GIFT OF the estate of Professor William F, Keyer rx-^^o/ / ^.^- Digitized by the Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/differentialinteOOsnydrich THE MODERN MATHEMATICAL SERIES LUCIEN AUGUSTUS WAIT . . . General Editor (SBNIOB PEOFE880K OF MATHEMATICS IN OOENELL UNIVEKSITT) The Modern Mathematical Series, lucien augustus wait, (Senior Professor of Matbematics in Cornell University,) GENERAL EDITOR. This series includes the following works : ANALYTIC GEOMETRY. By J. H. Tanner and Joseph Allen. DIFFERENTIAL CALCULUS. By James McMahon and Virgil Snyj)er. INTEGRAL CALCULUS. By D. A. Murray. DIFFERENTIAL AND INTEGRAL CALCULUS. By Virgil Snyder and J. I Hutchinson. ELEMENTARY ALGEBRA. By J. H. Tanner. ELEMENTARY GEOMETRY. By James McMahon. The Analytic Geometry, Dififerential Calculus, and Integral Calculus (pub- lished in September of 1898) were written primarily to meet the needs of college students pursuing courses in Engineering and Architecture ; accordingly, prac- tical problems, in illustration of general principles under discussion, play an important part in each book. These three books, treating their subjects in a way that is simple and practi- cal, yet thoroughly rigorous, and attractive to both teacher and student, received such general and hearty approval of teachers, and have been so widely adopted in the best colleges and universities of the country, that other books, written on the same general plan, are being added to the series. The Differential and Integral Calculus in one volume was written especially for those institutions where the time given to these subjects is not sufficient to use advantageously the two separate books. The more elementary books of this series are designed to implant the spirit of the other books into the secondary schools. This will make the work, from the schools up through the university, continuous and harmonious, and free from the abrupt transition which the student so often experiences in changing from his preparatory to his college mathematics. DIFFERENTIAL AND INTEGRAL CALCULUS BY VIRGIL SNYDER, Ph.D. (gottingen) AN© JOHN IRWIN HUTCHINSON, Ph.D. (chicago) OF CORNELL UNIVERSITY 3jO' (x) changes its sign in passing through zero or infinity 93 59. Second method of determining whether <^' {x) changes sign in passing through zero 95 60. Conditions for maxima and minima derived from Taylor's theorem 97 61. The maxima and minima of any continuous function occur alternately 98 62. Simplifications that do not alter critical values ... 99 63. Greometric problems in maxima and minima .... 100 CHAPTER Vn Rates and Differentials 64. Rates. Time as independent variable 105 65. Abbreviated notation for rates 108 66. Differentials often substituted for rates 110 CHAPTER Vni Differentiation of Functions of Two Variables 67. Definition of continuity 113 68. Partial differentiation 114 69. Total differential 116 70. Language of differentials 119 71. Differentiation of implicit functions ...... 120 72. Successive partial differentiation ...... 121 73. Order of differentiation indifferent . . . • . .121 XU CONTENTS CHAPTER IX Change of the Variable ARTIOLK PAOB 74. Interchange of dependent and independent variables . . 124 75. Change of the dependent variable 125 76. Change of the independent variable 126 APPLICATIONS TO GEOMETRY CHAPTER X Tangents and Normals 77. Geometric meaning oi-^ - % ^29 78. Equation of tangent and normal at a given point . . . 129 79. Length of tangent, normal, subtangent, subnormal . . . 130 Polar Coordinates 80. Meaning of p^ 133 81. Relation between -~ and p-r 134 ax ^ dp 82. Length of tangent, normal, polar subtangent, and polar sub- normal 135 CHAPTER XI Derivative of an Arc, Area, Volume, and Surface OF Revolution 83. Derivative of an arc 138 84. Trigonometric meaning of — , -r- 139 85. Derivative of the volume of a solid of revolution . . . 140 86. Derivative of a surface of revolution 140 87. Derivative of arc in polar coordinates 141 88. Derivative of area in polar coordinates 142 CHAPTER XH Asymptotes 89. Hyperbolic and parabolic branches 143 90. DefinitiGD of a rectilinear asymptote 143 C0:N TENTS XIU Determination of Asymptotes ARTICLE PAGE 91. Method of limiting intercepts 143 92. Method of inspection. Infinite oi'dinates, asymptotes parallel to axes 144 93. Method of substitution. Oblique asymptotes .... 147 94. Number of asymptotes 149 95. Method of expansion. Explicit functions .... 150 CHAPTER Xin Direction of Bending. Points of Inflexion 96. Concavity upward and downward . . . . . . 152 97. Algebraic test for positive and negative bending . . . 153 98. Analytical derivation of the test for the direction of bending . 156 99. Concavity and convexity towards the axis .... 157 CHAPTER XIV Contact and Curvature 100. Order of contact .159 101. Number of conditions implied by contact .... 160 102. Contact of odd and of even order 161 103. Circle of curvature 163 104. Length of radius of curvature; coordinates of center of curvature 163 105. Direction of radius of curvature 164 106. Total curvature of a given arc ; average curvature . . . 166 107. Measure of curvature at a given point 166 108. Curvature of osculating circle 167 109. Direct derivation of the expression for k and E in polar coordinates 169 EVOLUTES AND INVOLUTES 110. Definition of an evolute . 170 111. Properties of the evolute 172 CHAPTER XV Singular Points 112. Definition of a singular point 179 113. Determination of singular points of algebraic curves o . 179 XIV CONTENTS ARTICLE PAGB 114. Multiple points 181 115. Cusps 182 116. Conjugate points 184 CHAPTER XVI Envelopes 117. Family of curves 118. Envelope of a family of curves .... 119. The envelope touches every curve of the family 120. Envelope of normals of a given curve 121. Two parameters, one equation of condition 187 188 189 190 191 INTEGRAL CALCULUS CHAPTER I General Principles of Integration 122. The fundamental problem 195 123. Integration by inspection 196 124. The fundamental formulas of integration .... 198 125. Certain general principles 199 126. Integration by parts 203 127. Integration by substitution 205 128. Additional standard forms 209 129. Integrals of the form f M^ + -g)^£ 210 *" y/ax^ -{-bx+ c 130. Integrals of the form f- '^ ^ ... 212 JiAx + B) Vax2 -i-bx-^c CHAPTER n 131. Reduction Formulas 215 CHAPTER m Integration op Rational Fractions 132. Decomposition of rational fractions 223 183. Case L Factors of the first degree, none repeated . . 225 CONTENTS XV ARTICLE - PAGE 134. Case II. Factors of the first degree, some repeated . . 226 135. Case III. Occurrence of quadratic factors, none repeated . 228 136. Case IV. Occurrence of quadratic factors, some repeated . 229 137. General theorem on the integration of rational fractions . 230 CHAPTER IV Integration by Rationalization 138. Integration of functions containing the irrationality Vax -\- b 231 139. Integration of expressions containing Vax^ -\-hx + c . . 232 140. General theorem on the integration of irrational functions . 236 CHAPTER V Integration of Trigonometric and Other Tran- scendental Functions 141. Integration by substitution 238 142. Integration of i sec^" x dx, j cosec^** xdx . . . . 238 143. Integration of J sec* x tan^" +'^xdx, \ coseC" x cot^™ ^'^xdx . 239 144. Integration of \ tan« x dx, \ cot" xdx 240 145. Integration of j sin^a: cos"a:c?a: 242 244 246 146. Integration of ( ; , i t—. — . . . , J a-\-o cos X J a -{• sm x 147. Integration of J e«=* sin nx dx, \ e"^ cos nxdx . ' . CHAPTER VI Integration as a Summation 148. The definite integral 248 149. Geometrical interpretation of the definite integral as an area 253 150. Generalization of the area formula. Positive and negative area 255 151. Certain properties of definite integrals 256 152. Definition of the definite integral when f(x) becomes infinite. Infinite limits 257 XVi CONTENTS CHAPTER VII Geometrical Applications ARTICLE PAGB 153. Areas. Rectangular coordinates 260 154. Areas. Second method 260 155. Precautions to be observed in evaluating definite integrals . 263 156. Areas. Polar coordinates 267 157. Length of curves. Rectangular coordinates .... 269 158. Length of curves. Polar coordinates 271 159. Measurement of arcs by the aid of parametric representation . 273 160. Area of surface of revolution . 274 161. Volume of solid of revolution * . .277 162. Miscellaneous applications . . 279 CHAPTER VIII Successive Integration 163. Successive integration of functions of a single variable . . 286 164. Integration of functions of several variables .... 288 165. Integration of a total differential 289 166. Multiple integrals .292 167. Definite multiple integrals 293 168. Plane areas by double integration 294 169. Volumes 295 DIFFERENTIAL CALCULUS 3j»iC CHAPTER I FUNDAMENTAL PRINCIPLES 1. Elementary definitions. A constant number is one that retains the same value throughout an investigati6n in which it occurs. A variable number is one that changes from one value to another during an investigation. When the varia- tion of a number can be assigned at will, the variable is called independent; when the value of one number is determined by that of another, the former is called a dependent variable. The dependent variable is called a function of the indepen- dent variable. E.g., 3 x% 4 Va; — 1, cos x, are all functions of x. Functions of one variable x will be denoted by the sym- bols /(a;), (x)y •••; similarly, if 2 be a function of two variables a?, ^, it will be denoted by such expressions as z =f(P^^ y')^ 2J = F(x, y) '". When a variable approaches a constant in such a way that the difference between the variable and the constant may become and remain smaller than any fixed number, pre- viously assigned, the constant is called the limit of the variable. There is nothing in this definition which requires a vari- able to attain the value of its limit, or not to attain it. The 1 2 DIFFERENTIAL CALCULUS [Ch. I. examples of limits met with in elementary geometry are usually of the second kind ; i.e. the variable does not reach the limit. The limiting values of algebraic expressions are more frequently of the first kind. E.q., the function has the limit 1 when x becomes zero : it has the limit when x becomes infinite. The function sin x has the limit when x becomes zero ; tan x has the limit 1 when x becomes y. 4 EXERCISES 1. Let xp (x, y) = Ax + By -\- C ; show that ij; (x, y) =0,\f/ (y, —x)=0 are the equations of two perpendicular lines. 2. If f(x) = 2 xVl- x\ show that ffain-^ = sin x =ffcoa-\ 3. If 4> (x) = ^^, show that (^)-(y) = E^LIL. 4. K f{x) = log f^, show that f{x) +f{y) =fl^±JL\. 1 -\- X \1 + xy/ 5. Given /(x) = Vl^^, find /( VF^^). 6. If f(xy) =/(x) +f(y), prove that /(I) = 0. 7. Given f(x + y)=f(x)+f(y), show that /(O) = 0, and that pf(^x) =f(px), p being any positive integer. 8. Using the same notation as in the last example, prove that /(mx) = mf(x), m being any rational fraction. 2. 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.g., the number (J)" is an infinitesimal when n is taken larger and larger ; and its reciprocal 2'» is an infinite variable. 1-2.] FUNDAMENTAL PRINCIPLES 3 From the definitions of the words " limit " and " infinitesi- mal" 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, then the constant is the limit of the variable. For convenience, the symbol = will be used to indicate that a variable approaches a constant as a limit ; thus the symbolic form x = a is to he read " the variable x approaches the constant a as a limit." The special form a; = oo is read "a? becomes infinite." The corollaries just mentioned may accordingly be sym- bolically stated thus : 1. li X = a, then x = a + a, wherein a = ; 2. li X = a -\- a, and a = 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 combined or transformed by the laws governing the equality of numbers ; and then Cor. 2 will serve to express the 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 differ- ence 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 val- ues it may pass through. It is frequently defined as an "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. 4 DIFFERENTIAL CALCULUS [Ch. I. 3. Fundamental theorems concerning infinitesimals and limits in general. The following theorems are useful in the processes of the calculus ; the first three relate to in- finitesimals, the last four to limits in general. Theorem 1. The product of an infinitesimal a by any finite constant k is an infinitesimal ; i.e.^ if a = 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 a = 0, yS= 0, •••, then a-}-/3-f---- = 0. For the sum of the n variables does not at any stage numerically exceed n times the largest of them, but this product is an infinitesimal by theorem 1 ; hence the sum of the n variables is either an infinitesimal or zero. Note. The sum of an infinite number of infinitesimals may be infinitesimal, finite, or infinite, according to circumstances. E.ff.f if rt be a finite constant, and if n be a variable that becomes infinite; then — , -, — , are all infinitesimal variables; but n2 n „i — + ^ + .•• to n terms = -, which is infinitesimal, n* n^ n while - + - + ••• to n terms = a, which is finite, n n and — + — + ••• to n terms = an^, which is infinite. 3-4.] FUNDAMENTAL PRINCIPLES 5 Theorem 3. The product of two or more infinitesimals is an infinitesimal. Theorem 4. If two variables x, y be always equal, and if one of them, x^ approach a limit a, then the other ap- proaches the same limit. Theorem 5. If the sum of a finite number of variables be variable, then the limit of their sum is equal to tlie sum of their limits ; I.e., lim (a; + ?/+•••)= lim a: + liin^ + •••. For, let x = a^ y = ^-> **•• Then a; = « + «, y = h-\-^, •-, [Art. 2, Cor. 1. wherein « = 0, yS = 0, •••; hence x + y^ .- =(a + J + •••) + (« + ^+ -•); but «4.^4.... = 0, [Th. 2. hence, by Art. 2, Cor. 2, lim(x-|- ?/+ •••)= a + h -\ =lim a: + lim y + •••. Theorem 6. If the product of a finite number of varia- bles be variable, then the limit of their product is equal to the product of their limits. Theorem 7. If the quotient of two variables rr, y be variable, then the limit of their quotient is equal to the quotient of their limits, provided these limits are not both infinite, or not both zero. 4. Comparison of variables. Some of the principles just established will now be used in comparing variables with each other. The relative importance of two variables that are approaching limits is measured by the limit of their ratio. 6 DIFFERENTIAL CALCULUS [Ch. I. Definition. One variable a is said to be infinitesimal, infinite, or finite, in comparison with another variable x when the limit of their ratio a : a; is zero, infinite, or finite. In the first two cases, the phrase " infinitesimal or infinite in comparison with " is sometimes replaced by the less pre- cise phrase " infinitely smaller or infinitely larger than.'* In the third case, the variables will be said to be of the same order of magnitude. The following theorem and corollary are useful in com- paring two variables : Theorem 8. The limit of the quotient of any two varia- bles a;, y is not altered by adding to them any two numbers a, y8, which are respectively infinitesimal in comparison with these variables; i.e., provided For, since it follows, by theorems 4, 6, that 1+- ,. x-\-a ,. X ,. X lim 7^ = lim — • lim r; ; y^^ y 1+^ y but, by theorems 7, 5, and hypothesis, lim 5=1; y therefore, lim ^ = lim -• y+/3 y ,. x-\- a y-\-^ lim-, y X y = 0. X ■\- a X 1 + ^ X y-\-^~y 1 ,^' 4-5.] FUNDAMENTAL PRINCIPLES 7 Cor. If the difference between two variables rr, y be infinitesimal as to either, the limit of their ratio is 1, and conversely ; i.e., if ^"^-0, then ^ = 1. y y For, since x-y^x y y hence - - 1 = 0, and - = 1. [Art. 2, Cor. 2. y y •■ Conversely, if ""^1, then^~^=0. y y For, by Art. 2, Cor. 1, ? 1^0;z...,^-^-0. y y 5. Comparison of infinitesimals, and of infinites. Orders of magnitude. It has already been stated that any two 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, Tc^ is of the same order as y8 when h is any finite number; thus a finite multiplier or 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, a, may be chosen as a standard of comparison as to magni- tude ; then a is called the principal infinitesimal of the first order, and a~^ is called the principal infinite of the first order. S DIFFEBENTIAL CALCULUS [Ch. I. To test for the order n of any given infinitesimal yS with reference to the principal infinitesimal a on which it depends, it is necessary to select an exponent n such that lim ^ _ r. a = ^ "" '*^' wherein ^ is a finite constant, not zero. When n is negative, ^ is infinite of order — n. An infinitesimal, or infinite of order zero, is a finite number. E.g., to find the order of the variable 3 x* — 4 x^, with reference to x as the principal infinitesimal. Comparing with x^, x\ x^, in succession : lim 3x^-4x3 ^ lim (3 ^2 _ 4 ^) = q, not finite ; x = X lim 3 x^ — 4 xS _ lim a; = r4 a; hence 3 x* — 4 x^ is an infinitesimal of the same order of smallness as x*; that is, of the third order. The order of largeness of an infinite variable can be tested in a similar way. For instance, if x be taken as the principal infinite, let it be required to find the order of the variable 3 2^ _ 4 2^. Comparing with a^ and a^ : lim Sa^-4a^^ lim (32;_4)=qo; Hm 3 2:* -4 2:3 lij^ X = 00 ^ X r«(3-g=3; hence 3 a:* — 4 a;^ jg ^n infinite of the same order of largeness !is 3^, that is, of tlie fourth order. The process of finding the limit of the ratio of two in- finitesimals is facilitated by the following principle, based 5-6] FUNDAMENTAL PRINCIPLED 9 on theorem 8 of Art. 4 : 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, respectively. lim 3 a;2 + a:* _ lim 3 a:2 _ 3 a; = E.g., 4:X^—2X^ x = 04^-2 4 From these definitions the following theorems are at once established : Theorem 1. The product of two infinitesimals is another infinitesimal whose order is the sum of the orders of the factors. Theorem 2. The quotient of an infinitesimal of order m by an infinitesimal of order n is an infinitesimal of order m—n. Theorem 3. The order of an infinitesimal is not altered by adding or subtracting another infinitesimal of higher order. 6. Useful illustrations of infinitesimals of different orders. lim sin Theorem 1. -1 . lim tan 6 6 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 BO perpendicular to OA, cutting OA in O. Let the angle AOB = 6 in radian measure, then arc AB = rO, CB<^rcAB(x)^ the function is said to be discontinuous at a;= x^ E.g., the function may become infinite, as ^^ , when x = 2; the function may be imaginary, as Vd — x^, when x^ > 9 ; the function may be indeterminate, as sin -, when a: = ; X finally, the value of the function may depend upon the manner in which the variable approaches the value x^, as in the function A^)= 1 2-3*. V 1-3^ when X = + A, /(x)= 1 ; when x = - A, /(- A) = 2 as A = 0. A continuous function actually attains its limit for any value of the variable within the region of continuity, and the variable may be substituted directly. 7-8.] FUNDAMENTAL PRINCIPLES 15 It may be shown as on p. 14 that any polynomial aaf^ + ^2:""^ H [w a positive integer. is continuous for every finite value of x. The ordinary functions involving radicals and ratios are continuous only for certain intervals. The trigonometric functions sin x and cos x are continuous for all real finite values of x ; the other trigonometric func- tions are rationally expressible in terms of sine and cosine. Show that tan x is discontinuous when x= ^ir. The exponential function a^ and the logarithmic function logo; are each continuous, the former for all finite values of a;, the latter for all finite positive values of x [D. C, p. 31]. 8. Comparison of simultaneous infinitesimal increments of two related variables. The last few articles were concerned with the principles to be used in comparing any two infini- tesimals. In the illustrations given, the law by which each variable approached zero was assigned, or else the two vari- ables 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 rr^, 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^ -i-h, y^-\- Jc^ in which the increments h, 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 16 DIFFERENTIAL CALCULUS [Ch. I. function of x^ its increment can be made as small as desired by taking the increment of x sufficiently small. The determination of tlie limit of the ratio of k to A, as h 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 a:^, y-^ be simultaneous values of the variables rr, y ; and when X changes to the value x^ + A, let y change to the value y^ + h. Then y^j^k = Qx^+ A)2 = x^ + 2x^h^ A2. hence k=2 x-Ji + h^. 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 be 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, -=2x^-\-h\ hence, A^O^=^^i- This result may be expressed in familiar language by saying that when x increases through the value x^^ then y increases 2 x^ times as much as x ; and thus when x continues 8.] FUNDAMENTAL PRINCIPLES 17 to increase uniformly, 1/ increases more and more rapidly. For instance, when x passes through the value 4, and y 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 ^ through 25, the limit of the ratio of their increments is 10, and ^ is changing 10 times as fast as x. The following table will numerically illustrate the fact that the ratio of the infinitesimal increments A, Jc approaches nearer and nearer to some definite limit when h and Jc both approach the limit zero. Let x^, the initial value of x.he 4. Then 1/^, the initial value of t/, is 16. Let A, the increment of a;, be 1. Then k, the corresponding increment of ^, is found from^ 16 + A: = (4 + A)2; thus Ar=9, and y = ^' Next let h be successively diminished h to the values .8, .6, .4, •••. Then the corresponding values of Jc Jc and of - are as shown in the table : Ji x = 4 + A y = 16-\-k k 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 + A 16-f 8A + A2 8^ + A2 8 + A 18 DIFFERENTIAL CALCULUS [Ch. I. 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 Then y,^ = x,^ hence, by expansion and subtraction, 2 y^^ + ^2 _ 3 x2Ji + 3 ^.^^2 + ^8^ *(2y, + k)=h (dx^^+SxJi + A2), k^ dx,^-{- dx,h + h^ h 2yi+k Therefore lim f = lim ^^i'+ ^^'^ / ^' , as A = 0, * = 0, and, by Art 4, theorem 8, lim^ = 5^^ h 2y, The " initial values " of x, y^ have been written with subscripts 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 incre- ments depends on the particular values through which the variables are passing, when they are supposed to take these increments. With this understanding the subscripts will hereafter be omitted. Moreover, the increments A, k will, for greater distinctness, be denoted by the symbols Aa:, Ay, read "increment of x," "increment of y." Ex. 1. If x2 + 2/2= a\ find Hm^. I^t the initial values of the variables be denoted by x, y, and let the variables take the res}>ective increments Ax, Ay, so that their new values x + Ax, y •¥ ^y shall still satisfy the given relation. Then (x + Ax)2 + (y + Ay)2 = a«. 8.] FUNDAMENTAL PRINCIPLES By expansion, and subtraction, 2 a: . Aa: + (Aa:)^ + 2 y • Ay + (Ay)2 = hence Ax (2 or + Aa:) = - Ay (2 y + Ay), and Therefore 19 0, Ay_ 2a; + Aa: Aa: 2y + Ay lim Ay _ lim 2 a: + Aa: Aa; = OAx~ Ax = 02^+ Ay~ a: 2^ The negative sign indicates that when Aa: 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 a;^ -f y2 == a^ (Fig. 5). Ex.2. If a;2 + y = y2_2a:, prove lim Ay^2a: + 2, ^ Aa:=OAar 2y-l Fig. 5. Similarly, when the relation between x and y is given in the explicit functional form then and hence y -{- Ay = cl>(x -{- Ax), Ay = (i>{x + Ax) — (x) = A<^(a;), lim ^ = lim '^(^ + A^)- is given, the limit of this ratio can be evaluated, and expressed as a function of x. This function is then called the derivative of the function (x) 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. 20 DIFFERENTIAL CALCULUS [Ch. I. 9. 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 increment 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. If (i>{x) be a finite and continuous function of a:, and Aa; a small increment given to x^ then the derivative of (x) as to X is lim Ax = (i>(x + Aa:) — (x') 1 ^ lim Aj>(a;) Aa; /-Aa: = ^^ It is important to distinguish between lim ^^ ^ and — - — ^^ ^ ; that is, between the limit of the ratio of two lim Aa; infinitesimals and the ratio of their limits. The latter is indeterminate of the form - and may have any value ; but the former has usually a determinate value, as illustrated in the examples of the last article. EXERCISES 1. Find the derivative of a;^ — 2 a: as to x. 2. Find the derivative of 3 x* — 4 a: + 3 as to x. 3. Find the derivative of — as to x. 4x 3 4. Find the derivative of x* — 2 H — as to x. /[^ 10. Geometrical illustrations of a derivative. Some con- ception of the meaning and use of a derivative will be afforded by one or two geometrical illustrations. Let 1/ = <^(a;) be a function of x that remains finite and continuous for all values of x between certain assigned con- 9-10.] FUNDAMENTAL PRINCIPLES 21 stants a and b ; and let the variables x, y be taken as the rectangular coordinates of a moving point. Then the rela- tion between x and y is represented graphically, within the assigned bounds of continuity, by the curve whose equation is y=^^{x). Let (a;^, y-^^ (x^^ y^ be the coordinates of two points Py, Then it is evident that the ratio Pg? oil this curve -y\ Xc — x^ is equal to tan a, wherein a is the inclination angle of the secant line P^^'^ to the a;-axis. Let P^ be moved nearer and nearer to coincidence with P^ so that x^ = x^ y<^ = y-^. Then the secant line PiP^ approaches nearer and nearer to coinci- dence with the tangent line drawn at the point the inclination-angle a of the secant approaches as a limit the inclination angle of the tangent line. Hence, tan a = tan . Py and Pi carg.y*) Thus X, when X, — -^ = tan <^, 2 — ^v Vi — yy —X Fig. 6. It may be observed that if x^ be put directly equal to x^ and y2 to y^ the ratio on the left would, in general, assume the indeterminate form -, as in other cases of finding the limit of the ratio of two infinitesimals ; but it has just been shown that the ratio of the infinitesimals y^ — y^, x^ — x^ has, nevertheless, a determinate limit, viz., tan <^. 22 DIFFERENTIAL CALCULUS LCh. 1. They are thus infinitesimals of the same order except when <^ is or — • If the differences x^ — ajj, y^ — y^ be denoted by Aa;, Ay, then x^ = x^-\- Aa:, 3^2 = ^i + ^^ 5 but, since y = <^(a:), it follows that ^j = ^^(^i)^ ^2 ~ *^(^2)» hence the ratio of the simultaneous increments may be written in the various forms Aa; x^ — x-^ x^ — x^ Ax In the last form x is regarded as the independent variable and Aa; as its independent increment ; the numerator is the increment of the function <^(a;), caused by the change of x from the value a;^ to the value x^ + Aa;. The limit of this ratio, as Aa; = 0, is the value of the derivative of the function (a;) when x has the value x^. Here x^ stands for any assigned value of x. Thus the derivative of any continuous function (x) is another function of x which measures the slope of the tangent to the curve y = S=EQ=Ai/. It the area OAPM be denoted by «, then z is evidently some function of the abscissas;; also if area OAQN be denoted by z + Az^ then the area MNQP is Az ; it is the incre- ment taken by the function z^ when X takes the increment Ax, But MNQP lies between the rectangles MR^ MQ ; hence Y S / R " A X M N Fig. 7. and 1/Ax (2; + Aa;) - (x)^ the result of the operation is another function of x. The latter function may- have properties similar to those of (^(a;), or it may be of an entirely different class. The operation above indicated is for brevity denoted by the symbol — ^-^? and the resulting derivative function by (^'(x)', thus, #(^)_ lim A0(a:)_ lim <^(a:4- Aa;)- <^(a;) C?a; ~ Aa; = Aa: Aa: = ^^ <\>\x^. The process of performing this indicated operation is called the differentiation of (x) with regard to x. The symbol * — , when spoken of separately, is called the differ- dx entiating operator, and expresses that any function written after it is to be differentiated with regard to rr, just as the symbol cos prefixed to Qx^ indicates that the latter is to have a certain operation performed upon it, namely, that of finding its cosine. The process of differentiating <^(a;) consists of the follow- ing steps : 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 increment of the variable. 4. Obtain the limit of this quotient as the increment of the variable approaches zero. * This symbol is sometimes replaced by the single letter D. 11-12.] FUNDAMENTAL PRINCIPLES 25 EXERCISES Find the derivatives of the following functions : 1. 5 3/8 - 2 2/ + 6 as to 3^. 3. 8 w^ - 4 w + 10 as to 2 m. 2. 7 /2 _ 4 ^ - 11 ^8 as to «. sX 4. 2 a;2 _ 5 a: + 6 as to x - 3. This process will be applied in the next chapter to all the classes of functions whose continuity within certain inter- vals has been pointed out in Art. 7. It will be found that for each of them a derivative function exists ; that is, that lim —^ — - has a determinate and unique value, and that the curve 1/ = (t>{x) has a definite tangent within the range of continuity of the function. A few curious functions have been devised, which are continuous and yet possess no definite derivative ; but they do not present themselves in any of the ordinary applications of the Calculus. Again, there are a few functions for which lim \} ^ has a certain value when Ax = from Aa; the positive side, and a different value when Aa: = from the negative side ; the derivative is then said to be non-unique. Functions that possess a unique derivative within an as- signed interval are said to be differentiahle in that interval. Ex. Show that the four steps of p. 24 do not apply at a discontinuity. 12. Increasing and decreasing functions. A good example of the use of the derivative is its application to finding the intervals of increasing or decreasing for a given function. A function is called an increasing function if it increases as the variable increases and decreases as the variable de- creases. A function is called a decreasing function if it decreases as the variable increases, and increases as the variable decreases. E.g., the function a;^ + 4 decreases as x increases from — oo to 0, but it increases as x increases from to + oo. Thus a:'^ + 4 is a decreasing 26 DIFFERENTIAL CALCULUS [Ch. I. function while x is negative, and an increasing function while x is posi- tive. This is well shown by the locus of the equation y=x^-\-^ (Fig* 8). Fig. 8. Again, the form of the curve y = - shows that - is a decreasing func- tion, as X passes from — co to 0, and also a decreasing function, as x passes from to + oo. When x passes through 0, the function changes discontinuously from the value — co to the value + oo (Fig. 9). Most functions are increasing functions for some values of the variable, and decreas- ing functions for others. Fia. 10. ^'9"> v/2 rx- x^ is an increasing function from a; = to a; = r, and a decreasing function from x = rio x=2r (Fig. 10). A function is, said to be an increasing or decreasing func- tion in the vicinity of a given value of x according as it increases or decreases as x increases through a small interval including this value. 12-13.] FUNDAMENTAL PRINCIPLES 27 13. Algebraic test of the intervals of increasing and de- creasing. Let 1/ = (a;) be a function of x^ and let it be real, continuous, and differentiable for all values of x from a to h. Then by definition i/ is increasing or decreasing at a point X = x^, according as 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 0(:?^) is an increasing or a decreasing function at the value x^, accord- ing as Ax ^= . lim f <^(^i + Ax) - cl>(x{) I ^ ^,^^^^ dx is positive or negative. Thus the intervals in which (i>(x) is an increasing function are the same as the intervals in which '(x) is positive. Ex. Find the intervals in which the function , f^{x) = 2 a:^ - 9 a:2 + 12 a; - 6 is increasing or decreasing. The derivative is <^'(a:)= 6a:2 - 18a; + 12 = 6(a: - l)(a: - 2); hence, as x passes from -co to 1, the derived function ^'(x), is positive and ^{x) increases from ^( — go) to f^(l), i.e., from = — ao to <^=— 1; as x passes from 1 to 2, '(x) is negative, and (x) decreases from ^(1) to <^(2), i.e., from — 1 to - 2 ; and as x passes from 2 to +qo, (x) increases from (x) is shown in Fig. 11. At points where '(x)=0, the function <^(x) is neither increasing nor decreasing. At such points the tangent is parallel to the axis of x. Thus in this illustration, at a: = 1, x = 2, the tangent is parallel to the x-axis. Fig. 11. 28 DIFFERENTIAL CALCULUS [Ch. I. EXERCISES 1. Find the intervals of increasing and decreasing for the function ^{x) = a;3 + 2 a;2 + a: - 4. Here <^'(a;) = 3 a:2 + 4 x + 1 = (3 a: + 1) (a: + 1). The function increases from x = —co to x = — 1; decreases from x = — 1 to a; = — I ; increases from a; = — ^toa; = oo. 2. Find the intervals of increasing and decreasing for the function y = a;8 - 2 a;2 + a: - 4, and show where the curve is parallel to the a;-axis. 3. At how many points can the slope of the tangent to the curve 3/ = 2a;8-3a:2+l be 1 ? - 1 ? Find the points. 4. Compute the angle at which the following curves intersect : 3^ = 3 a:2 - 1, y = 2 x^ + 3. 14. Differentiation of a function of a function. Suppose that y, instead of being given directly as a function of a;, is expressed as a function of another variable w, which is itself expressed as a function of x. Let it be required to find the derivative of y with regard to the independent variable x. Let y=f(u)'> in which tt is a function of x. When x changes to the value x + Aa;, let u and y^ under the given relations, change to the values w-f-Aw, y + Ay. Then A^ _ A^ Aw _ f(u-\-Au) — f(u') ^ Aw . Ax A?/ Ax An Ax hence, equating limits, ^y _ ^U ^^^ _ df(u) ^ du dx du dx du dx 13-14.] FUNDAMENTAL PRINCIPLES 29 This result may be stated as follows; 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 u, and the derivative of u with regard to x. EXERCISES 1. Given y-'^u^-l, w = 3a;2+4:; find ^ dx dy ^ du ^ du dx 2. Given y = 8 m2 _ 4 m + 5, m = 2 a:^ - 5 ; find $^ dx 3. Given ^ = i, m = 5 x2 - 2 a: + 4; find $• u dx 4. Given, = 3„^ + ^,„=^+|; find g. CHAPTER II DIFFERENTIATION OF THE ELEMENTARY FORMS In recent articles, the meaning of the symbol -^ was ex- plained and illustrated; and a method of expressing its value, as a function x^ was exemplified, in cases in wliich y was a simple algebraic function of a;, 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 differ- entiable functions. 15. Differentiation of the product of a constant and a variable. Let y=^cx. Then ^ + A?/ = (?(a; + Aa;), Ay = cQc -f Aa;) — ca: = cAr, therefore t- SO Ch. II. 15-16.] DIFFERENTIATION OF ELEMENTARY FORMS 31 Cor. If y = cu^ where w is a function of x^ then, by Art. 14, ^, ^ dx doc The derivative of the product of a constant and a variable is equal to the constant multiplied hy the derivative of the variable. 16. Differentiation of a sum. Let y=f(x) + (x) + 'f{x). Then y + ^y =f(x + i^x) + (x + A:r) + -^(x + Lx), Ay ^ fCx+Ax}-f(x} ^ (f>(x-hAx}-(l)(x} Ax Ax Ax ^ '\lr(x + Ax}-ylr(x') Ax Therefore, by equating the limits of both members, g =/'(^) + 4,'Cx^ + ^'(x-). [Art. 3, Th. 5. Cor. 1. \i y = u -^ V + w^ in which u^ v, w are functions of X, then ^ ^ \, The derivative of the sum of a finite number of functions is equal to the sum of their derivatives. Cor. 2. \i y = u ■\- c^ c being a constant, then y -\- Ay = w + Au + c ; hence, Ay = Aw, and ■ dy^du dx dx The last equation asserts that all functions which differ from each other only by an additive constant have the same derivative. 32 DIFFERENTIAL CALCULUS [Ch. II. Geometrically, the addition of a constant has the effect of moving the curve y = u(x) parallel to the y axis ; this opera- tion will obviously not change the slope at points that have the same x, ^ ,-. dy du , dc trom(2), 3^ = T- + :r» dx dx dx but from the fourth equation above, dy _ du^ dx dx* dc hence, it follows that — - = 0. dx The derivative of a constant is zero. If the number of functions be infinite, theorem 5 of Art. 3 may not apply ; that is, the limit of the sum may not be equal to the sum of the limits, and hence the derivative of the sum may not be equal to the sum of the derivatives. Thus the derivative of an infinite series cannot always be found by differentiating it term by term. 17. Differentiation of a product. Let y=/(a^)(^)- Then ^ = /(y + Aa;)(a;) Ax Ax By subtracting and adding f(x)^Qx + Ax) in the numer- ator, this result may be rearranged thus : Now let Ax approach zero, using Art. 3, theorems 5, 6, and noting that the first factor (\){x -h Ax) approaches (x) is continuous (Art. 7). Then 16-18.] DIFFERENTIATION OF ELEMENTARY FORMS 33 Cor. 1. By writing u = (^{x)^ v=f(x)^ this result can be more concisely written, d{uv) _ dv du ^gv doc ~ doc doc ^ 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 hy the derivative of the first. This rule for differentiating a product of two functions may be stated thus : Differentiate the product, treating the first factor as constant, then treating the second factor as constant, and add the two results. Cor. 2. To find the derivative of the product of three functions uvw. Let y = uvw. / dv = w[ u—- \ dx du\ , dw dxj dx The result may be written in the form dCuvw) dw , du , dv ... By application of the same process to the product of 4, 5, '", n functions, the following rule is at once deduced: 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. 18. Differentiation of a quotient. Let y *(^) Then y + Ay=lp^, 84 DIFFERENTIAL CALCULUS [Ch. 11. fjx^^x-) fix-) Ay _ (x-{-Ax) (x) Ax" Ax ^ j^jx^fCx + Ax) ^f(x)( t>(x H- Ax-) Ax(t)(x)(t)(x-\-Ax) By subtracting and adding (\)(x)f{x) in the numerator, this expression may be written ... [ f(x-\-Ax)-f(ix) \ l^(x+A x')-{x) \ Ax (x)f{x) -f(ix)'(x) p. q Tha 6 7 di [f(x)Y "-Art. d, 1 hs. b, 7. This result may be written in the briefer form du dv d / '* \ _ dx dx i-gN dQc\v)~ v^ ' The derivative of a fraction^ the quotient of two functions, is equal to the denominator multiplied hy the derivative of the numerator minus the numerator multiplied hy the derivative of the denominator, divided hy the square of the denominator, 19. Differentiation of a commensurable power of a function. Let y = w^ in which w is a function of x. Then there are three cases to consider. 1. w a positive integer. 2. n a negative integer. 8. n a commensurable fraction. 1. w a positive integer. This is a particular case of (4), the factors w, v, w, ••. all being equal. Thus dx dx 18-19.] DIFFERENTIATION OF ELEMENTARY FORMS 35 2. wa negative integer. Let w = = -m, in which m\8 a. positive integer. Then y=„.=,*-»=i, and dx hence dx dx 3. w a commensurable fraction. P Let w=— , where jt?, q are both integers, which may be either positive or negative. p Then y = u'' = 'uP\ hence y^ = u^^ (ia; ^ dx Solving for the required derivative, dx q dx^ hence ^ = nw" -^ ~ (6) doc dx The derivative of any commensurable power of a function is equal to the exponent of the power multiplied hy the power with itB exponent diminished by unity, multiplied by the derivative of the function. •'■ * If two functions be identical, their derivatives are identical. 36 DIFFERENTIAL CALCULUS [Ch. II. These theorems will be found sufficient for the differentia- tion of any function that involves only the operations of addition, subtraction, multiplication, division, and involu- tion in which the exponent is an integer or commensurable fraction. The following examples will serve to illustrate the theo- rems, and will show the combined application of the general forms (1) to (6). 1. y = ILLUSTRATIVE EXAMPLES 3^'- 2. ^^^ dy X + 1 ' dx dy (x+l)£(B.-2)-(3x»-2)£(x + 1) by (6) dx (x+l)-^ |(^^^-2)=£(^^^>-|(2> by (2) = 6x. by (1). (0) i.(x+l) = ^ = l. dx dx by (2) Substitute these results in the expression for -^' Then dy _ (x + 1)6 x - (3 x^^ - 2) ^ 3 a:2 + 6 37 + 2 dx {x + 1)2 {x + 1)2 2. m = (3«2h-2)V1 + 552; find—. ds ^= (3«2 + 2)4 Vl + 5.2 + v/TfSTS. 4(3*2+2). by (3) ds ds ds 4vi + 5«2= 4(1 + 5*2)* €U ds = |(l + 5«2)-il(l + 5«2) by (6) 5» Vl + 5«2 4(8««4-2) = 6«. by (6) ds 19.] DIFFEBENriATlON OF ELEMENT AUY FORMS 37 Substitute these values in the expression for — • Then ds 3. y^VTJ^^+Vljz:^^^ fi^^ rf^. VI + x^ _ Vl - x^ ^a; First, as a quotient, dy dx dx~ (\/I+^^-V'n^)2 dx • Similarly for the other terms. Combining the results, dx x^ \ Vl — x^f Ex. 3 may also be worked by first rationalizing the denominator. EXERCISES Find the a:-derivatives of the following functions: 1. 2/ = xio. ^ y X 2. y = x~^. 3. y = cV^. 9. y 8/— 10. y=(x+ 1) Vx + 2. 4. V^ 3 5. y=^t^. 6. ?/ = (x + a)«. 7. y = a;« + a«. 11. y= ^«+^ Va + Vx --=Vr^- 38 DIFFERENTIAL CALCULUS [Ch. II. X TQ 3x8+2 13. y = 19. y= r x+Vl- x^ x{x^-\-\y 14. y = (2 J + x^) VJ77. ^20. y = ^x^ + 1)*(4 x^ - d). .„ 21. y=du^-7. 15. y=\ [ ' ^ I 1- x^ 16 17. y^ y/l _ a;2 > 22. ?^ = 4 m8 - 6 m2 + 12 M - 3. 23. y=(l-3w2+6M*)(l + u2)3. 24. y = wx. X«_4_l 25. 3^ = m2 ^ 3 a;ti2 ^ a;4. x«- 1 y = 18. y = -^ 1— . '"^"^'^ (a + xl"* (& + x)« 27. y = M%8w. 28. Given (a + x)^ = a^ + 5 a^x + 10 a^x^ + 10 a^x^ + 5 ax* + x^ ; find (a + x)* by differentiation. 29. Show that the slope of the tangent to the curve y = x* is never negative. Show where the slope increases or decreases. V 30. Given b^x^ + aY = a^^^ find -^ : (1) by differentiating as to x ; (2) by differentiating as to y; (3) by solving for y and differentiating as to X. Compare the results of the three methods. 31. Show that form (1), p. 31, is a special case of (3). ^ 32. At what point of the curve y^ = ax* is the slope 0? — 1? +1? r 33. Trace the curve iy = x^ + 3 x^ + x — 1. 34. V = '^ "' -^ "^ and M = 5 x2 - 1 ; find ^. V7 m2 + 5 rfa: y 35. At what angle do the curves y^ =: 12 x and y^ -\- x^ + Q x - QS = intersect ? 20. Elementary transcendental functions. The following functions are called transcendental func- tions : Simple exponential functions, consisting of a constant number raised to a power whose exponent is variable, as 4', a** ; 19-21.] DIFFERENTIATION OF ELEMENTARY FORMS 39 general exponential functions, involving a variable raised to a power whose exponent is variable, as x^^^ ; the logarithmic * functions, as log« x^ log^ u ; the incommensurable powers of a variable, as x^^, u^ ; the trigonometric functions, as sin w, cos u ; the inverse trigonometric functions, as sin~^ w, tan~^ 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. 21. Differentiation of loga oc and loga u. Let g = log„x. Then g + Ag = log^(^x -{- Ax}, ^y _ ^^^a (^ + Ax} - l0g« X Ax Ax 1 , (x^Ax\ -AxM X } For convenience writing h for Ax, and rearranging. Ax * The more general logarithmic function log„ w is not classified separately, as it can be reduced to the quotient -2E^. loga V ' / / 40 DIFFERENTIAL CALCULUS [Ch. II. X To evaluate the expression ( 1 + - 1 when A = 0, expand it by the binomial theorem, supposing - to be a large positive integer m. The expansion may be written l^W ~ »» 1-2 ™2+ 1.2.3 «t3+ ' which can be put in the form V^mJ ^ ^1 2 ^1 2 8 ^ 1 2 Now as m becomes very lar^e, the terms — , — , ••• become -^ ^ mm very small, and when w« = oo the series approaches the limit 1 + 1+ — + — + — +-. 2! 3! 4! The numerical value of this limit can be readily calculated to any desired approximation. This number is an important constant, which is denoted by the letter e, and is equal to 2.7182818...; thus lim m ^^ (l + -X = e = 2.7182818 = «>\ mj 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 • 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. 316 of McMahon and Snyder's '* Differential Calculus." ex- 21-22.] DIFFERENTIATION OF ELEMENTARY FORMS 41 subscript, as log x ; in other bases a subscript, as in log^ x, will generally be used to designate the base. The logarithm of e to any base a is called the modulus of the system whose base is a. X If the value, ;^^o(l+-l = e, be substituted in the pression for -~, the result is di/ 1 , More generally, by Art. 14, £l„„« = l.log„e.f. (7) 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 hy the function. 22. Differentiation of the simple exponential function. Let y =1 a^. Then log y = u log a. Differentiating both members of this identity as to a;, , l^=log«.f^, byforin(8), y ax ax dy 1 du ■dx^'^^'^y'Tx' therefore ^a'* = log a • a** • ~ (9) In the particular case in which a = «, eu = e** • -j— (10) doc dx 42 DIFFERENTIAL CALCULUS - [Ch. II. The derivative of an exponential function with a constant base is equal to the product of the function, the natural logarithm of the base^ and the derivative of the exponent. 23. Differentiation of the general exponential function. Let y = u^^ in which -m, v are both functions of x. Take the logarithm of both sides, and differentiate. Then logi/ = v log u, ydx dx udx dx \__ dx u dx] ' therefore dx^^ ~ ^^ ^^^ ^ d^^ ^^^^dx ^^^^ The derivative of an exponential function in which the base is also a variable is obtained by first differentiating^ regarding the base as constant^ and again, regarding the exponent as constant, and adding the residts. In the differentiation of any given function of this form it is usually better not to substitute in the formula directly but to apply the method just used in deriving (11), i.e., to differentiate the logarithm of the function by the preceding rules. Ex. y = (4 0:2 - 'jy+^^^a^ fi^^ % ax logy = (2 + Vz2-r5) log (4 x^ - 7). ^ = (4x^- 7)«+^'^« .^r log (4x^- 7) 8(2 + VJ^^Ts) -! *i^ I V^a^^ 4x»-7 J" \ \ 22-24.] DIFFERENTIATION OF ELEMENTARY FORMS 43 24. Differentiation of an incommensurable power. Let 1/ = w", in which n is an incommensurable constant. Then log 1/ = n log u^ \d]£_ n ^ du y dx u dx^ du y du dx u ax d n n-\ du dx dx This has the same form as (6), so that the qualifying word '' commensurable " of Art. 19 can now be omitted. EXERCISES Find the x derivatives of the following functions : 1. y = log(a:+a). i 16. y = e^+*. 2. y=zlog(ax + b). 3. y = log(4a;2-7a: + 2). ^^- ^ = i^PgT ^. y= log ii-^. 18. y = e''(l- x^). 1 — X 1 l + 3;2 19. y = - — 6. 2,= xlog^. 20. ./ = log(6^-.-»). 7. 2, = x-log:r. 21. 2, = log (^ + .^). 8. y = 3-^ logx- 22. y = x"a»=. ^ 9. y = log\/]r^^. 23. y = \og^±^' 10. y = y/x - log (Va: + 1). -i 11. y = log«(3x2-V2T^). 24. 3/ = j^- 12. 3/ = log,„ {x^ + 7 x). 25. 2/ = (log xy. 13. ?/ = log^ a. 26. 2/ = log (log x). 14. 2/ = e^^"- 27. y = xf'. 15. y = e^x+s^ 28. .y = a^og==. 44 DIFFERENTIAL CALCULUS [Ch. II. The followiug functions can be easily differentiated by first taking the logarithms of both members of the equations. 29. v = (^-1) (2:-2)^(a:-3)* 30. y = a;Vl-a:(l + x). 31. , = £lldL£!l. 32. y = x\a + 3 xy{a - 2 x)^. 33. y: ovy-'' Articles 25-31 will treat of the differentiation of the Trigonometric Functions. 25. Differentiation of sin u. Let y = sill u. Ay _ sin (u + Au) — sin 2^ A?* Then Ax Au Ax _ 2 cos |( 2 1^ + A?^) sill I- Au ^ Ai* A?^ Ax r . 1 A \ Sill J Au Au . s= COS (^ + i At*) r-^ • ^ ^ lAu Ax But, when Au = 0, cos (?* H- | Au) = cos w, and —^. = 1 by Art. 6 ; hence, passing to the limit. iAu d . du (12) The derivative of the sine of a function is equal to the prod- uct of the cosine of the function and the derivative of the function. 26. Differentiation of cos u. Let ^ = cosw = sinf ^ — wj- '^'>-|=f/Kl-")=-ct of the sine of the function and the derivative of the function. 29-32.] DIFFERENTIATION OF ELEMENTARY FORMS 47 EXERCISES Find the x derivatives of the following functions : 1. y = sin 7 x, 18. y = sin (m + li) cos (m — 6) / 2. y = cos 5x. 19 — s^^"* ^^ , ^ 3. y = sin x^. * cos« wa: 4. 3^ = sin 2 ar cos a:, 20. «/ = x + log cosf a; - | j- 5. y — sin^ x. i:j ^ 7. y=3m2 7i. 22. j, = sin (sin «). a 2/ = itan»x-tanx. 23. i, = sin^ e"". 9. y = sin'acosi. 24, y = sin «- • logx. 10. 2, = tanrt+8ec:t. 25. t, = ^^ii^. ' mx 12. , = tan(3-5.T. 27. , = csc» 4 x. - 13. ?/ = tan^x — logrsec^a;). _o ,. o\?! ^ •^ ^ ^ ^ 28. ?/ = sec (4 x — 3)2. ^14. y = log tan(ia: + iTr). or* « •> ^ ° vz ' 5 y 29. y = vers a;^. 15. y — log sin Vx. «^ .l 9 . /- -*> ^ ° J 30. y = cot x^ + sec vx. ^ ^ 16. y = tan a*. 31. ?/ = sin xy. \ « 4 17. y = sinna: sin«a:. 32. y = tan (x + y). •^ 32. Differentiation of Sin- li^. oM/l^>'^>^ Let ^ = sin~^w. Then sin y = u^ and, by differentiating both members of this identity, hence, dy du dx dx dy _ 1 du 1 du ^ dx cos y dx j. VI — sin^^ dx 6? . _i \ du i.e,j —-sin ^u=± — -— ' dx Vl — u^ ^^ The ambiguity of sign accords with the fact that sin""^ u is a many- valued function of u, since, for any value of u be- 48 DIFFERENTIAL CALCULUS [Ch. II. tween — 1 and 1, there is a series of angles whose sine is u : and, when u receives an increase, some of these angles in crease and some decrease ; hence, for some of them, du is positive, and for some negative. It will be seen that, when sin~^w lies in the first or fourth quarter, it increases with tfc, and, when in the second or third, it decreases as w increases. Hence, for the angles of the first and fourth quarters, -^sin-^^ = 4- ^ . ^sin-ii.^4- — 1— ^. (19) In the other quarters the minus sign is to be used before the radical. 33. Differentiation of the remaining inverse trigonoikfitric forms. The derivatives of the other inverse trigonometric- func- tions can be easily obtained by the method employed in the last article. The results are as follows : ^ co8-iw - -^ f" (in 1st and 2d quarters). (20) doc Vl _u'i^iio ^ ^ ^ /.**-"'" =, + «^^: (in all quarters). (21) ^'''^-x'^'u^Z 0" all quarters). (22) i^ s€c-»M - * "J** (in 1st and 3d quarters). (23) dx uVu^^ ^dx ^ ^ ^ ** CSC »t« - ~* ^^ (in 1st and 3d quarters). (24) ax uVu^ -idx ^ ^ vers-i w%: * ^** rin 1st and 2d quarters^ dx V2u-U'^^^ (26) The radicals in forms (20), (23), (24), (25) receive the opposite signs respectively when the angles are taken in the quarters other than those stated. 32-34.] DIFFERENTIATION OF ELEMENTARY FORMS 49 EXERCISES Find the a:-derivative of each of the following functions: 1. y = sin~^ 2 x^. 16. y = tan x • tan~^ x. I. y = cos- 1 VI - x^. *H\1. y = x sin-ix. ;6. y = Vl7. v = / 3. 2^ = sin- 1(3 a: - 1). 18. v = e tau-'x 4. ?/= sin-i(3x — 4x8). -. y< 19. y = csc-^ / 5. w=sm-i^ —• '^^ ^ / . ., ^20. 2/ = sec-i^-±-!^. 6. 3/ = vsin-ix. "^ x^—1 - 7. j, = tan-'«-. ^21. y = taD-'^ + l". S. y = cos-i log ar. 1 - Vaa: Ji_ V5S. 2.^ = 1 + 1-1. ^^ ^' ^" ' 16. , = eot-l±^^I±^. 3. y=(x-{- 5)V^^^. ^ X 4. ^ = xVcfi-x^. ^ 17. 2/ = tan'*a:-2tan2a;+log(sec4a;). 5. y = x log sin x. ^ ^g. y = ^M^ + log(l - x\ 1 — X 6. y = ^y/a'^- a:2. -^ 7. 2/ = -eV ' ""' "^ 5 + 3cosa: -^8. 3, = tan2z, 2 = tan-i(2x-l).i^20- 2^ = ^^^ (l^) -|tan-i^. V 9. y = .V«^ n = log sinz. . ; ■g^ •'- ^ j^g (^ + V^2i:^2)+ sec 19. v = cos-i§^Ll£2^. 5 + 3 cos X l + x\^ 1 10. y = log-. '22. y = e% t* = logo:. ^ .. ^;^ 1 _ a;2 23. y = log s^ + e«, s = sec a;. 11 Vl + a;2 24. x^ + yS _ 3 ^^-^ _ q. 12. y = e* cos X. 25. a:2y2 + ar^ + ?/8 _ q. 13. ?/ = vers-M-). 26. a:^ + a: = y + 3/8. 4 sin 3. 27. xy^-\-x^y = x + y. 14. v = tan-i-^^i5_£_. 3 + 5 cos a; 28. y = sm(2u- 7), m = log x^. tion? 29. For what values of x is the function —^ — - an increasing f unc- (l/jl n? a + a; ^^_^ / Vl + a:2 — 1\ 30. Prove that tan-^ ( j always increases with x. . Show that the a:-derivative of tan-^ A/ ~ ^^^ ^ is not a function ' 1 + cos a: 31 oix. '1 + 32. Find at what points of the ellipse — + ^ = 1 the tangent cuts off equal intercepts on the axes. ^ 33. Find the points at which the slope of the curve y = tan x is twice that of the line y = x. 34. Find the angle which the curves y = sin x and y = cos x make with each other at their point of intersection. jj^ CHAPTER III • SUCCESSIVE DIFFERENTIATION 35. Definition of nth derivative. When a given function 1/ = (^x) is differentiated witli regard to x by tlie rules of Cliapter I, tlien the result is a new function of x which may itself be differentiated by the same rules. Thus, dx\dxj dx Cpy The left-hand member is usually abbreviated to -y^, and the right-hand member to "(x)\ that is, Differentiating again and using a similar notation, and so on for any number of differentiations. Thus the d^y symbol -t4 expresses that y is to be differentiated with regard to x, and that the resulting derivative is then to be d^y differentiated. Similarly, ^ indicates the performance of 52 Ch. III. 35.] SUCCESSIVE DIFFERENTIATION 53 the operation — three times, T-fyf-^))- ^^ general, the ^"v/ *^ ctx\dx\dx// symbol -j— means that y is to be differentiated n times in succession with regard to x. Ex. 1. If y = a:^ + sin 2 x, 3^ = 4a:8+2cos2a:, dx g=12x2-48in2ar, ^ = 24 ar - 8 cos 2 ar, '-^ = 244-16sin2x. dx* If an implicit equation between x and «^ be given and the derivatives of i/ with regard to x are required, it is not necessary to solve the equation for eithef variable before performing the differentiation. Ex.2. Given x^-{-y^-\-4:a^xy=0', find ^. ±^(x^ + y* + ^a^xy) = 0, 4a:«+4/^ + 4a2a:^ + 4a2^ = 0. dx dx ^ The last equation is now to be solved for -^, dy x^ + a^y dx y^ + a^x Differentiating again, (1) G?a;2~ dx\y^ + a^xl (y^ + a^x) J- (x^ + a^y) - (x^ + a^y) -^ (?/« + a^x) (z/« + «2x) ^3 z2 + a2 ^) - (a:8 + a^y) (3 .^^ ^^ + a^^ 54 DIFFERENTIAL CALCULUS [Ch. III. The value of -^ from (1) is now to be substituted in the last equa- tion, and the resulting expression simplified. The final form may be written : dhf _ 2 a^xy - 10 a^:i^y^ - a\x* + j/*)- 3 a;V(x* + yQ In like manner higher derivatives may be found. 36. Expression for the nth derivative in certain cases. For certain functions, a general expression for the nth. derivative can be readily obtained in terms of w. Ex. 1. If y = ^, then -^ = e*, -fi^ = e^ ..., -^- e', ^ dx dx^ dx" where n is any positive integer. If ^ = e'*", -r-^ = a"e'^. Ex. 2. If y= sin x, -i^ = cos a; = sin ( X + - )> dx \ 2/ g=cos(..|) = sin(..V1, dx» \ 2 1 U 3/=sinax, — ^ = a» sin ( ax + n - V c/x" \ 2/ EXERCISES ON CHAPTER III 1. y=3a:*+5x2+3a:-9; find^. 6. y = e'\ogx', find ^. 2. y = 2x2+ 3a: + 5; find 7. y = xMogx; find ^ 3. y = 1 ; find g. / 8. y = sec^x ; find ^3. / 4. y = x«-i; find 0. 9. y = logsinx; find ^3. 5. y = c* ; find -t-|. 10. y = sin x cos x ; find ^. 35-36.] SUCCESSIVE DIFFEUENTIATION 55 . / 11. y = ,J1^ ; find '% ^19. y = cos mx ; find ^. 12. y = xMoga;2; find^,- . / 20. ?/=— J— -; find ^. 13. y = sin a; ; find ^,. "/2I. «/ = log (a + x)-; find ^. 14. y = log («- + e--) ; find ^. 22. y^=2px; find ^. 15. ^, = (x^-3x + 3).2.;findg. 23. ^'+ |-I= 1 ; find g. 16. y = xMoga;; find ^. 24. x^ + z/3 = 3 ax^/ ; find ^. 17. 2/ = e«* ; find |^. 25. e^+» = a:y ; find ^. 18. V = -ir; find ^' 26. «/ = 1 + a:e»; find ^. ^ a: - 1 ' rfx** ^ ' 6?a;2 rf^y dy 27. y = e* sin a: ; prove y| — 2-p + 2y = 0. 28. y=:aa:sina;; prove a:2 ^ - 2 a: ^ + (a:^ + 2) 3/ = 0. rf2?/ 29. y = aa;"+i + Ja;-*^ ; prove x^j~ = n(n+ 1) z/. 30. y= (sin-i a:)^; prove (1 " ^^) ^2 " ^ ^ = 2. 31. y = ^;±^; prove ^=l-y'- 33. y = a:-^ log x ; find ^. . 32. 2, = -r4-T; find ^. - 34. y = 1^; find ^f- ^ ix^-l dx'^ ^ 1 + a;' c?x« 35. y = xV; prove ^ = 2|^-'^^ + 2... 36. w = cos^ X ; find -r-^' rfv 1 d"^!! dy^ 37. From the relation -^ = -— -, prove that -^ = y-r-rg' CHAPTER IV EXPANSION OF FUNCTIONS It is sometimes necessary to expand a given function in a series of powers of the independent variable. For instance, in order to compute and tabulate the successive numerical values of sin x 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 met with in algebra. For example, by the binomial theorem, (« + xy = a" + na^-^x + ^^!^ ~ "^^ ^""'^ + '" » (^) J. * ^ and again, by ordinary division, 1 1-x =^\+x-\'x'^ + a^-\- .... (2) 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, the identity in (1) holds only for values of x between — a and + a when n is not a positive integer ; and the identity in (2) holds only for iralues of X between — 1 and + 1. In each of these ex- amples, if a finite value outside of the stated limits be given to a:, the sura of an infinite number of terms of the series will be infinite, while the function in the first member will be finite. 66 Ch. IV. 37.] EXPANSION OF FUNCTIONS 57 37. 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. Those values of x for which a series of powers of x is con- vergent constitute the interval of convergence of the series. For example, the sum of the first n terms of the geometric series * a + ax -^ aoc^ -\- ao^ -\- ••• IS s„ = -^- ^• 1 — X First let x be numerically less than unity. Then when n is taken sufficiently large, the term x^ approaches zero ; hence li"V^,^=^. Next let X be numerically greater than unity. Then rr" be- comes infinite when n is infinite ; hence, in this case Thus the given series is convergent or divergent according as X is numerically less or greater than unity. The condition for convergence may then be written -l I will be used to indicate that the expression on the left has respectively a smaller, or larger, numerical value than the one on the right. Any series of terms is said to be absolutely or uncon- ditionally convergent when the series formed by their abso- lute values is convergent. When a series is convergent, but the series formed by making each term positive is not convergent, the first series is said to be conditionally convergent.* *The appropriateness of this terminology is due to the fact that the terms of an absolutely convergent series can be rearranged in any way, without altering the limit of the sum of the series ; and that this is not true of a conditionally 60 DIFFERENTIAL CALCULUS [Ch. IV. E.g., the series ——— + — —••• is absolutely convergent; but the series \ — \ + \— ••• is conditionally convergent. 6. If there be any series of terms in which after some fixed term the ratio of each term to the preceding is numerically less than a fixed proper fraction ; then, (a) the successive terms of the series approach nearer and nearer to zero as a limit ; (J) the sum of all the terms approaches some fixed con- stant as a limit ; and the series is absolutely convergent. [Use 3, 4, 5.] Ex. 1. Find the interval of convergence of the series l+2.2a:+3.4z2+4.8a:3 4-5.16a:4+.... Here the nth term m„ is n2^-^x'^-'^, and the (n-f l)st term u„+i is (n + l)2"a;", hence Mn+i ^ (n + 1) 2»a:" ^ (n + 1)2 a: M„ - n2~-ix« 1 ~ n ' therefore when n = co, -^^ = 2x. It follows by (6) that the series is absolutely convergent when — 1 < 2 X < 1, and that the interval of convergence is between — | and + J. The series is evidently not convergent when x has either of the extreme values. Ex. 2. Find the interval of convergence of the series X x^ x^ £l_ 4. . (— 1)" 1.3 3-38 5.36 7-3^ (2n-l)32'-i ® w, '"^2n + 1 * 32«+i ' a:2«-i~ 2n + 1 ' 3«* hence --^-^ = :77,» when n = co w«+i . ar-* convergent series. Thus the numerical value of the series - + ••• 12 2* .3'^ is independent of the order or grouping ; but the value of the series \ — \-\- \ — \-\- •'• can be made equal to any nunjber whatever by suitable rearrangement. [For a simple proof, see Osgood, pp. 43, 44.] 38-39.] EXPANSION OF FUNCTIONS 61 thus the series is absolutely convergent when — < 1, i.e., when — 3 < a: < 3, 32 and the interval of convergence is from — 3 to + 3. The extreme values of X, in the present case, render the series conditionally convergent. E.. 3. Show that the seHes i(|)- |^(|)% J.4|)^-^(|)V ... has the same interval of convergence as the last ; but that the extreme values of x render the series absolutely convergent. 39. Remainder after n terms. The last article treated of the interval of convergence of a given series without reference to the question whether or not it was the develop- ment of any known function. On the other hand, the series that present themselves in this chapter are the developments of given functions, and the first question that arises is concerning those values of x for which the function is equivalent to its development. When a series has such a generating function, the differ- ence between the value of the function and the sum of the first n terms of its development is called the remainder after n terms. Thus if fQx') be the function, S„(^x} the sum of the first n terms of the series, and i2„(a;) the remainder obtained by subtracting S^C^") from /(a;), then in which S„(x'), Jl„(^x') are functions of n as well as of x, " „'L"'co^»C^) = 0, then J™^^,(:,)=/(^); thus the limit of the series S„(^x^ is the generating function when the limit of the remainder is zero. Frequently this is a sufficient test for the convergence of a series V" ^nC^^)- If a series proceed in integral powers of x—a^ the pre- ceding conditions are to be modified by substituting x — a for X ; otherwise each criterion is to be applied as before. 62 DIFFERENTIAL CALCULUS [Ch. IV. 40. Maclaurin's expansion of a function in a power-series.* It will now be shown that all the developments of functions in power-series given in algebra and trigonometry 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 — a^ of any assigned function which, with its successive derivatives, is continuous in the vicinity of the value x = a. The preliminary investigation will proceed on the hypothe- sis 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 = a. From this hypothesis the coefficients of the different powers of a: — a will be de- termined. 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 representa- tion of the generating function. Let the assumed identity be f(x)^ A 4- B(^x -d)+ C(x - ay -f- I)(x - ay + J57(a;-a)4H-..., (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 : • Named after Colin Maclaurin (1698-1746), who published it in his ♦'Treatise on Fluxions" (1742) ; but he distinctly says it was known by Stirling (1690-1772), who also published it in his " Methodua Differcntialis " (1730), and by Taylor (see Art. 41). 40.] EXPANSION OF FUNCTIONS 63 f{x)=B-{-2C(^x-a')+ SB{x-ay+ 4UCx-ay + '" f"(x)=^ 2(7 +2^'2D{x-a) + 4:-^ E(x-aY+-: f"'(x)= 3.2i) +^'2>'2E(x-a) +- If, now, the special value a be given to x^ the following equations will be obtained : /(a)= A, /'(«)= B, f\a)=^2C, f\a)= 3 • 2 D, .... Hence, ^ =/(«), 5 =/'(«), = ^^, I> = i^, -. Thus the coefficients in (1) are determined, and the re- quired development is /(ic) = /(a) + /'(a)(a5 - a) + ^^ (05 - a)2 + '^^^(a5 - «)» + -+^-^^(^ -«)" + -. (2) This series is known as Maclaurin's series, and the theo- rem expressed in the formula is called Maclaurin's theorem. Ex. 1. Expand log x in powers oi x - a. Here f{x) = \ogx, f(x) = I r(x)= - 1, f"(x) = ^ ••., Hence, /(a)=loga, /(«)=^, /"(«)=-i:2' /'"(«) = ^-' ^..)^,)^(-l)-Hn-l)! and, by (2), the required development is log X = log a + 1 (;r - a) - ^^(2: - «)2 + ^3(0: - a)8 - ... + i — i-i — (a: - a)« + .-.. 64 DIFFERENTIAL CALCULUS [Ch. IV. The condition for the convergence of this series is lim r {x-aY^^ . ( ^-«)" ]|^|i. n = ooL(n + l)a"+' * na'' J' ' ' I.e., • -|<|1, a x-a\<,a, 0 from the origin. Then at some point P between a and h the tangent to Pj^ j2 ^^® curve is parallel to the ic-axis, since by supposition there is no discontinuity in the slope of the tangent. Hence at the point P g =/'(..) =0. 43. Form of remainder in Maclaurin's series. Let the remainder after n terms be denoted by Rni^Xt «)? which is a function of x and a as well as of n. Since each of the succeeding terms is divisible by (x — a)", R^ may be con- veniently written in the form i2„ (x, a) = y^^ V ^ (x, a), n I The problem is now to determine ^(a;, a) so that the relation fix) =/(a) +/'(«)(:. - «) + £^ Cx - a)« + ] (n — 1)! n! 42-43.] EXPANSION OF FUNCTIONS 69 may be an algebraic identity, in which the right-hand mem- ber contains only the first n terms of the series, with the remainder after n terms. Thus, by transposing, fix-) -fia-) -fiaXx -a}- £^ (x - af - ... -fr^i^-ay-^-i^^ix-ay^O. (2) (» — 1) ! n: Let a new function, FQi), be defined as follows : J-CZ) =f{x-) -fiz) -fiz)(x - 2) - -f^ (X - Zy - ... (n—l)l nl This function F(^z) vanishes when^ 5ir as is seen by inspection, and it also vanishes when'^psa, since it then becomes identical with the left-hand member of (2) ; hence, by Rolle's theorem, its derivative F'^z') vanishes for some value of z between x and a, say Zy But -f"(2)=-/'(^)+/'(2)-/"(^)(a'-2)+/"(^)(^-z)--^J in-iy.^'' ^■> +(„_!)! ^"^ '■> ■ These terms cancel each other in pairs except the last two; hence ^'^'^ = %"-% ^'^^''' «)-/'"'-(^)]- Since F'(^z} vanishes when z = z^ it follows that <^(^,«)=/'">(z,). (4) In this expression z^ lies between x and a, and may thus be represented by z^^a + eCx-a-), AMrj\Ji\j ~i±-^ 70 DIFFERENTIAL CALCULUS [Ch. IV. where ^ is a positive proper fraction. Hence from (4) (^(a:, «)=/(«>[« + r \ f''^\a + e(x-a)'\ , .„ * and R^ (x, a) = ^ — > — -i—-^ ^ (x — a)". * The complete form of the expansion of f(x) is then finc>)=fia)+f'{a) (a? - a) + f^ (05 - a)^ + ... in which w is any positive integer. The series may be car- ried to any desired number of terms by increasing /i, and the last term in (5) gives the remainder (or error) after the first n terms of the series. The symbol /^"^ (a + ^(2; — a)) indi- cates that f(x) is to be differentiated n times with regard to a;, and that x is then to be replaced hj a ■\-6(x — a), 44. Another expression for the remainder. Instead of put- ting Rf^ (x, a) in the form (x — a)" , , N ^^ r-^(f>(x, a), n\ it is sometimes convenient to write it i2„ (x, a) = (a; - a) i/r (x, a). Proceeding as before, the expression for F'(z) will be F'(z) = - /"^^^^^ , (X - g)»-' + ir{x, a), (w - 1) ! In order for this to vanish when z = Zj, it is necessary that in which z^=a + 6(x — d), x — z^ = (x — a) (\ — 0). * This form of the remainder was found by Lagrange (1736-1813), who published it in the M^moires de TAcad^mie des Scieuces k Berlin, 1772. 43-44.] EXPANSION OF FUNCTIONS 71 Hence i/r (x, a) = /"^^ + 6>(a:- a)) ^^ _ Qy-w^ __ ^y-i^ (?^ — 1) ! "and i2„(^, a) = (l-^)«-i£!l£±%=^(^_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 a; — a, when x — a is negative, and also in the expansion of (a + a;)"*. 1. Find the interval of equivalence for the development of log a; in powers of a: — a, vv^hen a is a positive number. Here, from Art. 40, Ex. 1, hence /X. („ + ,(._„)),= |_£|r_IlL_, and,.,Art.43, ^•^(^,»^l = \;:^^^f^^J = ll[^^I^J- First let a: — a be positive. Then when it lies between and a it is numerically less than a + 0{x — a), since ^ is a positive proper fraction ; hence when n = oc r ?Lp^ — T^ 0, and i?„ (x, a) = 0. Again, when a; — a is negative and numerically less than a, the second form of the remainder must be employed. As before, hence i^.(., a)|H(l - ^)-- ■ ^^^^/^l^j. 1= 1(1 _ ^)n-i (a-x)- ._. r (a - x) - 6(a — x) !"*-^ a — x '~'L a-d(a-x) J 'a-e(a-x) * This form of the remainder was found by Cauchy (1789-1857), and first published in his "Legons sur le calcul infinitesimal," 1826. 72 DIFFERENTIAL CALCULUS [Ch. IV. The factor within the brackets is numerically 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 oi x — a are equiva- lent 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 arz in positive powers oi x — a holds for all values of x that make the series convergent; that is, when X lies between and 2 a. If the function is expanded in powers of a;, the complete form will be /(^) =/C0) +/'(0> + =^ ;^ + - + ^^ :^-> ^^-^^ (1) for the first form of remainder, and /W =/(0) +/'(0> + ^^'a? + - + -^^af-^ for the second form of remainder. Similarly, the complete form of Taylor's series (Art. 41) becomes /(a + ^)=/(a)+/'(a> + ^^r'+ - +^^^-' for the first form of remainder, and /(a + rr) =/(«)+/'(«> + =^^ a? + - +^--^^' (»-l)! ^^tS:^^^^-^)"-'-^ W for the second form of remainder. 44.] EXPANSION OF FUNCTIONS 73 Ex. Expand (a + x)"" in ascending powers of x, and determine the interval within which R^ has the limit zero. Here /(a + x) = (a + a:)"», hence f(x) = x»", and f'(x)= mx^-i, f"(x) = m(m - l)a;«-2, ..., /(«)(x)= m(m - 1) ••• (m-n + l)x'»-»; Aa)=a-^, f'{a)=ma-^-\ f"{a)= m{m - \)a-^-\ ..., f^^\a) = 7?i(m — 1) ••• (m — n + 1)0"^-". Therefore (a + x)*" = a"* + ma'^-^x + ^^^^~ ^^ a'"-^^;^ + ... ^ r/?(m-l)...(m-n + 2) ^«-«+i^n-i ^ ^^(^^ ^^^ (w - 1) ! in which, from the first form of remainder, R,(x, a) = Mm-\)'"(m-n + l) ^^ ^ Qx^-nx- n ! ^ m{m - 1) - (m - n + 1) ^^ ^ Sxyl-^Y^ n\ \a + dxf Consider the ratio 7t„ m — n + 1 a; When m is greater than - 1, the factor ^~ ^ is less than unity for every value of n greater than I (m+1), and when x lies between and a the second factor is also less than unity. Their product is there- fore less than some fixed proper fraction k for every such value of n. Hence Bnl(m + l). Since /?, is finite and ^'»-« can be made as small as desired by taking n sufficiently large, it follows that H"^ Rn(x,a) = n= CO "^ ' ^ when m>— 1, and 0i-t /.. / 1? TL. "', -/ .3. -2'3 ^V 1^, 2l, CHAPTER V INDETERMINATE FORMS 46. Hitherto the values of a given function f{x)^ 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 sma; for the value a;=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 /(:,)=^, and if, when x takes the special value a, the functions 4>{x) and ^(x) both vanish, then is indeterminate in form, and cannot be rendered determinate without further transformation. • 77 78 DIFFERENTIAL CALCULUS [Ch. V. 47. Indeterminate forms may have determinate values. A case has already been noticed (Art. 9) in which an ex- pression that assumes the form - for a certain value of its variable takes a definite value, dependent upon the law of variation of the function in the vicinity of the assigned value of the variable. As another example, consider the function _x^ — a^ ^ ~~ X— a' If this relation between x and y be written in the.forms y{x — a)= x^ — a\ (x — a)(^y — x — a) = 0^ it will be seen that it can be represented graphically, as in the figure (Fig. 14), by the pair of lines / X— a = 0, * y — X — a = 0. Hence when x has the value of a there — X is an indefinite number of corresponding points on the locus, all situated on the ^°- ^*' line x = a; and accordingly for this value of X the function y may have any value whatever, and is therefore indeterminate. When X has any value different from a, the corresponding value of y is determined from the equation y = x + a. Now, of the infinite number of different values of y corresponding to x = a, there is one particular value AP which is con- tinuous with the series of values taken by y when x takes successive values in the vicinity of x= a. This may be called the determinate value of y when x = a. It is ob- tained by putting x= a in the equation y = a: -{- a, and is therefore y = 2 a. 47.] INDETERMINATE FORMS 79 This result may be stated without reference to a locus as follows : When a; = a, the function 00^ — a^ is indeterminate, and has an infinite number of different values; but among these values there is one determinate value which is continuous with the series of values taken by the function as x increases through the value a ; this deter- minate or singular value may then be defined by lim ^- CL^ X = a X — a In evaluating this limit the infinitesimal factor x — a may be removed from numerator and denominator, since this factor is not zero, while x is different from a ; hence the determi- nate value of the function is lim X + a _ i) ^ Ex. 1. Find the determinate value, when x = 1, of the function x^+2x^-nx Sx^-dx^- X + 1' which, at the limit, takes the form — This expression may be written in the form (x^+^x)(x- 1 ) (3x2- l)(ar-l)' x^ 4- 3 x which reduces to jr— 5 r^. When x = 1, this becomes # = 2. 3 x2 - 1 ' ^ Ex. 2. Evaluate the expression x^ + ax^ + a^x + a» x^ + xb^ + ax^ + ab^ when X = — a. 80 DIFFERENTIAL CALCULUS [Ch. V. Ex. 3. Determine the value of a;3 _ 7 3:2 + 3 a: + 14 a;3 + 3 x-^ - 17 a; + 14 when X = 2. Ex. 4. Evaluate - — ^ — - when x= 0. x^ (Multiply both numerator and denominator by a + Va^ — r^.) 48. Evaluation by development. In some cases the com- mon vanishing factor can be best removed after expansion in series. Ex. 1. Consider the function mentioned in Art. 46, When numerator and denominator are developed in powers of x, the expression becomes 21 3! V 2\ S\ I X- 3,3 3! + • •• 2X + |;X3 + ... 2 ,.-,... 3.8 ^-31 + ... - 1 -i^-' which has the determinate value 2, when x takes the value zero. Ex. 2. As another example, evaluate, when a: = 0, the function X — sin-'a: sin^a: By development it becomes Removing the common factor, and then putting x = 0, the result is J. 47-49.] INDETERMINATE FORMS 81 In these two examples the assigned value of x^ for which * the indeterminateness occurs, is zero, and the developments are made in powers of x. If the assigned value of x be some other number, as a, then the development should be made in powers of x — a. Ex. 3. Evaluate, when x = -, the function cos a: 1 — sin X By putting a: — - = A, x = --\- h, the expression becomes eo,(|+*) . ^ -A + ^-... -1 + f \ 2 / — sm A 6 6 l-sin(|+^) 1 cos A '^_A!4- ^_A! 2 24 2 24 which becomes infinite when A = 0, that is, when a; = ^. TT hm cos X , ^ Hence „ . ^ : = ± co, •*^ — z 1 — sin a: according as h approaches zero from the negative or positive side. 49. Evaluation by differentiation. Let the given function be of the form ,; i ^ and suppose that /(a) =0, (a) = 0. It is required to find j^l^^^i^. As before, let/(2j), <^(a;) be developed in the vicinity of x = a^ by expanding them in powers oi x — a. Then fix) /(«)+/'(«)(^-«) + ^^(*-«)' + - ^^'"^ ~ <^(«) + f (a)(^ - a) + *^ (a; - «)2 + ... /'(a)(r.-a)+•^(a.-a)''+••. 82 DIFFERENTIAL CALCULUS [Ch. V. By dividing hj x—a and then letting a: = a, it follows that lim Ax) ^f{a) ^ = ^\a)' The functions /\'a), <^'(a) will in general both be finite. If /(a) = 0, <^'(a) ^ 0, then ^ = 0. If /(a) ^ 0, <^'(a) = 0, then ^^ = oo. If /'(a) and <^'(a) are both zero, the limiting^ value of f(x) \\ ^ is to be obtained by carrying Taylor's development one term farther, removing the common factor (^x — a)^, and then letting x approach a. The result is f,^ ^ - Similarly, if /(a), /(a), /'(«); (^(a), '(«), ''(«) all vanish, it is proved in the same manner that lim f(x}_ f"(a-) 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 numer- ator and denominator separately/ ; substitute the critical value of x in their derivatives^ and equate the qw)tient of tJie deriva- tives to the indeterminate form, Ex. 1. Evaluate ^"^P^^ when ^ = 0. Pat /(^) = l-cosd, ($) = e^. Then /'($) = sin 0, '(e) = 2 0, and /(O) = 0, <^'(0) = 0. 49.] INBETERMINATE FORMS 83 Again, f>(e)=oosO, <^"(^)=2, /"(0)=1, <^"(0)=2, hence /!"„ kl_^ = 1. Ex. 2. Find lim f!±iIl±2^2^^zJ. X - u 3,4 lim e" + e-'' + 2 cos a: — 4 _ lim e=' — e-» — 2 sin a? a;=:0 ^4 - x = 4 ^8 _ Mm 6» + e-^ — 2cosa: - X = 22 x2 _ lim e» _ e-x _^ 2 sin a; -x = 24x _ lim e» + e-*+ 2 cos a? "" a; i 24 Ex. 3. Find lip ^^-sinxcosar, a; = u ™8 Ex. 4. Find "P. a;^-2xa-4^^+9a:-4. + In this example, show that x—1 is a factor of both numerator and denominator. 1?^ R T?;^^ li°i Stana:- Sar-aH' ^ Ex. 5. Find ^ ^ -^ In applying this process to particular problems, the work can often be shortened by evaluating a non-vanishing factor in either numerator or denominator before performing the differentiation. Ex.6. Find "ro^^-^>'^^^^. The given expression may be written ^i°^ rr_4^2tana:_ lim , ..3 ^i^ tanar a: = ^^ *>' —^ -x = 0C^-*; a; = "^ = 16 . 1 = 16. 84 DIFFERENTIAL CALCULUS [Ch. V. In general, if fCx')= 'f(x)x(p^^^ ^^^ ^^ '^(^)= ^» xC^)"?^^' <^(a) = 0, then For lim ^(a^)x(^) _ lim ^. . . lim ^(^) _ ^,^^. . ^'(<^) Ex.7. Find lim sin x cos^ a; Ex.8. Find,^!P^/"-^)^^^^(^-^). * — ^ sin (x — 1) ^ p-i / EXERCISES ^ / Evaluate the following expressions : 9. Lz.£2i£ when a: = 0. 15. ^ "r e-^ -2 ^^^^^ ^^^ sin X x^ when a; = 0. , ^ tan a:- sin x cos a: sma: 16. "^"•^-^^"•^^"'^-^ when a: = 0. 11. £i=li when a: = 1. X —\ __ sin~^a: 17. -ii^i— ± when ar = 0. 12. ?l::ilwhen a: = 0. *^°"'^ 6- -1 , « sin aa: , «« a 18. ^' ^^° ^ ~ ^ ~ ^^ when a: = 0. s"i^ll a:2 + X log (1 - X) 14. (l±£)i:ii when X = 0. 19- ^"^l^^"^ when a: = 0. X x* There are other indeterminate forms than -• They are g,oo-oo,0«, r, ao». 00 50. Evaluation of the indeterminate form ^. oo Let the function ^rr^ become — when x^a. It is re- quired to find Jf'^lg. 49-50.] INDETERMINATE FORMS 85 This function can be written (x) 1 ' which takes the form - when x = a^ and can therefore be evaluated by the preceding rule. When a; = a, 1 (j() _ lira \.4>(^)J f(x) If both members be divided by , , { ^ the equation becomes -. ^ lim /(a:) <^'(a:) therefore l-JZ^l^^^^. (2) This is exactly the same result as was obtained for the form -; hence the procedure for evaluating the indetermi- nate forms -, — , is the same in both cases. When the true value of ~r-^ is or oo, equation (1) is satisfied, independent of the value of T77-T; but (2) still I' iCxi gives the correct value. For, suppose ^i" ,) ( = 0. Con- X — a ^ ^^^ sider the function 86 DIFFERENTIAL CALCULUS [Cb. V. which has the form — when x = a» and has the determinate 00 value {x) "^(x)^ such that <^(a) = oo, Vr(a) = 0. This may be written ^^ ^ , which takes the form - when a is substituted for a?, and therefore comes under the above rule. (Art. 49.) 52. Evaluation of the form oo — oo. The form oo — oo may be finite, zero, or infinite. For instance, consider y/a? + ax— x for the value x = cc. It is of the form oo — oo, but by multiplying and dividing by ^x^ 4- ax -f a; it becomes ^ , which has the form 00 1 ^x^ -{- ax -\- X 5o when a; = oo. Again, by dividing both terms by x, it takes the form ., which becomes ? when rr = oo. 4U Jl+' + l > X There is here no general rule of procedure as in the previous cases, but by means of transformations and proper 50-52.] INDETERMINATE FORMS 87 grouping of terms it is often possible to bring it into one of the forms -, — . Frequently a function which becomes oo — 00 for a critical value of x can be put in the form u t V V) in which v, w become zero. This can be reduced to uw — vt which is then of the form -• Ex. 1. Find 3.^5 (sec x — tan x). This expression assumes the form 00 — 00, but can be written 1 _ sin a- _ 1 — sin a: cos X cos X ~ cos X which is of the form -, and gives zero when evaluated. Hence ^ ^^ (sec x — tan a:) = 0. Ex. 2. Prove ^ ^^n (sec^a: — tan»a:) = 00, 1, 0, according as \^ EXERCISES ^ Evaluate the following expressions : 2. !^S^ when x = 0. 6. ^5£^ when a; = 5- cot x sec 5 a; 2 3. — when a: = 00. 7. (a^ — x^) tan — when x = a. - tan X „i V 0/1 +„^ ^\ o«« o «. T^kz^r. «._''' when ar = -• 8. (1 — tan ar) sec 2 a; when a: = -• tan5x 2 4 88 DIFFERENTIAL CALCULUS [Ch. V. y g 1-loga; ^YiQn x = 0. '^12. — ^ i— when x = l. e' — e X — 1 yr 10. ^-^ when :r = 00. N 13- csc^x - 1 when x = 0. J 11. -i ^ when a; = 1. /!*• r^ " r^ ^^^^ ^ = 1* ^ logx a:-l *^ logx loga; 53. Evaluation of the form 1*. Let the function u — [(a;)]'''^'^^ assume the form 1* when x — a. In order to evaluate this expression, take the logarithm of both sides. Then log u = ^(x) ■ log ^(x) = l2E|M. This expression assumes the form - when a; = a, and can be evaluated by the method of Art. 49. If the reduced value of this fraction be denoted by tw, then log w = m and u = g"*. Note. The form 1° is not indeterminate, but is equal to 1. For, let [^(x)] '/'(*) assume the form 1° when x — a. Put u = [(x)]<^(*). Then logu = ^(a:)log[<^(a;)], which equals zero when x ■=. a\ hence log u = 0, u — e^—\, 54. Evaluation of the forms oo^ 0^. Let [^^{xy^'''^ become oo^ when x — a. Put w = [. Then log u = y^(x) log 4>(x) = ]2K^^, 52-54.] INDETERMINATE FORMS 89 This is of the form — , and can be evaluated by the method of Art. 50. Similarly for the form 0^. Note. The form 0* is not indeterminate, but is equal to 0. For let u = [<^(x)]'l'(^> become 0"° when x = a. Then log u = i//(a:) log («) if '(«) is positive, and that ^{x) decreases through the value <^(«) if <^' («) is negative. Thus the question of finding whether ^{x) increases or decreases through an assigned value ^(a), is reduced to determining the sign of ' (a). 1. Find whether the function increases or decreases through the values ^(3) = 2, ^(0) = 5, ^(2)=|, ^(— 1)= 10, and state at what value of x the function ceases to increase and begins to decrease, or conversely. 56. Turning values of a function. It follows that the values of x at which ^{x) ceases to increase and begins to decrease are those at which <^' (x) changes sign from positive to negative ; and that the values of x at which {x) ceases to decrease and begins to increase are those at which <^' (x) changes its sign from negative to positive. In the former case, (^ {x) is said to pass through a Tyvaximum, in the latter, a nbinimum value. 91 92 DIFFERENTIAL CALCULUS [Ch. VI. Fig. 15. Ex. 1. Find the turning values of the function <^ (x) = 2 a;8 - 3 a;2 - 12 x + 4, and exhibit the mode of variation of the function by sketch- X ing the curve y = {x). Here ' (x)= Qx^ - Qx - 12 = Q(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) (Fig. 15) may- be inferred from the last statement, and from the following simultaneous values of x and y : x = -oo, -2, -1,0, 1, 2, 3, 4, 00. y = - 00, 0, 11, 4, - 9, - 16, - 5, 36, oo. Ex. 2. Exhibit the variation of the function (x) = ix-l)'-^2, especially its turning values. Since '(x) = ? , 3(x-l)^ hence '(x) changes sign at a:= 1, being negative when a: < 1, infinite when x = 1, and positive when x>l. Thus <^(1) = 2 is a minimum turning value of (x). The graph of the function is as shown in Fig. 16, with a vertical tangent at the point (1, 2). Ex. 3. Examine for maxima and minima th6 function (x) = (x- 1)^ + 1. Here '(x) never changes sign, but is always positive. There is accordingly no turning value. The curve y = (x) has a vertical tangent at the point (1, 1), since '-^ = 4*'(x) is infinit<* when x = l. (Fig. 17.) Fig. 16. Fio. 17. 1 56-58.] VARIATION OF FUNCTIONS 93 57. Critical values of the variable. It has been shown that the necessary and sufficient condition for a turning value of (x) is that '(^x) shall change its sign. Now a function can change its sign only when it passes through zero, as in Ex. 1 (Art. 56} , or when its reciprocal passes through zero, as in Exs. 2, 3. In the latter case it is usual to say that the function passes through infinity. It is not true, conversely, that a function always changes its sign in passing through zero or infinity, e.g., y = x^. Nevertheless all the values of re, at which (f>\x) passes through zero or infinity, are called critical values of a;, be- cause they are to be further examined to determine whether '(x) actually changes sign as x passes through each such value ; and whether, in consequence, 4>(x) passes through a turning value. For instance, in Ex. 1, the derivative (i>'(x) vanishes when a; = — 1, and when a: = 2, and it does not become infinite for any finite value of x. Thus the critical values are — 1, 2, both of which give turning values to (f>(x). Again, in Exs. 2, 3, the critical value is x = 1, since it makes '(x') infinite ; it gives a turning value to (^x) in Ex. 2, but not in Ex. 3. 58. Method of determining whether <{>'(») changes its sign ^ » in passing through zero or infinity. Let a be a critical value i^ of a:, in other words let (^'(a) be either zero or infinite, and let ^ be a very small positive number, so that a — h and a-\-h ^re two numbers very close to a, and on opposite sides of it. ^in order to determine whether <^'(a;) changes sign as x in- Acreases through the value a, it is only necessary to compare ^the signs of <^'(a + h) and <^'(a — h). If it is possible to ^ take h so small that <^'(a — h} is positive and <^'(a + h) V negative, then (j)'(ix) changes sign as x passes through the 94 DIFFERENTIAL CALCULUS [Ch. VI. value «, and (jc) passes through a minimum value <^(a). if <^'(a — h) and <^'(a + h) have the same sign, however small h may be, then <^(a) is not a turning value of <^(a;). Ex. Find the taming values of the function Here '(x)= 2{x - l)(a: + 1)8 + 3(x - V)\x + 1)« = (z-l)(a; + l)2(5a:-l). Hence 4*'(x) becomes zero at a: = — 1, |, and 1 ; it does not becdme infinite for any finite value of x. Thus, the critical values are — 1, |, 1. F Fio. 18. When X = — 1 — h^ the three factors of <^'(^) ^^^® ^-l^® signs — and when a: = — 1 + A, they become — thus ^'{x) does not change sign as x increases through — 1 ; <^(— 1)= is not a turning value of <^(x). When r = ^ — A, the three factors of ^'(•^) h*^® signs - and when x = \ + h, they become — thus '{x) changes sign from + to — as x increases through \y and ^(i) = 1 • 11052 is a maximum value of ' (x) have the signs — + + , and when x = 1 + h they become + + + ; thus '(x) changes sign from — to + as a; increases through 1, and <^(1)= 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 as follows, in which inc., dec. stand for increasing, decreasing, respectively : 1 +h + inc. The general march of the function may be exhibited graphically by tracing the curve y = (x) (Fig. 18), using the foregoing result and observing the following simultaneous values of x and y : X -l-Jl -1 -l + h \-^ \ l + h l-h 1 '{x) + + 4- - - (x) inc. inc. iiic. max. dec. dec. min. y = 1, 0, i, 1, 2, 00. 0, 1, 1.1..., 0, 27, 00. 59 Second method of determining whether '(i») 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 "(x) is nega- tive 2it X— a. On the other hand, if <\>\x^ changes sign from negative through zero to positive, it is an increasing function and (^"(a?) is positive at a: = a; hence : The function <^(a;) has a maximum value (f>(a'), when <^'(a) = and "(a^ is negative ; <^(a:) has a minimum value 4>(cl)-, when <^'(a)= and <^"(a) is positive. 96 DIFFERENTIAL CALCULUS [Ch. VI. It may happen, however, that " (ci) is also zero. In this case, to determine whether <\>(x) has a turning value, it is necessary to proceed to the higher derivatives. If (i>(x) is a maximum, "(x) ^^ 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 (t>"(x) I changes from increasing to decreasing as x passes through a. Hence "'(x') changes sign from positive through zero to negative, and it follows, as before, that its derivative <^'^(a;) is negative. Thus <^(a) is a maximum value of (x) if <^'(a)=0, <^'^(a)=0, '"(«)= 0, <^'''(a) negative. Similarly, <^(a) is a minimum value of Qc) if '(a) = 0, <^''(a) = 0, <^'^'(a) = 0, and '^(a) positive. If it happen that (^'^(a) = 0, it is necessary to proceed to still higher derivatives to test for turning values. The result may then be generalized as follows: The function <\>(x) has a maximum {or minimum') value at x= a if one or more of the derivatives <^'(a), "{a}, (i>"'{cL) vanish and if the first one that does not vanish is of even order ^ and negative {or positive), Ex. Find the critical values in the example of Art. 58 by the second method. <^"(:r) = (x+l)2(5a:-l) + 2(x-l)(x+l)(5x-l)+5(a:-l)(z+l)a, = 4(5x» + 3a:a-3x- 1), <^"(1)= 16, hence ^(1) is a minimum value of <^(a:), ^"(— 1) = 0, hence it is necessary to find '"(— 1) ; <^"'(x)=12(6x« + 2x-l), ^'"(— 1)=24, hence <^(- 1) is neither a maximum nor a minimum value of (j:)- Again, <^"(i) = H^ - l)(i + 1)* ^ negative, hence (^) is a maximum value of Qa) be a maximum value of <^(a;), it follows from the definition that («) is greater than either of the neighboring values, (« + A), or (^a — A), when h is taken small enough. Hence <^(a + A)— («) and <^(a — A)— («) are both negative. Similarly, these expressions are both positive if <^(«) is a minimum value of (f>(^x^. Let (x + K) and (f>Qc — K) be expanded in powers of h by Taylor's theorem. Then <^(:i:+A) = (2:)+<^'(a:)A+^^A2 + *^^A3+..., A . o I If X be replaced by a, and ^(a) transposed, the result is The increment h can now be taken so small that h(i>'(a') will be numerically larger than the sum of the remaining terms in the second member of either of the last two equa- tions, and its sign will therefore determine the sign of the entire member. Since these signs are opposite in the two equations, <^(a + A) — <^(a) and <^(a — A) — <^(a) cannot have the same sign unless <^'(a) is zero, hence the first condition for a turning value is <^' (a) = 0. 98 DIFFERENTIAL CALCULUS [Ch.VI. In case (a + K) — <^(«) and (f>(a — K)— (f>(cL) are both negative when "(a} is positive. Thus <^(«) is a maximum (or minimum) value' of {x) when <^'(a) is zero and " (a) is negative (or positive). If it should happen that " (a) is also zero, then and by the same reasoning as before, it follows that for a maximum (or minimum) there are the further conditions that "'(cL) equals zero, and that <^*^(«) 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 ? Ex. 2. Why was the restriction imposed upon '{x) that it should change sign by passing through zero, rather than by passing through infinity? 61. The maxima and minima of any continuous function occur alternately. It has been seen that the maximum and 60-62.] VABIATION OF FUNCTIONS 99 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 function of a single variable. For, let c/>((h^ be two maximum values of <^(a;), in which a is supposed less than h. Then, when x = a-\- h, the function is decreasing ; when x=h — h, 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. 62. Simplifications that do not alter critical values. The work of finding 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. When c is independent of x, any value of x that gives a turning value to c(\>(x) gives also a turning value to (t>(x); and conversely. 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-f- (a;); and conversely. 3. When n is independent of x^ any value of x that gives a turning value to [(2;)]" gives also a turning value to (/>(a;); and conversely. Whether these turning values are of the same or opposite kind depends on the sign of n^ and also on the sign of [^(a^)]""^- y\A iajL^ 100 DIFFERENTIAL CALCULUS [Ch. VI. EXERCISES Find the critical values of x in the following functions, determine the nature of the function at each, and obtain the graph of the function. / 1. M = ar(a;2-1). 6. M = ar(ar+1)2- ^ 2. M = 2 a:8 - 15 a;2 + 36 a: - -4. 1 7. M = 5 + 12 a: - J 3. M= (x- 1)8 (a; -2)2. 8. u = !2££. ^ 4. w — sin X + cos x. 5. «=(^^^. a-2x 9. X u = sin^ X cos8 X. ^' 10. Show that a quadratic integral function always has one maxi- mum, or one minimum, but never both. 11. Show that a cubic integral function has in general both a maxi- mum and a minimum value, but may have neither. 12. Show that the function (x — by has neither a maximum nor a minimum value. ^ ' 63. 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 maximized or minimized, being thus made a func- tion of a single variable, can be treated by the preced- ing rules. 62-63.] VARIATION OF FUNCTIONS 101 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 expression to be maximized is the area u = xy, (1) in which the variables x, y are subject to the stated condition 2a;+2?/=100, le,, y = ^0-x; (2) hence the function to be maximized, expressed in terms of the single variable x, is M = <^ (a;) = a; (50 - x) = 50 a; - x"^. (3) The critical value of x is found from the equation <^'(a:) = 50-2ar=0 to be a; = 25. When x increases through this value, <^'(x) changes sign from positive to negative, and hence ^ {x) is a maximum when x = 25. Equation (2) shows that the corresponding value of y is 25. Hence the maximum rectangle whose perimeter is 100 is the square whose side is 25. Ex. 2. 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 - 2 a:, and the depth of the box will be x. Hence the volume is v = x{a-2xy, which is to be made a maximum by varying «. Here ^= (a - 2ar)2 - 4a:(a - 2a:) d^ Fio. 19. = (a-2a:)(a-6a:). This derivative vanishes when x = -, and when x = -. It will be found, 2 by applying the usual test, that a: = ^ gives v the minimum value zero, and O 3 that x = - gives it the maximum value — ^. Hence the side of the 6 27 square to be cut out is one sixth the side of the given square. 102 DIFFERENTIAL CALCULUS [Ch. VI. Ex. 3. 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, 05 (Fig. 20); let2.T, 2y he the dimensions of an inscribed rectangle. Then the area is Fig. 20. u = 4:xy, (1) in which the variables x, y may be regarded as the coordinates of the vertex P, and are therefore subject to the equation of the ellipse t+t = l ft2 (2) It is geometrically evident that there is some position of P for which the inscribed rectangle is a maximum. The elimination of y from (1), by means of (2), gives the function of X to be maximized, (3) «=i*a:V^^^ By Art. 62, the critical values of x are not altered if this function be 4 A divided by the constant — , and then squared. Hence, the values of a? a which render u a maximum, give also a maximum value to the function Here {x) = x\a^ - x2) = a2x2 - x^, f(x) =2a^x -ix» = 2x(a^- 2x^, "(x) = 2a^- 12x2; 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 V2 y/f 63.] VABIATION OF FUNCTIONS 103 giving in each case the same maximum inscribed rectangle, whose dimensions are aV2, by/2, and whose area is 2 aft, or half that of the circumscribed rectangle. Ex. 4. 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 GAB ^^B (Fig. 21), and the inscribed cylinder be generated by the revolution of the rectangle AP. Let OA =hj AB = ttj and let the coordinates of P be (x, y). Then the function to be maximized is Try^Qi — x) subject to the relation - — t' - ^^' ^' This expression becomes V = h^ x\h - x). The critical value of a; is f A, and F = 27 ' EXERCISES ON CHAPTER VI 1. Through a given point within an angle draw a straight line which shall cut off a minimum triangle. Solve this problem by the method of the calculus, and also by geometry. [Take given lines as coordinate axes.] 2. The volume of a cylinder being constant, find its form when the entire surface is a minimum. 3. A rectangular court is to be built so as to contain a given area c^, and a wall already constructed is available for one of its sides. Find its dimensions so that the expense incurred may be the least possible. 4. The sum of the surfaces of a sphere and a cube is given. How do their volumes compare when the sum of their volumes is a minimum ? 5. What is the length of the axis of the maximum parabola which can be cut from a given right circular cone, given that the area of a parabola is equal to two thirds of the product of its base and altitude ? 6. Determine the greatest rectangle which can be inscribed in a given, triangle whose base is 2 6 and whose altitude is a. .ir . 104 DIFFERENTIAL CALCULUS [Ch. VI. 63. 7. The flame of a candle is directly over the center of a circle whose radius is 5 inches. What ought to be the height of the flame above the plane of the circle so as to illuminate the circumference as much as pos- sible, supposing the intensity of the light to vary directly as the sine of the angle under which it strikes the illuminated surface, and inversely as the square of its distance from the illuminated point ? y^ 8. A rectangular piece of pasteboard 30 inches long and 14 inches wide has a square cut out at each corner. Find the side of this square so that the remainder may form a box of maximum contents. 9. Find the altitude of the right cylinder of greatest volume in- scribed in a sphere whose radius is r. 10. Through the point (a, b) a line is drawn such that the part inter- cepted between the rectangular coordinate axes is a minimum. Find its length. ^ 11. Given the slant height a of a right cone ; find its altitude when the volume is a maximum. \ 12. The radius of a circular piece of paper is r. Find the arc of the sector which must be cut from it so that the remaining sector may form the convex surface of a cone of maximum volume. 13. Find relation between length of circular arc and radius in order that the area of a circular sector of given perimeter should be a maximum . 14. On the line joining the centers of two spheres of radii r, R, find the distance of the point from the center of the first sphere from which the maximum of spherical surface is visible. 15. Describe a circle with its center on a given circle so that the length of the arc intercepted within the given circle shall be a maximum. CHAPTER VII RATES AND DIFFERENTIALS 64. Rates. Time as independent variable. Suppose a par- ticle P is moving in any path, straight or curved, and let s be the number of space units passed over in t seconds. Then s may be taken as the dependent variable, and t as the inde- pendent variable. If As be the number of space units described in the addi- tional time A^ seconds, then the average velocity of P during As the time A^ is — ; that 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 velocity — is the same for all intervals A^. The actual velocity of P at any instant of time t is the limit which the average velocity approaches as A^ is made to approach zero as a limit. Thus t;= 1/^^^=^ ^t = ^M 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 ^ -}- 1. It may be observed that if the more general term, " rate of change," be substituted for the word "velocity," the above statements will apply to any quantity that varies with the time, whether it be length, volume, strength of current, 105 106 DIFFERENTIAL CALCULUS [Ch. VII. or any other function of the time. For instance, let the quantity of an electric current be C at the time ^, and C-{-AO at the time t + A^. Then the average rate of change of cur- AC rent in the interval A^ is ; this is the average increase in current-units per second. And the actual rate of change at the instant denoted by t is lim AO^dO At = /^t 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-\-l. Since, by Art. 14, dy _dy ^ dx dx dt dt hence -^ measures the ratio of the rates of change of y and ax of X, It follows that the result of differentiating y=f(?d (1) may be written in either of the forms !=/'(-), (2) 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 hQ x and y. Then — measures the rate of change of the a;-coordinate. Since velocity has been defined as the rate at which a point 64.] RATES AND DIFFERENTIALS 107 is moving, the rate — may be called the velocity which the Cit point P has in the direction of the a;-axis, or, more briefly, the rc-component of the velocity of P. It was shown on p. 105 that the actual velocity at any instant t is equal to the space that would be passed over in a unit of time, provided the velocity were uniform during that unit. Accordingly, the a;-component of velocity — - at may be represented by the distance FA (Fig. 22) which P would pass over in the direction of the a;-axis during a unit of time if the velocity remained uniform. Similarly -^ is the y-component of the velocity of P, and may be represented by the distance PB. ds The velocity — of P along the curve can be represented civ by the distance P(7, measured on the tangent line to the curve at P. It is evident that PC is the diagonal of the rec- tangle PA, PB. Since PC^ = PA^ + P&, it follows that m-m-m- <•> rdS. dt dx dt Fig. 22. Ex. 1. If a point describe the straight line 3 x + 4 ?/ = 5, and if x increase h units per second, find the rates of increase of y and of s. 2^ = 1 -far, dy _ S dx dt ~ 4:dt' dx dt it follows that ^ = -ih, ^ dt ^ dt Since hence When 108 DIFFERENTIAL CALCULUS [Ch. VII. Ex. 2. A point describes the parabola y"^ = 12 x in such a way that when a; = 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^=12x, then 2y^=12^, dt dt dy_^dx_ 6 dx, dt y dt Vl2 X dt ' hence, when x = = 3, and — =2, it follows that ^y dt dt ±2. ^^^'- {%'- {llT^ (I) ' ^-- I = ^^ '-' rer second. Ex. 3. A person is walking toward 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 the foot of the tower at time t, and y the. distance from the top at the same time. Then x^ + 60-2 = y\ and x^ = y^- dt ^ dt When a: is 80 feet, y is 100 feet ; hence if — is 5 miles per hour, -^ • . .. . ^^ dt ^ dt IS 4 miles per hour. 65. 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 dy, for the rates dt dt of change of the variables x and y. Thus the result of differentiating v ^^ n , /1^ /=/(a^) (1) may be written in either of the forms dy __ dx f'(x), (2) |=/'(.)|, (3) dy=f'ix-)dx. (4) 64-65.] BATES AND DIFFERENTIALS 109 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 dv, dx; for the derivative -^ has been defined as dx the result of performing upon «/ an indicated operation rep- resented by the symbol — ; and thus the di/ and dx of the 7 (XX symbol -^ have been given no separate meaning. The di/ dx and dx of equation (4) stand for the rates, or time deriva- tives, -^ and — occurring in (3), while the latter equation at ctt is itself obtained from (1) by differentiation with regard to t, by Art. 14. In case the dependence of ^ upon x be not indicated by a functional operation /, equations (3), (4) take the form dy _dy dx dt dx dt dy = -^ dx. dx In the abbreviated notation, equation (4) of the last article is written ds^ = dx^ + dy'^. Ex. 1. A point describing the parabola y^ — lpx is moving at the 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 f, ydy = pdx. But dx, dy also satisfy the relation rfa;2 + dy'^ = u^ ; hence, by solving these simultaneous equations, dx = — ^ V, dy — ^ y, in feet per second. 110 DIFFERENTIAL CALCULUS [Ch. VII. Ex. 2. A vertical wheel of radius 10 ft. is making 5 revolutions per second about a fixed axis. Find the horizontal and vertical velocities of a point oil the circumference situated 30° from the horizontaL Since a: = 10 cos 6, y = lO sin ^, then dx = -10 sin Odd, dy = 10 cos OdO. But f?^ = 10 TT = 31.416 radians per second, hence dx = — 314.16 sin ^ = — 157.08 feet per second, and dy = 314.16 cos $ = 272.06 feet per second. Ex. 3. Trace the changes in the horizontal and vertical velocity in a complete revolution. 66. Differentials often substituted for rates. The symbols dx, dy have been defined above as the rates of change of x and y per second. Sometimes, however, they may conveniently be allowed to stand for any two numbers, large or small, that are pro- portional to these rates; the equations, being homogeneous in them, will not be affected. It is usual in such cases to speak of the numbers dx and dy by the more general name of differentials; they may then be either the rates them- selves, or any two numbers in the same ratio. This will be especially convenient in problems in which the time variable is not explicitly mentioned. Ex. 1. When x increases from 45° to 45° 15', find the increase of logjo sin x, assuming that the ratio of the rates of change of the function and the variable remains sensibly constant throughout the short interval. Here dy = logjo^ . cot xdx = .4343 cot xdx = .4343 dx. Let dx = 15' = .004363 radians. Then dy = .001895, which is the approximate increment of log,o sin x. But log,o sin 45° = - J log 2 = - .150516, therefore log^o sin 46° 15' = - .148620. 65-66.] BATES AND DIFFERENTIALS 111 Ex. 2. Expanding logj^ sin {x + h) as far as A^ by Taylor's theorem, and then putting x = .785398, h = .004363, show what is the error made by neglecting the thii'd term, as was done in Ex. 1. Ex. 3. When x varies from 60° to 60° 10', find the increase in sin x. Ex. 4. Show that log^QX increases more slowly than ar, when x > logj^e, that is, X > .4343. Ex. 5. Two sides a, 6 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^ = a^ + b^ ~2ab cos C, regarding a, b as constant-! / Ex. 6. A vessel is sailing northwest at the rate of 10 miles per hour. At what rate is she making north latitude ? ^ Ex. 7. In the parabola y^ = 12 x, find the point at which the ordinate and abscissa are increasing equally. Ex. 8. At what part of the first quadrant does the angle increase twice as fast as its sine? /^ Ex. 9. Find the rate of change in the area of a square when the side b is increasing at a ft. per second. r Ex. 10. In the function y = 2 x^ -\- Q, what is the value of x at the point where y increases 24 times as fast as x ? l/ 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^ — 5 x^ -{■ 17 x and x^ — S X change at the same rate? / Ex. 13. Find the points at which the rate of change of the ordinate 7/ = x^ — 6 a:2 + 3 ar + 5 is equal to the rate of change of the slope of the tangent to the curve. jI Ex. 14. The relation between s, the space through which a body falls, and t, the time of falling, is s-16t^; show that the velocity is equal to 32 t. The rate of change of velocity is called acceleration ; show that the acceleration of the falling body is a constant. 112 DIFFERENTIAL CALCULUS [Ch. VII. 66. . Ex. 15. A body moves according to the law s = cos (nt + e). Show that its acceleration is proportional to the space through which it has moved. Ex. 16. If a body be projected upwards in a vacuum with an initial velocity Vq, to what height will it rise, and what will be the time of ascent ? ^ Ex. 17. A body is projected upwards with a velocity of a feet per second. After what time will it return ? v/ Ex. 18. If A be the area of a circle of radius x, show that the circum- dA ference is Interpret this fact geometrically. dx ^ Ex. 19. A point describing the circle x^ + y^ = 25 passes through (3, 4) with a velocity of 20 feet per second. Find its compon^it veloci- ties parallel to the axes. CHAPTER VIII DIFFERENTIATION OF FUNCTIONS OF TWO VARIABLES Thus far only functions of a single variable have been considered. The present chapter will be devoted to the study of functions of two independent variables x^ y. They will be represented by the symbol If the simultaneous values of the three variables a:, y, z be represented as the rectangular coordinates of a point in space, the locus of all such points is a surface having the equation 67. Definition of continuity. A function z oi x and ^, 2 = /(ic, ?/), is said to be continuous in the vicinity of any point (a, 5) when /(a, 5) is real, finite, and determinate, and such that however A and h approach zero. When a pair of values a, h exists at which any one of these properties does not hold, the function is said to be discon- tinuous at the point (a, 6). £.a., let 2 = ^^tl. x-y When a: = 0, then z = — 1 for every value of y ; when y = then a = + 1 for every value of x. In general, if y = wa:, 1 — m and z may be made to have any value whatever at (0, 0) by giving an appropriate value to m. 113 114 DIFFERENTIAL CALCULUS [Ch. VIII. 68. Partial differentiation. If in the function a fixed value y-^ be given to ?/, then is a function of x only, and the rate of change in z caused by a change in x is expressed by dz = ^dx, (1) dx in which — is obtained on the supposition that y is constant. dx To indicate this fact without the qualifying verbal state- ment, equation (1) will be written in the form d^ = ^Ux. (2) dx The symbol — represents the result obtained by differ- entiating z with regard to x^ the variable y being treated as a constant; it is called the partial derivative of z with regard to x. From the definition of differentiation. Art. 11, the partial derivative is the result of the indicated operation 50^ lim f(x + Aa:, .y) —f(x, y) . dx ^^ = ^ Aa: Similarly, the symbol — represents the result obtained by differentiating z with regard to y, the variable x being treated as a constant ; it is called the partial derivative of z with regard to y. The partial derivative of z with regard to y is accordingly the result of the indicated operation 68.] FUNCTIONS OF TWO VARIABLES 115 Bz_^ lim fix,y-\-Ai/)-f(x,^} dgZ = — dx is called the partial x-differential of 2, and ox dz dyZ = —-dy\^ called the partial y-differential of z. if Geometrically, the two equations define the curve of section of the surface z=f(x^y) made by the plane y — y^ The derivative — defines the slope of the tangent line to this curve. Similarly, when rr has a given constant value, x = x^^ the partial derivative — is the expression for the slope of the dy tangent to the curve cut from the surface z^fQc^y') by the plane x = x-^. The equations of these two tangent lines at the point (p^v Vv ^1) are y = y^, z-z^ = ^(x-x{), dz x = x^, z-z^=-A.(y-y{), and hence the equation of the plane containing these two intersecting lines is The plane is called the tangent plane to the surface ^ =f(p^^ y) at the point {xy, y^ z^. 116 DIFFERENTIAL CALCULUS [Ch. VIII. EXERCISES 1. Given u = x^ + Z ^V - 7 xy\ prove that a:^ + y ^ = 4 1*. dx dy 2. Given u = tan-i ^, show that a: ^ + j/ ^ = 0. a; dx ^ dy 3. M = log (e- + eO ; find ^ + ^. ^ 4. M = sinx3/; find ^ + ^. aa; dy ' 5. M = log (x + Va:2 + ?/2) ; find a: ^ + y ^. aa: ay 6. u = log (tan x + tan y + tan z) ; show that sin 2a:^ + sin 2^/^ + sin 23 ^ = 2. dx dy dz 7. M = log(a: + w); show that ^ + ^ = -• ^ ^^ ^^ dx dy e- 69. Total differential. If both x and y be allowed to vary in the function z — f{x^ y), the first question that naturally arises is to determine the meaning of the differential of 2. Let 2i =/(a^r Vx)-' and Zj + A2 = /(ajj + Aa:, i/j + A!/) be two values of the function corresponding to the two pairs of values of the variables Xy, y^ and x-^ + Aa:, y^ + Az/. The difference Az =/(a:i 4- Aa;, y^ + ^y)-f(x^, y{) may be regarded as composed of two parts, the first part beUig the increment which z takes when x changes from x^ to ajj + Aa:, while y remains constant (y = yj), and the sec- ond part being the additional increment which z takes when 68-69.] FUNCTIONS OF TWO VARIABLES 117 t/ changes from i/j to ^^ + Ay, while x remains constant (^x = x^ + Ax~). The increment Az may then be written Az =f(x^ + Ax, y^ + A^^) -f(x^ + Ax, y{) +/(^i + Aa:, yi)-fCx^, yO ^ /(^i + Ax, y^ + Ay) -/(a^i + Ax, y^ Ay ^ /(^l + Ax, y^-f(^x^, y^-) ^^ Ax From the theorem of mean value, Art. 45, the last equation may be written Az = — /(a^i + OAx, y{)Ax + —fC^i + Aa;, y^ + O^Ay^Ay. dx ay It represents the actual increment Az which the dependent variable z takes when the independent variables x and y take the increments Ax and Ay. In the preceding equation let Ax, Ay, Az be replaced by € • dx, € • dy, € ' dz respectively, in which dx, dy are entirely arbitrary. After removing the common factor €, let € approach zero. The result is d^ = ^f(^dx + ^^^^d2,. (1) The differential dz defined by this equation is called the total differential of z. It is not an actual increment of z, but the increment which z would take if the change con- tinued uniform while x changed from x-^^ to x-^ + dx and y changed from y^ to y^ + dy. Geometrically speaking this is the increment which z would take if the point (x, y, 2) should move from the position (x^, y^ ^i) in the tangent plane of the surface z = fix, y) instead of on the surface itself. 118 DIFFERENTIAL CALCULUS [Ch. VIII. Equation (1) may be written in the form dz = — dx -\ dy, dx dy ^ from which the following theorem can be stated : the total differential of a function of two variables is equal to the sum of the partial differentials. The same method can be directly applied to functions of three or more variables. Thus, if 2 be a function of the variables a;, ^, w, z = (z). Then dy_^dydz^^Mt(^^^dz^ dx dz dx dx dx^ dx (^'^43 126 DIFFERENTIAL CALCULUS [Ch. IX. g = r(.)(|)V^'(.)S. (4) The higher a;-derivatives of y can be similarly expressed in terms of a:-derivatives of z. 76. Change of the independent variable. Let t^ be a function of x^ and let both x and y be functions of a new variable t. It is required to express -^ in terms of -^, TO 1 -n and -T~ in terms of -^ and -=4* do!^ dt dt^ By Art. 14, dy__ dt dx ~ dx^ dt d^y dx cPy dt^ dt dh;d]i dt^ dt dx^ ~ fd x\^ (1) hence u^ u^^_u^ — u^i_u^^ ^2) 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 a: to 2 in the equation A. ^»+^l + 2' = ''' ''^"" ' = *•• dy dz dy dx~ dz dH. Hence x2^, + z^ + y = becomes ^i + y = 0. 7r>-76.] CHANGE OF VARIABLE 127 * 2. Interchange the function and the variable in the equation / 3. Interchange x and y in the equation R ' 4. Change the independent variable from a: to ^ in the equation Jd^y _dl fu _d^fdyy^ ^^ \dx^/ dx dx^ dx'^XdxJ *^ 5. Change the dependent variable from y to 2 in the equation d^y , 2n+y)fdyy ^ / 6. Change the independent variable from x to y in the equation x^ \- X 1- M = 0, when y = log x. dx^ dx 7. If y is a function of x, and x a function of the time t, express the ^-acceleration in terms of the a:-acceleration, and the x-velocity. Since dy^dy^Ix^ dt dx dt hence d^ ^d_yd^^lx , ^dJJ\ dt^ dx dt^ dt dAdxJ But ^(^Ie\ = jL(^]^I^ = ^^j dt\dxJ dxKdxl dt dx'^ dt hence d^^dj,d^ d^nix\\ dt^ dx dt^ dxAdt) In the abbreviated notation for ^derivatives, «4. 8. Change the independent variable from x to u in the equation d'^y 2 X dy y ^1 -t4 + ^ 5 -r- + ,, 0x0 = 0, when x = tan u. dx^ 1 -\- x2 t/a; (1 + x^y 128 DIFFERENTIAL CALCULUS [Ch. IX. 76. ^ 9. Change the independent variable from x to f in the equation ^^-'^'^S-'^i^^' ""^^^ :r = cos^ 10. Show that the equation remains unchanged in form by the substitution 2: = -• r 11. Interchange the variable and the function in the equation dx^ \dxi y\dxi " ^ 12. Change the dependent variable from y to 2 in the equation APPLICATIONS TO GEOMETRY CHAPTER X TANGENTS AND NORMALS 77. It was shown in Art. 10 that if -/(a;, «/) = be the dy equation of a plane curve, then -~ measures the slope of the tangent to the curve at the point x, y. The slope at a partic- ular point (a^j, ^j) will be denoted by -~=^ meaning that x^ is to be substituted for x^ and y^ for y in the expression for -^. (XX 78. Equation of tangent and normal at a given point. Since the tangent line passes through the given point (a^j, y-[) and has the slope -^, its equation is The normal to the curve at the point (x^^ y{) is the straight line through this point, perpendicular to the tangent. Since the slope of the normal is :zl=_^, [Art. 74, dy dy dx (2) its equation is dx-, r ^ y y,= ^(^ x,y i.e.. <^^-^i)+ii^2'-2'>>= 129 130 DIFFERENTIAL CALCULUS [Ch. X. 79. Length of tangent, normal, subtangent, subnormal. The segments of the tangent and normal intercepted be- tween the point of tangency and the axis OX are called, respectively, the tangent length and the normal length, and their projections on OX are called the subtangent and the subnormal. Fig. 23 a. Fig. 23 h. Thus, in Fig. 23, let the tangent and normal to the curve P(7 at P meet the axis OX in T and N, and let MP be the ordinate of P. Then TP is the tangent length, PN the normal length, TM the subtangent, JOT the subnormal. These will be denoted, respectively, by t, w, t, v. Let the angle XTP be denoted by <^, and write tan<^=m. 1 Then cos<^ = vr+ ; sin<^ = m m" VI + m'^ t = -^ = ^ — ; w = -^=yiVl H-TTi^; sin 9 m cos 9 T = j,,cot^ = y,^^J=^; r = y, tan <^=«/,^= my,. 79.] TANGENTS AND NORMALS 131 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 positive when the normal cuts OX to the right of the foot of the ordinate. Both are therefore positive or negative, according as (f) is acute or obtuse. The expressions for t, v may also be obtained by finding from equations (1), (2), Art. 78, the intercepts made by the tangent and normal on the axis OX. The intercept of the tangent subtracted from x-^^ gives r, and x-^^ subtracted from the intercept of the normal gives v. Ex. Find the intercepts made upon the axes by the tangent at the point (x^, ?/j) on the curve y/x + Vy = -\/a, and show that their sum is constant. Differentiating the equation of the curve, y/x y/y ^^ Hence the equation of the tangent is The X intercept is x^ + V^^p and the y intercept is y^ + Vx^~y[, hence their sum is If a series of lines be drawn such that the snm of the intercepts of each is the same constant, account being taken of the signs, the form of the parabola to which they are all tangent can be readily seen. EXERCISES 1. Find the equations of the tangent and the normal to the ellipse ^ + ^ = 1 at the point (xp y^) . Compare the process with that employ*^.d in analytic geonietry to obtain the same results. 132 DIFFERENTIAL CALCULUS [Ch. X. 2. Find the equation of the tangent to the curve x^{x-\-y)=a\x—y) at the origin. / 3. Find the equations of the tangent and normal at the point (1, 3) on the curve y^ = 9 x^. y 4. Find the equations of the tangent and normal to each of the following curves at the point indicated: (a) V = ) at the point for which a: = 2 a. (^) y2 _ 2 a:2 — z*, at the points for which x = 1. (y) ?/2 _ ^py.^ at the point (/>, 2p). 5. Find the value of the subtangent of y'^ = ^x'^—12 at a; = 4. Compare the process with that already given in analytic geometry. SI 6. Find the length of the tangent to the curve ^y^ — 2 a: at a; = 8. ~^ 7. Find the points at which the tangent is parallel to the axis of a:, and at which it is perpendicular to the x axis for each of the following curves : (a) ax^-\-2hxy+hy'^ = l. 08) y = ax (y) y^ = x\2a-x). -w 8. Find the condition that the conies ax'^ + hy^=l, a'x^ + l/y^= 1 shall cut at right angles. T 9. Find the angle at which x^ = y^ + 5 intersects Sx^-\-lSy^= 144. Compare with Ex. 8. 10. Show that in the equilateral hyperbola 2 xy = a^ the area of the triangle formed by a variable tangent and the coordinate axes is constant and equal to a^. 11. At what angle does y^ = Sx intersect 4 a:* + 2 y^ = 43 ? 12. Determine the subnormal to the curve y» = a*-^ x. 13. Find the values of x for which the tangent to the curve ?/8 = (x-a)2(x-c) is parallel to the axis of x. 14. Show that the subtangent of the hyperbola xy = a^ is equal to the abscissa of the point of tangency, but opposite in sign. / 15. Prove that the parabola y* = 4 aa: has a constant subnormal. 79-80.] TANGENTS AND NORMALS 133 16. Show analytically that in the curve x^ + y^ = a^ the length of the normal is constant. 17. Show that in the tractrix, the length of the tangent is constant, the equation of the tractrix being 2 ^e + V^^37 18. Show that the exponential curve y = ae" has a constant sub- tangent. 19. Find the point on the parabola y^ = 4:px at \v^hich the angle between the tangent and the line joining the point to the vertex shall be a maximum. POLAR COORDINATES 80. 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 make use of the angle between the tangent and the radius vector to the point of tangency. Let P, Q be two points on the curve (Fig. 24). Join P, Q with the pole 0, and drop a perpendic- ular PM from P on OQ. Let /?, be the coordinates of P; p+Ap^ 6+AO those of Q. Then the angle P0$ = A(9; Pil[/ = /3sinA6>; and MQ=OQ-OM=p + Ap-p cos Ad. Fig. 24. Hence tan MQP = p sin A^ p + Ap — p cos A^ 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 yjr. DIFFERENTIAL CALCULUS lira 134 Thus But p(l - cos Ae)=2p sin2 1 A^, [Ch. X. t^^r-A^-Op + Ap_pcosA^ hence ^^^i^ = Jlo p sin A^ A(9 . 1 ./J siniA^ , Ap Since . l^^ ^ ^^" ^ = i^ the preceding equation reduces to ^ , p dO tant = ^=P^-. dd Ex. 1. A point describes a circle of radius p. Prove that at any instant the arc velocity is p times the angle velocity, (It P dt dt Fia.2t). Fio. 27. Ex. 2. When a point describes a given curve, prove that at any instant the velocity ^ — has a radius component -^ and a com- ''* dt ^ dt ponent perpendicular to the radius vector p — , and hence that dt C08i^ = ^, 8mtf/ = p^, tan.Zr = p^. ds ds dp 81. Relation between ^ and ^• dx dp If the initial line be taken as the axis of X, the tangent line at P makes an angle (f) with this line. Hence 6 + yjr = ^; 80-82.] TANGENTS AND NOBMALS 135 82. 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, are called the polar tangent length and the polar normal length; their projections on this perpendicular are called the polar subtangent 'dudi polar subnormal. Fig. 28 a. Fig. 28 6. Thus, let the tangent and normal at P meet the perpen • dicular to OP in the points JV and M. Then PN is the polar tangent length, PM is the polar normal length, ON is the polar subtangent, OMi^ the polar subnormal. They are all seen to be independent of the direction of the initial line. The lengths of these lines will now be determined. Since PN= OP . sec OPN= psec^jr^ pyjp^f^J + 1 -'f.M%i dp hence polar tangent length = p -^\P^ + ( -^ dO 136 DIFFERENTIAL CALCULUS [Ch. X. Again, 0N= OP tan OPN= /o tan -f = p2 ^, dp hence polar subtangent = /o^ -—• dp PM= OP ' CSC (9PiV^= pGSGylr= \p^ + f^\ hence polar normal length = \p^ "*■ V;^) * 0M= OP cot OP]Sr= ^, hence polar subnormal = -^* The signs of the polar tangent length and polar normal length are ambiguous on account of the radical. The direc- tion of the subtangent is determined by the sign of p^ — . dS ^P When -- is positive, the distance ON^ should be measured dp 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. 13), and ylr is an acute angle (as dp ^Q in Fig. 28 h) ; when decreases as p increases, — is negative, and i/r is obtuse (Fig. 28 a). ^ EXERCISES 1. In the curve p = a sin ^, find \p. 2. In the spiral of Archimedes p = a$f show that tan \^ = and find the polar subtangent, polar normal, and polar subnormal. Trace the curve. 3. Find for the curve p^ = a*co8 2d the values of all the expressions treated in this article. 4. Show that in the curve pO = a the polar subtangent is of constant length. Trace the curve. 82.] TANGENTS AND NORMALS 187 5. In the curve p=a(l — cosO), find i/r and the polar subtangent. 6. Show that in the curve p = b • e^cota the tangent makes a constant angle a with the radius vector. For this reason, this curve is called the equiangular spiral. yC 7. Find the angle of intersection of the curves p = a(l + cos 6), p = 6(1 — cos^. 8. In the parabola p = a sec^ -, show that ^ + ^ = w. CHAPTER XI DERIVATIVE OF AN ARC, AREA, VOLUME, AND SURFACE OF REVOLUTION 83. Derivative of an arc. The length s of the arc AP of a given curve 1/ =/(a:), 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 s = (K^^)- The determination of the function (f> 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 -— = '(x)^ ax which is easily done by the methods of the Differential Calculus. Let PQ \)Q two points on the curve (Fig. 29); let x^ y be the coordinates of P ; a: + Aa:, y -\- ^y those oi Q\ s the length of the arc AP ; s + A« that of the arc AQ. Draw the ordinates MP, NQ ; and draw PR parallel — 2C_ to MN. Then PR= Ax, RQ= Ay; M N J.IQ29. arcP^=As. Hence Chord PQ = V(Aa:)2 + (Ay)2, Ax T..„,.„=.^.^.|iV..(g)' As _ As Ax PQ Ax PQ 138 Ch. XI. 83-84.] DERIVATIVES OF ARC, AREA, ETC. 139 Taking the limit of both members as Ax approaches zero and putting ^^"Iq-^^^ 1, by Art. 6, Th. 4, and Art. 4, Th. 8, Cor., it follows that %'Mf)'- Similarly Moreover, from Art. 65 or in the differential notation dx^ fdy^ dt \dt )'■ (1) (2) (3) (4) 84. Trigonometric meaning of ds ds dx dy Since PQ As PQ As it follows by taking the limit that dx , -— = cos 9, wherein . ds - - r ^ - ^^ Using the idea of a rate or dif- ferential, all these relations may be conveniently exhibited by Fig. 30. These results may also be de- rived from equations (1), (2) of Art. 83, by putting ^ = tan . dy Y dy y. I- ^1 X Fia. 30. 140 DIFFERENTIA L CALCUL US [Ch. XI. 85. Derivative of the volume of a solid of revolution. Let the curve APQ revolve about the a:-axis, and thus generate a surface of revolution ; let V be the volume included between this surface, the plane generated by the fixed ordinate at A, and the plane generated by any ordi- nate MP. Let A I^ be the volume gener- ated by the area PMNQ. Then A V lies between the volumes of the cylinders generated by the rectangles PMNR and SMNQ; that is, iry'^a^x < A r< 7r(y -h Ay)2Aa;. Dividing by Aa; and taking limits. F • ^ 1 -JL^ F r R X A { I V Fia. 30 a. dV — m-ll^ 86. Derivative of a surface of revolution. Let S be the area of the surface generated by the arc AP (Fig. 31), and AS that generated 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- erated by PQ' and P'Q\ i.e., 2 iryAs < AaS' < 2 irQy + Ay) A«. Dividing by A« and passing to the limit, dS M N Fio. 31. !;-'-'>• f-f-l— V^^ (1) (2) 85-87.] DERIVATIVES OF ARC, AREA, ETC. 141 87. Derivative of arc in polar coordinates. Let /3, 6 be the coordinates of P ; p -\- A/a, 6 + A^ those oi Q ; s the length of the arc KP ; As that of arc PQ ; draw Pilif per- pendicular to OQ. Then PM=p sin AO, MQ=OQ-OM==p^Ap-p cos Ae =p(l — GosA6) + Ap o^ = 2/)sin2iA<9-hA/). Hence PQ^=Cp sin Al9)2 -f (2 /o sin2 J A(9 + A/o)2, Replacing the first member by ( — -^ * t4 ) ' passing to the \ As Ad J limit when A^ = 0, and putting lim — ^ = 1, lim ^^^ - = 1, 1 A/? ^^ ^^ lim^i^4^= 1, it follows that ^Ad Fig. 32. ©■='■- gj *•'•' d0 Mtl In the rate or differential notation this formula may be conveniently written d»^ = dp^ + p^dff^. This relation may be readily deduced also from Fig. 26, Art. 80. 142 DIFFERENTIAL CALCULUS [Ch. XI. 88. Derivative of area in polar coordinates. Let A be the area of OKP measured from a fixed radius vector OK to any other radius vector OP ; ^x let A A be the area of OPQ. Draw arcs PM, QN, with as a center. Then the area POQ . lies between the areas of the sectors OPiltf and ONQ\ i.e.. Fig. 33. 1 /32A(9 < A^ < lip + Ap)2 A^. Dividing by A^ and passing to the limit, when A^ = 0, it follows that dA dd = \p'- For the derivative of the area of a curve in rectangular dA coordinates, see Art. 10. The result is —— = y. dx EXERCISES ON CHAPTER XI 1. In the parabola w^ = 4 ax, find — , - — , -— , — — dx dx dx dx 2. Find — and — for the circle x^-{-y^ = a\ dx dy ^ ds 3. Find — for the curve e^ cos x = \. dx 4. Find the x-derivative of the volume of the cone generated by revolving the line y — ax about the axis of x. 5. Find the ar-derivative of the volume of the ellipsoid of revolution, X^ 2/2 formed by revolving -^+ rj = 1 about its maior axis. ^ 6. In the curve p = a^ find -^- ^(ffi " ^ Im^^ de ds 7. Given p = a(H-co8^); find ^. d$ 8. In p« = a3 cos 2d, find d$ c^ CHAPTER XII ASYMPTOTES 89. Hyperbolic and parabolic branches. 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. On the other hand, the successive tangents may move farther and farther out of the field as in the case of 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. 90. 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 farther along any infinite branch of the given curve, then the fixed line is called an asymptote of the curve. 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 91. Method of limiting intercepts. The equation of the tangent at any point (x^, y{) being 143 144 DIFFERENTIAL CALCULUS [Ch. XII. the intercepts made by this line on the coordinate axes are 0) Suppose the curve has a branch on which x==oo and y = Qo. Then from (1) the limits can be found to which the intercepts rr^, i/q approach as the coordinates x^^ y-^ of the point of contact tend to become infinite. If these limits be denoted by a, 5, the equation of the corresponding asymptote is a 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, which 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. 92. Method of inspection. Infinite ordinates, asymptotes parallel to axes. When an algebraic equation in two co- ordinates X and y is rationalized, cleared of fractions, and arranged according to powers of one of the coordinates, say y, it takes the form ayn + (hx 4- c)5^"-^-f (c?r2 -f ex +f^y^-^+ ... + u„_,y 4- w„ = 0, in which w„ is a polynomial of the degree n in terms of the other coordinate a;, and w„_i is of degree n — 1. When any value is given to a:, the equation determines n values for y. Let it be required to find for what value of x the corre- sponding ordinate y has an infinite value. 91-92.] ASYMPTOTES 145 For this purpose the following theorem from algebra will be recalled : Given an algebraic equation of degree n a^» + /3?/"-^ + 7«/""' + - = 0. If a = 0, one root i/ becomes infinite ; if a = and /3 = 0, two roots 1/ become infinite ; and in general if the coefficients of each of the k highest powers of t/ vanish, the equation will have k infinite roots. Suppose at first that the term in y" 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 accordingly no infinite ordinates for finite values of the abscissa. Next suppose that a is zero, and 6, c, not zero. In this case one value of «/ is infinite for every finite value of x^ and hence the curve passes through the point at infinity on the ^ axis. There is one particular value of x, namely, x = ——, for which an additional root of the equation in t/ becomes infinite. For, when x has this value, the coefficient hx + o oi the highest power of ^ remaining in the equation vanishes. Geometrically, every line parallel to the i/ axis has one point of intersection with the curve at infinity, but the line bx + c = has two points of intersection with the i curve at infinity. A line having two coincident points of / intersection with a curve is a tangent to the curve, and ^ when the coincident points are at infinity, but the line Ij ^ itself not altogether at infinity, the tangent is an asymp- a tote. Hence an ordinate that becomes infinite for a defi- \ J" nite value of x is an asymptote. ( Again, if not only a, but also h and c are zero, there are 146 DIFFERENTIAL CALCULUS [Ch. XII. two values of x that make «/ infinite ; namely, those values of X that make dx^-{-ex-\-f = 0^ and the equations of the infinite ordinates are found by factoring this last equation ; and so on. Similarly, b}^ arranging the equation of the curve accord- ing to powers of a;, it is easy to find what values of 1/ give an infinite value to x. Ex. 1. In the curve 2 x^ + x^y + xy^ z= x^ - y^ - 5, md 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^ is zero. Arranged in powers of y it is f (x+1) + yx^ + (2 a:8 - a:2 + 5) = 0. I the equation for y becomes 0-1/2 + 2, +2 = 0, the two roots of which are y = 00, y = — 2 ; hence the equation of the infinite ordinate is x + JL=-0. — Xhe infinite ordinate meets the curve again in the finite poin^( — 1^ — 2 J ~^n5e~the~T«rm m^r'Tf^^TBSBtl^ there are no infinite values of x for finite values of y. Ex. 2. Show that the lines x = a, and y = are asymptotes to the curve a^x = y(x- ay (Fig. 34). ¥iQ. 34. Ex. 3. Find the asymptotes of the curve x"^ (y - a) + xy^ = o* 92-93.] ASYMPTOTES 147 93 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 /(^,^) = 0. (1) Consider the straight line y =mz + b, (2) and let it be required to determine m and b so that this line shall be an asymptote to the curve f{x, ?/) = 0. Since an asymptote is the limiting position of a line that meets the curve in two points that tend to coincide at infinity, then, by making (1) and (2) simultaneous, the resulting equation in x, fix, ma: + 5) = 0, is to have two of its roots infinite. This requires that the coefificients of the two highest powers of x shall vanish. These coefficients, equated to zero, furnish two equations, from which the required values of m and h can be deter- mined. These values, substituted in (2), will give the equation of an asymptote. Ex. 4. Find the asymptotes to the curve y^ = x^{2a ~ 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-{- by = x^(2a - x), or, arranged in powers of x, (1 + w8) x^ + (3 m% - 2 a) x2 + 3 b^x + &» = 0. Let m8 + 1 = and dm^b- 2a = 0. 148 DIFFERENTIAL CALCULUS [Ch. XII. Then hence -a: + 3 ' 2a 3 is the equation of an asymptote. The third intersection of this line with the given cubic is found from the equation 3 mb^x + &^ = 0, whence x = — • Y This is the only oblique asymptote, as the other roots of the equation for m are imaginary. Ex. 5. Find the asymptotes to the curve y (a* + x^) = a^(a — x). Fio. 3«. Here the line y = is a horizontal asymptote by Art. 92. To find the oblique asymptotes, put y = mx ■\- h. 93-94.] ASYMPTOTES 149 Then (mx + b) (a^ + x^) = a^ (a - x), i.e., mx^ + bx^ + (ma^ +a^)x + {aPh - a^) = ; hence m = 0, & = 0, for an asymptote. Thus the only asymptote is the line y = already found. 94. Number of asymptotes. The illustrations of the last article show that if all the terms be present in the general equation of an nth. degree curve, then the equation for determining m is of the nth degree and there are accord- ingly n values of m^ real or imaginary. The equation for finding h is usually of the first degree, but for certain curves one or more values of m may cause the coefficient of a^ and a;""^ both to vanish, irrespective of h. In such cases any line whose equation is of the form y = m^x + c will have two points at infinity on the curve independent of c; .but by equating the coefficient of a;""^ to zero, two values of h 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 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. This number must be reduced whenever a curve has a parabolic branch. Since the imaginary values of m occur in pairs, it is evi- dent that a curve of odd degree has an odd number of real asymptotes ; 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. 150 DIFFERENTIAL CALCULUS [Ch. XII. 95. 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 principle that a straight line will be an asymp- tote to a curve when the difference between the ordinates of the curve and of the line, corresponding to a common abscissa, approaches zero as the abscissa becomes infinite. 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. The principal value of the method of expansion is that it exhibits the manner in which each infinite branch ap- proaches its asymptote. Ex. Find the asymptotes of the curve a:— 3 TT n \ X/\ Xl Here y^ = :r^^ , Hence the oblique asymptotes are y = ±{x — 1) (Fig. 37). The sign of the next term shows that when z = + x), tlie curve is above the first asymptote and below the second; and vice versa when a; == — oo. 95.] ASYMPTOTES 151 The same method may be applied to cases in which x is an explicit function of y. This method can also be extended so as to apply to curves defined by an implicit equation, f(x, y) = 0. [See McMahon and Snyder's " Differ- ential Calculus," p. 234.] Fig. 37. EXERCISES ON CHAPTER XII Find the asymptotes of each of the following curves : 1. y(a^ - a;2) = b(2 x -\- c). ■' 7. (x + a)f = (y + h)x\ 8. x' x^+ X + y. \f 2 2^ q^(a:-a)(a:-3a) . ■ ^ x'^-2ax 3. a:Y ^ a\x'^ - y^). ^ 4. y = a+ . (X - C)2 / 5. / = x%a - x). ^ 6. y\x-l) = x^. 15. x^ + 2x^y - xy^-2y^ + 4:y^ + 2xy + y=l. 9. xy^ + x^y = a^. 10. 2/(^2 + 3 a2) = a;8. A 11. a:3 - 3 aar?/ + ^8 _ q. 12. x3 + ?/8 = aS. 13. x4 - a; V + a'^^^ + 6* = 0. CHAPTER XIII DIRECTION OF BENDING. POINTS OF INFLEXION 96. Concavity upward and downward. A curve is said to be concave downward in the vicinity of a point P when, for a finite distance on each side of P, the curve is situated below the tangent drawn at that point, as in the arcs -42), FH. It is concave upward when the curve lies above the tangent, as in the arcs DF^ HK. By drawing successive tangents to the curve, as in the figure, it is easily seen 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 downward concavity, negative bending. A point at which the direction of bending changes con- tinuously from positive to negative, or vice versa, as at F oi 102 Ch. XIII. 96-97.] DIRECTION OF BENDING 163 at D, is called a point 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 i>, #, H. 97. Algebraic test for positive and negative bending. Let the inclination of the tangent line, measured from the right- hand end of the a;-axis toward the forward (right-hand) end of the tangent, be denoted by . Then is an increasing or decreasing function of the abscissa according as the bend- ing is positive or negative ; for instance, in the arc AD^ the angle diminishes from + — through zero to — — ; in the arc i>jP, increases from — — through zero to + — • 2i 4 At a point of inflexion 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. 38, <^ is a maximum at JP, and a minimum at D and at S, Instead of recording the variation of the angle , it is generally convenient to consider the variation of the slope tan<^, w^hich is easily expressed as a function of x by the equation tan - 4, ^ is negative. dy Hence, at the point (— 4, 2), ~ d^v and -r4 are infinite. When j;<— 4, dx^ dh, 156 DIFFERENTIAL CALCULUS [Ch. XIII. 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. Ex. 3. Consider the curve y=x*. dx dx^ Fig. 41. At (0, 0), ^ is zero, but the curve has no inflexion, for — ^ never dx^ changes sign (Fig. 41). 98. Analytical derivation of the test for the direction of bending. Let the equation of a curve be i/ =/(2;), and let P, ^j, ^j), be a point upon it. Then the equation of the tangent at P is Suppose that when x changes from x^ to x^ + ^, the ordi- nate of the tangent change from ^j to y', and that of the curve from 2/i to y'' ; then it is pro- posed to determine the sign of the difference of ordinates y — «/' corresponding to the same ab- scissa x^ + h. By Taylor's theorem, and from the above equation of the tangent, Hence / = yi + ¥'(^i) = /(^i)+ ¥'(^i)' and it follows that y"-y = f/"(^i)+-- Fig. 42. P 97-99.] DIRECTION OF BENDING 167 When h is made sufficiently small, /''(a:j)+ ••• will have the same sign as /'^(^i); but the factor h^ is always positive, hence when f(x-^ is positive, y" — 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 f"{x{) is negative, the concavity is downward. This agrees with the former result. 99. 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. 43 a. Fig. 43 6. First, let the curve be convex toward the a^axis, as in the left-hand figure. Then if y is positive, the bending is positive and —^■ is positive ; but if y is negative, the bending is neg- ative and — - is negative. Hence in either case the product y-zTT, is positive. Next, let the curve be concave toward the a:-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- ing is positive and -j^ is positive. Thus in either case the product y-^ is negative. Hence: 158 DIFFERENTIAL CALCULUS [Ch. XIII. 99. In the vicinity of a given point (x^ y) the curve is convex or concave to the x-axis, according as the product y — ^ is positive . . dor or negative. EXERCISES ON CHAPTER XIII 1. Examine the curve ?/ = 2 — 3(a; — 2)5 for points of inflexion. 2. Show that the curve a^y = x(cfi — a:^) has a point of inflexion at the origin. 3. Find the points of inflexion on the curve y = ; — • TO 4. In the curve ay = i!^, prove that the origin is a point of inflexion if m and n are positive odd integers. 5. Show that the curve y =csin - has an infinite number of points of inflexion lying on a straight line. 6. Show that the curve y{x^ ■\- 0."^) = x has three points of inflexion lying on a straight line ; find the equation of the line. 7. If 3^2 =f(x) be the equation of a curve, prove that the abscissas of its points of inflexion satisfy the equation [f'(x)y = 2f(x)^f"(xy 8. Draw the part of the curve a^y = ^- ax^ + 2 a* near its point of 3 inflexion, and find the equation of the stationary tangent. CHAPTER XIV CONTACT AND CURVATURE 100. Order of contact. The points of intersection of the two curves are found by making the two equations simultaneous ; that is, by finding those values of x for which Suppose ic = « is one value that satisfies this equation. Then the point x — a^ q^ = (j) (^a} = yjr (^a) is common to the curves. If, moreover, the two curves have the same tangent a^ 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 of -^ are thus the same for both curves at the point in question, which requires that <^ (a) = i/r (a), • If, in addition, the value of -t4 be the same for each curve at the point, then <^"(«) = f"(a), and the curves are said to have a contact of the second order with each other. If <^(a) = i/r(a), and all the derivatives up to the nth. order inclusive be equal to each other, the curves are said 159 160 DIFFERENTIAL CALCULUS [Ch. XIV. to have contact of the nth order. This is seen to require n-^l conditions. 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 = '>^(x)', in which "^^Qc) proceeds in powers of x with undetermined coefficients, then n-\-l of these coefficients could be determined by requiring the second curve to have contact of the nth. order with the given curve at a given point. 101. Number of conditions implied by contact. A straight line has two arbitrary constants, which can be determined by two conditions ; accordingly a straight line can be drawn which touches a given curve at any specified point. For if the equation of a line be written i/=mx-\-b, then hence, through any arbitrary point x = a on a given curve «/ = (^(ic), 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 = (a), m = <^'(a), which are just sufficient to determine m and b. 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 example, the additional condition for second-order contact is = "(a)^ which is satisfied when the point a: = a is a point of inflexion on the given curve y = (x)' Thus the tangent at a point of inflexion on a curve has contact of the second order with the curve. 100-102.] CONTACT AND CURVATURE 161 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 specified point. As in the case of the tangent line, special points may be found for which these curves have contact of higher order. 102. 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 = (f>(x)^ yz='\^(x) have contact of the nth order at the point whose abscissa is a ; and let ^j, ^2 be the ordinates of these curves at the point whose abscissa is a + A. Then and by Taylor's theorem 2,1 = .^(a) + f (a) . A +^!^ . ^2 +... *^-^"+7;Sm^-«-^-- -^•^"-(^•^-'(«>-- 162 DIFFERENTIAL CALCULUS [Ch. XIV. Since by hypothesis the two curves have contact of the wth order at the point whose abscissa is a, hence and y^-y^= (^ + 1)! ^'^°^'^"^ + - " ^'^''^"'^ — '^ \ but this expression, when h is sufficiently diminished, has the same sign as 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 ti be even, y-^ — y^ 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. EXERCISES 1. Find the order of contact of the curves r/A' 4iy = x^ and y = x — 1. f jy*^ 2. Find the order of contact of the curves /» 'vtVr v'l^^t^^ x^f and x + y + 1 = ^,fifX^ fi^^^ 3. Find the order of contact of the curves 43^ = ^:2-4 and x^-2y = S-i/^ 4. Determine the parabola having its axis parallel to the y-axis, which has the closest possible contact with the curve ah/ = x^ at the point (a, a). 102-104.] CONTACT AND CURVATUEE 163 5. Determine a straight line which has contact of the second order with the curve y = a:8-3a;2- 9a; + 9. j6. Find the order of contact of y = log (x - 1) and x^ - Qx -{■ 2y -{■ S = at the point (2, 0). 7. What must be the value of a in order that the curves y =: X + 1 + a(^x — ly and xy = 'dx — 1 may h^pvB contact of the second order? 103. Circle of curvature. The circle that has contact of the closest order with a given curve at a specified 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. 104. Length of radius of curvature; coordinates of center of curvature. Let the equation of a circle be (X-«)2+(r-;S)2 = i2^ (1) in which R is the radius, and ct, /3 are the coordinates of the center, the current coordinates being denoted by X, T to distinguish them from the coordinates of a point on the given curve. — It is required to determine i2, «, yS, so that this circle may have contact of the second order with the given curve at the point (a;, y). From (1), by successive differentiation, it follows that (2) 164 DIFFERENTIAL CALCULUS [Ch. XIV. If the circle (1) has contact of the second order at the point (a;, ^) with the given curve, then when X=x it is necessary that Y=y. 1 dX dx dX^ d^ J Substituting these expressions in (2), (a:-«) + (2/-^)g = 0, (4) whence ^ \dx) d:^ \dx) J .. and finally, by substitution in (1), v = 6> + ^/r, hence "" c^s ds_ dd '* O-S) dO (•-^) [^m But tan -^ = p — , y^r — tan-^ dp je^ tlierefore, by differentiating as to 6 and reducing, (dp\^ cPp d^ \deJ Pd0^ which, substituted in (1), gives p'-Pw + \Te) (1) [Art. 87. ^-©7 Since «: = — , it follows that R E = ['■-s: de»^\d0) K = 170 DIFFERENTIAL CALCULUS [Ch. XIV. When M == - is taken as dependent variable, the expres- sion for K assumes the simpler form Since at a point of inflexion k vanishes and changes sign, hence the condition for a point of inflexion, expressed in in polar coordinates, is that u 4- -j^ shall vanish and change sign. EXERCISES Find the radius of curvature for each of the following curves : 1. p = a^. 3. /o = 2acos^-a. 5. p^cos2e=o^, 2. p? = aaco82^. 4. pcos2^^ = a. 6. p = 2a(l-co8^. 7. p6 = a. '\\J\^^^ E VOLUTES AND INVOLUTES oi 110. Definition of an evolute. When the point P moves along the given curve, the center of curvature Q 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 C is found by eliminating x and y from the three equations dx X — a = d^ 1 + y-P = W) ; therefore a = 2p-\-3Xf P = - 2p~^xk Fig. 47. The result of eliminating x between the last two equations is M^-^py=Kp^m t.e., 4(a-2py = 2^pl3^y 172 DIFFEEENTTAL CALCULUS [Ch. XIV. which is the equation of the evolute of the parabola, a, /3 being the current coordinates. Ex. 2. Find the evolute of the ellipse Here £+iL.^ = 0, f^^_^, a^ b^ dx dx a'^y y-x dy d^y fe2 ^ -dx -by b^xH -6^2 2. ;2 2N -^* dx^~ a^ " ~ -o o\./ I o. I— .s.9\"-a /— •> a> -by , b^xH -h\ .... 2, -6* a^y^V a^y j ahj^^ ^ ' a^y^ whence ^ ^ a%* \ b^ ^ a'r\ M ^ 2)-'r Therefore ^ B = ^—r^y^ (2) Similarly, a = ^—J^x^- (3) Eliminating x, y between (1), (2), (3), the equation of the locus described by («, p) is (aa)t + (bjS)^ = (a2 - J^)!. (Fig. 52) 111. 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, yS) of the center of curvature with the coordinates (a:, ?/) of the corresponding point on the curve are, by Art. 104, rr-« + (y-/3)g = 0, (1) By differentiating (1) as to x, consideriucf «, /Q, y as functions of rr. 110-111.] CONTACT AND CURVATURE 173 Subtracting (3) from (2), *! + ^^ = 0, (4) ax ax ax whence di^_dx^ da dy But -y- is the slope of the tangent to the e volute at (a, y8), dir and — ;t- is the slope of the normal to the given curve at (a;, y). Hence these lines have the same slope; but they pass through the same point (a, /8), therefore they are coincident. II. The difference "between two radii of curvature of the given curve^ \ p ^ which touch the evolute at the points Cy, C^ (^Fig. 4^), is equal to the arc O^Q^ of the evolute. Since B is the distance between points (x^ «/), (a, yS), hence (x-ay^+ or < bV2, or accord- ing as e^ > or < ^. 15. Show by inspection of the figure that four real normals can be drawn to the ellipse from any point within the evolute. CHAPTER XV SINGULAR POINTS 112. Definition of a singular point. If the equation f(x^ ^) = be represented by a curve, the derivative -5^, CLX when it has a determinate value, measures the slope of the tangent at the point (a;, ?/). There may be certain points on the curve, however, at which the expression for the derivative assumes an illusory or indeterminate form ; and, in consequence, the slope of the tangent at such a point can- not be directly determined by the method of Art. 10. Such values of x^ y are called singular values^ and the corre- sponding points on the curve are called singular points. 113. 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^ ^) = 0. This gives, by differentiation with regard to a;, as in Art. 71, , , , • ^+^^ = dx dy dx ' dl whence ^=-^. (1) dy In order that -^ may become illusory, it is therefore necessary that §^= ^' F=^- (^) 179 180 DIFFERENTIAL CALCULUS [Ch. XV. Thus to determine whether a given curve f(x^ ^) = ^ df df has singular points, put -^ and -^ each equal to zero and solve these equations for x and i/. If any pair of values of x and ^, so found, satisfy the equation /(a;, ^) = 0, the point determined by them 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 di/ _ dx _0 ^"""^■"0' by finding the limit approached continuously by the slope of the tangent when x^x^^ y = yv Hence dy^_ dx\dxj dx i_(^ dx\dy) ^ ^ dy dx^ dxdy dx r . .„ „ "- ay ^fdy [Arts. 49, 72. dx dy dy^ dx at the point (a^j, y{). This equation cleared of fractions gives, to determine the slope at (xy, ^i), the quadratic This quadratic equation has in general two roots. The only exceptions occur when simultaneously, at the point in question, Bt? ^' dxdy ' dy^ ' ^^ 113-114.] SINGULAR POINTS 181 in which case -:r- is still indeterminate in form, and must be ax evaluated as before. The result of the next evaluation is a cubic in -^, which gives three values to the slope, unless all the third partial derivatives vanish simultaneously at the singular point. The geometric interpretation of the two roots of equation (3) will 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 (: dx By) dx^ 5^2 is positive, zero, or negative. These three cases will be considered separately. 114. Multiple points. First let H be positive. Then at df df the point (x, y) for which ^ = 0, ^ = 0, there are two values ijx oy of the slope, and hence two distinct singular tangents. It follows from this that 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 (a^j, y-^ are and H(x-^, y{) > 0. Ex. Examine for singular points the curve 3 x^ - xy - 2 y^ + x^ - 8y^ = 0, 182 DIFFERENTIAL CALCULUS [Ch. XV. Here |f = 6a: - v + 3x2, ^= - a: - 4 v - 24 v^. dx oy ^ ^ The values x = 0, ?/ = will satisfy these three equations, hence (0, 0) is a singular point. Since 1^=6 + 6a: = 6 at (0,0), bxby * ^ = _4-48y=-4at (0,0), FiQ. 53. hence the equation determining the slope is, from (3), -(ir-(i)-«-. of which the roots are 1 and — f . It follows that (0, 0) is a double point at which the tangents have the slopes 1, — |. 115. Cusps. Next let 5"= 0. 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, the singular point is called a tacnode. Two distinct branches of the curve touch each other at this point. (See Fig. 54.) If both branches of the curve recede from the tangent in only one direction from the point of tangency, the point is called a cusp. 114-115.] SINGULAR POINTS 183 Here again there are two cases to be distinguished. If the branches recede from the point on opposite sides of the double tangent, the cusp is said to be of the first kind ; if they 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. f(x, y) = aY - «^^* + a:« = 0. dx dy The point (0, 0) will satisfy /(x, y)= 0, ^ = 0, ^ = ; hence it is a singular point. Proceeding to the second derivatives, - 12 a%2 + 30 a:* = at (0, 0), ^"f =0 dxdy * dy'' The two values of -r- are therefore coincident, and each equal to zero. dx From the form of the equation, the curve is evidently symmetrical with regard to both axes; hence the point (0, 0) is a tacnode. No part of the curve can be at a greater distance from the y-axis than ± a, at which points -^ is infinite. The maximum value of y corre- dx sponds to x = ±aV\, Between a; = 0, ar = aV| there is a point of inflexion (Fig. 54). Ex.2. /(a:,y)=3^2-.x8=0; |f=-3:r^, f =2^. dx dy ay dx^ Hence the point (0, 0) is a singu- lar point. Further, If,: ay dxby 6a:=0at(0,0); 0- ^-2 Fig. 54. 184 DIFFERENTIAL CALCULUS [Ch. XV. Therefore the two roots of the quadratic equation defining -^ are both dx equal to zero. So far, this case is exactly like the last one, but here no part of the curve lies to the left of the axis y. On the right side, the curve is symmetric with regard to the a:-axis. As x increases, y increases; there are no maxima nor minima, and no inflexions (Fig. 55). Ex.3. /(x, y)z=x^- 2ax'^y - axy^ + aV ^ q. The point (0, 0) is a singular point, and the roots of the quadratic defining dx are both equal to zero. Let a be positive. Solving the equation for y, When X is negative, y is imaginary ; when a: = 0, y = ; when x is positive, but less than a, y has two positive values, therefore two branches Pig. 66. Pro. 66. are above the a:-axis. When ar = a, one branch becomes infinite, having the asymptote x = a] the other branch has the ordinate \ a. The origin is therefore a cusp of the second kind (Fig. 56). 116. Conjugate points. Lastly, let H be negative. In this case there are no real tangents ; hence 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. 115-116.] SINGULAR POINTS 185 Ex. f(x, y) = ay^ — a;^ + hx^ = 0. a singular Here (0, 0) is a singular point of the locus, and dx 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 =±x4 Fig. 57. and hence, if a and b are positive, there are no real points on the curve between x = and x = b. Thus is an isolated point (Fig. 57). These are the only singularities that algebraic curves can have, although complicated combinations of them may ap- pear. In each of the foregoing examples, the singular point was (0, 0) ; but for any other point, the same reasoning will apply. Ex. f(x, 7/)= x^ -{- S y^ - U y^ - 4:x + 17 y - S = 0, ^=2x-4 ^: dx ' dy y2_2Qy + l\ At the point (2, 1), /(2, 1)= 0, %. = 0, ¥ = 0; hence (2, 1) is a singular point. ^ dV _ ^'^^ S = ^' ^y = '-^ W='''-''^ =-8 at (2,1). Hence --p- — ±\\ and thus the equations of the two tangents at the node (2, 1) are y - 1 = i(a: - 2), y - 1 = - K^ - 2). When H is negative, the singular point is necessarily a conjugate point, but the converse is not always true. A singular point may be a conjugate point when 11=0. [Compare Ex. 4 below.] 186 DIFFERENTIAL CALCULUS [Ch. XV. 116. EXERCISES ON CHAPTER XV Examine each of the following curves for multiple points and find the equations of the tangents at each such point : 1. a2x2 = ftV + ^ V- 2. ^ 2a-x 3. xt + yl =ai. 4. ^2(x2 _ a-2) = x\ 5. y z=za + X -\- hx^ ± cx'i. When a curve has two parallel asymptotes it is said to have a node at infinity in the direction of the parallel asymptotes. Apply to'Ex. 6. 6. (x^-y^)2-^y^+y = 0. ■ 7. a;4 - 2 a?/8 _ 3 a2^2 _ 2 a2a.2 ^ (j4 = 0. 8. y^ = x(x-\-ay. 9. a;8 - 3 axy -\-y^ = 0. 10. y^ = x*~{-x^. 11. Show that the curve y = x\ogx has a terminating point at the origin. CHAPTER XVI ENVELOPES 117. Family of curves. The equation of a curve, usually involves, besides the variables x and ^, 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 «/, 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 a be one of these "constants." Then if a be given a series of numerical values, and if the locus of the equation, corre- sponding to each special value of a be traced, 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 is called a family of curves. For example, if A, h be fixed, and p be arbitrary, the equa- tion Qy — k')'^ = 2p(x— K) represents a family of parabolas, each curve of which has the same vertex (7a, A;), and the same axis y=h^ but a different 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. The presence of an arbitrary constant a in the equation of a curve is indicated in functional notation by writing the 187 188 DIFFERENTIAL CALCULUS [Ch. XVI. equation in the form /(a;, y^ «) = 0. The quantity «, which is constant for the same curve but different for different curves, is called the parameter of the family. The equa- tions of two neighboring curves are then written f(x, y, a) = 0, f{x, y, a + h')= 0, in which A is a small increment of a. These curves can be brought as near to coincidence as desired by diminishing h. 118. Envelope of a family of curves. A point of inter- section of two neighboring curves of the family tends toward a limiting position as the curves approach coincidence. The locus of such limiting points of intersection is called the envelope of the family. Let f(x,y,a)=0, /(x, y, a+ h)==0 (1) be two curves of the family. By the theorem of mean value (Art. 45) f(x, y,a-hh^ = fCx, y, a)-\-h^Cx, y, a-^-OK), (2) da which, on account of equation (1), reduces to Hence, it follows that in the limit, when A = 0, is the equation of a curve passing through the limiting points of intersection of the curve /(a:, ?/, a) = with its consecutive curve. This determines for any assigned value of a a definite limiting point of intersection on the corre- sponding member of the family. The locus of all such 117-119.] -EN VEL OPES 189 points is then to be obtained by eliminating the parameter a between the equations /(a;, y, «)= 0, Z(a;, «/, «)= 0. da The resulting equation is of the form F(x^ y) = 0, and represents the fixed envelope of the family. 119. The envelope touches every curve of the family. I. Geometrical 'proof. Let A, B^ Q be three consecutive curves of the family ; let A^ B intersect in P ; B^ C 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 P(> is also the direction of the tangent to B at P, and accordingly 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, a) = da be solved for a, in the form a= (x, «/). Then the equation of the envelope is Equating the total rr-derivative to zero, dx dy dx dXj, i/-v, = ^-^ (1) 119-121.] ENVELOPES 191 The envelope of this line, when y^ takes all values, is required Differentiating as to y^ ~ Sp^ 2p Substituting this value for y^ in (1), the result^ 27 py^ = i(x - 2 py, is the equation of the required evolute. *• 121. 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^, y-^ 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 of the parameters a, regarding the other parameter yS as a function of a. This dB gives four equations from which a, y8 and -^ may be elim- da inated, the resulting equation being that of the desired envelope. Ex. 1. Find the envelope of the line a b the sum of its intercepts remaining constant. The two equations are X y ^ - + I = 1, a a+b = c. 192 DIFFERENTIAL CALCULUS [Ch. XVI. Differentiate both equations as to a ; 1 + ^ = 0. da Eliminate da Then — = ^^ which reduces to ^ I X y_ a b a b \ . , — , , — - = J- = 7 = -; whence a — y/cx, b = Vcy. a b a+b c'_ ^ Therefore Va; + Vy = Vc is the equation of the desired envelope. [Compare Ex. p. 131.] Ex. 2. Find the envelope of the family of coaxial ellipses having a constant Here 121.] ENVELOPES 193 For symmetry, regard a and b as functions of a single parameter t. Then ^da+^db = 0, bda •+ ac?6 = ; hence —- = ^ = --, a=±a:V2, b=±yV2, and the envelope is the pair of rectangular hyperbolas xy =±^ k^. Note. A family of curves may have no envelope ; i.e., consecutive curves may not intersect; e.g., the family of concentric circles x^ + y^=r^, obtained by giving r all possible values. If every curve of a family has a node, and the node has different positions for different curves of the family, the envelope will be composed of two (or more) curves, one of which is the locus of the node. Ex. Find the envelope of the system /= (y-\y + x^-x^ = 0, in which A is a varying parameter. Here -^ = — 2(y — A,) = ; by combining with /= to eliminate X, we obtain a;2 = 0, X - 1 = 0, x + 1 = 0. From Art. 114 it is seen that x = 0, y = X is a node on /; moreover, the various curves of the family are obtained by moving any one of them parallel to the y-axis. The lines a: — 1 = 0, a: + 1 = form the proper envelope, and a: = is the locus of the node. EXERCISES ON CHAPTER XVI 1. Find the envelope of the line x cos a + y sin « = jo, when ce is a parameter. 2. A straight line of fixed length a moves with its extremities in two rectangular axes. Find its envelope. 194 DIFFERENTIAL CALCULUS [Ch. XVI. 121. ^ 3. Ellipses are described with common centers and axes, and having the sum of the serai-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\ / 5. Find the envelope of the lines y — ^ = m(x — a) + rVl + m^, m being a variable parameter. 6. A circle moves with its center on a parabola whose equation is y2 = 4 axj and passes through the vertex of the parabola. Find its envelope. 7. Find the envelope of a perpendicular to the normal to the parabola y2 = 4 ax, drawn through the intersection of the normal with the x-axis. 8. Show that the curves defined by the equations ^ + ^=1, a + p = c, X y "^ in which a and j8 are parameters, all pass through four fixed points ; find them. 9. In the « nodal family '' {y - 2ay={x - aY + Sx^ - y\ show that the usual process gives for envelope a composite locus, made up of the "node-locus " (a line) and the envelope proper (an ellipse). INTEGRAL CALCULUS CHAPTER I GENERAL PRINCIPLES OF INTEGRATION 122. The fundamental problem. The fundamental prob- lem of the Differential Calculus, as explained in the preced- ing pages, is this : Given a function f (^x), of an independent variable x, to determine its derivative f'(x). It is now proposed to consider the inverse problem, viz. : Given any function f'(x), to determine the function fix) having f {x) for its derivative. The study of this inverse problem is one of the objects of the Integral Calculus. The given function f'(x) is called the integrand^ the function f(x) which is to be found is called the integral^ and the process gone through in order to obtain the unknown function f(x) is called integration. The operation and result of differentiation are symbolized by the formula , £/w=/'(^), (1) or, written in the notation of differentials, dfix-)=f'ix)dx. (2) 195 196 INTEGRAL CALCULUS [Ch. I. The operation of integration is indicated by prefixing the symbol j to the function, or differential, whose integral it is required to find. Accordingly, the formula of integration is written thus : Following long established usage, the differential, rather than the derivative, of the unknown function f(x) is written under the sign of integration. One of the advantages of so doing is that the variable, with respect to which the inte- gration is performed, is explicitly mentioned. This is, of course, not necessary when only one variable is involved, but is essential when several variables enter into the inte- grand, or a change of variable is made during the process of integration. 123. Integration by inspection. The most obvious aid to the problem of integration is a knowledge of the rules and results of differentiation. It frequently happens that the required function f{x) can be determined at once by recol- lecting the result obtained in some previous differentiation. For example, suppose it to be required to find / cos X dx. It will be recalled that cos x dx is the differential of sin x^ and thus the answer to the proposed integration is directly obtained. That is, cos xdx ss sin x. f' Again, suppose it is required to integrate j x"dx^ 122-123.] GENERAL PRINCIPLES OF INTEGRATION 197 where n is any constant (except — 1). This problem imme- diately suggests the formula for differentiating a variable affected by a constant exponent [(6), p. 49]. When this formula is written or, what is the same thing, it becomes obvious that /= x^dx = n + 1 An exception to this result occurs when n has the value — 1. For in that case it is apparent from (8), p. 50, that 'dx ix'^dx— \ — = log X. The method indicated in the above illustration may be designated as the method of integration hy inspection. This is in fact the only method of practical service available. The object of the various devices suggested in the subse- quent pages is to transform the given integrand, or to separate it into simpler elements in such a way that the method of inspection can be applied.* * When all has been done that can be accomplished in this direction, it will be found that a large portion of the field is yet unexplored and unknown, and that many functions exist whose integrals cannot be found. By .this we mean that such integrals cannot be expressed in terms of functions already known. To illustrate, let it be imagined that the integral calculus had been discovered before the logarithm function was known. It would then have I* (It been impossible to express the integral \ — in terms of known functions. This integral might in consequence have led to the discovery of the function log X. An exactly analogous thing, in fact, has happened in the attempt to integrate other expressions, and many important and hitherto unknown func- tions have been discovered in this way which have greatly enriched the entire field of mathematics. 198 INTEGRAL CALCULUS [Ch. I. 124. The fundamental formulas of integration. When the formulas of differentiation (l)-(26), pp. 49-50, are borne in mind, the method of inspection referred to in the preceding article leads at once to the following fundamental integrals. Upon these sooner or later every integration must be made to depend. 1^ I. \u^du = ^^^^* n + 1 II. f** = log«. III. frt''efw = -^^. J log a lY. (e^du = e^. T, f cos u du = sin u, VI. \^inudu = -eosu* Til. f sec^ udu = tan u, VIII. f cosec'-^ udu = - cot u, IX. f sec u tan w %2 -^ xV^« (ar-l)Va:2-2x 125-126.] GENERAL PRINCIPLES OF INTEGRATION 203 126. Integration by parts. If u and v are functions of a?, the rule for differentiating a product gives the formula d (uv) = V du + u dv, whence, by integrating and transposing terms, \udv = uv — \vdu. This formula affords a most valuable method of integra- tion, known as integration by parts. By its use a given integral is made to depend on another integral, which in many important cases is of simpler form and more readily integrable than the original one. Ex.1. Jlog xdx. Assume ' u - = log a:, dv = dx. Then du: dx V = X. By substituting in the formula for in itegration by parts, Jloga :dx = xlogx- -j-.. = a: log a: ■ -X = x (log X - -1) X = X (log X — log e) = a: log -. Ex. 2. ixe^'dx. Assume u = x, dv — eFdx. Then du = dx, v = e*, and j xe'^dx = xc* — i e'^dx = e*(a; — 1). Suppose that a different choice had been made for u and dv in the present problem, say u = e*, dv = X dx. 204 INTEGRAL CALCULUS [Ch. I. From this would follow du = e'dx, y = — , 2 and J xe'dx = ^ arV — \ ^e*dx. — e*dx is less simple in fomi than the original one, and hence the present choice of u and do is not a fortunate one. No general rule can be laid down for the selection of u and dv. Several trials may be necessary before a suitable one can be found. It is to be remarked, however, that as far as possible dv should be chosen in such a way that its integral may be as simple as possible, while u should be so chosen that in differentiating it a material sim- plification is brought about. Thus in Ex. 1, by taking u = log a:, the transcendental function is made to disappear by differentiation. In Ex. 2, the presence of either x or e* prevents direct integration. The first factor x can be removed by differentiation, and thus the choice u = X is naturally suggested. Ex. 3. Kx^a'dx. From the preceding remark it is evident that the only choice which will simplify the integral is u = x^, dv = a'^dx, qX Hence du = 2x dx, v = , log a and (x^a'dx = ^^ - -^ (xa'dx. J log a log a J Apply the same method to the new integral, assuming M = x, dv — a'dXf whence du = rfx, v = , log a and (xa' dx = -^^ - -^ f o-rfar J log a log a J logrt {\o%ay By substituting in the preceding formula, J logaU logo (log a)'' J 126-127. J GENERAL PRINCIPLES OF INTEGRATION 205 EXERCISES 1. (Birr'^xdx. 7. Kxcoi-'^xdx, J ^ ^ 8. J a: SI 2 3. \ x"^ cos X dx. 4. j x^ log X dx. 5. (x^t&n-^xdx. ^^' J '^ 6. sin 3 X dx, \ e* cos X dx, e" sin X dx. j sec a; tan a: log cos a: rfa:. 11. i cos a: cos 2 a; rfa:. 127. Integration by substitution. It is often necessary to simplify a given differential f'(x)dx by the introduction of a new variable before integration can be effected. Except for certain special classes of differentials (see, for example, Arts. 138, 139) no general rule can be laid down for the guidance of the student in the use of this method, but some aid may be derived from the hints contained in the problems which follow. Ex.1. J xdx Va2 - a;2 Introduce a new variable z by means of the substitution a'^ — x^ = z. Differentiate and divide by — 2, whence xdx = Accordingly The details required in carrying out this substitution are so simple that they can be omitted and the solution of the problem will then take the following form : r_^d^ = ((a^-x^r^xdx = - ^ ((a^-x^rH-2xdx) = - (a^-x^^. In this series of steps the last integral is obtained by multiplying inside the sign of integration by - 2 and outside by - J, the object being to 206 INTEGRAL CALCULUS [Ch. I. make the second factor the differential of a^ — x\ Thinking of the latter as a new variable, the integrand contains this variable affected by an exponent {— \) and multiplied by the differential of the variable, in which case formula I can be applied. Ex. 2. (^^^dx. J X Assume log a; = 2. Then — = dz, and |l2^rf. = pI = | = (!o|£l^ Here again it is not necessary to write out the details of the substitu- tion, as it is easy to think of log a; as a new independent variable and to perform the integration with respect to that. It is then readily seen that the expression to be integrated consists of the variable logx mul- tiplied by its differential — , and that the integration is accordingly X reduced to an immediate application of the first formula of integration. Thus Ex.3. fgtan-^x dx J 1 + X2 GoK^y logx ' d(logx)z= ^ Think of tan-^ a: as a new variable and apply formula IV. Thus Ex.4. r^iBli: + 'sin"^ X dx dx Think of sin-^ a: as a new variable and — — ^^^ as the differential of that variable. Apply formula I. ^^ ~ ^* Ex.5. j'(a;2+2a; + 3)(x + l)«?x. Multiply and divide by 2. The integral then takes the form i JCar^ + 2 ar + 3) • (2 x + 2)dx. Observing that (2 a: + 2)^a: is the differential of ar^* + 2 a: -|- 3, and think- ing of the latter expression as a new variable, it is seen that formula I is directly applicable, leading to the result 127.] GENERAL PRINCIPLES OF INTEGRATION 207 Ex. 6. flog cos (x^ + 1) sin (x^ +1)'X dx. Make the substitution ar2+l=2;. The given integral takes the form J j log cos 2; siw zdz. Make a second change of variable, cos z = y. Then sin zdz=: — dy. The transformed integral is -lyf^gydy, to which the result of Ex. 1, Art. 126, can be at once applied. It will be observed that two substitutions which naturally suggest themselves from the form of the integrand are made in succession. The two together are obviously equivalent to the one transformation, cos (x^ + 1) = 3/. Ex.7, f /^ . Either put x = az, or else divide numerator and denominator by a, and write in the form / <^ v-(iy Regarding - as a new variable, this comes under XI and gives the result C dx • ^ X , ^ •^ Va2 - x^ a = - cos-i- + C^. In a similar manner treat Exs. 8-10. Ex.8, r ^^ . J x'^ + a^ Ex.9, f— ^=. •^ xy/x^ - a2 Try also the substitution x = — z 208 INTEGRAL CALCULUS fCn. I. Ex.10, f- ^^ V2 ax - a;2 Try also the substitution 2: - a = x. Ex. 11. • ^^ /: Vx2 ± a2 Make the transformation From this follows, by differentiation, (1+ ^ - \dx = dz-r dx that is, (v^2±a2 + a;) ""^ = c?2, or, • = = — • Ex.12, r^^. Assume x — a _ ^ . ^^^ is, ar = a — i-?» a: + a 1 — « The reasons for the choice of substitution made in this and the pre- ceding example will be made clear in Arts. 133 and 139. Ex.13. jcosecx£?x. Multiply and divide by cosec x — cot x. It will be readily seen that rdz the integral then takes the form \ — Another method would be to use the trigonometric formula sin a: = 2 sin ^ cos |, aec^ -dl-\ whence I cosec xdx = I = I - • ' 2 sin ^ cos r *^ tan ? :. 14. J Ex. 14. i sec a: dx. Put « = « - J, and use Ex. 18. 127-128.] GENERAL PRINCIPLES OF INTEGRATION 209 Solve the problem also by means of substitutions similar to those used in the preceding example. ii^"^ Ex. 15. f ^£ = f i°^^ = 2f-5 P^ (s=2ax + 6) 2 tan-i ^«^ + A- , if 4 ac - 52 >0. V4 ac - 52 V4 ac - ¥• = — ^::::r=r log ' > if 4 ac — 6^ < 0. Vft-^ - 4 ac 2 aa; + 6 + V62 _ 4 ac Ex.16, f ^?£ l=f 2rf^ =f ^(2^) . ^2a;2+2a;+3 ^ 4^2+ 4a: + 6 ^(2a:+l)2+5 In this form it is ea^ to integrate by taking 2 a: + 1 as a new variable. Ex.17, f ^ . ^3a:2-2a:+5 Ex.18, f i^^ . Ex. 19. ^—^ ^ . V-9a;2+ 30ar-24 Ex. 20. f (3 a: - 2) cos (3 a: - 2) dx. Ex.21, f adx xVcfi + hx Substitute y/a^ + hx — z, and use Ex. 12. \\)J 128. Additional standard forms. The integrals in Exs. 7-14 of the preceding article, and in Exs. 15-16 of Art. 125, are of such frequent occurrence that it is desirable to collect the results of integration into an additional list of standard forms. 210 INTEGRAL CALCULUS [Ch. I. du Va2 _ ^2 a - a _„ r du ., 1 u XT. J— =:=r = 6iii-i-, or -cos XVI. f— z^^=log(i* + V^i2X^). XYII. f_^ = ltan-i^, or -icofi XIX. f — , ^^ , =^sec-*^, or -£:Cosec du 1 it* 1 1 «« ,.= = — sec-*— , or — cosec~* — t*Vw2_«2 a a a a XX. i ^ ==Yer8-^-- XXI. j tan udu = - log cos «* = log sec u, XXII. j cot u du = log sin u. XXI 11. f sec udu = log (sec i* + tan i*) = log tan (^ + t) • XXIY. f cosec udu = log (cosec u - cot u) = log tan ^ • 129. Integrals of the form r (Ax. + B)dx ^ y/axi^ -\-bx + c Integrals of this form are of such frequent occurrence as to deserve special mention. The integration is readily effected by the substitution of a new variable which reduces the radical to a simpler form. Two cases are to be considered according as a is positive or negative. Case I. a positive. In this case by dividing out the coefficient of a^ the radical may be written ^ a a ^\ 'la J 4a^ 128-129.] GENERAL PRINCIPLES OF INTEGRATION 211 The given integral then takes the form ( Az 4-B— \dz 1 r * (Ax-\-B)dx - ^ A ^^J ( _ ^ _ A r zdz V 2 , -nac z^-\- P + 2aB-bA 4:0,' C- dz ^ac — W = —Wax^ + bx-[- c-\ — log X + - — \'\x^ -\--x-\-- ^ 2aVa ""K 2a ^ a a Case II. a negative. When a is negative, by dividing out the positive number — a the radical becomes ^ a a 4a2 \ 2 a/ and in consequence the integral takes the form 1 y- (Ax + B)dx ^'^-hi-J V (Az+B-^)dz !_ A 2aJ — a*^ — -tao JblszA ^ ■ia' V 2a _ A r» zdz ^ -ia + 2 a^ - 6^ — 4:ac 2 2 aV <3 - 0^^ n dz \ A „1 V — a^ 4 a^ ac 2.2 a5 — bA ' _i 4^2 2a2 ac 2 z^ 2 a V — « V^^ — 4 J, second, and third, a reduction of type \A'\ . Can these reduc- tions be taken in any order ? The different possible arrangements of the order in which these three reductions might succeed each other are (1) [-4], [^], [D] ; (2) [^], [1>], [^]; (3) [1>], [^], [^], of which number (3) was chosen in the solution of the problem. Of the other two arrangements, (2) can be used, but (1) cannot. For, after first applying [^] (which would be done in either case), the new integral is ( x\a^ + x'^y^ dx. If [^] were now applied, it would be necessary to assume Ja:2(a2 + x"^)"^ dx = ^ f (a2 + x'^)~^-\- Bx{cfi + x'^)'^. This equation, when difEerentiated and simplified, becomes a:2 = ^ + 5a2, a relation which it is clearly impossible to reduce to an identity by equating coefficients of like powers of x, since there is no x^ term in the right member to correspond with the one in the left member. It will be observed that this is the exceptional case mentioned on page 218, in which m + np + 1 = 0. Ex. 5. Show thaib the integral f C^ -^ ) ^x can be integrated by four reductions. Prove that these can be arranged in six different orders, and determine those which can be used. 222 INTEGRAL CALCULUS [Ch. II. 131. Ex. 6. f — ^-— . (T-Ex. 13. (V^^Tadx. J (a:'-' + iy J Ex. 8. (•_^!*L_. ■^L^ .^..^ J ^ V^ y^' rt^x. 9. iV^^^^dx, Ov J^^ , .. Ex.16. f_^=. D>V •^ xWa' - x^ Ex.11, f-i^^ / Ex.17. r,^f ^o^« ' K^ Ex. 12. f (a2+ x2)^rfar. Ex. 18. f Vl - 2 x - x^ rfar. ^ ^' Ex. 19. Show that r ^^ = 1 r ^ +(2n-3)f ^-^ 1 ''^^' l> CHAPTER III INTEGRATION OF RATIONAL FRACTIONS 132. Decomposition of rational fractions. The object of the present chapter is to show how to integrate fractions of the form , ^ wherein (f)(^x) and '>jr(^x) are polynomials in x. The desired result is accomplished by the method of sepa- rating the given fraction into a sum of terms of a simpler kind, and integrating term by term. If the degree of the numerator is equal to or greater than the degree of the denominator, the indicated division can be carried out until a remainder is obtained which is of lower degree than the denominator. Hence the fraction can be reduced to the form ^ = a." + 5.«- + ... + ^, in which the degree oif(x') is less than that of '^(x)' As to the integration of the remainder fraction ^^ ^ , it is to be remarked in the first place that the methods of the preceding articles are sufficient to effect the integration of such simple fractions as x—a (x — ay^'' ' x^±d?'' (x^±a^y^^ ' x^-\-'mx-{-n Now the sum of several such fractions is a fraction of the kind under consideration, viz., one whose numerator is of 223 224 INTEGRAL CALCULUS [Ch. III. lower degree than its denominator. The question naturally arises as to whether the converse is possible, that is : can every fraction ^^^£2. he separated into a sum of fractions of as simple types as those given in (1) f The answer is, yes. Since the sum of several fractions has for its denomina tor the least common multiple of the several denominators, it follows that if •^^ ^ can be separated into a sum of simpler fractions, the denominators of these fractions must be divisors of 'yjr(x^. Now it is known from Algebra that every polynomial '^^(x) having real coefficients (and only those having real coefficients are to be considered in what follows) can he separated into factors of either the first or the second degree^ the coefficients of each factor heing real. This fact naturally leads to the discussion of four different cases. I. When '>^(x') can be separated into real factors of the first degree, no two alike. E.g. , '^^(2?) = (x — a)(x — 6) (x — c). II. When the real factors are all of the first degree, some of which are repeated. III. When some of the factors are necessarily of the second degree, but no two such are alike. U.g., ylr^x) =^Cx^ + a^Ca^-hx + l)(rr -hXx- c^. IV. When second degree factors occur, some of which are repeated. Kg., f(x) = (:t-2 + a2)V-^ + 0(^-^)- 132-133.] INTEGRATION OF RATIONAL FRACTIONS 225 133. Case I. Factors of the first degree, none repeated. When ylr(x) is of the form ^fr(x') = (x — d)(x —h}(x— 6) ••• (x — 7l), assume i|r(a;) X — a x — h x — c x — n in which A^ B^ C^ '"^ N are constants whose values are to be determined on condition that the sum of the terms in the right-hand member shall be identical with the left-hand member. Ex.l. (tpl^d^. Dividing numerator by denominator, -i— - = x , ^ ^ x2-3a: + 2 x^~^x+2 Assume -^ ^ = ^d_+ ^ (a:-l)(a;-2) x-1 x-2 By clearing of fractions, (1) x = A(x-2) + B(x-iy In order for the two members of this equation to be identical it is necessary that the coefficients of like powers of x be the same in each. Hence 1 = A-^B, = -2A-B, from which ^ = — 1, B=2. Accordingly the given integral becomes Kx + ^^-^)dx=:aL+\og(x-l)~2log(x--2)'rC A shorter method of calculating the coefficients can be used. Since equation (1) is an identity, it is true for all values of x. By giving x the value x=l the equation reduces to 1 = ^(— 1), or A=: — l. Again, assume x = 2. Whence 2 = B. 226 JtcVf ^ -^INTEGRAL CALCULUS ^^K '"^ [Ch. IH Ex. 2M ^^ Ex.10 ri?^±il^. V ^a;2-a2 J 2x2 + 3 3:- 2 Ex.3, flu^dor. Ex.11. f-M+^&l^^. lAL J x^-x X ^ a; (a-- -a) (a: + 6) ^ Ex.4 C ix^-\2)dx ^ Ex. 12. r_i£±ii^. \i Ja;2+4a:+3 J2x-a:2-a;8 , l^ Ex.6. f a:. Substitute for D the value just found, and transpose the correspondmg term. This gives b x'^ - Q X -t I = A {x - ly -{- Bx {x - \y + Cx {x - 1). It can be seen by inspection that the right-hand member of the result is divisible by a? — 1. As this relation is an identity, it follows that the left-hand member is also divisible by x—\. When this factor is re- moved from both members, the equation reduces to bx- \ = A(x-\y-\-Bx(x-l)+ Cx. Now put a; = 1. Then C = 4. Substitute the value found for C, transpose, and divide by a: — 1. The result is \ = A{x-l) + Bx. By giving x the values and 1 in succession, it is found that A = -l, 5 = 1. Accordingly, r (5x^-3x+l)dx ^ rf_l + _j_ + __^_ + _3_U J x{x-iy J\ X x-i (x-iy (x-iy) = log.^-l ^^-^ X 2(x - 1)2 Ex. 2. (—J^ J Ex. 9. r^£!±^^!±r«±l}^±«rf^. J(x^iyix+1) J x\a+x) Ex. 3. r C^^- 11^+26)^0: . ^^^^^ nx^-l)dx, ^'^' *• ^ {x^-a^y Ex.ll.'|(ax2+6;r8)-irfx.'\ Ex. 5. r(^2a:+l)rfa;, ^^ ^^ f ix^-x'^-r)dx . •^ x\x -f V2)a ' J (a; _ 1)2 Va:^ - 2 a: + 2 ^^- ^- f-^i^fl^^- [Separate ^Izz^iz-l into partial /Ex 7 r^{^i±_«Mi!^ J a:4 - 2 a%2 + a* yEx. B. I ^2rf. N^ fractions.] Ex.13. { ^' rfa?. -^ (a; — a)8 (2 -I- v^ - V^ a:)8 [Substitute a: - a = «.] 'm- 228 INTEGRAL CALCULUS [Ch. III. 135. Case III. Occurrence of quadratic factors, none repeated. ' • • J(x2+l)(a;2+2a;+2)* Assume (Vi 4a;2 + 5a: + 4 _ Ax ■{■ B Cx-{- D ^^ (a;2+l)(a:2 + 2x+2) x^^-1 x^-\-2x^-2 Then (2) 4a;« + 5a:4-4 = (^a: + ^)(a;2 + 2a: + 2) + (Ca: + 7))(a:2+l). By equating coefficients of like powers of x = J+C, 5 = 2^ + 2£ + C, 4 = 2^+5+ A 4 = 25 + A from which ^ = 1, 5 = 2, C = -l, i) = 0. Hence the given integral becomes C {x + 2)dx _C xdx ^2tan-ia;+tan-i(a:+l)+ilog /'+^ . To make clear the reasons for the assumption which was made con- cerning the form of equation (1), observe that since the factors of the denominator in the left member are a-^ + 1 and a;^ + 2 a: + 2, these must necessarily be the denominators in the right. member. Also, since the numerator of the given fraction is of lower degree than its denominator, the numerator of each partial fraction must be of lower degree than its denominator. As the latter is of the second degree in each case, the most general form for a numerator fulfilling this requirement (i.e., to be of lower degree than its denondnator) is an expression of the first degree such as Ax + J5, or Cx + B. Notice, besides, that in equating the coefficients of like powers of x in opposite members of equation (2), four equations are obtained which exactly suffice to determine the four unknown coefficients A, B,Cf D. 1 (Adx_, E^.6 C {^x-^)dx ^ ' J x^ + ^x J x^ + 2x^ Ex.2. Ex 3 f ^^^ Ex 7 f ^^^ " ^(a;+l)(a;2+l) ' ' J x^ -^ x'^ ■\- 1 Ex.5, f {a'-^')dx . E,.9. r {^ + ^^x + 2)dx . i\\x}^ 135-136.] INTEGRATION OF RATIONAL FRACTIONS 229 136. Case IV. Occurrence of quadratic factors, some repeated. This case bears the same relation to Case III that Case II bears to Case I, and an exactly analogous mode of procedure is to be followed. J (a;2 + 2)8 Ex Assume ,j. 2x^- x^ i- 8x^ + 4: ^ Ax -{- B Cx + D Ex + F ^ ^ (:r2 + 2)3 a:2 + 2 (a;2 + 2)^ ^ {x^ + 2)3* Whence, by clearing of fractions, 2 x^ - x^ + 8 x^ + 4: ={Ax + B){x'^ + 2)2 + (Ca:+ D){x^ + 2)+Ex-\- F. Instead of equating coefficients of like powers of x, as might be done, the following method of calculating the values of J., ^, C, ••• is briefer. Substitute for x^ the value — 2, or, what is the same thing, let x = V— 2. This causes all the terms of the right member to drop out except the last two, and equation (1) reduces to _ S\/^^ = EV^ + F. By equating real and imaginary terms in both members, - 8 = ^, ^ = F. Substitute the values found for E and F in (1), and transpose the corresponding terms. Both members will then contain the factor x^-^2. On striking this out the equation reduces to 2a:8-a:2+4a: + 2 = (^Ax + B^{xP- + 2)+ Ca: + 2). Proceed as before by putting x^ = — 2. Whence 4=CV^r2 + D, and therefore = C, 4 = Z). Substitute these values, transpose, and divide by a:^ + 2. This gives 2x-\ = Ax-\- B, whence ^ = 2, 5 = — 1. The given integral accordingly reduces to "^xdx J a:? 4- 2 J (x^ + 2)2 J {x (a;2 + 2)2 J {x^ + 2)8 230 INTEGRAL CALCULUS [Ch. III. 136-137. The first term becomes - The second, integrated by the method of reduction (Chap. II), gives ^ +_l.taii-i^ x^ + 2 V2 y/2 Finally, by applying formula I the last term integrates immediately Hence J {x^ +2)8 - ^ '^ ^ ^ ^ a:2 + 2 ^ (x2 + 2)2 Ex. 2. a^Y dx. Ex. 5. f(-^/ + V^. J\x2 + 1/ J a:2(x2+l>2 >v E^. 3. C{x + ay+a^ ^^^ ^x. 6. Cl^±l^^^^sU^ dx. V Ex.4, f ^^^ Ex.7, f ^'^^ . \ J(l + a:)(l + ar2)2 J (1 + a:2)8 The principles used in the preceding cases in the assump- tion of the partial fractions may be summed up as follows ; ^ach of the denominators of the partial fractions contains one and only one prime factor of the given denominator. When a- repeated prime factor occurs., all of its different powers must he used as denominators of the partial fractions. The numerator of each of the assumed fractions is of degree one lower than the degree of the prime factor occurring in the corresponding denominator. 137. General theorem. — Since every rational fraction can be integrated by first separating, if necessary, into simpler fractions in accordance with some one of the cases considered above, the important conclusion is at once deducible : The integral of every rational fraction can he found., and is expressihle in terms of algehraic, logarithmic, and inverse-trigo- nometric functions. CHAPTER IV INTEGRATION BY RATIONALIZATION At the end of the preceding chapter it was remarked that every rational algebraic function can be integrated. The question as to the possibility of integrating irrational func- tions has next to be considered. This has already been /touched upon in Chapter II, where a certain type of irra- tional functions was treated by the method of reduction. In the present chapter it is proposed to consider the simplest cases of irrational functions, viz., those containing ^ax + h and -\/ ax^ -\-hx -\- c^ and to show how, by a process of rationalization, every such function can be integrated. 138. Integration of functions containing the irrationality \/aa? + 6. When the integrand contains ^ ax -f- 5, that is, the wth root of an expression of the first degree in x^ but no other irrationality, it can be reduced to a rational form by means of the substitution Ex.1. le dx p + 3-1 ASSUDC that is, Then and V2a; + 3 = 2, 2a;+3 = z2. dx = z dz, C dx _Czdz -^^^^^^ 1^ •^V2a: + 3-l *^^-l = V2 a; + 3 + log ( V2 a; -f 8 - 1). 231 232 INTEGRAL CALCULUS [Ch. IV. Ex.2. J l-f^"-^"- v^ ^^^ ^. t:»_ « r 1 + a;6_— a;s — v^ a:* + a; It would appear at first sight that this integrand contains several irrationalities, viz., Vx, Vx, Vx, It is readily seen, however, that they are all powers of Vx, and hence the substitution Vx = z will rationalize the expression to be integrated. iryJ tK Ex.3, f ^^ .^y^^^^l ()^Ex.6. f— ^^ tvL^^^'' ^ Ex 4 f ^^ - Ex.7, r ^^ ^ ♦> A'^' Ex. 5. f ^? Ex. 8. f 4^^^- /tH When two irrationalities of the form Wax + 5, Vc^+^ occur in the integrand, the first radical can be made to dis- appear by the substitution } "Vax + 5 = 2. The second radical then reduces to ^ a r\ and the method of the next article can be applied. y^-^ Integration of expression s containing Vax^ -\-bx + c. Every expression containing ^ax^ + bx + c^ but no other irrationality, can be rationalized by a proper substitution. In order to make the necessary steps clearer, a geometrical interpretation of the problem will be very useful. To this end let the given radical be represented by y; that is, let i/^ = aa^-^bx + o, (1) 138-139.] INTEGRATION BY RATIONALIZATION 233 If now (a;, ^) be regarded as the rectangular coordinates of a point in a plane, equation (1) represents a conic (Fig. 60). Let (A, k'), or Q, be a given point on this curve. The equa- tion of any line through this point is i/-k = z(x-h), (2) X Fig. 60. in which z is the slope of the line. The line (2) will inter- sect the conic in a second point P. It is geometrically evident that the coordinates (rr, «/) of P depend on the value of 2, and in such a way that to each value of z corresponds only one pair of values x^ y. Consequently the variables x and y can be rationally expressed in terms of the variable z. This is done by treat- ing equations (1) and (2) as simultaneous, and solving for X and y in terms of z. For example, suppose it were desired to rationalize an expression containing Vic^ _ 5 ^j ^ g. Let ^2=a;2_5a.^.8, and select (1, 2) for the point Q, Then y-1 = z{x-V) represents any line passing through Q. In solving these two equations simultaneously for x and y^ the elimination of y gives z\x-Vf^-^z{x-r) = ^-bx^\, This quadratic equation in x has two roots, one of which should be a; = 1, since this is the value of x at C one of the 234 INTEGRAL CALCULUS [Ch. IV. points of intersection. The other root, corresponding to the variable point P, is 22—42 — 4 X = • 22-1 From this follows or y = V ;»^-5^ + 8 = -'^^'-^^-^ - Two particular cases of the method given above deserve to be noticed. (a) When the conic intersects the x-axis. In this case the quadratic expression aoc^ + hx-\-c has real factors, say, ax^ -\-hx-\- e = a(x — a) (a: — /3). The conic (1) intersects the a;-axis in the two points (a, 0) and (y8, 0), either one of which may be conveniently selected for the point Q. The equation of any line QP through the first point is y = z(x-a^, (^) and the equation of any line through the second point, y = z(x-^y (A') Either one of these equations, combined with (1), will effect the desired rationalization. (h) When the conic is an hyperbola. This case occurs when the coefficient of a^ is positive. The curve extends to infinity in two different directions, namely, the directions of the asymptotes. If one of the points at infinity on the curve be taken for the point Q, the lines QP passing through this point are parallel to that 139. J INTEGRATION BY RATIONALIZATION 235 asymptote which touches the curve at Q. The equations of the asymptotes are Accordingly the lines parallel to the one asymptote are and those parallel to the other Either of these equations used in place of (2) will serve equally well in expressing x and y{=-^a^ -\-hx-\- c) ration- ally in terms of a new variable z, Ex.l. ^ ^^ (x + V ar2 + 2 ar - 1)^ The conic y = y/x^ + 2 a: — 1 is an hyperbola and formula (B) can be applied. This gives Vx^ + 2 a: - 1 = x + Zy whence by squaring and solving for x, z^+1 ^-2(l-z)' and accordingly Vx +2x 1- 2^^_^^ When these expressions are substituted in the given integral, it becomes = i[-^ + 41og(l + .) + ^] = l(x-y/x^+2x-l) + +21og[H-Vrr2+2a;-l-a:]. l_a:+Va;2+2ar-l Since the conic y = Vx^ + 2 a: — 1 cuts the ar-axis, formula (A) [or (^')] could be used for the purpose of rationalization. 236 INTEGRAL CALCULUS [Ch. IV. Ex -•/: \/\ + xdx The denominator being rationalized, the integrand takes the form vT^^^ (1-xy The conic y = VI - a:2 intersects the a:-axis in two points (i 1, 0). If the point (1, 0) be chosen for Q, the equation of any line passing through this point is y = z{x- 1). The simultaneous solution of these two equations gives 22 _ 1 _2z whence fVl^^, ^ f- 2.^. = 2(— 2 + tan-^2) \ a: — 1 X — \ I Ex. 3. J /i^^n: = z.i 3. r.l£i+^Lzil^. •'(x2+l)2(xa+2)i 139-140.]^ 4. C ^ •^ X + y/x — 1 EC^MATION BY RATIONALIZATION 237 dx dx x + Vx^-1 -J dx 8. (a + a:)^ (2 - 3 a:-^) rfa: J a; — 3 376 + 5 a;i! v^ '•I Vx2 - lVv^TT + Va:-l 10 [Assume Vx + 1 + Va; — 1 = 2.] of a: J(a;2+a2) Va;2 - a2 [A ssume x = a sec ^.] 11. 1+V^ Vx r i + v Jl + v ^a:. 12. (/x + x (1 + a;)2 I Substitute ^-^ = ^8.1 L 1 + a; J CHAPTER V INTEGRATION OF TRIGONOMETRIC AND OTHER TRAN- SCENDENTAL FUNCTIONS 141. In regard to the integration of trigonometric func- tions, it is to be remarked in the first place that every rational trigonometric function can be rationally expressed in terms of sine and cosine. It is accordingly evident that such functions can be inte- grated by means of the substitution sin x=- 2. After the substitution has been effected, the integrand may involve the irrationality Vl — 252(= cos a;). This can be removed by rationalization, as explained in the preceding chapter, or the method of reduction may be employed. The substitution cos a; = 2 will serve equally well. It is usually easier, however, to integrate the trigonometric forms without any such previous transformation to algebraic functions. The following articles treat of the cases of most frequent occurrence. 142. jsec^**xdx, j cosec^^xdx. In this case n is supposed to be a positive integer. If sec^^a? dx be written in the form Bec^'^x . sec'a; dx = (l-\- tan'a;)"-^ (tan a;), (/ Ch. V. 141-143.] TRIGONOMETBIC FUNCTIONS 239 the first integral becomes C(i2iYi^x + l)«-i<:7(tan x). If (tanV + 1)""^ be expanded by the binomial formula and integrated term by term, the required result is readily obtained. In like manner, J cosec^"2J dx= \ cosec^"~^a; • cosec^a; dx = -C^cot^x + l)"-^^(cotrc). This last form can be integrated, as in the preceding case, ' by expanding the binomial in the integrand. (^ < The same method will evidently apply to integrals of thei ^ j foim \ f- rtan"*a;sec^«a;cZrr, Jcot"*a; cosec^^o: 6^ic, V^/V in which m is any number. \^ X "^^ dx K rn-cosxW^ cof*a: ^•1 2. J cosec^a; dx, "^ } ^^^^^ ^^^^^ (cos4^ _ sin^a:)^'"/^^ 3. (sec^xdx, ' ^' f . ^"^ (=(tan-^xsec^xdx). f J J sin^a; cos x -^ <\ J dx g r cos% dx XJ^^^^ 143. fsec'**ictan2»* + iicc?ic, J cosec"*a?cot2'» + ia5^iC. In these integrands n is a positive integer, or zero, so that ^ 2 w + 1 is any positive odd integer, while m is unrestricted./-- >w 240 INTEGRAL CALCULUS [Ch. V. The first integral may be written in the form J sec"*"^a; tan^"a; • sec x tan x dx = j sec"*"^a; (sec^a; — V)'*d (sec a:), which can be integrated after expanding (sec^ic — l)** by the binomial formula. Similarly, J cosec"*a; cot^"^^a; dx = j cosec^""^a; cot^"a; • cosec x cot x dx = — I cosec"*"^a; (cosec^a; — l)"c?(cosec2;). EXERCISES f^ 1. j sec% tan^a; dx. 5. \ tan^x dx. 2. fcosecSarcotSarda;. 6. (^^^^^^^=(sec''-^xta.n^xdx\ J J cos'*x •' o fsec ax -.^ ^ c , 4. J sin X Go\?x dx. 8. J cot x dx. y 144. (tun^^ocdx, KcoV^xdx, The first integral can be treated thus : j tan** a? dx= j tan""^ • tan^a; dx = J tan"-2 a; (sec^ a; — 1 ) ia; = ^.^^-Ct^n^-^xdx. When w is a positive integer, the exponent of tan x may be diminished by successive applications of this formula until it becomes zero (when n is even), or one (when n is odd). 143-144] TRIGONOMETRIC FUNCTIONS 241 In like manner, J cot" a? dx=\ GoV^-^x Qoi^x dx s= j cot'*~^a;(cosec2a;— V)dx = — — — I GoV'^x dx. n — 1 *^ Since tana; and cot a; are reciprocals of each other, the above method is sufficient to integrate any integer power of tan a;, or cot x. Another method of procedure would be to make the substitution tan x = z, whence 2" dz ft3in''xdx=f^ P If the exponent w is a fraction, say n = —^ the last integral can be rationalized -by the substitution z = u^. It is evident from this that any rational power of tangent or cotangent can be integrated. EXERCISES 1. J cot*xdx. 2. \ta.ii^(ixdx. 3. J (tan X — cot xy dx, 4. j'(taii«ar + tan«-2a;)da;. 5. I tan^ X dx. When w is a positive integer show that^ 6. f tan^n X dx = ^^^^^^ - *^B!!i!5 + •.•+(- l)«-i (tan x - x). J 2n-l 2n-3 v / v / 7. (t^n^n+i^dx = *^^^ - ^-^^+ ... + (- l)~-Kitan2a:+logcos:r). 242 INTEGRAL CALCULUS [Ch. V. 145. Tsinw* X COS" x dx, (a) Either m or n a. positive odd integer. If one of the exponents, for example m, is a positive odd integer, the given integral may be written gijjw-i ^ (3Qgn ^ gjjj xdx= — j (1 — cos^ x) ^ cos" a:c?(cos x). Since m is odd, w — 1 is even, and therefore ^ ~ is a positive integer. Hence the binomial can be expanded into a finite number of terms, and thus the integration can be easily completed. Ex. 1. j sin^a:Vcosarc?a:. According to the method just indicated this integral can be reduced to — J sin* a; Vcos x d(cos x) = — j (1 — cos2a;)2'(cosa;)^) w + ^ an even negative integer. In this case the integral may be put in the form /sin"* X C CQgm+» xdx= \ tan™ x sec'^™"^"^ x dx^ cos"* X ^ which can be integrated by Art. 142, since the exponent — (w + w) of sec a; is an even positive integer. 145.] TRIGONOMETRIC FUNCTIONS 243 Ex.7, f^^rfa:. COS^ X The integration is effected in the following steps : ^ Vcosa; cos^ x = I tan^ x (tan2 x + l)d (tan a;) = 2tanta:(^+ ^tan2a;). Ex. 8. i^-^^dx. Ex.11, f . /^ . -^ sin* a; •^ sin* a; cos^ x Ex. 9. f-;^. Ex.12, r sin^a; -^ Vsin^ a; cos^ x Ex.10. f5!:!?l^dx. Ex.13. fHH^dx. -^ sni^a: -^ cos^+^a; (c) Multiple angles. When m and n are both even positive integers, integration may be effected by the use of multiple angles. The trigo- nometric formulas used for this purpose are .2^ _ sma; cos a; = Ex. 14. j sin^a; cos* a: c?a:. 1 — cos 2 X sin^ X = ) 1 -h COS 2 X 2 ' sin 2 a; fsin^ X cos* xdx= \ (sin a: cos xY cos^ x dx _ r sin2 2a: 1 + cos 2 a; . ~~J 4 2 ^ = i f sin2 2xdx+ t^ f sin2 2 a; cos 2 a; c? (2 a:) 1 ri — cos 4 a: ,^ , i sin^ 2 x = -jJ^ a; — ^ sin 4 a: + j^ sin^ 2 a:. 244 INTEGRAL CALCULUS [Ch. V. Ex.15. J cos* ar sin 2 a; rfar. Ex.17, j sin* a; cos* arrfx. Ex.16, i silica: cos® a: rfa;. Ex.18. J (sin* a; — cos* x)*c?a;. Integrate the two following by the aid of multiple angles. Ex .19- f^ J sin* dx »a;-ncns«^). •^ • d?' + n^ By subtracting (1) from (2), the formula Ce-' cos rmdx^ """ <^ "^" ^^ + ^ ^^^ ^^^^ is obtained. EXERCISES ON CHAPTER V 1. Show that (1 + n) J 8ec'»+2 xdx = tan x sec" a: + n J sec a: dx. Integrate by parts, taking u = sec" x, dv = sec'* x dx. 146-147.] TRIGONOMETRIC FUNCTIONS 247 2. Show that (1 + n) I cosec^+2 x dx = — cot x cosec« a; + «n J cosec" x dx, J sin a; COS a; * J cos^ar l 4 C ^^ 8. fc^cos-rfa;. . [Put a; = cos ^.] 9. p^^^^dx, ^i ■ 5- J"^SiJF" "^ 10. j^'^sin" •sin a: c?a; xdx. dx 6. i :~;r~* ' 11 r^'' sin 2 a: sin a: dx. Jcoszsin^a; J [Suggestion. 2 sin 2 a; sin a; = cos x — cos 3 a:.] 12. Show that f sin aa: sin fta: ^a: = HLl^-^IL^ _ ££i^±*^ J 2(a-b) 2 (a + 6) Use the trigonometric formula sin a sin ^ = ^ [cos (a — P) — cos (a + ^)]. 13. Show that f sin ax cos 5a: ^a: = - "^,^ (^ " ^> - "",%(^ + ,^X ^ 2 (a - 6) 2 (a + 6) 14. Show that f cos ax cos 6ar dx = ^^" ^^ " ^>/ + ^^^ f ^ "^ \)^ . J 2(a-b^ 2(a + b^ 2(a-b) 2(a + b) . j sin« a: cos* a; (x)^x -f ?^ (i^xy + ^^^^(Ar)3 + .... From this, by transposing FQx')^ the increment (1) is ob- tained in the form of a series, viz., F{x-\-/:^')-F(x) = F(.)A. + A.[^A.4-^(A.y-H ...J n=/(a:)Aa;4-<^(a:)Aa;. (2) 248 Ch. VI. 148.] INTEGRATION AS A SUMMATION 249 In the last expression (/>(a;) has been written for brevity in place of the series in brackets, and f^x) is the equivalent of F'(x)^ since by supposition /(a;) is the derivative of F(x). Suppose now that the variable x starts with a given value a and increases until it reaches another given value h. The function F(x) will change accordingly, beginning with the value F(^a) and ending with F(h), The difference between these two, viz., F(h) - F{a) can be determined by the aid of (2) in the following manner. Let the variation of x from a to 5 be imagined to occur in successive steps, first from a to a-\- Ax^ then from a + Ax to a-\-2 Axy and so on. The increment which the function F(^x} takes at the first step of the change is F(a-\- Ax^ — F(a). Its value is found by giving x the value a in formula (2). That is, F(ia + Arr) - F^a} =f(a)Ax + 0(a)Aa;. The increment that F(x^ takes at the second step is F(a + 2 Ax} - F(a + Ax) =f(a + A2:)A2; + <^(« + Aa;)A2;, the right member of which is found by substituting x=a-\-Ax in (2). In like manner, by giving x the values a-\-2Ax, a + ^Ax, ..., a-{-(n — V)Ax, the additional equations are found: F{a + 3 A^^) -1^(^ + 2 Ax) =f(a + 2 A^^) Aa; + (/>(« + 2 A:r) Aa^, F(a -{-4: Ax)- FQa + 3 Ax) ^f{a + 3 A2;)Aa; + ^(a + 3 Aa;) Aa;, F{a-\-n Ax)-'F{a->rn-\ Ax)^ fia -\- n-\ Ax)Ax •f <^(a + 71—1 Aa;)Aa;. Assume a-\-nAx = 'b^ (3) 250 INTEGRAL CALCULUS [Ch. VI. and substitute in the first term of the preceding equation. The addition of the above n equations then gives = Aa:[/(a)+/(a + A:r)+/(a + 2Aa;)+ .•• +/(« + ^T^l Arr)] H-Aa;[(/)(a) + <^(a + A2;) + <^(a + 2A2:)H f-(« + /i— lAa:)]. This latter can be gotten rid of by taking its limit as Lx approaches zero. For, since ., . F"(x). , F'"{x^,. .2 , <^(^) = — ^Aa:+^^^(A2:)2+ ..., it follows that and hence, if ^ denote the numerically greatest term of the series <^(a)+(^(a + Arc)+ •-, then Ax [ \<\(h-a)^. But since, on account of (4), a"2o*=o. it follows that and hence lim = Ax'i [/(«)+/(« + ^^) +•••+/(<» + »- 1 Ax)]A2:. (5) 148.] INTEGRATION AS A SUMMATION 251 The second member of (5) is denoted for brevity by the symbol and is called the definite integral of fQc) between the limits a and h. Suppose one of the limits, say the upper limit 5, is regarded as variable, while the other has a fixed value. To emphasize this assumption concerning the variability of 5, let it be replaced by the letter x. Then equation (5) may be written lim •^(*) = Ail [/(«) +/(«> + Aa;) + - +/(a + n-16.x)-\^x + Fia). (6) Here the term FQa) has a fixed, although arbitrary, value depending on the particular choice that is made for the con- stant a. It may be regarded as a constant of integration. Formula (6) expresses in two steps the solution of the problem of determining the function F(x') : (1) Find the sum of the series of n terms fia), f{a + A:r), f(a + 2 Ao;), .-., /(a -V{n- l)A:i:), these being the values of the given function f{x) corresponding to the n equidistant values of x^ a, a + Arr, a + 2 Ax, •••, a -{- (n — 1) Aa;. (2) Find the limit of the product of this sum hy Ax, as Ax approaches zero while n increases to infinity, subject to the con- dition nAx = X — a. The addition of an arbitrary constant of integration makes the solution the most general possible. The method just formulated for determining the integral F(x) of a given function f(x) is not suitable for the actual 252 INTEGRAL CALCULUS [Ch. VI. work of integration, since, with few exceptions (cf. Exs. 1, 2 below), the summation of the series in the right-hand member of (6) presents insuperable difficulties. On the other hand, formula (5) admits of a very simple geometrical or physical interpretation in most of the applica- tions of the calculus, and herein lies one of its chief merits. It places before one a very convenient and useful formulation of many of the problems of geometry, mechanics, physics, etc., the final solution of which is most readily effected by the evaluation of the definite integral 'f(x)dx in the following manner. First obtain the function F(x) by integrating f{x)dx according to the methods already explained in the preceding chapters. Determine T'Q)) and JP(a) by substituting the limits h and a in the result. Finally subtract ^(a) from FQ)), This gives £f(x)dx = F(h^ - Fii^oL) as the value of the definite integral. Ex. 1. Given /(x) = e*, find F{x) by the method of summation. For the sake of brevity write Ax = A. Then formula (6) gives ^(^) = ^^^Q [e" + c*+* + e*-^" + - + e«+(«-i>*]A + F(a). The sum in the right member may be written gan ^. g» 4. g8» 4. ... 4. c(H-i)»]A = e«l^=^ • h 1 — e* (by the formula for summing a geometric series) 1 — c* = <-(^--i).-jf^- 148-149.] INTEGRATION A 8 A SUMMATION 253 As h is made to approach zero the factor becomes indetermi- e*— 1 nate. Its limit is found by the method of Chapter V (p. 77) to be lim ^ 4 1. A = e» _ 1 Hence ^^^^ e« [1 + e* + . . . + e(«-i)»]A = c« _ e«, and accordingly F(x) = \ e'^dx z= e'^ — e** -\- F(a) = e* -j- C, in which C{= F(a) — e«) may be regarded as an arbitrary constant of integration. Ex. 2. Given /(a:) = ax, find \ axdx by the method of summation. 149. Geometrical interpretation of the definite integral as an area. Let the values of the function f(x) be represented by the ordinates to a curve. Its equation would then be It is proposed to find an expression for the area bounded by this curve, the a;-axis, and two ordinates AP and BQ^ correspond- ing to two given values oi x^ x = a and a; = 5, respectively. Let the interval from J. to 5 be divided into n equal intervals AA^^ ^i-^g, • • •, An-\B each of magnitude Aa;, so that interval AB = h — a = nAx. At each of the points of division A, A^, - - -, B erect ordi- nates, and suppose that these meet the curve in the points P, Pj, • • •, Q, Through the latter points draw lines PB^^ PiB^, • • •, P„_i-B^ parallel to the a;-axis. A, A, A^^B FiQ. 61. 254 INTEGRAL CALCULUS [Ch. VI. A series of rectangles PA-^^ ^1^2.^ • • • is thus formed, each of which lies entirely within the given area. These will be referred to as the interior rectangles. By producing the lines already drawn, a series of rectangles SA-^<, ^1^2' • • • is formed which will be called the exterior rectangles. It is clear that the value of the given area will always lie between the sum of the interior, and the sum of the exterior rec- tangles, or, expressed in a formula, PA^ + P^A^ + ... + Pn-iB < area APQB < SA^ + S^A^ + ... + S^_,B. ^ (7) The difference between the sum of the exterior and the sum of the interior rectangles is SR^ + ^A + - + ^n-iBn = rectangle ^^1^= TQ . ^x. If the function /(a;) does not become infinite as x varies from a to 5, TQ will be finite and hence TQ • ^x will approach zero simultaneously with Aa;. Hence the limit of the sum of the exterior rectangles equals the limit of the sum of the inte- rior rectangles. From (7) it follows that the area is equal to the common limit of these two sums. To determine this sum observe that Rectangle APR^A^ = AP - AA^ =/(a) • Ax. Similarly A^P^R^A^ =/(« + Aa;) • Ax, A„_,P„_,R„B =fia + ir=l. Ax) . Ax, Adding, sum of rectangles = [/(«) +/(« + A^) + - +/(« + ^ir^Ax')-]Ax, Hence, by requiring Ax to approach zero, Area APQB = Ai'So[/(«) +/(« + ^^) + - +/('' + ^^^^^ Aa;)]Aa:. (8) 149-150.] INTEGRATION AS A SUMMATION 255 The expression just obtained for the area is identical with that occurring in the right-hand member of (5), and affords one of the simplest and most interesting of the geometrical interpretations of that formula. Thus Xb r*h f(x)dx = I ydo, (9) 150. Generalization of the area formula. Positive and negative area. Instead of taking the limit of the sum of the interior (or exterior) rectangles, a more general pro- cedure would be to take a series of intermediate rectangles. Fig. 62. Let x-^ be any value of x between a and a + Aa;, x^ any value between a 4- Aa; and a + 2 Lx, etc. Then f{x^Lx would be the area of a rectangle KLA^A (Fig. 62) intermediate between PA^ and SA^ ; that is, Likewise PA^ a), formula (8) or (10) gives a posi- tive sign to the area. On the other hand, the area is nega- tive if below the a;-axis. If the curve i/ =f{x) is partly above and partly below the jc-axis, the value of the definite integral (8) will be repre- sented by the algebraic sum of the positive and negative areas limited by this curve. 151. Certain properties of definite integrals. From the definition of the definite integral I f(x) dx as the limit of a particular sum [formula (5), p. 250], certain important properties may be deduced. (a) Interchanging the limits a and b changes the sign of the definite integral. For if X starts at the upper limit h and diminishes by the addition of successive negative increments ( — Aa;), a change of sign will occur in formula (5), giving F(ia^-Fih) = Jj(ix)dx. Hence j^f^^^ ^^ = "X'^^^^ ^^' ^^^^ (3) If c he a number between a and b (a(;r)cfo=/(?)(6-«), (13) in which f is some value of x between a and 5. This result is known as the Mean Value Theorem. (Compare Art. 45.) The theorem may be expressed in words as follows : The value of the definite integral ^Jix)dx is equal to the product of the difference between the limits by the value of the function f(x) corresponding to a certain value x= ^ between the limits of integration. 152. Definition of the definite integral when /(a?) becomes infinite. Infinite limits. In the preceding sections it has been assumed that /(a?) is always finite so long as x remains within the prescribed limits. It is now necessary to examine the cases in which /(a;) is infinite. Suppose, in the first place, that f(x^ becomes infinite at the upper limit x=b, but is elsewhere finite. In that event, 258 INTEGRAL CALCULUS [Ch. VI. take for upper limit a value x — x\ which is less than 5, a(x) •fa, Ex. 4. Prove that the area of the circle {x - hy ■\- {%f — kY — r^ is equal to /^A+r , _ ^ * 2\/r2- (x-hydx. Jn-r Ex. 5. Evaluate ♦'o < dx 32 4. a;2* F.TT 6. Evaluate dx Jx-1 Ex. 7. Evaluate V- dx (x - l)t CHAPTER VII GEOMETRICAL APPLICATIONS 153. Areas. Rectangular coordinates. It was shown in Art. 149 that the area bounded by the curve y =/(a;), the a:-axis, and the two ordinates a; = a, a: = 5, is represented by the definite integral jj(x)dx^j^ydx. ■ (1) In an exactly similar manner it can be shown that the area limited by the curve, the y-axis, and the two abscissas «/ = «, «/ = yS, is represented by xdy, (2) £• It was remarked at the end of Art. 150 that when h is greater than a the integral (1) gives a positive or negative result according as the area is above or below the a;-axis. Similarly, if /S>a, the integral (2) gives a positive or negative result according as the area which it represents is to the right or left of the y-axis. Whenever it is required to determine the area of a figure which is partly on one side and partly on the other side of the coordinate axis, it is necessary to calculate the positive and the negative areas separately and add the results, each taken with a positive sign. [Cf. Ex. 5, p. 262.] 154. Second method. Another method of determining the area is based on the result of Art. 10, p. 23. It was there shown that if z represents the area measured 260 153-154.] GEOMETRICAL APPLICATIONS 261 from a fixed ordinate AP (at a; = «) up to an ordinate MB' corresponding to a variable abscissa x, then the deriva- tive of area with respect to X is equal to the function f(x) ; that is or, in the differential nota- tion, dz = i/dx =f(x)dx,' The area z may accordingly be found by integrating/(a;). Hence z = \f(x)dx + C. Fio.64. The value of the constant of integration Q is determined by the condition that when x = a^ z must be zero, since in that event the ordinate MJSf coincides with the initial position. Ex. Find the area bounded by the curve y = log x, the ar-axis, and the two ordinates a; = 2, a; = 3. Axesi APNM= (log xdx-\- C = x(\ogx-l)-\-C. Since the area is zero when a; = 2, it follows that = 2 log2-2 + C, ■ whence C = 2-2 1og2. Accordingly X (log a; - 1) + 2 - 2 log 2 represents the area measured from the ordinate a: = 2 up to the variable ordinate MN. When ar = 3 the required area 1) + 2 - 2 log 2 = log -2;^ - 1. Fig. 65. is found to be 3 (log 3 262 INTEGRAL CALCULUS [Ch. VII. EXERCISES 1. Find the area bounded by the parabola y = 4 ax% the a:-axisj and the ordinate x = h. 2. Find the. area of the triangle formed by the line - + |= 2 and the coordinate axes. 3. Find the area between the a:-axis and one semi-undulation of the curve y = sin x. 4. Find the area bounded by the semi-cubical parabola y^ = ax^ and the line x = 5. 5. Find the area between the curve y — sin^ x cos x and the a;-axis, from the origin to the point at which a; = 2 tt. Fig. 66. An examination of the curve will show that the area is partly above and partly below the z-axis. The curve crosses the axis at x = -, and at a: = 2 The first portion of area, which is positive, is obtained by integrat- ing from to ^. The result is \. The next two portions of area are negative, and are calculated by integrating from - to — ^. The result ia 3^ |. The last portion, which is positive, is found, by integrating from to 2 TT, to be \. Hence total area = |4-f + | = f. 6. Find the area between the a;-axis and the curve y — a sin 4 x, from the origin to x = tt. 7. Find the area bounded by the cubical parabola y = x*, the ^-axis, and the line ^ = 8. 154-165.] GEOMETBICAL APPLICATIONS 263 8. Find the area bounded by the parabola y = x^ and the line y = x. [Cf . Ex. 3, p. 259.] 9. Find the area bounded by the parabola y = x^ and the two lines y = X, and y — 2 x. 10. Find the area bounded by the parabola y"^ = 4:px and the line x= a, and show that it is two thirds the area of the circumscribing rectangle. What is the area bounded by the curve and its latus rectum? 11. Find the area of the circle x^ + y"^ + 2 ax = 0. 12. Find the area bounded by the coordinate axes, the witch y = — , and the ordinate x = Xy By increasing x^ without limit, find the area between the curve and the ar-axis. 13. Find the area of the ellipse ^ + ^L = i. '14. Find the area of the hypocycloid x'^ + ys z= a*. 15. Find the area bounded by the logarithmic curve y = a*, the a:-axis, and the two ordinates x = x^ x = x^. Show that the result is proportional to the difference between the ordinates. . Precautions to be observed in evaluating definite integrals. The two methods just given for determining plane areas are essentially alike in the processes required, namely : (1) to find the integral of the given function f{x) ; (2) to substitute for x the two limiting values a and 5, and subtract the first result from the second. Erroneous results may be reached, however, by an in- cautious application of this process. In practical problems, the case requiring special care is that in which f(^x) becomes infinite for some value of x between a and h. When that happens, a special investiga- tion must be made after the manner of Art. 152. 264 INTEGRAL CALCULUS [Ch. vn. Ex. 1. Find the area bounded by the curve y(x- 1)^ = c, the coordi- nate axes, and the ordinate a; = 2. A direct application of the formula gives C^ cdx c ~|2 ^ area = \ — ^-^ — = — = — 2 c, Joix-iy x-Uo ]h is a sign of substitution, indicating that the values a ft, a are to be inserted for x in the expression immediately preceding the sign, and the second result subtracted from the first. This result is incorrect. A glance at the equation of the curve shows that/(a;)j ==—£—— becomes infinite for x = \. It is accordingly £C=2 Fia. 67. necessary to find the area OCPA (Fig. 67) bounded by an ordinate AP corresponding to a value x = x' which is less than 1. For this portion the area f(x) is finite and positive, and formula (1) can be immediately applied, with the result area Jo(x-iy (x-l)Jo x'-l If now x' be made to increase and approach 1 as a limit, the value of the expression for the area will increase without limit. A like result is obtained for the area included between the ordinates X = 1 and X = 2. Hence the required area is infinite. Ex.2. Find the area limited by the curve y^ (x^ - a^y = 8 x*, the coordinate axes, and the ordinate a; = 8 a. 165.] GEOMETRICAL APPLICATIONS 2x 265 Since /(a:) : . becomes infinite for x = a,it is necessary in (a;2 - a^)^ the first place to consider the area OP A (Fig. 68) and determine what B X Fia. 68. limit it approaches as ^P approaches coincidence with the ordinate X = a. Accordingly area OPA = C ^^^^ = d(x^ - a2)il"' = 3(a:'2 _ a2)i ^ 3 al, 0, y = b sin , complete arc = J Vl — e^ cos^ ^ d<^ = 2 ATT [1 - I .4_ ... 64 ], by expanding v 1 — e^ cos^ into a series and integrating term by term. y Ex. 7. Find the length of arc of the curve x — e^ sin 6, y = e^ cos 6 from ^ = to ^ = ^1. fi^ 160. Area of surface of revolution. Let ^^ be a con- tinuous arc of a curve whose equa- tion is expressed in rectangular coordinates x and y. It is required to determine a formula for the area of the surface generated by revolving the arc AQ about the ~ a;-axis. It has been shown in Art. 86, p. 140, that if 8 denotes the area of the surface generated by the rotation of AP (P being a variable point with coordinates (rr, y)), then Fig. 74. dS ds = 2 7r^. from which f— W..(|T. and f-'W.*(|)- 159-1*60.] GEOMETBICAL APPLICATIONS 275 Hence, by integrating these two expressions, surface = 2 rr,j^y^^lj^(^Jdx the limiting values of x and y being the coordinates of the points A and Q. That the result of integration is to be evaluated between the limits a and h (or a and y8) is readily seen by following the suggestions made in Art. 154. For, denoting the indefinite integral by (f> (x) + (7, since the area is evidently zero when a: = a (^.e., when the point P coincides with A) it follows that = as about either axis. Ex. 5. Find the volume obtained by the revolution of that part of the parabola Vx -f- y/y = y/a intercepted by the coordinate axes about one of those axes. J Ex. 6. Find the volume generated by the revolution of the witch y = 2^ — . about the x-axis. ^ a«+4a2 161-162.] GEOMETRICAL APPLICATIONS 279 Ex. 7. Find the volume generated by the revolution of the witch about the ?/-axis, taking the portion of the curve from the vertex (x = 0) to the point (x^, y^). What is the limit of this volume as the point (x^, y^ moves tov^ard infinity ? Ex. 8. Find the volume obtained by revolving a complete arch of the cycloid a; = a (^ — sin ^), y = a(l — cos &) about the ar-axis. Volume = TT f ''''y^dx = Tra^ f ''(I - cos dydO. Ex. 9. Find the volume obtained by revolving the cardioid p = a{l — cos 6) about the polar axis. Assume a; = p cos ^, ^ = p sin 6. Then dx=: d(p cos 6) = d[a{l — cos 0) cos $] " = asin^(-l + 2cos^) V1 + taii2 V 1+f^y \dx, From this follows log(y+-\/y^-a^) + a When the tangent is parallel to the a:-axis the ordinate itself is the perpendicular a. If this ordinate be chosen for the y-axis the point (0, a) is a point of the curve, and hence C = — log a. The equation can accordingly be written Fig. 78. (1) y + y/f From this follows, by taking the reciprocal of both members, = e «> or, rationalizing the denominator, (2) ^ ^ = e «. ^ ^ a 2 Adding (1) and (2) and dividing by -> * _x y = ^(e'a + e «), which is the equation of the catenary. Ex. 11. A right circular cone having the angle 2 ^ at the vertex has its vertex on the surface of a sphere of radius a and its axis passing through the center of the sphere. Find the volume of the portion of the sphere which is exterior to the cone. X^ ^2 Ex. 12. Find the volume of the paraboloid —-\-^=z cut off by the plane z = c. ^ 284 INTEGRAL CALCULUS £Ch. VII. EXERCISES ON CHAPTER VII 1. Find the area bounded by the hyperbola xy — a\ the a:-axis, and the two ordinates x = a, x = na. From the result obtained, prove that the area contained between an infinite branch of the curve and its asymptote is infinite. Cj 2. Find the area contained between the curves y^ = x and a^ = y, 3. Find the area of the evolute of the ellipse (ax)^ + (by)^ = (a"^ - b^)l 4. Find the area bounded by the parabola Vx + y/y = Va and the coordinate axes. 5. Find the area contained between the curve ^ a + x and its asymptote x = — a. [Hint. The integration may be facilitated by the substitution X = a cos $.'] 6. Find the area between the curve y\y^ — 2) = a: — 1 and the coordinate axes» V 7. Find the area common to the two ellipses 8. If (a, a) and (6, P) are two pairs of values of x and y, the formula for integration by parts gives J y dx = bp — aa ~ i xdy. Interpret this result geometrically in terms of area. 9. Find the area bounded by the logarithmic (or equiangular) spiral p = e«« and the two radii p^, p^ 10. Find the length of an arc of the spiral of Archimedes p — aO f between the points (pj, ^,), (pg, 6^. J 11. Find the surface of the ring generated by revolving the circle a:2 + (y - ky = a^ {k>d) about the a>axi8. 12. Find the volume of the ring defined in Ex. 11. 162.] GEOMETRICAL APPLICATIONS 285 13. Find the volume obtained by revolving about the a;-axis that X X portion of the catenary ^ ~^ (e" + e~*) limited by the points (—x^, y^) and (xj, y^, 14. Find the entire volume generated by the revolution of the cissoid a — X X / about its asymptote. i \ * [Hint. For the purpose of integration, assume tyt* x = 2a sin2 6, whence y = 1^1^. 1 cos 6 J 15. Find the surface generated by the rotation of the involute of the ^ circle x = a(cos t + t sin t), y = a(sin t — t cos t) about the a:-axis from t = 0\>o t = ty 16. Find the volume generated by the revolution of the tractrix (see Ex. 6, p. 281) about the positive ar-axis. 17. Find the area of the surface of revolution described in Ex. 16. , / 18. Find the length of the tractrix from the cusp (the point (0, o)) to the point (xj, y^). CHAPTER VIII SUCCESSIVE INTEGRATION 163. Functions of a single variable. Thus far we have considered the problem of finding the function y oi x when -^ only is given. It is now proposed to find y when its nth derivative -^ is given. The mode of procedure is evident. First find the func- tion ^_^ which has -y^ for its derivative. Then, by inte- d^'-^y grating the result, determine ■, n_2 , and so on until after n successive integrations the required result is found. As an arbitrary constant should be added after each integration in order to obtain the most general solution, the function y will contain n arbitrary constants. Ex. 1. Given ^4 = -«» ^^^ V' dx^ x^ ^ Integration of — with respect to x gives 3r H^o+C'^ dx^~ 2x2 Integrating a second time, dx 2x * * and finally y = i log a: + i C^x^ + CgX + C^ The triple integration required in this example will be symbolized by which will be called the triple integral of — with respect to x. Ch. VIII. 163.] SUCCESSIVE INTEGRATION 287 Ex. 2. Determine the curves having the property that the radius of curvature at any point P is proportional to the cube of the secant of the angle which the tangent at P makes with a fixed line. If a system of rectangular axes be chosen with the given line for a;-axis, it follows from equation (6), p. 164, and from Art. 10, that in which a is an arbitrary constant. This equation reduces to d^ y dx^ from which follows :^a, y = I Ja(rfx)2 =? a[|' + C^x + C2], C, and C2 being constants of integration. Hence the required curves are the parabolas having axes parallel to the y-axis. The existence of the two arbitrary constants Cj, Cg in the preceding equation makes it possible to impose further conditions. Suppose, for example, it be required to determine the curve having the property already specified, and having besides a maximum (or a minimum) point at (1, 0). Since at such a point -^ = 0, it follows that dx = a(l + Ci), whence Cj = — 1. Also, by substituting (1, 0) in the equation of the curve, = a(i-l + C2), from which Cg = i- Accordingly the required curve is Ex. 3. Find the equation (in rectangular coordinates) of the curves having the property that the radius of curvature is equal to the cube of the tangent length. [Hint. Take y as the independent variable.] 288 INTEGRAL CALCULUS [Ch. Vlll. Ex. 4. A particle moves along a path in a plane such that the slope of the line tangent at the moving point changes at a rate proportional to the reciprocal of the abscissa of that point. Find the equation of the turve. Ex. 5. A particle starting at rest from a point P moves under the action of a force such that the acceleration (cf. Ex. 14, p. Ill) at each instant of time is proportional to (is k times) the square root of the time. How far will the particle move in the time t'i 164. Integration of functions of several variables. When functions of two or more variables are under consideration, the process of differentiation can in general be performed with respect to any one of the variables, while the others are treated as constant during the differentiation. A repe- tition of this process gives rise to the notion of successive partial differentiation with respect to one or several of the variables involved in the given function. [Cf. Arts. 68, 72.] The reverse process readily suggests itself, and presents the problem : Griven a partial (^first^ or higher) derivative of a function of several variables with respect to one or more of these variables^ to find the original function. This problem is solved by means of the ordinary processes of integration, but the added constant of integration has a new meaning. This can be made clear by an example. Suppose u is an unknown function of x and y such that dx Integrate this with respect to x alone, treating y at the same time as though it were constant. This gives in which ^ is an added constant of integration. But since y is regarded as constant during this integration there is nothing to prevent from depending on it. This depend- 163-165.] SUCCESSIVE INTEGRATION 289 ence may be indicated by writing <^(y) in the place of (j). Hence the most general function having 2 a: 4- 2 ?/ for its partial derivative with respect to x is U = X^+2X7/ -^(l>(7/}, in which <^(y) is an entirely/ arbitrary function of y. Again, suppose dxdy Integrating first with respect to y^ x being treated as though it were constant during this integration, we find where '>^(x) is an arbitrary function of x^ and is to be regarded as an added constant for the integration with respect to y. Integrate the result with respect to a;, treating y as con- stant. Then Here («/), the constant of integration with respect to x^ is an arbitrary function of «/, while '^(x)=^'>\r(x)dx. Since '>^(x) is an arbitrary function of x^ so also is "^(x). 165. Integration of a total differential. The total differen- tial of a function u depending on two more variables has been defined (Art. 69) by the formula du=^^dx^^-^dy. dx dy ^ The question now presents itself: Given a differential expression of the form Fdx + Qdy, (1) 290 INTEGRAL CALCULUS [Ch. VIII. wherein P and Q are functions of x and y^ does there exist a function u of the same variables having (1) for its total differential P It is easy to see that in general such a function does not exist. For, in order that (1) may be a total differential of a function «*, it is evidently necessary that P and Q have the form P = ^, Q = ^. (2) dx By What relation, then, must exist between P and Q in order that the conditions (2) may be satisfied ? This* is easily found as follows : Differentiate the first equation of (2) with respect to y^ and the second with respect to x. This gives dP^^u_ dQ ^ d^u dy dydx dx dxdy from which follows dP^8Q (g. dy dx ^ ^ This Is the relation sought. The next step is to find the function u by integration. It is easier to make this process clear by an illustration. Given (2x + 2y-\-2')dx+(2y + 2x-\-2)dy, find the function u having this as its total differential. Since P=2a; + 2y + 2, Q = 2y + 2x + 2, it is found by differentiation that ^ = 2 and ^=2, dy dx and hence the necessary relation (3) is satisfied. From (2) it follows that |^=2a; + 2y + 2. dx 165.] SUCCESSIVE INTEGRATION 291 Integrating this with respect to x alone, u = x^-{-2xy + 2x + (l>(iy). (4) It now remains to determine the function <^(^) so that ^(=(?)=^^ + 2^ + 2. (5) Differentiate (4) with respect to y alone, whence dy where \y) denotes the derivative of <^C^) with respect to y. The comparison of this result with (5) gives 2y4-2r?:+2 = 2rr + Jo x^ + y^ ' 168. Plane areas by double integration. The area bounded by a plane curve (or by several curves) can be readily ex- pressed in the form of a definite double integral. An illus- trative example will explain the method. Ex. 1. Find by double integration the area of the circle (a: — ay + (y-6)2=r2. Imagine the given area divided into rectangles by a series of lines parallel to the y-axis at equal distances Ax, and a series of lines parallel to the ar-axis at equal distances Ay. The area of one of these rectangles is Ay • Ax. This is called the element of area. The sum of all the rectangles interior to the circle will be less than the area required by the amount X contained in the small subdivisions which bor- der the circumference of the circle. By a method exactly analogous to that used in Art. 149, it is easy to show that the sum of these neglected portions has a zero limit when Ax and Ay are both made to approach zero. To find the value of the limit of the sum of all the rectangles within the circle it is convenient to first add together all those which are con- tained between two consecutive parallels. Let P1P2 be one of these parallels having the direction of the x-axis. Then y remains constant Fig. 79. 167-169.] SUCCESSIVE INTEGRATION 295 while X varies from a — Vr^ — {y — b)'^ (the value of the abscissa at P,) to a + Vr- — (y — b)'^ (the value at Pg)- The limit as Ax approaches zero of the sum of rectangles in the strip from PJ^^ i^ evidently (1) Az/[limit of sum (Aa; + Aa; + •.•)] = A?/ C-+^^^^-^y-^ ^^^ Now find the limit of the sum of all such strips contained within the circle. This requires the determination of the limit of the sum of terms such as (1) for the different values of y corresponding to the different strips. Since y begins at the lowest point A with the value 6 — r, and increases to 6 + r, the value reached at B, the final expression for the area is \ dy ) , dx:=^ \ \ dy dx. »'h-r •^a-Vr2-(y-6)2 ^h-r -^ a-Vr2-{y—b)2 Integrating first with respect to x, •^a-v'r2-(y-6)2 Jo-s/r2-(y-6)2 This result is then integrated with respect to y, giving C'^''2Vr^-(y-bydy = (y -b)Vr^ - (y - by + r^ sin-i^^l ''*"'= irr^. Jh-r f- Jb-r If the summation had begun by adding the rectangles in a strip paral- lel to the 2/-axis, and then adding all of these strips, the expression for the area would take the form V X a+r rb+Vr2~(x-a)2 \ dxdy. r •^6-V'r2-(x-a)2 It is seen from this last result that the order of integration in a double integral can be changed if the limits of integration be properly modified at the same time. Ex. 2. Find the area which is included between the two parabolas 2^2 = 9 a: and 2/2 = 72 - 9 x. . Ex. 3. Find the area common to the two circles a;2 _ 8 a: + 2/2 - 8 2/ + 28 = 0, a;2 - 8 X + 2/2 - 4 y + 16 = 0. 169. Volumes. The volume bounded by one or more surfaces can be expressed as a triple integral when the equations of the bounding surfaces are given. 296 INTEGRAL CALCULUS [Ch. vni. Let it be required to find the volume bounded by the surface ABC (Fig. 80) whose equation is z=f(x^y^^ and by the three coordinate planes. Imagine the figure divided into small equal rectangular parallelopipeds by means of three series of planes, the first series parallel to the ^2-plane at equal distances A a;, the FiQ. 80. second parallel to the rca-plane at equal distances Ay, and the third parallel to the iry-plane at equal distances A2. The volume of such a rectangular solid is AxAi/Az; it is called the element of volume. The limit of the sum of all such elements contained in OABO is the volume required, provided that the bounding surface ABQ is continuous. 169.] SUCCESSIVE INTEGBATION 297 (The reader can easily show that the sum of the neglected portions is less than the volume of the largest plate formed by two consecutive parallel planes and that its limit is therefore zero.) To effect this summation, add first all the elements in a vertical column. This corresponds to integrating with respect to z (x and y remaining constant) from zero to f(x^ ^). Then add all such vertical columns contained between two consecutive planes parallel to the ?/2!-plane (x remaining constant), which corresponds to an integration with respect to y from y = to the value attained on the boundary of the curve AB. This value of y is found by solving the equation f(x^ ^) = 0. Finally, add all such plates for values of x varying from zero to the value at A, The final result is expressed by the integral ax ay az. in which ^{x) is the "result of solving the equation /(a;, ^) = for y^ and a is the a;-coordinate of A. Ex. 1. Find the volume of the sphere of radius a. The equation of the sphere is a;2 + y^ + 2:2 _ ^2^ or 2 = Va2 _ a;2 - y\ Since the codrdinate planes divide the volume into eight equal por- tions, it is sufficient to find the volume in the first octant and multiply the result by 8. The volume being divided into equal rectangular solids as described above, the integration with respect to z is equivalent to finding the limit of the sum of all the elements contained in any vertical column. The limits of the integration with respect to z are the values of z correspond- ing to the bottom and the top of such a column, namely, 2 = 0, and z — y/a^ — x^ — y\ since the point at the top is a point on the surface of the sphere. 298 INTEGRAL CALCULUS [Ch. VIII. 169. The limits of integration with respect to y are found to be y = (the value at the a;-axis), and y = Va^ — x^ (the value of y at the circumfer- ence of the circle a'^ — x^ — y^ = 0, in which the sphere is cut by the zy-plane). Finally, the limiting values for x are zero and a, the latter being the distance from the origin to the point in which the sphere intersects the X-axis. Hence F (= volume of sphere) = 8 \ I \ dxdydz. Integrating first with respect to 2, /•a /'VaZ— 12 , F=8j^j^ yJa^-x'^-y'^dxdy\ then with respect to y, V = 8 i''dx\y. yJa^-x^-y^ + "' " ^' sin"! ^ l^"^^ 47ra8 -r |(a2-x2)rfx= g Ex. 2. Find the volume of one of the wedges cut from the cylinder a:^ 4- y2 _ ^2 ijy tiig planes 2 = and z = mx. / Ex. 3. Find the volume common to two right circular cylinders of the same radius a whose axes intersect at right angles. Ex. 4. Find the volume of the cylinder (a: - l)^ + (y - 1)2 = 1 limited by the plane 2 = 0, and the hyperbolic paraboloid xy = 2. Ex. 5. Find the volume of the ellipsoid a^ b^ c^ Ex. 6. Find the volume of that portion of the elliptic paraboloid 2=1-^^-2? which is cut off by the plane 2 = 0. ANSWERS Page 20. Art. 9 6 a; -4. 3. -^. 4a;2 4. 4a:3_6 37* Page 25. Art. 11 14«_4-33«2. 3. 12m2_2. 4. 4x-5. 1. 2 a; -2. 1. 162/2-2. Page 28. Art. 13 2. Inc. from — co to ^ ; dec. from | to 1 ; inc. from 1 to + oo ; ^ and 1. 3. Two. +l3itx=l±V^; -I atx = l±V^. 4. ±tan-i^V Page 29. Art. 14 2. (6u-4)6x2. 3. -^(lOx-2). 4. (g n - A_^ (a.2 - i). Page 37. Art. 19 10. J4±A. 2Vx + 2 J, Va(Vx — Vq) ^ 2\/x(Vx + a)(Va + \/^)2 12. 1 13 1. 10ic9. 2. -8x-9. 3. c 2Vx 4. 1 1 5. -iV^. 6. w(x + a)'*-i. 7. MX«-i. Vl - x2 (1 - a;) 1 8. (a2 14. 2 xil - a;2) + VI - y/-^ ia^ + Sx^ iVx^a^ + x'^ 2-6a;-a;2 15, ^^ . (a;2 + 2)2 * ' a;Vl-a;2 299 300 AN 8 WEBS 16. 17. (1 - aj2)i(l 4- a:2)f ' dx 26. (2M + 6a;M)^+3M2 + 4a:8. (x«-l)2' wwn-i^ Oft dx nu^ 18 wi(6 + a;) + n{a-{-x) (a + a;)'»+i. (6 + a;)«+i in -2 x'^i:^ + 1)1 20. 66a;8(a;a + l)i 21. dx (a + x)'» (a + x)«+i 27. 2 wa;8w7 ^ + Wh^^ — + 3 waxa^;. dx dx jQ — &% __ 6x «'2^ «\/a^ - x=^ 82. (0,0), f-i-,~-§_V ^ V9a 27 ay* 22. 12(u2-t* + l)^. ^^" ^^«>' ^ ^ ^dx ^ (21t/8-i9zf)10x 23. muKl-^u^f^^ (7te2 + 5)t dx 35. At right augles at (3, ± 6). Page 43. Art. 24 1. -4- 12. Iogio6 2^ + 7 as + a x2 + 7a6 __« 18. 2^ ax + 6 xlogx 8x-7 14. ae*-. 4x2-7x + 2 15. 4e4«+« 2 _i_ r^* 16. -=-i^. (H-a;)2 4x 1 - X* 17. (1 + e*)2 6. logx + 1. 18 y_3a;2ex. 7. wx«-i logx + x«-i. 19. 1 - j/2. 8. 7ix"~^ log x« + mx""i. 20 e* + e~* ^ ^^--1 21. ^^''^ 1 x + e' 10. 2("v/x + 1) 22. wx"-! a*+ ai^a* log a. 11. log.6. 12xv/2T^-l . 23. ^:;« 2 V2 + x(3 X- - V2 + X) Vx(a - ») ANSWERS 301 24. - 25. 27. 10. 11. 12. 13. 14. 15. 16. 17. x([ogxy 2 log a; X xlogx x^^Qogx^ 1). X Page 47. 7 cos 7 «. — 5 sin 5 x. 2 X cos a;2. 2 cos 2 oj cos X — sin 2 a; sin x. 3 sin2 x cos X. 10 X cos 5 x^. 14 sin 7 a: cos 7 a;. sec-^ a; (tan2 a; — 1). 3 sin2 x cos^ x — sin* x. secx(tanx + secx). - 6 X (1 - 2 x2) sin (1-2 x2)2 cos (1-2 x2)2. -20 X (3 - 5 x2) sec2 (3 -5 x2)2. 2 tan X sec2 x — 2 tan x. secx. cot Vx 2Vx 1 ^ ^ — rloga • ««• sec2(a«). X2 w sin**-! X sin (n + 1) x. 31. Art. 19. -(a;_l)f(7a;2 4.30x-97) 12(x-2)i(x-3)V- 2 + X - 5 x2 2Vl -X 1 + 3 x2 - 2 X* (1 - x2)i 5x4(a + 3x)2(a-2x) (a2 + 2ax-12x2). (x — 2 g) Vx + q , (x-a)t 31 ?nn sin"*-i nx - cos (m — n) x cos'*+i mx 2 1 + tanx 21. csc^ dx 22. cos (sin u) cos w dx 23. 24. 25. 26. 27. 28. 29. 30. 31. 2 ae«* sin e"* • cos e«*. e* • cos e* • log X + 5HLi!. X cos x2 Vsin x2 'sin X + cosx log x^ — 8 csc2 4 X cot 4 X. 8(4 X - 3) sec (4 x - 3)2 tan (4 X - 3)2. 3 x2 sin x^. sec Vx tan y/x 18. cos 2 ?« dx - 2 X CSC2 X2 + ' y cos xy 1 — X COS xy — CSC2 (X + y). 2Vx 4x Vl-X2 Page 49. Art. 3. 33 V6 X - 9 x2 3 vr 302 ANSWERS 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 16. 16. -2 1+X2 17. sin-ix+- vT-^2 2 Vsin-i x Vl — a^ 1 xVl -(logx)2 sec^x Vl - tan2x 1 2 vT xV^-1 2 l + a;2* ^ Vl + CSC X. 1ft gtau z l + X^ 19 2 VI -x2 20. -2 x2 + l 21. 1 2Vx(x+ 1) 22. -2 e^ + e * 23. n C0S2x+ W2sftl2x 24. 2. 25 2sinx Vl-4cos2x 26. -1 2(1 + x2) 27. -1 sec2 X • tan-i x + tanx 1 +X2 Page 51. 6x + 15x2. -6 15 x8 X** 3x-l VT-i^ 28. 0. Exercises on Chapter 11 2Vir-~3 a2 - 2 x2 Va2 - x2 log sill X + X cot X. 9. — cotx. 2Vm 10. 1-loga. X 11. -C3x + x8) (1 + a;2)! 12. c*(cos X - sin «) -1 xWa^ - y? 2xg-2g + l ^ 2rx-aJ2)* 18. 14. xV2x-l 4 6 + 3 cos X 16. tan-i-^. 16. 2(1 + a^) AN S WEBS 303 17. 4tan5a;. 26 2 xy^ + 3 x^ _ \o3 — 2 g^xy XX. 12. (l-x)6 48 X 13. sin X. 14. 15. 8(ex_e-x) (ex + e-xy 8 x2e2x. 16. 4! X2* 17. a"e«^. 18. r-l)«w! (x - l)«+i (2/2 _ ax)3 25 -yr(^-l)^ + (y-l) !I. X2(?/ - 1)3 26. '^~ ^ . e^v. (2 - 2/)3 32. (- l)«.2"-i.»il [ 1 1 U2x-l)'*+i (2x+l)«+i 33. (^-^)^ X 34 2(-l)n.ri! (l+.x)«+i 36. 2»»-icos(2x+— y 304 ANSWERS Page 64. Art. 40 6. -8 + 4(y-3) + 3Cy-3)2. Page 67. Art. 41 1. x-\-— + —x^+—x- + B. 8. 1-^-^+B. 3 15 315 2 8 « , , a;2 a;3 3x4_llx5 , „ "^^ I"^ 8 30 9- l+2x + 2a;2 + 2x8+i?. 7. ^ + |- + |'+^+^- 10. x4 + 7x3 +11x2 -16 a; -41. Pages 75-76. Exercises on Chapter IV hi /,3 1. COS X — ^ sin X cos X + — • sin X + ^. ^ o ! 2. tan /i + X sec2 /^ + x^ sec2 ;^ tan A + — sec2 A (1 + 3 tan2 h) + B. 4. logx + ^-^ + ^-^ + i?. ' ^ X 2X2 3X3 4a;4^ 6. X6 + 5 X*y + 10 X3y2 + 10 x22/3 4. 5 xy4 + y6. Pages 79-80. Art. 47 2. 2 a2 a2 + 62 3. Pages 83-84. Art. 49 *-^- 8. f. 8. -4. 12. 12&^. 16. 1. log 6 4. 4. 9. 0. "f- 17. 1. 6. f 10. 2. n 18. -f 7. i. 11. 8. 15. 1. 19. h Pages 87-88. Art. 52 1. 1. 6. log a. 9. 0. 12. -i t 0. 6. -f 10. or 00 according As n > 0. 8. 0. 7. 4a« or<0. 18. h- 4. 6. 8. 1. 11. f 14. -1. ANSWERS 305 Page 89. Art. 54 1. 1. 8. e«*. 6. 1. 7. 1. 9. e«. 4. 1. 8_ X «• 1- Pages 89-90. Exercises on Chapter V 10. 0. 1. 1. 5. 0. 2. 0. 6. 0. 3. 00 if r>l, Oif r/2 2x — a 6. Vr^^ + 8in-ix. 7. v^^r=:¥2+|sin-i— ^ . ,, -, 1 , rV2+Vx2+2x4-3~l 8. Vx2+2x+3-log(x+l + Vx2+2x+3)--— log[ ^ ^_^^ J 9. Vx^ + x+l-^log[x + i + Vx'' + x + l]-log[ ^~'^'^^^'^'^^'''^^ ^ 1 ^ 14. -log(c-* + v^2*-l). 10. log-(x2+l + Vx* + x« + l). rV2W^T2-| 16. log • 11. c< 2V2 I « -• n. iiog^— ^. 18. ilog(6x» + 12x + 5). e* + l 18. ix5tan-ix-,«)yX* + TJff«a-Tifflog(x2 + l). 19. « - log (a? + 1) + 2 tan-» ». ANSWERS 311 20. -^— -^[2 + 2xloga+(xloga)2]. a* (log a) 3 21. tan^-sec0. 23 1 iog(acos2a;+6sin2a;). 22 -cot^. 2<^^-«> 2 24. ^[x — log(sinx + cosx)]. 25.' i(»^^ + 2)Vx2-l. 26. 6[^xH ix - ^x^ + -Jx* - 1 x^ + Ixi - x^ + log(xi + 1)]. 1 /I i 30. tan-i(logx). ' 27. -^vl -logx. 1 3j i 28. log Ve^' + 1. • 6 (a -ft tan x)' .0. Sin-. (-11^). »- i--[^-^^^S^J- 33. 2 Vx vers-i- + 4\/2 a - x. \ 34. -lV 3x2 + 2x + l +log P + ^+^-^^^+ -^^+l1. ^ X L a; J 85. -^.og(. + V^^-^r^+J^^. Pages 219-222. E:sercises on Chapter II 2 Vx'-^ - 2 X - 3 - 2 log (x - 1 + Vx2-2x-3). 3. ^-^V2ax-x^-f«-sin-i^::i«. 2 2 a 5. The arrangements which can be used are [5], [C], [5], [O], and [5], [^], [C], [CI 6. if— ^— +itan-i^l. 9. 5^'-v/^2Tr^ + ^sin-i?- , ^ I a;2 + 4^ 2j 2 2 a. 7. _l^.IlJ_+-±_tan-i2^Ill. 10. -^Z^. 3(x2-x+l) 3V3 \/3 «'^ 8. _^Va-rx-^ + f sin-i|. " "' 3 , ^.f^ ,)f + ^^^Sf^* 12. ^(2x2 + 5 a2) V^M^ +i|liog (X + Vx2 + «2). 8 o 13. ^ Vx2 + a + - log (x + Vx2 + a). 14. K2 aj2 - «a; - 3 a^) V2ax-x^ + ^ sin-i^^=-^. 15 (2 «x - x2)^ Ig vl + a;^ 3ax8 ' "2x2 312 ANSWERS J- 3(x + 2)8-5(x + 2) I 3 , x + 1 8(x2 + 4x + 3)2 ^16 ^x + 3 18. i(x+ l)Vl-2x-x2 + sin-i^^. V2 Page 226. Art. 133 2a x+a 2 2^x+l ^ ^^g x'cx"^- 1 ) • ^' x + \og(x-a)\x-b)\ 6. ?-±^ log (X - 2 - v/3) - ?—2^ log (X - 2 + V3). 2>/3 2V3 ^^ ^(2x + l)(x + 2) ° x-c 9. x + -^— [a2 1og(x + a)-62log(x + 6)]. 6 — a 10. log[(x + 2)V2x-l]. 13. ^_7x + 641og(x + 4) 11. log (^-«)Cx+ft) . !271og(x + 3). X 12. ^log ^ -. • 14. JLlog^^. * ^ (2 + X) (1 - x)6 2ab ax + b 15. sin-i-^ — sin-i ^^~^)^^ - log (x - 1)"^(2 - x + V2^=^). V2 v^ X - 2 16. log (X + 2 + y/x^ + 4 X + 7) - — log(x + 2)"\ V3 + Vx2 + 4x + 7) V3 + log (x + 3)"\l - X + 2Vx2 + 4 X + 7). Page 227. Art. 134 2. +^log^^^±-l. 9. ax-i + log 2(x-l) X— 1 X x + a 8. x-51og(x-3)-p^. 10. x+-^-K281og(x+3)-logx]. ^—^ 3x 2(a2-xa) 11- -a^og -1 X V2x(x + V2) l**- 2 log[x - 1 + Vx^-2x + 2] «• f-2x + ^^«+.og.(.+l)'. _.„,[- lW.»_-2x^2 -| 7. log(x3-a2)--,^^. +-^ Vx^-2x + 2. x* — a* X — 1 8. —7-^^ 18. log(x-a) + i^5!_=_i«?. 4(>/2 + l-x)2 ^^'^ ^ 2(«-a)« \ ANSWERS 313 Page 228. Art. 135 4. 3^,[log(a= + a) 7. -^log(a;2-ax+a2) tan- V3 2 a;2 + 1 , /o. i2a^-an 8. l_tan-i-. + V3tan-i— — ^J. 2a{x-a) 2 a^ a 6. -Itan-i^ + ltan-i?. 9. x-log ^'+^^ + ^ a a b b x—1 Page 230. Art. 136 2. tan-ia; + — ^^ • 6. _— ^— _ + log (x^ + a^) x2 + l 2(aj2 4.a2; ^^ ^ 3. Atan-i^+ ^-^^^ . -i-tan-i?. 2 a a 2 (x2 + a2) 2 a a 4. ilog ^^ + ^ + ^-1 . * (x+l)2 2(a;2+l) ^_l±2^_3tan-ia;. 2(x2 + l) Page 232. Art. 138 2. -2Vx-nx^ + 12 x^ + 6 log (x^ + l)-12tan-ixi 3. logX^LzJ:. 4. 2v^-3v^x4-6v^x-61og(v^+l). V X + 1 + 1 5. 21og(Vx^4-l) =Ji Vx - 1 + 1 6. 2tan-iV^r3^. 7. llog^^~^~A ^ Vx - a + 6 8. 14(xT^? _ 1 x7 + i a;T? - 1 x? + ^ xT?). Page 236. Art. 139 1) . 3. 21ogri-At^l + ^ L ^l-xj x-l-\- 4. _21og[V2+V^ff3j. \/l^^x2 314 ANSWERS 1. 3. 5. 6. 7. 8. 11. 12. Pages 236-237. Exercises on Chapter IV 2. |(a;-a)7- 2(a;2+ l)\/x2 + 2 ^[x2 - a;Vx2 - 1 + log (a; + Vx^ - 1)]. 4V2 V2(x-a)i+l ^a;(Vx2 + 2-Vx2+ l)+^log T3^(2x-3a)(a + x)i 61og(x3- 3x^ + 5). X + Vx=2 + 2 9. ti p -4 V Vx+ 1 +Vx-1 10. log Vx2 - a2 + a;V2 2v^a2 Vx2-a2-xV2 f x^ - 1 xs + f x^ + 2 x^ - 3 x^ - 6 x^ + 3 log (xi + 1) + 6 tan-i xi Page 239. Art. 142 \ tan^ X + tan x. 5. f cosec^ x — cot x — § cot* x. -^cot^x-cotx. 6. -64[cot4x + |cot84x]. tan X + I tan^ ic + i tan^ x. - 128 [cot 2 X + cot8 2 X + |cot62x+ }cot7 2x]. -— + log tan X. 2 tan2 X I C0t8 X — ^ COt^ X. Page 240. Art. 143 \ sec* X -- I sec2 x. 6. ^ sec* x — sec^ x + log sec x. I cosec'' X -\- 1 cosec^ x ^ cosec* X. 6. sec^^x sec*» ^^ 3. - {\ sec^ aa; — f sec* ax + sec ax). a 4. — (sin X + cosec x). - 1 « - 3 7. log sec X. 8. — log cosec a^ Page 241. Art. 144 1. — I cot' X + cot X + X. 2. — tan* ax log sec ax. 2a a 8. i(tan2x + cot2x) + 4 log (sin X cos x). ^ tan" ^x 71-1 * 6. \ tan' X - J tan» « + ^ tan» x — tan» + «. •1'- ANSWERS 315 Pages 242-244. Art. 145 2. - cos X + ^ cos3 X. 16. j^s (5x4-1 sin^ 2 x — sin 4 x 3. - ^ cos^ X + ^ cos^ X. — |sin8x). 4. log sin X — sin2 x + ^ sin* x. 17. ^i^ (3 x — sin 4 x + i sin 8 x). 5. fcos4x + 3costx-|cosix. 18- i(3x + sin 4 x + |sin 8x). ^ ^ ,, 3 „ ,^ 5 19. — icotx — icot^x. u 6. 4(1 -cos x)^ - 1(1 -cos x) 2. 2 ^ '\ , ^ ^^ 20. logtan2x. 8. -icot^x. ^ 21. tan X + ^ sin 2 X — f X, 22. 2cotx-^cot8x+|x+|sin2x. 9. — cot X — I cot^ X — I cot*^ X 10. - C0t5 X(i + } C0t2 X). 11. -4cot3x-2cotx + tanx. 23. L( ^^'^~^ ) . 12. I Vtan X (tan x — 3 cot x). ~ , «4 r ^ - , , ^ 24. ?(2x2-a2)Va2-x2+-sin-i?. -„ tan»-^x tan"+^x 8 8 a ^ ^+^ o^ Vx2-a2 1 ^ a; 25. sec^— 15. I X - s\ sm 4 X. 2 a2 a;'2 2 a^ a Pages 245-246. Art. 146 1. J_tan-if«i^Il^V ^tanfx-^V Va2 + 62 2V3 V3tan^x-^)4-l 6tan?-a-\/a2 + 62 - ^ in^ ^^"^^-^-^ log log ____. 2 VS tan X - 2 + V3 6 tan - - a + v'c2 + 62 3. itan-i(*-?|^y 4. itan-ihtan (7--)]" a(a tan X + 6) 7. -L tan-i (*^2^V V2 V V2 / Page 247. Exercises on Chapter V 8. e^fsin ?+cos?V 9. — ^ e-*(sin x + cos x). 1 5. itan2x(2 + tan2x). !<>• i e2x(2 - sin 2x - cos2x). 1 /^ _.\ 11. ie*(sinx + cosx — |sin3x «• -iif. + '^^'^d+i)- -icos3.). ^ , . „ . - - sin«+i X sin"+3 x 7. i tan2 X sin x 16. — • n+1 n -{- S 4frsin«-logtanf- + -l . ,» „ 2 « / ^L \2 4yj 16. f tan5x-2Vcotx. 316 ANS WEBS 17. -32cot2a;(l + |cot22x+ ^cot*2x). 18. ^tania;(H- f tan-2^x + ^tan^^x). Page 259. Art. 152 6. 2. 2a 4ct68 3 7. 00, Pages 262-263. Art. 154 2. 2a6. 3. 2. 4. 20 V5^. 6. 2 a. 7. 12. l 9. |. 10. faVop; |p2. n. ^a2. 12. 4a2tan-i-^; 4 Tra^. ^ a 13. 7ra6. 4. X 14. f7ra2. Page 266. Art. 155 6. 3 7ra2. 6. 4a2. 7. 00. Pages 268-269. Art. 156 Stto^^ 4. ^a(pi-p2). 15. a"^ - a^i log a 8. V- 9- ^<1' 5. 4a2 3 6. 7ra2. Pages 270-271. Art. 157 1. i)[\^4-log(l+V2)]. 61a 216* 6 a. 2irr. 6. |(e"-e '»). 6. 2 - \/2 + log ^ 4(a8-&8) a6 l4-\^ V3 * 8 a. 2ira Page 272. Art. 158 8. . 2arV6-2-V31og ^ + ^ 1 L \/2(2+V3)J aftan -sec- + logf tan - + secf^ 1*'. L 2 2 *V 2 2yjtf, a[-- vTT^ + log (d + >/rT^)T'. Pages 273-274. Art. 159 8 a. 8 ma Kxif + yit)i-|. 8. 6 a. 4. ia^i«. 7. >/2(«*>- 1). ANSWERS 317 Pages 276-277. Art. 160 ..(.-2)/^ 4.fC3V2-logCl.V2)3. (a) 2 7r6f6+ C0S-1-). (6) 2 ,ra2 + _^^^ log r«-+ ^«'-^'' Va2 _ 62. 4 7ra2. 9. 6. 7. (a) 7r6Va-^ + 62. «i«^. 8. (/3) 7raVa2 + 62. 3 Pages 278-279. Art. 161 1. 2. 4 7r«2 6 3 4 7rr3 3 3. TTk^; 00. - 32 7ra3 • 105 5. 6. 7ra3^ 15* 4 7r2a8. 7. TrfSa^ log 2a_ -4a2(2a-yi)]; oo. 8. 5 7r2a3. 3 8.a« 10. -'-^ fi. 3 6V2 Pages 280-283. Art. 162 1. f TT ahc. 2. 1 TT a6. 5. 41 cu. ft. 7. — cu. ft. V3 3. \Ah. 4. ■n abp 9, p= ea' ■ c 12. ^Tahc^. Pages 284-285. Exercises on Chapter VII 1. a21ogw. 2. 1. 5. a2(2+-j. 6. j%. 3. ^^i|!^l 4. |. 7. 4a6tan-^ 9. ^h!^ Sab 6 a 4 a 10. ^ r^ vT+T^ + log (e + vTT^)]^'. 13. 'La" [e a _ e a ] ^. ^ a2a;i. 14. 2 7r2a8. 11. 4 7r2aA;. 12. 2 7r2a2A;. 4 15. 2 IT a2 (3 sin h-Zh cos fi - h^ sin «i). 16. ^. 17. 2 7ra2. 18. alog^. 3 yi 318 ANSWERS Pages 287-288. Art. 163 3. xy = C2/2 + C'y + J. ^. y = kx(logx-l) + CiX-^C2. b. ^kA Page 292. Art. 165 1. xy + O. 2. - cos X cos 2/+ C. 6. st^ + y^ - S axy + C X X 3. Impossible. 4. log-+C. 6. tan-i-. 7. ^oi^ + x^y + 5x + ^y^-iy^-\-C. ^ Page 294. Art. 167 1. 1. 2. fa3. 3. 66». .4. Page 295. Art. 168 2. 64.. 8. ^-2 Vs. o Page 298. Art. 169 2. 2fl^. 3. Y^a. 4. ^. 6. iirabc. 6. ^^ INDEX (The numbers refer to pages) Absolute value, 59. Absolutely convergent, 59. Acceleration, 111. Actual velocity, 105. Arc, length of, 269. Area, by double integra- tion, 294. derivative of, 23. formula for, 255, 256. in polar coordinates, 268. in rectangular coordi- nates, 260. Asymptotes, 143. Average curvature, 166. Bending, direction of, 152. Binomial theorem, 73. Cardioid, area of, 268. Catenary, 168, 283. length of arc, 271. volume of revolution, 285. Catenoid, 276. Cauchy's form of remain- der, 71. Center of curvature, 163. Change of variable, 124. Circle, area by double in- tegration, 295. of curvature, 163. Cissoid, 168. area of, 266. Component velocity, 107. Concave, 152. toward axis, 157. Conditionally convergent, 59. Conditions for contact, 161. Conjugate point, 184. Conoid, 281. Constant, 1. factor, 31, 199. of integration, 200. Contact, 159. of odd and even order, 161. Continuity, 13, 113. Continuous function, 13. Convergence, 57. Convex, 157. to the axis, 157. Critical values, 93. Cubical parabola, 262. Cusp, 182. Cycloid, length of, 273. surface of revolution, 277. Decreasing function, 25. Definite integral, 251. geometric meaning of, 253. multiple integral, 293. Dependent variable, 1. Derivative, 19, 20. of arc, 138. of area, 23, 142. of surface, 140. of volume, 140. Determinate value, 78. Development, 56, 80. Differentials, 110, 196. integration of, 289. total, 117. Differentiating operator, 24. Differentiation, 24. of elementary forms, 49. Direction of curvature, 164. Discontinuous function, 14. Divergent series, 57. 319 Ellipse, area of, 263. length of arc of, 274. evolute of, 178, 284. Ellipsoid, volume, 280. Envelope, 187. Epicycloid, length of, 273. Equiangular spiral, 282, 284. Evaluation, 80, 81. Evolute, 170. of ellipse, 176, 271, 284. of parabola, 175. Expansion of functions, 56. Exterior rectangles, 254. Family of curves, 187. Formula for integration by parts, 203. Formulas of differentia- tion, 49, 50. of integration, 198, 210. of reduction, 217, 218. Function, 1. Hyperbolic branches, 143. spiral, area of, 269. Hypocycloid, area of, 263. length of arc of, 271, 273. volume of revolution of, 278. Implicit function, 120. Impossibility of reduc- tion, 218. Increasing function, 25. Increment, 13, 15. Independent variable, 1. Indeterminate form, 77. Infinite, 2. Infinite limits of integra- tion, 257. ordinates, 145. 320 INDEX Infinitesimal, 2. Integral, 195. definite, 251. double, 292. multiple, 292. of sum, 199. triple, 286, 292. Integration, 195. by inspection, 197. by parts, 203. by rationalization, 231. by substitution, 205, 238. formulas of, 198, 210. of rational fractions, 223. of total differential, 289. successive, 286. summation, 248. Interior rectangles, 251. Interval of convergence, 57. Involute, 170. of circle, 274, 285. Lagrange's form of re- mainder, 70. Lemniscate, area of, 268. Length of arc, 269. of evolute, 173. polar coordinates, 271. rectangular coordinates, 269. limit, 1. change of, in definite in- tegral, 295. Limits, infinite, for defi- nite integral, 257. Logarithm, derivative of, 39. Logarithmic curve, 263. spiral, length of arc, 272. Maclaurin's series, 63. Maximum, 91. Mean value theorem, 75, 267. Measure of curvature, 166. Minimum, 91. Multiple points, 181. Natural logarithms, 40. Non-unique derivative, 25. Normal, 129. Notation for rates, 108. Oblique asymptotes, 147. Order of contact, 160. of differentiation, 121. of infinitesimal, 8. of magnitude, 7. Osculating circle, 163. Osgood, 57. Parabola, 171. semi-cubical, 262. Parabolic branches, 143. Paraboloid, 283. Parallel curves, 175. Parameter, 188. Partial derivative, 114. Point of inflexion, 153. Polar coordinates, 133. subnormal, 135. subtangent, 135. Problem of differential calculus, 16. of integral calculus, 195. Radius of curvature, 164. Rates, 105. Rational fractions, inte- gration of, 223. Rationalization, 231, 233. Rectangles, exterior and interior, 254. Reduction, cases of impos- sibility of, 218. formulae, 217-218. Remainder, 61. Rolle's theorem, 67. Singular point, 179. Slope, 21. Solid of revolution, 140. Sphere, volume by triple integration, 297. Spheroid, oblate, 276, 278. prolate, 276. Spiral, of Archimedes, 136. equiangular, 137, 282, 284. hyperbolic, 269. logarithmic, 272. Standard forms, 198, 210. Stationary tangent, 153. Steps in differentiation, 24. Stirling, 62. Subnormal, 130. Subtangent, 130. Summation, 251. Surface of revolution, 140. area of, 274. Tacnode, 182. Tangent, 21, 129. Taylor, 62. Taylor's series, 66. Tests for convergence, 58. Total curvature, 166. differential, 117. Tractrix, 281. length of, 285. surface of revolution of, 285. volume of revolution of, 285. Transcendental functions, 38. Trigonometric functions, integration of, 238. Variable, 1. Volume of solid of revolu- tion, 277. Volumes by triple inte- gration, 295. Witch, area of, 263. volume of revolution of, 278, 279. 1 14 DAY USE ' RETURN TO DESK FROM WHICH BORROWED ASTRONOMY, MATffifAATlcT ^. ^, STATISTICS IfBRARY This book IS due on the last date stamped below, or on the date to which renewed. 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