UNIVERSITY OF CALIFORNIA AT LOS ANGELES THE CORNELL MATHEMATICAL SERIES LUCIEN AUGUSTUS WAIT - . . GENERAL EDITOB (SENIOR PKOFESSUK OF MATHEMATICS IN CORNELL UNIVERSITY) THE CORNELL MATHEMATICAL SERIES. LUCIEN AUGUSTUS WAIT, (Senior Professor of Mathematics in Cornell University,) GENERAL EDITOR. This series is designed primarily to meet the needs of students in En- gineering and Architecture in Cornell University ; and accordingly many practical problems in illustration of the fundamental principles play an early and important part in each book. While it has been the aim to present each subject in a simple manner, yet rigor' of treatment has been regarded as more important than simplicity, and thus it is hoped that the series will be acceptable also to general students of Mathematics. The general plan and many of the details of each book were discussed at meetings of the mathematical staff. A mimeographed edition of each vol- ume was used for a term as the text-book in all classes, and the suggestions thus brought out were fully considered before the work was sent to press. The series includes the following works : ANALYTIC GEOMETRY. By J. H. TANNER and JOSEPH ALLEN. DIFFERENTIAL CALCULUS. By JAMES 'McMAHON and VIRGIL SNYDER. INTEGRAL CALCULUS. By D. A. MURRAY. ELEMENTS DIFFERENTIAL CALCULUS BY JAMES McMAHON, A.M. (DUBLIN) ASSISTANT PROFESSOR OF MATHEMATICS IN CORNELL UNIVERSITY AND VIRGIL SXYDER, PH.D. (GOTTINGEN) INSTRUCTOR IN MATHEMATICS IN CORNELL UNIVERSITY . OF CALIFORNIA ATLOS ANSLLETS LIBRARY NEW YORK : CINCINNATI : CHICAGO AMERICAN BOOK COMPANY COPYRIGHT, 1898, BY JAMES McMAHON AND VIRGIL SNYDER. ELE. DIF. CAL. W. P. 3 ' ,' '- / " . . . AJ ;. .- '.v. 3if1& - : ' .' .. ,'rr- ',-,.,.. ' r > Library Q.A PREFACE THIS book is primarily designed as a text-book for the classes in the Calculus at Cornell University and other insti- tutions in which the object and extent of the work are similar. For the engineering students at Cornell, Differen- tial Calculus is taught during the winter term of the fresh- man year ; the students are then familiar with Analytic Geometry, and many properties of the conies can be sup- posed known. When use is made of Cartesian coordinates, they are always assumed to be rectangular. As an apology for adding still another work to a field in which the literature is already extensive, it may be said that probably no other book has just the scope of this one. Many of the works are too brief, and omit rigorous proofs as being too difficult for the average student, while the more extensive treatises have too much for a student to master in the allotted time. While the chapter on fundamental principles is a long one, nothing more is introduced than is necessary for sub- sequent parts of the work ; and it is hoped that the matter is so arranged that the student will not find it difficult reading. In the chapter on expansion of functions unusual stress is laid upon convergence and the calculation of the remainder ; and numerous examples are discussed to illustrate the prin- ciples. yi PREFACE The chapter on asymptotes is perhaps unusually long, as the subject is so presented that the form of any infinite branch can be readily determined from its approximate equation, and the process is fully illustrated both in this chapter and in a later one on curve tracing. Quite a full discussion is given to the form of a curve in the vicinity of a singular point, the method of expansion being extensively used. No list of " higher plane curves " has been prepared, since the subject, as usually given, is properly a part of Analytic Geometry. A chapter on that subject is contained in the Analytic Geometry of this series. The occasional marginal references [A. G.] are to this book. No specific acknowledgments to other works have been given ; for although various works have been consulted, the main inspiration has come from the class room and from extensive consultation with our colleagues. Many of the examples have been selected from other books, but a large number are new. When original exam- ples have been taken from recent works, acknowledgment of the source is made. We acknowledge our indebtedness to the other authors of this series for their hearty cooperation ; to our colleagues, Dr. J. I. Hutchinson and Dr. G. A. Miller, for the keen interest they have taken in the work and for their assistance in verifying examples and reading proof ; to Mr. Peter Field, Fellow in Mathematics, for solving the entire list of exercises ; and to Mr. V. T. Wilson, Instructor in Drawing in Sibley College, for drawing the figures. Every figure in the book is new, and drawn to scale, except that in some cases vertical ordinates are proportionately foreshortened to fit the page. CONTENTS INTRODUCTION ARTICLE PAGE 1. Number. 1 2. Operations 1 3. Expressions 3 4. Functions 4 5. Constants and variables 7 6. Continuous variable ; continuous function 7 CHAPTER I FUNDAMENTAL PRINCIPLES 7. Limit of a variable y 8. Infinitesimals and infinites ........ 10 9. Fundamental theorems concerning infinitesimals, and limits in general .11 10. Comparison of variables 17 11. Comparison of infinitesimals, and of infinites. Orders of mag- nitude . . . . . . . . . .18 12. Order of magnitude of expressions involving infinitesimals or infinites 20 13. Useful illustrations of infinitesimals of different orders . . 25 14. Continuity of functions 28 15. Comparison of simultaneous infinitesimal increments of two related variables ......... 33 16. Definition of a derivative 37 17. Geometric illustrations of a derivative ...... 38 18. The operation of differentiation . . . . . . .41 19. Increasing and decreasing functions ...... 43 20. Algebraic test of the intervals of increasing and decreasing . 45 21. Differentiation of a function of a function 46 22. Differentiation of inverse functions 47 vii yiii CONTENTS CHAPTER II DIFFERENTIATION OF THE ELEMENTARY FORMS ARTICLE PAGE 23. Recapitulation 50 24. Differentiation of the product of a constant and a variable . 50 25. Differentiation of a sum 51 26. Differentiation of a product 52 27. Differentiation of a quotient 53 28. Differentiation of a commensurable power of a function . . 54 29. Elementary transcendental functions 58 30. Differentiation of Iog x and log a u 59 31. Differentiation of the simple exponential function . . .61 32. Differentiation of the general exponential function . . .61 33. Differentiation of an incommensurable power . . . .62 34. Differentiation of sin u 63 35. Differentiation of cos u . . .63 36. Differentiation of tan u 64 37. Differentiation of cot u . . . . ' 64 38. Differentiation of sec u 65 39. Differentiation of esc u 65 40. Differentiation of vers u ........ 65 DIFFERENTIATION OF THE INVERSE TRIGONOMETRIC FUNCTIONS 41. Differentiation of sin -1 u 66 42. Differentiation of cos" 1 u 67 43. Differentiation of tan- J u 68 44. Differentiation of cot -1 u 68 45. Differentiation of sec" 1 u 69 46. Differentiation of esc- 1 69 47. Differentiation of vers- 1 ** 70 LIST OF ELEMENTARY FORMS 71 CHAPTER m SUCCESSIVE DIFFERENTIATION 48. Definition of nth derivative 73 49. Expression for the nth derivative in certain cases . . .74 50. Leibnitz's theorem concerning the nth derivative of a product . 75 51. Successive ^-derivatives of y when neither variable is independent 77 CONTENTS IX CHAPTER IV EXPANSION OF FUNCTIONS ARTirLE PAGE 52. Introductory 81 53. Convergence and divergence of series 82 54. General test for interval of convergence ..... 83 55. Interval of equivalence. Remainder after n terms . . .86 56. Maclaurin's expansion of a function in power-series . . .87 57. Development of f(x) in powers of x a 93 58. Remainder 95 59. Rolle's theorem c . 95 60. Form of remainder in development of /(x) in powers of x a. Lagrange's form . . . . . . . . .95 61. Another expression for the remainder. Cauchy's . . .97 62. Form of remainder in Maclaurin's series 99 63. Lemma. = 0, n = oo 100 n ! 64. Remainder in the development of a*, sin z, cos x . . . . 100 65. Taylor's series 101 Proof of the binomial theorem 102 Calculation of natural logarithms .. . . . . . 105 66. Theorem of mean value. Increment of function in terms of increment of variable 107 Increment of the increment 108 lim A/ rf*y 67. Illustration : to find the development of a function when that of its derivative is known 109 Series for tan -1 z, and calculation of TT . . . . . .110 Series for sin" 1 ^, and calculation of TT 112 CHAPTER V INDETERMINATE FORMS 68. Definition of an indeterminate form 115 69. Indeterminate forms may have determinate values . . .116 70. Evaluation by transformation 118 71. Evaluation by development 119 72. Evaluation by differentiation ....... 121 73. Evaluation of the indeterminate form ~ 125 oo X CONTENTS ARTICLK PAOB 74. Evaluation of the form oo ....... 126 75. Evaluation of the form oo oo . . . . . . . 126 76. Evaluation of the form 1" ....... 130 77. Evaluation of the forms 0, 00 . ...... 130 CHAPTER VI MODE OK VARIATION OF FUNCTIONS OF ONE VARIABLE 78. Review of increasing and decreasing functions . . . 132 79. Turning values of a function ....... 132 $0. Critical values of the variable ....... 134 81. Method of determining whether ' (a;) changes sign . . 134 82. Second method of determining whether ' (x) changes sign in passing through zero ..... . 137 83. Conditions for maxima and minima derived from Taylor's theorem ........... 139 84. Application to rational polynomials . ..... 141 85. Maxima and minima occur alternately ..... 143 86. Simplifications that do not alter critical values . . . 143 87. Geometric problems in maxima and minima .... 144 CHAPTER VII RATES AND DIFFERENTIALS 88. Rates. Time as independent variable ..... 151 89. Abbreviated notation for rates ....... 154 90. Differentials often substituted for rates ..... 156 CHAPTER DIFFERENTIATION OF FUNCTIONS OF MORE THAN ONE VARIABLE 91. Definition of continuity . ...... 158 92. Rate of variation. Partial derivatives ..... 159 93. Geometric illustration ........ 160 94. Simultaneous variation of x and y; total rate of variation of 2 161 95. Language of differentials ........ 164 96. One variable a function of the other ..... 165 97. Differentiation of implicit functions ; relative variation that z constant ......... 166 CONTENTS xi ARTICLB PAGE 98. Functions of more than two variables . . . . .168 99. One or two relations between the three variables x, y, z . . 169 100. Euler's theorem ; relation between a homogeneous function and its partial derivatives 171 CHAPTER IX SUCCESSIVE PARTIAL DIFFERENTIATION 101. Successive differentiation of functions of two variables . . 173 102. Order of differentiation indifferent ...... 175 103. Extension of Taylor's theorem to expansion of two variables . 178 104. Significance of remainder . . 180 105. Form corresponding to Maclaurin's theorem .... 180 CHAPTER X MAXIMA AND MINIMA OF FUNCTIONS OF Two VARIABLES 106. Definition of maximum and minimum of functions of two variables 183 107. Determination of maxima and minima ..... 183 108. Conditional maxima and minima ...... 191 Implicit functions ......... 193 CHAPTER XI CHANGE OF THE VARIABLE 109. Interchange of dependent and independent variables . . 198 110. Change of the dependent variable ...... 199 111. Change of the independent variable 199 112. Change of two independent variables 201 113. Change of three independent variables 204 114. Application to higher derivatives 205 Xli CONTENTS APPLICATIONS TO GEOMETRY CHAPTER XII TANGENTS AND NORMALS ARTICLE PAGE 115. Geometric meaning of -^ 208 dx 116. Equation of tangent and normal at a given point . . . 208 117. Length of tangent, normal, subtangent, subnormal . . 209 POLAR COORDINATES 118. Meaning of p^- ...... ,213 dp 119. Relation between ^ and p . . . ,213 dx dp 120. Length of tangent, normal, polar subtangent, and polar sub- normal 213 CHAPTER XIII DERIVATIVE OF AN ARC, AREA, VOLUME AND SURFACE OF REVOLUTION 121. Derivative of an arc 216 122. Trigonometric meaning of , 217 d$ dy 123. Derivative of the volume of a solid of revolution . . . 218 124. Derivative of a surface of revolution 218 125. Derivative of arc in polar coordinates 219 126. Derivative of area in polar coordinates 220 CHAPTER XIV ASYMPTOTES 127. Hyperbolic and parabolic branches 221 128. Definition of a rectilinear asymptote 221 DETERMINATION OF ASYMPTOTES 129. Method of limiting intercepts 221 130. Method of inspection. Infinite ordinates, asymptotes parallel to axes 224 131. Infinite ordinates are asymptotes 226 132. Method of substitution. Oblique asymptotes . . . 227 CONTENTS Xiil ARTICI.B PAGE 133. Number of asymptotes . 229 134. Method of expansion. Explicit functions .... 230 135. Method of expansion. Implicit functions .... 234 136. Curvilinear asymptotes 236 137. Examples of asymptotes of transcendental curves . . . 237 138. Asymptotes in polar coordinates 239 139. Determination of asymptotes to polar curves .... 240 CHAPTER XV DIRECTION OF BENDING. POINTS OF INFLEXION 140. Concavity upward and downward 243 111. Algebraic test for positive and negative bending . . . 244 142. Analytical proof of the test for the direction of bending . 247 143. Concavity and convexity towards the axis .... 248 144. Concavity and convexity ; polar coordinates .... 249 CHAPTER XVI CONTACT AND CURVATURE 145. Order of contact 252 146. Number of conditions implied by contact .... 253 147. Contact of odd and of even order 254 148. Circle of curvature 255 149. Length of radius of curvature ; coordinates of renter of curvature 255 150. Second method .......... 257 151. Direction of radius of curvature 259 152. Other forms f or R 260 153. Total curvature of a given arc ; average curvature . . . 261 154. Measure of curvature at a given point 261 155. Curvature of an arc of a circle 262 156. Curvature of osculating circle ' . . 263 157. Direct derivation of the expressions for K and R in polar coordinates .......... 265 EVOLUTES AND INVOLUTES 158. Definition of an evolute 267 159. Properties of the evolute 269 xiv CONTENTS CHAPTER XVH SINGULAR POINTS ARTICLE PAOK 160. Definition of a singular point ..... . 275 1(51. Determination of singular points of algebraic curves . 275 162. Multiple points . 277 163. Cusps ......... . 278 164. Conjugate points . 281 CHAPTER XVIII CURVE TRACING 165. Statement of problem . 283 166. Trace of curve x* - y* + 6 xy* = . . 284 167. Form of a curve near the origin . . 288 -1687 Another proof ........ . 292 169. Oblique branch through origin. Expansion of y in ascend- ing powers of x . 292 170. Branches touching either axis ..... . 295 171. Two branches oblique ; a third touching r-axis . 297 172. Approximation to form of infinite branches . . 299 173. Curve tracing; polar coordinates .... . 303 CHAPTER XIX ENVELOPES 174. Family of curves . . 307 175. Envelope of a family of curves .... . 308 176. Envelope touches every curve of the family . . 309 177. Envelope of normals of a given curve . 310 178. Two parameters, one equation of condition . 311 . 314 .327 335 INTRODUCTION IN this general introductory chapter some terms of fre- quent use in subsequent work will be briefly recalled to mind and illustrated, and their meaning somewhat extended. 1. Number. Mathematics is concerned with the study of numbers and their relations to each other. Numbers are represented by letters, as a, 6, *)> tnen ( Ar t- 3) is a symmetric func- tion of x and y. If y = F(x), x is often denoted by F- l (y~), and is called the inverse ^-function of y. This notation is illustrated in connection with "inverse operations" in Art. 2. EXERCISES 1. If f(x, y) = ax* + bxy + cy*, write f(y, x) ; f(x, x) ; f(y, y). 2. What relation must exist between the coefficients in exercise 1 to make it a symmetric function ? 6 DIFFERENTIAL CALCULUS [Ixr. 3. If (x, y) = l j-ij + x' 2 + x + 4 y - 7, show that ( 1 , 2) = (2, 1 ) . Does this prove < (r, //) to be a symmetric function? 4. Let i/-O, //) = Jj; + 5# +C. Show that i/ (, y) 0, if/ (y, x) = are the equations of two perpendicular lines. \Yhat curves are represented by 5. If f(x) = 2x Vl x 2 , show that /(sin - j = sin. (x\ cos - I 6. What functions satisfy the relations (*' + .'/) = <# C r ) ' ^> (.'/) - ; exponential functions. /(*) + /(//) = f(*y) ' logarithms. ./. (-2 r) = -J ^ O) \/l - [// ( x) ] a ? sine. -, 2 1 - [xW] 2 7. If /(a;) = z 2 + 3, and F(a:)= 2 - Var, find /[F (./)], and 8. In the last example, find /'[/(.')] or f*(x), /''[F^)] or 9. With the same notation, calculate / 2 + 2/F - :} F 2 . 10. If ^(z) = *nl, show that 1 + (.r 11. Given /(x) = log 1^*, show that /(*) +/(/) =/(.J/.) . 1 + x \ 1 + ./// / 12. If / ( X ) = VI - a: 2 , what is /( Vl - a- 2 ) V Does it follow that if (x) = y, (#) = x'i Give examples of cases in which this is not true; in which it is true. 13. If /(x/y) =/(*) +/O), prove that /(I) = 0. 14. If/(i- + y) = /W +/(y), show that/(0)=0, and p/(x) = /(/^). where p is any positive integer. 15. Using the same function as in the last example, prove that f(mx) mf(x), where m is any rational fraction. 16. If y Iog 6 (x + VI + .r 2 ), express a: as a function of y. 4-6.] INTRODUCTION . 7 17. (iiven/(j:)= z , find /(), /(I), /(O) ; show that in this func- tion if (t)Y =/('>x). 18. Given xy 2 x + y = n ; show that # is not a function of x when n = 2. 19. If y = 4>( = !^ 1-1, show that x = <(y), and that ar = < 2 (z). 20. If y =f(x) = x and z =/(y), find 2 as a function of x. 5. Constants and variables. Usually, during an investiga- tion, some of the numbers that enter into it preserve their values unchanged ; while all the other numbers take a series of different values. A constant number is one that always remains the same throughout the investigation. A variable number is one that changes its value, so that at different stages it requires different numerals to express it. The word number will usually be omitted, and the words constant and variable will be used alone, in problems where it is necessary to distinguish between them. If y be expressed in terms of x by the relation y = <(af), then, if a numerical value be given to x, the corresponding value of y may be computed ; and if another value be given to #, a new value can be found for y, and so on. In this equation, both x and y are variables, but if the value of x at any instant be given, the resulting value of y is known. In such a case, x is called the independent variable, and y the dependent variable. The argument is the independent varia- ble, the function is the dependent one. 6. Continuous variable ; continuous function. When the variable, in passing from one value to another, passes through every intermediate value in order, then it is continuous. 8 DIFFERENTIAL CALCULUS [!NT. 6. A function /C*0 of a continuous variable x is called a con- tinuous function in the interval from x a to x = 6, if it lias the following properties : It remains real and finite when x takes any real value in the assigned interval. For each value of x, the function has either a single value or any number of determinate values. As x changes from m to n (two arbitrary numbers within the interval), the function f(x), if single-valued, changes from /(w) to f(n) by passing through every intermediate value, in order, at least once ; and, if /(z) is multiple-valued, each value of /(#) changes from a particular value oif(m) to a corresponding par- ticular value of f(ji)t in such a way as to pass through every intermediate value, in order, at least once. If, at a value x = A, any one of these conditions fail, the function is said to have a discontinuity at x = k. The increment taken by a variable, in passing from one value to another, is the difference obtained by subtracting the first value from the second. An increment of x will be expressed by the symbol A#. It is implied in the definition of a continuous function that for any small increment of the variable, the increment of the function is also small, and that to the variable an in- crement can always be given, so small that the correspond- ing increment of the function shall be smaller than any number that may be assigned, no matter how small. E.g., if y = f(x) is a continuous function of x in the vicinity of the value x = x v then corresponding to any number c previously assigned, another number 8 can be assigned, such that when Aa; remains numeri- cally less than 8, then Ay = f(x l + Az) /(z,) shall remain numerically less than c. (For illustrations see Art. 14.) CHAPTER I FUNDAMENTAL PRINCIPLES This chapter treats of the fundamental ideas of a limit and of an infinitesimal, and uses them to lead up to the notion of a derivative, with which the Calculus is so largely concerned. 7. Limit of a variable. If a variable take successive values that approach nearer and nearer to a given con- stant, so that the difference between the variable and the constant may become smaller than any number that can be assigned, then the constant is called the limit of the variable. This definition applies whether the variable be always greater or always less, or sometimes greater and sometimes less than the constant. E.g., the circumference of a circle is the limit of the perimeter of an inscribed polygon, and also the limit of the perimeter of a circumscribed polygon when the length of the sides is made less than any assigned number. Similarly the area of the circle is the common limit of the areas of the inscribed and circumscribed polygons. The slope of a tangent to a curve is the limit of the slope of a secant, when two points of intersection of the secant with the curve approach coincidence. Thus far the illustrations apply to either the first or second case, in which the variable is either always less or always greater than its limit. An illustration of the third case, wherein the variable may be sometimes greater and sometimes 9 10 DIFFERENTIAL CALCULUS [Cn. I. less than the constant, is furnished by a decreasing geometric progression with a negative common ratio. For instance, consider the series 1, |, + J, J, , in which the number of terms is infinite. The sum of n terms of this series approaches f as a limit, when n is taken larger and larger. The first term is 1 ; the sum of the first two terms is % ; the sum of the first three is f; of the first four, f ; and so on; and these successive sums are alternately greater and less' than , but any one of them is nearer f than any sum preceding. By taking terms enough, a sum can be reached that will differ from f by less than any number that may be assigned; for the sum of n terms of this geometric series is - = 1 [1 -(-"]. hence s n - f = - f (- $), which can evidently be made less than any assigned number by sufficiently increasing n. 8. Infinitesimals and infinites. A variable that approaches zero as a limit is an infinitesimal. In other words, an infini- tesimal is a variable that becomes smaller than any number that can be assigned. The reciprocal of an infinitesimal is then a variable that becomes larger than any number that can be assigned, and is called an infinite variable. E.f/., the number ()" which presents itself in the last illustration is an infinitesimal when n is taken larger and larger; and it's reciprocal 2" is an infinite variable. From these definitions of the words " limit " and " infini- tesimal " the following useful corollaries are immediate inferences. COR. 1. The difference between a variable and its limit is an infinitesimal variable. COR. 2. Conversely, if the difference between a constant and a variable be an infinitesimal, the:i the constant is the limit of the variable. 7-9.] FUNDAMK.\TAL PRINCIPLES 11 For convenience, the symbol = will be placed between a variable and a constant to indicate that the variable approaches the constant as a limit ; thus the symbolic form x = a is to be read " the variable ;r approaches the constant a as a limit."' The corollaries just mentioned may accordingly be sym- bolically stated thus : 1. If x = , then x = a + , wherein = ; 2. If x = a -f , and = 0, then x = a. It will appear that the chief use of Cor. 1 is to convert given " limit relations " into the form of ordinary equations, so that they may be at once combined or transformed by the laws governing the equality of numbers ; and then Cor. 2 will serve to express the final result in the original form of a limit-relation. In all cases, whether a variable actually becomes equal to its limit or not, the important property is that their difference is an infinitesimal. An infinitesimal is not necessarily in all stages of its history a small number. Its essence lies in its power of decreasing numerically, having zero for its limit, and not in the smallness of any of the constant values it may pass through. It is frequently defined as an u infinitely small quantity," but this expression should be interpreted in the above sense. Thus a constant number, however small it may be, is not an infinitesimal. 9. Fundamental theorems concerning infinitesimals, and limits in general. This article will be devoted to a rigorous treatment of the theory of limits so far as necessary to furnish a logical basis for the process of differentiation to which the chapter leads up. Theorems 1-3, which are special theorems relating to infinitesimal variables, are deduced 12 DIFFERENTIAL CALCULUS [Cu. I. immediately from the definition of an infinitesimal ; and are then used in conjunction with the corollaries of Art. 8, to establish the general theorems 4-9 relating to the limits of any variables. THEOREM 1. The product of an infinitesimal by any finite constant, k, is an infinitesimal; i.e., if = 0, then ka = 0. For, let c be any assigned number ; then, by hypothesis, a can become less than - ; hence ka can become less than c, the k arbitrary, assigned number, and is, therefore, infinitesimal. THEOREM 2. The algebraic sum of any finite number (n) of infinitesimals is an infinitesimal ; i.e., if =0, = 0, ..., then + +.-. = 0. For the sum of the n variables does not numerically ex- ceed n times the largest of them, but this product is an infinitesimal by theorem 1 ; hence the sum of the n variables is an infinitesimal. NOTE. The sum of an infinite number of infinitesimals may be infinitesimal, finite, or infinite, according to circum- stances. E.g., let a be a finite constant, and let n be a variable that becomes infinite; then _, _, - , are all infinitesimal variables; but + + to n terms = - , which is infinitesimal, 2 2 while | 1- ... to n terms = a, which is finite, n n and - + + ... to n terms =an% which is infinite. 9.] FUNDAMENTAL PRINCIPLES 13 THEOREM 3. The product of two infinitesimal variables is an infinitesimal ; i.e., if = 0, /3 = 0, then a = 0. For, let c be any assigned number < 1 ; then a, $, can each become less than c ; hence a/3 can become less than c 2 , which is less than c, since c < 1 ; thus a/3 can become less than any assigned number, and is, therefore, infinitesimal. NOTE. From theorems 1-3, it follows that, if a = 0, /3 = 0, 7 = 0, then aa + b/3 -f- cy + d/3y + eya + /a/3 + <7/37 = when a, 6, c, c?, e, /, .] FUNDAMENTAL PRINCIPLES 15 COK. If the sum of a finite number of variables (x,y,z, ...) be constant, then this constant (c) is equal to the sum of their limits ; i.e., if x- + y 4- z + ... = y+fi y provided - = 0, ^=0. x y , , . X + a X For, since -^= y+& y hence, by theorems 4, 8, '. x + a x lim -= = hm - hm y + & y y 18 DIFFERENTIAL CALCULUS [Cn. I. but, by theorems 9, 5, arid hypothesis, 1+f lira 7. = 1 ; i.+= 9 therefore, lim ^- = II in COR. If the difference (8) between two variables (x, y) be infinitesimal as to either, the limit of their ratio is 1, and conversely; * P & V r\ , 1 jC * -( i.e., it = 0, then = 1. y y For, since x y = 8, then x = y + 8, and lim - = lim ^ = lim (1 + -)=!. y # \ y/ Conversely, if - = 1, then ^ = 0. y y For, by Art. 8, Cor. 1, y y 11. Comparison of infinitesimals, and of infinites. Orders of magnitude. It has already been stated that any t\v<> variables are said to be of the same order of magnitude when the limit of their ratio is a finite number; that is to say, is neither infinite nor zero. In less precise language, two variables are of the same order of magnitude when one variable is neither infinitely larger nor infinitely smaller than the other. For instance, k$ is of the same order as ft when k is any finite number; thus a finite multiplier or 10-11.] FU.\ DAM K NTAL PlUXCIl'LES 19 divisor does not affect the order of magnitude of any variable, whether infinitesimal, finite, or infinite. In a problem involving infinitesimals, any one of them, , may be chosen as a standard of comparison as to magnitude; then is called the principal infinitesimal of the first order of smallness, and its reciprocal u~ l is the principal infinite of the first order of largeness ; 2 is called the principal infini- tesimal of the second order of smallness, and its reciprocal u~- is the principal infinite of the second order of largeness. In general, n is called the principal infinitesimal of order w, when n is either a positive integer or a positive fraction; but when n is any negative number, " is the principal infinite of the corresponding positive order ( n). An infinitesimal or infinite of order zero is a finite number. Besides the principal infinitesimal (") of the wth order, there are many other infinitesimals of the same order of smallness, for instance, any infinitesimal of the form &", in which k is any finite multiplier. To test for the order (n) of any given infinitesimal (/3) with reference to the standard infinitesimal () on which it depends, it is necessary to select an exponent w, such that im H. = k, some finite constant. a. = (J (< n E.g., to find the order of the variable Sar 4 4 a- 8 , with reference to t as the base infinitesimal. Comparing with x 2 , x 3 , x 4 , in succession : lim 3 x 4 4 x 8 lim ,., 4 N A , - ., x = - ^ - = z ^ Q (3x2 _4x)=0, not finite; lim 3X 4 4 x 8 lim x = - it" =^ i0 lim 8*-4* lim _ 4\ w not finite x ; hence 3x 4 4x 3 is an infinitesimal of the same order of smallness as that is, of the third order. DIFF. CALC. 3 20 DIFFERENTIAL CALCULUS [Cn. I. The order of largeness of an infinite variable can be tested in a similar way. For instance, if x be taken as the base infinite, let it be required to find the order of the variable Comparing with 3? and x* : lim -: lim = > -- -- =x*>(*x- 4)=oo; lim 3a* 4:e 8 _ lim hence 3a^ 4a^ is an infinite of the same order of large- ness as a?, that is, of the fourth order. The process of finding the limit of the ratio of two in- finitesimals is facilitated by the following principle, based on theorem 10 of Art. 10: The limit of the quotient of two infinitesimals is not altered by adding to them (or subtract- ing from them) any two infinitesimals of higher order, re- spectively. j-, lim 3 a: 2 4 x* _ lim 3z 2 _3 9'' x = 0~ ? ~ x = 4~~a ~ 4" This principle is sometimes called " the fundamental theo- rem of the Differential Calculus," owing to its use in the "fundamental problem" stated in Art. 15. 12. Order of magnitude of expressions involving infinitesi- mals or infinites. THEOREM 11. The product of two infinitesimals is another infinitesimal whose order is the sum of their orders. For, let , 7 be infinitesimals of orders m, n, with refer- ence to the base infinitesimal ; then, by definition, = 6, V 1 ^ 3- =. c, where 6, c are finite ; ,,m a. U ~n 11-12.] FUNDAMENTAL PRINCIPLES 21 hence, multiplying and using theorem 8, mi PJ_ _ fr c ^ a f| n ite number, ;u a + therefore fty is an infinitesimal of order m + n. COR. 1. The product of two infinites is another infinite whose order is the sum of their orders. for, let /?, 7 be infinites of orders m, n ; then, since the principal infinites of these orders are ~ m , ~", lim /3 _ 7 Km 7 a = o^;- :0 ' a = o^;- c ' therefore V 11 ,, ^ = be, a finite number ; a = U -(m+) a hence @y is of the same order as ~ (m+n) , that is, an infinite of order m + n. COR. 2. The product of an infinitesimal of order m by an infinite of order n is an infinitesimal of order m n when m > n ; but it is an infinite of order n m when n>m, hence therefore when m > n, fiy is an infinitesimal of order m w, and when n > m, fty is an infinite of order n m. THEOREM 12. The quotient of an infinitesimal of order m by an infinitesimal of order n is an infinitesimal of order m n when m > n ; but it is an infinite of order n m when n>m. TT< lira /8 T lim y r or, since _^o m = ' 0' Ci m CC n 22 DIFFERENTIAL CALCULUS [Cii. I. therefore, dividing and using theorem 9, lim 7 ^ lim 7 o hence is an infinitesimal of order m w, when m > n, and 7. it is an infinite of order n m, when n > m. COR. 1. The quotient of an infinite of order m by an infinite of order n is an infinite of order m w, when m > n ; but it is an infinitesimal of order n m when n > m. COR. 2. The ratio of two infinitesimals is finite, infini- tesimal, or infinite according as the antecedent is of the same order, a higher order, or a lower order, than the conse- quent. Con. 3. The ratio of two infinites is finite, infinitesimal, or infinite according as the antecedent is of the same order, a lower order, or a higher order, than the consequent. THEOREM 13. The order of an infinitesimal is not altered by adding or subtracting another infinitesimal of higher order. For, let /3, 7 be two infinitesimals of order w, w, in which m < n, then lim ^ _ 7 lim 7_ _ a = - 7 - a = n ~ hence lim 2 = lim 4- lim -^ but -^ is an infinitesimal of order n ?n, by theor. 12; thus lim -^ = 0, K m 12.] FC \DAMK.\TAL PRINCIPLES 23 + 7 0r and hm - = liiu ~- = 6, a'" a ' therefore /3 + 7 is an infinitesimal of the same order as /3. NOTE. The order of an infinitesimal is not altered by adding, but may be altered by subtracting, another infini- tesimal of the same order and sign. For instance, let ft = 8 2 -f 4 3 , of second order, 7 = ; > ? 2 3 , of second order, then ft 4- 7 = <> 2 + 2 3 , of second order, but /3 7 = <> 3 , of third order. COR. The sum of a finite number of infinitesimals of the same sign is an "infinitesimal of an order equal to the lowest order among the infinitesimals summed. THEOREM 14. The order of an infinite is not altered by adding or subtracting another infinite of lower order. NOTE. The order of an infinite is not altered by adding, but may be altered by subtracting, another infinite of the same order and sign. (Proof and illustration as above.) COR. The sum of a finite number of infinites of the same sign is an infinite of an order equal to the highest order among the infinites summed. THEOREM lo. The limit of the finite sum of any num- ber of infinitesimals is not altered by replacing any infini- tesimal by another that bears to it a ratio whose limit is unity. For, let ! + 2 -f " + Pni so related to the first set that lim^- 1 = 1, lim^ = 1, lim^ = 1, i 2 a n then and 0! = x + !!, $j = 2 + 6 2 tt 2' "' ftn = hence Next let rj be an infinitesimal that is numerically equal to the largest of the e's, then ejttj + e 2 2 + + e n a n \<\ rj (ctj + 2 + "' + )>* hence Taking limits and remembering that, by hypothesis, lim (j + 2 + " + a ) is finite, and lim 17 = 0, it follows that lim (ft + /3 2 + - + &) = lim (a x + 2 + + ). NOTE. This theorem may sometimes be conveniently stated as follows : the limit of the finite sum of infinitesi- mals is not altered if these infinitesimals be replaced by others which differ from them respectively by infinitesimals of higher order, f * The symbol |<| stands for " is numerically less than." (See Art. 54.) t This is called the " fundamental theorem of the Integral Calculus." 12-13.] FUNDAMENTAL PRINCIPLES 25 13. Useful illustrations of infinitesimals of different orders. lim sin 6 -, . lim tan B -. THEOREM 1. ^ ^~1; With as a center and OA = r as radius, describe the circular arc AB. Let the tangent at A meet OB produced in D; draw BG perpendicular to OA, cutting OA in C. Let the angle AOB = in radian measure, then arc AB = rB, B D CA FIG. 1. by geometry, .e. OB < arc AB < AD, r sin B < r 6 < r tan B, sin B < B < tan B. By dividing each member of these inequalities by sin B, 1 < - < sec but when 6 = 0, sec 6 = 1, Similarly, by dividing the inequalities by tan#, cos 6 < hence tan0 lim 6 _ i i lim tan _ 1 o 7i ' anci A j_ (\ r i COR. 1. The numbers 0, sin 6, tan are infinitesimals of the same order. COR. 2. The expressions sin 9 6, tan 9 B are infini- tesimal as to 0. 26 DIFFERENTIAL CALCULUS [Cn. I. THEOREM 2. If one angle 6, of a right triangle, be an infinitesimal of the first order, then the hy pot hen use r and the adjacent side x are either both finite, or they are infinitesimals of the same order ; and the opposite side y is an infinitesimal of order one higher than that of r and x. y For = cos 0, which approaches the value 1 as = 0; r hence x, r are infinitesimals of the same order; which may be the order zero. Also y = r sin 0, and sin is of order 1 ; therefore y is of order one higher than r. COR. In the same case, if be of the first order, and if / and x be of the order n, then the difference between r and x is an infinitesimal of order n + 2. r 2 sin 2 For r 2 x 2 = y z = r 2 sin 2 6, r x = r + .r but the orders of r 2 , sin 2 #, r + , are respectively 2w, 2, n ; .-. r x is of order 2n + 2-n = n + 2. THEOREM 3. The difference between the length of an infinitesimal arc of a circle and its chord is of at least the third order when the arc is the first order. For, let CD be the arc, and CB, DB, tangents at its ex- tremities ; then chord CD < arc CD < DB + BC. Let the angle BOD = Q be taken as the principal infini- tesimal ; then, since arc(7Z) = 2r#, and r is finite, arc CD is of order 1, 13.] FUND. [ MKXTA L PRINCIPLES 27 Again, since AD is of order 1 (Th. 1, Cor. 1), and angle ADB 6 is of order 1, hence DB is of (T order 1, and DB - DA is of order >j (Tli. '2. C'or.); . . arc CD chord CD is of order, at least, three. THEOREM 4. The difference between the length of any infinitesimal arc (of finite curvature), and its chord, is an infinitesimal of, at least, the third order. NOTE. The curvature is said to be finite when the limit- ing ratio of the length of a small chord to the angle between tin- tangents at its extremities is finite, and not zero. I'll us. in the present case, the chord PQ and the angle y.V/' are, by hypothesis, infinitesimals of the same order.* Let the angle TSP be the principal infinitesimal ; then, since TSP = 8QR + RPS. it follows that the greater of the latter two angles, say RQS, is of the first order, while the other may be of the first or a higher order. Also, the greater of the two segments RQ, PR, say the latter, is of the first order, while RQ may be of the first or higher order. * If TSP wriv of higher order than PQ, the curvature would be zero; if of lower order, the curvature would be infinite ; the former is the case at an inflection, the latter at a cusp. Fio. 4. 28 DIFFERENTIAL CALCULUS [Cn. I. Again, by theorem 2, QR, QS are of the same order, and PR, PS are of the same order. Now arc QP - chord QP < QS + SP - QP, [geom. i.e., <(QS - QR) + (SP - RP); but since QS - QR = QS (1 - cos /3) = 2 QS sin 2 1, 2 and, similarly, SP - RP = 2 SP sin 2 1, ,4 and, since each of these products is, at least, of the third order, hence arc QP chord QP is of, at least, the third order. EXERCISES 1. Let ABC be a triangle having a right angle at C; draw CD per- pendicular to AB, DM perpendicular to CB, EF perpendicular to DB, FG perpendicular to EB; let the angle BAG be an infinitesimal of tlie first order, AB remaining finite. Prove that : CD, CB are of order 1 ; DB, DE are of order 2 ; EB, EF, (CB - CD) are of order 3 ; FB, FG, (DB - DE) are of order 4. 2. Of what order is the area of the triangle ABC1 BCD'} CDE1 3. A straight line, of constant length, slides between two rectangular straight lines, CA A', CB'B; letAB, A'B' be two positions of the line. Show that, in the limit, when the two positions coincide, AA' _ CB BB' ' CA ' 14. Continuity of functions. From the foregoing theorems on limits, and the definition of a continuous function, the following theorems relating to continuity are easily derived, and applied to the ordinary classes of functions. THEOREM 1. If a variable approach a constant, as a limit, according to any given law, then any function of the variable 13-14.] FUNDAMENTAL PRINCIPLES 29 approaches the same function of the constant as a limit if the function be continuous for values of the variable in the vicinity of the constant. Let f(x) be a continuous function of a?, for values of x near a ; then when x = a from either side /O) = /O), regard being had to correspondence of multiple values, if any. For, let x = a + A, where A = 0; then f(a + 1i) /(a) can be made less than any assigned number from the defini- tion of a continuous function (Art. 6) ; hence /(a + /0 = /<>) = /(). Ex. Prove lim f(x) =/(lim z), i.e., the operators /, Urn, commutative. COR. Conversely, any function, /(#), is continuous in the vicinity of x = a, if, when x = a, f(x) remains real and = /(a), a finite constant. THEOREM 2. If y = /(#) be a continuous function of x in the vicinity of x = a, then the inverse function is a continuous function of y in the vicinity of the value y =/()= b. For y = f (x) can be represented by a curve which is continuous at (, 5), and is represented by the same curve in the vicinity of (a, 5).* COR. If /=/(a), then one value of x approaches the limit a. * A rigorous algebraic proof of the continuity of an inverse function will be found in the appendix. 80 DIFFERENTIAL CALCULI '* [Cn. I. THEOREM 3. Jf two functions be continuous at x = a, then their sum, difference, and product are continuous func- tions at x a, and also their quotient, provided the denomi- nator does not vanish at x= a. This follows from Th. 8, 9, Art. 9. COR. 1. The product of any finite number of functions, each of which is continuous at x a, is continuous at x = a. COR. 2. If , i and 1 + h > a", hence a x 1 < h. Thus the exponent x has been chosen so small that a? 1 is less than the assigned number, i.e., a* 1=0, and of = 1, when .r = from the positive side. The proof for the nega- tive approach follows from the identity a~ r a x = 1, and' theorem 9, p. 16. THEOREM 5. The exponential function a x is a continuous function of z, when x is not infinite, provided a is positive, i.e., a r+A ^ = 0, when h = 0. For a* +h a* = a 1 (a A - 1), but a h 1 = 0, when h = 0, by lemma, hence a* +h a* = 0, when A = 0. and of is a continuous function of x. THEOREM H. The function log a z is continuous when x lies between zero and positive infinity where a is positive. For, let y = log a z, then x = a"; hence, by theorem 5, # is a continuous function of y. when y lies between x and -f x, that is x between and +oc. Therefore, by theorem 2, y is a continuous function of x, when x lies between and -f x. 32 DIFFERENTIAL CALCULUS [Ca. I. Con. 1. If w, v be two continuous variables, then u v = a b when u = a, where a is positive, and v = b. For log u = log a, and, since v = ft, hence v log w = ft log a, that is, log w" = log a 6 , therefore w" = a*, when M = a, v = ft. [Th. 2, Cor. COR. 2. If M, w be continuous functions of x, u v is a con- tinuous function of #. (Th. 1, Cor.) COR. 3. If x be a continuous variable, 3? is a continuous function of x, when n is either commensurable or incom- mensurable. This corollary is a generalization of theorem 4. THEOREM 7. The functions sin a;, cos 2; are continuous for all finite values of x ; i.e., sin (x + A) sin x = 0, when h = 0, for sin (x + A) sin x = 2 cos (x + ^ h) sin \ A, but sin ^h = when A = 0, and cos (x + % h) is not infinite, hence sin (x -f- A) sin a; = when A = 0, that is, sin z is continuous. Similarly for cos x. EXERCISES 1. Prove that tan z, secx are continuous functions of x for all values except z = ^(2n + l)7r, n being any integer. 2. Prove that cot a;, esc x are continuous functions of x for all values except x = mr, n being any integer. 3. Find the bounds of continuity of each inverse trigonometric func- tion. Draw the graph, and show the continuity of each of the multiple values. i 4. Show that 2 X is not continuous at x = 0. Let x successively approach zero from positive and negative values. 14-15.] FUNDAMENTAL PRINCIPLES 33 15. Comparison of simultaneous infinitesimal increments of two related variables. The last few articles were con- cerned with the principles to be used in comparing any two infinitesimals. In the illustrations given, the law by which each variable approached zero was assigned, or else the two variables were connected by a fixed relation ; and the object was to find the limit of their ratio. The value of this limit gave the relative importance of the infinitesimals. In the present article the particular infinitesimals com- pared are not the principal variables (x, y) themselves, but simultaneous increments (A, k) of these variables, as they start out from given values (x r y^) and vary in an assigned manner ; as in the familiar instance of the abscissa and ordinate of a given curve. The variables x, y are then to be replaced by their equiva- lents X-L + A, y^ + k ; in which the increments A, k are them- selves variables, and can, if desired, be both made to approach zero as a limit ; for since y is supposed to be a continuous function of x, its increment can be made as small as desired by taking the increment of x sufficiently small. The determination of the limit of the ratio of k to A, as A approaches zero, subject to an assigned relation between x and ?/, is the fundamental problem of the Differential Calculus. E.g., let the relation be let XY y\ be simultaneous values of the variables #, y ; and when x changes to the value x l + A, let y change to the value y l + k ; then y +k = (x + A^ 2 = x 2 4- 2x h + A 2 ; hence k 2 x^h + A 2 . 34 DIFFEKEXriAL CALCULUS [Cn. I. This is a relation connecting the increments h, k. Here it is to be observed that the relation between the infinitesimals A, k is not directly given, but has first to In' derived from the known relation between x and y. Let it next be required to compare these simultaneous increments by finding the limit of their ratio when they approach the limit zero. By division, ^ = 2^ + ^; n hence, by Art. 9, theorem 5, lirn /" r> 16V '*! This result may be expressed in familiar language by saying that when x increases through the value r r then y increases '2x 1 times as much as x; and thus when x continues to increase uniformly, y increases more and more rapidly. For instance, when x passes through the value 4, and // through the value 16, the limit of the ratio of their incre- ments is 8, and hence y is changing 8 times as fast as x; but when x is passing through 5, and y through 25, the limit of the ratio of their increments is 10, and y is changing 10 times as fast as x. The following table will numerically illustrate the fact that the ratio of the infinitesimal increments h, k approaches nearer and nearer to some definite limit when h and k both approach the limit zero. Let x v the initial value of x, be 4 ; then y r the initial value of y, is 16. Let A, the increment of x, be 1 ; then &, the corresponding increment of y, is found from 15.] FUNDAMENTAL PRINCIPLES 35 k thus k = 9, and - = 9. Next let h be successively diminished to the values .8, .6, .4, ; then the corresponding values of k k and of - are as shown in the table : h x = 4 + h y = lQ + k ft k h 4 + 1 25 9 9 4 + .8 23.04 7.04 8.8 4 + .6 21.16 5.16 8.6 4 + .4 19.36 3.36 8.4 4 + .2 17.64 1.64 8.2 4 + .1 16.81 .81 8.1 4 + .01 16.0801 .0801 8.01 4 + h 16 + 8 h + A 2 8* + * 8-M Thus the ratio of corresponding increments takes the successive values 8.8, 8.6, 8.4, 8.2, 8.1, 8.01, , and can be brought as near to 8 as desired by taking h small enough. As another example let the relation between x and y be f = x 3 , then y? = x*, O/i + *)' 2 = (*, + *), hence, by expansion and subtraction, 2 yjc + fc 2 = 3 z*h + 3 xjfi + *, t (2 y, + k) = h (3 x* + 3 Xl h + A 2 ), k _ 3 a:, 2 + 3 Xl h + A 2 k 3 x 2 + Therefore Km T = lira - h 2 ?/, + k and, by Art. 10, theorem 10, r * _ 3 *i* . . , as h = 0. Jb 0, I>IFF. CALC. 4 36 DIFFERENTIAL CALCULUS [CH. 1. The " initial values " of x, y have been written with sub- scripts to show that only the increments (A, k) vary during the algebraic process, and also to emphasize the fact that the limit of the ratio of the simultaneous increments depends on the particular values through which the variables are pass- ing, when they are supposed to take these increments. With this understanding the subscripts will hereafter be omitted. Moreover, the increments h, k will, for greater distinctness, be denoted by the symbols A#, A?/, read "incre- ment of #," "increment of /." The symbol A is derived from the initial letter of the word difference, as the increment of a variable, in passing from one value to another, is ob- tained by subtracting the first value from the second. Ex. 1. If x 2 + y 1 a?, find lim -~. Let the initial values of the vari- ables be denoted by x, y, and let the variables take the respective incre- ments Ax, A?/, so that their new values x + Ax, y + Ay shall still satisfy the given relation, then (x + Ax) 2 + (y + Ay) 2 = 2 . By expansion, and subtraction, 2 x Ax + (Ax) 2 + 2 y Ay + (A?/) 2 = 0, hence Ax (2 x + Ax) = Ay (2 y + A?/), Ay _ _ 2 x + Ax Ax 2 y + Ay lim Ay _ lirn 2 x + Ax _ x Ax = AT and Therefore Ax = o y + A?/ ?/ The negative sign indicates that when Ax, and the ratio x : y, are positive, Ay is negative, that is, an increase in x produces a decrease in y. This may be illustrated geometrically by drawing the circle whose equation is x 2 + y 2 = a 2 (Fig. 5). prove Ex. 2. If x 2 + y = y 2 - 2 x, lim AM 2 x + 2 - 15-16.] FUNDAMEXTAL PRINCIPLES 37 Similarly when the relation between # and y is given in the explicit functional form y = (*), then y + Ay = (a: + A#), and Ay = (x + Az) <(V) = A (z)> Ay (> + Ax) <(V) hence Inn - - = Inn - A# Ax When the form of < is given, the limit of this ratio can be evaluated, and expressed as a function of x; and this function is then called the derivative of the function (#) with regard to the independent variable x. The formal definition of the derivative of a function with regard to its variable is given in the next article. 16. Definition of a derivative. If to a variable a small increment be given, and if the corresponding increment of a continuous function of the variable be determined, then the limit of the ratio of the in- crement of the function to the increment of the variable, when the latter increment approaches the limit zero, is called the derivative of the function as to the variable. Let <$>(x} be a finite and continuous function of #, and Az a small increment given to #, then the derivative of (x) as to x is lira (x) be a function of x that remains Unite and ' continuous for all values of x between certain assigned con- stants a and b ; and let the variables .r, y be taken as the rectangular coordinates of a moving' point; then the relation between x and y is represented graphically, within the assigned bounds of continuity by the curve whose equa- tion is jf .'(*). Let (x v # x ), (x v y^) be the coordinates of two points P v P 2 on this curve; then it is evident that the ratio x wherein a is the inclination angle of the secant line P^Py to the #-axis. Let P 2 be moved nearer and nearer to coinci- dence with P v so that x 2 = x r y^ = y\\ then the secant line P^PI approaches nearer and nearer to coincidence with the tangent line drawn at the point P p and the inclination angle 16-17.] FUNDAMENTAL PRINCIPLES 39 () of the secant approaches as a limit the inclination angle (0) of the tangent line. Hence, by theorem 7* and Ex. 1, Art. 14, tan = tan rf>. Thus x a x, = tan FIG. 6. when z 2 =*r y^ = y r This can also be seen from the similar triangles KSP l and The proportion MP 2 KS is true, whatever be the position of P 2 . When _P 2 ap- proaches to coincidence with P r P^M= 0, MP Z = 0, but their ratio approaches 1 , which is tan . TS It may be observed that if x z be put directly equal to x v and ?/2 t y r the ratio on the left would, in general, assume the indeterminate form -, as in other cases of find- ing the limit of the ratio of two infinitesimals ; but it has just been shown that the ratio of the infinitesimals y^ y r x 2 x l has, nevertheless, a determinate limit measured by tan 0. They are thus infinitesimals of the same order except when $ is or If the differences x 2 x v y% y-^ be denoted by Aa:, Ay, then x = x,+ Az, y* = y* + &y ; 40 btFFERENTlAL CALCULUS [CnA but, since y = (re), 2/i = 0<>i)' 2/2 = 002); hence the ratio of the simultaneous increments may be written in the various forms Aa; 2 ! 2 1 In the last form, x is regarded as the independent variable, and A# its independent increment; and the numerator is the increment of the function (#), caused by the change of x from the value x l to the value x^ -j- Arc. The limit of this ratio, as A# = 0, is the value of the derivative of the function (x), when x has the value x r Here x l stands for any assigned value of x. Tims the derivative of any continuous function < (x) is another function of x which measures the slope of the tangent to the curve y $ (#), drawn at the point whose abscissa is x. 2 Ex., Find the slope of the tangent line to the curve y = at the point (1, 2). 2 _2_ lim (x + Ax) 2 z 2 Here tan d> = A ^ . n ^ t ' - Ao; = &x lira - 2 (2 x + Ax) _ 4 ~Aa: = x 2 (x + &x) 2 ' x 8 Hence tan = 4, when x = 1 ; and the equation of the tangent line at the point (1, 2) is y - 2-= - 4 (x - 1). [Cf. A.G., Art. 53. As another illustration, let the coordinates of P be (#, y), and those of Q,(x + Arc, y + A/) ; then MN=PR = &x, and PS=RQ = Ay. Let the area OAPM be denoted by 2, then z is evidently some function of the abscissa x; also let area OA QN, = z -f As, then area MNQP = Az, is the 17-18.] FUNDAMENTAL PRINCIPLES 41 increment taken by the function z, when x takes the incre- ment Az; but MNPQ lies between the rectangles MR, MQ, hence /Az < Az < (y + Az and y<< &M0 Therefore, when Az, A#, Az all = 0, - Az hm = y. Az Thus if the ordinate and the area be each expressed as functions of the abscissa, the derivative of the area function with regard to the abscissa is equal to the ordinate function. Ex. If the area included between a curve, the axis of y, and the ordinate whose abscissa is x, be given by the equation = * find the equation of the curve. Here y = lim ** = A M ft ( +'*)'-*' = Az = t 3 x2 + 3 x ^ x + C^*)*] = 3 ^ 18. The operation of differentiation. It has been seen in a number of examples that when, on a given function $(z), the operation indicated by lim (j> (x is performed, the result of the operation is another function of x. This function may have properties similar to those of <(z), or it may be of an entirely different class. The above indicated operation is for brevity denoted by 42 DIFFERENTIAL CALCULUS [Cn. I. the symbol ^ , and the resulting derivative function by $'(x) ; thus x) _ lim (. x The process of performing this indicated operation is called the differentiation of <(#) with regard to x. The symbol * , when spoken of separately, is called the differ- d^c entiating operator, and expresses that any function written after the d is to be differentiated with regard to x, just as the symbol cos prefixed to <(V) indicates that the latter is to have a certain operation performed upon it ; namely, that of finding its cosine. The process of differentiating < (x) consists of the follow- ing steps : 1. Obtain $(x + &x) by changing x into x + kx in 2. Find A< (#) by subtracting (#) from (x -f 3. Divide this difference A(rc) by A. 4. Find the limit of the quotient ^ - when A# = 0. This series of steps should be memorized. In words, these four steps can be expressed as follows : 1. Give a small increment to the variable. 2. Compute the resulting increment of the function. 3. Divide the increment of the function by the incre- ment of the variable. 4. Obtain the limit of this quotient as the increment (Fig. 9). Most functions are increasing functions for some values of the variable, and decreas- ing functions for others. E.g., V'2 rx z 2 is an increasing function from x = to x = r, and a decreasing function from x = rtox = 2r (Fig. 10). A function is said to be an increasing function in the neighborhood of a given value of x if it increases as x increases through a small interval including this value ; similarly for a decreasing function. 19-20.] FUNDAMENTAL PRINCIPLES 45 20. Algebraic test of the intervals of increasing and de- creasing. Let y = $(x) be a function of x, and let it be real, continuous and differentiable for all values of x from a to b ; then by definition y is increasing or decreasing at a point x = x v according as i) is positive or negative, where Ax is a small positive number. The sign of this expression is not changed if it be divided by Ax, no matter how small Ax may be ; hence <(x) is an increasing or a decreasing function at the value x v accord- ing as dy _ iim \ (x) is an increasing function are the same as the intervals in which <'(x) is positive. E.g., to find the intervals in which the function <(>) = 2 x 8 - 9 x 2 + 12 x - 6 is increasing or decreasing. The derivative is '(x) = 6x 2 - ISx + 12 = Q(x - l)(x - 2); hence, as x passes from oo to 1, the derived function '(-r)> is positive and <(/) increases from ^>( oo) to <(O'i i.e., from < = to = 1; as x passes from 1 to '2, <'(x) is negative, and (x) decreases from <(!) to <('2); i.e., from 1 to 2 ; and as x passes from 2 to + oo, <'(ar) is positive, and $(x) increases from (2) to <(x); i.e., from 2 to +00. The locus of the equation y = <(z) is shown in figure 11. At points where <'(#) = 0, the function <(-r) is neither increasing nor decreasing. At such points the tangent is parallel to the axis of x. Thus in this illustration, at x = I, x = 2, the tangent is parallel to the ar-axis. FIG. 11. 46 DIFFERENTIAL CALCULUS [Cii. I. EXERCISES 1. Find the intervals of increasing and decreasing for the function (x) =x* + 2z 2 + x-4. Here '(x) = 3z 2 + 4x + 1 = (3* + l)(x + 1). The function increases from x = oo to x = 1 ; decreases from x = 1 to x | ; increases from x = | to x = GO. 2. Find the intervals of increasing and decreasing for the function y x s 2 x 2 + x 4, and show where the curve is parallel to the x-axis. 3. At how many points can the slope of the tangent to the curve ?/ = 2x 3 -3x 2 +l be 1 ? - 1 ? Find the points. 4. Compute the angle at which the following curves intersect : y = 3x 2 - 1, y = 2x 2 + 3. [Cf. A.G., p. 164. 21. Differentiation of a function of a function. Suppose that y, instead of being given directly as a function of x, is expressed as a function of another variable M, which is itself expressed as a function of x ; and let it be required to find the derivative of y with regard to the independent variable x. X,et y =/(w), in which u is a function of x. Suppose that x passes through an assigned value x l ; and let u pass through a corresponding value w x ; and y, in consequence, through a value y r When x changes to the value, x l + Az, let u and y, under the given relations, change continuously to the values u 1 + AM, y l + Ay ; then Ay_Ay Aw _f(u + Aw) /() AM A# ~ AM Aa; ~~ AM Az' hence, equating limits, dy _dy du _ df(u) du dx du dx du dx 20-22.] FUNDAMENTAL PRINCIPLES 47 in which the combination of values (x = x r u = u v y = y x ) is to be substituted. The derivative of a function of u with regard to x is equal to the product of the derivative of the function with regard to w, and the derivative of u with regard to x ; each derivative being estimated at the same combination of corresponding values of the three variables. The given functions may be multiple-valued, such as y Va 2 u' 2 , u = sin 'a:. Then when any assigned value x x is given to x, the functions u and take multiple values ; let one of the branches of u be specified ; dx and let MJ be the value of w on this branch, corresponding to x = x v When the value w t is given to w, the functions y and -^ take multiple du values; let the value of y on a specified branch be y r Then, by the theorem, one of the values of taken at x = x v multiplied by one of T the values of ~* taken at (x = x v u = MJ), will give one of the values of i (1U -3- taken at (x = x v y = y t ), and these are the respective unique values of the three derivatives taken at the specified combination (z = x v u = u v y = yj). This combination is represented geometrically in three dimen- sions by one of the points of intersection of the plane x = x v with the intersection-curve of the two surfaces that represent the given functions. Ex. 1. Given y = 3 w 2 - 1, M = 3ar 2 + 4 ; find ^. ax dy du -^- = 6w, -j- = Gx; du dx = . = 86 * = dx tin dx Ex. 2. Given v = 3u 2 -4 + 5, = 2a^-5; find dx 22. Differentiation of inverse functions. Relation between du dx -^- and -=-. When y = f (JK) is a continuous and differen- dv tiable function of a:, the symbol -- stands for the numerical 48 DIFFERENTIAL CALCULUS [C'n. I measure of the limit of the ratio of an increment of y to an assigned increment of x. Next, let y be taken as the inde- pendent variable ; then the inverse function x f~ l (y} is a continuous function of y ; and if a small increment be given to y, it is required to find the limit of the ratio of the resulting increment of x to the assigned increment of y. Let #, y have the initial values x v y v and let the variables change, subject to the given relation, so as to assume the values x l + &x, y l + Ay; Ay Az then, since - - = 1, Ax Ay hence, by the theory of limits (Art. 9, Th. 8, Cor.), dx^ = \_ dy~ dy^ dx in which the two corresponding values, x = x v y = y v are understood to be substituted. Thus if y f '(#) be a differentiate function of #, the inverse function x = f~ l (y) is a differentiate function of y, and the derivative of x with regard to y is the reciprocal of the derivative of y with regard to x, each derivative being estimated at the same pair of corresponding values of x and y. NOTE. Either variable may be a multiple-valued function of the other, as in the familiar relation, x 2 + y 2 = a 2 . dy When any value x 1 is given to #, the functions y and -r- take multiple values; and, when the corresponding value y l ftf* is given to y, the functions x and - - take multiple values. dy 22.] FUNDAMENTAL PRINCIPLES 49 d'U else One pair of values of -^- and of -=- will be reciprocal, and ax ^y * these will be their respective values for the combination (x = x v y = yj). In geometrical language, they will belong to the same point (x-p y-i) of the representative curve. Ex. From Ex. 1, p. 36, find the values of -^, at the four points dx dy ( 3, 4) on the circle x 2 + y 2 ' = 25 ; and write down the equations of the four tangents. MISCELLANEOUS EXERCISES 1. Find lira f** asx = oo; 1; 2;0. 2(z 2 + 3x + 2) 2. If n = oo , show whether the theorems of limits apply to : - H --- H (to n terms) = a ; n n i i " x a n x (to n factors) = a; 1 JL i an* x a 2 x (to n factors) = a; 111 n+l an 2 x a" 2 x a 2 x x a* = n z ' 3. Draw graphs of a", log z, log (x' 2 x), tanx. Show discontinuities. 1 1 4. What kinds of discontinuity have a x , sin -, at x = ? 5. What locus has its area proportional to the square of the abscissa? 6. Show that the perimeter of an inscribed regular n-gon equals f sin n f n 2nrsin ='2trr = 2 irr, as n oo. I n \ 7. Prove that the derivative of a constant is zero. CHAPTER II DIFFERENTIATION OF THE ELEMENTARY FORMS 23. In recent articles, the meaning of the symbol -^ dx was explained and illustrated ; and a method of expressing its value, as a function of x, was exemplified, in cases in which y was a simple algebraic function of #, by direct use of the definition. This method is not always the most convenient one in the differentiation of more complicated functions. The present chapter will be devoted to the establishment of some general rules of differentiation which will, in many cases, save the trouble of going back to the definition. The next five articles treat of the differentiation of alge- braic functions and of algebraic combinations of other differentiable functions. 24. Differentiation of the product of a constant and a vari- able. Let y = ex ; then y + &y = c (x -f Az), A# = c (x -f- Are) ex = cAr, dy _ v 50 therefore ^ = c. [Art. 9, Th. 9. dx CH. 11.23-25.] DIFFERENTIATION OF ELEMENTARY FORMS 51 COR. 1. If y cu, where u is a function of x, then, by Art. 21, d(cu) _ du m dx dx' The derivative of the product of a constant and a variable is equal to the constant multiplied by the derivative of the variable. COR. 2. The operator and the constant multiplier c are ax commutative operators. Is this true of the operators A and c ? 25. Differentiation of a sum. Let y =/(x) + (x) + i/r (x), then y -f Ay =/(x 4- Ax) + ) t x -^a; Arc x A . Ax ; + Arc) Ax therefore, by equating the limits of both members, & =/'(x) + 4>'(z) + VT'(X). [Art. 9, Th. 7. dx COR. 1. If y = M + y -f w>, in which M, v, w, are functions of x, then -f (w + v + M , )= ^ + ^ + ^. (2) >/ / '/./ '/./ + ^ - /(a?) (# + A#) in the numer- ator, this result may be re-arranged thus : Equating limits, as A# = 0, using Art. 9, theorems 7, 8, and noting that the first factor (ft (x + A#) = < (x) since (a;) is by hypothesis continuous (Art. 14), it follows that COK. 1. By writing u = (V), v =/(%*), this result can be more concisely written, The derivative of the product of two functions is equal to the sum of the products of the first factor by the derivative of the second, and the second factor by the derivative of the first. This rule for differentiating a product of two functions may be stated thus : Differentiate the product, regarding the first factor as constant, then regarding the second factor as constant, and add the two results. 25-27.] DIFFERENTIATION OF ELEMENTARY FOKM* 53 Since (MV) ^ u u, the operator is not commutative ax ax ax with a variable multiplier. COR. 2. To find the derivative of the product of three functions jr *() '()*(*) Let /(*)=* ( then # = hence ^=/ *'(*)+ ^OO/'GO, cte but /' (a;) = (a:) i/r' (a;) + ^ (a:) 0' (a:) ; hence, substituting the values for /(#),/'(:), 0' (ar) ^'(x) + 0() + (a:) f and so on, for any finite number of factors. This result can also be written in the form uv . + vw + tvu . (4) dx dx dx dx The derivative of the product of any finite number of factors is equal to the sum of the products obtained by multiplying the derivative of each factor by all the other factors. Ex. Show that the operators A and are not distributive as to multiplication. 27. Differentiation of a quotient. Let = -?Q then 54 DIFFERENTIAL CALCULUS [Cn. II. /(s + As) f(x) Ay (ftQ As) -/(a?) (x) Hence, by equating limits, dy_ = 4>& Another form of this result is du v d lu\ dx dx ,KV The derivative of the quotient of two functions is equal to the denominator multiplied by the derivative of the numerator minus the numerator multiplied by the derivative of the de- nominator, divided by the square of the denominator. 28. Differentiation of a commensurable power of a function. Let y = w n , in which u is a function of x ; then there are three cases to consider. 1. n a positive integer. 2. n a negative integer. 3. n a commensurable fraction. 1. n a positive integer. This is a particular case of (4), the factors w, v, w, all being equal. Thus dii __i du -2- = nu n J ax dx 27-28.] DIFFERENTIATION OF ELEMENTARY FORMS 55 2. n a negative integer. Let n = 7W, in which wz is a positive integer ; then 1 = u n = u = > u m and dy_ = -mu^^du b (5) , and Cage (1) dx u 2m dx - ote dy __! C?M hence = nu n dx dx 3. n a commensurable fraction. P Let n = - , where jt?, q are both integers, which may be either positive or negative ; then y = u n = u 9 ; hence y 9 = ?/ p , ff _i c?y _! du ^.e., qy q 1 -&- = pu p dx * dx Solving for the required derivative, dii p PI du " - J_ iyj du 5 * C3 *- + 2) 3- = Vl + 5 s 2 V -I -f t -f V A J,- - , UO _^ ; find -f- Vl + x 2 - Vl - x 2 dx Vl + 5 a a First, as a quotient, by (5), '-/- = (Vl+x 2 - VT^x 2 ) ( Vl + x 2 + VI -x 2 ) - ( Vl + x 2 + VT^ 2 ) ( VT+7 2 - Vl - dx +X 2 -Vl - ^ 2 ) = VT dx dx = y- (1 + ^ 2 )^ = 5 (1 + ^"^(l + * a )- rfx 2 rfa: y(2) >y(6) by (2) and (6) Similarly for the other terms. Combining the results, dx x Ex. 3 may also be worked by first rationalizing denominator. EXERCISES Find the x-derivatives of the functions in 1-10. 1. = cVx. 2- = 5. = 1-x 3. , = ( + *) Va + x Va + Vx T. y = (-2 a% + xi) \a^ + xi 8. y = (x- n)(x - 6)(x - c) 2 . 58 DIFFERENTIAL CALCULUS [On. II. 11. Given, (a + a:) 5 = a 6 + 5 a*x + 10 a 8 * 2 + 10 a 2 * 8 + 5 ax* + x 6 ; find (a + x) 4 by differentiation. 12. Show that the slope of the tangent to the curve y = x s is never negative. Show where the slope increases or decreases. 13. Given i 2 * 2 + crty 2 = a 2 6 2 , find ^ : (1) by differentiating as to x ; (2) by differentiating as to y; (3) by solving for y and differentiating as to x. 14. Show that form (1), p. 51, is a special case of (3). 29. Elementary transcendental functions. Functions that involve operations other than addition, subtraction, multi- plication, involution (with integer exponent), and evolution (with integer index) are transcendental functions [Art. 4], The most elementary transcendental functions are : Simple exponential functions, consisting of a constant number raised to a power whose exponent is variable, as 4% a**; general exponential functions, involving a variable raised to a power whose exponent is variable, as x Binx ; the logarithmic * functions, as log a rr, Iog 6 u ; the incommensurable powers of a variable, as x-si, u" ; the trigonometric functions, as sin w, cos u ; the inverse trigonometric functions, as sin" 1 M, tan -1 x. There are still other transcendental functions, but they will not be considered in this book. The next four articles treat of the logarithmic, the two exponential functions, and the incommensurable power. * The more general logarithmic function log c u is not classified separately. as it can be reduced to the quotient & aU . log.t> 28-30.] DIFFERENTIATION OF ELEMENTARY FORMS 59 30. Differentiation of log a ic and log a w. Let y = log a ar, then y + Ay = log a (a; + A#) Ay __ loga (# + AaQ log a a; For convenience writing h for A#, and re-arranging, Ay 1 a;, .*. To evaluate the expression fl-f-1 when A = 0, expand it V xj by the binomial theorem, supposing ^ to be a large positive integer m. The expansion may be written i m 1-2 ?w 2 1-2-3 which can be put in the form i_ : f -- f ~-- "" 1 2 Now as m becomes very large, the terms > ^ become 171 f fH\> very small, and when m = QO the series becomes lim /- 1Y"- 1 1 1 60 "DIFFERENTIAL CALCULUS [CH. II. The numerical value of the sum of this series can be readily calculated to any desired approximation. This sum is an important constant, which is denoted by the letter e, and is equal to 2.7182814 ..., thus H . m A + 1Y 1 = e = 2.7182814 ....* m = co V / The number e is known as the natural or Naperian base ; and logarithms to this base are called natural or Naperian logarithms. Natural logarithms will be written without a subscript, as log x ; in other bases a subscript, as in log a 2;, will generally be used to designate the base ; but the common logarithm, Iog 10 #, is often written Log x. The logarithm of e to any base a is called the modulus of the system whose base is a. If the value, i . A(! + -p = e, be substituted in the ex- i \ XJ pression for -f-, there results [Th. 6, p. 81 ; Ex. p. 29. ax dx x More generally, by Art. 21, 1 1. 1' u doc In the particular case in which a = e, The derivative of the logarithm of a function is the product of the derivative of the function and the modulus of the system of logarithms, divided by the function. * This method of obtaining e is rather too brief to be rigorous ; it assumes that is a positive integer, but that is equivalent to restricting Aa; to Ax approach zero in a particular way. It also applies the theorems of limits to the sum and product of an infinite number of terms. The proof is completed on p. 315. 30-32.] DIFFERENTIATION OF ELEMENTARY FORMS 61 31. Differentiation of the simple exponential function. Let y = a u ; then log y = u log a. Differentiating both members of this identity as to x, 1^ = log a - , by form (8)., ydx dx dy , du -f- = log a . y ; dx dx therefore ^ a = log a a" ^, (9) antl do) The derivative of an exponential function with a constant base is equal to the product of the function, the natural loga- rithm of the base, and the derivative of the exponent. 32. Differentiation of the general exponential function. Let y = w", in which u, v are both functions of x. Take the logarithm of both sides, and differentiate ; then logy = vlogw, 1 dy dv i . v du = loM _ fi dv v du~\ \_ dx u dxj ' dx therefore The derivative of an exponential function in which the base is also a variable is obtained by first differentiating, regarding 62 DIFFERENTIAL CALCULUS [CH. H. the base as constant, and, again, regarding the exponent as constant, and adding the results. 33. Differentiation of an incommensurable power. Let y = u n , in which n is an incommensurable constant ; then logy = nlogu, 1 dy n du -- 2- _ . - , y dx u dx dy y du -2. ftO 4* *^** f19\ U dx -- COSM dx = sin M (13) dx ~ tan u- dx . (H) cot u ,__2 . dtA /"1E\ = - CSC Z M- (15) dx d dx = secwtanw^. (16) ~ CSC U ft/V* . / .. ^ dx d dx 1 (fit dx sl COS"' U VI M^ dx 1 du dx ~ tan" 1 u dx d * t t-i ,/ VI u* dx 1 du 1 du dx coi u d 1 + u* dx 1 d flrg* \M/vU d uVu 2 -l d -1 d dx 1 72 DIFFERENTIAL CALCULUS [Cn. II. (18) (19) (20) (21) (22) (23) (24) , vers-it* = A - ~ (25) te V2M-M 2da5 MISCELLANEOUS EXERCISES In Ex. 1-10 find ^: dx 1. ,y = log (e x + e-*). g _ x (x\ nx 2. # = I - ) 6. y e-^cosx. 7 ,, _ r sin-'2x 3. y = log cot x. , Q I X 4. #=(z-3)e 2 * + 4a:e a! + a: + 3. ' 2z 2 - 1 9. y = sin (2 M 7) ; u = log x 2 . 10. y = e u ; u = log'x. 11. y = logs 2 + e*; s = sec<; find & rf< 12- # = -. For what values of x is y an increasing function ? a^ -f x* 13. Prove that tan- 1 ( -J- J always increases with x. 14. Show that the derivative of tan~' \/ ~ COS3: is not a function of x. 1 + cos x 15. Find at what points of the ellipse ^ + #_ = 1, the tangent cuts off equal intercepts on the axes. 16. Find -^ from the expression x^y* y*x 5 + 6 x 2 5 y -f 3 = 0. CHAPTER III SUCCESSIVE DIFFERENTIATION 48. Definition of nth derivative. When a given function y = (f>(x~) is differentiated with regard to x by the rules of Chapter II, then the result defines a new function which may itself be differentiated by the same rules. Thus, d (dy\ d .,, , ) $(%) dx\dxj dx The left-hand member is usually abbreviated to *L and the right-hand member to <"(V); thus, Differentiating again and using a similar notation, and so on for any number of differentiations. Thus the symbol \ expresses that y is to be differentiated with dx 2 regard to #, and that the resulting derivative is to be differ- entiated again ; or, in other words, that the operation dx is to be performed upon y twice in succession. Similarly, 73 74 DIFFERENTIAL CALCULUS [Cu. III. \ indicates the performance of the operation three dz? dx times upon t/, and so on. Thus the symbol -^ j s equiva- / d Y . lent to ( y. It is called the nth derivative of y with \dxj regard to x. Ex. If y = x* + sin2x, ax = 24 + 16sin2z. dx 4 49. Expression for the nth derivative in certain cases. For certain functions, a general expression for the wth derivative can be readily obtained in terms of n. Ex 1 If -- y ~ e ' dx ax - Ex. 2. If y = sin x, -^ = cos x = sin ( z + ) , dar \ 2 / If y = sin ax, ' = a" sin [ ax + n } . dx" \ II 48-50.] SUCCESSIVE DIFFERENTIATION 75 50. Leibnitz's * theorem concerning the nth derivative of a product. Let y = wv, where M, v are functions of x ; then du dv . du du dv -2- = u -f v = uv l + u-,v, where , - ax ax ax ax ax are replaced by u v v 1 for convenience ; d^u again, -&- = uv 2 + Z u 1 v 1 + w 2 w ' d 8 y . o o c? r wl _-^ = MV 4- 3 M v + 8 M v + u v. u,. = dx* \_ ' dx r ] These subscripts and coefficients thus far follow the same law as the exponents and coefficients of the binomial series. To test whether this law is true universally, assume its truth for some particular value of w, . hence, by (2), = - dt Thus, if y = (t) and x =f(t), _ _ da* Expressions similar to (1) and (3), but more complicated, can be obtained for the higher derivatives. Next let y be given directly in terms of x, and x in terms of t ; then can be first expressed in terms of a;, and the dx result in terms of t by elimination. Thus if y = 0O), =/(0 then =>'* EXERCISES 1. y = x* - 4 * + 6 x 2 - 4 x + 1 ; find ^- 2. y = (x- 3) e 21 + 4 a-e" + x ; find ^- rfx 2 3. ?/ = x 6 ; find ^L rfx 6 4. ^ = x 8 logx; find dx* 5. T/ = log (e* + e~ x ) ; find 61.] SUCCESSIVE DIFFERENTIATION 79 6. w = log x - ; find ^-. x dx* 7. y tan 2 x + 8 log cos x + 3 x 2 ; find y^- 8. y = c ai sin bx ; prove - 2 -^ + (n 2 + // 2 ) y = 0. 23. y 3 + x 3 - 3 nxy = : find '^ 24. y = -r-^- ; find '&. 4 x 1 dx" 80 DIFFERENTIAL CALCULUS [On. III. 51. 25. y = l\og x -^; find ^2. b x--a' rfz 26. y = ^ x ; find ^- dx n 27. = arV ; find ^ rfx 28. # = x 2 log.r; find ^- rfx" 29. = ^- 30. w^e^sinx; find rfj;6 31. Show that the members of equation (3), p. 78, become identical when t is replaced by x. 32. Replacing t by y, show that dx* (dx\*' \dy) Also derive this relation independently. 33. Verify this relation when y = sin x. 34. Find when the slope of the curve y = tan x increases with x; and when it decreases as x increases. 35. Show that the slope of the curve y =f(z) changes from increas- ing to decreasing when f" ( J?) changes its sign. Apply to the curves y = sin x, y = sin 2 x. CHAPTER IV EXPANSION OF FUNCTIONS 52. It is sometimes necessary to expand a given function in a series of powers of one of its variables. For instance, in order to compute and tabulate the successive numerical values of sin a: for different values of x, it is convenient to have sin x developed in a series of powers of x with coeffi- cients independent of x. Simple cases of such development have been seen in algebra ; for example, by the binomial theorem, (a + #)" = a" + na n ~ l x + - a n ~ 2 x 2 -}-', (1) and again, by ordinary division, - -.^ + .... (2) l-x It is to be observed, however, that the series is a proper representative of the function only for values of x within a certain interval ; for instance, it is shown in works on algebra that when n is not a positive integer, the identity in (1) holds only for values of x between a and + a, and that the identity in (2) holds only for values of x between 1 and +1. In each case, if a finite value outside of the stated limits be given to x, the sum of an infinite number of terms of the series will be infinite, while the function itself will be finite. In both of these examples the stated interval 81 82 DIFFERENTIAL CALCULUS [Cn. IV. of equivalence of the series and its generating' function is the same as the interval of convergence of the series itself. The general theory of the convergence and divergence of series, so far as necessary for the present purpose, is briefly outlined in the next two articles. 53. Convergence and divergence of series.* An infinite series is said to be convergent or divergent according as the sum of the first n terms of the series does or does not approach a finite limit when n is increased without limit. For example, the sum of the first n terms of the geometric series a + ax + az? + ax* + a(\ -x n ~) is *= i -' \ x First let x be numerically less than unity ; then when n is taken sufficiently large, the term x" = ; hence s n = , when w = oo. 1 x Next let x be numerically greater than unity ; then when n = oo , x n == oo ; hence, in this case s n = QO , when n = oo . Thus the given series is convergent or divergent according as x is numerically less or greater than unity. The condi- tion of convergence may then be written 1 l ' e '^ therefore the series A is convergent. DEFINITIONS. The absolute value of a real number x is its numerical value taken positively, and is written \x\. The equation | x \ = \ a \ indicates that the absolute value of x is equal to the absolute value of a. When, however, x and a are replaced by longer expressions, it is convenient to write the relation in the form x =\ a, in which the symbol | = | is read "equals in absolute value." Similarly for the symbols IS n (x) from f(x~), then in which S n (x), R n (x) are functions of n as well as of x. * In some discussions of convergence of series without any reference to a generating function, the phrase " remainder after n terms" is occasionally used in a sense different from that given above, which is the recognized usage in treating of the equivalence of a function and its development. 54-60.] . EXPA.\SIO\ OF FUNCTIONS 87 A sufficient condition for the convergence of the series is that R n (x) approach a finite limit when n =x; ; for in that case S n (x), =/(#) 7?, ( (a - ), = a finite number, when W = QC. Thus the interval of convergence extends over those values of x that make n ""^ R n (x) equal to any number not infinite, and/(#) itself not infinite. On the other hand, the interval of equivalence of the series and its generating function extends only over those values of x that make n i" 1 ^ R n (x) = ; for it is only in that case that S n (x)=f(x)^ when n =00. Thus the interval of equivalence may possibly be nar- rower than the interval of convergence. It will appear later, however, that in the case of all the ordinary functions, n "^ R H (x) will be zero for certain values of x, and infinite for all other values of x ; and that thus the intervals of convergence and of equivalence are identical. 56. Maclaurin's expansion of a function in power-series.* It will now be shown that all the developments of functions in power-series which were studied in algebra and trigo- nometry are but special cases of one general formula of expansion. It is proposed to find a formula for the expansion, in ascending positive integral powers of x, of any assigned function which, with its successive derivatives, is continuous in the vicinity of the value x = 0. * Named after Colin Maclaurin (1698-1746), who published it in his "Treatise on Fluxions" (1742) ; but he distinctly says it was known by Stirling (1690-1 7 72), who also published it in his " Methodus Differentialis " (1730), and by Taylor (see Art, 65). 88 DIFFERENTIAL CALCULUS [C H . IV. The preliminary investigation will proceed on the hypoth- esis that the assigned function f(x) has such a development, and that the latter can be treated as identical with the former for all values of x within a certain interval of equiva- lence that includes the value x = 0. From this hypothesis the coefficients of the different powers of x will be deter- mined. It will then remain to test the validity of the result by finding the conditions that must be fulfilled, in order that the series so obtained may be a proper representation of the generating function. Let the assumed identity be f(x) = A + Bx+Cx i + Dx* + Ex* + , (1) in which A, B, (7, are undetermined coefficients indepen- dent of x. Successive differentiation with regard to x supplies the following additional identities, on the hypothesis that the derivative of each series can be obtained by differentiating it term by term, and that it has some interval of equivalence with its corresponding function : f"(x)= 3 2Z> + 4 3 2 Ex + in which, by the hypothesis,* x may have any value within a certain interval including the value x = 0. * The hypothesis here made with regard to each series would not be admissible in a process of demonstration. This preliminary investigation is for the purpose of discovering what the development is, if any exists. The validity of the development is fully tested in Arts. 60-65. It may be of interest to refer to Professor Osgood's " Introduction to Infinite Series," pp. 54, 01, for a proof that within its interval of convergence a power-series is a continuous and differentiate function of x, and that its 56.] EXPANSION OF FUNCTIONS 89 The substitution of zero for x in each identity furnishes the following equations : /(0) = A, /(0)= 5, /"(0)= 2 (7, /'"(0)= 3 . 2D, ... ; " " hence A =/(0), *= The unknown coefficients of (1) are thus expressed in terms of known indicated operations ; and substitution in (1) gives the form of development sought: /O)= Here the symbol /"(O) is used to indicate the operation of differentiating /(#) with regard to x, n times in succession, and then substituting zero for x in the expression for the nth derivative. The resulting constant, when divided by n !, is the required coefficient of the nth power of the variable in the assumed development of the function. It remains to examine what are the conditions that must be fulfilled in order that the series so found may be a proper representative of the function. This question can only be fully answered when the expression for R H (x), the remainder after n terms, has been obtained. This expression will be derived after another series, which may be regarded as a generalization of (2), has been established. There are, however, certain preliminary conditions that true derivative, can be obtained, within the same interval, by differentiating the series term by term. This theorem is, however, not necessary to the demonstration of Mac- laurin's or Taylor's theorem, as the series treated in Art. 60 consists of only a finite number of terms. 90 DIFFERENTIAL CALCULUS [Cn. IV. are easily seen to be necessary in order that the series may give the true value of the function. First, the functions /(#), /'(#), /"(#), must all satisfy the condition of being continuous in the vicinity of x = ; otherwise some of the coetncients/(0j, /*(()), /"(Oj, would be infinite or indeterminate, and the series would have no definite sum for any value of #, showing that the given func- tion f(x) could have no development in the form prescribed. 5 J Ex. Show that the functions ]ogx, x*, - cannot be developed in powers of x. e* + 1 When this condition is satisfied, it is further necessary for the equivalence of the function and its development that the values of x be restricted to lie within a certain interval not wider than the interval of convergence of the series. The method of computing the coefficients of the successive powers of x in the development of a given function, will be illustrated by a few examples. Ex. 1. Expand sin x in powers of x, and find the interval of conver- gence of the series. Here f(x) = sin x, /(O) = 0, /"(*) = - sin z, /'(0) = 0, /"(z)=-cosz, /'"(O) =-1, = sin x, 7^(0) = 0, = cosx, Hence, by (2), sin 2 thus the required development is sin x = + 1 x + x 2 - X s + x* + 3! 5! 56.] EXPANSION OF FUNCTIONS 91 To find the interval of convergence of the series, use the method of Art. 54, then _ = (2 n + 1) l' (-2 n - 1) ! ~ (2 n + 1)2 n' and this ratio approaches the limit zero, when n becomes infinite, how- ever large be the constant value assigned to x. This limit being less than unity, the series is convergent for any finite value of x, and hence the interval of convergence is from x> to + *. Assuming, for the present, that the value x = .5, for example, lies within the interval of equivalence of sin x and its development, the numerical value of the sine of half a radian may be computed as follows: = .5000000 - .0208333 + .0002604 - .0000015 + .0000000 sin(.o)= .4794256 Show that the ratio of u s to 4 is ?fa; and hence that the error in stop- ping at u 4 is numerically less than w 4 [^fa + Gir) 2 + ]> < ZST u v Ex. 2. Show that the development of cos x is X 2 j.4 T 6 f _ l\nl x 2n2 cos x = 1 - + .__+... + i L? + .. 2 ! 4 ! 6 ! (2 n - 2) ! and that the interval of convergence is from GO to +00. Ex. 3. Develop the exponential functions a x , e 1 . Here /(x) = ffr, f(x) = a* log a, /"(*) = ^(log ) 2 , -f"(x) = ^(log a)", hence /(O) = 1, /'(O) = log a, /"() = (logo) 2 , /"(O) = (log n), and * = 1 + (log ) As a special case, putting a = e, the Naperian base, then log a = log e = 1, and ^ = l + , + |^ + ^ + ... + ^ + .... These series are convergent for every finite value of z. 92 DIFFERENTIAL CALCULUS [Ca. IV. Ex. 4. Find the development of tan x. Let f(x) tan x, then f'(x)= sec 2 a;, f" (a:) = 2 sec 2 a; tan x, /'" (a*) = 4: sec 2 x tan 2 a; -f- 2 sec 4 ar, /i v(x) = 8 sec 2 x tan 8 z + 16 sec 4 ar tan x, f v (x) - 16 sec 2 a; tan 4 a; + 88 sec 4 a; tan 2 x + 16 sec e ar, / VI (a-) = 32 sec 2 x tan 8 a; + 416 sec 4 x tan 3 x + 272 sec 6 a: tan x, Hence /(O) = 0, /'(O) = 1, /"(O) = 0, /'"(O) = 2, /'-(O) =0, /v(0.) = 16, /v.(0) = 0, /v"(u) = 272, ... 9 1ft 979 therefore tan x = x + a; 8 H -a; 5 + ^-^a: 7 + ... 3! 5 ! 7 ! , 17 7 , + - a; 7 + .... 315 Here, as in many other cases, the law of succession of the coefficients is very complicated, and it is not possible to express the coefficient of x n directly in terms of n. Thus the interval of convergence of the series cannot be obtained by simple methods. Ex. 5. Develop e sinx in powers of x. Let f(x)=e sinx , then /' (a-) = e sin x cos x, f"(x) = e sin - r (cos 2 a; sin a;), f"'(x) e sin - r (cos 8 x 3 sin a: cos a? cos a;), f lv (x~) = e* in T (cos 4 x 6 sin a; cos 2 a; 4 cos 2 a; + 3 sin 2 a; + sin a:), f v (x) = e sin * (cos 5 a: 10 sin x cos 8 x 10 cos 8 x + 15 sin 2 x cos x + 7 sin x cos x + cos a;), hence = 0,/"(0)= - 3,/v(0) = _ 8, therefore e sin z = l + a; + - ar 4 - x 6 + 2! 4! 5! There is no observable law of succession for the numerical coefficients, and the coefficient of x n is not expressible as a simple function of n. 56-57.] EXPANSION OF FUNCTIONS 93 57. Development of /(#) in powers of & - a. It was seen in Art. 56 that if /(#) or any of its derivatives be discontinuous in the vicinity of x = 0, then /(#) has no development in powers of x. It will be shown, however, that if these successive func- tions be continue/us in the vicinity of some other value x = a, then f(x) will have a development in powers of x a, which will be a true representative of the function for values of x within a certain interval in the vicinity of x = a. First, to find the form of such development, let /(*) = A + B(x - a) 4- C(x - a) 2 + D(x-ay* + E(x-a)*+- (1) be regarded as an identity, the coefficients A, B, C, being independent of x. With the same hypothesis for the vicinity of x a as was before made for the vicinity of x = 0, differen- tiation furnishes the additional identities : = 2(7 +3-2D(a;-a) + 4 = 3-2D +4.3. If, now, the special value a be given to x, the following equations will be obtained : /(a) =4, /(a) =5, /"(a) = 2<7, /'"(a)= 3-2D, . hence, Thus the coefficients in (1) are determined, and the re- quired development is 94 DIFFERENTIAL CALCULI'S [Cii. IV Ex. 1. Expand log x in powers of x a. Here f(x) = log x,f'(x) = -,/"(<) = - //'"(*') = -^ "" ./" hence, /(a) = log ,/'() = -./"() = - ^. ./""(") = -7" and, by (2) the required development is log x = log a + 1 O - ) - -^ (x - ay + _L(* -)_... The condition for the convergence of this series is that liin [" (x q)" +1 (x fi) n 1 , , , . n = ooL( n + l)rt+' : na n J l< ar-o|;r 7 in powers of r 1. Ex. 5. Develop %y' 2 14y + 7 in powers of y 3. 67-60.] EXPANSION OF FUNCTIONS 95 58. Remainder. The second restriction imposed upon the series in order that it may be a correct representative of the generating function, is that the remainder after n terms may bo made smaller than any given number by taking n large enough. lief ore getting the general form for this remainder it is necessary to prove the following lemma. 59. Rolle's theorem. If /(#) and its first n -f 1 deriva- tives are continuous for all values of x between a and 5, and /(a),/ (6) both vanish, then f'(x) will vanish for some value of x between a and b. By supposition f(x) cannot become infinite for any value of #, such that a < x < b ; and if /'(V) does not vanish, it must always be positive or always be negative ; hence, /(a;) must continually increase or continually decrease (Art. 20). This is impossible, as/(a)= and/(>)= 0, hence at some point x between a and 6, f(x) must cease to increase and begin to decrease, or cease to decrease and begin to increase. This point x is defined by the equation f'(x)= 0. To prove the same thing geometrically, let y =/(#) be the equation of a continuous curve, which crosses the #-axis at distances x = a, x = b from the origin ; then at some point be- tween a and b the tangent to the curve is parallel to the a>axis, since by supposition there is no discontinuity in the direction of the tangent. Hence 60. Form of remainder in development of /(a?) in powers of x - a. Let the remainder after n terms be denoted by 06 DIFFERENTIAL CALCULUS [Cu. IV. R n (x, a), which is a function of x and of a as well as of n. From the form of the succeeding terms, R n may be con- veniently written in the form R n (x, a) = ( x ~a> ^(^ a ) ? and then the problem is to determine (#, a), so that the following may be an algebraic identity : /O) =/< +/()(* - a) +?-- (x - a)" + ... in which the right-hand membej contains only the first n terms of the series, with the remainder after n terms. Thus f"(a^ /(*) -/(a) -/(a)O - a) - 7 - ( x _ a) a _ ... __ ( , (n - 1) i ( " ^rr Let a new function F(z) be defined as follows : _ __ (n 1) ! w ! in which the right-hand member is obtained from (2) by replacing a by the variable z in every term except <(#, a). This function -F(z) vanishes when z = x, by inspection ; and it also vanishes when z = a, by (2) ; hence, by Rolle's theorem, its derivative F'(z) vanishes for some value of z between x and a, say z r But x _ 2 y-i 4 . ^C g * a > ' ^ 60-61.] EXPANSION OF FUNCTIONS 97 and these terms cancel each other off in pairs except the last two ; hence ' then since F'(z) vanishes when z = z v it follows that *(*,a)=/"(*i), (4) wherein z l lies between x and a, and may thus be represented by z l = a + 0(x ), where # is a positive proper fraction. Hence from (4) and R n (. x -> a ) = n (^ " )"* w ! The complete form of the expansion of f(x) is then /(OB) = /() + /'(a) (a; - a) + ^^ (ac - a) 2 + ... (n 1)! w! in which n is any positive integer. The series may be car- ried to any desired number of terms by increasing n, and the last term in (5) gives the remainder (or error) after the first n terms of the series. The symbol /"(a + Q(x a)) indi- cates that f(x) is to be differentiated n times with regard to #, and that x is then to be replaced by a 4- 6(x a) . 61. Another expression for the remainder. Instead of putting R n (x, a) in the form * This form of the remainder was found by Lagrange (173(5-1813), who published it in the M^moires de I 1 Academic des Sciences & Berlin, 177'2. 98 DIFFERENTIAL CALCULUS [Cn. IV. it is sometimes convenient to write it in the form R n (x, a) = (x a)-fy-(x, ). Proceeding as before, the expression for F'(z) will be F ' (z) = ~ (SjT^ - 2) "~' + *^ a) ; and this is to vanish when z = z l ; hence in which Zj = a + 9(x a), x z 1 = (a: a)(l 8) ; thus *(*, a)G /-(^-")) (1 _ t , r -. (j _ 0) -,, and <*, ) = (! - g)"^ a + 0< - x ~ a)) ( - )". (n 1 ) . An example of the use of this form of remainder is fur- nished by the series for log x in powers of x a, when x a is negative, and also in Art. 64. Ex. 1. Find the interval of equivalence of logx and its development in powers of x a, when a is a positive number. Here, from Art. 57, Ex. 1, hence /( + *(, - fl) ) N and .(x,a)| = | __ ^ " fl >" _ = ![ x " " 1". '( + ^(.r - )) n . I rt + ^(r - ) I Xow the interval of equivalence is not wider than the interval of con- vergence; hence, by Art. 54, the first condition of equivalence is that x a be numerically less than a. First let x a be positive, then when * This form of the remainder was found by Oatichy (1789-1857), and first published in his " Lecons sur le calcul infinitesimal," 1826. 61-02.] EXPANSION OF FUNCTIONS 99 it lies between and a it is numerically less than a + d(x - a), since 6 is a positive proper fraction ; hence when n = oo Again, when x a is negative, and numerically less than a, the second form of the remainder must be employed. As before, (a " .[ 'L ' a _0(a-z)' The factor within the brackets is always less than 1, hence the (n l)st power can be made less than any given number, by taking n large enough. This is true for all values of x between and a. Therefore, log x and its development in powers of x a are equivalent within the interval of convergence of the series, that is, for all values of x between and 2 a. Ex. 2. Show that the development of x z in positive powers of x a holds for all values of x that make the series convergent ; that is, when x lies between and 2 a. 62. Form of remainder in Maclaurin's series. The above form of remainder is at once applicable to Maclaurin's series by putting a 0. The result is The remainder formula will now be used to show that the interval of equivalence of any one of the ordinary functions, and its development in DIFF. CALC. - 8 100 DIFFERENTIAL CALCULUS [Cn. IV. powers of a;, is co-extensive with the interval of convergence of the development itself. The following lemma will be useful in several cases. 63. Lemma. When x has any finite value, however great, and n is a positive integer, then x n = when n =00. For, let u r = , then u r+l = , r ! (r + 1) ! u r+l x and i- 1 = - -. u r r + 1 Now, however large be the assigned value of #, it is possi- ble to take r so great that r + 1 where k is some proper fraction, and then for terms subse- quent to w r , the ratio of each term to the preceding term will be less than the fixed proper fraction k ; hence, by 6 (a) of Art. 59, these successive terms approach nearer and nearer to zero as a limit. 64. Remainder in the development of a x , sin x, cos x. If f(x)= a x , then /<= a*(loga) w , f*(0x)= 9j (log a) n , and R n (. x } = a ex (\og a) n - = a ex but C a:lo g a )"j = o, when n = oo, by Art. 63 ; and a 91 is finite, n ! when z is any finite number, however great ; hence R n (x)= 0, when n = oo. 62-65.] EXPANSION OF FUNCTIONS 101 Again let f(x) = sin z, then f*(x) = sin (a; + ^) by Art. 49; hence /\fe)- Wfe + !\ and /U*) = sin/W V ^ / \ but sirif 03 + - p) never exceeds unity for any value of x or \ * / of w, hence, by Art. 63, _#,,(V)=0, when w = oo. Similarly, if /(#) = cos x, f n (x) cos ( x + - j-Y and as before, 72 n (V)=0, when w = oo. Hence the developments of a*, sin #, cos z, hold for every finite value of x. Ex. 1. If f(x) = sin x, compute /? 3 (x), when x = $ IT radians. Ex. 2. Expand sin ax by Maclaurin's theorem, and determine the remainder after 7 terms, counting the terms that have zero coefficients. Ex. 3. Show that the absolute error in stopping the series for sin x, cos x, at any term, is less than the next term of the series. Ex. 4. Show that the relative error in stopping the series for e z , at any term, is less than the next term of the series; the relative error being the ratio of the absolute error to the true value of the function to be computed. Ex. 5, Prove by expansion that e */=\ x + e ^\ x - o cos a:, e^^ x e-*^ 1 = 2V- 1 sin a:; and, hence by addition, e^-^ x = cos x + V 1 sin x. Ex. 6. From the last example, prove De Moivre's theorem : (cos x + V 1 sin x) m = cos mx + V 1 sin mar. 65. Taylor's series.* It will next be shown how to write down the development for a function of the sum of two *So named from its discoverer, Dr. Brook Taylor (1685-1731), who pub- lished it in his " Methodus Incrementorum," 1715 ; but the formula remained almost unnoticed until Lagrange completed it by finding an expression for the remainder after n terms (Art. 60). Since then it has been regarded as the most important formula in the Calculus. 102 DIFFERENTIAL CALCULUS [Cn. IV. numbers in ascending powers of either number, and also an expression for the remainder after n terms of the series. If in the identity of Art. 60, which gives an expression for f(x) in powers of x a, the letter x be everywhere replaced by x + a, then x a will be replaced by #, and the identity will assume the form =/() +/'(a> + in which x, a are any two numbers, n is any positive integer, and 6 is some positive proper fraction, which may not, how- ever, be independent of the values of the other letters. If the second form of remainder be used, the last term on the right will be replaced by (1 - 6>)"~ 1a + 6n In the identity (1) the letters x and a may be interchanged, hence the expansion for f(x + a) in powers of a is /(* + a) =/< and the second form of remainder is Rjfr ) = (!- 0)-' / ' (a? a-. Ex. 1. Expand (a + :c) w in ascending powers of #, and find the interval of equivalence. Here / (a + x) = (a + x) m , hence ) "* 65.] EXPANSION OF FUNCTIONS 103 and f'(x = mx m -\ f"(x) = m(m-l)x?*-\~., therefore (a -f x) m = a m + ma m ~ l x + " > " a ^ ; + m(m-l)...(m-n + 2) ani _ n+1:gn _ 1 + jR (w-1)! in which, from the first form of remainder, w It will first be shown that the factor m(m 1) Tw w + 1) , y ^oo when n =00 ; w! for, if it be denoted by w n , then and this ratio can, by taking w large enough, be made as near unity as may be desired; but it can never exceed unity, hence the successive values of u n will approach the limit zero or a finite number, when n = oo (Art. 54, 6 (a)). Next, the expression (x V - ) =0 when w=ao. a + 0xj if x be positive and less than a. Hence R n (a, :c) = 0, when w=oo, if x be positive and less than a. 104 DIFFERENTIAL CALCULUS [CH. iv. Since the interval of convergence is, by Art. 54, from x= a to x = a, it remains to examine the value of R n (a, x) when x is negative and numerically less than a. For this purpose it is necessary to use the second form of remainder, a + to)*" t 1)1 (a But when - is negative and less than 1, the expression a -t a is a proper fraction, hence its (n l)st power ap- \ + 6 X - a proaches zero as a limit ; and it can be shown as before that the factorial expression is not infinite. Hence J2 n (a, a;) = 0, when w=co, if x lies between a and + a. This is therefore the required interval of equivalence. Ex. 2. Expand log (x + a) in powers of x, and find the interval of equivalence. Here f(x + a) = log (x + a), , hence log (* + ")= loga + ?- - C-D-* 65.] EXPANSION OF FUNCTIONS , 105 This expansion could also be obtained from the develop- ment of log x in powers of x a, in Art. 57. Similarly, x 1 1 x n * log (a x) = log a --- - -z 2 - re 3 --- - - a 2 a 2 3 a 3 (w-l)^- 1 a* n(a 6x) n When a = 1, these series become I \r\rf f\ , _. sy* I , sv* (1 + ftp)* n ~ l x n in which, as in Ex. 1 of Art. 61, the remainder R n (x) = 0, when n = oo, if 1< x < 1. Also, by subtraction, ' which can be used for computation when x is numerically less than unity. This identity can be thrown into a form suitable for the numerical calculation of the logarithm of any number ; for, put h . , vuvu ^ , 1 # w 2w + A from which This is an identity for all positive values of n and A, since the original condition rc|<|l is replaced by - - - |<| 1, 2 n + h 106 DIFFERENTIAL CALCULUS [Cn. IV. and the latter condition is always fulfilled when n and h are positive. Suppose it is required to find log 10. This could be done by putting n = 1, Ti = 9 in the last equation, but the series thus obtained would converge too slowly to be of practical value. Let log 2 be first calculated by putting both n and h equal to 1 ; thus , ofl.L 1 !_!_! 1..1 1 1 loor 2 = 2\ I 1 ^ ! [_3 3 3* + 5 3 5 + 7 3^ J Next, put n = 8, A = 2, , 1 1 , 1 1 , 1 1 r - + -- + -. _+ The numerical work can be greatly facilitated by proper arrangement of terms. The result correct to 8 places of decimals is log 10 = 2.30258509. The student should bear in mind the distinction between theoretical arid practical convergence. Here, only theoreti- cal convergence has been considered. To make a series practically useful, R n should be so small that after ten or twelve terms it could be neglected without affecting the desired numerical approximation. Sometimes, however, the expression for R n does not lend itself easily to a numerical estimate of the error made in stopping the series at a given term. The method of comparison with a descending geo- metrical progression, stated in (6) of Art. 54, and illustrated in Ex. 1 of Art. 56, and in Exs. 1, 2, 3 of Art. 67, will be found very useful in practice. Ex. 3. Expand sin (x + y) in ascending powers of y. Hence verify that sin (x + y) = sin x cos y + cos x sin y. 65-66.] EXPANSION OF FUNCTIONS 107 66. Theorem of mean value. Increment of function in terms of increment of variable. An important special case of Taylor's theorem is f(x + *)/(*) + hf(x + 0A), (1) which is obtained by putting n = 1, in equation (2) of Art. 65, and replacing a by A. If f(x) be transposed, and Ax be written for A, the identity may be written A/O) = Ax -f'(x + 6 - Ax), (2) which expresses that : The increment of the function is equal to the increment of the variable multiplied by the value of the derivative taken at some intermediate value of x. This theorem is true whether the increments be large or small. It has a simple geometrical interpretation. Since f(x) is continuous, it can be represented by a curve whose equation is y =f(x). In Fig. 13, let x = ON x + Ax = OR f(x) = NH, f(x + Ax) = RK, then A/(V) = MK, and I . HM hence, f'(x + 0- Ax~) = tan MHK. But f'(x + 6' Ax) is the slope of the tangent at some point S between If and K\ thus the theorem of mean value expresses that at some point between H and K the tangent to the curve is parallel to the secant HK. This is self- evident, geometrically ; and has already been mentioned in 108 DIFFERENTIAL CALCULUS [Cn. IV. Ex. 1. Verify the theorem of mean value for the function f(x} = a; 2 . Here f(x + h) - (x + h)* = x 2 + h . 2 (x + Oh), which is evidently a true identity when 6 = \. In most cases the exact numerical value of the proper fraction 6 is not so apparent. When the given increment of x is small and the increment of the function is desired, it is sometimes sufficiently accurate in practical computation to replace f'(x + 0- &x) in equa- tion (2) by its approximate value f'(x), then A/ (x) = Az -/'(a:), (3) in which the error is, by Taylor's theorem, a term of the second order of smallness. A second approximation to the value of A/ (x) is given by A/< = Az -f'(x) + } (Az) 2 ./"(*), (4) in which the error is \ (Az) 3 -/'"(x + 6 Az), of the third order. The third approximation is obtained by adding the term '"^), and the error will then be Ex. 2. Compute the first, second, and third approximations to the increment of log x when x changes from 10 to 10.1. Ex. 3. Show how to compute the difference for one minute in a table of natural sines. Increment of the increment. Let y =/(#) be a function which can be developed in the vicinity of x = x 1 ; and let x have the three successive equidistant values x l A, a^, x 1 + h. When x changes from x l h to #j, let y take the increment A^ = f(x^) f(x 1 7i) = h - f (ajj) - 1 A 2 /" Oi - OK) ; and 66-67.] EXPANSION OF FUNCTIONS 109 when x changes further from x 1 to x l + A, let y take the increment let the difference of these successive increments of y be written A (Ay) or A 2 # ; 12 then ^y = ^-\y= t {_f"(x l + d'h}+f'\x l -eJi)}. (5) This result may be expressed in words thus : The incre- ment of the increment of the function, corresponding to successive equal increments of the variable, is equal to the square of the latter increment multiplied by half the sum of the values of the second derivative taken at intermediate values of the variable on each side of its middle value. This may be called the theorem of mean value for the second derivative. Ex. 4. Prove that A 2 # is an infinitesimal of the same order as (Ax) 2 . Ex. 5. Show how to compute the change in the difference for one minute in exercise 3. Limit of the ratio of A 2 ^ to (Aa:) 2 . In equation (5), replace h by Az, divide by (Aa:) 2 , and take the limit of both members as Aa: = 0, then lira A 2 y _ d?y "' 67. To find the development of a function when that of its derivative is known. Development of the anti-trigonometric functions. The derivative of an anti-trigonometric function being an algebraic binomial, it is easy to expand it by the binomial theorem ; it is now proposed to show how to use the develop- ment of the derivative to determine the coefficients in the 110 DIFFERENTIAL CALCULUS [C i. IV. development of the given function, so as to avoid the labor of successive differentiation. 1. Power-series for tan" 1 #. Assume, within an interval including x = 0, the identity tan-ia; = A + Bx + O? + Drf + Ex* + - (1) With the same preliminary hypothesis as" in Art. 56, differentiation furnishes the identity = + 2Cx + 3D3? + 4Urf+>.., (2) but, within the interval from 1 to + 1, the left member is identical with hence, within a certain interval including x = 0, there exists the identity 1 x z +x^ aP+-"=JB+2 Cz+SDz z +4:JiJ'j?-)- .... (3) therefore B=\, (7=0, D = - , E = 0, F = |, .... The first coefficient A is found to be zero by putting x = in (1), hence 4~OT""~'l /y /* _ , 1 /yO I 1 /v& 1 *7l J. / 4- (.(ill / ~^ */ "S" ^ ^^ "Y" *t^ ~^f >^ * V / The interval of convergence of this series, found by the usual method, is from x = lto#=l. To show that this is also the interval of equivalence of the function and the series, and thus to establish the validity of the development, let R n (x) denote the remainder obtained by subtracting the sum of the first n terms from the func- tion, then n ( x \ (5) 67.] EXPANSION OF FUNCTIONS 111 hence by differentiation with regard to 2;, therefore R n '(x) is the remainder after n terms obtained by dividing 1 by 1 + re 2 , ~2m *'(*> =1^5- (7) By the theorem of mean value, Art. 66, R n (x) = 7? n (0) + xRJ(Ox), [0 < < 1 but, from (5), #(())= ; and when x is less than 1, 0x is less than 1, hence, by (7), R n '(0x) 0, when n = oo ; therefore n 1 V n co ^(aj) = 0. Thus the interval of equivalence is from 1 to -f 1. Ex. 1. Compute tan -1 J, tan- 1 , tan" 1 1 ; and hence the value of ir. tsn-4 = i - = + .5 - .0416667 + .0062500 -.0011162 + .0002170 - .0000444 + .0000095 - .0000023 + .0000005 - .0000001 .4636473 + radians. = + .3333333 - .0123457 + .0008230 - .0000653 + .0000056 - .0000005 .3217506+ radians. 112 DIFFERENTIAL CALCULUS [On. IV. To find tan -1 1, use the formula : tan- 1 - + tan- 1 - = tan- 1 2 3 hence ? = .4;}:J6473 + .3217506 = .7853979 + ., 4 and TT = 3.1415916 .... In the first series, to estimate roughly the error made by stopping at the tenth term, it may be observed that the ratio of any term to the pre- ceding is numerically less than , and approaches \ as a limit ; hence, if all the terms after M IO were positive, their sum would be less than the geometric series io(z + i + i +-), which is less than 10 ; moreover, since the alternate terms are negative, it follows that the error made in stopping at u llt is really much less" than w 10 , and is thus less than one unit in the seventh decimal place. Similarly, in the second series, the error is much less than (* + * + & + )> i.e., less than ^6, or less than 2J units in the seventh place. Thus the error in the value of is less than 3| such units, and the error in the 4 value of TT is less than 1J units in the sixth place. Therefore the numerical value of TT lies between 3.1415916 and 3.1415931. 2. Power-series for siri 1 a:. Proceed as before, and use the development (1 -*)-* = 1+1^ + 1. 3^ ..., - _! , 1 v? . I 3 z 5 , 1 3 5 x 7 . then sm .,. + -. _ + i ... - + _._._._+..., (2) in which the interval of convergence is from 1 to 1. Let R n (x) be the remainder after n terms in (2), then by differentiation, R n '(x) is the remainder after n terms in (1), but R n (x) = R n (Q) + xR n '( z) = z/CO*), [0 67.] EXPANSION OF FUNCTIONS 113 and, by Art. 65, Ex. 1, R' n (ex) = when n = , if x |<| 1 ; hence R n (x) = ; and the interval of equivalence in (2) is from 1 to 1. Ex. 2. Compute sin" 1 ^), and hence obtain the numerical value of TT. | = sin-(i) = 1 + I i (D 3 + i i i(i) 6 + 1-1-1 -Hi) 7 + - = .5000000000 -f .0208333333 + .0023437500 . + .0003487723 + .0000593397 + .0000109239 + .0000021183 + .0000004262 + .0000000881 + .0000000186 .5235987704 ; hence TT = 3.1415926224+. Here each term may be used to obtain the next by applying as a factor the corresponding term in the series of ratios : 1 .1 3-3 5-5 7-7 9.9 11.11 2.3-4' 4.5-4' 6.7-4' 8-9-4' 10- 11 -4' 12- 13- 4' To determine the maximum error made by stopping at the tenth term, it is evident that the ratio of each term to the preceding is less than \, and approaches \ as a limit; therefore the sum of the remaining terms is less than that is, less than \ u, . Hence the error in the value of \ TT is less than 63 units in the tenth place, and the error in the above value of TT is less than 378 units in the tenth place. Thus the numerical value of TT lies between 3.1415926224 and 3.1415926602.* Ex. 3. Show that the error made by stopping at any term in the series for log 10, Ex. 2, Art. 65, is less than ^ of the last term used. * Both of these formulas for v were found by Montferier. The correct value to ten places is 3.1415926536. By various methods mathematicians have carried the approximation to a much larger number of places. Mr. Shanks, of Durham, England, published tbe value of ir to 607 places in 1853. No other constant has so much engaged the attention of mathematicians. See "Famous Problems in Elementary Geometry," by Professor Klein, translated by Professors Beman and Smith, 1897. 114 DIFFERENTIAL CALCULUS [Cn. IV. EXERCISES Derive the following expansions: 2. , r> 3. cos 2 x 1 x 2 + - --- - + R. 4! b! T r 4. e'cos a: = 1 + x - - - - + /?. 6. r 4 O r 6 7. sin 2 x = a; 2 - + - - + R. o 3 2 5 8. VI +4 x + 12 x- = 1 + 2 x + 4 ar 2 + R. a 2 a 3 9. cos (x + a) = cos a: a sin ar -- cos x + sin x + R. 10. log sin (x + a) = log sin x + a cot x - esc 2 x + ^^- + .R. 2 3 sin 8 a: 9 r8 11. e 1 sec x = 1 + x + x 2 + =f- + #. o 12. log(l + **) = Iog2 + | + g - ^ + ^. 13. cot- 1 * = \TT-X + AX S - Jx 5 + [x; 14. cofc-ij:=i--L + _L-.... [a- x 3 a: 8 o x 6 . ,1 1,1 1,13 1 16. csc^ 1 x = sm- 1 -=- + -- i + o'7'^-7+ . ar a: 23 x 8 24ox 5 17. Expand cos^a; in powers of x\ see" 1 x in powers of ar* 1 . CHAPTER V INDETERMINATE FORMS 68. Hitherto the values of a given function / (a;), corre- sponding to assigned values of the variable x, have been obtained by direct substitution. The function may,- how- ever, involve the variable in such a way that for certain values of the latter the corresponding values of the function cannot be found by mere substitution. For example, the function e* - e-* sin a; for the value x = 0, assumes the form -, and the correspond- ing value of the function is thus not directly determined. In such a case the expression for the function is said to assume an indeterminate form for the assigned value of the variable. The example just given illustrates the indeterminateness of most frequent occurrence ; namely, that in which the given function is the quotient of two other functions that vanish for the same value of the variable. Thus if /(*) = -M, and if, when x takes the special value a, the functions (x) and -^r (x) both vanish, then 2i r \ cosf + A j \^ / . sin A 6 . b 1-cosA A 2 A 4 , A A 3 which = T oo according as A=0 from the positive or negative side, hence lim cos a; __ ^ 72. Evaluation by differentiation. Let the given function be of the form * ^ x ' , and suppose that/(a) = 0, (a) = 0. It O) is required to find u fel. a 0(^) As before, let f(x), (j>(x~) be developed in the vicinity of x = a, by expanding them in powers of x a, then 122 DIFFERENTIAL CALCULUS [Cn. V. By dividing by x a and then letting x = a, it follows that lim /Cap _/'O) *-*(*) '()' The functions /'(), <#>'() will in general both be finite. If /'(a) = 0, <'O) * 0, then ^ = 0. Pvv If /'(a) ^ 0, <'< = 0, then ^- = oo. If f'(a) and <'(<*) are both zero, the limiting value of f /Vt J ^ ' is to be obtained by carrying Taylor's development . one term further, removing the common factor (x a) 2 , and ftf(a) then letting x = a. The result is Similarly, if /(a), /'(a), /"(a); <#>(a), ^'(a), "() all vanish, it is proved in the same manner that and so on, until a result is obtained that is not indeterminate in form. Hence the rule: To evaluate an expression of the form -, differentiate nume- rator and denominator separately ; substitute the critical value of x in their derivatives, and equate the quotient of the deriva- tives to the indeterminate form. Ex. 1. Evaluate 1 -c 8 ^ w i, en = 0. Put then /'(0)=si and /'(0)=0, 72.] INDETERMINATE FORMS 123 Again, /" (0) = cos 0, " (0) = 2, hence lim 1- Ex, 2. Find X U = 0* H " g* + e lira e* e~ x 2 sin z ~~x = 4^8 _ lim e z +e~* 2 cos a: ~a: = 12 x 2 lim e 1 e~ J + 2sin x ~ x = U 04 x lim e z + e~*+ 2 cosx = -, in whichever wav ? = 0. Ex. 3. Find Ex. 4. Find " x- = ! x In this example, show that x 1 is a factor of both numerator and denominator. TTV R T?^^ li m Stanx 3z Ex.5. Fmd ^^o -- ___ In applying this process to particular problems, the work can often be shortened by evaluating a non- vanish ing factor in either numerator or denominator before performing the differentiation. Ex.6. Find Mm (*-4)tan* x = x _ lim (a; - 4 ) 2 sec 2 a: + 2 fa; 4)tan a; ~~x = ^ = 16. 124 DIFFERENTIAL CALCULUS [Cn. V. The example shows that it is unnecessary to differentiate the factor (x 4) 2 , as the coefficient of its derivative vanishes. In general, if/(V) = i/r (V)%(V), and if <(#) = (), then lim /O) f or i/r(a) = 0. Otherwise thus : lim Y(x)X(. x ) _ lim x N t lira YW _ /-^ E, p. , lim sin a: cos 2 a: Ex. 7. Find . -^ ^7=. Ex.8. Find m = 1 There are other indeterminate forms than - ; they are -, QO oo, 0, 1*, 00. The form is not indeterminate, the value of the function being evidently zero. The form QO GO may be finite, zero, or infinite. For instance, consider Vz 2 -f ax x for the value x = GO ; it is of the form QO GO , but by multiplying and dividing by Va; 2 + ax + x it becomes ax -- , which has the form V# 2 + ax 4- x when x = QO . QO Again, by dividing both terms by re, it takes the form and this becomes when x = QO , X 72-73.] INDETERMINATE FORMS 125 73. Evaluation of the indeterminate form ^. Let the function -L- become when a; = a ; it is required (p( 2? ) QO to find j 1 -^?- This function can be written 1 /CO which takes the form - when x = a, and can therefore be evaluated by the preceding rule. When x = a, *(*; JP Dividing through by ,, ^, it becomes therefore =. (2) This is exactly the same result as was obtained for the form - ; hence the procedure for evaluating the indetenm- QO . ,, . , ,, nate forms ;:> 1S * ne same in both cases. oo When the true value of ^ is or oo, equation (1) is -pi f \ satisfied, independent of the value of * ^2; but (2) still rGO 126 DIFFERENTIAL CALCULUS [Cn. V gives the correct form ; for suppose x ^-ZA^Z = ; and con- sider the function /OO , _/00+*00 which has the form when x = a, and has the determinate 00 value c, which is not zero ; hence by (2) Jim /OQ + Kx) /(a) + c'(a) /QO. * = a ( <'(a) <'() therefore, by subtracting c, lira /00 = /'OQ * = <) ^(a)' If 1^ /M = oo, then Jil"^ = 0, which can be treated 00 B vw as the previous case. 74. Evaluation of the form oo 0. Let the function be <(V) * "^(XX such that <() = oo. f(a)=0. This may be written ^^ which takes the form i(x) when a is substituted for x, and therefore comes under the above rule. (Art. 72.) 75. Evaluation of the form oo oo. There is here no general rule of procedure as in the previous cases, but by means of transformations and proper grouping of terms it is often possible to bring it into one of the forms -, Frequently a function which becomes oo oo for a critical value of x can be put in the form u t v w^ 73-75.] INDETERMINATE FORMS 127 in which v, w become zero ; and this equals uw vt vw which is then of the form -. Ex. 1. Find h "\ (sec x - tan *). This expression assumes the form oo GO, but can be written 1 sin x _ 1 sin x cos x cos x cos x which is of the form -, and gives, when evaluated, Ex. 2. Prove l , (sec n x tan"x) =ce, 1, according as n>2, = 2, <2. EXERCISES Evaluate the following expressions, both by expansion and also by differentiation; examine both modes of approaching limits: 1. o g x when x 1 2. sinx log (2x*-l) tan (x 1) lo sin x _ ^ (7T-2X) 3 ~2 5. - 15x 2 + 24x-10 (Evaluate also without the use of derivatives.) e* - e -* - 2 x o. ^ x sin x 7. *- 2 128 8. DIFFERENTIAL CALCULUS [On. V when x = 0. x = Q. x sm-'x 11. 13. x tan x x x sinx 1 x + log x 1 - V2 x - x 2 m sin x sin mx x (cos x cos mx) x 2 1 cos mx a x x = 0. x = 0. 16. 17. 18. Vl + x - Vl + x 3 V'2 sin x cosx log sin 2 x '1 -x 20. 21. tan x a; secx sec 3x 22. esc (ma~ x ) 23. I = 4 ' = 2- INDETERMINATE FORMS 129 24. e z sin- when x = co. 25. J^"-^ __ir 26. a* 2 a f x + sinx 4 sin - ) ( 3 + cos x 4 cos - j tan 5 x 2 x = a. I 27. e i(l -logx) z = 0. 28. log (x a) tan (x a) x = a. e Unx ~~ 2' 30. (1 - x) tan S or = 1 8L *(*+-)_ 2." =Q> (e - I) 3 (e z - 1) O 3- 32. - - cot - x = 0. x 33. = x = l. x 1 logx 34. cot 2 a: x = 0. sin 3 x 35. *- ^ x = 0. 4x 2 *(-* + !) 36. Prove that if /(a) = 1, <^(a) = 1, lim log/(X) /'(a) 37. 2*sin^- x = oo. 38. Vo 2 " 39. 4 ^ x = 0. 130 DIFFERENTIAL CALCULUS [Cn. V. 76. Evaluation of the form 1*. Let the function u = [$(x)'Y' (x) assume the form 1 when x = a. To make the exponent a multiplier, take the logarithm of both sides ; then logw = +&) -log* CO = This expression assumes the form - when x = a, and can be evaluated by the method of Art. 72. If the reduced value of this fraction be denoted by w, then log u = m and u = e m . NOTE. The form 1 is not indeterminate, but is equal to 1. For, let [<] * ( ^ then log u = yjr (x) log [$ (V)], which equals zero when x = a ; hence log u = 0, u = e = 1. 77. Evaluation of the forms 0, 00. Let [<}> (x)]^ become 00 when x = a. Put M = *<*>,- then log u = ^ (x) log < < = too This is of the form , and can be evaluated by the method 00 of Art. 72. Similarly for the form 0. NOTE. The form 0* is not indeterminate. For, let u = [<^>(a;)] 1/ ' (;r) become ao when x a, then log u = -fy(x) log (x) = T oo, and u = e* x = or oo. This completes the list of ordinary indeterminate forms. 76-77.] INDETERMINATE FORMS 131 The evaluation of all of them depends upon the same principle, namely, that each form (or its logarithm) may be brought to the form -, and then evaluated by differentiating numerator and denominator separately. In finally letting x = a, the two directions of approach should be compared, so as to reveal any discontinuity in the function. EXERCISES Evaluate the following indeterminate forms: 1. (cos ar)""* 1 when x = 0. 2. (cos x) C8C ** x = 0. 4. (1 - *)S 5. x^ 6. I 7. (1 a;)* x = co. 11 i i 9. (x a) z ~" when x = a from either side. BIFF. TALC. 10 CHAPTER VI MODE OF VARIATION OF FUNCTIONS OF ONE VARIABLE 78. In this chapter methods of exhibiting the march or mode of variation of functions, as the variable takes all values in succession from oo to +00, will be discussed. Simple examples have been given in Art. 19 of the use that can be made of the derivative function '(#) for this purpose. The fundamental principle employed is that when x in- creases through the value a, (#) increases through the value <() if <'(#) is positive, and that <(#) decreases through the value $() if $'(#) is negative. Thus the question of finding whether (x) increases or decreases through an assigned value <(), is reduced to determining the sign of '(). Ex. 1. Find whether the function <(V)=.r 2 -4:r + 5 increases or decreases through the values (3) = 2, $(0) = 5, $(2) = 1, (^ 1) = 10, and state at what value of x the function ceases to increase and begins to decrease, or conversely. 79. Turning values of a function. It follows that the values of #, at which $(#) ceases to increase and begins to decrease are those at which <'(V) changes sign from positive to negative ; and that the values of x, at which $(2;) ceases to decrease and begins to increase, are those at which '(#) changes its sign from negative to positive. In the former case, (x) is said to pass through a maximum, in the latter, a minimum value. CH. VI. 78-79.] VARIATION OF FUNCTIONS 133 FIG. 15. Ex. 2. Find the turning values of the function (x) = 2 x 3 - 3 z 2 - 12 x + 4, and exhibit the general march of the function by sketch- ing the curve y = (x~). Here <'(*) = 6 x" - 6 x - 12, = 6(x + 1) (x - 2), hence '(x) is negative when x lies between -1 and +2, and positive for all other values of x. Thus (x) increases from x = -co to x = - 1, decreases from x = - 1 to x = 2 and increases from x = 2 to x = oo. Hence (- 1) is a maximum ralue of <(.t), and <(2) a minimum. The general form of the curve y = (x) (Fig. 15) may be inferred from the last statement, and from the following simultaneous values of x and tj : x = - oo, - 2, - 1, 0, 1, 2, 3, 4, oo. y = - oo, 0, 11, 4, - 9, - 16, - 5, 36, oo. Ex. 3. Exhibit the march of the function especially its turning values. P i Since 1 . Thus ^>(1) = 2 is a minimum turning value of <(#); and Fio. 16. the graph of the function is as shown in Fig. 16, with a vertical tangent at the point (1, 2). Ex. 4. Examine for maxima and minima the function Here I \ S J n> 3 (X-I)I hence <'(x) never changes sign, but is always positive. Thus there is no turning value. The curve y = (x) has a vertical tangent at the point (1, 1), since -^ = '(x) is infinite when x 1. (Fig. 17.) X FIG. 17. 134 DIFFERENTIAL CALCULUS [Cn. VI. 80. Critical values of the variable. It has been shown that the necessary and sufficient condition for a turning value of <(#) is that '(x) passes through zero or infinity, are called critical values of #, be- cause they are to be further examined to determine whether <'(#) actually changes sign as x passes through these values ; and whether, in consequence, <$>(x) passes through a turning value. For instance, in Ex. 2, the derivative (f>'(x) vanishes when x = 1, and when x = 2, and it does not become in- finite for any finite value of x. Thus the critical values are 1, 2 ; and it is found that both give turning values to <(#). Again, in Exs. 3, 4, the critical value is x= 1, since it makes '(x) infinite, and it gives a turning value to $(x) in Ex. 3, but not in Ex. 4. 81. Method of determining whether <'(V) changes its sign in passing through zero or infinity. Let a be a critical value of x, in other words let <'() be either zero or infinite, and let A be a very small positive number ; then a h and a + h are two numbers very close to a, and on opposite sides of it ; thus in order to determine whether '(x) changes sign as x increases through the value a, it is only necessary to compare the signs of '(a + 7t) and <'(a A). If it is possible to take h so small that '(a h) 80-81.] VARIATION OF FUNCTIONS 135 is positive and '(a + h) negative, then (f>'(x) changes sign as x passes through the value a, and $(x) passes through a maximum value (). Similarly, if <'(a h') is negative and <'(a + Ji) positive, then (x) passes through a minimum value $(a). If <'(a A) and <'( + A) have the same sign, however small h may be, then <() is not a turning value of ${x). Ex. 5. Find the turning values of the function Here <'(*) = 2 (x - l)(x + I) 3 + 3 (x - \)\x + l) a '= (x -!)(*+ 1) 2 (5 x-1), hence <'( J; ) P* 8868 through zero at a: = 1, 1, and 1 ; and it does not become infinite for any finite value of x. Thus, the critical values are !,,!. When x = 1 h, the three factors of '(x) take the signs + , and when x = 1 + A, they become + ; thus 4>'(x) does not change sign as x increases through 1 ; hence ( 1) = is not a turning value of < (j:). When x = $ A, the three factors of '(x) are + , and when x = i + A, they become 4- +; thus <'( x ) changes sign from + to as x increases through , and () = 1 1 is a maximum value of (x). Finally, when x= 1 A, the three factors of <'(x) have the signs h +, and when x = 1 + A they become + + -f ; thus <'(*) changes sign from to + as x increases through 1, and <(!) = is a minimum value of $(x). The deportment of the function and its first derivative in the vicinity of the critical values may be tabulated thus : I-M X -1-A - 1 -1-f A $-* i i+ A 1-A 1 *'<*> + + + - - (f)(x) inc. infl. inc. inc. max. dec. dec. rain. 1.1 The general march of the function may be exhibited graphically by tracing the curve y = (x) (Fig. 18), using the foregoing result and also the following simultaneous values of x and y : x = - oo, - 2, - 1, 0, J, 1, 2, oo. jy = -co, - 9, 0, 1, 1^, 0, 27. oo 136 DIFFEREN TIAL CALC UL US [Ca. VI. FIG. 18. Ex. 6. Show the march of the function '(x) = 2 sin x cos 2 x sin 8 a: = sin x (2 cos 2 a; sin 2 ar), hence the critical values of x are found from the equations sin x = 0, and 2 cos 2 x sin 2 x = 0, or tan x = Thus the critical values of x are x = 0, x IT, x = 2ir and x = a, TT a, 27r a, ... where a = tan" 1 A/2 = .85 radians. When x = h, the factors of ^'(a:) are , +, 3 = 0, 0, +, x= + h, +, +; thus <'(#) changes from to + as a; increases through zero, and < (0) = is a minimum value of < (x). When x TT h, the factors of ^'(^ are +, +, x = TT, 0, + , x = TT + h, , + ; thus '(z) changes from + to at x IT, and < (?r) = is a maximum value of < (x). Similarly, '(x) changes from to + at x = 2ir, and < (27r) =0 is a minimum value of <>#. 81-82.] VARIATION OF FUNCTIONS 137 Again, when x = a h, the factors of '(x) are + , +, a? = o, +,0, x -a + h, + , - (Observe that when x increases to a + h, cos x diminishes, and sin x increases; thus the zero factor at x = a becomes negative at x = a + h. Similarly, it becomes positive at x = a h.) Thus <'(z) changes from + to at x = a, and (x) has a maximum value at When x = TT a h, the factors of '(x) are +, , * = TT - a, +,0, x = TT a + h, +, 4-. "(Observe that since IT a is in the second quarter, diminishing TT a increases the sine and diminishes the cosine numerically, and thus changes the zero factor to negative.) Thus '(x) changes from to -f as x increases through IT a, and <(TT a) is a minimum value of <(#) It may be shown in the same manner that <(TT + a) is a minimum, (2 IT a) a maximum, and so on. Combining the two sets of results, the form of the curve is found to be that of the accompanying figure (Fig. 19). FIG. 19. 82. Second method of determining whether <|>'(a?) changes sign in passing through zero. The following method may be employed when the function and its derivatives are continu- ous in the vicinity of the critical value x = a. Suppose, when x increases through the value a, that <'(a;) changes sign from positive through zero to negative. Its change from positive to zero is a decrease, and so is the change 138 DIFFERENTIAL CALCULUS [CH. VI. from zero to negative ; thus <'(#) is a decreasing function at # = a, and hence its derivative, "(x), is negative at xa. On the other hand, if $ (x) changes sign from negative through zero to positive, it is an increasing function, and $"(x) is positive at x = a; hence : The function $(x) has a maximum value <(#), when'(a) = and (f>"(a) is negative; (#) has a minimum value (#), when <'(a)= and $"(a) is positive. It may happen, however, that "(#) is also zero. In this case, to determine whether <(#) has a turning value, it is necessary to proceed to the higher derivatives. If <(V) * s a maximum, "(x) is negative just before vanish- ing, and negative just after, for the reason given above ; but the change from negative to zero is an increase, and the change from zero to negative is a decrease ; thus ''(x) changes from increasing to decreasing as x passes through a. Hence <>'"(V) changes sign from positive through zero to negative, and it follows, as before, that its derivative, < IV (a;), is negative. Thus <() is a maximum value of (x) if $'() = 0, <"(a)=0, <^ / "(a.)=0, < IV O) negative. Similarly, <(a) is a minimum value of (x) if '(a)= 0, <"()= 0, <'"(a)= 0, and $ IV () positive. If it happen that $ IV (a) = 0, it is necessary to proceed to still higher derivatives to test for turning values. The result may then be generalized thus : The function $ (x) has a maximum (or minimum) value at x a if one or more of the derivatives '(), "(), <"'() vanish and if the first one that does not vanish is of even order, and negative (or positive). 82-83.] VARIATION OF FUNCTIONS 139 Ex. 7. Find the critical values of Ex. 5 by the second method. "(!) = 16, hence <(!) * s a uiiniiaum value of <(*: <"( 1)=0, hence it is necessary to find <'"( 1), <'"( 1)=24, hence <( 1) is neither a maximum nor a minimum value of " (-} = 5 (- l\(- + I } is negative, hence (-) is a maximum value of <(z). Ex. 8. Examine similarly the critical values of Ex. 6. In this case the second derivative reduces to "(x) =cos x(2 cos 2 * -7 sin 2 a:), hence <"((') i s positive, <^"(TT) is negative ; thus <(0) is a minimum and <(TT) a maximum value ol <(x). Again, <" (a) = cos a ('2 cos 2 a 7 sin 2 a), but a satisfies the equation 2cos' 2 a sin' 2 a=0, hence <"(a) is negative and <(u) is a minimum value of <(#) Also "(7T a) = cosa(2cos 2 a 7 sin^) is positive, and (ir a) a minimum value of (x). Similarly for the other critical values of Ex. 6. 83. Conditions for maxima and minima derived from Tay- lor's theorem. In this article, as in the preceding, the function and its derivatives are supposed to be continuous in the vicinity of x = a ; otherwise the method of Art. 81 must be used. Let <() be a maximum value of <(X); then it follows from the definition that <>() is greater than either of the neighboring values, ( + A), <( ^), when h is taken small enough. Flence $(a + ^) (a) and <( A) <() are both negative. Similarly, these expressions are both positive if () is a minimum value of (x). 140 DIFFERENTIAL CALCULUS [Cn. VI. Let <(# + A), $(xh) be expanded in powers of h by Taylor's theorem ; then (a) and < (a A) <(#) cannot have the same sign unless <'() be zero, hence the first con- dition for a turning value is '(a) = 0. In this case and A can be taken so small that the first term on the right is numerically larger than either of the second terms, hence <>(a + A) $(a) and <(a A) $() are both negative when <"() is negative, and both positive when <"(#) is posi- tive. Thus $(a) is a maximum (or minimum) value of <(#) when '() is zero and <"(a) is negative (or positive). 83-84.] VARIATION OF FUNCTIONS 141 In case it should happen that "(#) is also zero, then <"'() M , < IV ( + ^)i4 (a + Ji) - (a) = y y A 3 4- ~ v . z & 4 , and by the same reasoning as before, it follows that for a maximum (or minimum)- there are the further conditions that <'"() equals zero, and that lv (a) is negative (or positive). Proceeding in this way, the general conclusion stated in the last article is evident. Ex. 1. Which of the preceding examples can be solved by the general rule here referred to 1 Ex. 2. Why was the restriction imposed upon '(x) that it should change sign by passing through zero, rather than by passing through infinity ? 84. Application to rational polynomials. When (f> (x) is a rational polynomial, its derivative <'(#) is of similar form. Let the real roots of 'the equation '(#) = be a, #, c, Z, arranged in descending order of algebraic magnitude ; sup- pose, first, that no two of them are equal ; then '(a;) has the form $(x) = (x - a) (x - 6) (x - c) - (x - P, (1) in which P is the product of the imaginary factors of the polynomial <'(#) This product will have the same sign for all values of x, and by giving the coefficient of the highest power of x in <'(#) a positive value, P will always be positive, by the theory of equations. Differentiating (1) with regard to x, and putting x = a, it follows that 142 DIFFERENTIAL CALCULUS [Cu. VI. but a 5, a c, are all positive, hence $"() is positive, and therefore <>(#) is a minimum value of $(). Again, <"(&) = (6 - a) (5 - c) - (b - l)P, but J a is negative, and the remaining factors are positive; hence <"(#) is negative, and <(>) is a maximum value of Also <"O) = (e - a) O - by - (c - in which the only negative factors are c a, c b ; hence "(('(z) = has 2n real roots, all of which are distinct, the function (:r) has n maxima and n minima occurring alternately ; if '(x) has 2w + 1 dis- tinct real roots, then $>(x) has n + 1 minima and n maxima, the latter being situated, respectively, between successive minima. Next, suppose that two of the roots are each equal to a ; then <' O) = (x- a) 2 ^ (35), and 0" O) = O - a) 2 -f' (x) + 2 (a - a) ^ (a), '" O) = (x - a) 2 f O) + 4 (3 - a) y (x) + 2 ^ ( ; hence <' (a) = 0, 0" (a) =. 0, <'" (a) = 2 -f () ; therefore <> (a) is neither a maximum nor a minimum. If three of the roots of ' (x) are each equal to a, it is proved similarly that < (a) is a maximum or minimum ac- cording as i/r (a) is negative or positive. These conclusions may be extended to the cases of n equal roots, in which n is even or odd, respectively. An illustrative example was given in Art. 81. 84-86.] VARIATION OF FUNCTIONS 143 85. The maxima and minima of any continuous function occur alternately. It. has been seen that the maximum and minimum values of a rational polynomial occur alternately when the variable is continually increased or diminished. This principle is also true in the case of- every continuous {unction of a single variable ; for, let (a ), () be two maximum values of (#), in which a is supposed less than b ; tl it'll when x = a + A, the function is decreasing ; when x- = b A, the function is increasing, h being taken suffi- ciently small, and positive. But in passing from a decreas- ing to an increasing state, a continuous function must, at some intermediate value of x, change from decreasing to increasing, that is, must pass through a minimum. Hence, between two maxima there must be at least one minimum. It can be similarly proved that between two minima there must be at least one maximum. 86. Simplifications that do not alter critical values. The work of rinding the critical values of the variable, in the case of any given function, may often be simplified by means of the following self-evident principles. 1 . Any value of x that gives a turning value to c (x) gives also a turning value to (V), and conversely, when c is independent of x. These two turning values are of the same or opposite kind according as c is positive or negative. 2. Any value of x that gives a turning value to c + (x) gives also a turning value of the same kind to (a;), and conversely, provided c is independent of x. 3. Any value of x that gives a turning value to [ (#)]" gives also a turning value to < (#), and conversely, when it is independent of x. Whether these turning values are of the same or opposite kind depends on the sign of w, and also on the sign of [6 144 DIFFERENTIAL CALCULUS [Cu. VI. EXERCISES Find the critical values of x in the following examples, and determine the nature of the function at each, and obtain the graph of the function. 1. u = x a + 18x' 2 + 105z. 2. u = (x - l) 3 O-2) 2 . 3. u = x(x - 1)2 (x + I) 3 . 4. u = Ax 2 + Bx + (7; show that u cannot have both a maximum and a minimum value, for any values of A, B, C. 5. u = 3 x 8 2 x + 4. Show that a cubic function has in general both a maximum and a minimum value. 6. u = 2 x + 4 x 8 . Compare the graph of this function with that of exercise 5. 7. u = 8. u = . 10. u = sin 2x - x. x 11. Show that the function b + c (x o)s has neither a maximum nor a minimum. 12. u = sin 2 x cos 8 x, 14. w = a; + tanx. 13. u '= sin x + cos 2 x. 15. u = +e~ 2z . x 87. Geometric problems in maxima and minima. The theory of the turning values of a function has important applications in solving problems concerning geometric maxima or minima, i.e., the determination of the largest or the smallest value a magnitude may have, while satisfying certain stated geometric conditions. The first step is to express the magnitude in question algebraically. If the resulting expression contains more than one variable, the stated conditions will furnish enough relations between these variables, so that all the others may be expressed in terms of one. The expression to be maxi- mized or minimized can then be made a function of a single variable, and can be treated by the preceding rules. 86-87.] VARIATION OF FUNCTIONS 145 Ex. 1. Find the largest rectangle whose perimeter is 100. Let x, y denote the dimensions of any of the rectangles whose perimeter is 100. The magnitude to be maximized is the area M = xy, (1) in which the variables x, y are subject to theTstated condition 2 x + 2 y = 100, i.e., y = 50 - x, (2) hence the function to be maximized, expressed in terms of the single variable x, is = < (x) = x (50 - x) = 5Qx - x 2 . (3) The critical value of x is found from the equation <'(.r) = 50-2* = 0, and is x = 25. When x increases through this value, <'( x ) changes sign from positive to negative, and hence (x) is a maximum when a: = 25. Ki liiation (2) shows that the corresponding value of y is 25. Thus the maximum rectangle whose perimeter is 100, is the square whose side is 25. Ex. 2. The sum of the three dimensions of a rectangular box is 10, the total surface is 34 ; find its dimensions so that its volume may be a maximum. Here the function = xyz 0) is to be maximized, the three variables being subject to the two condi- tions x + y + z = 10, (2) xy + xz + yz = 17. (3) Equation (2) multiplied by z, subtracted from (3) and transposed, gives a# = 17-10z + z 2 , by means of which the variables x and y can be eliminated from (1), giving = (17 - 10z + z 2 )z. Hence the function to be maximized 'by varying z is #(2) =z 3 -10z 2 + 17 z, then <'(z) = 3z a - 20 z + 17 = (z - l)(3z - 17), "(0=62-20; DIFFERENTIAL CALCULUS Leu. vi. hence the critical value 2 = 1, which makes <'(z) zero and <"(z) negative, gives to < (2) the maximum value 8. The other two dimensions, found from (2) and (;i), are 8 and 1. The second critical value, z = 5f, makes "(z) positive, and <(z) an algebraic minimum. The corresponding dimensions are 5f, 1^, 5|, a result not applicable to the special problem in question. Thus the required dimensions are 8, 1, 1. Any change of these dimensions subject to the given conditions will lessen the volume. Ex. 3. If, from a square piece of tin whose side is a, a square be cut out at each corner, find the side of the latter square in order that the remainder "may form a box of maximum capacity, with open top. Let a; be a side of each square cut out, then the bottom of the box will be a square whose side is a 2x, and the depth of the box will be x, hence the volume is which is to be made a maximum by varying x. Here =(--> x) 2 -4 x(a-2x) dx FIG. 20. This derivative vanishes when x= ^, and when #= 3- It will be found by applying the usual test, that x= ^ gives v the minimum value zero, and a 2 a 8 that x = ^ gives it a maximum value -^=-, hence the side of the square to be cut out is one sixth the side of the given square. Ex. 4. Find the area of the greatest rectangle that can be inscribed in a given ellipse. An inscribed rectangle will evidently be sym- metric with regard to the principal axes of the ellipse. Let a, b denote the lengths of the semi-axes OA, OB (Fig. 21); let 2 a:, 2y be the dimensions of an inscribed rectangle; then the area is FIG. 21. (1) 87.] VARIATION OF FUNCTIONS 147 in which the variables x, y may be regarded as the coordinates of the vertex P, on the curve, and are therefore subject to the equation of the ellipse =\. (2) ft 2 It is geometrically evident that there is some position of P for whicli the inscribed rectangle is a maximum ; for let P be supposed to take in succession all positions between A and B; then just as P moves away from A the rectangle begins by increasing from zero, and when P comes to B the rectangle ends by decreasing back to zero; hence there must be a change from increasing to decreasing, i.e., a maximum, for at least one intermediate position. The elimination of y from (1), by means of (2), gives the function of x to be maximized, = x Va 2 - z 2 . (3) a By Art. 86, the critical values of x are not altered if this function be divided by the constant , and then squared. Hence, the values of x a which render u a maximum, give also a maximum value to the function (x) = x 2 (a 2 - x 2 ) = a 2 * 2 - x*. Here <' (x) = 2 a 2 * - 4 X s = 2 z(a 2 - 2 a; 2 ), <"(x) =2 2 - 12 x 2 ; hence, by the usual tests, the critical values x = -^- render (x), and \/2 therefore the area u, a maximum. The corresponding values of y are given by (2), and the vertex P may be at any of the four points denoted by x = y=X V2 V2 giving in each case the same maximum inscribed rectangle, whose dimen sions are av/2, bVl, and whose area is 2 aft, or half that of the circum- scribed rectangle. Ex. 5. Find the cylinder of maximum volume that can be cut from a given prolate spheroid. Let the spheroid and inscribed cylinder be generated by the figure of Ex. 4 revolving about OA ; then the volume of the cylinder is = 2y a , (1) DIFF. CALC. 11 148 DIFFERENTIAL CALCULUS and this is to be maximized subject to the condition [Cn. VI. hence v = r x (a 2 z 2 ), and by Art. 86, when this function is a maximum, so is the function x (a 2 - a 2 ), which, according to the usual tests, has its maximum when x -. The corresponding value of y, from (2), is -; hence, from (1), the maximum volume is i 3V3 or of the volume of the prolate spheroid. Ex. 6. Find the greatest cylinder that can be cut from a given right cone, whose height is h, and the radius of whose base is a. Let the cone be generated by the revolution of the triangle OA B (Fig. 22) ; and the inscribed cylinder by that of the rectangle A. P. Let OA = h, AB = a, and let the coordinates of P be (x, t y); then the function to be maximized is TT ?/ 2 (A x) subject to the relation ^ = -. X ft Ex. 7. Find the area of the greatest rectangle that can be inscribed in the segment of the parabola y 2 - = px, cut off 1))' the line x = a. Ex. 8. What is the altitude of the maximum cylinder that can be inscribed in a given segment of a paraboloid of revolution ? Ex. 9. Find the greatest right-angled triangle that can be constructed on a given line as hypothenuse. Ex. 10. Given the vertical angle of a triangle, and its area. Find when its base is a minimum. Ex. 11. A Norman window consists of a rectangle surmounted by a semicircle. Given the perimeter; required the height and breadth of window when the quantity of light admitted is a maximum. FIG. 22. *>"] VARIATION OF FUNCTIONS . 149 Ex. 12. The diameter of a cylindrical tree is a. Find the strongest beam that may be cut from it, assuming that the strength is proportional to the breadth multiplied by the square of the thickness. Ex. 13. An open tank is to be constructed with a square base and vertical sides. Show that the area of the entire inner surface will be least if the depth is half the width. Ex. 14. The sum of the perimeters of a circle and a square is fixed. Show that when the sum of the areas is least, the side of the square is double the radius of the circle. Ex. 15. What should be the ratio between the diameter of the base and the height of a cylindrical fruit can in order that the amount of tin used in. constructing it may be the least possible? Solve the same problem when the top is open. Ex. 16. The top of a pedestal which sustains a statue c feet in height is b feet above the level of a man's eyes. Find his horizontal distance from the pedestal when the statue subtends the greatest angle. Ex. 17. A high vertical wall is to be braced by a beam which must pass over a parallel wall a feet high, and b feet distant from the other. Find the length of the shortest beam that can be used for the purpose. Ex. 18. Determine the cone of minimum volume that can be de- scribed about a given sphere. Ex. 19. Find the shortest distance from the point (2, 1) to t"he parabola y 2 = 4 x. Ex. 20. 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 the length of the crease is a minimum. Ex. 21. A tangent is drawn to the ellipse whose semi-axes are a and b, such that the part intercepted by the axes is a minimum ; show that its length is a + b. Ex. 22. A person being in a boat 3 miles from the nearest point on the beach, wishes to reach in the shortest time a place 5 miles from that point along the shore; supposing he can walk 5 miles an hour, but row only at the rate of 4 miles an hour, find the place where he must land. Ex. 23. A slip noose in a rope is thrown around a large square post, and the rope drawn tight in the direction as shown in the figure. At what angle does the rope leave the post ? Fig. 23. 150 DIFFERENTIAL CALCULUS [Cn. VI. 87. Ex. 24. Show that just before and after a turning value the function passes through equal values. Apply this principle to give geometrical solutions to Exs. 22, 23. Ex. 25. Show that in the vicinity of a turning value A/(x) is an infinitesimal of an even order when Aa: is of the first order. When is A/ (x) of the third order ? Ex. 26. A rectangular court is to be built so as to contain a given area, and a wall already constructed is available for one of the sides; find its dimensions so that the least expense may be incurred. Ex. 27. The work of driving a steamer through the water being pro- portional to the cube of her speed, find the most economical rate per hour against a current running a knots per hour. p Ex. 28. Assuming that the current in a voltaic cell is C = , E r + R being electromotive force, r internal resistance, R external resistance, and that the power given out is P = RC 2 , prove that P is a maximum when r = R. Trace the curve that shows the variation of P, as R varies. [Perry's Calculus for Engineers.] CHAPTER VII RATES AND DIFFERENTIALS 88. Rates. Time as independent variable. Suppose a par- tirle P is moving in any path, straight or curved, and let s be the number of space-units passed over in t seconds; then * may be taken as the dependent variable, and t as the in- dependent variable. Let As be the number of space-units described in the additional time At seconds ; then the average velocity of P As during the time A is > the average number of space-units described per second during the interval. The velocity of P is said to be uniform if its average As velocitv, > is the same for all intervals A. The actual A velocity of P at any instant denoted by t is the limit which the average velocity, for the interval between the time t and the time t -f A, approaches as A# is made to approach zero as a limit. Thus =*:=*" A ' = A/ 1 dt is the actual velocity of P at the time denoted by t. It is evidently the number of space-units that would be passed over in the next second if the velocity remained uniform from the time t to the time t + 1. It may be observed that if, for the word " velocity," the more general term, "rate of change," be used, the above 151 152 DIFFERENTIAL CALCULUS [Cn. VII. statements will apply to any quantity that varies with the time, whether it be length, volume, strength of current, etc. For instance, let the quantity of an electric current be G at time t, and G + AC" at time t + A ; then the average rate A(7 of change of current in the interval A is , the average E*v increase in current units per second ; and, as before, the actual rate of change at the instant denoted by t is A < = A* ~ dt' This is the number of current-units that would be gained in the next second if the rate of gain were uniform from the time t to the time t -f- 1. Since & = %L : ^, [Art. 21 dx dt dt hence -$- measures the ratio of the rates of change of y and of x. It follows that the result of differentiating y=/(*) 0) may be written in either of the forms (2) t =/'<*> t- The latter form is often convenient, and may also be obtained directly from (1) by differentiating both sides with regard to t. It may be read : the rate of change of y is f'(x) times the rate of change of x. Returning to the illustration of a moving point _P, let its coordinates at time t be x and y : then measures the rate 9 dt 88.] BATES ASD DIFFERENTIALS dt dt of change of the ^-coordinate, v and may be called the velocity of P resolved parallel to the cr-axis, or the ^-component of the velocity. Similarly, ^ is the y-compo- 77 nent of velocity. These three rates of change are connected by the equation FIG. 24. it dt) 00 Ex. 1. If a point describe the straight line 3x + 4y = 5, and if x increase h units per second, find the rates of increase of y and of s. Since y \ \ x, hence and when dy_ = _% dx_ dt~ 4 dt 1 dx i fly ^ Ex. 2. A point describes the parabola y 2 = 12 x, in such a way that when x = 3, the abscissa is increasing at the rate of 2 feet per second : at what rate is y then increasing? Find also the rate of increase of s. Since y 2 = 12 x, tlx 6 dx, dt y dt Vl2a: & hence, when x = 3, and = 2, ^ = 2. dt tit Again, * S = Y + Y, hence = 2 Vl feet per second. 154 DIFFERENTIAL CALCULUS [Cn. VII. Ex. 3. A person is walking towards the foot of a tower on a horizontal plane at the rate of 5 miles per hour ; at what rate is he approaching the top, which is 60 feet high, when he is 80 feet from the bottom ? Let x be the distance from foot of tower at time t, and y the distance from the top at the same time ; then x 2 + GO 2 = y*, x d - = y$L. dt y dt When x is 80 feet, y is 100 feet ; hence if is 5 miles per hour, - is 4 miles per hour. 89. Abbreviated notation for rates. When, as in the above examples, a time derivative is a factor of each member of an equation, it is usually convenient to write, instead of the symbols , -^, the abbreviations dx and c?v, for the rates dt dt of change of the variables x and y. Thus the result of differentiating y=/00 (1) may be written in either of the forms dy=f(x)dx. (4) It is to be observed that the last form is not to be re- garded as derived from equation (2) by separation of the symbols dy, dx\ for the derivative ^ has been defined as dx the result of performing upon y an indicated operation rep- resented by the symbol ; and thus the dy and dx of the dy_ dx The dy and dx of equation (4) stand for the rates or time symbol ^ have been given no separate meaning. dx 88-89.] RATES AND DIFFERENTIALS 155 derivatives -* and -- in (3), which is itself obtained from dt at (1) by differentiation with regard to , by Art. 21. In case the dependence of y upon x be not indicated by a functional operation/, equations (3), (4) take the form dy _ dy dx dt dx dt -. dx In the abbreviated notation, equation (4) of the last article is written d = dx z + dy*. Ex. 1. A point that is describing the parabola y' 2 = 2px is moving at time t with a velocity of v feet per second ; find the rate of increase of the coordinates x and y at the same instant. Differentiating the given equation with regard to t, ydy = pdx, but dx, dy also satisfy the relation log 10 e, that is, x > .4343. Ex. 5. Two sides, a, b, of a triangle are measured, and also the in- cluded angle C ; find the error in the computed length of the third side c due to a small error in the observed angle C. [Differentiate the equation c 2 = a 2 + 6 2 2 ab cos C, regarding a, b as constant.] 90.] KATES AND DIFFERENTIALS 157 Ex. 6. In a tangent galvanometer the tangent .of the deflection of the needle is proportional to the current. Show that the relative error in the computed value of the current, due to a given error of reading, is least when the angle of deflection is 45. Ex. 7. The error in the area A of an ellipse, due to small errors in the semi-axes, is approximately given by = - -\ - A a b Ex. 8. The side of an equilateral triangle is 24 inches long and is increasing at the rate of two inches per day ; how fast is the area of the triangle increasing? Ex. 9. Find the rate of change in the area of a square when the side b is increasing at a ft. per second. Ex. 10. In the function y = 2 x s + 6, what is the value of x at the point where y increases 24 times as fast as x ? Ex. 11. A circular plate of metal expands by heat so that its diameter increases uniformly at the rate of 2 inches per second ; at what rate is the surface increasing when the diameter is 5 inches? Ex. 12. What is the value of x at the point at which x z 5 x' 2 + 17 x and x 3 3 x change at the same rate ? Ex. 13. Find the points at which the rate of change of the ordinate y = x 3 6 x 2 + 3 a: -f 5 is equal to the rate of change of the slope of the tangent to the curve. Ex. 14. The relation between'*, the space through which a body falls, and t, the time of falling, is s = 16 * 2 ; show that the velocity is equal to 32 /. The rate of change of velocity is called acceleration; show that the acceleration of the falling body is a constant. Ex. 15. A body moves according to the law s = cos (nt + e) ; show that its acceleration is negative and proportional to the space through which it has moved. CHAPTER VIII In the previous chapters the dependence of one variable upon another, called the independent variable, has been discussed. The mode of dependence of one variable upon two others will next be considered; and the relation between the dependent variable z and the independent variables x and y will be expressed in the form *=/(*,y). (i) Examples of such dependence have been seen in coordi- nate geometry of three dimensions ; for instance, from the equation of a sphere referred to its center as origin 3? + y* + 2 2 = a 2 , any one of the variables may be expressed as a function of the other two ; thus z = Va 2 a? y 2 . Conversely, any relation of the form (1) can be exhibited graphically by taking x, y as coordinates of a point on a horizontal plane, and drawing at the point an ordinate to the plane to represent the corresponding value of the func- tion z ; the form of the surface of which (1) is the equation will represent the mode of variation of the function. 91, Definition of continuity. A function/^, y) is said to be continuous in the vicinity of the values x = a, y = b ; when 168 Cn. VIII. 91-92.] FUNCTIONS OF TWO VARIABLES 159 /(a, >) is -real, finite, and determinate (whether unique or multiple- valued) ; and when the difference f(a + A, b -}- &) /(a, 6) can be made less than any assigned number ??, by taking h, k small enough, independent of the ratio of k to h ; in other words, when no matter in what way h and & approach their limits. It is implied that, when the function is multiple-valued, attention is to be paid to the correspondence of the multiple values in the two members of this limit-relation. In geometrical language the f unction /(#, y) is continuous at x = a, y = b, when the ordinate of the surface z = f(x, y) drawn at the point (a + A, 6 + &) approaches as a limit the ordinate drawn at the point (a, 6) irrespective of the direc- tion in which the point (a -f A, b -f &) moves to coincidence with the point (a, i). [Cf. Exs. 7, 9, p. 182.] 92. Rate of variation. Partial derivatives. The most important question concerning the variation of a continuous function z is : what is the rate of change of z when x and y vary at given rates ? It is convenient to consider first the simpler question : what is the rate of change of z when x varies at a given rate, and y remains constant ? * In this case z is a function of the single variable x, and its rate of change is dz_ _ dz_ dx_ s-t-^ dt dx dt dz in which it is to be understood that the operation - ' is per- dx dz formed on the supposition that y is a constant, and that - dt is the rate of change of z in so far as it depends on the change of x. To indicate these facts without the qualifying 160 DIFFERENTIAL CALCULUS [Cn. VIII. verbal statements, equation (1) will be written in the form djZ _ dz < dx , 9 ~dt ~ dx dt' in which -- stands for the ^-derivative of z when y is kept dx constant, and is called the partial derivative of z with regard to x, and -^- denotes the rate of change of z in so far as it dt> depends on the change of x. Thus, by Art. 18, the partial derivative is the result of the indicated operation cte _ lim A,,, _ lini f(x + Ax, y ) / (x, y) to~ A - r = Az A * = &x Similarly, the rate of change of z when x is kept constant and y varies at a given rate is measured by ^? = cte.^/, , 3) dt dy dt in which -^- is the rate of change of z in so far as it dt dz depends upon the change of y, and denotes the partial 7 derivative of z taken, with regard to /, that is, the result of the operation indicated by dz _ lim A,,g _ lim f(x,y + Ay) / (x, y) dy Ay=0 Ay Ay = Ay 93. Geometric illustration. Let the function f(x, y) be represented graphically by the ordinate of a surface whose equation is z=f(x, y) and let a vertical section be taken parallel to the plane (z,x~) at a given distance y = y\ from that plane ; then if a point P be supposed to describe on the surface the contour of the section, the ^-coordinate will remain constant, and the value of the varying ordinate z will 92-94.] FUNCTIONS OF TWO VARIABLES 101 be given by the equation z = f(x,y 1 ). If the rate of varia- tion of x at any instant be known, the corresponding rate of variation of z is given by d^z _ dz dx _ df(x, y_^ dx dt dx dt dx dt which may be called the rate of variation of the ordinate in the ^-direction. The partial derivative is the ratio of the rates of in- to crease of z and #, and is represented geometrically by the slope of the tangent drawn to the contour at P. Ex. 1. A point P on the surface z = x^y + 2 xy 1 moves in the plane y 1 ; the x-rate is 10 feet per second ; find the rate of change of z, when P is passing through the point for which x = 3, and also the dii'ection and velocity of the motion of P. Differentiating the given identity with regard to t, y being kept con- stant, ^ = (2 xy + 2 y 2 ) = 20 = 200 feet per second, and the slope dt dt dt of the tangent at P in the plane of motion is 20. The velocity of P in the curve is VlO' 2 + 200 2 = 200.25 feet per second. Similarly, if P move on the surface in the plane x = x r the rate of change of z will be given by dyZ = dz dy _ df(x, y) dy dt ~ dy dt~ By dt ' and , the ratio of the rates of change of z and y, will dy measure the slope of the tangent at P in the plane of motion. Ex. 2. Find for the same surface as before, at the point for which x = 3, y = 2 the rate of change of z in the y-direction, if y be changing at the rate of 5 feet per second, x being kept constant. Here '& = (x 2 + 4a;v) ^ 7 = 33 ^ = 165 feet per second, and the slope dt , dt tit in the direction of motion is 33. 94. Simultaneous variation of a? and y ; total rate of varia- tion of z. It will now be shown that when x and y van 162 DIFFERENTIAL CALCULUS [Cn. VIII. simultaneously, the total rate of change of z is the sum of its separate rates of change as x and y vary alone; that is, dz = d**,d j , m dt dt dt or dz = dzdx dzdy (i) dt dx dt by dt For, let z =/(x, y), and let x, y start at the values x r y r and take increments Ax, Ay; then the initial value of z is /(#!, y^), and its final value is f(x + Ax, y^ + Ay); hence the total increment of z is /(! + Ax, y x + Ay) -/(!, y x ). By subtracting and adding the intermediate value /(! + Ax, y x ), in which x alone has varied from its original value, the total increment of z may be written as the sum of two partial increments in the form Az = /(* + Ax, y + Ay)-/(x + Ax, the latter being the increment of /(x, y) as x changes from Xj to Xj + Ax, y remaining constant, and the former being the further increment of the function as x remains at the value x x + Ax while y changes from y x to y 1 + Ay. The result of dividing by A, the increment of t, may be written Az = /(x t + Ax, y 1 + Ay) -f(x l + Ax, y^ Ay A ~ Ay A Ax 94.] FUNCTIONS OF TWO VARIABLES 163 Taking limits as A, A#, Ay, Az, all approach the limit zero, and remembering that by Art. 92, /* + A*, -/ *, y 1 + Ay) Ay ^ taken at ^ = ^ y = yi? t ^ = == , taken at x = #,, it follows that, at any values of x and y, for which the func- tion and its partial derivatives are continuous, dz __dz dx dz dy dt dx dt dy dt In the abbreviated rate notation, equations (2), (3) of Art. 92, and (1), (2) of 4rt. 94, are respectively, j dz j j dz j d.jz = dx, a v z = a y, Bx dy y dz = d,z + dyZ = dx H -- dy. dx dy ' Ex. 1. A particle moves on the spherical surface a; 2 + y 2 + z 2 = a 2 in a vertical meridian plane inclined at an angle of 60 to the plane (z%). If the x-component of its velocity be ^a per second, when x = ^a, find the ^-component, the z-component, and the resultant velocity. Since z = Va 2 x' 2 y 2 , , _ x dx y dy Va 2 - x 3 - y* Va 2 - x 2 - .y 2 but since dx = ^a, and the equation of the given meridian plane is y = x tan 60, hence dy = dxV3 = \/3, and y = $~, Therefore 10 4 dz = - -^_ _ 4lL = _ ^ = _ .115 a in feet per second. 2V8 2 15 a\/3 Also, ds - Vrfx 2 + dy 2 + dz 2 = 2J_ = 0.23 a in feet per second. 7.5 DIFF. CALC. 12 164 DIFFERENTIAL CALCULUS [Cn. VIII. 95. Language of differentials. The results of the preced- ing articles may be stated thus : The partial z-differential due to the change of x is equal to the ^-differential multiplied by the partial re-derivative. The partial z-differential due to the change of y is equal to the ^-differential multiplied by the partial ^-derivative. The total 2-differential is equal to the sum of the partial 2-differentials. One advantage in keeping the equation in the differential form is that it may be divided when necessary by the differ- ential of any other variable s, to which x and y are related, and then, remembering that the ratio of two differentials (or rates) may be expressed as a derivative, the equation becomes dz_dzdxdz dy ds dx ds By ds Ex. 1. Given z = axy 2 - + bx*y + ex 8 + ey, dz = (ay 1 + 2bxy + 3 cx*)dx + (2 axy + 6x 2 + e)dy. Ex. 2. Given z = x", d x z = yx y ~ J dx, d y z = x log x dy, dz = yx*~ l dx + x* log x dy. Ex. 3. Given u = tan-' I du = ^J^. x x 2 + y 2 Ex. 4. Assuming the characteristic equation of a perfect gas, vp Rt, in which v is volume, p pressure, t absolute temperature, and R a con- stant ; express each of the differentials dv, dp, dt, in terms of the other two. Ex. 5. Being given that in the case of air, R = 96, when p is measured in pounds per square foot, v in cubic feet, and t is centigrade ; and letting t = 300, p = 2000, v = 14.4; find the change in p when t changes to 301, and v to 14.5, supposing that p changes uniformly in the Interval. [Perry's Calculus for Engineers, p. 138.] Since vdp + pdv = Rdt, dv = .1, dt = 1 ; heuce dp = - 7.22. 95-96.] FUNCTIONS OF TWO VARIABLES 165 The actual increment of p will be a little different from this, and is easily found by direct computation to be 7.17. The difference in the results is analogous to the difference between the ordinate of a surface and the ordinate of its tangent plane, taken near the common ordinate of the point of contact. 96. One variable a function of the other. When there is a definite relation connecting the variables x and y, the equation , dz j , dz j dz = ax -\ ay ox dy may be divided by the differential of either variable, then ** + *&. (1) ax ox oy ax It is here well to note the difference between and . ox dx The former is the partial derivative of the functional ex- pression for z with regard to #, on the supposition that y is constant. The latter is the total derivative of z with regard to #, when account is taken of the fact that y varies with x. It is to be observed that the implied assumption in Art. 94, that the variables x and y have at any instant some definite numerical rate of change, is only equivalent to assuming that they vary in some continuous manner. They need not on that account be expressible as definite func- tions of the time, or have any fixed relation of dependence upon each other. On the other hand, a fixed relation of dependence is not pre- cluded, for Art. 94 only assumes that x, y take the increments Ar, Ay in the time A<, without inquiring whether one of the increments may not be determined by the other, or whether they may not both arise from the increment of some other hidden variable. The supposition that the letters x, y are independent in forming the partial derivatives is only a convenient algebraic rule or artifice for obtaining the coefficients of the differentials dx, dy; and does not imply the physical independence of the magnitudes denoted by these letters. Thus the " independence " is formal, or operational, rather than physical. 166 DIFFERENTIAL CALCULUS [Cn. V11I. The value of -&- is to be obtained by differentiating the ijLjL functional relation between x and y. If this relation ex- presses y as an explicit function of #, the right hand mem- ber of (1) can then be expressed in terms of x alone ; and the result will be the same as if z had been first reduced to the form of a function of the single variable #, and then differentiated with regard to this variable. Ex. 1. A point moves on the surface z = f(x, y) in the curve deter- mined by the cylindrical surface y = < (x) ; express dz in terms of dx. Ex. 2. If z = tan -1 -#-, and 4 x* + y' 2 = 1, find 2 x dx Ex.3. A point moves on. the curve of intersection of the surfaces 2 = (x, y), z = f(x, y) ; find the mutual ratio of the rates dx : dy : dz at any point x, y, z. For shortness, denoting partial derivatives by subscripts, dz = f^lx + f 2 dy - fadx + , f 2 r 97. Differentiation of implicit functions ; relative variation that keeps z constant. An important special question is how to vary x and y so as to keep f (x, y) from varying. If 2 =f(x, y) = constant, then dz = -- dx + dy = 0, ox dy ' hence the relative rates of change of x and y are given by the equation df dy _ dx dx-~W dy Ex. 1. If x pass through the value 2 at the rate of 5 units per second, at what rate must y pass through the value 3 in order to keep the func- tion x 2 y + 3xy 2 constant? Since d (.r' 2 .y + 3 xy*) = (2 xy + 3 y 2 ) dx + (x 2 + 6 xy) dy = 0, hence 39 dx + 40efy = 0, dy= 4j units per second. 96-97.] FUNCTIONS OF TWO VARIABLES 167 Ex. 2. Defining the elasticity of a gas as the limit of the ratio of an increment of pressure to the corresponding relative decrement of volume, find e, the elasticity of a perfect gas under constant temperature. "all and by differentiating vp = Rt, keeping t constant, vdp + pdv = 0, -/- = , hence e = p. dv v As a geometrical illustration, let a section of the surface z =/(#, y) be made by the plane z = e, then for all points on the contour of the section z =/O, y) = c, and if a point describe the contour, the #-rate and the #-rate will be in the ratio : ; and this ratio will measure dy ox the slope of the tangent to the contour with reference to the plane zx. Ex. A particle is moving on the ellipsoid + 2- + ^- = 1 at the point , a 2 o 2 c 2 x = ", y = -, z -^; find the relative rates of x and of y so that the 2 ' 2 V2 rate along z may be zero. Since xdx yd^ = bence dy = _bjn = _b a 2 b* dx a?b a Similarly, if a point whose coordinates are x, y move in a plane so as to keep the function f(x, y) constant, then it describes a curve whose equation is f(x, y) = c?, hence the differentials dx, dy are connected by d dx + d fdy = 0, (1) dx dy and the slope of the direction of motion is given by dy = _(x, y, 2) = c r \j/(x, y, 2) = c 2 ; find rfu in terms of dx. Differentiating, and denoting partial derivatives by subscripts for shortness, du =f { dx + / 2 dy + f 3 dz, = 2 dy + 3 dz, = \{/ 1 dx + {f/ 2 dy + i/'j dz ; hence, by elimination of dy, dz, ^s) = ^ C/i (^s - ^2) Geometrically speaking, the point (x, y, z) moves on the curve of intersection of two surfaces and has therefore only one degree of freedom. "99-100.] FUNCTIONS OF THREE VARIABLES 171 Thus the variation of a single independent coordinate is sufficient to determine the variation of the other coordi- nates, and of the function u itself. 100. Euler's theorem. Relation between a homogeneous function and its partial derivatives. Let u =/(a% y, z) be a homogeneous function of x, y, z, of degree w; then Bu . du , du x T + ^T' + z ^~ = nu - dx oy dz For, let u=Ax a y f) zv + Bx^'y^z^' + > where a + /3 + 7 = ' 4- /3' + 7' = = n. = x = Similarly, - dy z -^ = dz Adding these three equations, du . du . du x T + y^- + z ^- ox dy dz = n (.x-yzi + xy^z^ -f ) = nu. The theorem can be extended to functions of any number of variables. If a function, homogeneous in several variables, be differ- entiated partially with regard to each of them; then each partial derivative be multiplied by the variable with regard 172 DIFFERENTIAL CALCULUS [On. VIII. 100. to which the derivative was taken; and all these products added ; the result is n times the original function ; where n is the degree of the function. This is known as Euler's theorem.* EXERCISES Verify Euler's theorem for the following expressions : 1. x* + 3xy-7x/. 4. x 3 - 3 x 2 y - y* x ^ ^P + x2 Sm z' Prove the following identities : dx By 2. U = Qy Qz x+y + z 3. =: 4. u = sin -1 (xy2), = tan 2 u sec u. dx dy d z 5. = log(tan x + tan^ + tan z), sin 2x + sin2 y +sin2 z =2. 6. = e*siny + e>'sinx, (|^) 2 + (^y=e 2l + e 2 +2e I +sin(x+y). 7. = ] xlogy 9. u = log y', du = log ?/ dx + - dy. y 10. u = i * Leonard Euler (1707-1783), one of the most eminent mathematicians of the eighteenth century. CHAPTER IX SUCCESSIVE PARTIAL DIFFERENTIATION 101. Successive differentiation of functions of two variables. Let z=f(x, #), in which a;, y are functions of another vari- able , which may conveniently be thought of as time. As the rate of change of z is usually variable, it is sometimes useful to have an expression for the rate of change of this rate. Just as is the rate of change of 2, so [ ] is the dt dt\dt) dz d?z rate of change of , and it is written - . This rate of dt dt* the rate may conveniently be called the acceleration. It will now be shown that the expression for the z-acceler- ation involves the ^-acceleration, the ^-acceleration and also the squares and products of the x-rate and the y-rate, each with a certain coefficient. It was proved in Art. 94 that the 2-rate is dz _dz dx dz dy ^ ~di~dx~di + fy dt' Differentiating each term of this identity with regard to t, d?z _ d_f^\dx dz_ d?x d_ (d_z\dy , ^2 d*y . ^^\ dt z ~ dt\dx)~dt dx~di? dt\dy) dt By ~dt 2 ' r)z but, since is itself a function of two variables, hence, by ox Art. 94, f^\ = f?!L\fa 4. f-?.\^ t dt \dx) ~~ dx \dx) dt dy (dx) dt ' 173 174 DIFFERENTIAL CALCULUS [Cn. IX. dfdz\ d fdz\dx , d fdz\dy also [ )= [ - )-- dt\By) dx\dy) dt dy \dy) dt hence (2) becomes, by substitution and slight re-arrange- ment, N d d // N T- -si/te y) = r~ rvt^ y) 3y oa; oa; 5y or, since /(a;, y) = 2, 3y 5a; dx dy COR. 1. It follows directly that under corresponding conditions the order of differentiation in the higher partial derivatives is indifferent. In other words, if u and all its partial derivatives are continuous, the operations dx dy are commutative. T-, E.g., dx dy dx dx 2 dy dy dx z COK. 2. Equation (3) may now be written in the simple form dx dy d z u fdy^ dx z \dt dxdydtdt ,djid^x_d_u_d^y_ to^ 2 " a^^"' or, if the independent variable t is not expressed, (5) 102.] SUCCESSIVE PARTIAL DIFFERENTIATION 177 If x be taken as independent variable, then t is to be re- d?x placed by x ; and since - = 0, the equation becomes da? d?u = d z u , o d 2 ^ dy d*u fdyV du d z y dx 2 dx 2 dx By dx dy 2 \dxj dy dx 2 ' Similarly, if y be taken as independent variable, and x be a function of y, then d?u _d 2 ufdx^ cy d^u dx d 2 u du d?x ~~ ~dy 2 ~dxdf EXERCISES 1. Verify that -^- = -&, when = zV. dxdy Byte 2. Verify that -^- = -^" when u = a?y + xy*. dy 2 ox 3. Verify that -- = --, when u = y log (1 + xy). dx dy dy dx 4. In Ex. 3 are there any exceptional values of x, y for which the relation is not true ? 5. Given =(x 2 + # 2 )s verify the formula 6. Given u = (a: 3 + y a p, show that the expression in the left member of the equation in Ex. 5 is equal to 4 7. Given = (x 2 + y + 2 2 )~^ ; prove that ^ + d + <& = 0. o-r 2 3^ 2 3z 2 8. Given u = sec (# + ax) + tan (y ax) ; prove that - = a 2 SJL ox (jy d*u d*u d*u 9. Given = sin x cos y ; verify that = = ," 2 2 dx 2 dy 2 10. Given M = (4 ab - c 2 )'^ ; prove that ^ = 178 DIFFERENTIAL CALCULUS [Cn. IX. 11. If u = sin v, r being a homogeneous function of degree n in x and w, determine the value of x + y dx dy 12. If u = tan-* xy - -- , show that -^- = (1 + * 2 + # 2 )~$ and Vl + x 2 + y* 103. Extension of Taylor's theorem to expansion of func- tions of two variables. Taylor's theorem, as developed in Chapter IV, relates only to functions of one variable, but it can be readily extended to functions of any number of variables in a manner first shown by Lagrange. Let f(x, y) be a function of the two variables x, y, which, with its first 2 n partial derivatives, as to x and y, is finite and continuous for all values of x, y within a certain por- tion of the coordinate plane. It is required to expand f(x -f A, y + &) in a series of ascending powers of h and of k. Using an auxiliary variable t, let x' = x + ht, y' = y + let, (1) then /(>' , y'} =f(x + ht, y + kt) = ^(0, say ; (2) the development of F(f) in powers of t is, by Maclaurin's theorem, 2 ! (*-!)! + ^ W y, [0<0<1. (3) W ! whence, putting t 1, T TTT " l C^) 102-103.] SUCCESSIVE PARTIAL DIFFERENTIATION 179 To express JF(0), jP(0) in terms of h, k, first find J?"(t), F'"(f), ' by successive differentiation of (2); then but, from (1), = A, -- = A;, hence Ctt> Ctv likewise, --7 dt\dx'J dt\dy'. = ,_ a 2 / (#, y) has a maximum value < (a, 5) for x = a, y = b if , as the variables pass in any manner through the values a and 5, the function hitherto increasing, ceases to increase and begins to de- crease ; the function has a minimum value (a, 6) if it ceases to decrease, and begins to increase, for every varia- tion of x and y through the values a and b. This fact is expressed analytically thus : <> (a, 6) is a maximum or minimum value of (x, y) according as the increment (a -f A, b + &) (a, 5) preserves a negative or a positive sign for all values of the increments h and k which are numerically less than a given small number m. If the function be represented by the ordinate of a sur- face, then a maximum (or minimum) ordinate $ (a, 6) is greater (or less) than every neighboring ordinate (a + A, b + &) drawn at any point (a + A, b + &), irre- spective of the signs and relative magnitudes of h and k. 107. Determination of maxima and minima. It was shown in Art. 79 that the necessary and sufficient condition that a function of one variable may have a maximum or a mini- 183 184 DIFFERENTIAL CALCULUS [Cn. X. mum for a given value of the variable is that its first deriva- tive change its sign as the variable increases through the given value. Similarly for a function of two variables, its differential must change its sign at a maximum or minimum, independent of the mode of variation of the variables through these values. Since dz = -2 dx + dy, dx ay ' and since either x or y may be varied alone, the first neces- sary condition is that the coefficients , change signs {jjC (ju separately ; otherwise it would be possible to find a mode (or direction) of variation in which dz does not change sign ; for instance, if does not change sign, then dz pre- dx serves its sign when dy is zero and x increases through a. Hence the critical values are those at which ^ = 0, $ = 0, dx dy or at which ^s ^ become infinite. dx dy To determine whether these values of x, y will give a maxi- mum or minimum value to z, it is usually impracticable to test the signs of 22, ^ for all neighboring values of #, y. dx dy It is consequently necessary to proceed to the higher deriva- tives. Usually, those values which make , infinite, dx dy will also make successive derivatives infinite ; hence such values will be excluded from the present mode of investiga- tion. As an example of a function which has a minimum, and yet has no partial derivatives, consider z = (x* + y*)l. 107.] MAXIMA AND MINIMA IN TWO VARIABLES 185 When x = and y = 0, then z = 0; but for every other value of x and of y, z must be positive ; hence z = is a minimum ; but = , d = at * = 0, y = 0. 5* fy First expand the function (a + 7i, 6 -f- &) in the vicinity of (a, 5) by Taylor's theorem ; thus <(a + A, J + &)_<( a , ft) = A^+fc^ dx By 2! I to 2 tody VI + higher powers of A, but.^=0, ^=0; hence dx By Criteria. To distinguish between a maximum and a mini- mum, at both of which -2=0, -5=0. it is usually suffi- Bx By cient to consider the sign of the expression involving terms of the second degree in A, k ; for A, k can generally be made so small that this expression numerically exceeds the sum of all the subsequent terms ; hence its sign will determine the sign of (a -f 7t, 6 + k~) (a, 6). When z = a, y = 6, let ^=^, -^-=5, ^=tf, a^- 2 dz fy fy 2 then the quadratic expression can be written in either of the forms \ (Ah* + 2 Bhk +Ck*) = - (^ 2 /i 2 + 2 ABJik + A Ck*) = JL [(Ah + Bk? + (AC- &) A: 2 ]. -i . i The first term of the numerator of the last form is always positive or zero ; the second term has the same sign sis 186 DIFFERENTIAL CALCULUS [Cn. X. AC S 2 . If the latter expression is positive, the numera- tor is positive for all values of h and k ; but if it is negative, the sign of the fraction will depend upon the values of h and &, and hence there can be no maximum nor mini- mum ; for instance, the numerator is positive when k = 0, and negative when h and k are so taken that Ah-\-Bk = Q. The second indispensable condition for a maximum or minimum is, therefore, & (a, b~) is a mini- mum ; if A is negative, (a, 6) is a maximum. It follows from the condition (1) that, since IP is posi- tive, A arid O must have the same sign. The whole process may be summarized as follows : to determine whether (#, y) has either a maximum or a minimum, equate its first partial derivatives to zero, and solve the resulting equations -2=0, -^ = 0, for #, y. Sub- stitute these critical values in the three second derivatives r-?i , ??.J then if 2-JjL ^ have the same sign, and ox* ox dy dy* ox* ay* f \dx dyj dx 2 dy* rflfh rfleh there is a maximum when the common sisrn of -4 and -? dz 2 dy* is negative, and a minimum when it is positive. It is instructive to examine the form of the representative surface in the vicinity of the critical point, especially when some of the conditions for a complete maximum or minimum are not satisfied. The geometric meaning of all the condi- 107] MAXIMA AND MINIMA IN TWO VARIABLES 187 tions (except the one regarding the sign of IPAC) is immediately evident by considering the conditions that the ordinate may have a turning value in each of the vertical sections parallel to the coordinate planes. The deportment of the ordinate in the intermediate sections depends on the sign of B l AC, the discriminant of the quadratic expres- sion in A, &, as will be illustrated in the examples. Special cases can arise in which J. = 0, 1?=0, f=0, or when IP A (7 = 0. It is then necessary to consider the higher degree terms. Instead, however, of finding general test formulas for such cases, it is better to work special examples independently. The higher degree terms can in many other cases be made to give useful information regard- ing the deportment of the function in the vicinity of the criti- cal value, especially in cases of incomplete maxima or minima. The method of Art. 167 will be helpful (Note C, p. 318). EXERCISES 1. Find the maximum and minimum values of < (* y) = 3 axy - x 3 - y*. Here dx dy The critical values are therefore x = a, y=a; z=0, y = 0. d* 2 dx dy dy* At x = 0, y = 0, A = 0, B = 3 a, C = 0, hence (0, 0) is neither a maximum nor a minimum. At x = a, y = a, A = -6a, B = 3a, C = - 6 a. In this case both 2_? and ~_r are negative, and B 2 < A C, hence (a, a) dx' 2 dy 2 has a maximum value a 8 . 2. Exhibit graphically the deportment of the function 2 = 1 - 4 * 2 + 21 x;i -'5 ?/ 2 f x 8 + y 3 in the vicinity of the critical point (0, 0). 188 DIFFERENTIAL CALCULUS [CH. X. It is here unnecessary to find the derivatives, as the function is already expanded in the vicinity of the point (0, 0), the letters x and y taking the place of the increments h and k. The absence of the first degree terms shows that the point (0, 0) is a critical point. As the discriminant of the second degree terms, 21' 2 4-4-5, is positive, the quadratic expression has real factors, and can therefore be made to change its sign for different ratios of y to x ; hence there is no complete maximum nor minimum. To distinguish the sections that have a maximum ordinate at this point, from those that have a minimum ordinate, write the equation in the form z = 1 - 5 - - 4x) + z 3 + y\ which shows that the second degree expression is zero when the ratio of y to x is either I OF 4 ; positive when this ratio lies between ^ and 4 ; and negative, for all other values of the ratio. Hence, all vertical sections y i within the acute angle between the directions - = - and -=4 have a X O X minimum ordinate at (0, 0) ; and all vertical sections within the obtuse angle have a maximum ordinate at this point. In the first transition y direction - = P . hence the increment of the ordinate is positive when x is positive, and negative when x is negative; and there is neither a maximum nor mini- KIG. 25. mum, but an inflexion, in the transition section. Similarly for the other transition direction. The two horizontal inflexional tangents in these 107.] MAXIMA AND MINIMA IN TWO VARIABLES 189 vertical sections are also tangents to the contour of the section made by the horizontal tangent plane through P (Fig. 25). Some idea of the form of the cubic surface at the critical point P is given in the figure. It shows the vertical sections XPX', YPY' in the coordinate planes, in both of which OP is a maximum; the transition sections APA', CPC', the contours of which bend upwards in the first quarter, and downwards in the third quarter; and an intermediate section BPB', in which OP is a minimum. If the third degree terms were absent, the transition contours A PA', CPC' would be straight lines, the surface would be a hyperbolic parabo- loid, and XYX'Y 1 would be a parallelogram. 3. Examine the deportment of the function z = - 70 + 38 a; - QOy - 10 x 2 + 12 xy - 15y 2 -f- 2x* - y* in the vicinity of the critical point (1, 2). Differentiation and substitution give dz _ A dz _ a% _ d*z- _ 3% _ dx dy dx* Bxdy d Hence the expansion of the function in the vicinity of the point (1, 2) is + h, -2 + ) = <(!,- 2) - 4 A 2 + 12 A- 9 2 + 2 A 3 - 8 . This is one of the exceptional cases referred to above, in which the discriminant B' 2 A C vanishes, and the terms of the second degree form a complete square. Thus, (\ +h, -2 + )-<(!, -2) = -(2A-3t) 2 + 2A-P, hence the increment of the function is negative for all small values of h and k, unless when k = ; and thus the ordinate <(!, 2) = 4 is a maximum in every vertical section but one. In this section the incre- ment of the function is A< = 2 A 8 (| h~) 3 = If A 3 , hence the contour of the section bends upwards in the first quarter and downwards in the third quarter. In Fig. 26, XPX' and YPY are the contours of the sections parallel to the coordinate planes, and A PA' is the contour of the vertical section in the intermediate direction k = | h. This may be regarded as a limiting case in which the two transition directions coincide. The hori- 190 DIFFERENTIAL CALCUL US [CH. X. zontal tangent plane at P cuts the surface in a curve which has a cusp at that point; the cuspidal tangent coinciding with the inflexional tan- gent to the vertical section just mentioned. FIG. 26. 4. Find the transition directions in exercise 1 for the critical point (0, 0), and show the form of the surface in the vicinity of the point. 5. Examine the function z = x 1 6 xy* + cy* at the point (0, 0). Show that if c > 9 there is a minimum ; and if c ~$> 9, neither maximum nor minimum. Draw graph. 6. Show that xe y+xsla " has neither a maximum nor a minimum. 7. Divide a into three parts such that their continued product may be a maximum. 8. Find the minimum surface of a rectangular parallelepiped whose volume is a 3 . 9. What value of x, y will make - $ ' * a maxmum or a minimum? 10. Find the values of x and y that make sin x + sin y + cos (x + y) a maximum or a minimum. 11. Find the maximum of (a x) (a y) (x + y a). 12. The electric time constant of a cylindric coil of wire is mxiiz = - r- - > ax + by + cz where x is the mean radius, y is the difference between the internal and external radii, z is the axial length, and m, a, b, c are known constants. The volume of the coil is nxyz = g. Find the values of x, y, z to make a minimum if the volume of the coil is fixed. (Perry's Calculus.) 107-108.] MAXIMA AND MINIMA IN TWO VARIABLES 191 108. Conditional maxima and minima. Maxima and minima of implicit functions. In certain problems the maximum or minimum values of a function of two variables are desired, when the mode of variation of x and y is re- stricted by an imposed condition. Let the function be z =f(x, i/), and let the assigned con- dition be (x, y) = ; then it is required to find the maxi- mum or minimum values passed through by the function 2, when x and y vary consistently with the relation (x, #) = 0. This problem may also be stated in the following geomet- rical form : A point moves on the surface 2 =/(#, y) in the curve of intersection made by the cylindrical surface $(X y) = > nn( l the maximum and minimum values of its height above the horizontal coordinate plane. Since the variables x and y always satisfy $(2:, y) = 0, hence their rates of change are connected by the relation but, since 2 is at a turning value, its rate of change vanishes, hence *-0| (2) .therefore, by elimination of dx and dy, "> ^\ -> > ox dy oy dx This equation, together with (x, y) = 0, will determine the critical values of x and y. The value of the function 2, corresponding to a critical value, will be a maximum or minimum according as d?z is negative or positive ; but 192 DIFFERENTIAL CALCULUS [Cn. X. W + d*x+d*y=Qi (5) BxBy By* Bx By to eliminate cPx, d*y, multiply (4) by -^ and (5) by SL, By By subtract, and take account of (3) ; then in which the subscripts 1, 2, indicate differentiation with regard to #, #, respectively. The sign of the right hand member of (6) is not changed by dividing by dx 2 , and then replacing -j- by ^, from (1); hence the sign of d?z at a #

2 -a;r), / n = 6a:, / 12 =-3a, / 22 = fi.y, ^ 1 = 2x, ^ 2 -2y, ^ u = 2, ^ 18 = 0, < 22 = 2. The critical values are found from ^> K /" 2 ^/j = and x 2 +y 2 = 8 a 2 ; and one pair is easily found to be x 2 a, y = 2 a. At this critical point / t = 6a 2 , / 2 = 6a 2 , / u = 12a, / 18 =-3a, / 22 =12a, ^ = 4:0, <^> 2 = 4a, < n = 2, <^) 12 = 0, ^> 22 =2; and the sign of the discriminating expression above is found to be posi- tive, showing that z is a minimum. Ex. 2. Show that the maximum and minimum of the function x 2 +# 2 , subject to the condition ax*+2 hxy + by* = l, are given by the roots of the quadratic equation 103.] MAXIMA AND MINIMA 1 IN TWO VARIABLES hence show how to find the axes of the conic defined by the above equation of condition. Ex. 3- Find the minimum value of a: 2 + # 2 , subject to the condition -+!=! a b NOTE. When the equation (x, y) = can be solved for one of the variables, the method of Art. 81 can also be used. IMPLICIT FUNCTIONS Let y be defined as a function of the single variable x by the implicit relation /(#, /) = ; it is required to find at what values of x the function y' passes through a maximum or a minimum. By successive differentiation, leaving the independent variable at first arbitrary, From (1) 3* = - ^; dx df hence the values of #, y at which ^ changes sign satisfy dx one of the equations Thus the first set of critical values of a, with the corre- sponding values of y, are to be found from the simultaneous equations, /(a,y) = o, = 0, independent variable in (2) and putting - = 0, ~- = 0, it 194 DIFFERENTIAL CALCULUS [Ch. X and the second set of critical values from /(*,y) = o, | = o. These two sets may or may not have values of x, y in common. Those of the first set that do not belong to the second set make 3- = 0, ^- = 0, and hence make -~ = 0. ox oy dx d\i To test whether -j- changes its sign, in passing through ctzc zero, the method of Art. 82 is available. Taking x as the indep gives ay <&y_ = _dx^ dx z df' dy Hence for the critical values under consideration, y is a maximum or a minimum according as * ~- have the same dx 2 dy or opposite signs. Those of the second set of critical values that do not be- df f)f long to the first set make -i- = 0, -~ = 0, and hence make ' By dx dV r> , 2aL infinite. ax To find whether ^ changes its sign, the second derivative ax is not available, since it and all subsequent derivatives are infinite ; but methods of trial may be resorted to, in which assistance can often be derived from the graphical represen- tation of the function. The critical values that are common to the first and the second set make -^- = 0, -i- = 0, and hence render -^ inde- dx dy dx 108.] MAXIMA AND MINIMA IN TWO VARIABLES 195 terminate in form. When numerically evaluated it is either zero, infinite, or finite. In the last case -*- cannot change ax its sign and there is no turning value of y. In the first two cases the question whether -^ changes its sign as x passes through the critical values, and y changes correspondingly, is to be decided by trial. Ex. 4. Given (x 2 + # 2 ) 2 - 2 y (a; 2 + # 2 ) - x* = ; find the turning values of y, and the corresponding values of x. dy _ 2x (x 2 + ?/ 2 ) - 2 xy - x dx~ the first set of critical values are found from (* 2 + Z/ 2 ) 2 - 2 y (^ + y 2 ) - x 2 = 0, (2) z[2(z 2 + ^)_2*,-l]=0. (3) Equation (3) is satisfied by x = 0, which, substituted in (2), gives y = 0, or y = 2. Equation (3) is also satisfied when 2(z 2 + # 2 )-2y-l = 0, i.e., when x 2 = y 2 + y + 5, which substituted in (2) gives y = }, whence x = .43 . Thus the first set of critical values of (x, y) is composed of the four pairs : (0, 0), (0, 2), (.43, - .25), (-.43, -.25). The second set is found from (2) and the equation 2y(x* + y*)-x*-3y 2 = Q, (4) which, on eliminating a: 2 , gives y = .75 or 0, whence x = 1.3 or 0. Thus the second set of critical values of x, y is composed of (0, 0), (1.3, .75), (-1.3, .75); the values (0, 0) being common to both sets. To test the remaining critical values of the first set, use the second derivative . flV rf 2 __6j_ 6 x 2 + 4 y 2 -*??/-! rfx 2 ~ df~ 2y(x* + y 2 ) - x 2 - 3 y* dy DIFF. CALC. - 14 196 DIFFERENTIAL CALCULUS [Cn. X. which, for (0, 2) is negative, and for (.43, -.25), (- .43, - .25) posi- tive ; hence, when x passes through 0, the function y passes through a maximum value 2, and when x passes through .43, .43, y passes through a minimum value .25. It is to be observed that in the latter case the function has three other values (or branches), real or imagi- nary, that do not pass through turning values when x passes through .43. To test the critical values (0, 0), for which equation (1) becomes inde- terminate, evaluate the function in the usual way, by replacing both numerator and denominator by their respective total x-derivatives. This gives dy_ _ __ _ dx, dx (4ay-2*)+(2*+6*-6)4l! dx a quadratic equation in - Now put x = 0, y = ; the two roots of the equation become infinite, hence ^ = oo. In the present case it is easy to flit ^ find by trial whether 2Z changes sign ; for in the vicinity of the values (0, 0) equation (1) may be written in the approximate form dx x 2 + 3 y 2 in which only the important terms are retained ; hence -^ changes sign from + to as x increases through zero, and thus y passes through a maximum. The values (0, 0) could also be shown to give a maximum without the use of derivatives, by observing that in the vicinity of the values (0, 0) equation (2) can be replaced by x 2 + 2 y 8 = 0. When x is small, and either positive or negative, y must be negative ; but when x = 0, then y = ; hence y = is a maximum value of the function. It is not easy to test the other critical values at which -^ becomes dx infinite without anticipating the methods of curve tracing. It will appear by the methods of Chapter XVIII that the graph of the function is as 108.] MAXIMA AND MINIMA IN TWO VARIABLES 197 in the accompanying figure, and that the critical values last mentioned are neither maxima nor minima values for y. FIG. 27. Ex. 5. Given x* + 2 ax*y ay 8 = ; find the maximum and minimum values of y, and of x. Ex. 6. If x 8 + y 3 3 axy = 0; find the maxima and minima of y. Ex. 7. If 3 a 2 # 2 + xy 9 + 4 az 8 = ; find the turning values of x, y. Ex. 8. Show that in the vicinity of a maximum or minimum value of f(x, y), the increment A/ (a:, y) is an infinitesimal of an even order, when Aa: and Ay are of the first order. When is &f(x, ?/) of the third order? CHAPTER XI CHANGE OF THE VARIABLE 109. Interchange of dependent and independent variables. It has already been proved in Art. 22, as the direct conse- quence of the definition of a derivative, that if y=$(x*), then = _ dx dx dy (1) This process is known as changing the independent varia- ble from x to y. The corresponding relation for the higher derivatives is less simple, and will now be developed. rr- To express as to #, *- in terms of , -, differentiate (1) dx 2 dy dy 2 cPy _ d 1 d 1 dy d 1 1 dx 2 dx dx ~ dy dx dx dy dx dx .dy j dy ; dy dy but dy dx dy hence dx 2 dy 198 (2) CH. XI. 109-111.] CHANGE OF VARIABLE 199 In a similar manner, dx d*y_ dfdy \df dx* 110. Change of the dependent variable. If y is a function of z, let it be required to express -^, #, ... in terms of 7 ax ax* Let y = $(2), then = yz_ = ,, ( dz dx dx* ^_d_(dy\ f dz_ = d^(.,.dz\ m dz dx 2 dz\dx) dx dzV dx) dx - -- dx) J dz\dx) dx but the second term can be expressed directly as (b'(z) , , dx 2 hence ,. (4) The higher ^-derivatives of y can be similarly expressed in terms of ^-derivatives of 2. Ex. Show that (4) may be regarded as a special case of (6), Art. 102, in which one of the variables is replaced by a constant. 111. Change of the independent variable. Let y be a function of #, and let both x and y be functions of a new variable t . It is required to express ^- in terms of -- ; and d*y . f dy,d*y dx 4 in terms of -jL and ^. dx 2 dt dt* 200 DIFFERENTIAL CALCULUS [CH, XI. By Arts. 21, 51, dy dy _dt n Tx~^ dt d?y dx d 2 x dy & ~di?~dt~W ~dt dt If x be given as an explicit function of t in the form =/(0, then l - may be written x=f(f), then -=/'(^), - =/"(), and the last equation dt at* (3) [/CO] In practical examples it is usually better to work by the methods here illustrated than to use the resulting formulas. EXERCISES 1. Change the independent variable from x to z in the equation dx* dx dy dy ^ = 2 t dx dz dx*~~dz* 6 ~ dz 6 Hence x* =3L + x '-& + y = becomes 22! 4. ?/ = o. dx* dx dz* 2. Interchange the function and the variable in the equation d*y c dx~* 111-112.] CHANGE OF VARIABLE 201 3. Interchange x and y in the equation R = rfz 2 4. Change the independent variable from a; to y in the equation = 0. *l dxdx* dx 2 \dx 5. Change the dependent variable from y to z in the equation + y 2 6. Change the independent variable from x to y in the equation 9 d 2 u . du n , a; 2 - + Z -- 1- = 0, when w = loga:. dx 2 dx 7. If y is a function of x, and a: a function of the time t, express the //-acceleration in terms of the x-acceleration, and the a;-velocity. Since dt dx dt but dt\dx) dx\dxl dt dx* dt In the abbreviated notation for ^-derivatives, Compare this result with (4), Art. 110, and with (6), Art. 102. 112. Change of two independent variables. Let u=f(x, y) be a function of the two variables x, y which are themselves functions of two new variables w, 2 ; it is required to ex- du du du du press , in terms of - , dx dy dw dz 202 DIFFERENTIAL CALCULUS [Cn. XL I. The variables #, y. explicit functions of w, z. Let w=/O, y); x = 1 (w, 2); y = < 2 (w, 2). Since w is the function of w and 2, dw df dx . df dy , TJ , N = -^ --- f--^._!Z- (2 regarded as constant). dw dx dw dy dw df o?/ The values of , -^ are to be found from x = * , df 3<6o Similarly, = f- -fl + ^- - d2 02; 02 02/ 02 J (1) (w regarded as constant). O?/ In the expression for , 2 is to be regarded as constant, dw and w as variable ; and re, y as functions of w. o?y In the expression for , w is to be regarded as constant, and 2 as variable ; and re, y as functions of 2. If re, y be called the old variables, and 2, w the new variables, then it appears from the above expressions that when the old variables are explicit functions of the new variables, the new derivatives , - are explicit functions dw dz f\ ^ of the old derivatives , The last two equations may, ox dy when desired, be solved for -^, -- dx dy II. The variables w>, 2 explicit functions of re, y. Let 2 = ^(re, ?/); w = ^(^ #)' ,i du du dw . du dz ,- i j ,N then = 1 (y regarded as constant), dx dw dx dz dx du du dw . du dz , j -, ,^. = 1 (x regarded as constant). dy dw dy dz dy 112.] CHANGE OF VARIABLE 203 a u i-i i- 1.1. i r dt# dtfl ^2 62 ,. Substituting the values of , , , trom z = -dr., dx dy dx dy w = T|r 2 , the last equations become du _ du cty 2 dw di/^ da; dw dx dz dx ' (2) du __ du o>/r 2 dw oy 1 3y 3w 9y dz dy i j^ These equations may, when desired, be solved for , . dz dw In this case the new variables are explicit functions of the old ones, while the old derivatives are explicit functions of the new ones. Ex. Let u = x 2 y 2 , x pcosO, y = psinO. Find , 2^ by the method of I. dp d& l . 2 + 2H 2U. (^ regarded as constant), dp dx dp dy dp = 2x cos 2 y sin 0, = 2pcos 2 0-2psin 2 0, = 2pcos20, which agrees with the result of direct substitution. = 2 xp sin 2 yp cos 0, = 4 p 2 cos 6 sin 0, = -2p 2 sin20, which also agrees with the result of direct substitution. Next suppose the new variables p, 6 are expressed in terms of the old variables x, y in the form p = Vx 2 + y 2 , 6 = tan- 1 ^ ; find |^, ^ by the method of II. " Here 204 DIFFERENTIAL CALCULUS [Cn. XI. but &=2x = <^& + <^M , regarded ox dp dx dv dx hence 2pcos0 = <* cos0 - ^ . Op OP p also, ^? = _2#=^^ + ^"<^ (x regarded as constant), dy dp dy 30 dy hence - 2psin 6 = ^ sin0 + & i (2) op do p Now, solving (1) and (2) for f, f-s it follows that op 06* op the same results as were obtained before. III. The relation between #, y and w, z denned by implicit equations. Let /!<>, y, z, w') = 0, / 2 (>, y, z, w~) = 0. *\ *-| In the first place, to find , -*L required in I (1), differ- oz dz entiate the two given equations partially with regard to 2, then , dz dx dz dy dz (w regarded as constant) & + ^^+ 2 (>, w, w), z = < 3 O, v, M>). (1) df It is required to express -^- in terms of w, v, w. O2/ df df dx df By df dz , ^ = ^ - + -^-JL + ^- (v, w regarded as constants), du dx du dy du dz du df df dx df dy dfdz , ^ = ^- _ 4--^^. + ^ (u,w regarded as constants), dv dx dv dy dv dz dv df df dx df dy df dz , . N = - -^- 4- - (w, v regarded as constants). dw dx dw dy aw dz dw From (1), , , , 2", can be found ; their values du dv div du .j/. are to be substituted in the equations for ~- , , and the re- *\/* * / ^ w suiting equations solved for -i-, X, . dx dy Similarly for the case in which w, v, w are explicit func- tions of #, z. 114. Application to higher derivatives. The second and higher derivatives can be obtained in the same way. As the general formulas become too complicated to be of much use, it is better to work out special examples independently. Ex. Express 2 -- (- _ i n terms of p, 0, given dx 1 dy 2 x = p cos 0, y = p sin 0. (1) The general formula is 5" _ du dp_ , 5" 9^ . (2) dx dp Qx dO dx ' in which ^, ^ are to be obtained from (1), by differentiating and dx dx solving. 206 DIFFERENTIAL CALCULUS [CH. xi. Thus I=^cos0-psin0^, dx dx (y regarded as constant) ; hence & = cos 80 = _ sin_0 (3) dx dx p IT. J3/3 Similarly ^, - can be obtained from (1) : By dy dy dy (x regarded as constant) ; . , dy dy hence & = sin0, M = *1. (4) dy dy p Substitution from (3) in (2) gives ^ = du c0 s0-^^. (5) dx dp dO p A repetition of this process gives d 2 u sin cos dMsin 2 d 2 cos sin dx 2 dp 2 dO dp p ^P P dO dp p d 2 u sin 2 du cos sin du sin cos xgv W>~p T ~ dO ~?~ 50 p^"" The expression, similar to (2), for 5p combined with (4), leads to ^ = ^sin0 + ^^, (7) dy dp d6 p ' and when this step is repeated, there results, &lL = d^. 8 i n 2 . d 2 u sin cos du cos 2 3 2 u cos sin d.y 2 dp 2 dOdp p dp p dOdp p d 2 u cos 2 d u cos sin gn sin cos . xg^ 60' p 2 50 p 1 " 50 /^~ and the addition of (6) and (8) gives the required identity 5 2 d 2 M = d 2 u 1 3 1 5 2 u ' 114.] CHANGE OF VARIABLE 207 EXERCISES i. Given x = p cos 0, y = p sin 0, y being a function of x, show that _ >I6 dx* dO 2. Given x = a(l cos /), y = a(n< + sin <); prove that rf' 2 y _ _ n cos < + 1 rfa; 2 a sin* t 3. If | = a: cos a y sin a, 77 = x sin a + y cos a, prove that 4. Given z = p cos 0, y = p sin ^, show that X 5_y5u = 5; x 5 + y5 a. 3z y dy dO dx y dy p dp 5. If x = p cos 0, y = p sin 0, show that the expression APPLICATIONS TO GEOMETRY CHAPTER XII TANGENTS AND NORMALS 115. It was shown in Art. 17 that if f(x, y) = be the equation of a plane curve, then -^ measures the slope of ax the tangent to the curve at the point x, y. The slope at a particular point (x r y^) will be denoted by -^, meaning ax \ Qf that x 1 is to be substituted for #, and y^ for y in ^ = -- ax oj after the differentiation has been performed. ' 3"^ 116. Equation of tangent and normal at a given point. Since the tangent line goes through the given point (x v y) and has the slope -^ , its equation is tt#i The normal to the curve at the point (x v y^) is the straight line through this point, perpendicular to the tangent, 208 CH. XII. 115-117.] TANGENTS AND NORMALS dx 1 Its equation is, since . = by Art. 22, dy dy dx 209 i.e. (2) 117. Length of tangent, normal, subtangent, subnormal. The portion of the tangent and normal intercepted between the point of tangency and the axis OX are called, respec- tively, the tangent length and the normal length; and their projections on OX are called the subtangent and the sub- normal. FIG. 28 a. FIQ. 286. Thus, in Fig. 28, let the tangent and normal at P to the curve P 7 meet the axis OX in 2* and N, and let MP be the ordinate of P, then TP is the tangent length, PN the normal length, TM the subtangent, MN the subnormal, which will be denoted, respectively, by , w, T, v. Let the angle XTP be , then tan < = m, say : m Vf sn < = m* 210 DIFFERENTIAL CALCULUS [Cn. XII. The subtangent is measured from the intersection of the tangent to the foot of the ordinate ; it is therefore positive when the foot of the ordinate is to the right of the intersec- tion of tangent. The subnormal is measured from the foot of the ordinate to the intersection of normal, and is posi- tive when the normal cuts OX to the right of the foot of the ordinate. Both are therefore positive or negative, ac- cording as < is acute or obtuse. The expressions for r, v may also be obtained by finding from equations (1), (2), Art. 116, the intercepts made by the tangent and normal on the axis OX. The intercept of the tangent subtracted from x l gives T, and x-^ subtracted from the intercept of the normal gives v. EXERCISES 1. In the curve y (x 1) (x 2) = x 3, show that the tangent is parallel to the axis of x at the points for which x = 3 V2. 2. Write down the equations of the tangents and normals to the curve y = at the points for which y = - a* + x* 4 3. Find the equations of the tangents and normals at the point (x } , y^) on each of the following curves : (a) x 2 + y 2 = c 2 , (c) xy (x + y) = a 8 , (i) xy = k 2 , (d) e = sin x. X 4. Prove that - -f - = 1 touches the curve y = be~ at the point in a b which the latter crosses the axis of y. 5. Find the points on the curve -r=<*-J><*-2)(*-8) at which the tangent is parallel to the axis of x. 117.] TANGENTS AND NORMALS 211 6. Find the intercepts made upon the axes by the tangent at (x p y x ) to the curve Vx + Vy = Va, and show that their sum is constant. 7. In the curve x 2 y 2 = a?(x + #), the tangent at the origin is inclined at an angle of 135 to the axis of x. 8. In the curve x? + yi = a, find the length of the perpendicular from the origin on the tangent at (x r y^) ; and the length of that part of the tangent which is intercepted between the two axes. (A. G., p. 323.) 9. Show that all the curves represented by the equation when different values are given to n, touch each other at the point (a, b). 10. Show that all the points of the curve # 2 = 4a (x + asin-Y at which the tangent is parallel to the axis OX lie on a certain parabola. 11. Prove that the parabola y 1 4 ax has a constant subnormal. 12. Prove that the circle x 1 + y 2 = a? has a constant normal. 13. Show that in the tractrix, the length of the tangent is constant ; the equation of the tractrix being / n. x = Vc' 2 - y' 2 + log 2 c + Vc 2 - y' 2 X 14. Show that the exponential curve y = ae e has a constant subtan- gent. 15. At what angle does the circle x* + y z = 8ax intersect the cissoid ~3 16. Find the subtangent of the cissoid y 2 = -- 2 a x x _n 17. Find the normal length of the catenary y = ^(* 4- e ). 18. Show that the only Cartesian (j, .y) curve in which the ratio of the subtangent to the subnormal is constant is a straight line. 19. Show that the equation of the tangent to the curve f(x, y) = at the point (x,, y^ may be written DIFF. CALC. 15 212 DIFFERENTIAL CALCULUS [Cn. XII. 20. Prove that the equation of the tangent to the curve x 3 - 3 axy + y 3 = may be written xfa ax r y axy l + POLAR COORDINATES 118. When the equation of a curve is expressed in polar coordinates, the vectorial angle 6 is usually regarded as the independent variable. To determine the direction of the curve at any point, it is most convenient to express the angle between the tangent and the radius vector to the point of tangency. Let P, Q be two points on the curve (Fig. 29). Join P, Q with the pole 0, and drop a perpen- dicular PM from P on OQ. Let />, 6 be the coordinates of P ; p + A/o, + A0 those of Q ; then the angle and -p cos A0 ; p sin A0 FJS. 29. hence tan M QP = p + A/a p cos A0 When Q moves to coincidence with P, the angle MQP approaches as a limit the angle between the radius vector and the tangent line at the point P. This angle will be designated by i/r. Thus lim p sin Afl but hence p(l tani/r= A /)^ A -> u p + A/> p cos A0 cos A0) = 2 /> sin 2 1 A0, /a sin A0 A0 117-120.] but Therefore TANGENTS AND NORMALS lim sin A0 _ . Hm sin | A0 _p_ d0 d6 213 Examples on dynamical interpretation. Ex. 1. A point describes a circle of radius p; prove that at any instant the arc velocity is p times the angle velocity; i.e., dt dt Ex. 2. When a point describes any curve, prove that at any instant the ve- locity has a radius component f -& and J dt dt a circle component Jft , and hence that dt do i fid , i dO = -*-, sin y = p , tan y = p ds ds dp = 1. FIG. 30. Fiu. 31. 119 Relation between ^ and . If the initial line be taken as the axis of x, the tangent line at P makes an angle with this line by Art. 117. Hence t.e., \dpj dx FIG. 82. 120. Length of tangent, normal, polar subtangent, and polar subnormal. The portions of the tangent and normal inter- cepted between the point of tangency P and the line through the pole perpendicular to the radius vector OP to the point of tangency, are called the polar tangent length and the polar 214 DIFFERENTIAL CALCULUS [Cn. XII. normal length ; and their projections on this perpendicular are called the polar mbtangent and polar subnormal. M FIG. 336. Thus, let the tangent and normal at P meet the perpendic- ular to OP in the points N, M. Then PN is the polar tangent length, PM is the polar normal length, ON is the polar subtangent, OM is the polar subnormal. They are all seen to be independent of the position of the initial line. The lengths of these lines will now be consid- ered. Since PN= OP sec OPN = p sec ^ = p\p z (~}\ 1 \dpj hence polar tangent length = p-\p 2 + ( -^ ) . dp \dv/ jn Again, ON '= OP tan OPN= p tan ^ = p*j-, hence polar subtangent = p 2 - dp 120.] TANGENTS AND NORMALS 215 PM = OP csc OPN = p esc f = X/3 2 + * hence polar normal length = 04f = OP cot OPN =^ pdv hence polar subnormal = - do The signs of the polar tangent length and polar normal length are ambiguous on account of the radical. The di- rection of the subtaugent is determined by the sign of p 2 : when is positive, the distance ON should be meas- dp dp ured to the right, and when negative, to the left of an observer placed at and looking along OP; for when increases with p, is positive (Art. 20), and ty is an acute dp -IQ angle (as in Fig. 33 b~) ; when 6 decreases as p increases, is negative, and \fr is obtuse (Fig. 33 a). EXERCISES 1. Show that the polar subtangent is constant in the curve pO = a. 2 Show that in the curve p = a e flcot , the tangent makes a constant angle with the radius vector. For this reason, this curve is called the equiangular spiral. (A. G., p. 330.) 3. For the same curve as in Ex. 2, find the polar subtangent and polar subnormal. 4. Find the angle of intersection of the curves p = a (1 + cos 0), p = b (1 - cos 0). 5. In the circle p = a sin 0, find \f/ and <. 6. In the curve p = aO, show that tan ^ = 6, and that the polar sub- normal is constant. (A. G., p. 325.) A 7. In the parabola p a sec 2 -, show that < + ^ = TT. CHAPTER XIII DERIVATIVE OF AN ARC, AREA, VOLUME AND SURFACE OF REVOLUTION 121. Derivative of an arc. The length of the arc AP of a given curve y =/(#), measured from a fixed point A to any point -P, is a function of the abscissa x of the latter point, and may be expressed by a relation of the form *() The determination of the function when the form of / is known, is an important and sometimes difficult problem in the Integral Calculus. The first step in its solution is ds to determine the form of the derivative function = $'(2)' ax which is easily done by the methods of the Differential Cal- culus. Let PQ be two points on the curve (Fig 34); let #, y be the coordinates of P; x + Az, Fie. 34. y + Ay those of Q ; 8 the length of the arc AP ; s + As that of th arc AQ. Draw the ordinates MP, NQ ; and draw PR parallel toMN; thenPJ8 = Aa?,J2^ = Ay; arc PQ = As. Hence Chord PQ = V(Az) 2 +(Ay) 2 , 216 CH. XIII. 121-122.] DERIVATIVES OF ARC, AREA, ETC. 217 r , A* As PQ As /T, /Ay\ a Therefore , = -- s , = A I -f I ) Ax P# Ax PQ^ \bx) Taking the limit of both members as Ax = and putting = 1, by Art. 13, Th. 4, and Art. 10, Th. 10, Cor., s ^ , x - = \ 'I + ) dy \dyj it follows that a- -i i Similarly and i.e. s = i + y 122. Trigonometric meaning of Ax Since = Aa; As . PQ As it follows, by taking the limit, as As = 0, that ^ ON (2) (4) -' dy = esc wherein <, being the limit of the angle RPQ, is the angle which the tangent drawn at the point (x, y) makes with the x-axis. Similarly, -^ = sin ; whence - = sec ; ds dx Using the idea of a rate or dif- ferential, all these relations may be conveniently exhibited by Fig. 35. These results may also be de- rived from equations (1), (2) of _ dy Art. 121, by putting -/- = tan . {,(, T IS. SO. 218 DIFFERENTIAL CALCULUS [Cn. XIII. 123. Derivative of the volume of a solid of revolution. Let the curve APQ revolve about the z-axis, and thus gen- erate a surface of revolution ; let V be the volume included between this surface, the fixed initial plane face generated by the ordinate AB, and the terminal face generated by any ordinate MP. Let AF be the volume generated by the area PMNQ; then A V lies between the volumes of the cylinders gener- ated by the rectangles PMNR and SMNQ ; that is, Try 2 Ax < A V< TT(J/ + Ay) 2 Az. Dividing by Ax and taking limits, dV = TT y\ dx 124. Derivative of a surface of revolution. Let S be the area of the surface generated by the arc AP (Fig. 36) ; and that by the arc PQ, whose length is As. Draw PQ', QP' parallel to OX and equal in length to the arc PQ; then it may be assumed as an axiom that the area generated by PQ lies between the areas gen- _K. erated by PQ' and P'Q; i.e., M N Fl - 36 - 2 Try As < AS < 2 7r(y + Ay) As. Dividing by As and passing to the limit, dS n ^ -T- = 2 try, (1) as dx ds dx --"- ' l ' ' 123-125.] DERIVATIVES OF ARC, AREA, ETC. 219 125. Derivative of arc in polar coordinates. Let p, be the coordinates of P ; p + Ap, 6 + A0 those of Q ; the length of the arc KP ; As that of arc PQ. Let PM be perpendicular to OQ; then = p(l cos A0) + A/a Hence P Q 2 = (p sin A0) 2 + (2 /> sin 2 A0 ... p sm Replacing the first member by f '-} , passing to the \ taAS LAC7/ limit when A# = 0, and putting lim - = 1, lira - 3 = 1, i \a As A0 T sin .i Ac/ ., ., r n ,1 Inn - 2- = 1, it follows that In the rate or differential notation this relation may be conveniently written and its dynamic interpretation is shown in the figure of Art. 118 (Fig. 31). 220 DIFFERENTIAL CALCULUS [Cn. XIII. 126. 126. Derivative of area in polar coordinates. Let A be the area of OKP measured from a fixed radius vector OK Q to any other radius vector OP; let A-4 be the area of OPQ. 'N Draw arcs PM, QN, with as a center; then the area POQ L lies between the areas of the sectors 0PM and ONQ ; i.e., ;-i-(> + A/>) 2 A0. Dividing by A# and passing to the limit, when A0 = 0, it follows that For the derivative of the area of a curve in #, y coordi- nates, see Art. 17. The result is - = y. ax EXERCISES 1. Given *- a + 2_ a = i ; find f*L, * *2> *L . a 2 ft 2 dx dx dx dx 2. Similarly for the parabola y 2 = 4 ax. 3. In the curve e(e* - 1) = e x + 1, sho ~ j. 4. If < be the eccentric angle of the ellipse h ^- = 1, prove that t 2 fe 2 = o Vl e 2 cos 2 <, e being the eccentricity. [dx = a sin d, dy = b cos d(f>, ds" (a 2 sin 2 + b 2 cos 2 ^>)rf^> 2 , etc.] 5. Given P = acosO; find ^, ^- da dd 6. In p 2 = a 2 cos 2 0, show that ~ = <> dv p 7. Given p = a (1 + cos ^), prove ^= V2ap. CHAPTER XIV ASYMPTOTES 127. When a curve has a branch extending to infinity, the tangents drawn at successive points of this branch may tend to coincide with a definite fixed line as in the familiar case of the hyperbola ; or, on the other hand, the successive tangents may move further and further out of the field as in the parabola. These two kinds of infinite branches may be called hyperbolic and parabolic. The character of each of the infinite branches of a curve can always be determined when the equation of the curve is known. 128. Definition of a rectilinear asymptote. If the tangents at successive points of a curve approach a fixed straight line as a limiting position when the point of contact moves farther and further along any infinite branch of the given curve, then the fixed line is called an asymptote of the carve. This definition may be stated more briefly but less pre- cisely as follows: An asymptote to a curve is a tangent whose point of contact is at infinity, but which is not itself entirely at infinity. DETERMINATION OF ASYMPTOTES 129. Method of limiting intercepts. The equation of the tangent at any point (x^ , y^) being 222 DIFFERENTIAL CALCULUS [CH. xiv the intercepts made by this line on the coordinate axes are fy\ (1) dx y\- Suppose the curve has a branch on which x = oo and y = oo ; then from (1) the limits can be found to which the intercepts # , y approach as the coordinates x l , y l of the point of contact tend to become infinite. If these limits be denoted by a, 6, the equation of the corresponding asymptote is Ex. 1. Find the Asymptotes of the curve y 2 = 4 x 2 + 2 x + G. Since dx hence , = ,- = dy 4 x + 1 = > and this = 4:X + 1 4 when x = oo. du y* -4x* -x y y x 3- = 2 . dx x + 6 y Fl. 39. V4 a: 2 + 2 z + 6 To evaluate this expression, square both terms, and then apply the rule of Art. 73. The value of the square is \; thus, y = \. Hence the asymptotes are l, y = - 2 * - J. 129.] ASYMPTOTES 223 Ex. 2. Find the equations of the asymptotes of the curve * 2 + 3 xy + 2 y 2 + 3 x - "2 y + 1 = 0. Here- ^/y = _ 2x + 3,y + 3. dx 3 x + 4 y 2 hence substituting in (1), and omitting the subscripts throughout the right-hand member, y = 3 x + 4 y - 2 Replacing z 2 + 3 zy + 2 y 2 by -3ar + 2y + l from the given equa- tion, this becomes ty\ 2 r .,- . n " + 2( 3+4 ()_! Va;/ a: Next, to find the limit of - as y = co, x = co, observe that the terms 3ar, 2^, 1 are infinities of a lower order (1 is an infinite of order 0) than x 2 , xy, y 2 ; hence, for large values of x and y, the terms of the second degree would have most effect in fixing the form of the curve ; and in the limit, when x = GO and y = -x>, the smaller terms can be neglected. Then the equation becomes Hence, on one branch ^ = , and on the other, " = _ 1. x 2 x Using these limiting values for ^ in the values of y n , yn .-3 + 2(-i) * apd ^-S+^-l) 3 + 4(_) 3 + 4(_i) on the respective branches. Similarly for the z-intercept, after reduction, - 3x + 2y -2 5, when - = 1 ; and = - 8, when - = = 224 DIFFERENTIAL CALCULUS [CH. XIV. The equations of the asymptotes are therefore .e, x + y = 5, x + 2y + 8 = 0. Except in special cases this method is usually too compli- cated to be of practical use in determining the equations of the asymptotes of a given curve. There are three other principal methods, of which at least one will always suffice to determine the asymptotes of . curves whose equations involve only algebraic functions. These may be called the methods of inspection, of substitution, and of expansion. 130. Method of inspection. Infinite ordinates, asymptotes parallel to axes. When an algebraic equation in two coordinates x and y is rationalized, cleared of fractions, and arranged according to powers of one of the coordinates, say y, it takes the form ay" + (bx + e)y n ~ l + (d& + ex +/)y~ 2 + - + *-# + u n = 0, in which u n is a polynomial of the degree n in terms of the other coordinate x. When any value is given to x, the equation gives n values to y. Let it be required to find for what value of x the corre- sponding! ordinate y has an infinite value. Suppose at first that the term in y n is present ; in other words, that the coefficient a is not zero. Then when any finite value is given to x, all of the n values of y are finite, and there are thus no infinite ordinates for finite values of the abscissa. Next suppose that a is zero, and i, c not zero. In this case one value of y is infinite for every finite value of a;, and 129-130.] ASYMPTOTES 225 thus one branch of the curve lies entirely at infinity. It is shown in projective geometry that this branch always has the form of a straight line. In this work no account will be taken of such branches, and the wording of the theorems will in no case refer to them. There is one particular value of x that gives one additional infinite value to ^, namely, the value x = - ; for this makes o bx + c (the coefficient of the highest power of y) zero, and hence from the theory of .equations one corresponding value of y must be infinite; and this value is finite when x 3= The equation of the infinite ordinate is bx + c = 0. Again, if not only a, but also 6 and c, are zero, there are two values of x that make y infinite ; namely, those values of x that make dz z + ex+f = 0, and the equations of the infinite ordinates are found by factoring this last equation ; and so on. Similarly, by arranging the equation of the curve accord- ing to powers of x, it is easy to find what values of y give an infinite value to x. Ex. 3. In the curve 2 x s + x*y + xy* = x* - y* - 5, find the equation of the infinite ordinate, and determine the finite point in which this line meets the curve. This is a cubic equation in which the coefficient of y 3 . is zero. Arranged in powers of y it is y 2 (x + 1) + yx* + (2 x* - z 2 + 5) = 0. When x = 1, the equation for y becomes . y* + y + 2 = 0, the two roots of which are y = co, y = 2 ; hence the equation of the infinite ordinate is x + 1 0. The infinite ordinate meets the curve again in the finite point ( 1, 2). Since the term in x 3 is present, there are no infinite values of x for finite values of y. 226 DIFFERENTIAL CALCULUS [Cn. XIV. Ex. 4. In the curve x*y + 5xf + 2 x 2 = 3 x*y + 6, find what values of x make y infinite and what values of y make x infinite. 131. Infinite ordinates are asymptotes. Applying to the general equation of the last article the method of Art. 129, the slope of the tangent at (x, y) is -^ dx Now, the first condition that y may become infinite for a finite value of #, is a = ; but when a is zero, x finite, and y infinite, the numerator is an infinite of higher order than the denominator, hence ^=00, when x = and y = cc. dx o Therefore the inclination of the tangent approaches nearer and nearer to 90, and the tangent approaches to coincidence M with the ordinate through the point x = - ; and thus this line is an asymptote parallel to the y-axis. Similarly, if the value y = k gives an infinite value to #, then the line y k is an asymptote parallel to the a>axis. Thus* to determine all the asymptotes parallel to the /-axis, equate to zero the coefficient of the highest power of y, if it be not a constant. If this equation be of the first degree, it represents an asymptote parallel to the y-axis. If it be of higher degree, it may be resolved into first degree equations, each of which represents such an asymptote. Similarly, to determine all the asymptotes parallel to the a>axis, equate to zero the coefficient of the highest power of a;, if it be not a constant. 130-132.] ASYMPTOTES 227 Ex. 5. In the curve a 2 x = y(x a) 2 , the line y = is an asymptote coincident with the z-axis, and the line x = a is an asymptote parallel to the y-axis. FIG. 40. Ex. 6. Find the asymptotes of the curve x z (y a) + xy 2 = a 8 . 132. Method of substitution. Oblique asymptotes. The asymptotes that are not parallel to either axis can be found by the method of substitution, which ' is applicable to all algebraic curves, and is of especial value when the equation is given in the implicit form /O, y) = o. (i) Consider the straight line y = mx + 5, (2) and let it be required to determine m and b so that this line shall be an asymptote to the curve /(#, y) = 0. Since an asymptote is the limiting position of a line that meets the curve in two points that tend to coincide at in- finity, then, by making (1) and (2) simultaneous, the result- ing equation in #, f(x, mx + 6) = 0, is to have two of its roots infinite. This requires that the coefficients of the two highest powers of x- shall vanish. DIFF. CALC. 16 228 DIFFERENTIAL CALCULUS [CH. XIV. These coefficients, equated to zero, furnish two equations, from which the required values of in and b can be deter- mined ; and these values, substituted in (2), will give the equation of an asymptote. Ex. 7. Find the asymptotes to the curve y* = x 1 (2 a x). In the first place, there are evidently no asymptotes parallel to either of the coordinate axes. To determine the oblique asymptotes, make the equation of the curve simultaneous with y = mx + b, and eliminate y, then (mx -f J) 8 = a; 2 (2 a x), or, arranged in powers of x, (1 + m 3 ) x s + (3 m*b - 2 a) x 2 + 3 b*mx + b a = 0. Let in 8 + 1 = and 3 rri*b - 2 a = 0, 1 * 2a then m = 1, b = ; hence y = x + is the equation of an asymptote. The third intersection of this line with the given cubic is found from the equation 3 mb 2 x + b s = 0. Y Fio. 41. 132-133.] whence ASYMPTOTES 2a 229 This is the only oblique asymptote, as the other roots of the equation for TO are imaginary. Ex. 8. Find the asymptotes to the curve y(a 2 + z 2 ) = a 2 (a z). T FIG. 42. Here the line y = is a horizontal asymptote by Art. 130. To find the oblique asymptotes, put y = mx + b, then (mx + ft) (a 2 + x 2 ) = a 2 ( - x) ; i.e., mx* + bx 2 + (ma 2 + a 2 )* + (a 2 6 - a 8 ) = 0, hence m = 0, 6 = 0, for an asymptote. Thus the only asymptote is the line y = 0, already found. 133. Number of asymptotes. The illustrations of the last article show that if all the terms be present in the general equation of an wth degree curve, then the equation for determining m is of the nth degree, and there are accordingly n values of m, real or imaginary. The equation for finding b is usually of the first degree, but for certain curves, when y has been replaced by mx -f- 6, one or more values of w, say T/IJ, may cause the coefficient of re" and x"" 1 both to vanish, irrespective of b. In such cases any line whose equation is of the form y = m^x + c will satisfy the definition of an asymptote, independent of c ; but by equating the coefficient of a^~ 2 to zero, two values of b can be found such that the resulting lines have three points at infinity in common with the curve. These two lines are parallel ; and it will be seen 230 DIFFERENTIAL CALCULUS [Cn. XIV. that in each case in which this happens the equation defining m has a double root, so that the total number of asymptotes is not increased. Hence the total number of asymptotes, real and imaginary, is in general equal to the degree of the equation of the curve. It is to be observed, however, that in special cases (i.e., for certain special values of the given coefficients) two or more of these lines may coincide, and moreover that some of these n "tangents at infinity" may be situated entirely at infinity and thus be improperly called asymptotes. Since the imaginary values of m occur in pairs, it is evident that a curve of odd degree has an odd number of real asymp- totes ; and that a curve of even degree has either no real asymptotes or an even number. Thus, a cubic curve has either one real asymptote or three ; a conic has either two real asymptotes or none. 134. Method of expansion. Explicit functions. Although the two foregoing methods are in all cases sufficient to find the asymptotes of algebraic curves ; yet in certain special cases the oblique asymptotes are most conveniently found by the method of expansion in descending powers. It is based on the following principle : a straight line will be an asymptote to a curve when the difference between the ordi- nates of the curve and of the line, corresponding to a com- mon abscissa, approaches zero as a limit as the abscissa becomes larger and larger. It will appear from the process of applying this principle that a line answering the condition just stated will also satisfy the original definition of an asymptote. Suppose that the equation of the given curve can be solved for y in the form of a descending series of powers of x, 133-134.] ASYMPTOTES 231 beginning with the first power, arid let the equation then be The line whose equation is y = V + i (2) is an asymptote to the curve represented by (1) ; for the difference between the ordinate of the curve arid line, corre- sponding to the same abscissa x, is 2 3 , x x 2 which approaches zero when x = . It is also evident that the line (2) satisfies the original definition of an asymptote ; for, from (1), the slope of the tangent at the point whose abscissa is x, is dy _ _2 and the intercept made by the tangent on the #-axis is y ~ X dx = ai+ ^ + '"' hence when x = 00, the slope approaches the limit a , and the intercept = a x ; thus the equation of the asymptote is y = a x + a v Ex. 9. Find the asymptotes of the curve x 1 The line x = 1 is an asymptote parallel to the z/-axis. To obtain the oblique asymptotes, write the equation in the form a.2 ^___^__^^^ - - /, 1\- 1 x[ 1 -- 1 -- \ xl x DIFFERENTIAL CALCULUS [CH. xiv. 2x Sx* IGx* y= (* + i + i + !I + \ \ 28x16 a; 2 y Hence the two oblique asymptotes are FIG. 43. 31 "i The sign of the term - - shows that when x = + co, the curve is above 8 x the first asymptote, and below the second, as in figure; and that when z = co, the curve is below the first asymptote, and above the second. 134.] ASYMPTOTES 233 The principal value of the method of expansion is that it exhibits the manner in which each infinite branch ap- proaches its asymptote. Ex. 10. Find the asymptotes of the curve x-3 Here FIG. 44. Hence the oblique asymptotes are The same method may be applied to cases in which x is an explicit function of y. 234 DIFFERENTIAL CALCULUS Ex. 11. Find the asymptotes of [Ctt XIV; Here Hence the asymptotes are x = (y + 2). The next term shows that 2 y when y = + oo, the curve is to the right of the first asymptote, and to FIG. 45. the left of the second ; and vice versa when y == oo. The form of ohe equation shows that the curve has a horizontal asymptote y = 0. 135. Method of expansion. Implicit functions. It was shown in Art. 132 that the direction of each oblique asymp- tote is determined by equating each factor of the terms 134-135.] ASYMPTOTES 235 of highest degree, in the equation of the curve, separately to zero. The subsequent procedure will be shown by an example. Ex. 1. Determine the asymptotes to the curve y* - x* - 2 az 2 ^ - b*x = 0, and the manner in which the corresponding branch of the curve ap- proaches each. The terms of highest degree are y* x*, and this expression has but two real linear factors, hence the curve cannot have more than two real asymptotes ; and these are parallel to the lines y x = 0. To find the asymptote parallel to y x = 0, arrange the equation of the curve thus : (1) When y, x becomes infinite, - = 1 ; hence X *>+ KW If T& 4 --i- 1 i t I and the equation of the asymptote is _ x x 2 - 2a a ^ } (3) To obtain the next term in the equation of the curve, use (3) as a first approximation, which gives y~ = 1 + , correct as far as the order - x 2x x 2? = ( 1 + V= 1 + -, to the same order; (4) a; 2 \ 2xJ x I 4 x) 236 DIFFERENTIAL CALCULUS [Cn. XIV. These values substituted in (1) give as a second approximation Hence the curve approaches the lower side of the asymptote on the right, and the upper side on the left. Similarly the equation of the branch approaching the direction y + x will be found to have the successive approximations , a , a , 1 a 2 *=-*+ r^-*>2 + ' and thus on the right the curve approaches the upper side of the asymp- tote, and on the left, the lower side. If the term in - should happen to disappear from the result, a third x 1 approximation may be obtained by keeping the terms of order in the a: 2 equations that correspond to (1), (4), (5), (6). Ex. 2. y s - x*y + 2 y 2 + 4 y + x = 0. Ex. 3. x s -i- 2 x*y - xf - 2 z/ 3 + 4 y 2 + 2 xy + y = 1. Ex. 4. y 8 = x 9 + a?x. 136. Curvilinear asymptotes. When two curves are so situated that the difference between their ordinates corre- sponding to the same abscissa approaches zero as a limit when the common abscissa is made larger and larger, then each curve is said to be an asymptote of the other. This definition will also apply if the words " ordinate " and "abscissa" be interchanged. E.g., suppose that the equation of a given curve can be brought to the form y = :z 2 + fo; + c + - + -^ + ^+-, X 3? 3? then it follows from the definition that the curve y = ax* + bx + c c + -, x 135-137.] ASYMPTOTES 237 is a second degree asymptote to the given curve ; and y ax 2 + bx i.e., xy = 0,3? + by? + ex + d is a third degree asymptote, and so on. Ex. Find the second and third degree asymptotes to the curves of examples 8-11, Arts. 132-134. 137. Examples of asymptotes of transcendental curves. 1. Consider the curve y = log x. Here, when z=0, and ax = 00 x FIG. 46. hence the line x = is an asymp- tote, by Art. 131. 2. The exponential curve y = e x . In this case, when x = oo, y = 0, -^ = 0. Hence y = is an asymptote. dx O FIG. 47. 238 DIFFERENTIAL CALCULUS [Cn. XIV. 3. Find the asymptotes to the curve 1 + y = e x . When x approaches zero from the positive side, y = + oo, and -^ = 4- oo ; but when x approaches zero from the nega- 7 tive side, x = 0, and -^ = 0. Hence the line x = is an ax FIG. 48. asymptote at y = -4- oo on the positive side of the ?/-axis. Again, when x = GO, y = ; hence the line y = is an asymptote both at x = -f oo and oo. 4. The probabil- ity curve, y = e-* 1 . FIG. 49. 5. The curve 2/ 2 = IG. 50. 137-138.] ASYMPTOTES 239 EXERCISES Find the asymptotes of the following curves : 1. (x + a)f =(>j + i)x 2 . 8. (x - 2 a)# 2 = x 8 a 8 . 2. xV + ax (x + #) 2 - 2 a 2 / 2 a 4 = 0. 9. y a = x t(2 a x) . 3. x 4 y 4 ("x 2 w 2 ) 2 -'- w 2 1 = 0. in j/^na i ^2^ _ n i( n r\ j.u. y(a -i- x j a {a x). \ U ) y ' ^ H xw 2 ~4~ wx 2 /i 8 5 r 2 fr w^ 2 n 2 fr 2 4- w 2> 4 + #;-U- 12 . (x 2 + a 2 )x 2 =(a 2 -x 2 ) V 2. \ J \ / 3 1 o 22 -3 i 14. xV = (a + y) 2 (6 2 - y 2 ). 7 v ' / y v i/ y 15. wfx - ?/)8 = vfx 6. v = x 2 + 3a 2 7. 2 = 2 a - x 138. Asymptotes in polar coordinates. When a curve defined by an equation in polar coordinates has an asymptote, this line must be paral- lel to the radius vector to the point at infinity on the curve. In Fig. 51, consider the curve KP'P, hav- ing the asymptote PT. The radius vector to (jf-' ^ the point at infinity must be parallel to the asymptote, for these two lines must inter- sect at infinity ; and, moreover, the asymptote, according to the definition in Art. 128, must pass within a finite distance of this radius vector. The polar subtangent OM, being by definition perpendicu- lar to the radius vector OP, will, when P passes to infinity, become a common perpendicular to the radius vector OP FIG. 51. 240 DIFFERENTIAL CALCULUS [Cn. XIV. and to the asymptote MP ; hence the measure of the com- mon perpendicular is lim / 2 d0\ P = V dp)' 139. Determination of asymptotes to polar curves. To de- termine whether a given curve has asymptotes, first find for what values of #, the vector p becomes infinite ; then substi- tute each of these values of 6 in the expression for the polar subtangent. If the result of any such substitution is finite, there is a corresponding asymptote. To construct the asymptote, look along the direction of the infinite radius vector from the pole, and turn through a right angle, to the right if V 11 ^ p 2 be positive, and to the ctp left if it be negative (Art. 120). Measure a distance from the pole in this perpendicular direction equal to V p 2 - , and through its extremity draw a line parallel to the infinite radius vector ; this line will be the required asymptote. Circular asymptotes. In some cases it may happen that when 6 is made larger and larger without limit, the value of p may approach a definite limit a; thus Q^^P a - The circle whose equation is p a is then called an asymptotic circle. E.g. The curve p = + sm has an asymptotic circle p = 1; the 6 + cos 9 curve being exterior to the circle from the middle of the first quarter to the middle of the third quarter, and interior for the remainder of the circle ; it approaches nearer to the circle with every revolution of 0. Ex. 1. Find the rectilinear asymptotes to the curve p = -. sm0 When = 0, p = a ; but when = nir, n being any positive or nega- tive integer, p becomes infinite. Since d$ sin 2 hence p 2 = dp sin 6 - 6 cos 138-139.] ASYMPTOTES When 6 nir, this expression becomes amr or anir, according as n is odd or even, and may therefore be written in the form ( \) n ^ l an^. There are thus an infinite number of asymptotes, all parallel to the initial line, and situated at intervals air from each other. When 71 is positive, the asymptotes are above the initial line ; when n is negative, they are below it. There are no circular asymptotes. FIG. 52. In many problems it shortens the work to substitute - for p in the equation of the curve, and then to find what values of will make u vanish. The expression /o 2 for dp the length of the polar subtangent then becomes ; and r / rlff\ ^ hence . n f ], taken for any of the values of iust found, U V duj measures the distance of the corresponding asymptote from the pole. Ex. 2. Find the asymptotes to the curve p sin 4 a sin 3 0. Put p - , then u sin 4 6 = au sin 3 0, 242 DIFFERENTIAL CALCULUS [Cn. XIV. 139. and 11 = 0, when , , 42 , 4 By differentiation, 4 cos 4 & = a ^ sin 3 + 3 aw cos 3 6, dv du 4 cos 4 6 + 3 aw cos 3 This expression becomes corresponding asymptote is 4\/2 a sin 3 6 when = ; hence the distance to the 4 To construct the asymptote, look 4V2 from the pole along the direction of rt f, 45, measure a distance units to 4V2 the right, perpendicular to this radius vector ; then draw a line through the end of the perpendicular, parallel to the infinite radius vector (Fig 53). The student should determine the number and position of the remaining asymptotes. FIG. 63. EXERCISES Find and draw the asymptotes to the following curves : 1. The reciprocal spiral pO = a. 2. p cos 6 = a cos 2 0. 3. p = b sec aO. 6. Show that the curve p = 4. p cos 2 = a sin 3 6. 5. p(e<> - 1) =a(e + 1). has no asymptote. 1 cos 7. Show that the initial line is an asymptote to two branches of the curve p 2 sin 6 = a 2 cos 2 9. 8. Find the rectilinear and circular asymptotes of the curve P ~ I'-l 9. Which of the curves in 1-7 have circular asymptotes? CHAPTER XV DIRECTION OF BENDING. POINTS OF INFLEXION 140. Concavity upward and downward. A curve is said to be concave doivnward in the' vicinity of a point P when, for a finite distance on each side of P, the curve is situated FIG. 54. below the tangent drawn at that point, as in the arcs AD, FH. It is concave upward when the curve lies above the tangent, as in the arcs DF, HK. It is evident, by drawing successive tangents to the curve, as in the figure, that if the point of contact advances to the right, the tangent swings in the positive direction of rotation when the concavity is upward, and in the negative direction when the concavity is downward. Hence upward concavity may be called a positive bending of the curve, and down- ward concavity, negative bending. A point at which the direction of bending changes con- tinuously from positive to negative, as at F, is called a point I>IFF. CALC. 17 243 244 DIFFERENTIAL CALCULUS [Cu. XV. of inflexion, and the tangent at such a point is called a stationary tangent. The points of the curve that are situated just before and just after the point of inflexion are thus on opposite sides of the stationary tangent, . and hence the tangent crosses the curve, as at D, F, If. 141. Algebraic test for positive and negative bending. Let the inclination of the tangent line, measured from the right- hand end of the #-axis toward the forward (right-hand) end of the tangent, be denoted as usual by <, then (f> is an in- creasing or decreasing function of the abscissa according as the bending is positive or negative ; for instance, in the arc AD, the angle diminishes from + -^ through zero to - ; in the arc DF, <> increases from - through zero to 7T rr ; in the arc FH, decreases from H through zero to ; and in the arc HK, increases from ^ through j L zero to + 4 At a point of inflexion (j> has evidently a turning value which is a maximum or minimum, according as the concavity changes from upward to downward, or conversely. Thus in Fig. 54, is a maximum at F, and a minimum at D and at H. Instead of recording the variation of the inclination <, it is generally convenient to consider the variation of the slope tan , which is easily expressed as a function of x by the equation tan < = ^. dx Since tan is always an increasing function of <, it follows that, according as the concavity is upward or downward, the 140-141.} DIRECTION OF BENDING 245 slope function -* is an increasing or a decreasing function ax ot x, and hence that its ^-derivative is positive or negative. Thus the bending of the curve is in the positive or nega- tive direction of rotation, according as the function \ is d$s positive or negative. At a point of inflexion the slope -/- is a maximum or , and then testing whether f'(x) changes its sign as x passes through the critical values thus obtained. To any critical value a that satisfies the test, corresponds the point of inflexion (a, /(a)). Ex. 1. For the curve y = (** - I)', find the points of inflexion, and show the mode of variation of the slope and of the ordinate. Here ^/ = 4 z (z 3 - 1), ax hence the critical values for inflexions are x = -- = = .58 approxi- V3 246 [CH. XV. mately ; and x = + .58. It will be seen that as x increases through .58, the second derivative changes sign from positive to negative, hence there is an inflexion at which the concavity changes from upward to down- ward. Similarly, at x + .58 the concavity changes from downward to upward. The following numerical table will help to show the mode of variation of the ordinate and of the slope, and the direction of bending. As x increases from oo to .58, the bending is positive, and the slope continually increases from oo through zero to a maxi- mum value, 1.5, which is the slope of the stationary tangent drawn at the point (-.58, .44). As x continues to increase from - .58 to +.58, the bending is negative, and the slope decreases from +1.5 through zero to a mini- mum value, 1.5, which is the slope of the stationary tangent drawn at the point (+ .58, .44). Finally, as x increases from + .58 to + 00, the bending is positive, and the slope increases from the value 1.5 through zero to + oo. The values x = 1, 0, +1, at which the slope passes through zero, correspond to turning values of the ordinate. Ex. 2. Examine for inflexions the curve x + 4 = (y - 2) 8 . X y dy d*y dx dx 2 00 + 00 00 + -2 + 25 -24 + -1 + -.58 + .44 + 1.5 1 + .58 + .44 -1.5 1 + + 00 + 00 + 00 + FIG. 56. I f In this case y = 2 + (x + 4)i, X FIG. 56. Hence, at the point (-4, 2), and ( ^- are infinite. When ar< 4, \ is positive, and when x > 4, ^ is negative. dx* dx* 141-142.] DIRECTION OF BENDING 247 Thus there is a point of inflexion at (4, 2), at which the slope is infinite, and the bending changes from the positive to the negative direction. F Ex. 3. Consider the curve y = x*. ax is ' ^) and from the above equation of the tangent, y - y\ = hence y' = y, + and it follows that =f(xj + 248 DIFFERENTIAL CALCULUS [Cn. XV. As h is made smaller and smaller, /"(^ + OK) will have the same sign as/"^); but the factor A 2 is always positive, hence when/' '(ij) is positive, y #' is positive, and thus the curve is above the tangent, at both sides of the point of contact, that is, the concavity is upward. Similarly when /"(%) is negative, the concavity is downward. This agrees with the former result. 143. Concavity and convexity towards the axis. A curve is said to be convex or concave toward a line, in the vicinity of a given point on the curve, according as the tangent at the point does or does not lie between the curve and the line, for a finite distance on each side of the point of contact. FIG. 59 a. . FIG. 59 &. First, let the curve be convex toward the #-axis, as in the left-hand figure ; then if y is positive, the bending is positive and -3| is positive ; but if y is negative, the bending is neg- *^^ J2 ative and ^ is negative. Thus in either case the product ja dor d?y ... y^ is positive. GC2/ Next, let the curve be concave toward the aj-axis, as in the right-hand figure ; then if y is positive, the bending is negative and ^ is negative ; but if y is negative, the bend- dx* 142-144.] DIRECTION OF BENDING 249 d?y ing is positive and *| is positive. Thus in either case the d?u product y ^ is negative. Hence : In the vicinity of a given point (, y) the curve is convex or dPu concave to the x-azis, according as the product y ^- is positive C13/ or negative. EXERCISES x s 1. Show that the curve y = - has a point of inflexion at the a 2 + x 2 origin, and also when x = aV3. 2. In the curve y (a 4 6 4 ) = x (x a) 4 xb 4 , there is a point of inflexion at x ^ Examine the points at which x = a. o 3. Find the points of inflexion of the curve 4. Show that the curve y (x z + a 2 ) = a 2 (a x) has three points of inflexion on the same straight line. 5. Find the points of inflexion on the curve y 2 (x 1) = x 2 . 6. Show that the curve 6ar(l r)y = l + 3x has one point of inflexion, and three asymptotes. 7. Show why a conic section cannot have a point of inflexion. 8. Draw the part of the curve a?y = - ax 2 + 2 a 8 near its point of inflexion. 144. Concavity and convexity ; polar coordinates. A curve referred to polar coordinates is said to be concave or convex to the pole, at a given point on the curve, according as the curve in the neighborhood of that point does or does not lie between the tangent and the pole. Let p be the perpendicular from the pole to the tangent at the point (/>, 0). Then when the curve is concave to the pole, p evidently increases with p, as in the arc AB, and dt) diminishes with p, as in the arc BO (Fig. 60 a); hence -j- is positive (Art. 20). 250 DIFFEREN TIAL CALCUL US [Cn. XV. Again, when the curve is convex to the pole, p increases when p diminishes, as in the arc DE (Fig. 60 J), and p diminishes when p increases, as in the arc EF '; hence -f- dp is negative. FIG. 60 a. FIG. 60 b. Thus the curve is concave or convex to the pole at the point (p, 0), according as -~ is positive or negative. ctp To express this condition in terms of ^-derivatives of p, use the equation p = p sin -^, i i . i , . , . , i r, i AMn rl s because tan ty = /y by Art. 118. This may be simplified by putting - = w, p = -, whence 7 -1 7 P U -?/ and equation (1) becomes dff * 9 i -= W 2 + 2 (2) Differentiation as to u gives 2 dp _ 9 ~du d 2 u dd 5 ' ~7 " % ~T~ - 1 7/1 ' 7/y> * ~? } dp -JL 144.] DIRECTION OF BENDING 251 dp du dp du p 2 du hence, from (3), -~- = p s u 2 f w -f -^ V Since p is always taken positively, hence The curve is concave or convex to the pole at the point (/>, 0), according as u + - - is positive or negative, do* EXERCISES Trace the following curves near their points of inflexion : 1. p = ^ (find its asymptotes). 2. p = . 3. p = bO 2 . 4. In the curve defined by the two equations x a (1 cos <), y = a (n< + sin <), show that there is an inflexion at the point where cos 6 = n 5. Locate the inflexions on the curve p = - (See Fig. 52.) sin 9 6. Find the coordinates of the inflexion in Fig. 40. 7. In Fig. 41, show that the inflexional tangent is vertical. 8. Show that there are three real inflexions in Fig. 42. 9. How many inflexions are there in Figs. 44, 45? 10. In the logarithmic curve, the curvature is always negative; and in the exponential curve it is always positive. (Figs. 46, 47.) 11. Locate the points of inflexion in Figs. 48, 49, 50. CHAPTER XVI CONTACT AND CURVATURE 145. Order of contact. The points of intersection of the two curves y = 4> O)> y = ^ O) are found by making the two equations simultaneous ; that is, by finding those values of x for which $(x) = ty(x). Suppose xa is one value that satisfies this equation, then the point x = a, y = (#) = ^(a) is common to the curves. If, moreover, the two curves have the same tangent at this point, they are said to touch each other, or to have contact of the first order with each other. The values of y and dy dx question, and this requires that of -^ are thus the same for both curves at the point in ax If, in addition, the values of ^ be the same for each ctor^ curve at the point, then *"00=*"00. and the curves are said to have a contact of the second order with each other at the point for which x = a. If $ (a) = i|r (a), and all the derivatives up to the wth order be equal to each other, the curves are said to have contact of the wth order. This is seen to require n + 1 con- 252 CH. XVI. 145-14(5.] CONTACT AND CURVATURE 253 ditions ; hence if the equation of the curve y = (x) be given, and if the equation of a second curve be written in the form y ty(x), in which "^(x) proceeds in powers of x with undetermined coefficients, then n + 1 of these coeffi- cients could be determined by requiring the second curve to have contact of the nth order with the given curve at a given point. 146. Number of conditions implied by contact. A straight line has two arbitrary constants, which can be determined by two conditions ; thus, a straight line can be drawn which touches a given curve at any specified point. In general no line can be drawn having contact of an order higher than the first with a given curve ; but there are certain points at which this can be done. For instance, if the equation of a line be written y = mx + 5, then dy cPy A 3. = rn, = ; dx dx 2 hence, through any arbitrary point x = a on a given curve y = (f)(x), a line can be drawn which has contact of the first order with the curve, but which has not in general contact of the second order ; for the two conditions for first order contact are ma -\-b= <(), m = <#>'(), which are just sufficient to determine m and J; and the additional condition for second-order contact is = <"(a), which is satisfied whenever the point x = a is a point of inflexion on the given curve y = <(V). Thus the tangent at a point of inflexion on a curve has contact of the second order with the curve. 254 DIFFERENTIAL CALCULUS [Cn. XVI. The equation of a circle has three independent constants. It is therefore possible to determine a circle having contact of the second order with a given curve at any assigned point. The equation of a parabola has four constants, hence a parabola can be found which has contact of the third order with the given curve at any point. The general equation of a central conic has five inde- pendent constants, hence a conic can be found which has contact of the fourth order with a given curve at any given point. As in the case of the tangent line, special points may be found for which these curves have contact of higher order. 147. Contact of odd and of even order. THEOREM. At a point where two curves have contact of an odd order they do not cross each other ; but they do cross where they have contact of an even order. For, let the curves y <(V), y = ^(X) nave contact of the nth order at the point whose abscissa is a ; and let y v y z be the ordinates of these curves at the point whose abscissa is a + h; then y v =4>(a + A), y a = -f (a + k), and by Taylor's theorem 146-149.] CONTACT AND CURVATURE 255 Since by hypothesis the two curves have contact of the nth order at the point whose abscissa is , hence <>"<= and -^ but this expression, when h is sufficiently diminished, has the same sign as A n + 1 [ +1 O)--f" +1 O)]; hence, if n be odd, y^ y^ does not change sign when h is changed into A, and thus the two curves do not cross each other at the common point. On the other hand, if n be even, y l y z changes sign with h ; and therefore when the contact is of even order the curves cross each other at their common point. For example, the tangent line usually lies entirely on one side of the curve, but at a point of inflexion the tangent crosses the curve. Again, the circle of second-order contact crosses the curve except at the special points, noted later, in which the circle has contact of the third order. 148. Circle of curvature. The circle that has contact of the closest (i.e., second) order with a given curve at a speci- fied point is called the osculating circle or circle of curvature of the curve at the given point. The radius of this circle is called the radius of curvature, and its center is called the center of curvature at the assigned point. 149. Length of radius of curvature ; coordinates of center of curvature. Let the equation of a circle be )+(r-) = JZ, (1) 256 DIFFERENTIAL CALCULUS [Cn. XVI. in which R is the radius, and , /3 are the coordinates of the center, the current coordinates being denoted by JT, y, to distinguish them from the coordinates of a point on the given curve. It is required to determine R, , yS, such that this circle may have contact of the second order with the given curve at the point (z, y). From (1), by successive differentiation, (2) If the circle (1) has contact of the second order at the point (x, y~) with the given curve, then the common abscissa x = X makes dX dx dX 2 dx 2 ' . hence, from (2), (x - a) + (*/ - 0) ^ = 0, dx whence x _ a = dxl\dx^^ dx 2 and finally, by substitution in (1), dx 2 149-150.] CONTACT AND CVRVATJJEE 257 If, for shortness, TW, n be written for -^-, ^, then the ax dor coordinates of the center and the radius of the circle of curvature are given by the equations . -. = '"(!+"'); ,_ /3= _i^ ; R . a + *)*. w w w. 150. Second method. The osculating circle is sometimes defined as the limiting position of a circle passing through three points on the curve when two of these points move towards the third as a limit. It is proposed to find the equation of this circle, and thus to show that the two definitions lead to the same result. Let x h, x, x + h be the abscissas of three points on the curve, and y k. y. y + k' the corresponding ordinates, in which k' is not in general equal to k. Let these three points lie on the circle whose equation is (x - ) 2 +(y -) = #, (1) then O - h - ) 2 + (y - k - /8) 2 = ^, ( X + k - )2 + (y + k 1 - ) 2 = B*. Subtracting the second and third from the first, 2A(*-)-A* + 2*(y-)- P = 0,i - 2 AO - ) - A 2 - 2&'<> - /3) - &' 2 = 0,1 whence by adding, and solving for y ft, 2 (*-*') To find the limit of this fraction as h = 0, let y = (x) be the equation of the given curve, then y - k = <$>(x A), y + k' = $(x + A), 258 DIFFERENTIAL CALCULUS [Cn. XVI. whence, by Taylor's theorem, - k = X - z + <>"x - <, y + V = < + ty'CO + < and & = h'(x) - |^ <"<> ~ hence, when h = 0, Equation (3) may now be written \ 2 1 k'-k therefore, by (4), //7\a ! + ['()]'_ 7 " AJ _ d ^ As == ~A : TT~' and may be regarded as the rate of deflection of the arc from the tangent estimated per unit of length ; or again, as the ratio of the angular velocity of the tangent to the linear velocity of the point of contact. To express K in terms of a;, y, and their derivatives. Since tan (f> = -^-i dx then = tan" 1 &> dx j dd> d f. _, dy\ and ~f. = - tan a -^ ds ds \ dx) d f , _j dy\ dx dx\ dx) ds <&_ J_ A 2 ' di" :) dx therefore /c, = ^, = .. [Art. 121 155. Curvature of an arc of a circle. In the case of a cir- cular arc the normals are radii ; 154-150. J CONTACT AND CURVATURE 263 hence As = r A, = -, (1) As r thus K = r Thus the average curvature of all arcs of the same circle is constant and equal to - radians per unit of length. r For example, in a circle of 2 feet radius the total curva- ture of an arc of 3 feet is | = 1.5 radians, and the average curvature is .5 radian per foot. It also follows from (1) that in different circles, arcs of the same length have a total curvature inversely propor- tional to their radii. Thus on a circumference of 1 meter radius, an arc of 5 decimeters has a total curvature of .5 radian, and an average curvature of .1 radian per decimeter; but on a circumference of half a meter radius, the same length of arc has a total curvature of 1 radian and an average curvature of .2 radian per decimeter. 156. Curvature of osculating circle. A curve and its oscu- lating circle at P have the same measure of curvature at that point. For, let /c, K' be their respective measures of curvature at the point of contact (z, y} ; then from Art. 154, dx* K = and from Art. 149, K' =- = 7, hence K = K'. R 264 DIFFERENTIAL CALCULUS [Cn. XVI. It is on account of this property that the osculating circle is called the circle 'of curvature. This is sometimes used as the denning property of the circle of curvature. The radius of curvature at P would then be denned as the radius of the circle, whose measure of curvature is the same as that of the given curve at the point P. Its value, as found from Art. 154 and Art. 155, accords with that given in Art. 149. EXERCISES 1. Find the order of contact of the two curves y = x 8 , y = 3x*-3x + l. 2. Find the order of contact of the parabola y 2 = 4 x, and the straight line 3 y = x + 9. 3. Find the order of contact of 9 y = x s - 3 x 2 + 27 and 9 y + 3 x = 28. 4. Find the order of contact of y = log (x - 1) and a: 2 - 6 x + 2 y + 8 = at (2, 0). 5. Show that the circle (x-Y+(y-Y=~ and the curve Vx + Vy = Va have contact of the third order at the point x = y = - 4 6. What must be the value of a in order that the parabola y = x + 1 + a (x - I) 2 may have contact of the second order with the hyperbola xy = Bx 11 7. Find the order of contact of the parabola (x-2a) 2 +(y-2a^ = 2xy, and the hyperbola xy = a 2 . EXERCISES ON CURVATURE 8. In the curve y = x* 4x 3 18x 2 , the radius of curvature at the origin is &. 9. Show that the two radii of curvature of the curve a x at the origin are a \/2; and that R = \ a at ( a, 0). 15(3-157.] CONTACT AND CURVATURE 265 Find the radius of curvature in each of the following curves: 10. The parabola y 2 = 4 ax. 11. The ellipse 12. The catenary a 2 " 1 "** 2 i*+ 13. The exponential curve y = ae e 14. The parabola Vx + Vy = 2 Vo. 15. The hypocycloid x t + ^f = a*. 16. The curve y = logsecar. Catenary of uniform strength. 17. Derive the formula -L = (** R 2 \ds* + 157. Direct derivation of the expressions for K and R in polar coordinates. Using the notation of Art. 119, hence ^8 afc d0 ds d0 But tan ty = p , -\lr = tan" 1 c?/? [Art. 124 266 DIFFERENTIAL CALCULUS [Cn. XVI. therefore, by differentiating as to 6 and reducing, which, substituted in (1), gives K = HI)' and the relation R = - then reproduces the result obtained K in Art. 152 by transformation of coordinates. When u = is taken as dependent variable, the expres- P sion for K assumes the simpler form K = Since at a point of inflexion K vanishes and changes sign, hence the condition for a point of inflexion, expressed in d?u polar coordinates, is that u + -j^ shall pass through zero do and change its sign. See Art. 144. EXERCISES 1. Show that the radius of curvature of the curve p = a sin nO at (0, 0) is ^ M 2. Find the radius of curvature of p m = a m cos m0. Find the value of R in each of the following curves : 3. The circle p a sin 0. 157-158.] CONTACT AND CURVATURE 267 4. The lernniseate p 2 = o 2 cos 2 6. 5. The logarithmic spiral p = e a . 6. The trisectrix p = 2 a cos a. 7. The equilateral hyperbola p 2 cos 2 = a 3 . 8. For any curve prove the formula R = __ ** \ + rf0/ EVOLUTES AND INVOLUTES 158. Definition of an evolute. When the point P moves along the given curve, the center of curvature describes another curve which is called the evolute of the first. Let f(x, y) = be the equation of the given curve, then the equation of the locus described by the point is found by eliminating x and y from the three equations dx I \dx dx 2 dx 2 and thus obtaining a relation between a, /9, the coordinates of the center of curvature. No general process of elimination can be given ; the method to be adopted depends upon the form of the given equation f(x, y) = 0. 268 DIFFERENTIAL CALCULUS Ex. 1. Find the evolute of the parabola y- = 4 px. Since y = 2/M ^ =/,!*-*, ^ = - V*~ f , ), and y - /$ = (1 -f px~ l ) 2p~?x% therefore a = 2p + 3x, ft [H. XVI FIG. 64. when, by eliminating x, -fa (a 2j) 8 = (/> 5 /?) 2 is the equation of the evolute of the. parabola, in which a, {3 are current coordinates. Ex. 2. Find the evolute of the ellipse (1) 158-159.] CONTACT AND CURVATURE 269 Here ^ + .^ = , & = -?, a a 6 2 dx dx a 2 y dy ft* y X dx_-V, ^\-b^ -6* Jfe5~ a* y* aV I* + a'V ~ ay ^ y * ) ~ rfh? whence (*/ + ft** 8 )?/ /a 2 !/ 2 a: 2 \ /aV --- Therefore - /8 = y 8 . (2) o* Similarly, ^a 2 -^ ^ 3 ^ 4 Eliminating a:, y between (1), (2), (3), the equation of the locus de- scribed by (a, ft) is (aa)S + (&)f = (a 2 - 6 2 )1 (Fig. 69.) 159. Properties of the evolute. The evolute has two im- portant properties that will now be established. I. The normal to the curve is tangent to the evolute. The relations connecting the coordinates (a, y8) of the center of curvature with the coordinates (#, /) of the correspond- ing point on the curve are, by Art. 149, From these equations , /9 may be considered functions of x; hence, by differentiating (1), regarding , $, y as func- tions of x, __ \dx &* dx dx dx 270 DIFFERENTIAL CALCULUS [Cu. XVI. Subtracting (3) from (2) gives da df3 dy _ dx dx dx ^ ' whence d/3 dx dy ft fi but- -j- is the slope of the tangent to the evolute at (, /3); and is the slope of the normal to the given curve at dy (x, T/). Hence these lines have the same slope ; but they pass through the same point (, y8), therefore they are coincident. II. The difference between two radii of curvature of the given curve, touching the evolute at the points C r O z (Fig. 65), is equal to the arc CjGj of the evolute. Since R is the distance between points (z, y), (, yS), hence Fio. 66. (x - a) 2 + O - ) 2 = R*. (5) When the point (#, y~) moves along the given curve, the point (a, /8) moves along the evolute, and thus a, /3, 72, y are all functions of x. Differentiation of (5) as to x gives ,A, da\ (a: -a) 1 --,- j \ dx) hence, subtracting (6) from (1), dB\ n dR ,n^ - -- \= R (6) dx) dx dx 159.] CONTACT AND CURVATURE 271 Again, from (1) and (4), da d X - a y- Hence, each of these fractions is equal to dx) dx /CPV =^r5 = ' \ v ) in which a is the arc of the e volute. Next, multiplying numerator and denominator of the first member of (8) by x a, and those of the second member by y #. and combining new numerators and denominators, it follows that each of the fractions in (8) is equal to / N doc. , Q ^ dQ ( *_)_ +(3 , _)_!? which equals -- - , by (7) and (5). Whence, by (9), = dx dx that is, 4( <7 ^)= 0: dx therefore a- R = constant, wherein er is measured from a fixed point A on the evolute. Now, let Cj, (7 2 be the centers of curvature for the points P r jP 2 on the given curve ; let P 1 O 1 = .Bj, P 2 ^2 = ^2 ' let the arcs A O r A (7 2 be denoted by or or < J. 15. Show by inspection of the figure that four real normals can be drawn to the ellipse from any point within the evolute. CHAPTER XVII SINGULAR POINTS 160. Definition of a singular point. If the equation dii f(x, /)=0 be represented by a curve, the derivative ~, when it has a determinate value, expresses the slope of the tangent at the point (#, y). There may be certain points on the curve, however, at which the expression for the deriva- tive assumes an illusory or indeterminate form ; and, in consequence, any line whatever drawn through such a point may be regarded as a tangent at the point. Such values of x, y are called singular values, and the corresponding points on the curve are called singular points. 161. Determination of singular points of algebraic curves. When the equation of the curve is rationalized and cleared of fractions, let it take the form f(x, y)= 0. This gives, by differentiation with regard to x, as in Art. 96, d + dfdy = ^ dx dy dx V dy dx whence -/- = -^y (1) dx df dy In order that -* may become illusory, it is therefore dx df df necessary that = 0, -^- = 0. (2) ox dy DIFF. CALC. 19 275 276 DIFFERENTIAL CALCULUS [Cn. XVII. Thus to determine whether a given curve f (x, y^) has singular points, put -*- and -*- each equal to zero and solve dx dy these equations for x and y. If any pair of values of x and y, so found, satisfy the equation f (a?, y) = 0, the point thus determined is a singular point on the curve. To determine the appearance of the curve in the vicinity of a singular point, (x^ y-^) evaluate the indeterminate form dy _ dx _ = ~ = by finding the limit approached continuously by the slope of the tangent when x = x v y = y v thus dy_ = _dx\dx dx djt dx\dy; dy | dxdydx dy dxdy dy 2 dx This equation cleared of fractions gives, to determine the slope at (x v 2/j), the quadratic This quadratic equation has in general two roots. The only exception is when simultaneously, at the point in question, g.o, V..O, ^ = 0> (4) dx z dx dy dy 2 161-162.] SINGULAR POINTS 277 dtj in which case -/ is still indeterminate in form, and must be ax evaluated as before. The result of the next evaluation is a dy cubic in -p, which gives three values of the slope, unless all the third partial derivatives vanish simultaneously at the point. The geometric interpretation of the two roots of equation (3) Avill now be given, and similar principles will apply when the quadratic is replaced by an equation of higher degree. The two roots of (3) are real and distinct, real and coin- cident, or imaginary, according as ~ is positive, zero, or negative. These three cases will be con- sidered separately. 162. Multiple points. First let H be positive. Then at the point (#, y) for which -^- = 0, = 0, there are two values dx dy of the slope, and hence two distinct singular tangents ; thus the curve goes through the point in two directions, or, in other words, two branches of the curve cross at this point. Such a point is called a real double point of the curve, or simply a node. The conditions, then, to be satisfied at a node (2^, y^) are that and that ff(x r y^) be positive. Ex. Examine for singular points the curve 3z 2 - xy - 2y* + z 8 - 8y 8 = 0. 278 DIFFERENTIAL CALCULUS [Cn. XVII. Here ^ = 6 x y -\- 3 x 2 , -^- = x 4 y 24 y 2 . The values x = 0, y = will satisfy these three equations, hence (0, 0) is a singular point. Since ^L Q + Q X Q a t (0, 0), fOf -^4- = - 1 = - 1 at (0, 0), M - - 4 - 48 y = - 4 at (0, 0), hence the equation of the slope is, from (3), of which the roots are 1 and f . Thus (0, 0) is a double point at which the tangents have the slopes 1, f. Fro. 70. 163. Cusps. Next let JT=Q. The two tangents are then coincident, and there are two cases to consider. If the curve recedes from the tangent in both directions from the point of tangency, it is called a point of osculation; and two branches of the curve touch each other at this point. If 162-163.] SINGULAR POINTS 279 both branches of the curve recede from the tangent in only one direction from the point of tangency, the point is called a cusp. Here again there are two cases to be distinguished. If the brandies recede from the point on opposite sides of the double tangent, the cusp is said to be of the first kind ; if the,y recede on the same side, it is called a cusp of the second kind. The method of investigation will be illustrated by a few examples. Ex. 1. /(*, y) = oV - a V + x 6 = 0. - Qx By The point (0, 0) will satisfy f(x, y) = 0, -J- = 0, -j- = ; hence it is a o x ay ngular point. Proceeding to the second derivatives, + 30 z* = at (0, 0), _ - ' The two values of -J- are therefore coincident, and each equal to dx zero. From the form of the equation, the curve is evidently symmet- rical with regard to both axes; hence the point (0, 0) is a point of osculation. No part of the curve can be at a greater distance from the #-axis than a, at which points ^ is infinite. The maximum value of y dx corresponds to z = aV$. Between x = 0, a; = aVf there is a point of inflexion (Fig. 71). 280 DIFFERENTIAL CALCULUS [" XVII. Ex.2. f(x,y) = y*-3* = Q; Hence the point (0, 0) is a singu- lar point. Again, |/ = _ 6 a: = at (0, 0) ; Fio. 71. Bxdy Therefore the two roots of the quadratic equation defining are both dy equal to zero. Thus far, this case is exactly like the last one, but here no part of the curve lies to the left of the axis of y. On the right side, the curve is symmetric with regard to the x-axis. As x increases, y in- creases ; there are no maxima nor minima, and no inflexions (Fig. 72). Ex. 3. f(x, y)=x*-2 ax z y - axy* + <*V = 0. The point (0, 0) is a singular point, and the roots of the quadratic defining *- are both qual to zero. dx Let a be positive. Solving the equation for y, When x is negative, y is imaginary ; when x = 0, y = ; when x is positive, but less than a, y has two positive values, therefore two branches FIG. 72. FIG. 73. are above the z-axis. When x = a, one branch becomes infinite, having the asymptote x = a; the other branch has the ordinate Ja. The origin is therefore a cusp of the second kind (Fig. 73). 163-164] SINGULAR POINTS 281 164. Conjugate points. Lastly, let H be negative. In this case there are no real tangents ; hence at the point in question, no points in the immediate vicinity of the given point satisfy the equation of the curve. Such an isolated point is called a conjugate point. Ex. f(x, y) = ay 2 - x* + bx 2 - 0. Here (0, 0) is a singular point of the locus, and . dx a both roots being imaginary if a and b have the same sign. To show the form of the curve, solve the given equation for y, then y = hence, if a and b are positive, there are no real points on the curve between a;=0 and x = b. Thus is an isolated point (Fig. 74). FIG. T4. These are all the singularities that algebraic curves can have, though complicated combinations of them may appear. In all the foregoing examples, the singular point was (0, 0); but for any other point, the same reasoning will apply. Ex. f(x, y) = x* + 3 y 9 - 13 y 2 - 4z + I7y - 3 = 0, 17. At the point (2,1), /(2, 1) = 0, = 0, = 0; hence (2, 1) is a singular point. Also ay_ 9 . = 0; - 26, = - 8 at (2, 1). s of the two node (2, 1) are y - 1 = 2(ar - 2), y - 1 = - 2(x - 2). Hence = 2; and thus the equations of the two tangents at the fix 282 DIFFERENTIAL CALUULUS [Cn. XVII. 164. When, at a singular point, H is negative, the point is necessarily a conjugate point, but the converse is not always true. A singular point may be a conjugate point when #=0 (cf. Ex. 9). Transcendental singularities. A curve whose equation involves a transcendental function may have a stop-point, at which the curve ter- minates abruptly (Fig. 48) or a salient point at which two branches of the curve meet and stop without having a common tangent. In the first case there is a discontinuity in the function; in the second, a discon- tinuity in the derivative. They are usually discovered by inspection in tracing the curve. EXERCISES Find the multiple points, and the direction of the tangents at them, in the following curves : 3. (x* + ifY = 0. 4. 2 = x* + 5. If ay 2 =(x a) 2 (a; b), show that, when x =a, there is a con- jugate point if a be less than b, a double point if a be greater than b, and a cusp if a be equal to b. 6. Show that the curve y 3 = (x a)\x c) has a cusp of the first kind. 7. Draw the curve x 9 4- y 8 = x 2 + if- in the vicinity of the origin. 8. Prove that the curve x* 2 ax z y axy 2 + 2 ^/ 2 = has a cusp of the second kind at the origin. 9. What change in the coefficient of x^y in the last example will make the origin a conjugate point? Show that the tangents at this point are still real and coincident. 10. Trace the curve x* + 2ax 2 y ay s = for points near the origin. i 11. In the curve y(\ + e x )= x, show that if x = from positive side, y y - = 0; if from negative side, - = 1 ; hence a discontinuity in slope, i.e., a salient point. CHAPTER XVIII CURVE TRACING 165. Tracing a curve consists in finding its general form when its equation is given. Three kinds of equations present themselves. 1. Cartesian equations : (a) algebraic ; (6) transcendental. 2. Polar equations. There is no fixed method of procedure applicable to all cases. A few general suggestions for Cartesian equations will be given, and then some examples worked out in detail. Find -&- ; this will give the direction of the curve at ax any point, and will serve to locate maximum and minimum ordinates. Examine for asymptotes, and construct them. Deter- mine on which side of each asymptote the corresponding infinite branch is situated. Find y~ ; this will give the direction of bending at any dx^ point, and will determine the points of inflexion. Examine algebraic curves for singular points, and deter- mine whether they are nodes, cusps, or conjugate points. If the minute configuration of a curve at any particular point is desired, it is often expeditious to transform the 283 284 DIFFERENTIAL CALVULUS [Cn. XVIU. origin to that point, and then neglect the higher powers of x and y, as relatively unimportant. This principle will be used and discussed in some of the examples that follow. 166. Illustration. Trace the curve This curve goes through the origin ; and it is symmetric with regard to the a>axis, for the equation is not changed when y is changed to y ; but it is not symmetric with regard to the /-axis. Putting x = gives y* = ; and putting y gives x* = ; hence the curve does not intersect either of the coordinate axes, except at the origin. Since = , ox oy hence dx (6 x 2 y^y which becomes indeterminate only for x = 0, y = 0. Thus the origin is a singular point of the curve. The second partial derivatives are n o oar which all vanish at the origin ; hence those of the third order must also be obtained : 93/ -24* ^ -0 9Sf -12- ^-0 "5 5 *'* * . oT " J Z ^ o - ** } T Q - " ClIJ The general equation determining -=^, derived similarly da? to that in Art. 149, is 166-160.] CURVE TRACING 285 which becomes in this case + = 0. j 0( \dx Thus two values of -^ are 0, and the third root is infinite ; showing that the x-axis is tangent to two branches, and the y-axis to a third branch. To obtain the form of the first branches in the vicinity of the origin it may be observed that since on these branches y is evidently an infinitesimal of a higher order than #, hence y^ may be neglected in comparison with the other terms, and there results X s = 6 ?/ 2 , as the equation of a curve approximately coinciding with the two branches in question near the origin. This curve, and hence also the given curve, has obviously a cusp of the first kind lying to the left of the axis of y. Similarly, in the case of the branch that is tangent to the y-axis, y may be neglected, and the resulting curve is y* = 6 #, which is a parabola situated on the right side of the #-axis. Thus, the third branch is parabolic in form near the origin. By solving for y, y = 3 x + \/9 x 2 + rf, in which only the positive sign is to be retained before the inner radical, as the negative sign would give imaginary values to y. Any line parallel to the y-axis will therefore meet the curve in only two points. Again, regarding y as given, the resulting equation in x has one positive root between and y because /(O, y) is negative, and/(y, y) is positive, and similarly one negative 286 DIFFERENTIAL CALCULUS [CH. XVIII. root numerically greater than y ; the others being imagi- nary. Thus no branch of the curve crosses the lines x=y, except at the origin. To examine for asymptotes. Put y mx + 5, then X* (mx + 6) 4 + 6 x(mx + b) 2 = ; i.e., 1 - ra 4 + - 4 ra 3 6 + 6 w 2 ^ + - 6 w 2 6 2 + 12 Let l-w 4 = and - then m = 1, 6 = |, thus y = x + ^ y ~ x ~ f are the asymptotes. The other two asymptotes are imagi- nary. To find the finite points in which this asymptote cuts the curve, put m = 1, b = | in the above equation for x ; it then becomes of which the four roots are oo, oo, +|V2, -|V2; hence the approximate values of the finite roots are 1.06. The manner in which the infinite branches approach their asymptotes is best shown by the method of expansion, in which y is expressed in a series of descending powers of x. Write the equation in the form then = ar x 166.] CURVE TRACING 287 Hence -^- I ~* r\ I r This verifies the equations of the asymptotes already found ; and, moreover, the sign of the third term shows that the curve is above the first asymptote for large positive FIG. 75. values of x, and below it for large negative values. On the other hand, the curve is below the second asymptote for 288 DIFFERENTIAL CALCULUS [Cn. XVIII. large positive values of x, and above it for large negative values. The form of the second derivative ^ is too complicated ax 2 to be of practical use in determining the direction of bending. Since each infinite branch is convex to its asymptote for large values of x, hence on the upper right hand branch the concavity is ultimately upwards. Near the origin the con- cavity is downwards, hence there must be a point of inflexion on this branch, and also on the branch symmetrical to it. On the left hand branches there are no points of inflexion, for if there were one on either branch there would be two on that branch, and it would then be possible to draw a line cutting the given fourth degree curve in more than four points. 167- Form of a curve near the origin. In the above ex- ample, in the vicinity of the origin, the curve approaches the form of an ordinary parabola on one side of the ^-axis (which is the tangent at its vertex), and has a cusp of the first kind on the other side, the axis of x being the cuspidal tangent. In the first case x was neglected in comparison with y, since ^" n - = ; while in the second case y is neglected in * ~~ u v 1 1 TY> 7/ comparison with x, since _._ n ^ = 0. y -" x In many cases it is not so obvious which terms can be rejected, especially when the lowest terms in the expression are of high degree. Before proceeding to the more difficult curves, a few ele- mentary type forms will be given. The branches of every algebraic curve approximate to combinations of these forms in the vicinity of any assigned point as origin. 166-107.] CURVE TRACING 289 1. Trace the curve t/ 2 = x. FT dti 1 Here -~= -, dx 9 ~t FIG. 76. dx* 4 x \ ' hence the slope is infinite at the origin, and diminishes to zero at in- finity, showing that the curve becomes more and more hori- zontal ; the bending is negative on the upper branch, and positive on the lower. 2. The curve y = x*. Here dy_ = dx hence the slope is zero at the origin, and becomes infinite at infinity, showing that the curve becomes more and more vertical ; the bending is always positive. 3. The curve y = X s . In this case -f- = 3 a? 8 , dx FIG. T7. Fiu. 78. hence the slope is zero at the origin, is elsewhere always positive, and becomes infinite at infinity. The bending changes sign where the curve passes through the origin, the a>axis being the inflexional tangent. 290 DIFFERENTIAL CALCULUS [Cn. XVIII. 4. Show the form of the curve y* = x*. i dy , 3 i d?u ,3-i Here y = x*, -f- = -x*, ^ = 7^ dx 2 dy? 4 The curve is symmetrical with regard to the rr-axis ; ana since the slope is zero at the origin, the axis of x is tangent to the upper and the lower branch. Since a negative value of x makes y imaginary, the curve does not extend to the left of the origin, hence there is a cusp of the first kind at this point. The slope increases numerically to infinity when x becomes infinite, and the bending is always positive on the upper branch, and negative on the lower. This curve is called the semicubical parabola because the ordinate is pro- portional to the square root of the cube of the abscissa. In each of these fundamental types, if the sign of either member of the equation be changed, the curve is simply turned over, and if x and y be interchanged, the curve is revolved through 90 degrees. Some more complicated cases will now be taken up. The following general principles will be of use. I. When the equation of an algebraic curve is rationalized and cleared of fractions, if the constant term be absent the origin is on the curve ; and the terms of the first degree, equated to zero, give the equation of the tangent at the origin. II. If the constant term and terms of the first degree be absent, the origin is a double point ; and the terms of the second degree, equated to zero, give the equation of the pair of nodal tangents. 167.] CUEVE TRACING 291 III. If all the terms below the third degree be absent, the origin is a triple point ; and the terms of the third degree, equated to zero, furnish the equation of the three tangents at the multiple point. Similarly, in general. For, let the equation be of the form f(x, y) = ax + by + (ex* + dxy + ey 2 ) + = 0, (1) then the tangent at the origin will be represented by the equation in which ~ is to be obtained from the relation ax . Bx dy dx~ dii hence, eliminating -^- between the last two equations, the equation of the tangent at the origin becomes but at the point (0, 0), hence the equation of the tangent at this point is ax+by = 0. (6) Again, if the constants a, b be zero, the expression for -^, given by (3), is indeterminate, and the slope at the origin (M? is to be obtained from the quadratic (Art. 161), dZf ( d y\ + 92f - o m Wdy\fo) + dx*~ DIFF. CALC. - 20 292 DIFFERENTIAL CALCULUS [Cn. XVIII. hence eliminating -^ between (2) arid (7), there results which is then the equation of the pair of tangents at the origin ; but at the point (0, 0), -i9 i ,. ,, , ) dar da; dy dj/ z hence the equation of the pair of tangents at the origin is ex 2 + dxy -f ey 2 = (9) Similarly proceed in general. 168. Another proof. The equation that gives the abscissas of the intersections of the line y = mx with the given curve is (a 4- bm)x + (c + dm + em 2 )x 2 + = 0. There will be two intersections at the origin if a + bm = 0, that is, if m = - Hence the tangent at the origin is a , = --*. Again, if a = 0, 6 = 0, every line through the origin will meet the curve in two coincident points ; and in this case the origin will be a double point. If, moreover, m be so taken that c + dm + em 2 = 0, the line y = mx will meet the curve in three coincident points at the origin ; hence the equation of the pair of nodal tangents is to be found by eliminating m between y = mx and c + dm + em 2 = 0, and is therefore ex 2 -f d xy + ey 2 = ; and so on. 169. Illustration. Oblique branch through origin. Expan- sion of y in ascending powers of x. Given the equation y* = 0, 167-169.] CURVE TRACING 293 to expand y in ascending powers of x, and thence to trace the locus in the vicinity of the origin. The first approximation to the value of y, obtained by omitting terms of order x 2 , is y = 2 x, which is the equation of the tangent, and gives the direction of the curve at the origin. In approaching the origin along the curve, the variables x and y are infinitesimals of the same order, and their ratio ^ = 2. x To obtain the second approximation to the value of y, test for the order and value of the infinitesimal y 2 #, by com- paring it with z 2 , thus = 9 when x= 0, y = 0, ^ = 2, x hence y = 2 x + 9 x*, with an error above the second order of smallness. Since the second-order term 9a^ is positive, the curve is situated above the tangent y = 2 x on both sides of the origin. If desired, the third-order term can be obtained by substituting 2 x + 9 x 2 for y in the second and third order terms of the given equation, and collecting the coefficient of X s . The third approximation is then y = 2x + 9x* + 130 X s . This shows that the curve is above the parabola y = 2x + 9a^ on the right of the origin, and below it on the left. These two curves have contact of the second order at the origin. (2) Trace in the vicinity of the origin the curve xy + */ 2 + X s - 2/ + a* - 2 a% = 0. Here the origin is a double point, at which the tangents, obtained by factoring 2 x 2 -f xy + y z = 0, are y x = 0, 294 DIFFERENTIAL CALCULUS [Cn. XVIII. y + 2 x = ; hence on one branch ^ = 1, and on the other x y~ = 2 ; thus the branches are obKque, and on each branch x x and # are infinitesimals of the same order. The second approximation to the equation of each branch is to be obtained by taking account of the third-order terms in the given equation, thus then, on the first branch the comparison of y x with x* gives y - X* hence the branch has the approximate equation y = x -f- 1 # 2 , which shows that it lies above the tangent y = x on both sides of the origin. The third approximation, obtained by writing the given equation in the form 11 x =^ 7; , y + 2x substituting for y the second approximation and dividing as far as z 3 , is y = x + ^ x 2 + f f X s ; which shows that the first branch is above the parabola y x + ^x 2 on the right, and below it on the left of the origin. On the second branch the com- F IO . go. parison of y + 2 x with a? gives y axis is a double one ; thus the origin is a triple point. To determine the form of the curve near the origin, the method of Art. 169 will not apply, as x, y are not infinitesimals of the same order on either branch. Here a method of trial will be employed. Suppose the terms xy* and y* are of the same order on one branch, then x and y 2 are of the same order, i.e., y is of the same order as x*, hence the terms in the given FJQ * 81 equation are of the respective orders 2,31,2,5; thus the terms selected are of the lowest order, and are there- fore the controlling ones near the origin, showing that there is a branch having the approximate equation xy* y* = 0. Removing the factor y z , the equation of this branch is x y z = ; and the next term in its equation is given by 2 T^ - (Y^ / >l hence the branch is situated to the left of the parabola x y 2 = above the 2>axis, and to the right below that axis. 296 DIFFERENTIAL CALCULUS [CH. XVIII. Next suppose there is a branch for which y^ and 2 x 6 are infinitesimals of the same order ; then y has the same order as x*, and the four terms have the orders 31, 4J, 5, 5; hence there is no branch for which the two terms selected are the controlling ones. Once more, suppose there is a branch on which xy z and y?y are of the same order; then dividing by xy, it follows that y is of the same order as z 2 , and the orders in x of the four terms are 5, 5, 8, 5. Therefore there is a branch on which the first, second, and fourth are the controlling terms, and its approximate equa- tion is xy z + x*y - 2 y*> = 0, which reduces to + x*y 2 x 4 = 0, i.e., \ Hence the part of the curve in question consists of the two parab- olas y = x 2 , y = 2 x 2 . Writing the given equation in the form the first of these branches has the equation y- = J/ 4 x(y = (approx.) 170-171.] CURVE TRACING 297 hence the curve gets steeper than the approximate parabola on one side and flatter on the other. Similarly, the approximate equation of the other branch is Combining these two sets of results, the form of the curve in the vicinity of the origin is as given in Fig. 83. Fis. 83. 171. Two branches oblique ; a third touching jr-axis. Trace, in the vicinity of the origin, the curve z 4 4- a?y y^ = 0. Since there are no terms below the third degree, the origin is a triple point, and the three tangents represented by the equation *fy - f = y (? - y} ( * + y) = have the separate equations y = 0, y = x, y = -x. To show roughly, without resorting to expansion, how the curve is related to these three lines, write its equation in the form First consider points near the origin on the branch that touches the line y = 0. Here ll j n n ^= 0, hence y is infini- u x tesimal as to z, and the factor (y z z 2 ) is negative, but the term on the right is positive, hence the other factor on the left, /, is negative ; thus the curve is below the line y = on both sides of the /-axis. 298 DIFFER EN TIA L CALC UL US [CH. XVIII. Next consider points on the branch that touches the line y x. When x is positive, y is positive, hence the factor y 2 x 2 is also positive ; thus y is greater than cc, and the curve lies above the tangent y = x in the first quarter. In the third quarter both x and y are negative, hence y 2 x 2 is negative, and y is numerically less than x ; thus the curve is above the tangent y = x in the third quarter. Lastly, consider points on the branch that touches the line y = x. Here again (y 2 x 2 ) has the same sign as /, hence in the second quarter y is numerically greater than x, and the curve is above the tangent ; but in the fourth quarter y is numerically less than x, and the curve is again above the tangent. The position of the three branches can, however, be ascer- tained with greater accuracy from their approximate equa- tions, obtained by the method of expansion : y = x 2 X* ; y= x 4- \$ fz 3 + ; y = % + %v* + fz 3 "-- The form of the infinite branches will be considered later, and it will appear that the branches in the first and second quarters are the only ones that ex- tend to infinity (Fig. 84). Form of curve in vicinity of point (, 6). The form in the vicinity of any point (a, i) can be found by first transforming to (a, 5) as origin, and proceeding as in Arts. 169-171. This is equivalent to expanding the Fio given function f(x, y) in powers of x a, y b ; and then expressing the small number y b in ascending powers of the small number x a. Remark on expansion of implicit functions. The methods 171-172.] CURVE TRACING 299 of Arts. 139-171 are often practically useful in purely alge- braic operations. They may evidently be applied to any implicit relation between x and y, when the object is to ex- press either variable explicitly in terms of the other, in the vicinity of two given corresponding values (x = a, y = 5), to any required degree of approximation. EXERCISES Examine the following curves in the vicinity of the origin and find two or three terms of the expansion of y in ascending powers of x : 1. y 2 - - x* + x* ; 3. y~ = 2x* + x*; 5. y* - x z = x* -2x 2 y - x 9 ; 2. y* = 2x*y + x*; 4. y a = x a -x*; 6. 9xy = 2x* + 2 y 3 . 7. In No. 6, for the vicinity of the point (1, 2), expand y 2 in as- cending powers of x 1. 172. Approximation to form of infinite branches. It has been shown that in the vicinity of the origin, the approxi- mate form of each branch of the curve could be obtained by examining the different suppositions regarding the rela- tive orders of the infinitesimals x and y, in consequence of which two or more terms of the equation should become infinitesimals of like order, and compared with these all the other terms should be of higher order, and could therefore be neglected in writing down the first approximation to the value of y in ascending powers of x. On a similar principle the approximate form of each branch of the curve at great distances from the origin can be obtained by examining the different suppositions, regard- ing the relative orders of the infinites x and y, in conse- quence of which two or more terms should become infinites of the same order, and in comparison with these all the other terms should be infinites of lower order, and could therefore 300 DIFFERENTIAL CALCULUS [CH. XVIII. be neglected in writing down the first approximation to the value of y in descending powers of x. Ex. 1. Take the curve traced near the origin in Art. 171, y(x 2 - y 2 ) + x* = 0. Here the supposition that y is an infinite of the same order as x* makes the terms y z and x* infinites of order 4, and the term yx 2 an infinite of order 3; thus there is an infinite branch which has the approximate equation y z = x*, and hence passes out of the field in the manner shown in Fig. 85. Again, the supposition that y is of the same order as x' 2 makes yx 2 and z 4 infinites of order 4, and the term y z an infinite of order 6, which cannot be neglected in com- parison. Hence there is no infinite branch on which y is approximately proportional to x 2 . Similarly the third supposition does not correspond to an infinite branch. Ex. 2. Consider the curve that was traced near the origin in Art. 170, xy 2 + x s y - y* - 2 x 6 = 0. The supposition that y is of the same order as x* makes the four terms of the orders 3J, 4J, 5, 5 ; hence there is an infi- nite branch whose approximate equation is y* + 2 x 5 = 0. The form of the curve is shown in Fig. 86. Hyperbolic and parabolic branches. Expansion in descend- ing series. On an infinite branch the coordinates x and y may behave as follows : 1. One of the coordinates may approach a finite number, and the other become infinite. The branch has then a hori- zontal or vertical asymptote (Art. 130), and is thus a hori- zontal or vertical hyperbolic branch (Fig. 40). FIG. 86. 172.] CURVE TRACING 301 2. The coordinates may become infinites of the same y order. Then - = ?w, a finite number ; hence there is, in general, an oblique asymptote, that is, the infinite branch is, in general, an oblique hyperbolic branch. [In a special case it is an oblique parabolic branch. See Exs. 4, 5.] 3. The coordinates may become infinites of different or- ders. If y is an infinite of higher order than #, there is a parabolic branch on which the tangent tends to become ver- tical (Figs. 85, 86), called a vertical parabolic branch. If y is of lower order than #, there is a horizontal parabolic branch (Fig. 81). The test for Case (1) has been given in Art. 131. Case (2) comes under the head of oblique asymptotes ; but it may be conveniently treated along with Case (3) by the method of this Article. The test for Case (2) is to observe whether there are two or more terms of the highest degree in x and y. If so, the supposition that y is of the same order as x makes these the controlling terms. Ex. 3. Test for oblique infinite branches the fourth-degree curve 3* + x*y + 2 y* = x* + 3 x z - y\ Here there are two fourth-degree terms, and the supposition that y and x are infinites of the first order makes these the controlling terms : hence there is an oblique branch on which " = 1. On putting the first x approximation, y= x, in the third-degree terms, and dividing by z 8 , there results, for the second approximation, y = x + 3; and this, when used 1 fi in the same way, gives the third approximation, y = x + 3 --- h x Thus the branch is hyperbolic, having the oblique asymptote y= z+3. There is also a pair of vertical parabolic branches, on which v/2 302 DIFFERENTIAL CALCULUS [Cn. XVIII. Ex. 4. Test in the same way the cubic curve (y - 2 xy 2 (y + z) = 5 z 2 + xy + 5 y* + 3 x - 7 y + 8, in which the terms of the third degree have a square factor. Corresponding to the single factor y + x there is, as before, a hyper- 7 bolic branch whose equation is w = z + 1 + . 9 x The equation of the branch corresponding to the square factor is given by (v o y = y + x The first approximation, y = 2 x, used on the right, gives (y 2 a;) 2 = 9x; and the second approximation, y = 2x 3z2, used in the same way, gives y = 2x 3 xz + 2 + in descending powers of x?. Hence the branch on which ?-= 2 has no linear asymptote. The curvilinear asymp- x tote of lowest degree is the second-degree curve (y 2x 2) 2 = 9 x. There are thus two oblique parabolic branches. Ex. 5. When the terms of highest degree have a factor repeated three times, show that the corresponding expansion of y descends in powers of x*, and that the asymptote of lowest degree is a cubic curve. The method of successive approximation in descending series can also be used in Case (3), when once the first ap- proximation has been obtained by the method of comparison given above. Ex. 6. In the curve of Fig. 85, the first approximation is y = X*. Substituting this on the right of y* = x 4 + x*y, and taking cube root, the second approximation is y x$ + \x* + , in descending powers of x$. 4 2 For the third term it is easiest to let y = x* + | x* + p, substitute in y s x 4 + x 2 y, and determine p so that the coefficients shall be equal as fa , t Z7 , y + \X* ^f + . Ex. 7. In Ex. 5 of Art. 171 show that on two branches the controlling terms are y 2 + 2 x*y - z 4 ; that is, [y - x 2 (V2 - 1)] [y -f ar 2 (\/2 + 1)], and that the equations of these branches are 172-173.] CURVE TRACING 303 Remark on implicit functions. By this method, when any implicit algebraic relation between x and y is given, the value of either variable for large ,values of the other can be computed by descending series, with small relative error. Transcendental Cartesian Curves. A number of figures of important transcendental curves are shown on pp. 237-238, and in A. G., p. 211 ff. They are traced by tabulating y, with assistance from ^, &- dx ax 2 EXERCISES Apply the methods of this article to the equations at end of Art. 171. In No. 6 compute the value of y when x 20, by descending series. 173. Curve tracing : polar coordinates. In tracing curves defined by polar equations there is, as in the case of Cartesian equations, no fixed method of procedure. If, as usually happens, the equation can be solved for p, successive values may be given to 0, and the corresponding values of p computed and tabulated. In constructing the table it is useful to record at what values of the radius vector p has turning values. The critical values of 6 for this purpose are, as usual, determined from the equations ifl = ^' ^ = ' anc ^ are se P ara tely tested by observing whether the derivative changes its sign. Next should be noted the asymptotic directions, which correspond to those values of 0, if any, at which p passes through an infinite value. The distance of the asymptote from the infinite radius vector is given in magnitude and sign by the corresponding value of the polar subtangent a- = p z . Again, if p tends to a definite limit, as 6 becomes dp infinite, there is a circular asymptote. 304 DIFFERENTIAL CALCULUS [Cn. XVIII. On sketching the path of the point (p, 0) from the tabu- lated record, greater accuracy in the direction of the curve at any point may be obtained by computing the slope of the tangent line to the radial direction, from the relation tan i/r = p -. The same result can be achieved by tabulating dp the values of to vl. The lines 6 = JTT, 6 = |TT are axes of symmetry. Ex. 2. Transform to polar coordinates the equation (x 2 + y 2 ) 2 - 2 a#(a: 2 + # 2 ) = a 2 or 2 , and then trace the curve. On putting x = p cos 0, y = p sin 0, dividing by p 2 , and solving the quadratic for p, there results p = a (sin 1). First take the upper sign ; then dO and the following table is easily computed. The figure is shown in Art. 108. If the lower sign be taken, the same curve will be traced in a different order. The line 6 = i TT is an axis of symmetry. 306 DIFFERENTIAL CALCULUS [Ca. XVIII. 173. p dp d tan i// a a 1 ITT 1.7 a .71 a 2.41 i* 1.87 a .5 a 3.73 ITT 2a + o GO t 1.7 a -.71 a -2.41 7T a a -1 1* .29 a -.71 a -.41 ** + .29 a ~0 .71 a .41 VTT .5 a .87 a .58 27T a a 1 p a maximum, ij/ = | TT. p a minimum, if/ = 0, origin a cusp. EXERCISES Trace the following curves : _ x ' -' 4. y\x - a) = (x + o) 2. ?/ 2 = 2 x 2 + z 3 . 3. ?/ 2 = or 4 + x 5 . 5 x^f = a\xi - v/ 2 ). 6. Show that the curve y z = x s x* has two branches which are both tangent to the axis of x at the origin. 7. Determine the direction of the curve y s = x\x o) at each point where it crosses the axis of x. 8. Trace the curve y 8 axy b z x = in the neighborhood of the origin. 9. Show that the curve p = 1 + sin 5 consists of 5 equal loops. 10. Trace the curve p cos 2 = a. Find its asymptotes and lines of symmetry. 11. Trace the curve p = a (tan 1). 1 _ i _ 12. Trace the curves v = e x . y = e x , - _ = e x ~ 3 . y y 13. Find the points of inflexion of the curve y = e~ x *. This curve is known as the probability curve (Fig. 49). 14. /3=a + sinf0. 15. p = a(l-tau0). CHAPTER XIX ENVELOPES 174. Family of curves. The equation of a curve, /(*, y) = o, usually involves, besides the variables x and y, certain coeffi- cients that serve to fix the size, shape, and position of the curve. The coefficients are called constants with reference to the variables x and y, but it has been seen in previous chapters that they may take different values in different problems, while the form of the equation is preserved. Let be one of these "constants" ; then if a be given a series of numerical values, and if the locus of the equation be traced, corresponding to each special value of a, a series of curves is obtained, all having the same general character, but differing somewhat from each other in size, shape, or position. A system of curves so obtained by letting one of the constant letters assume different numerical values in the fixed form of equation f (x, y) = is called a family of curves. Thus if A, k be fixed, and p be arbitrary, the equation (y k) z =2p(x7i) represents a family of parabolas, having the same vertex (^, F), and the same axis y = k, but having an arbitrary latus rectum. Again, if k be the arbitrary constant, this equation represents a family of parabolas having parallel axes, the same latus rectum, and having their vertices on the same line x = h. . CAI.C. 21 307 308 DIFFERENTIAL CALCULUS [Cii. XIX. The presence of an arbitrary constant a in the equation of a curve is indicated in functional notation by writing the equation in the form f(x,y, ) = 0. The quantity a, which is constant for the same curve but different for different curves, is called the parameter of the family. The equations of two neighboring members are then written f(x, y, a) = 0, /(>, y, a + 1i) = 0, in which A is a small increment of a ; and these consecutive curves can be brought as near to coincidence as desired by diminishing h. 175. Envelope of a family of curves. The locus of the points of ultimate intersection of consecutive curves of a family, when these curves approach nearer and nearer to coincidence, is called the envelope of the family. Let /(>, y, ) = 0, /(>, y, + A) = (1) be two curves of the family. By the theorem of mean value (Art. 66) /(re, y, a + A) =f(x, y, ) + A|^>, y, + 0A), (2) [0 < < 1 but the points common to the two curves satisfy equations (1), f) f and therefore also satisfy ~(x, y, 4- #A)= 0. Hence, in rl f the limit, when h = 0, it follows that (x, y, a) = is the da equation of a curve passing through the ultimate intersec- tion of the curve /(#, y, ) = with its consecutive curve. This determines for any assigned value of a definite point of ultimate intersection on the corresponding member of the family. The locus of all such points is then to be obtained by eliminating the parameter a. between the equations ftf /O, y, ) = 0, ~(x, y, )= 0. 174-176.] ENVELOPES 309 The resulting equation is of the form l?(x, y) = 0, and represents the fixed envelope of the family. 176. The envelope touches every curve of the family. I. G-eometrieal proof. Let A, B, O be three consecutive curves of the family ; let A, B intersect in P ; B, O inter- sect in Q. When P, Q approach coincidence, PQ will be the direction of the tangent to the envelope at P ; but since P, Q are two points on B that approach coincidence, hence PQ is also the direction of the tangent to B at P ; thus B and the envelope have a common tangent at P ; similarly for every curve of the family. II. More rigorous analytical proof. Let f(x, y, )= be solved for a, in the form a = (x, y} ; then the equation of the envelope is f(x, y, 4>(z, y))= 0. Equating the total ^-derivative to zero, .__ dx dy dx d\dx dy dx df f)f but ~ = -i- = 0, hence the slope of the tangent to the en- 09 da velope at the point (a?, y) is given by .-. dx "*" dy dx but the same equation defines the direction of the tangent to the curve /(#, y, )= at the same point. Therefore a 310 DIFFERENTIAL CALCULUS [Cn. XIX. point of ultimate intersection on any member of the family is a point of contact of this curve with the envelope. Ex. Find the envelope of the family of lines y = mx + ji> ' C 1 ) obtained by varying m. Differentiate (1) as to m, = *- (2) wi 2 Hence the line (1) meets its consecutive line where it meets (2). To eliminate m, solve (2) for m, substitute in (1), and square; then the locus of the ultimate intersections is the fixed parabola y 2 = kpx. 177. Envelope of normals of a given curve. T.he evolute (Art. 158) was defined as the locus of the center of curva- ture. The center of curvature was shown to be the point of intersection of consecutive normals (Art. 151), hence by Art. 175, the envelope of the normals is the evolute. Ex. Find the envelope of the normals to the parabola y* = 4 px. The equation of the normal at (x v y^) is or, eliminating x l by means of the equation yf = 4 px v The envelope of this line, when y l takes all values, is required. Differentiating as to y v Substituting this value for y^ in (1), the result, 27^/ 2 = 4(x-2/)) 3 , is the equation of the required evolute. 176-178.] ENVELOPES 311 178. Two parameters, one equation of condition. In many cases a family of curves may have two parameters which are connected by an equation. For instance, the equation of the normal to a given curve contains two parameters, x v y v which are connected by the equation of the curve. In such cases one parameter may be eliminated by means of the given relation, and the other treated as before. When the elimination is difficult to perform, both equa- tions may be differentiated as to one parameter , regarding the other parameter ft as a function of a, giving four equa- tions from which a, /3, and -/- may be eliminated, and the da resulting equation will be that of the desired envelope. Ex. 1. Find the envelope of the line 2+f-X a b the sum of its intercepts remaining constant. The two equations are - + ? =1 ' a b a + b = c. Differentiate as to a, ^_^ = , a 2 6 2 da da it eliminate , then = ^ therefore da a 2 2 x y x + y = ^ = ? * = !, hence a = a b a + b c therefore Vx + Vy = Vc is the equation of the desired envelope. 312 DIFFERENTIAL CALCULUS [Ca. XIX. Ex. 2. Find the envelope' of the family of coaxial ellipses having a constant area. Here ab = k\ For symmetry, regard a and b as functions of a single parameter t, then r 2 ift bda + adb ; hence = = a 2 b 2 2' a = and the envelope is the pair of rectangular hyperbolas xy = | k*. Y FIG. 89 NOTE. A family of curves with a single parameter may have no envelope ; i.e., consecutive curves may not intersect ; e.g., the family of concentric circles z 2 + y 2 = r 2 , obtained by giving r all possible values. 178.] ENVELOPES 313 EXERCISES 2 1. Find the envelope of the parabolas / 2 = ^- (x a), a being a parameter. 2. A straight line of fixed length a moves with its extremities in two rectangular axes; find its envelope. 3. Ellipses are described with common centers and axes, and having the sum of the semi-axes equal to c. Find their envelope. 4. Find the envelope of the straight lines having the product of their intercepts on the coordinate axes equal to k 2 . 5. Find the envelope of the lines y fi = m(x a) + rVl + m 2 , m being a variable parameter. 6. What is the evolute of the envelope of Ex. 5? 7. Circles are described on successive double ordinates of a parabola as diameters; show that their envelope is an equal parabola. Find what part of this system of circles does not admit of an envelope. 8. Show that the envelope of = 9. Find the curve whose tangents have the general equation y = mx -v/ai 2 + bin + c. 10. Prove that the circles which pass through the origin and have their centers on the equilateral hyperbola x*-y* = a* envelop the lemniscata (x 2 + y 2 ) 2 = 4 2 (x 2 y 2 ). 11. If in Ex. 10 the locus of the centers of circles passing through the origin be the parabola y 2 = 4 ax, the envelope will be the cissoid 12. Show that a family of curves having two independent parameters has no envelope. 13. In the " nodal family" (y - 2 a:) 2 = (z - x) 2 + 8 x* - y 9 , show that the usual process gives for envelope a composite locus, made up of the "node-locus" (a line) aud envelope proper (an ellipse). Generalize. APPENDIX NOTE A (P. 29) Let y=f(x) be a function which is continuous and increasing from x = a to x = b ; and let /(a) = A, f(b) = B. Let the inverse function be written x = $ (y) ; then it is pro- posed to show that <(y) is a continuous function of y from y = A to y = B. Let h be any assigned positive number numerically less than b a ; then, since f(x) is an increasing function, preserves its sign unchanged when x and x + h both lie anywhere in the interval from a to b. Let the smallest value that this dif- ference can take for the assigned value of h be /(* + *)-/(*) = *;. (1) then when x' > x + ^, /(*') -/(*)> fc. (2) Consequently, if /(aO -/(*)<*, ( 3 ) then #' must be less than a? + A, i.e., ' a; < h ; (4) or, putting /(a?) = y, /(*') = y', a? = < (y), ' = < (y 1 ), (3) and (4) may be written thus : if y '- y(y) < h, the assigned number. (6) 314 NOTES A-B.] APPENDIX 315 Hence, (y) is a continuous function of y throughout the stated interval. A similar proof applies to intervals in which f(x) is a decreasing function. Hence: For every interval in ivhich a function is continuous there exists an interval in which the inverse function is continuous. NOTE B (P. 60) To prove m l x (l H ) = e, when ra is a positive integer. \ m J The proof of Art. 30 can be readily completed by use of a method exemplified later in Art. 67. As shown in Art. 30 the problem is to prove rigorously that the limit, when m = oo, of the sum of the entire m + 1 terms of the series 11 ml m m 1 m m f + ~~~ +~~~ ~~ *~~~ ~~ is equal to the sum to infinity of the series without unduly applying the theorems of limits in the case of an infinite number of variables. For this purpose the remainder of series (1) after the first n terms will now be examined. Let the sum of the first ?i terms in (1) and (2) be denoted by S n and E n respectively, then 1_1 i_l ! n-2 ! m 1 m m and, evidently, when n is any finite number, ^00^ = ^. (5) Next let R a be the remainder of the series (1) after its first n terms, that is, the sum of the last m + 1 n terms ; then the sum of the series is S = S n +K n , (6) 316 DIFFERENTIAL CALCULUS [NOTE B. ,_, lim o lim ry lim r> / 7 \ and m = oo S ~ m = oo & = m = oo * ( 7 ) Now the first term in R n is the (n -f l)st term of series (1), and the ratio of this term to the preceding (which is the last term in (3)) is m but this ratio is less than -, and evidently the ratio of any subse- n quent term to the preceding one is still less than this, therefore f- + - + -\ [Of. p. 112. n 7i 2 w 3 / hence * R n < U.(--\ < ,_ * (8) It follows from (5), (7), (8) that lira T? ^ 1 /a\ j;_ O Ml n (\J ) and therefore that this difference can be made as small as desired by taking n large enough. Thus the limit when m = oo of the sum of series (1) is equal to the limit approached by the sum of the first n terms of series (2) when n is infinitely increased; and this completes the proof of Art. 30, when m is a positive integer. To prove the theorem when m is unrestricted. If m is positive but not an integer, let it be supposed to lie between the two positive integers p and p + 1, i.e. p < m < p + 1 ; 11 1 1 A IV f. IV then -> , 1+->1H , 1 + 1 > 1 H > p m p m \ pj \ mj hence (l-f-].(lH ) >(lH ) (1) \ PJ \ m J \ mj A g ain > ;TTT < -' ( ! + r-r hence fl + _-L_) . .fl+JLV <(! + -) ( 2 ) NOTE B.] APPENDIX 317 Hence, from (1) and (2), / -\ \P+l/ 1 \m-p-l / 1 \m / INp/ 1 \m-p (1+- v) (1+-) <(1+ 1 ) <(!+-) (1+1) (3) v P + v V m J \ m J \ PJ \ J It will now be shown that when p, m, p + 1 all =00, the first and third members of these inequalities have the common limit e. For, since the exponents in p, m p 1 are finite, 1 \"*-P / 1 \m-p-l 1+i =1, 1+1 =1; but since p,p + l are infinite positive integers, / 1V / 1 V+ 1 fi+lj =e, (1+^3) =; hence e is the common limit of the first and last members of (3), and is therefore also the limit of the intermediate member, lim Finally, let m be any negative number, say p, f 1\ m / 1\~p /n 'P then (1+1) =( 1 - i ) = (- = \ mJ \ pj '\ P , Writing fc f or p 1, * +1 but when m = co and A: = + oo, therefore, by (4), lim m = o 318 DIFFERENTIAL CALCULUS [NOTE C. NOTE C (P. 187) On maxima and minima in two variables. In giving the criteria for maxima and minima in Art. 107 it was stated that it is in general unnecessary to consider terms above the second degree in h and k, as such terms are usually infinitesimals of an order higher than that of the second degree terms. The exceptional cases, in which some of the terms of higher degree may become of equal importance with the second degree terms, can be readily treated by the method of comparison illustrated so extensively in the later chapter on curve tracing. Using the notation of Art. 107, let A< = (a + h, b + k) (a, b) = u 2 + u 3 + u 4 -\ , (1) in which u r denotes a homogeneous polynomial in h and k of degree r; and, representing the function <(#, ?/) as usual by the ordinate of the surface whose equation is z = (x, y), let the origin be transferred to the critical point whose coordinates are a, b, < (a, 6) ; then the equation of the surface becomes z' = A4> = u 2 + w 3 + W4 + -", (2) in which h, k, z' are the new current coordinates. The equation of the tangent plane at the origin is then z' = 0, and the curve of section which it makes on the surface has the equation u 2 + u 3 + u 4 +-~=0. (3) The form of this plane curve in the vicinity of the origin will be a decisive test for a maximum or minimum. By Chapters XVII, XVIII, when the lowest terms are of the second degree, the origin is either a node, a cusp, a point of osculation, or a con- jugate point. If the factors of u 2 are imaginary, the origin is an isolated or conjugate point of the locus, hence, in the vicinity of the critical point, the surface is altogether at one side of the tangent plane, and has a maximum or minimum ordinate. If the factors of u 2 are real and distinct, the curve of section has two NOTE C.] APPENDIX 319 branches passing through the origin, hence part of the surface will be above the tangent plane and part below it, and there will thus be no complete maximum or minimum. In both of these cases it is unnecessary to examine the higher terms unless a minute knowledge of the deportment of the given function is desired. Lastly, let u 2 be a complete square of the form (Ah + Bk) 2 . In this case, the origin is usually either a cusp or a point of oscu- lation, as in Art. 163 ; but it may possibly be a conjugate point of the kind noticed in Ex. 9, Art. 164, at which the tangents are real and coincident. It is therefore necessary to examine the higher terms. For convenience transform the axes so that h' = Ah + Bk, k' = BhAk, then the equation of the curve takes the simple form A*+u', + u' 4 + ...=0. (4) When the method of comparison of Art. 170 is applied, suppose it is found that h 1 and k' m are of the same order, then all the terms of (4) that are of the same order as h' 2 will constitute a poly- nomial in h' and k' m , which can, as in Art. 170, be factored into the form (h 1 -f /zfc' m ) (h 1 + vk' m ). These will be the controlling terms ; hence, when p., v are imaginary, the origin is a conjugate point, and there is a maximum or minimum ordinate of the sur- face ; but when /x, v are real and distinct, the origin is a cusp or a point of osculation, according as m is or is not a fraction with even denominator, and there is no complete maximum or mini- mum. When (1, v are real and equal, the above process is to be repeated. For a simple illustration see Ex. 5, p. 190. Ex. 1. Show that when c < 1, unity is a turning value of Ex. 2. For different values of c, examine in the vicinity of the values x = 0, y = 0, the deportment of the function c (z 2 + 3 y)* - 4 (x 2 + 3 y) (a* + 5 y) + 2 (a; 3 + 5 y*). sinh x , i cosh x cosh a; 1 sinh x 1 cosh Ox- sinh x 320 DIFFERENTIAL CALCULUS NOTE ON HYPERBOLIC FUNCTIONS Definitions and direct inferences. For the present purpose the hyperbolic cosine and sine may be defined analytically in terms of the exponential function, as follows : cosh x = \(e* + e *), sinh x = i(e* e'*), (1) and the hyperbolic tangent, cotangent, secant, and cosecant are then defined by the equations (2) Among the six functions there are five independent relations, so that when the numerical value of one of the functions is given, the values of the other five can be found. Four of these relations consist of the four defining equations (2). The fifth is derived from (1) by squaring and subtracting, giving cosh 2 x sinh 2 x = 1. (3) By a combination of some of these equations other subsidiary relations may be obtained ; thus, on dividing (3) successively by cosh 2 a;, sinh 2 x, and applying (2), it follows that 1 tanh 2 x = sech 2 a;, ) f (4) Equations (2), (3), (4) will readily serve to express the value of any function in terms of any other. For example, when tanh a; is given, coth x = , sech x = VI tanh 2 x. tanh x 1 i tanh x cosh x = . sinh x = Vl tanh- a, VI tanh 2 a; Ex. 1. From equations (1) prove cosh ( *) = cosh a;, sinh ( a;) = sinh a;, coshO= 1, sinhO = 0, coshoo = oo, sinhco = oo. APPENDIX 321 Ex. 2. From equations (3), (4) show that cosh x > siuh x, coshx>l, tanhx cfar It thus appears that the functions sinh x, cosh x reproduce themselves in two differentiations, just as the functions sin x, cos x produce their opposites in two differentiations. In this connection it may be noted that the frequent appearance of the hyperbolic (and circular) functions in the solution of physical problems is due to the fact that they answer the question : What function has its second derivative equal to a positive (or negative) constant multiple of the function itself ? 322 DIFFERENTIAL CALCULUS Ex. 7. Eliminate the constants by differentiation from the equation y = A cosh mx + B sinh mx, and prove " m 2 w. dx 2 Ex. 8. Prove coth x = csch 2 x, sech x sech x tanh x. dx dx Expansions. By applying Maclaurin's theorem, using (5), (6), (8), or else by substituting the developments of e 1 , e~ x , in (1), the following series are obtained : x 2 x 4 x 6 + 2~! + 4~] + 6] + = . + ^ + ^ + ^ + 3! 5! 7! By means of these series, which are available for all finite values of x, the numerical values of cosh x, sinh x can be com- puted and tabulated for successive values of x* Derivatives of inverse hyperbolic functions. Let y = sinh" 1 x, then x = sinh y ; (10) dx = cosh y dy = Vl + x* dy ; hence sinh" 1 x = (11) dx vT~ Similarly, cosh a x = (12) Again, let y = tanh" 1 x, then x = tanh y, dx = sech 2 y dy = (1 tanh 2 y) dy = (1 x*) dy ; therefore tanh- 1 x = L- -1 - (13) dx 1 a^Jxo Similarly, -- coth- 1 a = (14) Ex. 9. Prove sech- 1 x = cosh- 1 - = -1 dx dx x xVl - 1 - csch- 1 x = sinh- 1 - = dx dx re * See Tables, p. 162, Merriman and Woodward's " Higher Mathematics." APPENDIX 323 Ex. 10. Prove * d sinh- 1 - = dx , d cosh-i - = - a Vx' 2 + a 2 . a Vx 2 a 2 a taiih-i * = -^-1 , cZ coth-i 5 = *L a tf x 2 J z < a x 2 a* Relation of hyperbolic functions to hyperbolic sectors. In the circle of -f- y 2 = a 2 , let yl be the area of the sector in- cluded between the radii drawn to the points (a, 0), (x, ?/); and let 6 be the included angle ; then, by geometry, 2 A=a?0=a? sin- 1 -^ = a 2 cos - 1 - a a Again, it is shown in Integral Calculus by means of the deriva- tives in Ex. 10, that in the hyperbola a? y 2 = a 2 , if A' be the area of the sector included between the radii drawn to the points (a, 0), (x, y}, then 2 A'= a? sinh~- = a 2 cosh" 1 '-. a a Thus the hyperbolic functions are related to hyperbolic sectors as the circular functions are related to circular (and elliptic) sectors, t Expansions of inverse hyperbolic functions. By the method of Art. 67, sinh~ 1 a;=a; - + . [ l 1, is obtained by writing the derivative in the form sinh- 1 * = (a* + 1) = da; as\ = Vi_ 1 i + 131 \ a;V 2^ 2 4 as* / hence, S inh-^^ + lo ga; +- +..., (16) * These derivatives will be found useful in the "Integral Calculus." t For a treatment of the hyperbolic functions from this point of view, see Merriman and Woodward. DIFF. CAI.C. 22 324 DIFFERENTIAL CALCULUS where A is a constant, which is shown later to be equal to log 2 ; (17) in i i , i 11 131 similarly, coslrt^+log *_-__- L ___ which is always available for computation, since cosh" 1 ^ is a real number only when x > 1. Ex. 11. Prove that tanh -1 x = always available when tanh~'x is real, i.e., when 1 < x< 1. -x 3 +-x 5 + , and that this series is 3 5 xcoshy, then Vs? 2 l = sinh?/, 05+ Va? 2 1 = cosh ?/-f- sinh y=e v , y= cosh" 1 ^, = log(a;+ V^ 2 1). sinh~ 1 o;=log(x+ Va^-f 1). - ! T = cosh" 1 - = log csch~ 1 a;=sinh~ 1 - = log x Logarithmic expressions for inverse hyperbolic functions. Let hence Similarly, Also Again, let therefore i.e., hence and c=tanhw= _ g-V 1 a; = log x 2 * x-l Ex. 12. Show from (18), (19) that, when x = , sinh- 1 x log x = log 2, cosh" 1 x log x = log 2, and hence that the constant A in (16), (17) is equal to log 2. (18) (19) (20) (21) (22) (23) APPENDIX 32o Graphs of hyperbolic functions. The student is advised to sketch the graphs of these functions from their definitions and fundamental properties. Aid is also obtained from the values of their first and second derivatives. Ex. 13. The curve y = sinh x has an inflexion at the origin, the slope of the tangent being unity ; the bending is upwards to the right and downwards to the left of the origin. Ex. 14. The curve y = coshx is symmetrical as to the //-axis, and has a minimum ordinate at x = 0. Ex. 15. Show that the curve y = tanh x has two asymptotes y = 1. Ex. 16. Using the graphs, give approximate solutions of the transcendental 3 equations, tanhx = x 1, coshx = x + 2, sinhx=-x, cos x cosh x=l. 2 *i /* Ex. 17. The equation of the catenary is - = cosh - ; show that the deriva- tive of the arc is = cosh -, and hence that - = sinh -. dx c c c Gudermanian function. When two variables x, y are so related that secy = cosh a?, then y is called the Gudermanian function of x, and is denoted by gd x. The angle whose radian measure is equal to gd x is called the Gudermanian angle of x. Ex. 18. Show that the six hyperbolic functions of x can be expressed as circular functions of gd x : e.g. , cosh x = sec gd x, sinh x = tan gd x, etc. Ex. 19. The curve y = gd x has asymptotes y = J *. Ex. 20. Prove gdx = sech x, gd~ J x = sec x. dx dx NOTE ON INTERPOLATION BY TAYLOR'S THEOREM Two ordinates given ; to compute an intermediate ordinate. In the curve y =/(&), let the ordinate at x = a be y lf and let the ordinate at x = a + h be y 2 . If y\, y 2 be given numerically, it is required to compute the ordinate y at the intermediate point x = a + fh, where e < 1. DIFFERENTIAL CALCULUS Consider the three equations, 2/i =/(), (1) y*=f(a + li)=f(a)+hf'(d), [neglecting ft 2 /"(a)] (2) y =/(a + eft) =/(o) + cV" (a) i (3) then from (1), (2), ft/'(a) = y 2 y 1 ; hence, by (3), The neglect of the term ft 2 / "(a) in (2) is justified either when ft 2 is very small, or when /"(a) is zero. The former is the case when the given ordinates are very close together. The latter is the case when f(x) is of the first degree, i.e., when the locus y =f(x) is a straight line; hence (A) gives accurately the ordinate of the straight line joining two given points on a curve. TJiree equidistant ordinates given ; to compute an intermediate ordinate. Let the ordinate at a ft be y lf at a be ?/ 2 , at a -f h be y 3 ; it is required to find the ordinate at a + eft, where !<l) 11. 2 xa l2 log a. 6. 1 - y 2 . i 19 /o 12. iZXVJ -j-X 1 tf & lo S 2(.3x 2 V2T^)V2~4^ X-l NO (1 + e*) 2 13. x z (l + logx). 8. y 3x 2 6 x . - o 14. -(x - l) 2 (7x 2 + 30x - 97) xlogx 12(x-2)^(x-3)V Art. 40 1. 10 x cos 5 x 2 . 7. 2 esc 2 x. 2. 14 sin 7 x cos 7 x. 8. n sin"- 1 x sin (M -f- l)x. 3. tan* x - 1. 9. tan 2 x. 4. 2 cos 2 x. 10. W ~dx 1 - ' 5. - ~ log a a x sec 2 (a z ). X 11. f sin x . i \ y \ r ^uo jj iug ju i V x y ,7.. 6. secx. 12. cos (sin M) cos M dx Art. 47 1 nn- 1 x in -2x Vl -x 2 XV. 1 + (x 2 - 5) 2 2. sec 2 x tan- 1 x + tana; 1 1 + x 2 . Vl - x 2 3. ] Vl 2 x x 2 12. ^ 1 2(l-x 2 ) ' 2 x V9 x - 1 1 -f 6 x 2 + x* 13. cos log x px X ' 1+e** 14. cotx. 6. 1 X COS X 1 * xVl (logx) 2 15. Vsin x 2 1 ~ . i COS^X Vl X 2 Ifi cos- f> X I in d. V * U . x 2 8 * x Vl -4x* gtan-'z 17- 91 1 + X V2 ax x 2 18. cos (cosx)sinx. ANSWERS 329 Page 72. Miscellaneous Exercises T ^ v 1 X 2 M^VYl+lo^V 4 cos (2 log x 2 - 7) UJ I *J 9- " ~^~ 3. 2 esc 2 x. 10. 1. 4. (2x-5)e 2l + 4(x + 1) e 1 + 1. n. 2 taiH + e eect sectt&nt. 5 e*(l x) 1 12. For all values. ( e * ' 15. x, ?/ are determined from 6. 2 xe-* 2 cos x e~ x2 sin x. ' 2 2/= ?> 2 a; and equation of curve. 21ogz \. dy_ 2x.y 3 -5x^y 2 + 12x /l-4xV ' rfcc Art. 51 1. 12(x 2 -2x+ 1). 7. 6tan*x. 2. 4[(x-2)e2* + (x + 2) e x ~\. 10 cos x 3. 0. (1 - sin x) 2 11 4fl3 4. - x . U ' (a 2 + x 2 ) 2 - 8(e* - e ) 13- 3fl2 . 5. f* g \g 4>/x(x - a)* ,.5x 10 k -T* lOf^ 1" ^ O. *C i^', As n^ M.fiUM x 16 4! o ~ 24 * d-x) ' (l + 2y) 1Y ^' 2 .V_4 ?/ /2 V 2 i~) (gz+y l)(x y) 3 v (e " + 1): ' 18 ' gfy oo ^ ' l- y (ev + x) 8 2(5 + 8z/ 2 + S?/ 4 ) ' (z/ 2 -ax) 8 , (- 1 )n-l( n _l)l | 1 1 -^ 330 ANSWERS ' ' 27. a .30. - Art. 57 3. e' + e'(*- 2)+ ^(*- 2)*+.. 4 ' -3+4(*-l) 4- (*- 21 5. -8+4(2/-;r>+3(j/-3) Art. 64 -V 2 ) Art. 69 Art. 70 a(\/3 + \/2) 2. ao. Art. 72 3. f . 4. 4. 5. I- 7- i 8. -4. Art. 75 1. 1. 11. -1. 20. ~ 2- 30. 2 2. 2. 12. m 21. -3. r 3. 4. 3 22. m. 31. \. 4. -i- 13. t m?' 23. 0. 32. 0. 5. 6. TV- 2. 14. V2 24. 25. 30. 5. 33. 34. 2' v/3+1 7. 1 15. -f- IT* n 16. 2. 26. ^i. 35. 8 : * 8. 9. ,o g 5. 17. 18. 19. 2 3' 1. -1 27. 28. IT 0. 0. 37. 38. a. 7T 10. 2. 2V2' 29. 0. 39. Ait 77 ft. -*. 8. 1. 5. 1 7. 1. a* 4. i. e 8. did". a n 2. C 6 6. 1. 9. 0, discont ANSWERS 331 Art. 86 1. 5, min. ; 7, max. 2. 2, min. ; f, max. 3. 1, j, min. ; |, max. 7. -, min. 8. e, max. 9. ~, min. e 4 10. (?i + )T, max.; (w |)TT, min. ; n any integer. 12. 2 nir, min., and also tan- 1 Vf for angles in 2 and 3 quarter. (2 n+ l)w, tau" 1 V|, 1 and 4 quarter, max. 13. (2 n + 1), min.; sin- 1 ^, max. 14. No max. nor min. i 15. Min. for value of x which satisfies the equation (x l)e 3 * 2x 2 = 0. It is between 1-fa and 1^. Art. 87 6. -fa ira 2 h = f vol. of cone. 9. Isosceles. 7. - A / a3 P. 10- Isosceles. 11. Radius of circle is equal to height 8. Half that of paraboloid = *feL of rectangle. 4 12. Breadth = -5L, thickness = ^. V3 V6 15. Height is equal to diameter of base. 16. V6(c + 6). 17. (* + &!)! 18. Sine of semi- vertical angle is J. 19. \/2. 20. 4 22. One mile from destination. 23. 30. 26. Side parallel to wall is double the other. 27. 2 Art. 90 3. .00145. 8. 24 V3. 9. 2nb. 10. 2. 11. STT. 12. 2. 13. 1 and 5. Art. 96 ^f' y) . dx + dftjf' y) ^'(x)dx ; substitute 0(x) for y. ^ = - 2 dx y Art. 97 ff. Ax 332 ANSWERS Art. 99 x Art. 105 2. x 4 + 4 x*y + 6 x 2 !/ 2 + 4 xz/ 8 + y*. 3. V^tany+^M+fcVisec^ + - ~ tan(y + 4. -25 + (x-2) 2 +(y-3) 2 +(s- I) 2 . Art. 107 4. x = 0, y = 0. 7. The three parts are equal. 8. 6 a 2 ; the parallelepiped is a cube. 9- - = y , = - , + " "^ ; with the upper sign there is a maximum ; a o a 2 + b* with the lower, a minimum. Q 10. x = y = -, min. ; x = y = -, max. 2 6 11. g- 12. ax = by = cz = j< Art. 108 a 2 6 2 3. ^-j-^j. 5. Min.,x = a. 6. Max., x = av^; min., x = 0. 7. Max. for x = 2 Art. 117 3\/3x a 8x , 41 3. (a) XX! + 2/2/1 = c 2 ; (6) x?/! + X K V = 2 fc 2 ; (c) (2 xij/ij- yi 2 )x + (x! 2 + 2 xiyi)y = 3 a 3 ; (d) y-yi = cotx^x - x t ). 5. as = 2 Vj. 8. P = v'ox^, a. 15. At (0, 0), 90. At the other points, 45. 16. 2 ax ~ x2 . 17. ' 3a x a Art. 120 3. Subtangent = p tan a ; subnormal = p cot a. 4. 90. 5. f = 0; $ = 20. ANSWERS 333 Art. 126 - , ffl 2 COS 2 9 5. a, 5 Art. 130 4. x = 0, y = 1. Art. 131 6. y = a ; x = ; and the oblique asymptote x + y = o. Art. 135 S.x y = 1, z-t-y=l, z + 2y = 0. 4. j/ = x. Art. 136 8. xy + a 2 = 0, x 2 y - a 3 = 0. 9, 10, 11 are given in text. Art. 137 1. x = a, y = b, y = x+ b a. 9. x + y = f a. 2. x = -2a, x = a. 10. y = 0. 3. x = 1, y = 1. 11. x = 0, y = 0, x + y = 0. 4. x = y 1, x + y = 1. 12. x = a. 5. x = a, x = jraV2. 13. x 2 = 0; two parabolic branches. 6. y = x. 7. x = 2 a. 14. y = 0. 8. x = 2 a, x + a = y. 15. y = 0, x = y, x = y 1. Art. 139 1. Parallel to initial line ; a units above it. 2. One, perpendicular to initial line, at distance a left of pole. 3. Their equations are p sin ( (2 k + 1) - ] - esc / (2 k + 1) - \ . 2a J a I 2 / 4. ^- = cos 6 sin 0. 5. p sin = 2 a. Art. 143 3. x =(f). a. 5. (4, |V3). 334 ANSWERS a m (w 8. - 2 Art. 156 1. Second. 2. First. 3. Second. 4. Second. 6. a = - 1. 7. Third. 10 2(.r+aO*. 13 (c2 + 2/ 2 Va "J 11 (a*y 2 + bW)* 14. (:c +-^I a 4 & 4 Va 2 15. 3 Vaxy. ' "c" 16. secx. Art. 157. Art 159 o(5 4cosg)' 9 - 6 cos 7 ^ 7> "^ 3. a = -- va 5. 1. (0,0), x y=0. 2. (a, 0), 2(za) Art. 164 , (0, -a). 3. (0, 0), x = 0, ?/ = 0. 4. (0, 0), x y = 0. 9. When it is made numerically smaller. Art. 178 1. & = 2. x% + 3. x^ + 9. (4 4. (x - y) 2 + 4 ky = 0. 5. (x - a) 2 + (y - )2 = 6. The point (a, /3). cx 2 )= 4 ac - 6 2 . (The numbers refer to pages) Absolute value, 84. Absolutely convergent, 84. Acceleration, 157. Actual velocity, 151. Algebraic expression, 3. operation, 1. Argument, 4. Asymptote, 221. Asymptotic circle, 240. Average curvature, 84. velocity, 151. Beman, 113. Bending, 243. Catenary, 211. Cauchy, 94. Center of curvature, 255. Change of variable, 1118. Circle, asymptotic, 240. of curvature, 255. Cissoid, 211. Commutative, 2. Comparison of infinitesimals, 21. Computation of it, 111. Concave, downwards, 241. upwards, 241. Conditionally convergent, 85. Conjugate point, 281. Constant, 7. Contact, '_'.">_'. Continuity of an algebraic function, 30. of a*, 31. of log*, 31. of sin x, cos x, 32. Continuous function, 7. variable, 7. Criteria for continuous function, 29. Critical value, 134. Curvature, 27. Cusp, 279. Decreasing function, 43. Definition of continuity, 8, 158. of curvature, 261. of nth derivative, 73. De Moivre, 101. Dependent variable, 7. Derivative, 37. of arc, 216. of area, 40. of surface, 218. of volume, 218. partial, 160. total, 160. Determinate value, 117. Differentiable, 43. Differential, 156. Differentiation, 41. of inverse function, 48. Discontinuity, 8. Distributive; 3. Elementary forms of curves, 289. Entire, 4. Envelopes, 308. Equiangular spiral, 125. Euler, 172. theorem, 171. Even contact, 171. Evolute, 267. Explicit function, 4. Exponential curve, 211. function, 58. Expression, 3. Family of curves, 307. Form of remainder, 95. Function, 4. Functional differentiation, 47. Fundamental problem, 37. theorem. 20. 335 336 INDEX General exp. func., 82. Generating function, 82. Geometric applications, 212. illus. of a der., 39. Hyperbolic branches, 221. functions, 318. Implicit function, 4. Incommensurable power, 58. Increasing function, 43. Increment, 8. Independent variable, 1. Indeterminate form, 115. Infinite, 10. Infinitesimal, 10. Inflexion, 243. Integral expression, 3. Interval of convergence, 82, 90. of equivalence, 82. Inverse function, 5, 58. operation, 1. Involute, 267, 272. Irrational, 4. Klein, 113. Leibnitz, 75. theorem, 75. Limit, 9. Logarithmic function, 58. Maclaurin, 87. theorem, 8L ^ Maximum, 132, 185. Mean value, 107. Measure of curvature, 261. Minimum, 132, 185. Modulus, 60. Montferier, 113. Multiple point, 277. Naperian base, 60. Natural base, 60. Newton, 75. Node, 277. Non-unique derivative, 43. Normal, 208. length, 209. Number, 1. Oblique asymptotes, 227. Odd contact, 254. Operation, 1. Order of contact, 252. infinite, li. infinitesimal, 19. Osculating circle, 255. Osgood, 82, 85. Parabolic branches, 221, 300. Parallel curves, 272. Parameter, 308. Partial derivative, 160. Perry, 150, 1(54, 190. Polar coordinates, 212. normal length, 213. subnormal, 213. subtangent, 213. tangent length, 213. Polynomials, 141. Principal infinitesimal, 19. Probability curve, 238, 306. Process of differentiation, 42. Radius of curvature, 255. Rate, 152. Rational expression, 4. Rectilinear asymptote, 221. Relative error, 101. Remainder, 86. Rolle, 85. s Semicubical parabola, 290. Series, 81. Shanks, 113. Simple exponential functions, 58. Simultaneous increments, 33. Singular points, 275. values, 117, 275. Slope of a line, 40. Smith, 113. Stationary tangent, 244. Stirling, 85. Subnormal, 209. Subtangent, 209. Successive differentiation, 73. operations, 2. Sum of a series, 83. Surd expression, 4. Symbol of approach, 11. of an increment, 8. for inverse functions, 5. of a function, 5. Symmetric expression, 4. Table of derivatives, 71. Tangeut, 208. INDEX 337 Tangent length, 209. Taylor, 87. Test for convergence, 84. Test for increasing function, 45. Theorems on infinitesimals, 12, 16. Total curvature, 2(51. Total differential, 164. Tractrix, 211. Transcendental, expression, 3. operation, 1. Transformed expression, 4. Trigonometric functions, 58. Turning value, 132. Unconditionally convergent, 89. Uniform velocity, 151. Unique derivative, 43. Variable, 7. Vectorial angle, 213. Typography by J. S. Gushing & Co., Norwood, Mass., U. S. A. McMahon - Elements of the differential calculus UNIVERSITY OF CALIFORNIA LIBRARY Los Angeles This book is DUE on the last date stamped below. 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