va* - e'-x:\ ax a^ ^in^, ^; (2) y' , t, - ^y— , ^Vi^^^r^; (3) 10,7J,8,4^] X a?- y/yi _ ^2 a ^yi _ a-2 a 8. Show that all the points of the curve y^ = 4:a(x -\- a sin - ) at which V «/ the tangent is parallel to the axis of x lie on a certain parabola. 9. (1) In the curve r=a sin^-, show that <(> = ix//. (2) In the lemnis- o cate r'^ = a^ sin 2 d, show that i^ =2 6, (x + eh), where ^ is a proper fraction. Show that it is possible that in this relation d may have more values than one. 13. If A is the area between the graph of f(x), the x-axis, a fixed ordi- fj A nate, and the variable ordinate /(cc), show that ^^=/(x). dx CHAPTER VI. 1. Find the ?ith derivative of the product of two functions of x in terms of the derivatives of the separate functions. 2. Find the fourth derivative of x^ cos^ x and the ?ith derivatives of 1 t'^ (1) x^ cos ax ; (ii) x* cos* x ; (iii) tan-^ - ; (iv) sin^ x cos^ x ; (v) ; (vi) e«^ sin bx. ^ x^ - I 3. Show that (i) D>»f^U(-i)^?i(^±ll.i:i-(^ + ^^-^)^; (ii)i>»(x^-ilogx)= (^-^)' ; (iii) 2>/i/Lzl^\ = ?IziI1!L?LJ; (iv) 2)Xesinx) = _gsm*cosxsina;(sinx + 3). \1 + Xj (1 "T X)'^ 4. If X = a(l - cos 0, 2/ =^ a(nt + sin t), then ^ = - ^L^-S>^±±1. dx^ a sin^ t 6. Derive the following : (i) If gy + »;«/- e=0, D^ u = y • (^ - y)e^ -h^x ^ (ii) If a;* + ?/* + 4 a^a;?/ = 0, (y^ + a2a;)3^| = 2 a^xy{xh/ + 3 a*). (iii) If da;2 (Aa; + by + fY 6. Prove the following: (i) If ?/ = sin (m tan-' a;), (l+a;2)2^ + 2 x(l + a;2) ^ + wi2^ = 0. (ii) Ii y = {x + Vx2~^=T)«, (a;^ _ l) ^^ + a:-^ - da: ,, TO dx? dx d^y dx^ 7i^y=0. (iii) If:/3; (ii) x = -2; (iii) x = 1, x = 2; (iv) x = ± -^, x = 0, x = ± V|. I ■\/2 J 16. Find the inflexional tangent of the curve y = x — x^ + x^. \_Ans. 27 y = 18x + l.] 17. Show that : (i) The cone of maximum volume for a given slant side has its semi- vertical angle = tan-i ^72; (ii) The cone of maximum volume for a given total surface has its semi-vertical angle = sin~i \. 18. Show the march of each of the following functions : (i) sin^x cosx; (ii) sin2x — x; (iii) x(a + x)2(a — x)^. 19. Examine the following functions for maxima and minima : ^.^ x{x^-\~) ^..^ x^+lx+_ll ^^ 1-x + x^ l + x + x\ ^ ^ X* - X2 + 1 ' ^ ^ X2 + 4 X -f 10 ^ ^ 1 + X - X2 ^ M - X + X2 ' (v) X Vax - x^ ; (vi) (x - 1)* (x + 2)3 ; (vii) (1 -f x)2 - (x - x^) ; (viii) secx — x; (ix) sinx(l + cosx) ; (x) asinx+6cosx; (xi) x-" ; (xii) — — Ans. C'l) Two max., each = I : two min., each =— ^ ■ ^ Mog X L ^ ^ " " (ii) max. =2, min. =§ ; (iii) min. =f ; (iv) max. =3, min. =^; (v) min. ^ 3j^^2 . (vi) min. = 0, max. = 12* • 9^ - 7^ ; (vii) max. = 0, min. = 8 ; 16 /- ■»/r 1 (viii) sin x= ; (ix) max. = 1.299; (x) max. = Va^^ ;?2^ min. = — Va^ + b'^ ', (xi) min. for x = - ; (xii) min. = e. | CHAPTERS VIII., IX. 1. What is meant by partial differentiation ? 2. State precisely the restrictions as to the function /(x, y) so that the theorem "-^ = ^ -' may hold, and prove the theorem. dxdy dy dx 22 Show that if /(x, y) = xy ^ ~ ^ , the theorem does not hold for x=0, y=0, and explain why. ^ -\- y 3. Explain the meaning of a partial derivative. In what sense may we logically speak of the partial derivative of c with respect to a, when c is a function of a and 6, and a and b are both functions of x ? 4. Prove Euler's theorem for a homogeneous function 4> of x, y, z : QUESTIONS AND EXERCISES, 387 5. If w be a homogeneous function of the nth degree in any number of variables x, y, z, •••, then x^ + ?/^ + z— + ••• = nu. dx dy ds 6. Verify that A(d^l] = A(^\ in the case of each of the following functions : sin {x^j), cos ( f ^^ V 'og /^!±l!\ ^/^V 7. Verify the following : (i) If w == sin-i - + tan-i ^, x ^ + ?/ ^ = 0. y X dx dy (ii) If wzzz(4a6-c2)4 4^ = -^^. (iii) If z=xHsin-'^ ^ - y^ tan-i^', ^'^ _ ^ ' ac2 dadb ^ ^ X ^ y Sx dy -^. (iv) If y =J\y + ax)+{x) integral functions of x, and give some of the standard types for the integrals on which the value of / may be made to depend. Show how to integrate the fraction when the equation 0(x) = has repeated imaginary roots. 20. Show that if /(«, v) is a rational function of u and i?, / x, -%/ — '^^— \dx ax + b ^ ^cx + dj can be rationalised by means of the substitution — -i— = «". ex -\- d 21. What is meant by a formula of reduction for an integral ? Investigate formulas of reduction for the following : (i) i sin"' 6 dd in which m is an integer; (ii) | sin"' tf cos" ^ fW ; (iii) i ~ =dx ; (iv) I X" sin X dx. 22. Explain how it is that J cos2»+i rf^ = 0. dx p) Vrtx2'+ 2 6x + 23. Evaluate ( , by means of the substitution y{x — p) = \/ax?- 4- 2 &x + c. 24. Evaluate the following integrals, and verify the results by differentia- tion: f ^"'°"''^ , psin-iJIi^dx, T— ^V' f'^^^^'^^' ' •^ (1 + ^.2)1 ^0 ^a^x J^ sin e cos3 ^ J, ^^^^ ^ r d^ C x^ dx C dx C dx J a2 cos2 ^ + 62 sin 2 ^' J x^^ _ l' Jx(3 + 4x5)3' J a sin x + sin 2 x' f x*(« + x)*f?x, f ^^ + ^ — dx, Txs tan-i x dx, fe^^ sin2 x dx, -' J x2 — 4 X + 3 J ^ /• dx /* (x 4- l)dx J X V-x2 + 5x-6' J Vx2 + X + 1 x^ - ^ + ^ CHAPTERS XV., XVI. 1. Find an expression for the area bounded by a curve given in polar coordinates and two straight lines drawn from the pole. 2. Show how to find the length of the arc of a plane curve whose equa- tion is given (i) in rectangular Cartesian coordinates, (ii) in oblique Carte- sian coordinates, (iii) in polar coordinates. 390 INFINITESIMAL CALCULUS. 3. Investigate a formula for finding the superficial area of a surface of revolution about the axis of x. 4. Trace the curve f^ = a'^ cos 3 6, and find the area of one of its loops. 5. Show that in the logarithmic spiral, r = a^, the length of any arc is proportional to the difference between the vectors of its extremities. 6. Find the area of the curve r Va^ -\- b^ = (a^ + 62) cos 6 + a"^. 7. Find the surface of a spherical cup of height h, the radius of the sphere being E. 8. Find the average value of sin x sin (a — x) between the values and a of the variable x. 9. Find the volume bounded by the surface a/- + 'v/- + 'v/- = 1 and the coordinate planes. ^ 10. The axis of a cone is the diameter of a sphere through its vertex ; find, in terms of its vertical angle, the volume included between the sphere and the cone, and examine for vvhat angle it is greatest. 11. Determine the areas of each of the following figures : (i) The segment cut off from the parabola y'^ = 4 ax by the line 2x — Sy + 4ia = 0. (ii) The 2 2 curve /-V + ^^y = 1. (iii) The evolute of the ellipse («x)^ + (by)^ = (a2 _ b^)i. (iv) The figure bounded by the ellipse 16 x"^ + 25 y^ = 400, the lines x = 2, x = 4, and 2y -^ x = S. (v) The curVe (x^ + y^y = a^x^ + &V« (vi) The oval y = x^ -\- V(x — 1)(2 — x). (vii) The loops of the curve a^y^ = x2(a2 _ x^). (viii) The segment of the circle x'^ + y^ — 25 cut oif by the line x-{-y = 7. (ix) The area common to the ellipses b^x^ + a-y^ = a^b^, a^x'^ + 6V ^ ^2^2. Vjins. (i) I a". (ii) | -Kob. (iii) f tt ^^^ ~ ^'^^^ L ab (V) ^(«^ + &2) . ^^i>) ir_ ^^j.^ -gg^^j^ 2(j2. (viii) 2^5 sin-1 ^3 - |. (ix) 4 a6 tan-ill 12. Find the volume and the area of the surface generated by the revolu- tion of the cardioid r = a(l — cos 6) about the initial line. [Area = -^^ 7r«-.] 13. Show that the volume enclosed by two right circular cylinders of equal radius a whose axes intersect at right angles is ^^ a^, and the surface of one intercepted by the other is 8 a^. 14. Show that the volume included between the surfaces generated by the revolution of a hyperbola and its asymptotes about the -transverse axis and two planes cutting this axis at right angles is the same, no matter where the sections. are made, provided that the distance between the planes is kept constant. QUESTIONS AND EXERCISES. 391 15. The parabola y'^ =6x intersects the circle x"^ -\- y"^ = 16. Show that if the larger area intercepted between the curves revolves about the ic-axis, the volume generated is 60 w cubic units ; and show that if the smaller area intercepted revolves about the ?/-axis the volume generated is ^i^VS^^ cubic units. 16. An arc of a circle of radius a revolves about its chord. Show that if the length of the chord is 2 aa, volume of the solid = 2 7ra3(sin « — | sin^ « — a cos a), surface of the solid = 4 7ra"^(sin a — a cos a). 17. Find the area of the segment cut off from the semi-cubical parabola 27 ay^ = 4 (x — 2 ay by the line x = 6 a. Also find the volume and the area of the surface generated by the revolution of this segment about the a^-axis. ^Ans. 2^4 «2, ^^2 I Z^ _,. 3 log ( V2 + 1) I .] 18. A number n is divided at random into two parts. Show that the mean value of the sum of their squares is f n^. 19. Show that the mean of the squares on the diameters of an ellipse, that are drawn at points on the curve whose eccentric angles differ successively by equal amounts, is equal to one-half the sum of the squares on the major and minor axes. 20. Prove that the mean distance of the points of a spherical surface of radius a from a point P at a distance c from the centre is c H or a -f — , according as P is external or internal. CHAPTER XVII. 1. Define curvature of a curve. Find an expression for the radius of curvature of a curve whose equation is in the form y =f(x). 2. Show that the curvature at any point of the curve given by x = ^(O, y — \p{t) is ^^ — ~r^ ^ where accents denote differentiations with respect to t. (n{^) nniay be true. ^ S^^ 394 INFINITESIMAL CALCULUS, 2. Write' a note on the conditions under which (1) the integral, (2) the differential coefficient of an infinite series, may be obtained by integrating or differentiating the series term by term. 3. Prove that if /(x) be a continuous function of x, then f{x + h) = f{x) + hf'{x + eh), where 0<^< 1. Show clearly how this proposition may be applied to prove Taylor's theo- rem, and specify the circumstances in which the theorem as you state it is true. 4. Prove Taylor's theorem for the expansion of f(x + h) in ascending powers of h, carefully specifying the conditions which f(x) must satisfy. Find an expression for the remainder after n terms of the series have been written down. 5. State Maclaurin's theorem, and give the conditions under which it is applicable to the expansion of functions. Derive the theorem, 6. Expand in series of ascending powers of x the functions : (i) cos mx. (ii) tan-i(«4-x). (iii^sin(?7isin-i x). (iv) (1 + ?/)*, where y<,l. (v) e"«^ + e-"*\ (vi) e^^^+A, 4 terms. 7. Expand the following functions in powers of x : (i) e^i^ ^. (ii) tan-i x. (iii) cot-la;. IAiis. (i) 1 + x + | x^ - i x* - ^V x^ + •••. (ii) For values of x from x — — 1 to x = I, x — ^ x^ -\- I x^ — ^ x"^ + -" ; for | x | > 1, --- + ^: — + -'. (iii) For |xl/cj/2-y+ ^ hycos-J(2cy-l) + Ci. c 2cVc ^ (20) 2 X = log(!/2 + ci) + C2. (21) 15 Ci2?/ = 4(cix + a2)t + CoX + Cg.] 2. Find the singular solutions of : (1) x2p2_3 xyp+2 y^-{-x^=0. (2) xp2-2 yp + ax=0. (3) Solve equation (2). r Solutions : (1) x^iy^-ix^) = 0. (2) y^ = ax\ (3) 2 1/ = cx2 + ^-l MISCELLANEOUS. 1. How far does the symbol — obey the fundamental laws of algebra ? dx 2. Prove that if D denote — , and /(Z>) be any rational algebraic func- tion of 2), i\\enf{D)uv = uf{D)^ + Diif'(D)v + ^ f"(D)v + —. 3. If

f— ( e«^ • ?n1 = e''''(a -\ jw. 6. Show how to find the values of -^ and —^ when x and y are con- dx dxr nected by the equation /(x, y) = 0. 896 INFINITESIMAL CALCULUS. 7. If M =/(aj, y) and if x = (p(t), y = \p{t), state and prove the rule for obtaining the total derivative of u with respect to t. If x = r cos ^, y = r sin d, transform (x'-^ — w^) - — — -\- xy -r-^ — ^—5 into an expression in w^hich r and 6 are the independent variables. 8. Calculate the nth derivative of (sin~ia;)2. Show by the use of Mac- laurin's theorem that (sin-i x)2 = 2 — + - — + ^—^jL + 9. The curves u = 0, u' = intersect at (x, y) at an angle a. Show that d^t du[ _ du^ du doc dy dx dy tan a du du__ ^ drii^ d_u dx dy dx dy y2 ift. ffl r.fl Show that the curves —Ar , = 1 and ^— + ^ 1= 1 intersect at right angles 10. Show that the total surface of a cylinder inscribed in a right circular cone cannot have a maximum value if the semi-angle of the cone exceeds tan-i \, i.e. 26° 34'. 11. Through a diameter of the base of a right circular cone are drawn two planes cutting the cone in parabolas. Show that the volume included between 4 these planes and the vertex is — of the volume of the cone. 3 TT 12. Calculate the area common to the cardioid r — a{\ — cos ^) and the circle of radius f a whose centre is at the pole. 13. Find the area and the perimeter of the smaller quadrilateral bounded by the circles tP- ■\- y'^ ■=. 25, x^ -f ^2 _ ^44^ ^.^(j ^j^g parabolas, y^ _ g x, yi + 12 (X + 2) = 0. 14. Given the cardioid r = 4 (1 — cos 0) and the circle of radius 6 whose centre is at the cusp, find the length of the circular arc inside the cardioid and the lengths of the arcs of the cardioid which are respectively outside the circle and inside the circle. 15. If a curve be defined by the equations -— "=: — ^ = , find an ex- 0(0 V'CO /(O pression for the radius of curvature at a point whose parameter is t. 16. Expand (by any method) x^ cosec^ x in a series of powers of x as far as the term in x*. At what place of decimals may error come in by stopping at this term, when x is less than a right angle ? 17. Trace the curve x* -1- ?/* = d?-xy, and find the points at which the tan- gent is parallel to an axis of coordinates. Find the area of the loop. 18. Trace the curve x = a sin 2 ^ (1 + cos 2 0), 1/ = a cos 2 ^ (1 — cos 2 &). (a) Prove that Q is the angle which the tangent makes with the axis of x, and obtain the equation of the tangent to the curve. (6) Find the length of the radius of curvature in terms of Q. QtJt^STlONS AND EXERCISES. 897 19. Find ^ under each of the following conditions : {\) x^ = e dx (ii) y = e*"" tan-i x. (iii) e'' -{■ x = €« + y. (iv) y = • (v) sin (xy) -e^y-x'^y = 0. x + Vl-x^ 20. Four circles x^-{-y^ = 2 ax, x^ + y'^ = 2ay, x'^-\-y^ = 2 bx, x~ + y^ = 2 hy, form by their intersections in the first quadrant a quadrilateral ; prove that the area of this is (a- + 6') cot-i ^^ ^^ .^ - (a - 6)2. 21. Prove that the area of a sector of an ellipse of semi-axes a and h be- tween the major axis and a radius vector from the focus is — (0 — e sin 0), where is the eccentric angle of the point to which the radius vector is drawn. 22. Trace the curve xy^ = a^ ; and find whether the area between it, a given ordinate, and the coordinate ajces is finite. Show also that if the tangent at P meet the axis of x in T, then MT=SOM, where 3/ is the foot of the ordinate at P, and O is the origin. 23. If w be a homogeneous function of n dimensions in x and y, show that : (i):«-^f^+2xj,/l'-+y-^f^=»(,.-l)«. (ii)xfi^+,/|- = («-l)f. dx- dxdy dy- dx^ dxdy dx (iii) x^ + ypi^ = (n - 1) f . (iv) (x^ + yj-W = »%. dxdy dy^ dy V dx dy) 24. Prove the following : (i) If ?< = sin-^ {xyz), d^djidu^ ^an2 u sec u. dxdy dz (ii) If u = log (tan x + tan y + tan z), sin 2 x^ -\- sm2 y^ -\- sm2 z^ = 2. dx dy dz (iii) If u = log (x^ + y^ + z^-S xyz), ^ -H ^ -h ^ = —-1-—. (iv) If dx dy dz x-\-y + z u ^ tan2 X tan2 y tan2 z, du = 4 m f '^^ + ^■' + ^^ ^ • \ sin 2 a: sin2y sin2 zj 25. If b be the radius of the middle section of a cask, a the radius of either end, and h its length, show that the volume of the cask is -^-^ tt (3 a2 ^ 4 ab -t- 8 b'')h, assuming that the generating curve is an arc of a parabola. 26. 031 is the abscissa, MP the ordinate of a point P(xu y\) on the hyperbola — =1, (a^i, 2/i, hoth being positive). If A is the vertex nearest P, show that area AMP = \ Xiyi — ^ «&log ( ^ -h ^ ), and area sector OAP 27. Show that the mean of the squares on the diameters of an ellipse that are drawn at equal angular intervals is equal to the rectangle contained by the major and minor axes. 398 INFINITESIMAL CALCULUS. 28. Find the mean square of the distance of a point within a square from the centre of the square. 29. Through a diameter of one end of a right circular cylinder of altitude h and radius a two planes are passed touching the other end on opposite sides. Show that the volume included between the planes is (tt — ^)d^h. 30. Show that the integration of the expression /(x, y)dxdy may be per- formed in any order, provided the limits of x and y are indepen'dent of each other. 31. Evaluate 111 x"-y^zy dx dy dz taken throughout the space bounded by the coordinate planes and the plane x -\- y -\- z =\. 32. Prove geometrically or otherwise that x dy — ydx=r'dd, and show that the area of a closed curve is represented by ^ j (x dy — y dx) . 33. The equation to a curve being written in terms of the polar coordi- nates r and d, p being the perpendicular from the pole to the tangent and u=-, show that, -=u^-^ f — ^ '^ r p^ \dd , 34. If a is a first approximation to a root of the equation f(x) = 0, deter- mine graphically or otherwise the conditions under which a — ^y^ is a valid second approximation. ^ -^ ^^^ 35. If /(x) be a finite and continuous function of x between x = « and X = b, show that a value Xi of x, lying between a and b, may be found such that/'(xi) = {f(b) -f(a)} -^{b- a). If the function be x^'-l-cx, find the point in question when a = a and & = 2 a, and thence show that in this case Xi is such that ^ ~ ^ is constant for all values of a. o —x\ 36. Find the radius of curvature of the curves: (i) lima^on r='a cos d-\-b^ r = ^\ (ii)a2/2=(x where r = -\ (ii) ay"^ = (x— a)(x — by at (a, 0). Trace the curves. Ans. , (ii) ^^-^n 37. (1) Trace the curve r=a-t-& cos 0, a>6>0 ; find its area. (2) Find the area of the loop of y^ = (x — 1) (x — 3)2. (3) Find the area between the X-axis and one arch of the harmonic curve y=b sin -• Ans. \{2 a^-\-V^)Tr., ,- -, a i- 32 V2 15 ', 2a6.] 38. Trace the curve ^y"^ — (x -\- 7)(x + 4)2. Find the area and the length of the loop, ^nd the volume and area of the surface generated by the revolu- tion of the loop about the x-axis. [A7is. | V3, 4V3, f tt, Sir.'] QUESTIONS AND EXERCISES. 399 39. Find the limiting values of : (i) log 7^/^"/ , when (?=7r ; (ii) f l^i^\i, (ir'^ — ff^)d \ X J when a; = 00 ; (ill) — ^" "~ ^ , when x = I ; (iv) , when 1 — X 4- log X 2 x^ 2 ic tan r x 1^ x = 0; (V) /?i^V^ when X = J (vi) ^^:=-^ , when x = j (vii) ^—Z^"^ \ X J X x^ — a- when x = a. 40. Find the mass of an elliptic plate of serai-axes a and &, the density varying directly as the distance from the centre and also as the distances from the principal axes. 41. From a fixed point A on the circumference of a circle of radius a, the perpendicular ^F is let fall on the tangent at P. Prove that the greatest 3"\/3 area APY can have is a^. 42. A rectangular sheet of metal has four equal square portions removed at the corners, and the sides are then turned up so as to form an open rec- tangular box. Show that the box has a maximum volume when its depth is \(^a -\-b — y/d^ — ab + b'^), a and b being the sides of the original rectangle. 43. Two ships are sailing uniformly with velocities u, v, along straight lines inclined at an angle 6 : show that if a, b, be their distances at one time from the point of intersection of the courses, the least distance of the ships is equal to (av — bu) sin 6 (u^ + v''^ - 2 uv cos e)^ 44. A right circular conical vessel 12 inches deep and 6 inches in diameter at the top is filled with water : calculate the diameter of a spherical ball which, on being put into the vessel, will expel the most water. 45. A statue a feet high is on a pedestal whose top is b feet above the level of the observer's eyes. How far from the pedestal should the observer stand in order to get the best view of the statue ? [Ans. y/b{a -\- b) feet.] 46. The lower corner of a leaf, whose width is a, is folded over so as just to reach the inner edge of the page : find the width of the part folded over when (1) the length of the crease is a minimum, (2) when the area of the tri- angle folded over is a minimum. [^Ans. (1) fa; (2) fa,] 47. (1) Show that the cylinder of greatest volume for a given surface has its height equal to the diameter of the base, and its volume equal to .8165 of that of the sphere of equal surface. (2) Show that the cylinder of least surface for a given volume has its height equal to its diameter, and its surface equal to 1.1447 of that of the sphere of equal volume. 400 INFINITESIMAL CALCULUS. 48. Trace the graph of y ^ si" 2 x - sin a; ^ -p^^^ ^^^ 2iY\g\es, at which it cos X crosses the a;-axis, and show that its finite maximum distance from the a;-axis is (2i - l)t. 49. An ellipse, whose centre is at the origin and whose principal axes coin- cide with the axes of x and y, touches the straight line qx+py=pq ; find the semi-axes when the area of the ellipse is a maximum, and also the coordinates of its point of contact with the given line. 50. Find the volume of the greatest parcel of square cross-section which can be sent by parcel post, the Post-ofiBce regulations being that the length plus girth must not exceed 6 feet, while the length must not exceed 3 feet 6 inches. INTEGRALS. FOR EXERCISE AND REVIEW. The following list of integrals provides useful exercises in formal differentiation and integration. It will also afford some assistance in the solution of practical problems as a table of refer- ence. Those who have to make considerable use of the calculus will find it a great advantage to have at hand Peirce's Short Table of Integrals* (Ginn & Co.). GENERAL FORMULAS OF INTEGRATION. Formulas A, B, C, pages 173, 174 ; formula for integration by parts, page 177. FUNDAMENTAL ELEMENTARY INTEGRALS. Formulas L-XXVL, pages 172, 173, 180, 181. (These should be mem- orised.) REDUCTION FORMULAS FOR (x^^^^a + bx^ypdx. [Here X denotes (a + ftx").] 1. (x'^XP dx = ^"•""^^^^' - "CM-n + D C^m-nxP ax. J h{np -\- m ■\- \) b(np + in-\- 1)J 2. (x^XP dx = a^"^^^Xi>+^ _ Hm + n + np + l) C^m^nxP dx. J aCm + 1) . a(i»-l-l) •^ 3. ix^XPdx= «^*"^^^^ + ^^P (x^XP-^dx. J m-{- np +1 in-\- np + 1 •/ 4. ix'-XP dx = - oc«'*'XP*^ ^ m + n + np + 1 C^„.j^ph-i ^^. J an(p-\-l) an(p-{-l) J * There are two editions, the briefer edition of 32 pages and the revised edition of 134 pages. 401 402 INFINITESIMAL CALCULUS. J b7i(p + l) bn(p + l)J J 771 -\- 1 m + IJ 7 r ^a; 1 (m - n ■{■ np — l)b r dx J x"'Xp~ {m - l)ax"'-iXp-i (w - l)a J x"»-«X^ 8 T-^^^ — 1 , m — n + np — \ C dx _ J x'^Xp ~ an(p - 1)x'»-iXp-i an{p - 1) J tC^JTp-i' 9 (XPdx _ _ Xp+1 _ 6(m - ?i - wp - 1) f X^c^a; 10 fr^^ = Z^ I «J_^? C XP-^dx ^ J x^ {np - m + l)a:'»-i «j9 — wi + 1 J x"* -J r x"*c?x _ x"'-"+^ a(m - »i + 1) rx'^-^t^x^ J Xp 6(w - «p + \)Xp-^ h(jn - wp + 1) J Xp 12 r?^L^ x"*+i w 4- n — np + 1 r x"*(?x ' J Xp ~ an{p - \)Xp-^ an{p - 1) J Xp-^' 13. f ^ = ^- r - + C2n-3)f ^ ]. J (a + 6x2)" 2(71 - l)a L(a + &x2)«-i ^ J {a + hx^y-U Put a2 for a, 6 = 1, and compare with Ex. 3, Art. 118. 14 r ^^^^ — Zl^ 1 C dx J (a + &x2)«~2 6(n-l)(a + 6x2)«-i 2 6(w - 1) J (a + 6x2)"-i' 15 r___^___ = 1 r dx hC dx J x\a 4- 6x2)« a J x\a +' hx^y-^ aJ {a-\- hx^y * EXPRESSIONS CONTAINING Va + hx. Also see Ex. 10, page 191. C dx _ Va + bx b C dx J nri-^/n _L h^r ttX 2aJr 16. x2Va + 6x «^ 2a^xVa + 6x 17. f^^«+5idx = 2V^Tfei + af— ^^ J X J xy/a + bx INTEGRALS. 403 EXPRESSIONS CONTAINING Vx2 ± dK Also see Ex. 7, page 191. 18. f^g^ = log p + ^^±«"' V See XXIV., XXV., page 181. n 19. r(a:2 ± a2)2-^^ = ^(^±ji!}i ± Ji«L r(x2 i a4~'dx. J n + 1 71+lJ 20. f (a;2 ± a2)^dx = - VW^cfi ± — log (x + Vx^To^). •/ 2 2 21. f (x2 ± a2)|^a; = |(2 x2 ± 5 a^) y/x^ ± a^ + 3a_* iog(x + >/x2 ± a?-). Jo 8 22. f x2(x2 ± a2)^ dx = I (2 x2 ± a2) Vx2 ± a2 - ^ log (x + \/x2 ± a2) . •/ 8 8 23. f ^ dx = ± ^ (x2 ± a2)f a2Vx2 ± a2 2^ C x^dx ^ X ^^2 _j_ ^2 :p ^ log (^a; ^ Vx2 ^ a2). •^ (x2 ± a2)i 2 2 25. f ' a;2(?x ^_ x j^i^^^^ j^^^i _ ^^x •^(x2±a2)t Vx2-a2 26. r ^ = llog ? ; f ^ = lsec-i?. *^x(x2 + a2)i « a + Vx2 + a2 ^a:(x2-a2)i « « 27. r ^3; ^ -^ Vx2 d, a2 •^ x2(x2 ± a2)^ «^a; I a C ^x ^ Vx2 + a2 1 g + Va;2 + a2 ■^x3(x2 + a2)^ 2a2x2 2a3 6. r ^^__ = ^^^E«:% J_sec-i?. •^x3(x2_a2)^ 2a2x2 2a^ >. a. I ^ ~ = Vx2 ^ a^ - a log — ■ ■ J X ^ X (X2 - a2)i clX V x2 — a^ — a cos-i - • X 6. f(^^ «/ X J X'^ X 404 INFINITESIMAL CALCULUS. EXPRESSIONS CONTAINING y/a^ - x^. Also see Ex. 7, page 191. 31. f (g2 - x^ydx = ^^^' ~ f )' + -^ f(a2-a;2)2 'djc. J W + 1 W+lJ X2 34 35 J Va2 - x2 »^ m J Va2 _ x'-^ ^ m + 2 W 4- 2 J y'^2 _ jp2 <^»^ dx = — "^^^ ~ ^^ ' »'* - 2 J \/a2_x2 - x2 ^ m - 2 r 4 a,c. V62 - 4 ac 2 ex + 6 + V62 - 4 ac ■ J a + 6x - cx2 V62 + 4 ac V62 + 4 ac - 2 ex + 6 63. a. r <^^ — ^ J_ log (2 ex + 6 + 2 Vc Va + 6x + cx2). *' Va + 6x + cx2 Vc 6. f "-^ = 4 sin- ^'^^-^ . •^ Va + 6x — cx2 Vc V62 + 4 ac 64. a. f Va + 6x + cx^dx = ^ ^^ "^ ^ Va + 6x + cx2 J 4e - ^'^ ~ ] ^^ log (2 ex + 6 + 2 Vc Va + 6x + cx2). 8c^ 6. rVa + 6x-c.x2(^x = g^^:::^Va+6x-cx2 + ^^^±i^sin-i ^^^-^ • -^ 4 c g J V62+4 ac gg ^ I X(fx _ Va + 6x 4- ex2 i. a. J Va + 6x + cx'^ ^ ^ log (2 ex + 6 + 2 Vc Va + 6x + cx2). 2c^^ ^ r xc?x _ _ Va + 6x - cx2 _6_ ^.^.j 2 ex — 6 ^ Va + 6x - ex2 c ^ ^^ VPTToc N.B. Other algebraic integrals that are occasionally useful are given in Exs. 7-10, page 191, and in Exs. 4, 6, page 222. 66 INTEGRALS. 407 EXPONENTIAL AND TRIGONOMETRIC EXPRESSIONS. The most elementary of these are given in the integrals on pages 172, 180. a. fsinxcos^xt^x^-^^^^^. 6. fsin»a:cosa; = ^*^"^^^ J n+ 1 J n-\-l 67. a. fsin2xdx =-- ^sin2x b. f cos2xdx =|+ ^sin 2x. CO r ■ ■, sin«-ixcosx , n — I C • „ 9 j 68. \ sm'^xdx = h \ &n\^-^xdx. J n n J «rt C , cos^-i^sinx n — lC o -, 69. I cos^xdoj = 1 I cos"-2x(^a:. J n n J »t. C dx 1 cos X , n — 2 C dx 70. \ = — H I -: — -z- * J sm'^x n — 1 sin"-ix n — \J sm"-2/j; w-i C d^ _ 1 sin X n — 2 C dx J cos«x n — \ cos^-ix n — 1 J cos^-^^ 72. Csecnxdx = ^^'^ ^ '^""'"'^ + ^^-=^ fsecn-^xdx. (Cf. 71.) J W — 1 ?l — IJ 73. fcosecnxdx = -^^^-^^^^^^^^ + ^Lzi2rcosec»-2xc?x. (Cf. 70.) J n — 1 n — IJ 74. f tan« x dx = ^H1!L2^ _ ftan^-s x dx. 75. f cot** X dx = - ^2i!!li£ _ f cot«-2 x dx. J ?i — 1 ^ 76. f sm"» a; cos » x tta; = - ^i-""' '^^ «»s»+i a; V ill + /i + ^~^ f sin»»*- 2 a; cos** x dx. ni + nJ 77. rsin-«.cosn^^^..«m!!^il^J5^ J IW + 1 78. Jsln". X COS" »;b J a-\-bcoBx y/a^ — b^ \^a + b 2) y/b + a + Vb-a tan - — log , when a 6 J a + 6 sm x Va2 _ 52 Va^ _ 52 atan-+&-\/&2_a2 log , when a<6. v^^^^a^ atan5+6+V62_a2 2 85. f ^ = J_tan-if^i?^V J Or' cos2 ic + 52 sin2 X ab \ a ) 86. f e«^ sin nx dx = e"''(«siQ »^^ - ^ cos^^;; ^ ^g^^ ^^ ^g ^^.^ ^^g x ^ a^ + ^a ^ ' '' 87. fe-cosnxdx=2^1^?^i5^^^i^^^^i^. (See Ex. 6, Art. 106.) r \ \ 57? \ \ 2 / +1 / f 1 -1 V \ \ IT > \ ■z y ( ' % X I \ -TT y-'Snix y — coax TtFX y^sinz^x y -cos'^x 409 1 J y y TT ~2 0/ / 1 1 TT y=ta n a; /" ¥ X 27r 37r __ ^ ^ TT t-^ ^^-^^^ Jt '^^^ t/=tan _87r I- 'a?' 410 "X i/ = sec'a; 411 The Parabola x^-v y^ =a'' The Cubical Parabola o?y=x'^ The Astroid or Four-Cusped Hypocycloid, a; ^ + 1/ ^ = a ' Asymptote The Cissoid of Diodes ^3 The Witch of Agnesi l/ = x2-|-4o* 412 The Folium of Descartes O X The Catenary Asymptote ^O X The Exponential Curve The Cycloid a;=a (^-sln^),i/=a (1-cos ^) O a X 0\ X The Logarithmic Curve l/=log X Parabola r=asec- ^ The Cardioid r=a{l-cos6) 413 The Lemniscate, rLa^ cos 2 0, The Curve, 7- a sin 2^ The Parabolic Spiral Asymptote The Spiral of Archimedes, r=a(j The Hyperbolic or Reciprocal Spiral, r =^ Cb X The Lituus or Trumpet, Tlie Log-arithmic or Equiangular r' 6 -^a' Spiral, r-e**^ or log r-a >«<< CHAPTER I. Art. 4. 1. 45°, 0°, 63^ 26' 4", 71° 33' 54", 75° 57' 49", 78° 41' 24" 80° 32' 16", 82° 52' 30", 104° 2' 11", 99° 27' 44", 135°, 126° 52'.2, 110°33'.3 2. (.18, .033), (.29, .83), (.5, .25), (.87, .75), (5.72, 32.66), (-1.07, 1.15) (- .35, .12), (- .18, .033), (- .09, .008). 3. [The latter part.] (a) -- y \by \by y a^y fi) 9J^. 4. a. CO, ± .5774, ± .2582, 0, ± .4045, ± 1.8074 ; 90°, 30° and 150°, 14°28'.7 and 165°31'.3, 0°, 22° 1'.4 and 157°58.'6, 61°2'.7 and 118°57'.3. b. 27, 12, 3, 0, 6.75, 18.75; 87° 52'.7, 85° 14'. 2, 71°33'.9, 0°, 81°34'.4. 86°56'.8. c, 00, ± 1.4142, ± 1, ± .8165, ± .5774, ± .5,- 90°, 54°44'.l and 125° 15'.9, 45° and 135°, 39° 14' and 140° 46', 30° and 150°, 26° 34' and 153° 26'. d. 0, i.l937, ±.4330, oo, ±.0945, ±.3034; 0°, 10°57'.7 and 169°2'.3, 23°24'.8 and 156°35'.2, 90°, 5° 24' and 174° 36', 16° 52'.7 and 163°7'.3. e. 00, ±.8661, ±.8183, ±1.25, ±.9139; 90°, 40° 53'.8 and 139°6'.2, 39° 17. '6 and 140° 42. '4, 51°20'.4 and 128°39'.6, 42°25'.4 and 137°34'.6. 5. Where x = ± 2.57 ; where x = ± 2.78. CHAPTER IT. Art. 12. 1. 35.2426 or 26.7574, 29.9586 or 28.0614, SVsinx + -^ 1J. r^ sinx + 7sin2x + 2. 2. 68, 28, 14, 3 sin2x - 5 sinx + 21. 3. ^^ ~ '^ ^ . 4. 18 + 2-49X 8\/x + X, 4 + \/x2 + 2. 5. ay^ + bxy + cx2, (a + 6+ c)x% (a + 6 + c)2/2. CHAPTER III. Art. 22. 4. (a) 2x, 2 x, 2x; (6) 3x2, 33.2, 3x2. 5. 4^.3^ 2 x + 4, - -, - 4 - 3 + 4 X. 6. 6 ^ 12 ^2 _ 8 - ^. 7. 6 2/6, § « _ 8 + ^. x2 x2 ' ^2 ^ ' 2 ^ y2 Art. 26. 2. 2 wr times, r being the measure of the radius ; 1.51 sq. in. per second ; 2.83 sq. in. per second. 3. .866 a times, a being the measure of the side ; 25.98 and 51.96 sq. in, per second. 4. 4 irr'^ times, r being the measure of the radius ; 9.425 and 37.7 cu. in. per second. 5. 5l\ mi. per hour. 415 416 INFINITESIMAL CALCULUS. Art. 27, 3. Sx'^dx, dx, 2dx, Sdx, adx, 2xdx, lixdx, etc. 4. 1.6; 1.681. 6. 42.2 ; 43.696. Ex. 5.03 and 9.425 sq. in. Ex. 1.3 and 2.6 sq. in. CHAPTER IV. Art. 31. 6 x2 + 14 X - 10, 2 X - 17, - 2 x + 21. Art. 32. 4. (5 X* - 8 x3 + 21 x2 + 2 X - 2) dx, .... Art 33 1 3 X* - 14 x3 + 6 x^ 16x-21x-^- x^ - 2 x^ + 44 x - 96 (8x2-7x + 2)2' (x3 + 8)2 ' (2x-^-9x + 3)-'i ' (3 X* - 14 x3 + 6 x'-^) dx ^ 2 ^ _ J^ -_8 (3 x-^ - 7 X + 2)-2 ' "" ■ '^' 640' 245' Art. 35. 2. ii-Ciii:^. 3. ^ 4 M- 17 3 X + 7 Art. 37. 1. 2 w^\ 12 «3^— , 63 i<8— , 8 x^ 12 x^, 84 x", 27 x2 - 34 x + 10. dx dx dx 3. 240 x(5 x2 - 10)23, 120 x3(3 x* + 2)9, (432 x^ + 300 x^ - 168 x^ + 448 x - 50) (4 x2 + 5)7(3 x* - 2 X + 7)4. 4.-2 21-^ u\ - 7 ir^ u\ - 11 m-^'^ w', _ 7 x-s, -15X-6, -170X-11, -8x(x2-3)-5, -60x3(3x4+ 7) -6, 15 x*- 21x2 + - - ^ + -• 5. A u^ Du, - f zr* Du, I J Du, h x'K ^- x^, f x'", ^^ , x2 x3 x* o . . . V3x2-5 ^^ + '^ (2x2 + 7x-3)"3, 1 , _9(3x-7)~^, 6x-|x"^-x~^- 3 V2X + 7 2 x~i + /s x"i 6. V2 2/2-1 ^^'^ V3 x^3-i^ 5 V7 x^7-i, 2 V5(2 x + 5)^^-1^ \/3(6 X + 7) (3 x2 + 7 X — 4)^3-1, 7. — + c, and give c any three particular 4 constant values. 8. (In each of these expressions k is to be given any three /y6 1 2^ 2^ 62 particular constant values. ) — \- k, h k, ~-x^ -\- k, -x'^ + k, - x^ + - 6 X 3 5 5 X -2Vx + A:. 12. 6x2 + 34x-61, max"»-i- n&x-'*-! ^^ ~"^^ (1 - x2)2' (a + x)2 _12 + 5^-f_35^4, i « , _1^^, ^' 3 ' X2V1 + X2 (a_&x2)2 (l_x2)^ , mnx«-i(l + x«)"»-i, 12 6x2(a + 6x3)3, x^-\l - x)"-i • (I -X)\/1-X2 [m-(m + .)x], ^-'^^ . 14. a. «|^^^ 4f^f±^^' 2V^^::^ 2/2 -ax a(3 2/2-2x2) 9x2y- 8x- 14xy2-2y3 -(x + a)y2 ^ _ x^ _6^. 14x2?/ + 6x?/2_3x3- 16?/ (a + ?/)(&2_aj/_2«/2) + ?/(x + a)2' y a2y &0 - f , I, I, - f . 17. ?/ = x2 + ^', in which k is an arbitrary constant ; ?/ = x2 + 1. 18. 5 ft. per second. 19. 10 ml. an hour ; 8f ft. per second. 20. (4,8). 21. 3hr. ;60mi. 22. f ft. per second. 23. 36°52'.2. 24. 36°52'.2. ANSWERS, 417 Art 39 1 (6a; + 4)logae 6x + 4 .4.34(6x + 4) 11 • • 3x2 + 4a;-7' Sx^ + ^x-f 3x^-\-^x-f 16 log, a' ii, .29868. 2. i .144765. 3. -=^^, -1-, L_^, -— i=, 1 _ a;2 1 _ a:^ (1 - a:) Vx v^M^ — —, 1 + log X. 4. log (x2 + 3 X + 5) + c, log c (x3 - 7 X - 1), log Vkx, xlogx in which c and ^ are arbitrary constants. (Ex. Write each of these anti- derivatives with the arbitrary constant involved in other ways.) 6 (a) - (21^7 + 1877 x + 228 x^) Vx + 2 .^. 6(x^ - 2) 30(4x-7)^(3x + 5)t ' (x+l)Hx + 2)-^' .. ^91 x2 + 475 X + 450 Art. 41. 15 (2 X + 5)^ (7 X - 5)^ (x + 3)^ Art. 40. 1. 2xe^\ 2.303 (10^, 2.303(6 x • 10*'='), -^e"^^- 2. 2 e^ 2\/x 2.303 (2 t . 10*'), 2 ^e''+3, 4.606 (102«+7), 3. e' x"-i (x + wi). ««'" • a;"-i log a, "t~-V ' ('-^>*-^' (^^' ^"(^-^)- *• i'"-^'' ^^" + ''' I gSx+i 4- c, c being an arbitrary constant. L. 2. (3x+7)««r2xlog(3x+7)+^|^1, (3x+7)2xriog(3x+7)2 + 3!^], as last, Vi(l^^^|x-''.x-l(7^1ogx+l),6e'.e^ -^(^Vlogx, - - log a. X Art. 42. 1. — sin 2 w = cos 2 ?< • — (2 «) = 2 cos 2 « • — , 3 cos 3 u - Du, dx dx dx ^ cos I M . m', I cos f w — , Y COS J^ w . Du. 2. D sin 2 X = COS 2 X • i> (2 x) = 2 cos 2 X, 3 COS 3 X, | cos ^ x, 6 x cos 3 x^, 3 sin 6 x, 20 x* cos 4 x^, 20sin4 4xcos4x. 3. 5cos5^ «cosi«2. 4 2 cos2 xsin 3 x-3 sin 2xcos3x ^ sin2 3x sin2x + 2xcos2x, 2 xsin (x + -W x2cos [x + -V 5. 45° and 135°. 6. Where x = rnr ± • 9553, in which n is any integer. 7. 63° 26' and 116° 34'. 8. Where x = nir - -, in which n is any integer ; 54° 44'. 1 and 125° 4 15'. 9 ; where x = wtt + j, n being any integer. 9. n cos nx, wx"-i cos x**, n sin«-i X cos x, 2 x cos (1 + x^) , w cos (nx + a) , w6x"-i cos (a + 6x") , 12 sin2 4 X cos 4 x, ^^osx-sinx ^ cos(logx) ^^^ ^, ^^^ j smef . X2 X X 10. (a) sin x + c, i sin 3 x + c, ^ sin (2 x + 5) + c, ^ sin (x^ — 1) + c, in which c is an arbitrary constant. (6) | sin 2 x + c, | sin (3 x — 7) + c, \ sin x^ + c, in which c is any constant. 418 INFINITESIMAL CALCULUS, Art. 43. 3. Where x = wir, n being an integer ; where a; = (4 w — 1) ^ ± . 485, 2 WTT - • 485. 5 - - cot 6>. 6. cot f; 60°. 7. -2sin(2x + 5), — 15 cos^ 5 X sin 5 a:, 2 aj cos a; — x^ sin x, -^^p-; , — {m cos wa; sin mx 2 sin a; (1 + cosa;)^ + wcoswxsinwx), e'^o«*(l — a;sinx), e^'^Cacoswix — msin wix). 8. — cosx + c, — 2 cos I X -h c, — i cos (3 X — 2) + c, — ^ cos (x^ + 4) + c ; c being an arbi- trary constant. Art. 44. 3. 2 sec^ 2 u - Du, 3 sec^ 3 u • 2)i<, m sec^ wi?< . ?«', 2 w?( sec^ wm^ . u', 2sec2 2x, Jsec^ix, wisec^wx, 6xsec2 3x2, 12x2sec2 4x3, wmx'*-^ sec^ wix", 6 tan 3 x sec^ 3 x, 12 tan^ 4 x sec^ 4 x, wm tan«-i 9nx sec^ mx, | tan (| x + 3) sec"-^ (|x + 3), orcosecx. 4. tanx + c, |tan2x + e, |tan(3x + a) + c. sin X 6. When x is an odd multiple of - and dx is finite. Art. 48. 1. -2csc2(2x+3), isec(^x+3) tan (ix+3), -3csc(3x-7) cot (3 X- 7), 5sin(5x + 2), wsec^xtanx. 2. -6cot (3« + 1) csc2 (3« + 1), secH|«- l)tan(^«- 1), - |csc2 2(i + 5) cot f (? + 5), -18 «csc2 9f-^, 14(7 « - 2) sec (7 t - 2)2 tan (7 - 2)2. Art. 49. 2. ^^"-' 1 2 2 Vl - x2« Vl - 2 X - x2 1 + a^^ (1 - x2) Vl - 5 x2 ^Vl + CSC X. 4. sin-i x + a, sin-i x2 + «, Vl - X2 VI - X2 \ sin-1 x^ + a, in which « is an arbitrary constant, 2 wx"~i 2 a Art. 50. 3. Art. 51. 1. X2«+1 ' 1+X2' V2«X-X2 2 2 dy 2x _3j^ ^ g 4 1+4x2' l + 4y^dx' l + x^' 1 + y« dx* '1 + 16^2' 4 «3 6 X dx K 2 1 - x2 1 1 — • 0. 1 + «8' 1 + 9 X* d« ' 1 + X2' 1 + 3x2 + X*' Vn^' ^(1 + ^^) ' ^ „^^ . 7. tan-ix + c, tan-ix2 + c, itan-ix^ + c. 2(a + 2x)Vx(a + x) a^ + a;2 Art. 52. 2.^-^. Art. 53. 2. ^ "^ ^ "^ Art. 55. 1. x4-a* Xy/x^-l \/l-x2 Va2i:x2 a:Hl 2 l + x2 Art._56. 2. (3 x2?/2+3) dy-\- (2 xy8-|-2)dx, 3(y2_fl;a;) ^y^_3(a;2_^y) ^x, etc. 3 _J^ --^^ _(^Y(-Y~^ y tan X 4- log sin y ^ dx dy ^a;' ^x' \a) \yl ' log cosx - xcoty' ' 2Vx 3Vy' i(^ + ^V ^(?!!:i^^ + r^y ,(ytanx + logsint,)dx-(logcosx •^ ^ V X Vy ' \ a"» 0"* / — X cot ?/) dy. ANSWERS, 419 Page 82. 1. (i) 24 x^ + 15x^ + 124 x + 55, (ii) a-\-b-[-2x, (iii) (a + x)'«-i (6 + a;)-' [m(6 + x) + «(« + x)], (iv) ('"^ - "^ + "'6 - »«) (x + a)-i ^ (X +0)"+' (V) (rn + mx - nx)xr^-^ . a^ . 1 (yiii) ^(^-^) -, (ix) -l(l+ ^ V (X) _i^L_, 2v^A/a + a;(Va + Vx)2 a;3V Vl - x*/ xVl - ^2 aHa%^-4r^ ^^ 28x3 + 6x-17 j -2a2^ "« , Va2i:^2 ^^ 7a:*+3x2-17a; + 2^ a*-x*^ xVa^^^ (iv)secx, (v) 3. (i)20x*cos4x^ (ii) -7sinl4x, (iii) 6sec2xtanx, vT+x2 (iv) 8sec2(8x + 5), (v) x'»-i(l + wlogaj), (^vi) pqx^'^sinp-^x^cosx^, /■ ••\ / • N,. 1 • / . 1N / "-N /- • \ /• ^ cos (log nx) (vii) 7i(sinx)"-^sm(n + l)x, (viii) cos (sin x) • cos x, (ix) ^^— ^^ — ^, (x) 7icotnx. 4. (i) — i— , (ii) ^ , (iii) — — 6. (i) , ^ ^ ^ X* + 1 ^ ^ tan2x _ 1 ' ^ ^ 1 _ x4 ^ ^ e^+e-^ (ii) _i, (iii) -^ , (iv) ?i , (V) -^^'-^' , y/i _ a;2 cos2 X + ;i sin2 x a -\- b cos x 2 I (vi) e"* sin"*-! rx(<:< sin rx + ?«r cos ?'x), (vii) — ^ — — log a • a^, (iii)x«v!-t^^^^, (iv) e*V(l + logx), (v)x(-').x-{i + logx+(logx)2|, Cvi^ x=«'+ia + 21oex^ 8 m ^"^ + ^y + ^ m a; 2(x2 + y2) _ ^2 (VI) X (l + 21ogx). 8.(1) /^^ + ^,y^^, 00 y-2(x2 4-l/2) + a2' ..... 2xy* ,• N If , N 1 / N cos X (cos ?/ + sin ?/) (ill) — , ., a ■ ? (iv) -{msec (xw) — w}, (v) — -. ^ — -. r^^, ^ ^ 4: x-y^ ■\- cos y ^ ^ x^ \ if/ i^^^ \ y sin x (cos y — sin y) — 1 ^"^'^ eJ' + x' ^ ^ x2 - x?/ log x' ^ ^ x(l + »iy) ^- (1 + log x)2 10. (i)2y-|, (ii)8«-ll, (iii) sec x, (iv) - cot 2;, (v) ■ 11. (i) (12 x3 + 18 X + 5) (6 x2 + 3), (ii) (e**" « + 2 tan t) sec2 1, (iii) g, (iv) f^P^- 12- (i) 90% (ii) 73°41'.2, (iii) 90°, (iv) 2°21'.7, (v) 70''31'.7. 14. Speed of Q in inches per second is 116.82, 225, 7, 319.18, 390.9, 436, 451.39, 390.9, 225.7, respectively. CHAPTER V. Art. 59. 1. The lengths of the subnormal, subtangent, tangent, and normal, are respectively : (1) 3, 5^, 6|, 5 ; (2) 4, 4, 5.66, 5.66 ; (3) - ^, - ^ln^, _L V'(a2-Xi2Xa*-c2xi2), ^^^^~^^^i^ e being the eccentricity ; Xi axi a2 (4) sin xi cos xi, tan xi, tan Xi Vl 4- cos2 xi, sin xi Vl + cos2 xi ; (5) yi^, 1, 420 INFINITESIMAL CALCULUS. vT+y?, yi VI + yi^. 2. Where x is infinitely great. 3. Infinitely great. ----- -i -h i 6. xxi ^ + yyi ^ = a^. 7. xxi ^ + yy\ ^ = a^ 8. a sin 6, 2 a sin^ - tan -, 2 « sin -, 2 a sin - tan -• 12. 90°, 0°, cot-i 4^, i.e. 32° 12'. 5. 2' 2 2 Art. 61. 1. (1) a, a^^ a Vl + ^'^ rVIT^; (2) — , 2 rd, -^4:6 + 0-\ 2r 2 aV^OT+T^; (3) -^, -a, -^/a^~+l^, _VaM^; (4) ««^'-i, ^^^, a a n a^"-i Vw2 + d^, - VwM^. 3. ar, -, r V'l + a^ - VHTo"^. 4. (a) lA = n a a 34° 65'. 2, 0=:74°55'.2; ^p = 50°4V.9, 120°41'.9; (6) \i/ = 26°33'.9, 4> = 55° 12'.8. Art. 62. 1. In feet per second: 0, 4; 2.828, 2.828; 3.57, 1.79; 3.77, 1.33. Solution for x = 2: Where x = 2, the tangent to the parabola has a slope 1. Accordingly, the moving point is there going in a direction which is at angle 45° to the ic-axis. Hence, the speed of the x-coordinate (i,e. ^^ = ^ X cos 45° = 4 X — ; also ^ = 4 x — .1 2. 20 and 22.36 ft. V dt) dt V2 <^i V2 J per sec. Suggestion : Differentiation with respect to the time gives 2 y^ = 4— .1 3. .399 and - 9.97 ft. per sec. ; 9.7 and - 2.425 ft. per sec. 4. 442.82 and 161.6 ft. per sec. ; 199.15 and 427.08 ft. per sec. 6. (1) (2, 8), (- 2, - 8) ; (2) a, ^h), (- h - 3rh) ; (3) 300. Art. 64. 2. ^, h. 3. The tangent at the middle point of a parabolic arc is parallel to the chord of the arc. 4. — S ± V^ ; find the abscissa of the point where the tangent is parallel to the chord joining the points whose abscissas are 3 and 4. Art. 65. 3. 25.1 cu. in. ; ^i^. 4. 4 Trr^ • Ar; 50.3 sq. in., 502.7 cu. in.; ^h^ ^h^ ^- 5. 1.35 sq. in. ; 7:5 approximately. Art. 66. 2. (1) _ 1, - 1, I ; (2) - f, - I, _ 1 ; (3) 2, 2, 3, 4 ; (4) - ^, - I, 3, - I ; (5) 2, 2, - 3, - 3, 1. 3. n-r-^ = Ap»(n - 2)«-2. Art. 67. 4. 1.6, .4. 6. Jr2, i.e. 2 6'^- .0048, .035. 7. .0349, 0, .0025. 9. J «±^, J-«+^ ; ^. 10. 2.41, .1. 11. a Vrn"2, 1 y/WT^, ' X ' a ^x a 12. .078. 14. TTX^ Tzx^. 15. 5.03, 10.05. 18. 10.37, 5.06. 19. J^^^^=-^, ^ a^ — x^ ^Va2-x2, ^^'^(^^-^'^ ^^Va2_e2x2, e being the eccentricity. 20. ^, — —2 /I « a a a, r cosec a, V2ar. ANSWERS. 421 CHAPTER VI. Art. 68. 1. (i) -—^^ ; (ii) 8 + ^ + -1= ; (iii) ^°^ ^ (l + x--2)2' ^' 0!^ 4Vx^' ^ (l-sinx)2' («2 + ^2)2' sin^X (1_ 3,2)1 24(l-10x2 + 5x^). 4 ^. _4! _8e^sinx. 6. (i) -^; ^ ^ (1 + a;2)5 ^ ^ x2 ' ^ ^ ^ ^ aV ^^ - n T9.V0 ^- ^^ -^•^' -^•^^> <^"^ ^' -^20. 9. ^; -i- i^+^yy 4asin4^ , 2 12. 24 ic. 13. ^ = I x2 + 2 X + Ci, y = I x3 + x^ + cix + C2, in which Ci and dx C2 are arbitrary constants. 14. Sy = x^ — 9 x + 19. 15. y = 4 x^ 4- x. 16. (2) '— f ft. per sec' per sec. 17. In ' in. per sec' per sec : (i) 1152 tt^, (ii) 768 7r2, (iii) 384 7r2, (iv) 0. 18. s = igt^ + Cit + C2. 19. 15.5 sec, 3881.9 ft. 20. ^Vi)Sec Art. 69. 2. e^ a^(loga)»», a''e«^ ft^a*^ (loga)«. 4. cos^a; + — V « • / ■ W7r\ „ / , 7i7r\ - (-l)»-i(>i-l)! (-1)«-I2.(7i-1)! «» sin ax 4- — , a»cos axH )• 5. ^^ ^ ^^ —■, ^ ^ ^^ ^• V 2 y V 2 / x« (x-2)» g (- l)»;t! ^ (- \Yn\ ^ 2.n! (- l)^ac''(w + n-\)\ x«+i ' (1 + a:)"^i' (3 - x)«+i' (m - 1) ! (6 + cx)'«+« ' l(l + x)'»+i (l-x)«+i/ l(l-x)«+i (l+x)~+i/ Art 71 2 ^_±A22^, _ ^ + « cos ^ 6 sin ?)2 sjn^ ^ Art. 72. 2. (x4-120x2 + 120)xsinx-20(a;2-12)x2cosx. 3. (x + n)e*, 2«-i(Ai + 2x)e2*. Art. 73. 3. (1) y' = xy" ; (2) x^y" + 2y = 2xy'; (3) y' + 2 xy" = (4) (ix^-2y^)y^' -ixyy' -x'^ = 0; (5) yy' = x(yy" + y^-'). 4. (1) y' = (2) y=x?/'; (S)y" = 0; (4) y" = y ; (5) y" = m'^y ; (i6)y" + m^y = (7) y" + wi22/ = 0. 5. ^2(1 + 2/2') = r2 ; x2(l + y2') = r-f'' ', {1 + J/^'f = ry". CHAPTER VII. Art. 76. 4. A minimum ; neither a maximum nor a minimum. 8. See Ex. 3. 12. See Ex. 3. 13. (l)_Min. f or x = ^ ; max. for x = - 2. (2) Min. at -'^-^^ ; max. at i:l±y^. (3) Max. for x=0 ; min. for x = ll^^; 6 6 ^ ^ 12 min. for x = -^ — '— ; for x = 2, neither a max. nor a min. (4) Max. for 12 ' ^ ^ X = — 1 ; min. for x = I ; neither a max. nor a min. for x = 2. (5) Min. for X = 4. (6) Max. when x = — 4, and when x = 3 ; min. when x = — 3, and when X = 4. (7) Min. for x = 16 ; max. for x = 4 ; neither for x = 10. 422 INFINITESIMAL CALCULUS. (8) Max. for a; = — 10 ; min. for x=—2; neither for x = 2. (9) Min. value is — , i.e. —.3678. (10) Max. when x = e. (11) Max. value = 8 ; e min. value = 2. (12) Max. or min. when sin a; = Vf according as the angle X is in the first or the second quadrant. (13) Max. when ic = cotx. 16. (av'2, a ^4). Art. 77. 7. Each factor = Vthe number. 8. -• 9. A square. 10. (i) (a^ + b^)^; (ii) a + 2Vab + b; (iii) 2ab. 11. Let the perpen- diculars drawn from A and B to MN meet 3IN in B and ^S* respectively ; then (1) BG = CS; (2) BC = ^^^^^ 12. (i) f r; (ii) f r. 13. 19°28'. 14. (i) Vol. = .5773 vol. of sphere; (ii) height =rV2. 15. (i) Vol =^\Tr a^b ; (ii) height = ^ &. 16. 1. 17. 2s le. 114°35' 29".6. 18. fa. 19. 1:2. 22. 1| times the rate of the current. 23. — d, ^ d. 24. (^J + 6^)i (1) (0, 0) ; (2) (3, - 3) ; (3) (f , W) I (4) (2, f ) ; where x = 0, and where x =± V3 ; (7) where x = 0, 25. « . V2 Art. 78. 1 3^ 2, ); (6) and where x = ± 2 \/3. 2. (1) Where x = — ; (2) where x = — ; (3) where 5 4 x = ±-^; (4) (c, &); (5) (cm); (6) (^^, ^)- CHAPTER VIIL Art. 79. 2. 3 x2 + e' sin y, 4 ?/ + e* cos y — cos 2 sin ?/, 6 2 — sin z cos y. 3. (a) -:^ and -lli|, ; (6) -Zli^, and -^^4 . ^^^ ^20 ^^^ -4|^ 4V1I9 5VII9 3V'89 5\/89 3V47 4>/37 respectively. Art. 81. 3. Increasing 3^:: units per second. 4. Decreasing ~ units per second. 20 V 119 5V89 Art. 82. 3. .036; .036011. 4. (i) ^<^y-y^^^ . (ii) y-\ogy .dx-hxy^-Ulf/; y ^ +y /logy logx (iii) yxv-^ dx + xv log x-dy; (iv) - dx-\- logx- dy, (v) it I — ^ <^^ + — rp <^2/ 5. .025. 6. 2.2; 2.37. 7. .0017. 8. xy''-'^(yzdx+zx\ogx - dy+xy\ogx - dz). Art. 83. 3. 4.72 sq. in. CHAPTER IX. Art. 90. /sf^V-^^Uf^'' I [dyV dydy^i \dy, — 4asin -. 2 a. 4. - (a2 sin2 e -\- b"^ cos^ $) 2 ^ a6. ANSWEBS. 423 Page 147. 1. (i) ^i^-2y^ = 0. (ii) ^ + ^ = 0. 2. ||-2f^V = cos2 ^. 3. (i) ^ + M = 0. (ii) ^ + ?/ = 0. (iii) ^ = 0. (iv) ^ + a2y = 0. (V) g+y = 0. (vi) |^+2g+2/ = 0. 4. (i)tan«; (ii) — 3 sin* fcosi ; 3 sin^ i(4 — 5 sin^O. atcos^t CHAPTER X. Art. 97. 3. y = x^ ; y = x^ - ?A7 ; y = x^ + 514 ; y - k = x^ - h\ 4. ?/=:4x + c;?/ = 4x;«/ = 4ic — 5;y=:4x + 20. 5. y = 4 x^ + c ; y = 4 x'^ ; ?/ = 4 x2 - 2 ; ?/ = 4 x2 - 13 ; y = 4 X - 62. 8. 16 «2 ; 64 ; 256 ; 400 ; 16 «2+ 10, etc. ; 16 «2 4- 20. Art. 98. 3. |. 4. 2 ; 0. 5. 4 ; 0. Art. 100. 4. (a) 2 y = x'^,6 y = x^,24 y = X* ; (b) y = x""- ■]- 5x,6y -2 x^ + 15x2, 12y = a;* + lOx^; (^^ y = 1 —cosx^ y = x — s'mx, 2y = x--\-2cosx—2 ; (d) y = e' -\, y = e"" - X - 1, 2y = 2 e'^ - x"^ -2x -2. 6. y = 1, y = 2, y = cos X, y = e*. CHAPTER XI. Art. 104. 9. I x» + c, /^ »^'^ + c, tr ^" + c, - | x-is + c, - ^^^ x-i^ + c, -r^,+ c, -4 + <'. f^^ + c, —i — x'^^-^^ + c, |x^ + c, ^x^+c, 8Vx + c, 2 x^ X* \/2 + 1 - — + c, -_§- + c. 10. iv^ + c, ^M^ + c, ^+c, 12s^+c. v9. 14 X* 2^^4 . '+» m+n m+S 6+n 11. _£ZL_x"^+c, -l^t~^ + k, -J^v^ + c, ?:^^^+c. 12. logc^, m + >i m + 3 6 + w t -\- s logc(s + 2)2, - ilogc(7 -a^), logc(4«2_3^ + ll). 13. e« -j. c, | e^ + c, 2e^' + c, ^ h c, — ^^^ f-c. 14. - icos3x + c, isin7x+c, nan5x+c, log 4 2 log 10 - cos (x + «) + c, ^ sin (2x4- «) + c, f tan ( -^ + - j 4- c. 15. ^ sec 2 x + c, fsecf x+c, sin-i^ + r, ^sin-ix24-c, |sin-i5x4-c, f sin'^x^+c, log(l + vl+«2) + c, \ tan-i «2 _|. c^ tan-i 2 x + c, sec-^ t + c, sec-i 3 x 4- c, i sec-i x2 4- c, I vers-i 3 X 4- c or ^ sin-i (3 x — 1) 4- c, \ vers-i 4 x + c or ^ sin-i (4 x — 1) + c. 16. ^ - f «3 4- 16 < 4- c, a^x + V- «^x5 4- f a^x^ + j\ x''^' + c, -e""" + c, 1.^1 - sm «x cos nx + c. [In the following integrals the arbitrary constant of integration is omitted.] Art. 105. 11. I sine ^^ ^^I^ (3 + 2 tan2 x), - ^ tan (4 - 7 x), - ^ e-*'. 12. log(x + l)4- ,f^^+f^^ , f +3x+31ogx-l, f(x4-2)^(x-8),^5(x-2)^ 424 INFINITESIMAL CALCULUS. (2^ + 3). 13. f(x+a)3, Hm + nx)\ _ 3 Vs-^y^, i(44-5i/)l 14. Igm+nx^ -■^r~l-> log(tan-ix), -COS (log X). 15. r'5(«-l)2(3« + 2), 35 1 — (« + &2/)^, {(w + 0)*, f sin I a:. 16. |sinx(3-sin2x), ^tanx(tan2x+3), I cos3 a; - cos X - i cos^ x, n tan ( - J • 17. - -^ log (3 + 7 cos x), -^log(9-2sinx), _ | V4 - 3 tan x, _Lsin-i /^^^^^l^V ig. Va2 + x2, Va2 _ x2 X Art. 106. 7. ^(ax-1). 8. -(x + l)e-^. 9. ae^(x2 - 2 ax + 2 ^2). 10. xlo gx-x . 11. ^x2(logx-^). 12. ^x3(31ogx-l). 13. xtan-ix - log vTTx2. 14. i (1 + x2) tan-1 x - | x. 15. 2 cos x + 2 x sin x - x- cos x. 16. e^[x'» - mx"*-! + m{m - l)x'«-2 _... + (_ \)^-'^m{m - 1) ... 3 • 2 • x + (-l)'».w!]. 17. - ^xco6 2x + I sin2x. 18. — Vl - x^ . sin-i x + x. Art. 107. 7. ^- tan-i ^-±^ ; sin"! ^^ ; log (x + 3 + Vx-^ + 6x+10) . 2V2 2V2 V26 8. ilog^-±^^; sin-i ^^±^ ; log(2 x - 5 + 2 Vx2 - 5 x + 7). 9. -^ 1-^ _ V53 V33 ^^^ 2x + 5-V33 . _^iog2^±A.:-^; :iog(8x-3 + 4V4^233^T5). 2 X + 5 + V33 Vei 2 X + 5 + V61 10. ^tan-i^^^; lsin-i^^±^; _4^ ^^g Vl37 + 5 + 8 x. \/71 V71 13 V137 V137-5-8X 11. vers-i? and sin-i ^^=i ; 1 vers-i — and \ sin-i ^ ^ - ^ ■ 1 sec"!^. 4 4^ 9 ^ 9^^ 5 12. isec-i^^; i(xV9 -x2 + 9sin-i^V ^-i-ir. 13. xV9^^ + 9sin-i?; logtan^^ + ^V 1 log tan i^^. 14. |logsec(3x+«); 1 logsin(4x2+«2). |logtanfx + ^V 16. -^2^:11^; — ^=5 -^51ZZZ. ^ ° \ 4/ 75x8 ' 4V4T^' 6x Art. 108. 3. log (X + 3)2 + — i_. 5. log (x2 + 4)2 (x - l)^ ; ^^-^(f^T-^""1' '• '^^Iri?' '• l-«^(2x-f5)(x-7)a. 8. ^x2-2x + log ('^ + ^)' . 9. ix2 + log:^^^^^. 10. log ^ — + X - 1 X (X — 1)2 |log(2x+5). 11. log^^=£K^±^. 12. log(x-3)2(x+3)8(x-2)(x+2)6. X 13. log (X - 1) - -^. 14. log V4¥+^ + ^ . 16. log X + X - 1 4 (4 X + 5) ANSWEB8, 425 flog(2a;+5) + -- 16. log (x + 4y V3 x + 2 + -—f——- 17. log(x4-l)2 X 3(3x + 2) + ^"^^t - 18- log "^ - ^^ ^^1^"' -V- 19. I log (3 X - 2) - i log (x2 + 5) __Ltan-i— . 20. log X + 2 tan-i X. 21. x + ilog^^-±-^-\/3 tan-i— . V5 V5 ^ x2 V3 22. log x2 + \/3 tan-i -^. 23. log x3(x2 + 3)2. 24. 2 logx --- 2 tan"*?. V'3 ^ ^ 2 25. log ^^^^ + i tan-i ^^:^- 26. tan"! x + log VxM^ - x2 _ 2 X + 5 " 2 " x2 + 1 Art. 109. 4. e^ cos y ; x^ 4- 4 x2?/ + 4 x — 6 y. 5. cos x tan ?/ — sin x ; ?y2 xey — 2 xy + x2 ; 3 x - 2 x2 - xy — ^• 2 Page 190. I, JIL^ + c, ^x2(«+6) + c, -J-±l—zn+t+s^c, V2 + W + 1 w + « + 3 -f-2/'-« + c, -12x\ + 291ogf, ^H-8?;-flog(«2 + 3)-llV3tan-i-^ + c, rts^ ^ ^ V3 ^- 2x + f log (x2 - 2) - -A^ log ^z::^+ c, -1^ tan-i ^4- c, 2 2\/2x + v^ 6V5 2V5 _JL log ^ ~ ^ ^ + c, 7 a^ + H«- a^ + H-1-, ^sin-i- + c, i log (x^ + 4V3 Z + 2VS 3 Vx6 - 9) + c, i2+— — ^— -+ilog(2 2;-l) +c. 2. -log sec (wx 8 (2 2: — 1) - m + n) + c, ^ tan 3 X + t log (sec 3 x + tan 3 x) + 4 x + c, 00, 2.4288. 3. X cos-i X — Vl — x2 + c, X sec-i x — log (x + Vx'^ — 1) 4- c, x cot"i x X 4- i log (1 4- a;2) + c, x{(log x)2 _ 2 logx + 2} 4- c, - «e'«(x2 + 2 ax + 2 ^2) 4. c, - (x^ 4- 3 x2 4- 6 X 4- 6)e-* 4- c, cos x (1 - log cos x) 4- c, -^^^ log x ^- m + lV 1^4-1 4-c. 4. fx^-fVx + c, 18(|x^4-ia;^ + ix^4-a;'b +91og ^ ~ '^ 4- c, 11 qi 1 , , X* 4- 1 4 (3* - 2*) 4- 4 log ^-— ^, \/x2^n: 4- log (x 4- Vx2^^) + c. 5. .206 (the 2^-1 base being 10), 1 ^ - 1 V \{f-V), -^\%ir^ 6. - ^^ log (m 4- n cos 0) + c, log (sin e tan ^U c, J- log tan f ^ 4- ^U c, ^-^ ^^ log "^^^"^ V ^/ V2 V2 8/ 8(sec2x-4) ^^ ^secx4-2 + c (see result in Ex. 3, Art. 118), sin-i /l^IL^A 4. c, — log^ (ms + w) + c, V 2 / 3m _l-sec->«!+c, tan-ie. + c, }log?i^+c, J log I+i5!t|» + e, »n log a m e* + e-== 1 — tan 2 ^ 4 \/2 sin-i ( V2 sin - j 4- c, cos x cos y — ?/2 _|_ ^^ _)_ c^ cos x sin y + x — y 4- c. 426 INFINITESIMAL CALCULUS. CHAPTER XII. Art. 111. 5. (6) 76. 6. 18. 8. 5. 11. -2/ VS. 13. (a) 2 ; {d) 4. 16. .862025; 6.644025; .862; .401. 17. (1) -^ ; (2) lOf ; (3) 3.2; (4) 68^*3 ; (5) I a2 . (6) 12 V2 ; (7) No area is bounded ; (8) (a) log 7, i.e. 1.946 ; log 15, i.e. 2.708 ; log n ; A;2 log ^ 18. -W Vf . a Art. 112. 9. -W^TT. 10. iQj^iTT. 11. ^^'w. 12. (a) f(2v^-l)7r; (6) K4 ^2 - l)7r. 13. -V^3. ^. 18. 405 (| - 1) ,r, 225 ^| - ^) tt. Art. 113. 2. ?/2 = 48 x - 80 ; 24. 3. sc - 4 = 2 log ?/. 4. cc - 4 = 4 log ?/ ; 4. 5. 3 y2 = 16 X. 6. 5 1/2 = 48 x2 - 112 ; the conies if = kx^ -}- c, k and c denoting arbitrary constants. 7. 3 y = a;2 + 6 ; the parabolas y = kx^ + c, k and c being arbitrary constants. 8. ?/2 t= 7 x + 4 ; the parabolas 1/2 z= A:x + c, A; and c being any constants. 9. The circles r = c sin ^ ; r = 4 sin d. 10. r^ = ce^ ; r2 = 4 e^. 11. r = a(l — cos ^), in which a is an arbitrary constant. CHAPTER XIII. Art. 116. 1. f v^ic ( Vx - 3) + 4 tan-i ^x + c. 2. 2( Vx - tan"-iv^) +c. 3. i(3x-2)^-^-A___+c. 4. ^2^(2 + x)^(5x+17)+c. 3 v/3 X - 2 6. - I log (7 + 5 \/2 - X) + c. 6. X + 1 + 4 Vx + 1 + 4 log (Vx + 1- 1) + c. Art. 117. 5. ^\/4x2 + 6x+ll + | log (2 x + 3 + \/4x2 + 6x+ 11) + c. 6. -3V l2-4x-x2 -10sin-i^±^ + c. 7. JL ipg^gEg ^" v^^+^ 4- c. 4 2V3 \/6-8x + V6 + x 8. 3sin-i^+g-^log^^-^^-^^J:;+c. 9. log^-^+ ^ ^'^+^+1 + c. 4 \/3 V6-3X+ Ve+x x+l+Vx2+x+l 10. Vx2 + x + l4-ilog(x + ^ + Vx2 + x + l)-31og ^~"^+^'^' + ^+^ + c. ,, , 1X + 2 , X + 1 + \/x2 + X + 1 11. isec-1— ^^^ — ^f.^ 4 Art. 120. 2. ^ cos^ x — cos x + c ; sin x — | sin^ x + c ; f cos^ a; — ^ cos^ x -cosx+c. 4. (1) |cosix(cos2x-4)+c; (2) 5sin^x(|-^sin2x+5Vsin*x)+c; (3) 2Vsinx( 1 - I sin2 x + f sin* x) + c; (4) 3 cost jc (^J^ cos2 x - ^) -\- c. 7.(l)^tan8x+tanx + c; (2)-^cot8x-cotx+c; (3)^tan6x+|tan8x+tanx+c. 9. (1) j\ tanS X (3 tan2 x + 5) + c ; (2)2 tant x (| + ^ tan2 x + A tan* x) + c ; (3) f t anf X (^ + I tan2 x) + c ; (4) sec^ x(^ sec* x - ^ sec2 x + i) + c ; (5) ^ Vcsc x(6 - csc2 x) + c ; (6) - esc* xQ esc* x - § csc2 x + |) + c. ANSWERS. 427 Art. 121. 3. (l)\(^-sm2x-\-^^^^\+c; (2) ^JgCSx + 4 sin2 a; -ismB2x + fsm4x) + c; (3) ^ _ ^j^^jl _ sm^ 2 x ^ * ^ ' ^ ^ 16 64 48 ' (4) ^V cos 2 x(cos2 2 X - 3) + c ; (5) ^l^f 3 x - sin 4 x + ^^^^\ + c. Art. 122. 1. (1) _ s^na^cosx _^g_^^. ^g) _ ^sin2xcosx - | cosx + c ; (3) -^^^^-5^(sin2x+f) + fx+c; (4) -lsin4xcosx-i^^^(sin2x+2)+c. 4 15 2. (1) -cotx + c; (2) logtan--|cotxcscx-fc; (3) -i-^^ -|cotx + c. 2 sin^ X 5. (1) ^sinxcosx(2cos2x+3)+f x + c; (2) ^ sin x(cos*x + f cos2x+|)4-c ; (3)4 ^^^^ 4-|tanx+c; (4) ^tanxsec^x+f secxtanx+f log(secx+tanx) + c. cos^ X I v\ 6. (1) I tan X sec X + ^ log tan f J + - J + c ; (2) i tan x (sec2x + 2) + c ; (3) i tan X sec^x + f | tan x sec x + log tan (7 + -) } + c. 7. (1) ^ log tan - — I cot X cosec X + c ; (2) — ^ cot x (cosec2 ^ + 2) + c ; (3) — | cot x cosec^ x — I ( cot X cosec X — log tan - J + c. 11. (1) | tan^ x — log sec x + c ; (2) — icot^x+cotx+x+c; (3) ^ tan^x— tan x+x+c; (4) ^ tan^cc— ^ tan2a; 4-logsecx+c. 14. (1) |(sinxcosx+x) — ^sinxcos^x+c; (2) — ^sinxcos^x + 2V sin X cos3 x+ x^5 sin x cos x + j^^ x + c ; (3) - i ^^ (3 - cos^x) - — + c. 2 sm X 2 17. (1) -lcot7x-|cot5x+c; (2) ^tan^x+c; (3) -TVcot3x(3cot2x+5) + c. Page 222. 3. (1) 3 «i + ^ log ^^i^^ - V3 tan-i f^-^l+^W c ; t - 1 V V^ / (2)3(2. + 3)^^. (3)^tan-i(^^^Uc; (4)-I- log^^EM^VS ^^. 8\^7+l V5 \Vl-4x2/ 2\/5 Vl-4x2+\/5 2 V4x— x2 _i X (5^ _£.^15±II:L_vers-i-+c; (6)2Vx2+3x+5-21og(x+f +Vx2+3x + 5)+c; (7)21og(x + |+V x2+'3x + 5 ) + -Llog ^Q + 3^-^^^(^^ + '^^ + ^) + c; V5 aj ^^~Vi ^ ^T^: ^^^~^^n(x2-16)2 32(x2-16) ^.t.-.:4^n-^ C-)iS^2-^_^t.an-.|.c. ^^^^-^^-i^)^- (12)cos-.(^)-2Ve^,,. CHAPTER XIV. Art. 125. 2. 2525. 3. 3690 ; 3660 ; (true value = 3660). 6. 333 in 20,000. 7. .05075; 1509. 428 INFINITESIMAL CALCULUS, CHAPTER XV. Art. 129. 4. The parabolas ?/ = 3 x^ + cix + Cs, whose axes are parallel to the ?/-axis ; 2 «/ = 6 x2 + 11 x - 13 ; ?/ = 3 x'^ + 15 x + 22. 5. The cubical parabolas y = x^ + cix + C2 ; y = x^ -\- x; y = x^ — x + i. 6. The cubical parabolas y = cx^ + CiX + C2, in which c, Ci, c^ are arbitrary constants ; 6 ?/ = x^ + 11 X ; 5 1/ + x^ + 16 = 22 X. 7. The cubical parabolas x = Ciy^ + C22/ + C3; 120x = lly3-251?/ + 240; 7x + 4?/3 = 622/ - 85. 8. 15,528 ft.; 62.1 sec. 10. Half a mile. Art. 130. 4. (1) 37; (2) ^^i^a^; (3) Qa^; (4) -fa^^; (5) iTrafec; (6) l^raB; (7) ^'; (8) '^'; (9) \-.a^-^a\ Art. 131. 3. 5. Art. 132. 5. 1154.7 cu. in. 6. faHana. 7. K^r - f)a^ 8. 2720.3 cu. in. ; ^^ tan a. 2 Art.133. 4. f7r(rt2_62)f CHAPTEE XVI. Art. 135. 4. 301.6; ^Trahh. 5. 55|cu.ft. 6. faft^cota. 7. |(3 7r + 8)a3. 8. ^aVi. Art. 136. 2. '^I- 3. |; ^. 5. fTra^. 6. 11 tt. 7. fa2. Art. 137. 2. (1) 2 7ra; (2) (&) 2{\/2 + log (v^ + l)}a ; (3)4afcos^-cos^V8a; (4) « (/« - e"?), «f e -IV 3. i^^^^±«^±^. ^ ^ V 2 2/' ' ^ ^2^ ^'2V ey a+6 Art. 138. 2. (3) '-^ ; (4) (a) Z sec «, in which Z is the difference in length of the radii vectores to the extremities of the arc ; (4) (h) like (4) (a) ; (5) ^ r^2 VlT^^ - e^^We? 4- log h±}^l±Jl\ ■ (6) a tan ^ sec ^ + 2L ^j+Vm^J 2 2 a log tan [^ + -") ; 2 a [sec ~ + log tan - ttV Art. 139. 5. 47ra2. 6. ttOk -2)a^. 7. 2 irfta + 2 Trtft ^^^Z^. e 8. (1) 37ra2; (2) 5 7r2a3, ^ira'^; (3) 7r2a3, ^7ra2. 9. 2 7r2a2&, ^ir^ah. 10. 2 7ra2(l_iy 12. ^7ra3(22 + 37r); -J— Ta2(7r + 4). V e) 2V2 Art. 140. 2. 4(z2. 3. 47ra2. 4. Surface = 8af2 6sin-i -—^=:^ ■ 7,2 X \ y/¥^^' — a sin-i — - — ] ' «2 - b^l ANSWERS. 429 Art. 141. 3. 1341; 91. 4. 4.64. 5. (1) 2|, 5^; (2) f, 1.14, .94; (3) 5^, 9^. 6. (1) 9.425; (2) 15.71; (3) 1.571 &, 1.571a. 7. — , — • 9. InK 10. 1.273 a. 12. 1.132 a, 1.5 a2. 13. |a, Ja^. 14. 32.704°. 15. J a, fa. 16. fa, fa2. 17. fa, fa^. 18. 1.273 a, 2 a^ 19. .6366 a, ^ a^. CHAPTER XVII. Art. 143. 1. (1) First order at (1, 1) ; (2) second order at (2, 8). 2. ?/ = 5x2-6x + 3. 3.-1. 4. ^ = 3a;2-3x+ 1. 5. 1/ = x2 - 3x + 3. Art. 144. 1. 5.27 and (- 4, |) ; 2.635 and (- |, Jgi). 2. B = 145.5 ; (- 143, 20^2). Art. 148. 1. The curvature of y=x^ is one-half that of y = 6x2— 9x+4. 2. — ; i?=-88.4; (-87.5, -12.5). 125' ' ^ ^ Art. 149. 3. liP±^; f2p + 3x, - -^\; - 2p and (2p, 0). pi \ ^p'y 4. ^^_XMx^+aw!^ (o!,:^^. Centre at l^^^^x% -^^-=^yA. a%^ ab ' V «* b^ ^ J 5 a^ j> ^ i^'x^ + aV)^ ^ (e2x2-a2)l ^ / a^ + b\ ^ «^ + 6% .\ ^ ^ a4&* aft ' V a* ' 6* ^ A ^^> ^^ (*-.^. ^^-a- ''' 'i' {-'-'^^ '^)- (4) - 3 (iaxyy ; (x + 3 v^, 1/ + 3 v^). (5) 3 a sin cos ; (a cos^ « 3 1 + 3acos«sin2^, asin3^+3asin^cos2«). (6) (4q + 9x)^x^ . /^_o^^ Qa \ a 4y + |^y (7) -2a; (a, -fa). (8) ±4asin-; {a ■ 6 + sin ^, - y). 6. (1) (£±J}!. (2) («* + 9a;^)^ (3) csec?- (4) |a. (5) 2acosec3f. 2 Va 6 a^x c ^g^^ (a2sin2 + &2cos2 0)f ^^^ (a2tan2 + 62sec2 0)i asec2^, 2 a& ab i.e. ^• a Art. 150. 1.(1) a; 6. (2)^^- (3) | V2^. (4) - 4 (5) ± f^- V a ^ 3 r (6) rv/TT^. (7) ±<1 + ^. (8) ^a0"-Hn^ + ^)l Art. 151. 3. (1) (ax)^-(6?/)*=(a2 + 62)l. (2) (x-\-y)^ - (x -y)^ = (4 a)i [Suggestion : Show that a -¥ ^ = -(--^-Y, a-3 = ^(^--]\ 2\x a) 2 \x a) and deduce therefrom.] (3) (x + ?/)^ + (x — ?/) ^ = 2 a^' 430 INFINITESIMAL CALCULUS. CHAPTER XVIII. Art. 157. 5. (1) x2 + ?/2 = a2. (2) h^yi^ + a'^y'^ = a%\ (3) iay^-\-bxy -\-cx^ = 4.ac- h\ (4) 4 x^/ + a^ = 0. (5) 4 i/^ = 27 a'^x. (6) [x - a)^ + (y _ 5)2 =: ^2. 6. (1) a;2 + ?/2 = a2, (2) a;2_^2:=^2. (^s) (axy + (byy^ = (^2-52)1. 7. (1) The lines a; ± ?/ = ; (2) 27 cy^ = 4 x^ 10. A parab- ola ; 1/2 = 4 ax if the fixed point be (a, 0) and the fixed line be the jz-axis. Art. 158. 3. 4x?/ = a2. 4. ^jl _|_ ^1 = ^f 5. x^ + yt = ^^1 , Art. 160. 4. (l)x = a,y = b. (2) x = 2. (3) ?/ + 3 =0, 2x + 3 = 0. (4)2/ + l=0. 5. (2) (2, I). (3) (-1, -3), (-1, -V). (4)(-i, -1). 8. (1) x=0, y=0. (2) x=2a. (3) y=0. (4) x=±a, 2/=± &. (5) ?/=0, x = a. (6) x=0. 0) y=0. (8) ?/=0. (9) x=(±2 w + 1) -, in which w is any integer. Art. 161. 2. te±«y = 0. 5. (1) ?/ = x. (2) x + y = 1, x ~ y = 1. (3) x=2, |/+3=0, 2(^-x)^5. (4) x=2/±l, x + ?/=±l. (5) 6y = Sx+2. Art. 162. 2. (1) Lines parallel to the initial line and at a distance ± nair from it, n being any integer. (2) The line perpendicular to the initial line, at a distance a to the left of the pole. (3) The two lines which are parallel to the initial line and are at a distance 2 a from it. 4. r sin (^— 1) =1 ; r = l. Art. 165. 3. (1) Node at origin ; slopes there are ± 1. (2) Cusp at (—3, 1) ; slope there is 0. (3) Cusp at (2, 1) ; tangent there is parallel to the ?/-axis. (4) Double point at (0, 0) ; slopes of tangents there are 1, — |. (5) Cusp at (1, 2) ; slope of tangent there is 1. (6) A conjugate point at (3, 0). CHAPTER XIX. Art. 171. 3. (1) Convergent. (2) Convergent. (3) Divergent. (4) Divergent except^ V.ben p > 2. (5) Convergent if p >^'2y 4. (^);^^*<1, convergent; x'> 1, or .x,= 1^ divergent. -(2) Xbsolutely convergent if x2 <4, divetgent if x2 = 1, divergent if x2 > 1. (3) Absolutely convergent for all values of x. (4) x < 1, or x = 1, convergent; x>l, divergent. (5) Same as in Ex. 4. CHAPTER XX. Art. 176. 5. (a) cosx - ^sinx - — cosx + — sinx H ; (6) cos^ ^2„„..x3 2! 3! 2! 3! Art. 177. 4. e + e(x - 1) + ^ (x - 1)2 + Ji I ANS IVERS. 431 Art. 178. 10. (1) 1 + — + — + ^^+ •••; (2) ?^ + ^ + ^+.... ^ ^ 2 ! 4 ! 6 ! 2 12 45 12. (l)c + x + f-|f-?^-?^^?^-f...;(2)log^M^-a) 2.213.3! ' ' a -3 1-2.5 1.2.3-7 CHAPTER XXI. Art. 186. 2. 2/ Vl - x-* + x Vl - ^^ = c. 3. (y + 6)"(ic + a)"* = c. Art. 187. 1. x2 + i/2 = cy. 2. x2(x-2 + 2 y'^) = c*. 3. Xi/2 = c^(x + 2 ?/). 4. xy(x -y) =c. Art. 188. 1. xy = c. 2. x2y + 3 X + 2 y2 _ c. 3. gx gin y + x^ = c. 4. 3 axy - y^ = x^ + c. 7. a log (x^y) - y = c. 8. log — = — • y xy Art. 189. 3. vl — x"^ . y = sin-i x + c. 4. y = tan x — 1 + ce"'*"*. b. y = x2(l + ce^. 7. Sy^ = c(l - x^)^ - 1 + x2. 8. y^^x^ + 1 + ce'') = 1. Art. 190. 2. 2/2 = 2 ex + c^. 3. y = c - [i?2 + 2p + 2 log (p - 1)], X = c - [2p + 2 log (p — 1)]. 4. log (p — x) = -^^ H c, with the given p-x relation. 5. (x^ + y)^ {x^-2y) +2 x(x^ -Sy)c = cK 6. y = ex + -. 7. y = ex + a VH- e2. 8. 2/2 = cx2 + c2. ^ Art. 191. 2. x''« + y2 = a2; x2(x* - 4 2/2) = 0. 3. (1) 2/ = ex +c2, x2 + 42/ = 0. (2) (2/ + x-c)2 = 4x2/, x2/=0. (3) (x -y + c)^ = a{_x ^ y)^, X + 2/ = 0. Art. 192. 3. The concentric circles x2 + 2/^ = a2, 4, The lines y = mx. B. (1) The ellipses y^ + 2 x2 = e2 ; (2) the hyperbolas x2 - ^2 _ ^2 ; (3) the conies x2 + ny^ = c ; (4) the curves y^ — x^ = c^ ; (5) the ellipses x2 + 2 2/2 = c2 ; (6) the cardioids r = e(l + cos d) -, (7) the curves r** cos w^ = c" ; (8) the curves r^ = c^ sin nd ; (9) the lemniscates r^ = c^ sin 2 ^, whose axes are inclined at an angle 45° to the axes of the given system ; (10) the con- focal and coaxial parabolas r(l — cos ^) = 2 e ; (11) the circles x2 + 2/2-2 Ix + a2 = 0, in which I is the parameter. 10. The conies that have the fixed points for foci. 11. The conies that have the fixed points for foci. 12. The conies 62a;2 _[. a^y'2 = ^262. 13. The hyperbola 4 xy = a"^. 14. The parabola (x - 2/)^ - 2 a(x + 2/) + a2 = 0. Art. 193. 3. (1) 2/=e^(acos3x+6sin3x). (2) 2/=Cie2^+C2e^+C8e*'. (3) y = Cie*^-\-e-^(c2-^Csx). (4) 2/ = e2x(ci -f e2x) + e*'(e3 cos 5 x+e4 sin 5 x). 7. (I) 2/ = X (a cos log X 4- 6 sin log x). (2) y = x(ci -\- c^ log x). (3) 2/ = x2(ei + C2 log x). (4) y = CiX'^ + x(c2 cos logx + C3 sin log x). 9. 2/ = (5 + 2 x)2{ei(5 + 2 x)v'2 _,. 02(5 + 2 x)-v^2^ 432 INFINITESIMAL CALCULUS, Art. 194. 4. (1) y = cie«^ + C2e-«*. (2) e2cx _|. 2 ccie^-y = ci2. (3) t =^5^ { I (vers-i ?^ - tt^ - v/aa; - a;^ | . 6. The circle of radius a. 6. (1) ?/=Cia;+(c'iHl)log(a;-Ci)+C2. (2) y = Ci log x + cg. (3) 2(^-&) =:gx-«4.g-(x-a), (4) j,^cilog(l-fx)+^x-ia;Hc2. 8. (1) y2=a;2_|.Cia; + C2. (2) log^ = cie^+C2e-^. (3) {x-cxY=C2{y'^ + C2). (4) ?/ = logcos(ci-x) + e2. Page 351. (l)r=asin^. (2) xeJ'=c(l + x-|-y), (3) c(2?/2+2a:y-a:2)2>/3 = (V3+l)x + 2y ^ ^^>j x2 = 2 cy + c^. (5) |/ sec a; = log (sec x + tan x) + c. (l_V3)x + 2?/ (6) 3y = x2(l + a;2)^ + cx^. (7) 3x2 + 4xy + 6?/2 + 5x + ?/ = c. (8) (x - 2 c)y2 = c2x. (9) ?/(x2 + 1)2 = tan-J X + c. (10) 602/3(a; + 1)2 _ 10 x6 -f 24 x^ + 15 X* + c. (11) X = ^ {_c-\- a sin-^jo), y =- ap-\- Vl-i)2 — (c + asin~ip). (12) x + c = alog (jp + Vl +^^2), ^ = a Vl + P^ (13) ^2:^cx2--i^. (14) x = cx?/ + c2. (15) y ==8(^924.^3) + | log (2p- 1), c + 1 (16) ?/(l±cosx)=c. (17) ?/2+(x+c)2=a2. ^2=0^2. (Ig) ?/=cx+ V&2+a2c2; 62x2 -h aY = «^&^- (19) 9(2/ + c)2 = 4 x(x - 3 a)2 ; X = 0. (20) y = Cie«« + C2e-«* + C3 sin (ax + «). (21) y = (cie* + C2e-*) cosx + (cge* + 046-*) sin x. (22) 2/ = e2x(-ci + C2X) + cse-*. (23) y = CiX -{■ CsX-i. (24) 2^ = -^ + x^|c2Cos[ — logx) + cssin( — logxYl. (25) 2/ = Ci(x + a)2 + C2(x+a)3. (26) (cix 4- C2)2 + a = Ciy\ (27) 3 x = 2 a^(y^ - 2 c{) {y^ + Ci)^ + Co. (28) y = ci log X + ^ x2 + C2. (29) e-«y = CxX + C2. INDEX. [The numbers refer to pages.} Abdank-Abakanowicz, 169. Absolute, constants, 16; value, 14. Acceleration, 109. Adiabatic curves, 87. Aldis, Solid Geometry, 299. Algebra, Chrystal's, 14, 29, 67, 70, etc. ; ChrystSiVs Introduction ^o, 19 ; Hall and Knight's, 70, 303, etc. ; Hall's Introduction to Graphical, 19. Algebraic equations, theorems, 98, 99. Algebraic functions, 17, 61, 98. Allen, see * Analytic Geometry.^ Amsler's planimeter, 229. Analytic Geometry, Ash ton, 131 ; Candy, 5; Tanner and Allen, 19, 131; Wentworth, 131. Analytical Society, 45. Anti-derivatives, 50, 53, Anti-differentials, 50, 170, 171. Anti-differentiation, 148, 170. Anti-trigonometric functions, 17. Applications: elimination, 114; equa- tions, 98, 99; geometrical, 84; physical, 84 ; rates, 91 ; of partial differentiation, 141 ; of integra- tion, 192, etc. ; of successive inte- gration, 235, etc. ; of integration in series, 313 ; of differentiation in series, 313; of Taylor's theorem, 322, 331, 332. Approximate integration, 223. Approximations : to areas and integrals, 157, 223, 316; to values of func- tions, 49; to small errors and corrections, 96, 138. Arbitrary constants, 16. Arbogaste, 42. Arc: derivative, 102, 103; length, 245, 248 ; Huyghen's approximation. Archimedes, see ' Spiral.' Area : approximation to, 223, 225 ; deriv- ative, differential, 99, 101 ; me- chanical measurement, 228, 229; of curves, 192, 242, 244; of a closed curve, 198,245; of surfaces of revolution, 249; of other sur- faces, 253 ; precautions in finding, 198 ; sign of, 197, 245 ; swept over by a moving line, 245. Argument of function, 15. Ashtou, see 'Analytic Geometry.' Astroid, see ' Examples.' Asymptotes, 286; circular, 292; curvi- linear, 291 ; oblique, 290 ; parallel to axes, 288; polar, 292; various methods of finding, 291. Asymptotic circle, 292. Average value, 259. Beman, Famous Problems, 13. Bernoulli, 150, Binomial Theorem, 322. Bitterli, 169. Borel, divergent series, 305. Burmaun, 18. Byerly, see ' Calculus.' Cajori, History of Mathematics, 42, 46, 149, 204, 222. Calculation of small corrections, 96. Calculus, 1; differential, 11, 39, 149; integral, 11, 39, 50, 149; funda- mental theorems, 36; invention, 1, 149; notions of, 11. references to works on : Byerly, Diff\ 34, etc.; Byerly, Problems, 111, etc.; Echols, 41, etc.; Ed- wards, Integral, 213, etc.; Ed- wards, Treatise, 129, etc. ; Gibson, 29, etc. ; Harnack, 20, etc. ; Lamb, 433 434 INFINITESIMAL CALCULUS. 29, etc. ; McMahon and Snyder, Biff., 33, etc. ; Murray, Integral, 163, etc.; Perry, 12, 381, etc.; Smith, W. B., 135, 222; Snyder and Hutchinson, 156, etc. ; Taylor, 129, etc.; Todhunter, Diff., 70, etc. ; Integral, 163 ; Williamson, Diff., 70, etc.; Integral, 163, etc.; Young and Linebarger, 381. Candy, see 'Analytic Geometry.' Cardioid, see ' Examples.' Catenary, see ' Examples.' Cauchy, 304; form of remainder, 327. Centre of curvature, 269, 270; of mass, 373. Change of variable, in differentiation, 145; in integration, 175. Chrystal, see ^Algebra.' Circle, curvature of, 267 ; of curvature, 268 ; osculating, 264 ; see * Exam- ples.' Circular asymptotes, 292. functions and exponential functions, 327. Cissoid, see ' Examples.' Clairaut's equation, 339. Commutative property of derivatives, 133, 140. Comparison test for convergence, 307. Complete differential, 136. Compound interest law, 70. Computation of tt, 313, 314. Concavity, 260. Condition for total differential, 141. Conjugate points, 295. Conoids, 241. Constant: absolute, 16; arbitrary, 16; elimination of, 114; of integra- tion, 160, 162, 166, 335. Contact: of curves, 261, 280, 282; of circle, 263; of straight line, 262. Continuity, continuous function, see 'Function.' Convergence : kinds of, 304, 305, 311 ; interval of, 306; tests for, 307, 308; see ' Series,' ' Infinite Series.' Convexity, 260. Corrections, 96. Cos X, derivative of, 73; expansion for, 322, 325. Criterion of integrability, 188. Critical point, critical value, 118, 120. Crossing of curves, 263, 332. Cubical parabola, see ' Examples.' Curvature: average, 266; at a point, 266, 267; total, 266; centre of, 269, 270; of a circle, 267; circle of, 268 ; radius of, 268, 271. Curves: area of, 192, 242, 244; asymp- totes, 286; contact of, 261, 280, 282; derived, 44; differential, 44; envelope, 277; equations derived, 203; evolute, 272; family, 277; integral, 168, 169; involutes, 276; length, 245, 248 ; locus of ultimate intersections, 278 ; Loria's Special Plane, 2^; parallel, 276; see 'Ex- amples.' Curve tracing, 298. Curvilinear asymptotes, 291. Cusps, 280, 293, 294, 296,. 297. Cycloid, see 'Examples.' Decreasing functions, 116. Definite integral, sec ' Integral.' De Moivre's theorem, 328. Density, 372. Derivation of equation of curves, 203. Derivative : definition, 38 ; notation, 41 ; general meaning, 4(5; geometric meaning, 43; physical meaning, 45; progi'essive, regressive, 93. Derivatives: of sum, product, quotient, 51, 53-57; of a constant, 52; of elementary functions, 61-80 ; of a function of a function, 59; of im- plicit functions, 80, 1.39 ; of in- verse functions, 61 ; special case, 60; geometric, 99-106; successive, 107, 112; meanings of second, 108, 109. Derivatives, partial, 81, 130, 131 ; com- mutative property of, 133, 140; geometrical representation, 132; successive, 133. Derivatives, total, 136; successive, 143. Derived, curves, 44; functions, 40, 41. Descartes, 149. Differentiable, 41. Differential calculus, see ' Calculus.' Differential coefficient, see ' Derivative.' Differential, differentials, 47, 49; com- plete, 136; exact, 142; geometric, 99-106; infinitesimal, 155 ; partial, 136; successive, 112; total, 136, 137 ; condition for total, 141 ; in- tegration of total, 188. Differential equations, 114, 334 ; classifi- INDEX. 435 cation, 334; Clairaut's,339; exact, 336; homogeneous, 336; linear, 337, 34(i, 348 ; order, 334 ; ordinary, 334; partial, 334; second order, 349; solutions, 114, 335, 340 ; refer- ences to text-books, 115, 351, etc. Differentiation, 39, 170; general results, 51 ; logarithmic, 68 ; of series, 312 ; successive, 107; see 'Derivative,' ' Derivatives.' Discontinuity, discontinuous functions, see ' Functions.' Divergent series, see ' Series.' Double points, 280, 293, 294. Doubly periodic functions, 221. Durand's rule, 227. Echols, see ' Calculus.' Edwards, see ' Calculus.' Elementary integrals, 172, 180. Elimination of constants, 114. Ellipse, see 'Examples.' Ellipsoid, 235. Elliptic functions, 158, 221. integrals, 158, 221, 317. End-values, 155. Envelopes, contact property, 280, 282; definition, 278; derivation, '281, 284. Equations, derivation of, 203; graphi- cal representation, 19, 20, 130 ; roots of, 98, 99. Equiangular spiral, see ' Examples.' Errors, small, 9(j, 138; relative, 96. Euler, 142, 314, 328 ; theorem on homo- geneous functions, 142. Evolute, definitions, 272, 275, 283. properties of, 273. Evolute of the ellipse, see ' Examples.' Exact differential, 142. equations, 336. Examples concerning: adiabatic curves, 87. astroid (or hypocycloid), 86, 102, 198, 203, 251, 270, 273, .345, 365. cardioid, 91, 101, 244, 249, 252, 271, 345, 365, .376, 383, 390, .392, 393, 396. catenary, 201, 248, 25.3, 366, 383. circle, 86, 194, 202, 244, 249, 252, 271, 344, 375, 376, 378, 391, 393, 399. cissoid, 290, 393. cubical parabola, 92, 101, 102, 158, 166, 195, 198, 201, 270. cycloid, 87, 248, 252, 270, 273, 366. ellipse, 86, 106, 200, 203, 248, 257, 276, 290, 374, 385, 390, 391, 393, 397, 399, 400. evolute of the ellipse, 273, 390. exponential curve, 86, 392. folium of Descartes, 87, 244, 290. harmonic curve, 398. hyperbola, 87, 92, 270, 271, 273, 290, 291, 345, 383, 388, 390, 393. hypocycJoid, see ' Astroid.' lemniscate, 244, 271, 345, 383, 393. limaQon, 398. parabola, 86, 87, 92, 102, 104, 152, 159, 166, 195, 196, 198, 234, 249, 257, 270, 271, 273, 276, 283, 284, 290, 345, 366, 376, 383, 388, 391, 393. probability curve, 290. semi-cubical parabola, 86, 87, 159, 198, 270, 366, 391. sinusoid, 86, 159. tractrix, 3(Ki, 392. the witch, 87, 271, 290. Spirals : archimedes', 01, 101, 103, 249, 271. equiangular (or logarithmic), 91, ^14, 249, 271, 366, 390. general, 91, 271. hyperbolic (or reciprocal), 91, 244, logarithmic, see ' Equiangular.' parabolic (or lituus), 91. reciprocal, see ' Hyperbolic' Expansion of : cos X, 322, 325. log (l + x), logarithmic series, 315, 321. sin X, 322, 325. sin-ia;,314. e', exponential series. .326. tan"i X, Gregory's series, 228, 313. Expansion of functions : by algebraic methods, 326. by differentiation, 228, 315. by integration, 228, 313. by Maclaurin's series, 325. by Taylor's series, 321. Explicit function, 17. Exponential curve, see ' Examples.' function, 17; expansion of, 326; and trigonometric, relations between, 327. Family of curves. 377. Fermat, 122, 149, 247. Fluent, fluxion, 45. 436 INFINITESIMAL CALCULUS, Folium of Descartes, see ' Examples..' Forms, indeterminate, 367. Formulas of reduction, 213, 218. Fourier, 155. Fractions, rational, integration of, 184. Frost, Curve Tracing, 291, 293, 299. Function, 15; algebraic, 17, 61, 221; cir- cular, 221 ; classification, 17 ; con- tinuous, 25, 28, 41, 130; derived, 40, 41 ; discontinuous, 25, 27 ; elliptic, 158, 221; explicit, 17; exponential, 17, 70; graphical representation, 19, 20, 130; homo- geneous, Euler's theorem on, 142; hyperbolic, 183,221, 353; implicit, 17, 80; increasing and decreasing, 116; inverse, 18, 76; irrational, 206; logarithmic, 17, 67; march of a, 123 ; maximum and minimum values of, 117 ; notation for, 17, 18; of a function, 59, 60; of two variables, 130 ; periodic, 221 ; tran- scendental, 17 ; trigonometric and anti-trigonometric, 17, 71, 215; turning values of, 117; variation of, 116. Gauss, 304. General integral, see 'Integral.' spiral, see ' Examples.' Geometrical interpretation, a certain, 215. Geometrical representation of : derivatives, ordinary, 43. derivatives, partial, 132. functions of one variable, 19, 20. functions of two variables, 20, 130. function of a function, 60. integrals, definite, 163. integrals, indefinite, 166. total differential, 137. Geometric derivatives and differentials, 99-106. Geometry, Famous Problems in, 13. Gibson, see ' Calculus.' Glaisher, Elliptic Functions, 222. Graphical Alr/ebra, Hall, 19. Graph of a function, 19. Graphs, sketching of, 123. Gregory, 305, 313. Gregory's series, 313. Gyration, radius of, 377. Hardy, infinitesimals, 38. Harkuess and Morley, Analytic Func- tions, 15, 29, 303 ; Theory of Func- tions, 41. J Harmonic curve, 398. Harmonic motion, 83, 110. Harmonic series, 304. Harnack, see ' Calculus.' Hele Shaw, Mechanical Integrators, 229. Henrici, Report on Planimeters, 229. Herschel, 18, 46. Hobson, Trigonometry, 303, 314, 363. Homogeneous, differential equations, 336. functions, Euler's theorem, 142. linear equation, 348. Horner, Horner's process, 324, 333. Hutchinson, see ' Calculus.' Huyhen's rule for circular arcs, 326. Hyperbola, see ' Examples.' Hyperbolic functions, 183, 221, 328, 353. spiral, see ' Examples.' Hypocycloid, see ' Examples.' Implicit functions, differentiation, 8G> 139. Increasing function, 116. Increment, notation for, 4. Indefinite integral, see ' Integral.' Indeterminate forms, 25)6, 367. Inertia, centre of, 373. moment of, 377. Infinite numbers, 14, 30, 31. orders of, 31. Infinite series, 300; algebraic proper- ties, 304; differentiation of, 302, 312; general theorems, 305; inte- gration in, 227, 316; integration of, 302, 310; limiting value of, 301; questions concerning, 301; Osgood, article and pamphlet, 303, 306, 307; remainder, 306; study of, 303. See ' Series.' Infinitesimal, 1, 48, 49. Infinitesimal differential, 155. Infinitesimals, 30; Hardy, 38; orders, 31; summation, 150; theorems, 33, 35. Inflexion, points of, 118, 127, 261. Inflexional tangent, 129. Integral curves, 168, 169. Integral, definite, approximation, 223, 316; definition, representation of, properties, 164-158, 163, 164. INDEX. 437 Integral : double, 230 ; element of, 155 ; elementary, 172, 180 ; elliptic, 158, 221, 248, 317 ; general, 162. Integral, indefinite, 162; representa- tion of, 166. Integral: multiple, 231; particular, 162; precautions in finding, 198; triple, 230. See ' Calculus.' Integrand, 150. Integraph, 169, 228, 229. Integrating factors, 336. Integration, 148, 170 ; as summation, 154, 170 ; as inverse of differentiation, 160; constant of, 160, 162, 166; general theorems in , 173 ; succes- sive, 230, 232. Integration : by parts, 177 ; by substitu- tion, 175, 183, 215 ; by infinite se- ries, 227, 316; by mechanical devices, 228. Integration of : infinite series, 302, 310; irrational functions, 206 ; rational fractions, 184; total differential, 188 ; trigonometric functions, 215. See ' Applications.' Integrators, 228, 229. Intrinsic equation, 249, 363. Invention of the calculus, 1, 149. Inverse functions, 18, 61, 76. Involutes, 276. Irrational functions, integration, 206. Isolated points, 295, 296. Jacobi, 133. Kepler, 122. Klein, 13, 67. Lagrange, 42, 149, 326. Lagrange's form of remainder, 323. Lamb, see ' Calculus.' Laplace, 149. Legend re, 317. Leibnitz, 1, 42, 45, 149, 150, 282, 313. theorem on derivative of product, 113. Lemniscate, see 'Examples.' Lengths of curves, 245, 248. Lima^on, see ' Examples.' Limits, limiting value, 20, 23, 42; Hardy, 38; theorems, 35; in inte- gration, 155; of a series, 301. Linear differential equations: of first order, 337; with constant coeffi- cients, 346 ; honwJgeneous, 348. Linebarger, see ' Calci^lus.' Lituus, see ' Examples.' Locus of ultimate intersections, 278. Logarithmic, differentiation, 68. function. 17, 67. series, 315, 321. spiral, see ' Examples.' Loria, Special Plane Curves, 299. Machin, 314. Maclaurin, 327. theorem and series, 324, 328. Magnitude, orders of, 31. Mass, centre of, 372. Mathews, G. B., 305. Maxima and minima, 116; by calculus, 117-122; by other methods, 122; of functions of several variables, 122 ; practical problems, 123. McMahon, proof, 142. See 'Calculus.' Mean square value, 259. Mean values, 255. Mean value theorems : differentiation, 84, 94, 95, 318. integration, 165, 255. Mechanical integrators, 228. Mechanics, 372. Mellor, Higher Mathematics, 381. Mercator, 315. Minima, see ' Maxima.' Moment of inertia, 377. Morley, see ' Harkness.' Motion, simple harmonic, 83, 110. Muir, on notation, 133. Multiple, angles in integration, 217. integrals, 231. points, 293, 296. roots, 98. Neil, 246. Newton, 1,45, 149, 314. Nodes, 294. Normal, rectangular, 84. polar, 89. Notation for: absolute value, 14; de- rivatives, 41, 46, 107 ; differentials, 47; functions, 17; increment, 4; infinite numbers, 15; integration, 149, 162, 163, 233; inverse func- tions, 18, 61 ; limits, 24 ; partial derivatives, 81, 131, 133, 137 ; sum- mation, 155. 438 INFINITESIMAL CALCULUS, 1 Notation, remark on, 42. Numbers, 13; algebraic, 13; finite, in- finite, infinitesimal, 14,30; tran- scendental, 13, 67. e and tt, 67, 328. graphical representation, 13, 14. Oblique axes, 193. Order of, contact, 261. derivative, differential, 108, 333. differential equation, 334. infinite, 31. infinitesimal, 31, 333. magnitude, 31. Orthogonal trajectories, 341, 343. Oscillatory series, 304. Osculating circles, 264, 271. Osgood, W. F., article, 311, 313. pamphlet, 303, etc. Parabola, see ' Examples.' Parabolic rule, 225. spiral, see ' Examples.' Parallel curves, 276. Parameter, 277. Partial derivative, see ' Derivative.' Partial fractions, 184. Particular integral, see ' Integral.' Pendulum time of oscillation, 317. Periodic functions, 221. Perry, on notation, 133. See ' Calculus.' Picard, 156. Planimeters, 228, 229. Henrici, Report on, 229. Points, see 'Critical,' 'Double,' 'Iso- lated,' ' Multiple,' ' Salient,' ' Sin- gular," Stop,' 'Triple,' 'Turning.' Power series, 307, 310, 312, 313. Precautions in integration, 198. Probabilities, 326. Probability curve, see 'Examples.' Progressive derivative, 93. Radius of curvature, 268, 271. of gyration, 377. Rate of change, 11, 45, 46, 47. variation, 134. Rational fraction, integration, 184. Reciprocal spiral, see ' Examples.' Rectification of curves, 246. Reduction formulas, 213, 218. Regressive derivative, 93. Remainder after n terms, 306. Remainders in Taylor's and Maclaurin's series, 320, 323, 327. Ring, 202. Rolle, 99. Rolle's theorem, 84, 93, 95, 99, 319. Roots of equations, 98, 99. Rouche et Comberousse, 246. Rules for approximate integration, 223, 225, 227. Salient points, 295. Schlomilch-Roche's form of remainder, 327. Second derivative : geometrical meaning, 108. physical meaning, 109. Semi-cubical parabola, 246. See ' Examples.' Series, 70; absolutely convergent, 305; conditionally convergent, 305 ; convergent, 304; divergent, 304, 305; harmonic, 304; oscillatory, 304; uniformly convergent, 311, 312. See ' Convergence,' ' Expansion,' ' In- finite Series,' 'Power Series.' Serret, 320. Sign of area, 197, 245. Simpson, Simpson's rule, 225. Sin X, sin-ix, expansions, 314, 322, 325. Singly periodic functions, 221. Singular points, 293, 295. Singular solution, 340. Sinusoid, see ' Examples.' Slope, 5, 6, 11, 84,88, 129. Slopes, curve of, 44. Smith, C, Solid Geometry, 253, 299. Smith, D. E., Famous Problems, 13. Smith, W. B., Infinitesimal Analysis, 135, 2*22, 354.' Snyder, see 'Calculus.' Solution, see ' Differential Equation.' Speed, 2, 3, 4. Sphere, surface, 252, 254. volume, 203, 237, 238. Spiral, see ' Examples.' Stationary tangent, 1*29. Stirling, 327. Stop points, 295. Subnormal, rectangular, 84. polar, 89. Substitutions in integration, 175, 207, 215. INDEX. 439 Subtangent, rectangular, 84. polar, 89. Successive differentiation, 107. derivatives, 107, 112. differentials, 112. integration, 230, 232. of a product, 113. total derivatives, 143. Summation, examples, 150. integration as, 154. Surfaces, areas of, 249, 253. volumes, 199, 235, 238, 240. Tangent, 5; inflexional, 129; length, 84, 89; stationary, 129. Tanner, see ^Analytic Geometry.^ Taylor, F. G., see ' Calculus.' Taylor's theorem and series : applications : to algebra, 332 ; to cal- culation, 49, 137, 322, 323, 324; to contact of curves, 332 ; to maxima and minima, 331. approximations by, 49, 137, 322. expansions by, 321-324. for functions of one variable, 49, 96, 318, 319, 323, 328, 330. for functions of several variables, 327. forms of, 320, 321,323. historical note, 326. Test-ratio, 308. Time-rate of change, 45, 135. Todhunter, see ' Calculus.' Total derivative, 136, 143. differential, 136. rate of variation, 134. Tractrix, see 'Examples.' Trajectories, orthogonal, 341, 343. Transcendental functions, 17. numbers, 13. Trapezoidal rule, 223. Trigonometric functions, direct and in- verse, 17, 76. differentiation of, 71-80. integration of, 215. substitutions by, 207. Trigonometry, Hobson, 303, 314, 363. Murray, 76, etc. Triple points, 294. Turning points, values, 117. Undulation, poiuts of, 128. Uniform convergence, 311, 312. Value, see * Average,' ' Limits,' ' Maxi- mum,' ' Mean,' ' Mean Square,' ' Turning.' Value of TT, computation of, 313, 314. Van Vleck, E. B., 305. Variable, dependent, independent, 13, 15. change of, 145. Variation of functions, 116. total rate of, 134. Velocity, 92. Volumes, methods of finding, 199, 235, 238, 240. AVallis, 149, 246. Wentworth, see 'Analytic Geometry.' Whittaker, Modern Analysis, 15, 29, 304, 309. Williamson, see ' Calculus.' Witch of Agnesi, see ' Examples.' Wren, 247. Young, see ' Calculus.' 7 UNIVERSITY OF CALIFORNIA LIBRARY This book is DUE on the last date stamped below. § cen|« cm first day b\reirdue' , y? i jjIRH cents on fourth day overdiie , . '' '^%:?r^ e dollar on seventhday ove^li|e^':l;", -Jw^ OCT 8 194: JAN 29 SDec'SJjr 1J)48- ."•■.2 Cf RUG 3 1^^^' lAprS/Rr l7Nov'58WW >?ecD do n REC'D 03 REC'D LO LD 21-100rn-12,'46(A2012sl6)4120 r f^^SORi^S 9B3d3 THE UNIVERSITY OF CAUFORNIA LIBRARY