■.C':\ I S 6 Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.arGhive.org/details/elementsofdifferOOhardrich ELEMENTS DIFFERENTIAL AND INTEGRAL CALCULUS. METHOD OF RATES. BY ARTHUR SHERBURNE HARDY, I'H.D., Professor of Mathematics in Dartmouth College. UNJVfcft&i . BOSTON, U.S.A.: PUBLISHED BY GINN & COMPANY. 1890. Copyright, 1890, By ARTHUR SHERBURNE HARDY, All Rights Reserved. Ttpoorapht bt J. S. CusHiNG & Co., Boston, U.S.A. Pbesswobk by Ginn & Co., Boston, U.S.A. PEEFACE. This text-book is based on the method of rates, which, in the experience of the author, has proved most satisfactory in a first presentation of the object and scope of the Calculus. No comparisons have been made between this method and those of limits or of infinitesimals. This larger view of the Calculus, and of mathematical reasoning and processes in general, cannot readily be given with good results in the brief time allotted the subject in the general college course. The immediate object of the Differential Calculus is the measurement and comparison of rates of change when the change is not uniform. Whether a quantity is or is not changing uniformly, however, the rate at any instant is de- termined in essentially the same manner ; viz. by ascertaining what its change would have been in a unit of time had its rate remained what it was at the instant in question. It is this change which the Calcidus enables us to determine, however complicated the law of variation may be. This conception of the nature of the problem is simple, and seems to afford the best foundation for further and more compre- hensive study; while for those who are not to make a 183665 IV PREFACE. special study of mathematics it secures a more intelligent and less mechanical grasp of the problems involved than other methods whose conceptions and logic are not easily mastered in undergraduate courses. My thanks are due to Professor Worthen, my colleague, for valuable suggestions and assistance in the reading of proofs. ARTHUR SHERBURNE HARDY. Hamoveb, N.H., June 2, 1890. CONTENTS. Part I. — The Differential Calculus. . chapter I. — introductory theorems. ART. PAGE 1. Quantities of the Calculus 8 2. Functions :5 3. Classification of functions 4 4. Increments 6 5. Uniform change 6 6. Uniform motion 6 7. Varied change 7 8. Differentials 8 9. Distinctions between increment, differential, and rate n __ 8 10. Corresponding differentials and simultaneous rates . . . -p.,.,-.^ 9 11. Symbol of a rate 9 12. Corresponding differentials of equals are equal 9 13. Object of the Differential Calculus 10 14. Differentiation 11 chapter II. — differentiation op explicit functions. The Algebraic Functions. 15. Differential of a constant 12 10. Differential of x + z-v 12 17. Differential of mx 12 18. Differential of xz 13 19. Differential of xzv 13 20. Differential of - 13 z 21. Differential of ar». Examples 14 VI CONTENTS. ART. PAGE 22. Analytic signification of y- 18 23. Applications 19 24. Geometric signification of -7^ 24 25. Relations between — , — , ^ 25 dt dt (It 20. Expressions for ~ and -^ 26 (Is (Is 27. Applications 26 The Transcendental Functions. The Logarithmic and Expoiietitial Funrtio)is. 28. Differential of log.r 31 29. Differential of a*. Examples 32 30. Applications 36 The Trigonometric Functions. 31. Circular measure of an angle 37 32. Differential of sinr '. 38 33. Differential of cos a- 38 34. Differential of tan x 39 35. Differential of cot x 39 36. Differential of sec x 39 37. Differential of cosecx 39 38. Differential of vers x 40 39. Differential of covers .r. Examples 40 The Circular Functions. 40. Differential of sin-' x 41 41 . Differential of cos-i x 41 42. Differential of tan 1 x 42 43. Differential of cot-' x 42 44. Differential of sec -' t 42 45. Differential of cosec 1 .r 42 46. Differential of vers ' x 43 47. Differential of covers- ' x. Examples 43 48. Applications 45 ( CONTENTS. Vll CHAPTER III. — SUCCESSIVE DIFFERENTIATION. ART. PAGE 49. Equicrescent variable 49 50. Differential of an equicrescent variable 49 51. Successive derived equations 49 52. Notation 50 53. Remark on the equicrescent variable. Examples 50 54. Successive derivatives 52 55. Sign of the nth derivative. Examples 54 56. Derived functions which become oo 66 67. Notation 67 68. Change of the equicrescent variable. Examples 57 Applications of Successive Differentiation. Accelei-ations. 59. Accelerations 60 60. Signs of the axial accelerations. Examples 61 Development of Continuous Functions. 61. Limit of a variable 63 62. Geometrical limit 64 63. Two meanings of limit 64 64. A quantity cannot have two limits 65 65. Continuous functions 65 66. Series 65 67. Sum of a series 65 68. Development of a function 65 69. Maclaurin's theorem ". 66 70. Taylor's theorem 68 71. Completion of Taylor's and Maclaurin's formulaj 69 72. Applications 72 73. Failing cases of Taylor's and Maclaurin's formulai 80 Evaluation of lUusorij Forms. 74. The form -• Examples 81 75. The form — • Examples 84 76. Tlie form x «. Examples 87 77. The form oo — oo . Examples 88 78. The forms 00 0, 1*, 00. Examples 89 VIU CONTENTS. Maxima and Minima. ART. PAGE 79. Definition of maxima and minima values 91 80. Condition of a maximum or minimum value 91 81. Geometric illustrations 92 82. Examination of the critical values when /'(a-) = 93 83. General method 93 84. Abbreviated processes. Examples 96 85. Examination of the critical values when /'(a-) = 00. Examples.. 99 86. Geometrical problems 100 CHAPTEB IV. — FUNCTIONS OF TWO OR MORE VARIABLES. 87. Partial differentials Ill 88. Notation T Ill 89. Partial derivatives. Examples Ill 90. Total differential. Examples 113 91. Total derivative 114 92. Total derivative with respect to any variable. Examples 115 93. Implicit functions of two variables. Examples 117 94. Evaluation of the first derivative of an implicit function. P]x- amples 118 98. ~ in terms of x and y 124 CHAPTER v. — PLANE CURVES. Curvature. 95. Direction of curvature 121 9G. Points of inflexion. Examples 121 97. Rate of curvature 123 ds 99. Curvature of the circle 125 100. Radius of curvature 125 101 . Centre of curvature 126 102. Maximum or minimum curvature 127 103. Intersection of the curve and circle of curvature. Examples... 127 Evolntes and £nTelopes. 104. Evolute and involute 129 105. Equation of the evolute. Examples 129 CONTENTS. ix ART. PAOI 106. Envelopes 132 107. Equation of the envelope. Examples 133 108. The evolute is the envelope of the normals to the involute 137 109. Property of the involute and evolute 138 110. Mechanical construction of the involute 138 111. Orders of contact. Examples 138 ^ing^ular Points. 112. Multiple, cu.sp, and salient points 140 113. Conjugate points 141 114. Determination of singular points by inspection. Examples. . . . 141 Asymptotes. 115. Rectilinear asymptotes 140 110. Asymptotes parallel to the axes. Examples 147 117. Asymptotes oblique to the axes. Examples 148 Curve Tracing. 118. Examples 151 Polar Curves. 119. Subtangent and subnormal 150 120. Lengths of the subtangent and tangent 150 121. Lengths of the subnonnal and nonual. Examples 157 122. Curvature. Examples 158 123. Radius of curvature. Examples 160 124. Asymptotes. Examples 161 125. Tracing of polar curves. Examples 163 CONTENTS. Part IT. — Thk Integral Calculus. CHAPTER VI. — TYPE INTEGRABLE FORMS. ART. PAOE 12(5. Integral and integi-atioii I(i7 127. Symbol of integi-atioii 107 128. Constant of integi-ation 108 129. Integral of a polynomial 108 130. Cadx=(x), etc., read 'y a function of a.',' is used to denote that y is an ex- plicit function of x; and the notation f{x, y) = 0, "^'j etc.)] is called the first derived, or first differential, equation of y =f{x, z, v, etc.). The above is an immediate consequence of the definitions. For if a and 3 be any functions whatever, and a— & for all values of the variables involved, the rates of a and fi must be the same at any instant. Now these rates are what the changes in a and \vould be in a unit of time were the common rate to become constant at any instant ; and if the rate remained constant for any interval greater or less than the unit, the cor- responding changes would still be equal ; but these changes are the differ- entials. 13. The immediate object of the Differeyitial Calcidus is the determination and comparison of the rates of variables. The following problem will serve as an illustration. Suppose a wheel to revolve about a fixed axis through its centre, P being any point in the rim, and that we desire to compare the rate of P's ^^-—r^' motion in the arc AB with that of its / / \ ^ / / V '9' 2 motion vertically upward at any instant. / / \ This is equivalent to asking what are the c d a rates of change of the arc AP and its sine PD. Hence if AP=x, DP=y, the ' fundamental relation is y = sin a*. Now if, as will be shown, -^ = cos x — ? the rate of ^ ' ' dt dt y is seen to be cos x times the rate of x ; that is, at any instant the sine is changing cos x times as fast as the arc. If P is moving in the arc at the rate of 10 ft. per sec, then at A, where cos x = l, it is also moving upward at the same rate. INTRODUCTORY THEOREMS. 11 At P, where AP=SiVG of 60° and cos cc = cos 60° = ^, it is moving upward at the rate of 5 ft. per sec, or half as fast as it moves in the arc. At B, where cos x = cos 90° = 0, it is not moving iipward at all. 14. Differentiation. In the above illustration the rate of y is the rate of sin x ; and, in general, the determination of the rates of variables involves the determination of the rates of the functions on which they depend or in which they enter. Since the rate of a variable is the differential of the variable divided by the differential of t, the relation between the rates of variables will be known when the relation between their differ- entials is known. The process of determining the differential of a function is called differentiation. We now proceed to determine rules for the differentiation of the several algebraic and transcendental functions. CHAPTER II. DIFFERENTIATION OF EXPLICIT FUNCTIONS. THE ALGEBRAIC FUNCTIONS. 15. The differential of a constant is zero. This is evident since a constant admits of no change, and therefore has no increment, whatever the interval. Properly- speaking, such expressions as ' the differential of,' or ' rate of a constant ' involve a contradiction of terms. But for uniformity of expression it is usual to say that both are zero. 16. The differential of a polynomial is the algebraic sum of the differentials of its several terms. Let y = X -i- z — V. If the changes of x, z, and v, at any in- stant, that is, at any of their simultaneous values, become uni- form, the change of y at that instant will also become uniform ; and therefore, if dx, dz, dv, dy, be corresponding differentials of the variables and the function, dy=dx-\-dz—dv (Art. 12). The above is evidently true of a polynomial of any number of terms. Cor. ^ = ^ _,_ ^ _ ^, or the rate of the sum of any num- dt dt dt dt ber of variables is the sum of the rates of the variables. Since the relation between the rates is always the same as that between the differentials, it will not be necessary to repeat this inference in the cases which follow. 17. The differential of the product of a variable and a constant factor is the differential of the variable multiplied by the constant factor. 12 ^Q Fig. 3. B P THE ALGEBRAIC FUNCTIONS. . 13 Let y=x-^z-\-v-\-etc. From Art. 16, dy=dx-\-dz-{-dv+ etc. Hence if x=z=v=etc., and m be the number of terms, y=mx and dy = dx + dx -\- etc. = mdx. 18. The differential of the product of tivo variables is the sum of the products of each into the differential of the other. Let y = xz. Then y is the area of a rectangle whose sides are x and z. Let a, b, be any two simultaneous values of x and z ; then at the instant when x=a = AB and 2 = 6 = AD, we have y = ^1 iS area ABCD. Let BP represent what would be the change in x in the inter- val dt if at this instant its change were to become uniform, and DR the corre- sponding change in z were its change also to become uniform at the same instant. Then BP = dx, DR = dz. The change of y would then also become uniform, and for the interval dt would be dy = BPQC -{- DRSC = bdx + adz. But a and 6 are any simultaneous values of x and z. Hence, in general, at any instant, dy = zdx + xdz. 19. The differential of the product of any number of variables is the sum of the products of the differential of each variable into all the others. Let y = xzv, and o:z = u. Then y = uv. But dy = vdu -f- udv (Art. 18), and du — zdx + xdz. Substituting in the former the value of du from the latter, and of w = xz, we have dy = zvdx -f- xvdz + xzdv. In the same manner the theorem may be proved for the product of any number of variables. 20. Tlie differential of a fraction is the denominator into the differential of the numerator, minus the numerator into the differential of the denominator, divided by the square of the denominator. 14 THE DIFFERENTIAL CALCULtlS. Let y=' • Then x = yz. But dx = zdy -\- ydz (Art. 18). _^ , dx dz dx xdz zdx — xdz Hence dit = y— = s- = — —r, Z "^ Z Z Z' z- CoR. 1. If x = a, a constant, then dx = (Art. 15), and dy = 2" I ^^' ^'^^ differential of a fraction loliose numerator is constant is minus the numerator into the differential of the de- nominator, divided by the square of the denominator. dec Cor. 2. If z = a, a constant, then dz = 0^ and dy = — , as it 1 " should be, since y is then -x (Art. 17). 21. The differential of a variable having a constant exponent is the product of the constant exponent, the variable ivith its ex- ponent diminished by one, and the differential of the variable. I. Let the exponent be positive and integral. Then y z=x"' = xxx • • • to n factors. Hence (Art. 19), dy = x"~^dx + x'^'^dx + ••• to ?i terms, or dy = nx"~^ dx. II. Let the exponent be a positive fraction. Then y = a;", whence i/" = a^" The differentials of the two numbers of this equation being equal (Art. 12), we have, by I., n?/""^ dy ■= maf~^ dx, , , m cc""^ , m ^-1 whence dv = ; do; = — x" dx. ^ n 2/"~ w III. Let the exponent be negative. Then y — x'" = —, n being fractional or integral. Clearing a;" of fractions, yx" = 1 ; whence, differentiating the product and remembering that the differential of a constant is zero. THE ALGEBRAIC FUNCTIONS. * 15 Of" cly + ynx"~^ clx = 0, or ay = — - — = — nx~"'^ax. ^ x" - dx Cor. 1. If ?i = ^, ?/ = Vi», and dy — — -=, or tlie differential of 2wx the square root of a variable is the differential of the variable divided by twice the square root of the variable. Special rules might be framed for n = i, 7i = \, etc., but in such cases the general rule is preferable. Remark. The above method of proof depends upon the resolution of the power into equal factors, and is therefore inapplicable to the case of a variable having an imaginary, or an incommensurable, exponent. The rule, however, as will subsequently appear, holds good for these cases also. Examples. Differentiate : 1. ?/ = a^ -f 8 a; — 4 ar". dy = d(a^ + 3 a; - 4ar") = cZ(ar') + d(3'x) - (Z(4ar'') [Art. IG = 2xdx + 3 da; - 12 x^dx = {2x-\-^ -Viy?) dx. [Arts. 21, 15 2. y = a + mx'^ — lnx^. dy = (m-af~^ — 14:7ix)dx. 3. y- = 2px. Although an implicit function, it may be differentiated directly without first reducing to an explicit form. Thus, P d(y^) = d(2pa;), whence 2ydy = 2pdx, or dy = —dx. 4. a-y- ± b-x^ = ± a^b-. dy = ^ — dx. d-y 5. 2/ = (l4-a^)(l-2ar5). dy = (1 - 2ar')cZ (1 + a;^) + (1 + ^)d (1 - 2ar=) = (1 - 2ar^) 2a;da; - (1 + ar) Ga^^da; = 2(a; — 3a;^— ox*)da;; or we may first expand and then differentiate. '16 THE DIFFERENTIAL CALCULUS. 6. y = {a + b3iP)K dy = d[ (a + ha?) ^] = i(a + hxY^-dia + ha?) [Art. 21 = ; or, more expeditiously by the special rule for the square root of a variable (Art. 20). 1. y = {ax) ^ + hx^. dy = f~ yj^ + ^ -|) ^•'^'• 8. y = {l-\- x) Vl — X. dy = — ^ — "^ — dx. 2VI — a; 9. y = Vl + V^. dy = = — • 4Vx Vl + Vx Put the expression in the form x^z'^v. Then , nvx'^''^dx 2x"vdz , afdv 11. y = Va.-2;-t'J. 7 _ d^xzH"^) _ z^v- dx -\- 2xv'^zd z + ^ccz-t; ■'dt; 2Va-22'y* 2aj^2v^ 2^4 do; , 1 /- , , z-\/xdv = ^ 4- i'^ yxdz -\ — - • 2V« 4i;* HO 1 7 dU; 12. y = - dy = -~- X xf- HO a 7 "dx 13.2/ = —:. ^^y = -r^- Voj 2a;:i H , VaJ , do; 14. ?/ = -^ • dy = 2 ' 4V a; H ^ 1 + a; 7 2 dx- lo. y = z~ — <^^y = -r, r»- ^ 1-x 0--^) (1 - a;)d(l + a;) - (1 + a;)d (1 - a;) dy = ^^^y ^ a-x ^ {a-xf THE ALGEBRAIC FUNCTIONS. (1 + x)" f ^ Y Vl - x) 2abx' (1 + ax^y^ 2 + mx + x^ dy = {l + xy+' ■dx. dy = 2xdx a-xy d>f = 2abx\A- a3?)dx 17. y= ''■' 18. y = 19. « = (1 + aa^)2 on 2 + WIX + X^ -, c( l\j 20. y = — ! ! dy = 21 X ]dx. x \ x^J 2 Put the expression in the form y = - + m -\-x-. 21. y= , ' dy = Put the expression in the form 2/ = 2(l — ar)~^ and differentiate as a power rather than as a fraction. oo Vl + ar , dx 22. y = ^ — dy = - « X- Vl + ^ 23. y = dy= ^^^^ {1-^y (1-a^)' 24. y=J^. dy = - '^"^ \T + ^* (l + a;)Vr^^ o- L , a , 6 , 1 ax + 2b 2o. y = ^1 + - + -. dy = - — - ^ \ X or 2ar -w/'i.2i „^_i ..^ da;. X a^ 2x^ Va;2 + ax + 6 r>p _ Va +x , _ Va( Vx — Va)c?x Va + Vx 2 V.x Va + aj ( Va + Vx)^ 27. y = ^^ dy/ = I ^'^ "^^•^' + 3a^ I dx, Vl + 03^ — X <- Vl + ar^ ^ -ar nationalize the denominator before differentiating. 28. v^Vl-'^-^. eZ, = 2J ^"^l-^-^^lLUa^. Vl-x2^^ C (l-2x2)Vl-x'^ 29. 1/ = ^^. dy = - ^-t^ dx. Vx 2x2 18 THE DIFFERENTIAL CALCULUS. 30. v=_l_ + __i dy = l( ^ ^ \dx. Vl+x ^/l-x 2V(i_a;)' (l+a;)V 31. y = ^=- dy=(l-\-—^:^)dx. X — -Vaf — c \ Va^ — cj 32. y = - -^^^ dy = -a^^(^ + ^') + ^^^^-^^)da;. OQ „, 1 ^ if — Vl — a^ 7 d3. ?/ = • dy = — ^ — ^z=:- dx. x+^l-a? Vl-x2(l + 2a;Vl-a;') 35. y =(2a*+a;^)(a^+a;^)i d^ = ^a- + Ja;- _^^^ 4:X^- {a^- + x^Y 36. y = {x- VT^^^)". dy = w (« - Vl^^) " ^ ^^-^'+^' (^3.. Vl— a^ 37. y = — ^zi=r^ . dy = , 1 H == dec. Vl + a'-Vl-a^ ar'V Vl-W 22. Analytic siffnification of the ratio — • rfa? Let 2/=/ (a;). The only variable which enters the function being x, the function y will change only as its variable x changes, and the rate of change of y will depend upon the rate of change of x. Let k be the ratio of these rates at any instant. Then dy ^ dx . dy -Tr = K-rr, whence -V- = a;. dt dt dx Hence the ratio of the differential of the function to the dif- ferential of the variable is the ratio of the rate of change of the function to that of the variable. It is evidently independent of the interval dt. THE ALGEBRAIC FUNCTIONS. 19 As derived by differentiation from the function y =f (x), this ratio is called its first derived function, or simply its first derivative ; also, being the factor k by which the rate of the variable is multiplied to obtain the rate of the function, it is called the first differential coefficient. CoR. If ^ is positive, y is an increasing function of x; and if negative, y is a. decreasing function of x (Art. 11, Cor.). 23. Applications. 1. Compare the rates of change of the ordinate and abscissa of the parabola whose parameter is 4. Which is changing most rapidly at the point x = d? Where are they changing at the same rate ? If at the point x = 16 the abscissa is increasing at the rate of 24 ft. a second, at what rate is the ordinate then changing ? 2 , dy 2 .. . . . ^ dy 2 dx ^ =^^' ••• dx = y^ ^^^^^' "^^^ ^^ '^''^^''''dt=ydt' °^' "' general, the ordinate changes - times as fast as the abscissa. 2^1 For x = 9, y = ±6, and - becomes ± o , showing that the or- y -^ '^ dinate is increasing, or decreasing, k as fast as the abscissa at the points (9, 6), (9,-6). When the ordinate and abscissa are changing at the same rate, we must have -^ = -=1, .: y = 2, which, in the equation of the curve, gives x = l. This is as it should be, for (1, 2) is the extremity of the parameter, at which point the generating point is moving in the direction of tlie focal tangent, whose inclination to the axis of X is known 2 1 to be 45°. For a; = 16, ?/ = ± 8, and - becomes ± -j, or dy Idx , ., , y . ^ jt = ± J TT ; hence if at x = 16 the abscissa is increasing at the rate of 24 ft. a second, the ordinate in the first angle is increasing, and in the fourth angle decreasing, at the rate of ^ X 24 = 6 ft. a second. 2. Compare the rates of change of x and y in the ellipse ahf -j- 6V = a^6l Is y an increasing or decreasing function of 20 THE DIFFERENTIAL CALCULUS. ic? Compare the rates at the extremities of the axes. If the (5 axes are 6 and 4, at what point is y changing 2-y/- as fast as a;? -^ = s- , and is negative when x and y have like signs ; dx a^y ° ' hence the function is decreasing in the first and third, and increasing in the second and fourth, angles. At the extremi- dy ties of the transverse axis, y = 0, and ~ = oo, or y is changing infinitely faster than x. To determine the point where y \5 b^x 4x fS changes 2\ - as fast as x, we have = =2 v-, which, ~3 a-y *dy ^3 with the equation of the curve 9i/^ + 4a^ = 36, gives four points 3 .— 1 whose coordinates are numerically j V lo and ?,• 3. The altitude of a right triangle increases at the rate of 10 in. a second. At what rate is the area increasing ? Let h = base, x = altitude, y = area. Then y = — , .-. -^ = -, 2 dx 2 which is a constant ; therefore the area increases uniformly at the rate of - X 10 = 5 6 sq. in. a second. Li 4. A spherical balloon is being filled with gas at the rate of m cub. ft. a second. At what rate is the diameter increasing when its length is 6 ft.? Let y = diameter, x = volume. Then X = ^/, or j, = f^J; .-, f? = {l^. 6 VV (^'^ \97rar When y = Q, x = 36 tt, and ^ = •^ ' dx IStt Hence when the diameter is 6, it is increasing at the rate of q-5— ft. a second. loTT 5. A rectangle whose sides are parallel to the axes is in- scribed in the ellipse a^y^ + b-ay' — o?b^. Compare the rates of change of its area and side. THE ALGEBRAIC FUNCTIONS. ^"-^ 21 Let X and y be the half sides and z the area. Then 2; = 4 ccy. To eliminate y, and so obtain z a function of a single variable, we have from the equation of the ellipse, 46 ,-^0 i dz Ab a- -2a? ?/ = -Va— ar, .•.z = — -Va-ar — x*, — = a a dx In a similar manner we may find the ratio a a dx a -^Jq^ _ jp2 dz dy' 6. The radius and altitude of a right cone vary, the slant height remaining constantly 25 ft. Compare the rates of change of the volume and altitude. When the altitude is 4 ft. and changing 3 ft. a second, how fast is the volume changing ? Let s = slant height, x = altitude, z = radius of base, and y = volume. Then y = To eliminate z we have the con- o dition z^ + 0? = ^, .: z- = s- — xr and y = ^ {^x — x^). Whence -f- = Z (^— 3af)= ^ (625 — 3x^) ; that is, the volume is increas- ing ^ (625 — So?) times as fast as the altitude is in linear feet. 577 When a; = 4, this becomes tt, and at that instant the volume o is changing at the rate of 577 tt cub. ft. a second. The student will observe that he may eliminate before or after differentiation. Thus, differentiating first, dy = '^{2zxdz + z'dx), .■/^ = l(2zx'^+z'). o dx 3 dx But from z^-\-a? = s% — = ; substituting this value with dx z those of z and z-, ^-^ = - (^ — 3ar), as before. dx 3 7. A point P moves from ^ at a uniform rate in the direc- tion of AP, at right angles to AB. A light C, whose intensity at a unit's distance is 125, is vertically over B. If AB = 10, BC = 5, compare the rates of the motion and illumination of 22 THE DIFFERENTIAL CALCULUS. P (understanding that the intensity of a light at any point is its intensity at a unit's distance divided by the square of the distance), when AP= 10. Let AP = X. Then tlie illumination at „ 125 125 GP- 10- + 52 + a;2 d//^ 250 a; '■ dx~ (125 + 0^)2' which, when x = 10, becomes — ^j, or the rate of change of the illumination of P is ^ times the rate of change of AP, and is decreasing. 8. If, in Fig. 4, BC is a lamp-post 10 ft. high, and a man whose height is 6 ft. walks from B in the direction BA at the rate of 3 miles an hour, show that the extremity of his shadow moves at the rate of 7^ miles an hour. 9. In Ex. 8 show that the length of the man's shadow is in- creasing at the rate of A^ miles an hour. 10. In Fig. 4 show that if the man walks from A towards B, y he is approaching B - times as fast as he is approaching C, where BA = x, CA = y. 11. A ship is sailing northeast at the rate of 10 miles an hour. At what rate is it making north latitude ? Ans. 5V2 miles an hour. 12. The area of a circular plate of metal is expanded by heat. Find the rate of change of the area when the radius is 5 in. and increasing .01 in. a sec. Ans. -^^ir sq. in. a sec. 13. If the thickness of the plate of Ex. 12 increases one- half as fast as the radius, find the rate of increase of the volume when the radius is 5 in. and the thickness .5 in. THE ALGEBRAIC FUNCTIONS. 23 Let V = volume, x = radius, y = thickness. Then v = njc^y ; whence dv o dx , ^ dy ,o , , ,v dx 7 , . — = ZttX)! h irxr -^ = CzTTxy + iTrar) — = — n cub. in. a sec. dt dt dt dt 40 14. Two ships, on courses whose included angle is 60°, are sailing away from the intersection of the courses with veloci- ties of 6 and 4 miles an hour. Find the rate at which they are separating when 10 and 15 miles respectively from the intersection. Let z = distance between the ships, x and y their distances from the intersection, 6 and 4 being the rates of x and y respectively. Then z- = x^ -{-y- — 2xy Go%Q(f = 3? + y- — xy, dz dt ^ -^^ dt ^ -^ ^ dt 55 21- Vl75 "' X^ 15. Find the rate of separation in Ex. 14 under the suppo- sition that the ships start together from the intersection of the courses, with the velocities 6 and 4. 22 = (60' + (40'-24f-, .-. -=V28. 16. C is any point without a circle whose centre is 0, and OC cuts the circle at A. Find the relative rates of departure from C and OC of a point P moving from A in the arc of the circle. Let P be the position of the point at any instant, y = PM, the perpendicular on OC, PC=x, OC=a, OA = B. Then pc=v<;.'M' + PM% or x=-^(a-VW^yy + f, xVR' - y' 24 THE DIFFERENTIAL CALCULUS. 24. Geometric signification of — • dx Since to every equation y=f(x) there corresponds some plane locus, the ratio -^ is evidently capable of geometric interpretation. Let M'N' be the locus of y =f(x), and P the position of the generating point at any instant.- Then dx, dy, being corre- sponding differentials of x and y, are what would be the changes in x = OD and y = DP during any interval if at its beginning their rates of change should become constant. But this will evidently be the case if at P the motion of the generating point should become uniform along the Fig. 5. tangent at P. Hence PQ, QB, being what would be the corresi)Onding increments of x and y in any interval dt if the change of each became uniform at the instant considered, are corresponding differentials of x and y, and |^ = ^ = tanXrP=tana. (1) PQ dx ^ ^ The tangent of the angle made by any straight line with the axis of X is called the slope of the line. As the tangent at any point of a curve has the direction of the curve at that point, the slope of a curve is that of its tangent ; hence the dy value of -j-, ft^ ctwy instant^ that is, for any simidtaneous values of X and y, measures the slope of the curve at the corresponding point. In the figure, y is an increasing function of x, a is an acute dy . . . angle, and tan a, or -^^ is positive. In the vicinity of M', how- ever, y is a. decreasing function of x, a is an obtuse angle, and tan a, or -p? is negative, as already seen in Art. 22. THE ALGEBRAIC FUNCTIONS. 25 It is evident that the slope will in general vary from point to point, and the first derivative will therefore be in general a function of x; but that for any particular value of x it has a definite value independent of dt, that is, independent of dx, since from the similar triangles PQR, PQ'E', -j- remains con- stant, whatever the interval. 25. Relations between the velocities in the path and along the axes. Let s = distance passed over by the generating point, esti- mated from any point in its path, that is, the length of the path. Since, when the changes in x and y (Fig. 5) become uniform, the generating point moves in the direction of the tangent PR, PR = ds, PQ = dx, QR = dy, are corresponding differentials of s, x, and y ; and from the right triangle PQR, ds'- = dx^ + dy\ (1) Hence if y=f{x) be the path of a moving point, — is the rate of change of the distance, or the velocity of the point in its path; and, for like reasons, — > -^, are its velocities in the dt dt directions of the axes. By differentiating y=f(x) we can compare the horizontal dor (In and vertical velocities, and substituting either — or -^ from ' ^ dt dt the differential equation of the path in dt \[dtj [dtj we can compare the velocity in the path with either the hori- zontal or vertical velocity. dx dv Since -j- and -— are distances, namely, the distances which etc uz the point would pass over in a unit of time in the directions of the axes if its velocity in each direction became uniform, they 26 THE DIFFERENTIAL CALCULUS. are positive or negative according as each is in the positive or negative direction of the corresponding axis. 26. The following relations will be found of use hereafter. Let PN (Fig. 5) be the normal at P. Then cosa= sin<^ = — 1 (1) ds dy sm a = — cos (^ = -j-- (2) 27. Applications. 1. To find the general equation of a tan- gent to any plane curve. Let y = /(a-') be the equation of the curve and {x', y') the point of tangency. The equation of a straight line through dv (x', y') is y — y' = m (x — x'). If we form ~ from the equation of the curve, and substitute in it the coordinates of the given point of tangency, we have the slope of the curve at this point (Art. 24). But the slope of a curve at any point is that of its cly dv tangent at that point ; hence, representing by -—, what -y- be- dy' comes for the point (x', y'), and substituting -^, for m, 2. Deduce the equation of the tangent to the ellipse ay + h'x" =.a'b^ From the equation of* the ellipse, -^ = ^j the general expression for the slope. For the particular point (x', y') this b'x' becomes -— > and the equation of the tangent is, therefore, dy b^x' y — y' = -— (x — x'). Clearing of fractions and svibstituting a-y' for a^y'- + b'x'^ its value a^6^, the equation assumes the simpler form a^yy' -|- b'xx' = aH)^. THE ALGEBRAIC FUNCTIONS. 27 Show that the equation of the tangent to : 3. The hyperbola aV — ^'^ = — «"^^ is a^yy' — b-xx' = — a-b^. 4. The parabola y- = 22JX is yy' = p(x + x'). 5. The circle y^ -{-x^= R- is yy' + xx' = R-. 6. The circle y^=2 Rx — a^ is y — y' = — ~ — (cc — x'). 7. The hyperbola xy = in, referred to its asymptotes, is 8. The cissoid f = ^ is y-y' = ± ^•''(3«-^') (x^x'). 2«-^ (2a-a;')' 9. The curve a^y" -f- 6''ar' = orW is y — y' = — (x — x'). a^y'^ 10. Find the slope of y^ = 2px at the vertex ; at the extremi- ties of the parameter. Is the generating point ever moving in a direction parallel to X ? cly p From y^ = 2px, -^ = -, which is oo for y = 0. Hence the tangent is perpendicular to X at the vertex. For y = ±p, -;-= ±1, the slope of the focal tangents, which therefore make ctx 1 civ angles of 45° and 135° with X Since -^ is zero only when y = Ri5"- 22. Find the subtangents and subnormals of the conic sec- tions. ♦ SCBTANQBNT. Subnormal. Ellipse, x^ - a? x' a? Hyperbola, a;'2 - a? x> b^x' a' Circle, y" x'' -x'. Parabola, 2x'. P' The signs may be neglected if lengths only are required. The sign will, however, indicate the direction if the subtangent and subnormal be reckoned respectively from T and D, Fig. 5. 23. Prove that the subtangent of the hyperbola xy = m is the abscissa of the point of contact, and that the subnormal varies as the cube of the ordinate. ,dx' , ,dy' y'^ ^ dy' dx' m 30 THE DIFFERENTIAL CALCULUS. 24. Prove that the subtangent of the serai-cubical parabola if = a'3? is two thirds the abscissa of the point of contact, and that the subnormal varies directly as the square of the abscissa. 25. A point moves with a constant velocity m in the arc of the parabola 2/^= 8 a;. Pind the velocities in the directions of the axes when cc = 8. From ?/- = 8 a;, we have — = — - , and by condition — = m. dt y dt ^ dt Substituting these values in ds di =m'^($} we obtain dx _ my ^ dy _4: dx _ Am ^^ V/-I-16' " dt~ y dt ~ -yjy-i _|_ le For X =%,.-. y = 8, these become — - and — - ; hence at the V5 V5 point (8, 8) the horizontal and vertical velocities are as 2 to 1, 2 1 and are — ^ and — r. times that in the path. V5 V5 26. The orbit of a comet is a parabola, the sun occupying the focus. Compare the velocity of the comet with its rate of approach to the sun. The distance of any point of the parabola from the focus is rz=x+^, .'. — = — , or its rate of approach to the sun is the 2 dt dt same as its horizontal velocity. But, as shown in Ex. 25, — = — ^ — • Hence, in general, its rate of approach to dt V2/'+p' f^^ 5 & » i^i- the sun is — ^ times its velocity. At the vertex, y = ^ and ■ — = — = 0, or, at the vertex, it is not approaching the dr 1 ds sun at all. When y =p, — = — - — • When at a distance dt -^2 <^i THE TRANSCENDENTAL FUNCTIONS. 31 from the sun equal to the parameter of the orbit, r = 2» = x + ", .-. a; = -» and w=Vo», and — =-V3 — ^2 T ^ ^ dt 2 dt 27. A point moves in the arc of the circle a? -\-y- = 25, and has a velocity 10 in passing through the point (3, 4). Show that its velocities in the directions of the axes are 8 and G, numerically. THE TRANSCENDENTAL FUNCTIONS. The Logarithmic and Exponential Functions. 28. The logarithmic function. Let x = ny, (1) n being any arbitrary constant. Then log„x = log„w + log<,?/, (2) in which a is the base of the logarithmic system. Differentiating (1) and (2), dx = ndy, d (log^x) = d (log^y) ; and, by division, iO^^^LO^. (3) ' •' ' dx ndy ^ ' Eliminating n from (3) by substituting its value from (1), dx dy ^ X y clcc or the ratio of d (log^x) to — is the same as that of d (log^y) dy „. . ^ to — Since n is arbitrary, the ratios in (4) are constant. Let m be this constant. Then d(log„x) = m^^ X 32 THE DIFFERENTIAL CALCULUS. Now the only quantities involved in any logarithmic system are the number, its logarithm, and the base. Since of these the two former are variable, while m is constant, m must de- pend upon the base. The value of m corresponding to any base is called the modulus of that system. Hence The differential of a logarithm of a variable is the modulus of the system into the differential of the variable divided by the vari- able. The relation between the modulus and the base of any sys- tem will be established later ; but as the only system employed in analytic investigations is that whose modulus is unity, called the Naperian system, the above rule becomes : The differential of the Napenan logarithm of a variable is the differential of the variable divided by the variable. Unless otherwise mentioned, by log x will hereafter be meant Naperian logarithm of x. The base of this system is represented by the letter e, and its value will be shown to be 2.718281. 29. The exponential function. I. When the base is constant. Let y = a". Then, in the system whose base is b, log^y = xlog,,a. Differentiating both members, •m— =log^adx, y . , a'' logft adx whence a?/ = > -- ^ m or the differential of an exponential function whose base is constant is the function into the logarithm of the base into the differential of the exponent, divided by the modulus of the system. For the Naperian system, m = 17 and we have dy = a" log adx. THE TRANSCENDENTAL FUNCTIONS. 33 If the exponential base is also the base of the logarithmic system, log a = 1, and dy = a* dx, or, e being the Naperian base, the differential of y = e"^ is dy = ef'dx. II. When the base is variable. Let y = af. Then log?/ = z log a;. Differentiating both mem- bers, dy dx , -^ = z h logical, y X whence dy = x'z-^+ -^"^Jog xdz = zx''^ dx -{■ x' log xdz, or the differential of an exponential function whose base is variable is the sum of the residts obtained by differentiating first as if the exponent were constant and then as if the base were constant. If z = x, y=xi', and rf^ = af (1 +loga;) da*. Examples. Differentiate : /7 ,1,, /j*2 1. y = log (3 ax -f x'^). dy = 3 — — — -dx. 2. y = loga^. dy = 3. y = (loga;)^ dy = 2 log x 4. ?/ = log(loga;). dy = a-log x 5. y = x log X. dy = (log a; + 1 ) dx. • y = \ — r dy = Zax + x^ 2dx X dx x_ dx loga^ ""' a; (log 03^)^ 7. y = los(l-\-x^y, or 21og(l + a^). ixdx dy = l+ar« 34 THE DIFFERENTIAL CALCULUS. xdx S. y = log Vl + a^- d}j = d. y = log {Vl + x' + Vl - X-) 1+a^ dx. 10. y : log^-+^-^', or log (1 + -Vx) - log (1 - V.0- 1- Va d!/ = —= dx ■\/x (1 — ic) 11. 2/ = log[Vl-a;(l + a;)], or ^ log (1 - a;) +log (1 + a;), (1-Sx)dx 12. y 13. // 14. y 15. v/ 16. y 17. y 18. 2/ 19. 2/ 20. y 21. y 22. y = log„4.TT. = e^(l-cc-). dy = dy = 2(1 -a^) mdx 4:X dy = e^(l — 2x — x^)dx. dy — af (log X + l)dx. dy = x^ e^ —L ^ dx. X dy = e^a;«^(log x + l)dx. = x^. .log J dy = x^x log a; (logo; + 1) 4- dy z= 2 x^"^'' log X dx -.{logxy. = log {e" — e"). dy = (log xy log (log x) -f logcc dx. dx. dy dx. e' — e' e^ — e' dy= - 4dx {e-e-'f dy = ('j flog--l]dx. THE TRANSCENDENTAL FUNCTIONS. 35 23. y = (^. dy = (fT(log^^ + l\dx. Algebraic functions may sometimes be differentiated with greater facility by first passing to logarithms, but it is usually more expeditious to differentiate directly. Differentiate the following by passing to logarithms. 24. y = x^l-x{l+x). log y = log x + \ log(l - x) + log(l + x), y \x 2{l-x) 1-^xJ ' 2 + X-5.V' .-. dy = icVl —x(l-\-x\ — "^^^ ^-^ dx -^ ^ ^2x{l-x){\+x) = 1±^IzMdx. 1 — X 1 —x^ 26. y = ^•(^ + ^). dy = l+Mzil^i^d.. y/l-x" ■(l-.x-)l 27. y — a^. dy = (0"%" log a log hdx. 28. 2/ = J. dy = ^!(ilzM^da.. 29. y = log '^ ^ ~ dy= - Vl + ic — Vl — X a; Vl — ix^ 30. y = a"''^\ dy=a}°^'' log a—' dx 31. y = log( Va; — a + ^x — ?;). f??/ 2^{x-a){x-h) 32. ?/ = log {x — Va;" — «-) . dy— — dx Vic'^ — ci^ Qo a; , e'(l — x) — 1 , 33. y = - — -. dy=^— — -^ dx. 36 THE DIFFERENTIAL CALCULUS. 30, Applications. 1. Compare the rates of change of a num- ber and its logarithm. a; = log„y, whence — = — , or the logarithm (x) changes dy y faster or more slowly than the number {y), according as the number is less or greater than the mq^ulus of the system. Since m = 1 in the Naperian system, the Naperian logarithms of proper fractions change faster than the fractions. 2. Compare the rates of change of a number and its loga- rithm in the common system, where the number is 534. The modulus of the system where base is 10 will be shown to be .434294, .-. - = 1^5^?^ = .00081, which will be found by ex- y 534 amination of the tables to be the tabular difference correspond- ing to the number 534. Since — changes with y, the relative rate of change of a number and its logarithm varies with the number. If we assume that for an increase of say .1 in the number there will be a proportional increase in the logarithm, the quantity to be added to the logarithm of 534 to obtain the logarithm of 534.1 will be .1 X .00081 = .000081. This, in fact, is the manner of using the tabular difference of the tables, and is equivalent to the supposition that — remains constant while the number 534 changes to 534.1, a supposition which, although not strictly true, gives results sufficiently accurate within the limits of practice. 3. Find the tabular difference corresponding to the number 3217. Ans. .000135. 4. Prove that the rule for the differentiation of a power applies when the exponent is incommensurable. Let y = a;", n being incommensurable. Passing to logarithms (first squaring, as y may be negative, and negative numbers have . , ^ , , dy dx no logarithms), log y = n log x, .-. — = ?i — ? or dy = ± nx" 'ax. y ^ THE TRANSCENDENTAL FUNCTIONS. ' 37 5. Prove in the same manner that the rule applies when the exponent is imaginary. 6. Find the slope of the logarithmic curve at the point where it crosses the axis of Y. X = loga y, .: -^ = ~i which for x = (whence y = l) becomes — Since y = 1 when a; = 0, whatever the base, the slopes of all logarithmic curves at their common point on the axis of Y vary inversely as the moduli of the systems. In the Naperian system m = 1, hence the slope of x = log y is the ordinate of the point of contact. 7. Find the equation of the tangent to x -— log y. Ans. y — y' = y'{x — x'). 8. Show that the subtangent of a; = log^ y is constant and equal to the modulus of the system. Also find the subnormal. . ,dy' , dx' y"' Ans. y' ^—, = m: v 3-, = " — •^ dx' ' ^ dy' m 9. Compare the rates of change of x and its ccth power when a; = 1. Ans. The rates are equal. 10. Compare the rates of change of x and its iKth root when Ans. ~- = 0. dx The Trkjonometric Functions. 31. Circular measure of an angle. Any angle AOB, measured in degrees, may also be measured by the ratio of its arc to the radius of its arc, since for any given angle this ratio is constant whatever the radius of the arc. If the arc 6 be de- scribed with a radius equal to the linear unit, then, since x = r6 (Fig. 6), - = B, or, by this method, the angle is measured by the arc intercepted at a unit's distance. To express the angle 38 THE DIFFERENTIAL CALCULUS. n° in circular measure, we have - = = 2 tt for the circular r r 2-77 IT measure of 360° ; hence the circular measure of 1° is — — = -— -, 360 180 and of 71° is ^^ ; or the circular measure of an angle is expressed 180' by multiplying the number of degrees by t^- 180 Since - = 1 when x = r, the unit of circular measure is the angle whose arc equals its radius; or, making — ^ =1, n = i-FTrt r» 1 loO IT = 57°.3 nearly. 32. Differential of since. Let the point P move in the circular path AB, x being the length of the path, estimated from A, at any instant when the generating point is at P. Then PD = y = sin x. If at this instant the motion of P should TV^g'?^' become uniform along the tangent at P, the changes in AP and PD would also become uniform. Hence if PQ, RQ, are what the increments of x and y would be in any interval dt, PQ = dx and liQ = dy = d (sin x) . But BQ = PQ cos AOP. Hence dy = cos xdx, or the differential of the sine of an angle is the cosine of the angle into the differential of the angle. 33. Differential of cos x. In Fig. 7, SD = BP, being the decrement of OD simultane- ous with BQ and PQ, is the differential of cos x. Hence, if OD = y = cos X, dy = BP = — PQ s in AOP= — sin xdx. Otherwise : ?/ = cos ic = Vl — sin^ x, whence — 2sina:d(sin x) sin aj cos a^dx . , dv = ^ - = — = — sm xdx, 2Vl-sin2aj cosa; . or the differential of the cosine of an angle is minus the sine of the angle into the differential of the angle. THE TRANSCENDENTAL FUNCTIOl^S. 39 34. Differential of tan x. sin X -_,, Let V = tan x = inen ^ cos a; , cos a;d(sinx) — sin.Trf(cos.r) cos'a; -l-sin'^a; , dy = 5^ '— ^ '- = \ dx Q,Q'S>^x COS'' a; dx _ . 2 cos^a; sec^a^dx, or the differential of the tangent of an angle is the square of the secant of the angle into the differential of the angle. 35. Differential of cot x. Let y = cot x = tan [ ^ — re j . Then dy = sec^ I- — x\{ — dx) = — cosec^ xdx, a result whicli may • COS x also be obtained by differentiating y = cot x = Hence sin X The differential of the cotangent of an angle is minus the square of the cosecant of the angle into the differential of the angle. 36. Differential of sec x. Let y = sec x = Then COSiC , d(cosa;) sina/'d.« , , dy= ^^ — - — - = ——, — = sec x tan xdx, cos- X cos- X or the differential of the secant of an angle is the secant of the angle into the tangent of the angle into the differential of the angle. 37. Differential of cosec x. Let y = cosec x = sec (- — x). Then V2 J dy = sec [ ^ — x\ tan[ - — a; j ( — dx) = — cosec x cot xdx, or the differential of the cosecant of an angle is minus the cosecant of the angle into the cotangent of the angle into the differential of 40 THE DIFFERENTIAL CALCULUS. 38. Differential of vers x. Let y = vers x=l — cos x. Then dij = sin xdx, or the differential of the versine of an angle is the sine of the angle into the differential of the angle. 39. Differential of covers x. Let y = covers x =1 — sin x. Then dy = — cos xdx, or the differential of the coversine of an angle is minus the cosine of the angle into the differential of the angle. Examples. Differentiate : 1. y = sin 6 x. dy = 6 cos 6 xdx. 2. y = cos XT. dy = — 2x sin x-dx. y 3. y = cos^ x^ dy = — sin 2 xdx. 4. ?/ = tan (3 - 5 x^) \ dy = - 20 a; (3 - 5 x") ^ec\3 - o x^f dx. 5. y = sin2 ^^ _ 2x^y-. dy = - 8a;(l- 2ic2)sin 2(1- 2x'y-dx. ^ 6. 2/ = (sin a; cos x) ^ dy = sin 2 a; cos 2 xdx. '^ _ 7. 2/ = sin 2 a: cos 2 a;. dy = 2 cos 4:xdx. '^ S. y = sin2 (1 - a.-^)^. dy = -Sx{l-x-) sin (1 - x^dx. V 1 ^ ^ . J 1 + sin a; J ' 9. V = tan X + sec x. dy = — '—^ dx. COS'' X 10. y = X -j- sin x cos x. dy = 2 cos^ xdx. O^ll. y = tanVl — a;-. . dy = — sec^ Vl - x^ • \. Vl — x^ -'-^'- , , cos (log x) , ^.12. y = sin (log a;) . dy = ^^-^^ dx. '''da; 13. 2/ = log (cot a;). dy = --^-^. 14. y = m sin" ax. d?/ = a??m sin" "^ ax cos axda;. 15. t/ = sin'a;. dy = sin''a;(log sin a; 4-x cot a;) dx. THE TRAXSCENDENTAL FITNOTIONS. 41 16. ?/ = vers -• dy = - sin - dx. 17. y = sin e"^. d?/ = e^ cos e' dx. 18. ?/ = .^•^ cos 0/*^. dy = 2 a; (cos cc- — a^ sin a;-) d;f. io • " 7 tt d , 19. w = sin - • dw = -„ cos - da\ ^ X -^ x^ X 20. y = log (sin a-) , d// = cot xda;. 21. _?/ = sin «a; sin''a;. dy = a sin"~ '.'c sin (ax + x) dx. J. 1, oo 4- X 7 sec- a* log a • a* da; 22. y = tn,nce. dy = ^ • ar 23. y=zx'"'\ dy = a; »'" * /sin^ _,_ j^g ^j cos a;"] da;. 24. y = (sina;)'"°". dy = (sin a;)™** |cot x cos a; — sin x log sin a; j d-r. (sin na;)"* , ?nn(sin na;)'""^ cos (mx— ?ia;) , 2i>. y = -7 T7- d?/ = ^^ ^ r-xi -dx. ^ (cos ma;)" "^ (cosmx)"+^ The Circidar Functions. 40. Differential of sin~'ic. Let y = sin~^T. Then x = sin y. But dx = cos ydy, hence d _ ^^ _ da; _ dx ~cosy~ Vl-sin^y ~ VH^^' or i/te differential of an arc in terms of its sine is the differential of the sine, divided by the square root of 1 minus the square of the sine. 41. Differential of COS" ^aj. Let y = cos^a;. Then x = cos y. But da; = — sin ydy, hence , _ _ f?.i7 _ da; _ _ dx si" y Vl - cos^y Vl-a;' 42 THE DIFFERENTIAL CALCULITS. or the differential of an arc in terms of its cosine is minus the differential of the cosine, divided by the square root of 1 minus the square of the cosine. 42. Differential of tan^^a?. Let y = t'An~^cc. Then x = timy. But dx = sec- ydy, hence , dx dx dx (^y = — V- = rTT — ~ = 1— , — ■>' sec^y 1 + tan- y 1 + ar or the differential of an arc in terms of its tangent is the differen- tial of the tangent, divided by 1 plus the square of the tangent. 43. Differential of cot"' x. Let y = cot~^ x. Then x — cot y. But dx = — cosec'' ydy, hence 7 _ ~ '^^•^" _ _ '^^^ _ _ ^^ cosec- y 1 + cot- y l-\-x^ or the differential of an arc in terms of its cotangent is minus the differenticd of the cotangent, divided by 1 pilus the square of the cotangent. 44. Differential of sec ~^ 35. Let y = sec~^ x. Then x = sec y. But dx = sec y tan ydy, hence dx dx dx dy = sec y tan y gee 2/ Vsec- ?/ - 1 re Va^ - 1 or the differential of an arc in terms of its secant is the differential of the secant, divided by the secant into the sqxiare root of the square of the secant minus 1. 45. Differential of cosec "^£c. Let ?/=cosec~^a;. Then a; = cosec y. But c?a7=— coseca;cota;(7x, lience ^^^ dri = > THE TRANSCENDENTAL FUNCTIONS. 43 or the differential of an arc in terms of its cosecant is minus the differential of the cosecant, divided by the cosecant into the square root of the square of the cosecant minus 1. 46. Differential of vers"^ic. Let y = vers'^cc. Then x = vers y. But dx = sin ydy, hence , dx dx dx dy = -, — = sin y -y/i _ cos^ y Vl — (1 — vers y)'^ dx dx or the differential of an arc in terms of its versine is the differential of the versine^ divided by the square root of twice the versine minus the sqxiare of the versine. 47. Differential of covers"^ a?. Let y = covers"' x. Then x = covers y. But dx = — cos ydy, hence , , dx dy=- V2x'-x2 or the differential of an arc in terms of its coversine is minus the differential of the coversine, divided by the square root of tivice the coversine minus the sqxiare of the coversine. Examples. Differentiate : ^xdx VI -4 a;* dx VI 1. y = sin~' 2.x*^. dy = 2. y = cos 'Vl — a^. dy = S. V = sin~' -^ — dy = ^ 1 + a^ ^ 1+x 1 4. y = tan 'a*. dy = — X- 2dx 1 a^ lo» adx x" (l -j- a') 44 THE DIFFERENTIAL CALCULUS. 5. 2/ = tan-V. dy = dx e^ + e' 6. 2/ = sin-i(tancc). dy = - ^^^-^^~ ~ . Vl — tan'' X 7. y = cos-\2 cos x). dy = ?smxdx ^ Vl — 4oos^x S. y = cos'^(]ogx). dy= — 9. ^ = log(cos-ia;). dy = — dx Vl — log^ a; cos Iojy'I — a^ 10. 2/ = tan--^. dy = ^(^-^)^^. 1+a^ ^ l + (jx' + x* 11. 2/ = a? sin-i X - Vl - x^. dy=(sm-^x-\ ~ — \dx. \ Vl - x'j 12. x = r versin"^^ — V2 rv — v^ dx= y^y . V2 r?/ - 2/2 13. y = (sin~^a;)^ f?.y = (sin 'a;)-' j s in-^a^logCsm-^a;) Vl - a;'' + a; ) ^^^ <. Vl - a.-^ > 14. 2/ = a?"""'^ ^2/ = ^^"""■'K I ''^^"'^ + ^Qg^- 1 dx i X Vl - af' ^ 15. 2/ = siu-i-. dy = —^^~-. IG. 2/ = cos-i?. d2/= '^''' '■ Vr^ - ar* a' .7 ?TZa; 17. 2/ = tan-i-- c?2/ = 5.2 _|_ ^2 18. 2/ = cot-i^. dv=- ^'^^ . r ^ y~- ^_ ar* THE TRANSCENDENTAL FUNCTIONS. 45 19. y = sec -• cly = '>' x^ar r 20. y = cosec"''— dy= — r a; Va^ - 21. y = vers~^— dy = '• -^2rx — cc 22. 1'/ = covers"^ -• dy= — >' -yj^rx-^ 23. 2/ = tan-\ -— -^^^. dy = \dx. \ 1 + cos X' 24. 2/ = sin~^ Vsm^. dy = ^Vl + cosec x dx. 25. 2/ = log('^y4-itan-^x-. ^^2/ = ^,- 26. ?/ = sec~'- dx 27. 2/ = sin-»?-±^ +1 V2 cZ?/ = vr — af' d?/ = dx vr -2x- -x" dt/ = (^20; tan-i^ r + r ^da;. rfy = ndx cos^ x-i-n^i sin- a; dy = -2dx. 28. 2/ = 0-2-|-a^)tan-^-. ?' 29. y = tan~^(9itana;). 30. ?/ = cos~'(cos2a;). 48. Applications. 1. A wheel revolves about a fixed axis through its centre. Compare the velocity of a point on the rim with its velocity in a horizontal direction. The horizontal velocity is evidently the rate of change of the cosine of the arc described by the point ; hence, if the arc be denoted by «, y = cos x, whence dy= — sin xdx, which is also the relation between the rates of y and x. The point is there- fore moving in a horizontal direction sin x times as fast as it 46 THE DIFFERENTIAL CALCULUS. is moving in the arc. At the highest point, where x = 90°, sina;=l, and dy = — dx, the rates being equal. At a; = 30°, sin x = ^, and at this point the horizontal velocity is one-half that in the arc. 2. Compare the horizontal and vertical velocities of a point on the rim of a wheel which rolls without sliding with a constant velocity m on a horizontal line. In this case the path of a point on the rim is a cycloid whose equa- -1 V2?-?/-2/^, tion IS X = r vers' whence — = — dy ,fidt (!)• ON dt -^2 ry — y Since the wheel has a constant velocity m in a horizontal direction, and its centre C is always vertically over Z>, this velocity is the rate of change of OD = r vers~^ - • d Hence r vers ' - r dt V2ry dy Substituting this value in (1), dx y — - = - m. dt r Hence At O, At B, At E, '^y ds l/dxY fdyV /2i dt-\{dt)+[i) = '^\i- dy y = 0, and dx dt = ^ = 0. dt dt ^ dx ds ^ dy ^ 2/ = 2r, and -- = - = 2m,^ = 0. dt dt dt ds ^ dx dy y^r, and - = ^- = m,^^ tV2. y I, f) THE TRANSCENDENTAL FUNCTIONS. 47 3. Find the subnormal of the cycloid. y''^ = y' V27y- y;^ ^ ■V2i'y'-y'-\ But (Fig. 8) ■V2ry'-y'-' = PM = ND, or the normal passes through the foot of the vertical diameter of the circle when in position for the point P. Hence, also, the tangent passes through the upper extrem- ity T. Therefore to draw a tangent and a normal at any point P, put the generating circle in position and join P with the extremities of its vertical diameter. Also, to draw a tangent parallel to a given line, draw BQ parallel to the given line, and PQ parallel to the base. Then P is the required point of tan- gency. 4. A man walks in a direction AB. Compare the rate of change of his distance from a point with the rate of his angular motion about 0. Let fall the perpendicular 0Z> = 2:> upon AB, and take for the pole, OD for the polar axis. Then the equation of AB is p ■ , dr p sin $ d6 ^ /, a f^** a r r=-^, whence ^t=- — ^r?r -rr- For ^ = 0, -,^ = 0; for cos 6 dt cos- 6 dt ' dt ' 6 = 90°, - = 00. dt 5. An elliptical cam making two revolutions a second about a horizontal axis through one focus, gives motion to a bar in a vertical direction through the centre of revolution. The trans- verse axis being G and the eccentricity f, find the velocity of the bar when the angle between the vertical and the trans- verse axis is 60° ; 90°. a{l — e-) , dr a(l — e-)esmO dd ... „ r = z ~, whence -j- = yz — jr-r- -77' which for 1— ecos^ dt {1—ecos 6y dt a = 3, e=|, and — = 47r, becomes -tt. When ^ dt ' {S-2cosOy = 60°, ^=-5V37r; when = 90°, ^' = -12^. dt dt 9 6. The crank of a steam engine is one foot in length and makes two revolutions a second. If the connecting rod is 5 48 THE DIFFERENTIAL CALCULUS. feet in length, find the velocity of the piston when the crank makes angles of 45°, 135°, 90°, with the line of motion of the piston rod. Let a, b, x, represent the crank, connecting rod, and variable side of the triangle, respectively, and 6 the angle between a and x. Then x = a cos $ + V&" — a^sin^ 6, whence dx ( ■ a , a^ sin cos ] dO — = — a sm 6 H — [ — ? dB which for a = 1, ?> = 5, — = 47r, becomes dt f . /) , sin 6 cos 6 ) \ sin ^ H - y ( ■\/25 - sin2 ) V25 - sin^ 6 Ans. -if-^Tr; --y_V27r; -47r. 7. Find the slope of ?/ = sin x at the points where the curve crosses X. Ans. ± 1. 8. Find the angle at which y = sin x crosses y = cos x. Ans. tan-i2V2. 9. Find the length of the normal to the cycloid. Ans. -y/'^ry'. CHAPTER III. SUCCESSIVE DIFFERENTIATION. 49. Equicrescent variable. A variable lohich changes uni- formlJ^, fTiat is, whose rate is constant, is said to be equicrescent. 50. The differential of an equicrescent variable is constant. For, if X be equicrescent, its rate -^ is constant. But dt is constant ; hence dx is also constant. It is evident that, if — is not constant, dx is a variable. dt The above is a direct consequence of the definitions ; for the differen- tial of a variable is what would be its change during any interval were its rate of change to remain throughout the interval what it was at its begin- ning. If the rate varies from instant to instant, differentials correspond- ing to equal intervals also vary; while if the rate remains the same, these differentials are equal. 51. Successive derived equations. Let y=f(x). Then d>j=f'(x)dx, in which f'{x)=-^, the first derivative. Now, in general, dy, or /' {x) dx, is a variable. For dx is a variable unless x is equicrescent ; and /' (x) is a variable unless f{x) is linear, in which case it can be reduced to the form dv y = mx -f b, whence -r-=f' (x) = m, a constant. Hence, unless the function is linear and x is equicrescent, dy =f'(x)dx is variable, and, being true for all values of x, can be differenti- ated, thus forming a second derived equation which may in its turn be differentiated, a repetition of this process leading to 49 50 THE DIFFERENTIAL CALCULUS. successive derived equations called the fii^t, second, third, etc., in order. Since differentiation introduces no function which has not been already treated, the successive derived equations are ob- tained by the rules already established. 52. Notation. The second differential of a variable x is represented by the symbol cBx, read ' second differential of a;,' the exponent being a symbol of operation indicating how many times the variable has been differentiated. The student will observe the different meanings of the forms cZ-a;, doi?, and d (x')l Illustration. Given if = 2px. The first derived equation is 2ydy = 2pdx, or ydy = pdx. Differentiating again, we have yd{dy) + dyd{y) =pd {dx), or, in the above notation, yd-y + dy^ = pd-x, which is the second derived equation. Differentiating again, yd{d'y) + dhjd{y) -f 2 dyd{dy) =pd{d'x), or yd^y + d-ydy + 2 dydnj = pcZ%, whence yd^y -\- 3 dyd^y =^?d''a;, which is the third derived equation. If X were equicrescent, the successive derived equations would be much simplified. For when x is equicrescent, dx is constant, and, since the differential of a constant is zero, all the successive differentials of x after the first would vanish. Thus, in the above illustration, d'x = d^x = etc. = 0, and the successive derived equations become ydy = 2^dx, yd^y + dy^ = 0, yd'y + 3dyd'y = 0. 53. Remark. It is important to observe that in most cases it is permissible to consider the variable equicrescent and thus SUCCESSIVE DIFFERENTIATIOX. 51 secure the simplicity above noted. For example, let y =f{x) be the equation of any plane curve. The assumption that x is equicrescent, or that — is constant, implies that the velocity of the generating point in the direction of the axis of X is con- stant. Now, so far as the geometrical properties of the curve are concerned, these being independent of the velocity of the generating point, we are at liberty to make any assumption re- garding the velocity which will facilitate their investigation. We therefore assume the velocity-law in the curve such that the motion in the direction of the axis of X is uniform. Again : suppose a right cylinder is inscribed in a right cone, the problem being to find, of all right cylinders so inscribed, that one whose volume is the greatest. If the radius of the base and altitude of the cone are h and a, and those of the cylinder x and y, we have h : a : : X : a — y, whence x = - {a — y)\ and if V is the volume of the cylinder, F= iryxr = TT - y{a - yy. cr Now in determining the gred,test value of V, it is evidently immaterial whether we regard y equicrescent or not, since the cylinder of greatest volume is independent of the law of change oiy. In functions of a single variable, unless mention is made to the contrary, the variable will hereafter be regarded equicrescent. Examples. Regarding x equicrescent, form : 1. The second derived equations of aV + Wa? = a^S a-yd^y + a-df- + b'dx^ = 0. f + x^ = R^, yd?y + df' + daf = 0. xy = m, 2 dydx -f xd^y = 0. y =x^ log X, d^y = 2 log xdx -f- 3 dx^. A 52 THE DIFFERENTIAL CALCULUS. 2. The fifth derived equation of y — x^ log x. 24 d^y = — dx^. X 3. The fourth derived equation oi y = X \-x 24 d'y = —^^^±—dx\ 4. The third derived equations of : y = tan x, d^y = 2(3 sec^ x — 2) sec^ xdx\ y = e*, d'^y = e'^da;^. y = -, d^y = dx\ X X* y = COS X, dhj = sin xdx\ 5. Prove that the lith derived equation of y = a' is d"y = (log a)"a'(te". 6. If 2/ = log sin X, prove that d^y = 2 darl sin^a; 7. If y = sin~^Va;, prove that d-y = ^ dx^. 4{x-x'y^ 8. If y = m cos^mcc, prove that d^y = —m*\ (cos mx')"* — (m — 1) (cos wa;)"*"^ sin^ mxldaf. 9. If 2/ = a«^, show that d^y = 0. 54. Successive derivatives, or differential coefficients. Let y=f{x), in which x is equicrescent. The first deriva- dv d,\ f(x)~\ tive oif{x) has been defined as -j-= "-^ ^ , and is the ratio of the rates of change of the function and its variable. Since the first derivative is variable except when f(x) is linear, it is in general a function of x and may be denoted by f'(x), or dv — =f'(x) ; it may therefore be differentiated in its turn, and a second derivative formed by dividing d\_f'{x)'] by dx, and SUCCESSIVE DIFFERENTIATION. 53 this process may evidently be continued until a derivative is reached which is constant. The successive derivatives thus obtained are called in order the first, second, third, etc., deriva- tives, and are denoted hj f'(x),f"(x),f"'(x), etc. Since each derivative is obtained from the preceding one in the same manner that f'(x) is obtained from f(x), it follows that: 1. The nth derivative of f{x) is the ratio of the rate of change of the (n — l)th derivative to that of the variable. 2. The nth derivative off{x) may be obtained either by differ- entiating the (n — l)th derivative and dividing by dx, or by divid- ing the nth derived equation by dx". Illustratiost. Given y = a -j- bx^. The first derivative is ^y = Sba^=f'(x).^-^^ "■'' • ^*" ''' dx J \ / Differentiating, remembering that dx is constant, — ^ = 6 bxdx, dx ' whence the second derivative g = 6to=/"(x). • v: Differentiating again, d?v ^=Gbdx, whence the third derivative cPy dx" z=6b=f"'{x).-Uc^. U-i{ M ^i ' ^^ Here the process ends, since the third derivative is constant. Otherwise, differentiating y = a+ ba? successively three times, the successive derived equations are dy = 3 bx^dx, d^y = 6 bxdx^, d^y — 6 bda?, 54 THE DIFFERENTIAL CALCULUS. and, dividing the last by dx"^, as before. • "^ 55. Sign of the nth derivative. Since f'(x) is positive or negative as f{x) is an increasing or decreasing function (Art. 22), and since /"(x) is the first derivative of f'(x), f"'(x) the first derivative of/"(aj), etc., therefore /"(a;) will be positive or negative asf"~^{x) is an increasing or a decreasing function. Examples. 1. If y = mx"', prove that ~, ovf"'(x), is m^(m — 1) (m — 2) 03""^ f'(x) =m^a;'"~\ f"{x) = m^{m — 1)0^-^ f"'lx) = m\m-l){m-2)x"'-^ 2. If 2/ = e* sm x, prove that — ^ = 2 e* cos x. f'(x) = e* cos x-^-e" sin x = e'' (cos x + sin a;) . f"(x) = e''(— sin x + cos cc) + e"'(cos x -f sin ic) =2e' cos ic. 3. If 2/ = log cos X, prove that /'"(x) =—2 sec^ic (3 sec^a; — 2). 4. If 2/ = Vl — ar^, prove that — = • dx- if 5. li y = e""'', prove that f"'{x) = e^^'^'Q.o&x (cos^ a; — 3 sin a; — 1). 6. If y"^ = sec 2 x, prove that /"(a;) = 3y^ — y. 7. If y = a''^, prove that f\x) = h^ log^a • a*^ 8. If ?/^ = 2px, and y is equicrescent, prove that — = - • dy^ p The following first and second derivatives, being of frequent use hereafter, may be here established for future reference. In all implicit functions of two variables, x will be regarded as the equicrescent variable unless otherwise mentioned. SUCCESSIVE DIFFERENTIATION. 55 9. The ellipse, a-y'^+ &V= a-b-. 10. The circle, ^f+x'= R-. f{x) = -^=T , ^ • f\x) = -^. 11. The hyperbola, a^ — Ira? = — a~b^. /'(a.)=^=±^-^==:. /"(x)=— ^. 12. The hyperbola, ir?/ = m. a; a- ir a^ 13. The parabola, y'^ = 2px. /'(..) =?;=±-^. /"(a.) =-4 14. The cubical parabola, 'f = o?x. f(x\--^--^. f"(x\- 2^' 15. The semi-cubical parabola, ay'' = ar\ 2a?/ 2 \a 4tty 16. The witch, x^y = i a\2 a - ij) . fi(^\^_ 2a7/ 16 a" a. x2 + 4a- (iK2-j-4a2)2 f"(x) = 2y ''^^'~'^f„ w 17. The cycloid, x = r versin"^- — V2?'?/ —y-. 66 THE DIFFERENTIAL CALCULUS. 18. The cissoid, ->/ = — 2a — X f'(x) = ± Xi 3^^-^ /" (,;) = ± 1^ {2a -x)^- x'-{2a-xy^ 19. The hypocycloid, x^ -f y^ = a*. 1 . 2 /'(^•) = -V /"(^)=o-r~4- a: X 20. The catenary, v/ = - (e" + e "). ^ ft 21. The logarithmic curve, x = logy. f'(x)=f"(x) = y. 22. The sinusoid, y = sin a;. /'(a;) = coscc. f"(x) = — su\x = — y. 56. Remark. If a function becomes infinite for ajinite value of the variable, its derived functions also become infinite. For if the function be an algebraic one, it can become infinite for a finite value of the variable only by having the form of a fraction whose denominator vanishes for that value, and, in differentiating to form the derived functions, this denominator never disappears. So that if f{x) = co when x = x',f'{x), f"{x), etc., also become infinity when x = x'. Examination of the transcendental functions leads to the same conclusion. Thus log X becomes infinity when x = 0, as do also all its deriv- 11 - atives -, -, etc. : and a'', tan x, sec x, illustrate the same fact. X XT This is not necessarily true when f{x) becomes infinity for an mfinite value of the variable. Thus, log a? = oo when a; = oo ; but f\x) = - becomes zero for x= cc. SUCCESSIVE DIFFERENTIATION. 67 57. Notation. To denote what a function becomes for a particular value of the variable, the variable is replaced by its particular value. Thus, /(a), /(O), f{x'), represent what f(x) becomes when x = a, a* = 0, x = x', respectively. The particu- lar value may also be written as a subscript in either of the following ways : 1] =0, 1] =0, XJ a; = CO XJ ^ ( 1 read - equals zero when x is infinity.' 58. Change of the equicrescent variable. In forming the successive derivatives of y =f{x) wo have considered x equicrescent, that is, dx constant, and hence drx = cfx = etc. = 0. If x is not equicrescent, dx is a variable, and am \dxj dxd-y — dy(Px dx dx^ (1) which is the general form of the second derivative when neither x nor y is equicrescent. Differentiating (1), regarding dx and dy as variables, we have fdxd^y — dyd-x\ \ daf J ((f'ydx — d^xdy)dx — 3(d'ydx — d?xdy)d?x dx dx x^v which is the general form of the third derivative when neither X nor y is equicrescent. The general forms of the third, fourth, etc., derivatives may be found in like manner. If in (1) and (2) x is equicrescent, d?x — d?x = 0, and we have ^, and ?y, (3) while if y is equicrescent, d-y = d^y =0, and we have dyd^x 3((^xydy — d^xdydx ,.. dx' dxP 58 THE DIFFERENTIAL CALCULUS. Thus the forms of the successive derivatives, after the first, differ, according as the variable, the fvmction, or neither, is considered equicrescent. To transform a differential expression which has been formed on the hypothesis that x is equicrescent into its equivalent in which neither x nor y is equicrescent, we have only to re- place the successive derivatives by the general forms (1), (2), etc. To change the equicrescent variable from x to y, we replace the successive derivatives by (4) directly, or by the general forms, and then make dry = cfy = etc. = 0. To transform a differential expression formed on the hypoth- esis that either a; or y is equicrescent into its equivalent in terms of a new equicrescent variable 9, we first replace the successive derivatives by their general forms when neither x nor y is equi- crescent, and then substitute for x, y, dy, dx, d^y, d\ etc., their values in terms of 6. Examples. 1. Change the equicrescent variable from xio y in the expression y— + —+1 = 0. dx^ dxr -r, T • d"y 1 dyd^x , dyd^x , dy^ , -, ^ Replacing -^ by - -^~-, we have - y ^^ +-^- + 1 = 0, dxr dor aor dx- or, dividing by dif and multiplying by da^, d^x _ dx^ _ ^^ _ dy^ dy^ dy in which the position of dy indicates that y is the equicrescent variable. 2. Change the equicrescent variable from x to z in the equa- d^v ■ 1 tion x*-^ + a^y = 0, having given x = -- dx^ z Replacing ^„ by ^^^^ - ^J/^'^ ^e have, after substituting ^ ^ dx? ^ daf ' ' J dz ^ ,o 2 dz^ dhi , 2dy , « ^ dx= and d^x = -, — ^ -\ ^- + a-y = 0. z^ ^ dz- zdz SUCCESSIVE DIFFERENTIATION. 59 3. Change the equicrescent variable from a; to ^ in the equa- tion — ^ -\ ^ + ?/ = 0, having given x^ = 4t. dor X dx From x^ = 4it, dx = — -_, d^^ = :• Hence, replacing d^v' ^^ • ^ ^^^ d'v dv — ^ by the general form as in Ex. 2, we find t^ -i- -^ + w = 0. dx" ^ ^ ' de dt ^ 4. Change the equicrescent variable from a; to ^ in d?y X dy ^ ^ =0 dx^ 1 — x^dx 1 — X- having given x = sin 6. dx = cos 6d9, d^x = — sin 6d6^, 1 — a;- = 1 — sin^ $ = cos^^. Hence dxd^y — dyd^x _ x dy y _ cos 6d$d^y 4- sin 6d £^dy dx? 1 — XT dx 1 — a^ cos'^ dd6^ sin 9 dy , y ^ d^y , ^ ^ — = 0, or — ^ + « = 0. cos^'e cos Odd cos'' 6 dO' ^ 5. Change the equicrescent variable from x to 6 in the ex- pression — < — ^^ , having given x = a cos 6, y — h sin 6. d^ ^^^ (a^sin^O + b^cos^O)'^ ah C. If (a^ — xr) — 2=0, show that x' — - — 2 = 0, dxr X dx dy- having given x- + 2/^ = a". We have from x- + y^ = a-, cZa; = — •" cZy, (Z^a; = — - — —• X ar Replacing — by ^^^^ ~ ^^^^^'^ , substituting the above values dxr dx' of dx and cZ^a;, and for a-—x^ its equal ?/-, the given expression becomes a; z = 0. cZ/ 60 THE DIFFERENTIAL CALCULUS. 7. Change the equicrescent variable from « to ^ in the ex- pression dry dx- having given y = r sin 0, x = r cos 6. dy = sin 6dr + r cos 6d6, dx = cos Odr — r sin 9d6. (Jpy = sin 6drr + 2 cos OdBdr — r sin OdO-. d-x = cos ^d-)- — 2 sin ^cZ^cZr — r cos 6d^l Substituting these values, we find d-t/da; — d-a;cZ?/ d'yda; — d^xdy « fZ^'^ _ . d^r APPLICATIONS OP SUCCESSIVE DIFFERENTIATION. Accelerations. dh 59. Acceleration. Signification of — • Velocity has been defined (Art. 6) as the rate of change of the distance passed over by a moving point ; hence if s be the distance and v the velocity, ds '' = dt' The rate of change of v is called the acceleration. Now the rate of change of v is fds\ dv \ dt J d^s dt dt df ' fjpg hence — measures the acceleration of the point in its path. df Being the rate of v, the acceleration is the amount by tchich the APPLICATIONS OF SUCCESSIVE DIFFERENTIATION. 61 velocity woxdd change in a unit of time were the velocity-rate to become constant at the instant considered. Thus, if at any in- stant the acceleration is said to be 5, it is meant that at that instant the velocity is changing at the rate of (5 feet per sec- ■ end) per second, or (5 miles per hour) per hour, according as the second or hour is the unit of time, and the foot or mile the unit of distance. Cor. 1. If -y = -TT is constant, -tto = 0, or in uniform mo- dt dt- tion there is no acceleration. doe dn CoR. 2. Since — , -j-, are the velocities in the directions of ^ d?x d^y the axes, -^, -~, are the corresponding accelerations. 60. Signs of the axial accelerations. — may be plus or minus, and the sign is interpreted as fol- lows : When plus, the velocity — is accelerated in the positive dx direction of X. Thus, suppose — is negative, or the point ^^ d-x moving in the negative direction of X; then if — ^ is positive, the velocity is being accelerated in the direction -|- X, that is, it is algebraically increasing, although numerically diminish- ing, till the motion is reversed, after which it increases numeri- cally. In other words, the ± signs of the accelerations — -, j^, must be interpreted as an algebraic increase or decrease of the corresponding velocity whether the latter be positive or negative. Examples. 1. A point moves in the arc of the parabola y^= 2px with a constant velocity m. Find the accelerations in the directions of the axis. From the equation of the path we have dy^2ldx^ ^-j^^ dt y dt' 62 THE DIFFERENTIAL CALCULUS. and, by condition, dy Substituting in this the value of -~ from (1), we have which in (1) gives V^ j)^ ^ . dx _ my " ,f,, fdt' •'' di ~ V^^+7' dy _ mp dt Vj^Tp (3) Differentiating (2) and (3), and dividing by dt to obtain their rates, we have cPx _ mp'' dy _ m^p^ dry _ _ mpy dy _ m^p^y df-~ {y^^p2)^dt~ {y'-\-py d-x . Since — is always positive, the velocity along X is always • d^v increasing algebraically. — ^ is negative in the first angle and positive in the fourth, hence the velocity along Y is decreasing algebraically in the first angle and increasing algebraically in the fourth. These remarks are true when the point describes the arc of the parabola in either direction. d^X Wi^ d'V TTV^ At y = p, — ;,=-;—, — ^ = — — , or at the extremity of the dr Ap dr 4p focal ordinate the velocities are changing at the same rate. 2. A point moves in the arc of a circle, its horizontal veloc- ity being 9. Find the accelerations in the path and along Y at the point x = 3, the radius of the circle being 5. From ar2_L 7/2— 7?2 dy__xdx__9x__ dx dt y dt y ^E^-x" APPLICATIONS OF SUCCESSIVE DIFFERENTIATION. 63 since by condition — = 9. Differentiating and dividing by dt, ^ = t^'y ^ 81 J?^ ^ 81 R' , df ' df 'f (72^-0^)1' ^^ or the acceleration along X is zero, as it should be since the motion in this direction is uniform, and that along Y is de- creasing or increasing algebraically as y is positive or negative. To find the acceleration in the path, we have dt \[(ltj [dtj \ y'^dt y' whence d^s ^ _9Rdy ^SlRx^ _S1R^^ dt- y- dt f (i2-'_a^)i Making a; = 3, i2 = 5, in (1) and (2), ^ = -31.+ ^=19.+ dt^ df 3. A point moves in the arc of a parabola, the velocity in (ilT ft S the direction of Y being constant. Find — , and — dt dt^ dy dx my ds m /-o—, — a d-s m^y -s. = rn, — = — ^, — = — Vi? -hy , — 5 = ■■ ' dt dt p dt p dv p^p^^y^ The Development of Continuous Functions. 61. Limit of a variable. The limit of a variable is that value which it constantly ap])roaches hut never reaches. Thus, the limit of a; = 1 + | + ^-f i-H is 2. The statement that 2 is the limit of x implies a particular law of in- crease. If X increases by the successive additions of |^ to 1, 2 is not the limit of x = 1 + J + J + •••, for by the law of its increase x can be made to exceed 2 in value. But x=\-\-^-\-\-\-\-\---- can never become equal to 2, since by the law of its change each increment is but half the difference between 2 and the value of x at any instant. So if a circle be circum- scribed about a regular polygon, its area is not the limit of the area of the 64 THE DIFFERENTIAL CALCULUS. polygon if the polygon changes by the motion of its vertices along the pro- duced radii ; for in that case the area of the polygon may become greater than that of the circle. But if the number of sides of an inscribed polygon be indefinitely increased, its vertices remaining in the circle, the area of the circle is the limit of that of the polygon, since no inscribed polygon,' however many its sides, can coincide with the circle. I It is evident that if we conceive the law of change of a vari- able to continue indefinitely in operation, the variable may be made to approach as nearly as we please to its limit. Hence the difference between a variable and its limit is itself a variable whose limit is zero. 62. The term limit is also applied to a magnitude of varying position as well as to one of varying value. Thus, OT, the tangent to MN at 0, is said to be the limit of the secant OP, since the secant, having at least two points in common with the curve by definition, can never coincide with the tangent ; or, more properly, 6 is the limit of ^ as P approaches 0. Observe that, as in the previous illustrations, if P approaches without condition, $ is not the limit of ^; but if we affix the condition ' OP remaining a secant,' then 6 is the limit of <^, P being made to approach as near as we please to but not coinciding with it. 63. The term limit is frequently used with another meaning which must be carefully distinguished from that above ex- plained. Thus ± R are said to be the limiting values of x and y in the equation a;- + y^ = Pr. To distinguish such limit- ing values of a variable from one which the variable approaches but never reaches, the latter is often written x = 2, ^ = 6, which in the illustrations of Arts. 61 and 62 are read ' x ap- proaches 2 as a limit,' as the number of terms of the series increases indefinitely, '^ approaches ^ as a limit,' as P ap- proaches (Fig. 10). APPLICATIONS OF SUCCESSIVE DIFFERENTIATION. 65 64. It is evident from the definition that a quantity cannot approach two limits simultaneously. Thus, if 2 is the limit of a; = l+:^ + ^-j , X can be made to approach 2 in value as near as we please, and therefore no value less than 2 can be its limit; nor can any value greater than 2 be its limit, since it can never equal 2 and therefore cannot be made to approach any value greater than 2 as near as we please. 65. Continuous functions. A function of a variable is con- tinuous between certain values of the variable when it has a finite value for every intermediate value of the variable and changes gradually as the variable so changes from one value to the other. Thus, in y = mx + 6, y is a continuous function of x for all values of x ; in iC-y^ + h-'j? = arb^, y is continuous between x = ±a; in aV — ^^^ = — <*"^^ V is discontinuous between X = ± a, and continuous for values of aj > a numerically ; in xy = m, y is discontinuous for x = 0. And, in general, if ?/ is a continuous function for all values of x, y—f{x) represents a curve of unbroken extent. _^ m 66. Series. A succession qj terms ivhicfi follow each other iticcording to some law is called a series. When known, the law enables us to determine any term of the series. A series is finite or infinite as the number of its terms is limited or unlimited. 67. The sum of a finite series is the sum of its terms. The sum of an infinite series is that finite limit whose value the sum of its terms continually approaches as the number of terms increases. If there be no such finite value, the series is diver- gent ; if such a value exists, the series is convergent. 68. To develop a function is to find a series whose sum is equal to the function. The development of a function is there- fore a finite or an infinite converging series ; in the former case 66 THE DIFFERENTIAL CALCULUS. the function being the sum of the terms, and in the latter the limit of the sura of the terms. When the series is converging, the difference between the function and the sum of the first n + 1 terms of the series is called the remainder after n + 1 terms, and the limit of this remainder as n increases must evidently be zero. Illustrations. A function may be developed by involution when its exponent is a positive integer. Thus (1 + x)^= 1 + 3 a; + 3 x^ + «^, a finite series, whose sum is equal to the fimction, and which is therefore its development. A function may be developed by division if the indicated division can X* — 1 be completed. Thus, = x' + x+ 1, a finite series. When the divisor X — 1 is not exactly contained in the dividend, division leads to an infinite series, as =l + x4-x2-fx^H — , and the process also furnishes the remain- 1— X der after n + 1 terms. Since this remainder, when added to the tenns already found, must equal the function, it must decrease as n increases, and its examination will discover whether the series is or is not converg- ing, that is, whether it is or is not the developaient of the function. Thus, in the above case, the remainder after n + 1 terms is - — , which decreases 1-x as n increases, only when x < 1. Hence if x < 1, the series is converging, and we may virrite = 1+ x + x^+x^-l- •••, understanding that the 1 — X second number approximates more closely in value to the first as the series is extended; while if x > 1, the series is diverging, and cannot be equal to the function, or is not its development. Other processes of deriving a series from a function do not afford the remainder, and tlnis do not indicate whether the series diverges or con- verges. Thus evolution, or the extraction of the root of a polynomial, fiu-nishes in general an infinite series, but no remainder. No imiversal criterion for determining whether a given series is converg- ing or diverging has been found. 69. Maclaurin's theorem. The object of Maclaunn's theorem is the development of a function of a single variable into a series arranged according to the ascending powers of the variable with finite and constant coefficients. The proposed development will be of the form f(x) =A + Bx + Cx' + X>.^-' + Ex*+--; ( 1 ) APPLICATIONS OF SUCCESSIVE DIFFERENTIATION. 67 in which A, B, C, etc., are finite and independent of x. It is required to find such values for A, B, C, etc., as will satisfy (1) for all values of x, that is, render the series either finite, or, if infinite, then converging. Since (1) is to be true for all values of x, it must be true for ic = ; whence A =f(x) when a; = 0, or the first term of the series is what the function becomes when x = 0. Differentiat- ing (1), the successive derivatives are /' (a;) =B + 2Cx-\-3 Dx" + 4 Ex- -f • • -, /" (X-) = 2 C + 2 . 3 Z).« + 3 • 4 ^^2 + ..., /'"(a;) = 2 . 3 i) + 2 . 3 • 4 ^o; + • • •, etc., which, being true for all values of x, are true for a; = 0. Hence representing by /(()),/' (0),/"(0), etc., what/(a;),/'(a)),/"(a;), etc., become when x—0, we have i^=/'(0), 2C=/"(0), .-. = -^"^^^ 2-3Z)=/"'(0), .-. D = etc., and substituting these values in (1), f{x) =/(0) +/'(0)a: +/"(0) | +/"'(0) t + ..., (2) and the theorem may be thus stated : The first term of the series is lohat the function becomes when a; = ; the second term is what the first derivative of the function becomes when a; = 0, into x; the third term is what the second derivative of the function becomes ivhen x = 0, into x' divided by factorial 2; and, in general, the {n + \)th term is what the nth derivative of the function becomes when aj = 0, into aj" divided by factorial n. 68 THE DIFFERENTIAL CALCULUS. If the resulting series is finite, it is equal to the function, the two members of (2) are identical, and the development is effected. If the resulting series is infinite, it is necessary to determine whether it is convergent. 70. Taylor's theorem. The object of Taylor's theorem is the development of a f auction of the algebraic sum of two variables into a series arranged according to the ascending powers of one of the variables, with finite coefficients depending upon the other and the constants which enter the function. The proposed development will be of the form f{x + y) = P+ Qy + Rf + Sf + :., (1) in which P, Q, R, etc., are functions of x, and independent of y. It is required to find such values of P, Q, R, etc., as will satisfy (1) for all values of x and y, that is, render the series finite, or, if infinite, then converging. Since (1) is to be true for all values of x and y, it must be true when y=0; in which case P=f{x), or the first term of the series is what the function becomes when 2/ = 0. Let a be any value of x, and P', Q', R', etc., the correspond- ing values of the coefficients, which are functions of x. Then (1) is true for x = a, and we have /(a + y) = P' + Q'y + R'-f + S'f + • • •, (2) whose successive derivatives are /' (a + 2/) = Q' + 2P'.v + 3.Sy..., f'{a + y)=2R' + 2-'SS'y:., /'"(a + 2/). = 2. 3^'..., etc. Since these equations must be true for all values of y, they are true for y = 0. Hence f"(a) f"'(a) p'=f{a), Q'=f\a), i^' = •^^ ^' = -^' ^*^- APPLICATIONS OF SUCCESSIVE DIFFERENTIATION. 69 Substituting these values of P', Q', R', etc., in (2), /(a + y) =f{a) +f>(a)y+f"{a) t +/"'(a) | ••., in which the coefficients are what f{x), f'{x), f"{x), etc., be- come when x = a. But a is any arbitrary value ; hence, what- ever the value of x, f{x + y)=f(x)+f'{x)y+f"{x)y^ + f"'{x)t...^ and the theorem may be thus stated : The first term of the series is what the function becomes when y=Q', the second term is the first derivative of the function when 2/ = 0, into y ; the third term is the second derivative of the func- tion ichen 2/ = 0, into y^ divided by factorial 2 ; and, in general, the (n-}-l)th term is the nth derivative of the function when y = 0, into t/" divided by factorial n. As before, if the series thus obtained is infinite, it is neces- sary to determine whether it is convergent. 71i Completion of Taylor's and Maclanrin's Formolae. Since the use of infinite eeriea as the equivalents of the functions is inadmissible unless the series are converging, it is necessary to determine the remainder after m + 1 terms in the preceding formula;, and to examine this remainder in any particular case to see if its limit is zero as n increases. I. Iff{.x) becomes zero when x=a and x = h, and is continuous between these val- ues, and iff'{x) is also continuous between these values, then f'{x) will be zero for some value ofx between a and b. For, since /{x) = for x= a and x =b, as x changes from a to b, /(a;) must either first increase and then decrease, or first decrease and then increase. But the iirst deriva- tive is positive when the function is increasing and negative wlien it is decreasing (Art. 22), and therefore in either case it changes sign between the values x=a and x=b; and being continuous, it cannot become infinite, and therefore must pass through zero. II. First form of the remainder. Resuming Taylor's formula, a)Ax + y)=Ax)+f(.x)y+f(x)y^+J-(x)'^...+f'>(x)'^+": Writing x + y= X, whence y= X -x, and representing by B the remainder after n + 1 terms, we have (2) /(X) =/(«) +f(x)(X-x) +f(x) ^^~^^' +/"W ^^~^^' - 70 THE DIFFERENTIAL CALCULUS. (A' — a;)"+^ Writing the remainder in the form P-^^. ^ — » P being a function of X and x ti be determined, substituting this value of R, and transposing,. we obtain (3) AX) -f{x) -f(x)iX-x) -fix) H^' _/■'(«) ^^§^'- •' ^ ^ \_n \n + l Representing by F{z) the function of z which (3) becomes by substituting z for x, (X—z)^ (X—z)3 (4) Fiz)=/{X) -/(«) -f{z){X-z) -/"(g) ^ |/ -/"'(g) ^ ^ ••• I » I w + 1 If 2= a: in (4), it becomes identical with (3) and therefore = 0. It also becomes zero if z= X, for every term then contains a zero factor. Therefore, by I., its derivative F'(,z) must be zero for some value of z between x and A'. If S be a proper fraction, z= x-i- 9(X — x) will represent such iutermcdiate value. Differentiating (4) to obtain F'{z), we have F\z) = 0-f{z)+f(,z)-f(z)(X~z)+f{z)iX-z) -f(Z)'-^+f(z)'—^-.:+riZ)'-^^-^ !_n (_M whose terms vanish in pairs, except the last two, giving ^,.,._,«,„«^.,<^. Substituting the value z = x + d(_X—x) for which F' (z) is zero, we have, after can- (X—z)^ celline the common factor , [n- or -f'^+^lx + e{X-x)] + P=0, y p=/n+i[x + e(A'-a;)], in which all we know of is that its value lies between and 1. Hence the remainder after n + 1 terms is I « +1* / Ji-P ^'^;f r' =/"+^[^ + KX-X)^ (^^Z|)^1'^/n+l(^ + gy)^j^ (5) fix + y)=fix)+fix)y +fix) y^... +/»(a;) ?J+/»+i(a;+9y) ^. and, substituting in (1), the completed form of Taylor's theorem ia Making a; = 0, and changing y to x, (6) /(a;) =/(0)+/(0)a; +/"(0) ^' ... +/»(0) g +/»+l(to) j^. the completed form of Maclaurin's theorem. "We thus have in both cases the remainder after n + \ terms, which is found by dif. APPLICATIONS OF SUCCESSIVE DIFFERENTIATION. 71 ferentiating the function »t + 1 times and changing a; to a; + 9y, or to 0x, in the (n + l)th derivative, and multiplying this result by "^ j, or by , — --^- If this remainder is zero, the series is finite ,• if its limit is zero as n increases, the series is convergent: if it is neither zero nor lias zero for a limit, the formulas fail. III. Second form of the remainder. Writing the remainder in the form It = I\(X- x), (3) would become (7) /(A-) ~Ax) -f{x) (-Y- x)-f '{x) ^-^^^' - -f'\x) ^^^^f^- A(-V- X) = 0. Kex>reBenting by F{z) what (7) becomes by substituting z for x, F(z)=f(X)-f(z)-f(z)(X- z)-f"{z) —^ --/"(s) ^^^^ - AC-V-s), in which, if a = a; or «=T, jPCs) = as before ; and therefore F {z) = (iiov z= x + 6(^X—x). Differentiating to find F'{z), the terms vanish in pairs except the last two, giving F'(.z) = -r'+\z) ^' ^' +p„ and, substituting the value of z for which F\z) = 0, V ..n+lr-i _ g\n and R= Pi(X- X) =f"+^(x + Oy) ^ ^ —• Substituting in (1), a second completed form of Taylor's formula is (8) A^ + y) =/(«) +fWy +/• C^) |f - +/"(^) ^ +/"+'(a; + -7a\ • 4.5.6.7a2. .3.4.5.6.7a. .3.4.5.6.7. [2 [3 li + 3. 4.5.6. 7a2 - + 2.3. 4. 5. 6.7a- [5 [6 + 2.3.4.5-6.7 iZ = a' + 7 a^x + 21 a^a;^ -j, 35 a'x^ + 35 aV + 21 aV + 7aa;'' + a;^ APPLICATIONS OF SUCCESSIVE DIFFERENTIATION. 73 Being finite, the series is the development of the function, as will evidently be the case so long as the exponent of the binomial is a positive integer. rvtS /y^ rv*t 2. sm. = .-- + ---. .. Making a; = 0, /(O) = 0. The successive derivatives are . /' (cc) = cos X, whence /' (0) = 1, /" (rr) = - sin x, " /" (0) = 0. /'"(a;) = - cos X, « /'"(O) = - 1. f" {x) = sin X, " f\^) = 0. Since f"{x) is the original function, these values will recur in sets of four, and we have a^ , 0^ x^ since = x .. [3 |5 \l 3 0*^ '>•'* "7*" . cosa;=l , — . [2 li L5 Since the (n + l)th derivatives of sinx and cos a; are finite whatever the value of n, the formula develops these functions (Art. 71, FV.), and the error may be made as small as we please by taking a sufScient number of terms. By means of these series we may compute the natural sine or cosine of any arc, but few terms being necessary as the series convel-ge rapidly. Thus, if x = — = .174533 be sub- 18 stituted for x in the series of Ex. 2, sin x = sin 10° = .173G5+. 4. a' = 1 + log a ■ x + log^ a — \- log*a — h •••• [2 [3 Making x = 0, /(O) = a" = 1. The successive derivatives are f(x) = a'loga, f"{x) = a'\og^a, f"(x) = a'log'a, etc. ; whence /'(O) = loga, /"(0) = log2a, /"'(O) = log«a, etc.. 74 THE DIFFERENTIAL CALCULUS. which, substituted in Maclaurin's formula, give the above series. Making a = e, whence log e = 1, it becomes />•- /yJ o** |2 [3 [4 ' and if x = 1, e = 1 + 1 +,^ + ,4 + ,4- =2.718281+ [2 [3 U the Napierian base. These are the exponential series. The (»i + l)th derivative of o^ is (loga)"+^a*, and hence •^ ^ |W + 1 j W + 1 But a"-^ is finite, and (r lo g a)""^ _ a; log a a: log a a; log a r» + l ~ i 2 '" n + 1 ' which approaches zero as Ji increases ; therefore the formula develops a'. K /i I \m 1 , 1 wi(m — 1) o, m(m— l)(m — 2) , -I- ^(^-l)---(^-^ + l) a;n I .. To determine for what values of x the formula develops (1 + ar)*". The (n + l)th derivative, when 9x is written for x, is m(,m - 1) — (TO - n) (1 + ea:)"'"''"\ which bec6mes zero if m is a positive integer when n= in. Hence the series is finite, and is the development of (1 + x)'" when m is a positive integer. If m is negative or frac- tional, the series >s infinite. The ratio of its nth term to the one immediately before it is m — n + 1 /TO + 1 ,\ x = \ T-JX, n ^ n ' whose absolute value, as n increases, will eventually become and remain greater than unity if x is numerically greater than 1. Hence (Art. 71, V.) the series is divergent, and cannot equal (1 + a;)"* when x is numerically greater than 1. The remainder after n + 1 terms is b"+^ rTO(?»-l) ••• (to — 7!) „,i-| 1 R ^fn+H6x) fZL r m(TO- )-(TO-7») ^„^n ■' n+1 L n + 1 J (1 + 6x)" When X lies between and 1, the last factor becomes less than 1 as n increases. In- creasing n by 1 multiplies the first factor by — x, or ( ~ ^x, which approaches —a: as n increases; that is, a quantity numerically less than 1. Hence to increase n indefinitely is to multiply by an infinite number of factors each less than 1 ; the product therefore decreases indefinitely, and the formula develops (1 + a.')" for values of X between and 1. By means of the second form of the remainder we have APPLICATIONS OF SUCCESSIVE DIFFERENTIATION. 75 rOT (OT-l)-(m-n) ,1-1 / l-9 \"+i (1 + to)"» = L [n JU + ea;; 1-9 ' When X lies between and -1, the last factor is finite; ( ^— ^ — )" approaches zero VH- ex' as n increases ; increasing n by 1 multiplies the first factor by ^ ~ " ~ x, which ap- n + \ proaches — x as w increases. Hence, as before, the formula develops (1 + ar)™ for values of X between and — 1. Since (a + x)"* may be written in either of the forms «'»(i.3-, x".(i.^)» and as one of those can be developed, whatever the relative values of a and x, the Binomial formula holds good for fractional and negative exponents. When m is a positive integer, the series is finite, and the formula holds good for both the above forms. °^ ^ 2 3 4 Making x = 0, /(O) = log 1 = 0. The successive derivatives are 1+x •' ^ ^ {l + xf f"(x) = ? , fUx) = ll^ etc. ; whence/'(0) = l,/'(0)=-l,/"(0) = 2,/-(0) = -2.3, etc., and these in Maclaurin's formula give the above series. The ratio of the nth term to the preceding one is ( ~^M" " '■' x or — (1 — l)x, which, n \ «/ if xis numerically greater than 1, becomes and remains greater than unity as n increases; hence (Art. 71, V.) the series is divergent if x is numerically greater than 1. The (n + l)th In derivative is (—1)" '— , and, using the first form of R, (1 + x)''+i •' ^ |n + l n + 1 ^\ + ex' If X lies between and + 1, — ^— is a proper fraction, and R approaches zero as n 1 + ex increases. If x lies between and —1, the series becomes ~x — - — ..., and the second form 2 3 of R gives, numerically, ' ^ ' [n \\-ex' i-ex 76 THE DIFFERENTIAL CALCULUS. For valneB of x between and 1, / ^~ ^ y is a proper fraction, and approaches zero as n increases, while the last factor is finite. Ilence the formula develops log (1 + x) when X lies between + 1 and —1. 7. log.(l + x) = ™(x-| + f-| + f...); (1) if a = e, we have, as in Ex. 6, /y*~ /y^ /y*'* O*^ log (l+x) = x- -+--- + --, (2) ^^^ 2345 ^ ^ which are the logarithmic series. As they diverge, if a; > 1, they are not suitable for the computation of logaritlims. To adapt them to this purpose, substitute — x for a; in (1), and we have / rt*2 /yH> /}*4 ™,5 \ log,(l-.) = ™(_.-|-|-|-|...). (3) Subtracting (3) from (1), log„ (1+a^)- log,. (l-rr) = 2m |a; + | + | + y + •••}• Let X = : then x is less than 1 for all positive values 22 + 1' '■ of z, and log„ (1 + a) - log„ (1 - x) = log„-±^ = log„^-±- 1 — X z = log„(2; + l)-log„2; = 2m\-^A ^ + ^ I, (4) or, if a = e, whence m = 1, logCz+l) -log2 = 2 j -^— -\ ^ H ^ \ . ''^ ^ '' (22+1 3(22+1)3^5(22+1)^ j From this series, which converges rapidly, we may compute the Naperian logarithms of numbers. Thus, if 2 = 1, log 1=0, and we have log2 = 2|- + -^ + -^+^-,+ "- !- =.693147+, ^ (3 3-3=' 5-3^ 7-3^ i ' when six terms are taken. APPLICATIONS OF SUCCESSIVE DIFFEKENTIATION. 77 Making z = 2, log3=log2+2|l+^+Jj, + ^+...}=1.0986m. log 4=2 log 2 = 1.386294+. Making x = 4, log5=log4+2{--f^+-^+-^,+ ---| = 1.6094379+. In like manner, the Naperian logarithms of all numbers may be computed. Cor. 1. The Naperian logarithtn of the base of the common system is ^^^ ^^ = log 5 + log 2 = 2.302585+. Cob. 2. From (4), b being the base of the system, and m' the corresponding modulus, ^* z 12^ + 1 3(2;z + l)^ i ^ ^ Since (4) and (5) are true for all positive values of z, writ- ing X for , we have z log^x m (6) logj X m' or the logarithms of the same number in different systems are proportional to the moduli of the systems. CoR. 3. If in (6) b = e, then m' = 1, and log„ ic = m log X. (7) Having then computed, as above, a table of Naperian loga- rithms, the logarithms in any system may be found by multiply- ing their Naperian logarithms by the modidus of the system. Con. 4. Since log„ a = 1, if a; = a in (7), 1 m= , log a 78 THE DIFFERENTIAL CALCULUS. or the mochihis of any system is the reciprocal of the Naperian logarithm of its base ; which is the relation between the mod- ulus of a system and its base referred to in Art. 28. CoR. 5. In the common system a = 10, hence 1 1 m — = .434294+, log 10 2.302585 the modulus of the common system. rjM'i nfl^ rv** 8. tan~^a; = rc 1 . 3 5 7 Making a? = 0, /(O) = 0. The first derivative is = 1 — ic^ -f a;'' — «'"' + a/* — oj"" • • • by division ; hence the successive derivatives are /' {x) = l-x' + x^- x" + x^- a;'" ••., /" (a;) = - 2x' + 4 ar^ - Ga:* + 8a;' - 10a;»..., /'"(a;) = - 2 + 3 • 4a^ - 5 • 6a;^ + 7 . 8a* - 9 • lOa^ ..., /' (a;) = 2 . 3 .4a; - 4 . 5 . Gar* -f G . 7 . 8ar' - 8 . 9 • 10a;' ..., /' (a;) = 2.3-4-3.4.5-Ga;^ + -.-, from which /'(0)=1, /"'(0) = -2, /^(0) = 2.3.4, /"(0) = 0, /'^(0) = 0, etc., and these in Maclauriu's formula give the series above. Since a series whose terms are alternately plus and minus converges if each term is numerically less than the preceding, the series converges for x =1, whence tan~'l= 45° = -, and we have whence the value of tt. 9. sin (x -\-y) = sin x cos y -f- cos x sin y. This being a function of the sum of two variables, we use Taylor's formula. APPLICATIONS OF SUCCESSIVE DIFFERENTIATION. 79 Making y = 0, f(x) = sin x, whose successive derivatives are f{x) = cosx, f"(x) = — sinx, f"'{x) = — cosx, /'''(a.')= sina;, and so on in sets of four. Hence, substituting in Taylor's formula, .... . V^ tf . V* ff ^vi\{x-\-y) = sin 0;+ cos x-y — sm a;-^ — cos x'^ + sm a;"^ +cos x.-p • • • =sin^|l-|' + g-...|+eos.{,,y-| + |-...| = sin cc cos ?/ 4- cos a; sin ?/ (Exs. 2 and 3). 10. cos (x +y) = cos a; cos ?/ — sin a; sin y. 11. sin [x — y) = sin x cos y — cos x sin y. 12. cos (x — y) = cos x cos y + sin x sin y. 13. Deduce the Binomial formula by Taylor's theorem from (a; + ?/)•». Making y = 0, fix) = a;"*, whose successive derivatives are /' (a;) = mx'"""', f"(x) = m {m — V) x'^~-, etc., hence (a; + ?/)•" = a-"* + mx'^-^y + m{m — l)x'"'^'^ + etc. Main T ^ I I »^ O 3/ , O t*/ O it/ ^[2 |4 ^ |5 |6 1 2 ^ 1 4 I G J IG. tana; = xH 1 . 3 15 17. secaj=l + - + — •••• 2 24 18. -^ = l + a^ + a;2 + ?|!-f^.... cos a; 3 2 19. a;V=x-2-|-ar^-|-- + -.... [2 [3 80 THE DIFFERENTIAL CALCULUS. 20. e"'"'=l + x2 + -. 3 .v3 T ^■t 21. etan-'x = i + a; + -- --—.... 2 6 24 feav -r^; t,a -r 1^ 2a^ Sar* 4x* 23. a'-^^ = a=^|l + loga.2/4-log==«|' + log«a^...|. y , ^ 24. sin~'(a;4- ?/) = sin~'.r H " f- [3 (1-a^)' 25. (a^ - eV) J = a - — - ^'^* ^ ^"^^ 2a 2.4a''' 2.4.6a° 73. Failing cases of Madaurin's and Taylor's formulce. It has been seen that the above formulae often lead to diverg- ing series and therefore fail. The following exceptions are also to be noted. Since the proof that the formulae develop any function de- pends upon the condition that the derivatives of the functions are continuous, no one of them becoming infinite for a finite value of the variable, if log x be the function, whose first deriv- ative f'{x) = - becomes oo, as do all the succeeding derivatives, when x = 0, the coefficients /'(O), /"(O), etc., of Maclaurin's formula become infinite, the series has no determinate value, and log X cannot be developed in powers of x. The same is ^ I true of x", a', cosec x, cot x, etc. Again, from (x-\-y-\-ay, we have, for y = 0, f(x) = {x + a)-, whence f'(x) = -> which is finite for all values of x 2{x + a) except x = — a. For this value of x, f'(x) = co, as are all tlie APPLICATIONS OF SUCCESSIVE DIFFERENTIATION. 81 successive derivatives. Hence the coefficients f'{x), f"(x), etc., of Taylor's formula become infinite for x = — a, and the function (x + y-\-ay can be developed in powers of y for all values of x except x = — a. Evaluation of Illusoi'y Forms. 74. The form ^• It frequently happens that for a particu- Thus lar value of the variable a function assumes the form — Sill 3y • = - when X = 0. How is this result to be interpreted ? X Let X, y, be the coordinates of P, x and y being functions of z, and let MN be the curve the coordinates of whose points are the simultaneous values of x and y as 2 changes. Then - = ^ ' Since by hypoth- esis X and y become zero for some value of «, the curve MN passes through the origin. Let a be the value of z which renders' x and y zero. Then as z approaches a, x and y approach zero, and P approaches 0, so that y the value of - when z = a is the limit of X tan , (f> being the angle which the secant makes with X. But the limit of tan as P approaches is tau^, OT being the tangent at ; hence tan^ = ^^" Fig. II. ~1 = Xjs=a Therefore, to find the value of ■}: , we find that of ' ,^ , since these are equal. /'(«) If TTT"^ is also j^j then since /'(z) and '{z) may be regarded as new lunctions of z whose ratio is - when z = a, ' 82 THE DIFFERENTIAL CALCULUS. • <^'(a)-'{x) ~ 2x _ ^ . /"(*) _ cos x' o~0' "{x) ~ 2 6* —^ 6~* 3. when a; = 0. Ans. 2. log(H-x) /7* ■ fi"^ ft 4. when a; = 0. Ans. log-- X b 5. when x = l. Ans. -• X" — 1 ri 6. -^ — — ^^^^ when x = 3. Ans. ^. x^ — X- — ox — 3 The successive differentiations will be facilitated by evalu- ating a factor in any result when possible. Thus : APPLICATIONS OF SUCCESSIVE DIFFEKENTIATION. 83 7. X — sill ' X sin^a; when cc = 0. fix') V1-X--1 _ Vl-x=^-l '^\^) 3 sin^ it- cos a; Vl - »- cosa;Vl-a^ Ssiir'a; the first factor of which becomes 1 when cc = 0. Proceedmg fUx\ 1 X with the second factor, • „ ) [ = -, the 6 cos x* Vl — V? sin a; first factor becoming — \ when a; = 0. From Ex. 1 the value of the second factor when a; = is 1. Hence X — sin~' X sin^ X """^ --^ when x=\. 9. ar^-3a; + 2 x^-Ca^' + Saj-S when x=-\. Ans. 0. ^ns, 00. 10. I : — when x — 11. log sin X Vajtana) (e^-1)^ when a; = 0. Ans. a log a. Ans. 1. Write in the form ^ ^ — and evaluate the \e^ — 1 X e' — l factors separately. ^ o tan X — sin x , a 12. when x = 0. 13. af* tan a; — a; X — sin x when a; = 0. Ans. Ans. 2. 14. e' — e~ {e'-iy when a; = 0. Ans. 84 THE DIFFERENTIAL CALCULUS. 75. The form — When f{x) and {x) x=a CO 1 /(^•)J {x) = ^- Hence the form — can be reduced to 00 the form - and treated as already explained. Thus sec 3 a; _ 00 -r> i. sec X _ cos X \ 00 sec 3 a; 1 cos 3 a; Hence, by the process already established, cos 3 a:"! ■J; cos a; f{x) ^ -3 sin 3a;' '{x) = -, if we treat the latter by the ^ or ix) L<^(a;)J '^'{^) /'(^) f(x) c^'(a;) <^(^) /'(^) (1) (2) APPLICATIONS OF SUCCESSIVE DIFFEKENTIATION. 85 whence -^^ ^ = -^ } ( ; cfy{x) <\>\X) and the form — can be treated directly in the same way as the form — Since all the derivatives of a function which becomes oo for 0. finite value of the variable also become infinite (Art. 56), this process would appear to lead to no result except when the f(x) f'(x) given value of the variable is infinite, ,,, . , ,,,, { , etc., be- ° ' cf> (x) {xy coming in turn — This is true, but , may, by changing f(x) its form, be more easily evaluated than . , , . Thus logx x CO co' 1 ^ . = m which also becomes — for ic = 0, but it may (.^) /(«) preceding, ^^ -' = — ^, or •') ( =;^^^, and the process holds m this /(o) /(«) ^(«) '(«) case also. Examples. Evaluate : ^ tan cc 1 TT 1. w^nena; = -- tan 5x 2 sin a; tana; cos a; sin a; cos 5 a; tan 5 X sin 5 a; sin 5 x cos a; cos 5 a; When a; = 7TJ the first factor is 1, and the second factor be- comes - • Evaluating the latter by Art. 74; we find tan X " tan 5 a; irX . = 5. 2 sec"^ when x = l. log (1 - x) TT TTX . TTX TTX - sec — tan — tan -r: f'{x) 2 2 2 ^^^ 2 and differentiating once we find its value to be — 1. Hence sec ~2 log(l-a;). = GO. loga; when X = oc. Ans. 0, or go, as »i > or n < 0. 4. ^Qg^^"^^^whenx = 0. log tan X Ans. 1. 76. The form ox qo. When, for x=a,f(x)=0 and (l>(x) = , f{x) c}>{x) =Ax)^ = I, or f{x) ^(x) = -j- {x) = ^. Hence, by introducing the reciprocal of one of the fac- tors, the function may be reduced to one of the two forms -, ~, as is most convenient, and treated as before. GO Examples. Evaluate : 1. (1 — x) tan^ when x = l. z TtX \—x (1 — ic) tan -^ = cot — 2. Hence, by Art. 74, \-x - .irX cot — 2_ TT o TTX ■ -cosec- — 2 2. 2. e "(1 — logic) when a; = 0. -Li 1 X 1— logic"! GO e '(1 - logx) = j-^- = -. - Jo °^ 88 THE DIFFERENTIAL CALCULUS. Hence, by Art. 75, l-loga; 1 Q^ 3. e* sin — when x = cc. e' . « _ . a e* e'' sm — = e' e' Alls. ft. 4. (a''— l)a; when a; = oo. ^>i,s. log a. 77. The form — cio. When, for x = a, /(x) = oo and <}}(x) = cc, f(x)-^(x) = -^ ^ ^{x) f(x) 1 ~0' and may be treated when thus transformed by Art. 74. Examples. Evaluate : log X log x when a; = 1. Here /(a^) = rrrr.' '^(^) = log a; log X 1 1 log a; 1 —^ log X Hence ^<-^ t > n \ - APPLICATIONS OF SUCCESSIVE DIFFERENTIATION. 93 neither a maximum nor a minimum value of y. The change of sign of f'{x) from + to — , and from — to +, in passing through infinity is shown in Fig. 16, the tangent at P being perpendicular to X and its slope infinity. 82. Examination of the critical values when /'(a?) = 0. Since /'(x) changes sign from + to — as /(a;) passes through a maximum value, it is a decreasing function, and its first deriv- ative /"(a;) must be negative (Art. 22). Also, since f'{x) changes sign from — to + as f(x) passes through a minimum value, it is an increasing function, and its first derivative /"(a;) must be positive. Hence, to examine f(x) for maxima or minima values, observe whether f" {x) is negative or positive for critical values of x, that is, for values derived from the equation /'(a;) = 0. As the second derivative may become zero for a critical value of X, the above test may fail. To provide for such case we have the following more general rule. 83. Let y =f(x), and y^ =f(xi) in which x^ is the value of X which renders y = y^=.a. maximum or a minimum. Let y'=f(Xi — h) and y" =f(xi + h) be the values immedi- ately preceding and succeeding the maximum or minimum value 2/1, Xi — h and x^ + h being the corresponding values of x. Developing y' and y" by Taylor's formula, we have y' =f{x, - h)=f{x,) - f\x,)h + fix,) 'I If y" =f{x^ + h) =f{x,) + f\x,)h + f"(x,) +f"'ix^)^+f"(^i)~ 94 THE DIFFERENTIAL CALCULUS. But f(x) I = 2/1, and since Xi corresponds to a maximum or a minimum, /'(a;i) =0. Hence, transposing, 2/' -2/i=/"(^i)|-/"'(^i)|'+/^(a:i)^'-, (1) y - y, =/"(a^i)| +r{x,) 1 4-/n^i) I -. (2) Now the signs of the second members of (1) and (2) will be those of their first terms, that is of /"(a^i), if h be taken suffi- ciently small ; and since h approaches zero as the function approaches its maximum or minimum, we are at liberty to make h as small as we please. Hence if /"(cKj) is positive, the first members are positive, and both ?/' and y" greater than y-^, which is therefore a minimum; while if/"(.Ti) is negative, the first members are negative, both y' and ?/" are less than y^, and 2/1 is a maximum. This accords with what has already been said. If /"(a^i) is zero, then y"-y,= /'"(a;0^+/^(a;Og-, in which, whatever the sign of /'"(Xj), the first members will have opposite signs, and y' and y" cannot both be greater than 2/1, nor both less. Hence neither a maximum nor a minimum can exist unless f"'{Xi) = 0, If this condition be fulfilled, there will be a maximum or a minimum according as /'"'(it'i) is nega- tive or positive. We have therefore the following rule : To determine whether a function has maxima or minima val- iies, form its first derivative and place it eqiial to zero. Tlie roots of this equation contain the values of the variable which correspond to either maocima or minima values of the function. Find the first derivative which does not become zero for one of these critical val- ues of the variable. If this derivative is of an odd order, there is APPLICATIONS OF SUCCESSIVE DIFFERENTIATION. 95 neither a maximum nor a minimum; if of an even order, the function is a maximum or a minimum according as the derivative is negative or positive. Each critical value must of course be examined in turn. Illustration. Examine o(^ — 5x* + 5x' + l for maxima and minima values. f\x) =5x*-20a^ + 15x' = 5a^(a^-4:X + 3) = 0. The roots of this equation are a; = 0, a^ = 1, x = 3. f"{x) = 20 ar^ - GO a^ + 30 a; = 10 a; (2 .r-' - G a; + 3). Substituting x =3, f" (x) = -\- 90 ; hence x — 3 renders the function a minimum, and substituting this value of x in the function we find f{x) = — 2G, the minimum. Substituting a; = 1, f"{x) = — 10 ; hence a; = 1 renders the function a maximum, which we find to be 2. As f"(x) = for a; = 0, we form f"'(x) =GOx^- 120 a.' + 30, which does not vanish for a; = and is of an odd order. Hence a; = corresj)onds to neither a maximimi nor a minimum. 84. Abbreviated processes. I. Since the essential characteristic of a maximum or mini- mum value of a function is a change in the sign of its first de- rivative, it will be sufficient, when possible, to observe whether for a critical value of the variable such change actually takes place. Thus, from {x — a)* + b, f'{x) = 4 (a; — a)'' = 0, the critical value being x = a. Now in passing through x= a, f'{x) changes sign from — to + ; hence x=a renders the function a minimum, namely b. Again, from (x — ay -\- b, f(x) = 3 (x — a)^ = 0, which cannot change sign for any value of x; hence the function has no maxima nor minima values. II. Since if yl is a constant factor, Af{x) increases and decreases with /(a;), a constant factor may be omitted in the search for maxima or minima values. 96 THE DIFFERENTIAL CALCULUS. III. Since ±A -\-f{x) increases and decreases with/(a;), we may substitute /(x) for ±A-\-f(x) in searching for maxima or minima values. If A—f{x) is the given function, we may substitute f{x), provided we reverse the conclusions, as A —f(x) increased when f(x) decreases, and decreases when f(x) increases. IV. Since — - decreases as f(x) increases, and conversely, the reciprocal of the function may be substituted for the func- tion, provided the conclusions are reversed. V. Since log [/(a^')] increases and decreases with f{x), the number may be substituted for the logarithm of the number in the search for maxima and minima values. VI. If f{x) is positive, [/(a;)]" is also positive, and there- fore increases and decreases with f(x) ; or any power of a posi- tive function may be substituted for the function. If f(x) is negative, [/(>>/•)]" will have the same sign as f{x) if n is odd, but the opposite sign if n is even ; or any power of a negative function may be substituted for the function, provided the conclusions are reversed if n is even. We are thus enabled to omit the radical sign in the search for maxima and minima values of any positive radical ; also when the radical is negative, if we reverse the conclusions. Examples. Examine the following functions for maxima and minima values. 1. ^-3-90^ + 15a' -3. Omitting the constant term (III., Art. 84), f{x)=x^-9x^ + 15x. f'{x) = Sx^ — 18 a; -j- 15 = 0, whence the critical values x= 5, X = 1. f'{x) = 6a; - 18, which is 12 for a; = 5 and - 12 for x = l. Hence x = 5 renders the function a minimum, and x = l ren- APPLICATIONS OF SUCCESSIVE DIFFERENTIATION. 97 ders it a maximum. Substituting x = 5 and x = 1 in the func- tion, its minimum and maximum are found to be — 28 and 4, respectively. 2. b + c(x — d)\ f(x) = (x-a) 3 (Art. 84, II. and III.). f'{x) = ^(x — ay = 0, whence x = a; and as f'(x) changes sign from — to + for increasing values of x as x passes through a, & is a minimum value of the function. 3. x^-5x*-i- 5x^-6. 4. Examine the circle y^-\-x^=R^ for maxima and minima ordinates. The function to be examined is y = ± Vii' — o?. Omitting the radical (Art. 84,VI.),/(x) =R-- x', whence /'(a;)= -2x=0, OT x = 0', and as /'(a*) changes sign from -f to — as a; passes through 0, a; = corresponds to a maximum. If we take the negative value of the function, then, in omitting the radical, we raise the function to an even power and must reverse the con- clusion ; hence when y is negative, x = corresponds to a mini- mum. 5. (x-iy{x-\-2y. f\x)=A{x-iy(x-h2y + s(x-iy(x + 2y = (x — ly {x -\- 2y (7 X -{- 5) ; whence the critical val- ues X = 1, X = — 2, X = — ^. Since /'(a;) is — if x is a little less than 1, and -f- if a; is a little greater than 1, it changes sign from — to -}- as x passes through 1 ; hence a; = 1 corresponds to a minimum. f"(x)=3ix-iy(x-{-2y{7x-{-5)+2{x-iy{x-\-2){7x + 5) -^7(x-iy(x + 2y = (x-iy(x + 2)\3(x + 2){7x+5)+2(x-l)(7x+5) -^7(x-l)(x+2)\ = 6{x - ly (x -f 2) (7x'-\- 10a; + 1). 98 THE DIFFERENTIAL CALCULUS. When ic = — f , the first two factors are positive, and the sign will depend upon that of the third factor, which is — for a;= — |; hence x = — ^ corresponds to a maximum. Since/"(x) =Oforx= -2,f"'{x) =6{x-iy (7 ar'+lOrc+l) + other terms which contain (a; + 2), and which therefore vanish when cc = — 2, while the term 6(a; — 1)^(7 a^ + 10a; + 1) does not. Hence f"'(x) does not become zero for a; = — 2, and this value of X corresponds to neither a minimum nor a maximum. 6. xi^ — 3x^ + 6x + 7. The critical values are imaginary. 7. Sin^ X cos x. f'(x) =3 sin^ X cos^ x — sin* x = 3 sin^ x(l — sin^ a;) — sin* x = 3 sin^ X — 4: sin* x = 0; whence sin^a;(3 — 4sin^.^•)=0, and the critical values are sina; = 0, sina;=-— , or a; = 0°, a; = 60°. Since f'{x) evidently changes sign from + to — as sin x passes through the value — , x = 60° corresponds to a maximum. If x is a little greater or less than 0°, 4sin^a/'<3 and /'(x) is +; hence a; = 0° corresponds to neither a maximum nor a minimum. 8. a+ V4ar'-2a.-3. Omitting the constant term, radical sign, and factor 2 (Art. 84, III.,VI., II.), we have 2a;'- x^; whence /'(a;) = 4a; - Sa;^ = 0, or a; = 0, a; = |. f"(x) =4 — 6a;, which is -f for a; = and — for a; = |. Hence the function is a minimum when a; = and a maximum when X = ^. 9. Divide a into two factors, the sum of which shall be a minimum. Let X = one factor ; then - = the other, and the function is X ' a; + - ; or f'{x) = 1 — -^ = 0, whence a;= Va, and the factors a; ' -^ ^^ ~ x" are equal. APPLICATIONS OF SUCCESSIVE DIFFERENTIATION. 99 10. The difference between two members is a. Prove that the greater = twice the less when the square of the greater divided by the less is a minimum. 11. Find a number a; such that its x'th root shall be a maxi- mum. Ans. X = e. 12. To determine the number of equal parts into which a must be divided in order that their continued product may be a maximum. T in 1 « . a a a Let X = number oi parts : then - = one part, and - • - • — • / \:r. X ^ XXX /ct\ to X factors = ( - ) is to be a maximum. \xj log I ]= x([og a - log x) =f(x). /'(a;) = logct — logx — 1 = 0, or log - = 1, whence - = e, or a X = -' e Arithmetically the problem would not be possible unless a was a multiple of e, otherwise x would not be an integer. The general solution belongs to the statement : to find a number x such that the ccth power of - shall be a maximum. 1--^ 13. — ^ — r(x)= — ^^i^ 1 + a; tan a; (1+cctana;)^ A maximum when x = cos x. . , sin X ,,, ^ 1 — tan^ x 14. zr—: /'(^•) = 1 + tan a; (1 + tan x)'' cos X A maximum when x — 45°. 85. Examination of the critical values when /' {x) = oc . Since, when f'{x) =go for a particular value of x, f"(x), f"'{x), etc., also become infinity (Art. 56), the function cannot 100 THE DlFFEllENTIAL CALCULUS. be developed by Taylor's formula, and the results of Art. 83 are inapplicable. In such cases we may examine the first deriva- tive directly to see if it changes sign as the variable passes through its critical value. Examples. 1. b+{x — ay. f'(^x) = ^(x — a)~^ = =00, whence x — a = 0, or 3(x-a)^ x = a. It is readily seen that f'{x) changes sign from — to +, and that x = a therefore corresponds to a minimum. 2, (^ + 2)« {x-sy (0^ + 2)^(0.-13) /w- (^x-'sy y (a;) = Ogives x = — 2 and o; = 13. f'(x)=oo gives a; = 3. o; — 13 is negative if o; is a little less or greater than 3, while (o; — 3)^ is negative if o; < 3 and positive if o; > 3. Hence /'(a;) changes sign from + to — at o; = 3, which gives a maximum. a; = — 2 and a; = 13 may be examined in like manner ; the latter gives a minimum, and the former neither a maximum nor a minimum. 3. (^-^>^ (a; + 1)3 f'(x) = gives a; = 1 and x = 5, the former corresponding to a minimum, and the latter to a maximum. /' (a;) = oo gives a; = — 1, which corresponds to neither. 86. Geometrical Problems. In the following problems F= volume, A = area, S = sur- face, and the substituted function obtained after omitting constant factors, radical sign, etc. (Art. 84), is designated by an accent. \ 1. Determine the rectangle of greatest area which can be inscribed in a given circle. APPLICATIONS OF SUCCESSIVE DIFFERENTIATION. 101 Let R = radius. If we take x, y, to represent the half -sides of the rectangle, then the equation of the cir- cle gives the relation af -\-y^ = It% by means of which we can eliminate y from the expres- sion for the area A = 4:xy, obtaining _ ^ _ 4:X^B^ — x^, or ^^m^af — x*, thus reducing the function to be examined to one of a single variable. Omitting the factor 4 and the radical, we have the substituted function A' = i2V — x*, whence f'{x) = 2E^x-4:x' = 0, or ic=0 and x = ^- V2 f"{x) = 2R'-12x% which becomes -AR' for x = — ; V2 hence corresponds to a maximum. Substituting x = — ::- V2 V2 in w- -(- ar^ = Rr, we find y = — - ; hence x = y, the rectangle is V2 a square, and its area A = 4:xy = 2 R-. Before proceeding to the remaining examples the student will observe : 1°. As the point P moves from A to B, the area of the rectangle increases from 0, passes through its maximum, and decreases again to 0. Whenever, then, the conditions of the problem are such that the existence of a maximum value is clearly seen, it will be unnecessary to test the critical value. 2°. The solution consists in first finding an expression for the quantity to be examined, as 4a;y in the above case. If this is a function of two variables, the next step is to eliminate one by means of some relation between them furnished by the conditions of the problem, as in the above case y^ -\-a? = R^. 3°. This elimination may be effected before or after differen- tiation. In the above case y was eliminated before finding the derivative ; but we might have proceeded as follows : A = 4:xy; A' = xy; f\x) = x^ + y = 0. ax 102 THE DIFFERENTIAL CALCULUS. From X- + y^ = R^, -^ = , hence f'{x) =. — x'- -\- y = (), or dx y y 3i? = y^, as before. Eliminating now y by substituting y^ = a^ in x^ + 2/" = R') we have x = — -. It is frequently preferable thus to eliminate after differentiating. 2. Determine the rectangle of greatest area which can be inscribed in a given ellipse. With the notation of Ex. 1, A = 'ixy, the auxiliary relation being a^ + ^'•'^ — ^'^'} ^^^ equation of the ellipse. Hence A = 4:-x^a^ — af, A' = a-x^ — x*, f'{x) = 2a'X — 4:x'^ = 0, and x = — -, or 2x = a-\/2, which, substituted in the equation of V2 the ellipse, gives 2y = by/2, the sides of the rectangle. 3. Determine the rectangle of greatest area which can be inscribed in a given segment of a parabola. Let OA = a, and y = the half-side AB. Then A = 2y{a — x), or, since y- = 2px, A = 2^2px{a — x) = 2^2py/x{a — xy, o whence A' = a^x -2ax^ + x\ f'{x) = a? -4tax + ^x- = 0, from which we find a; = -• Therefore a — a; = f a = oue side, and 2y = 2^2px = 2-yP^= the other side, and Y / 4. Find the cylinder of greatest volume which can be in- scribed in a given sphere. With the notation of Ex. 1, V = 2iry-x = 2ivx{R''-;i?), APPLICATIONS OF SUCCESSIVE DIFFERENTIATION. 103 whence V = R'x - x\ f\x)= R- - 3 ar = , R 2 R X = — -, or 2 ic = — ::: = altitude. V3 V'S 5. Find the cylinder of greatest convex surface which can be inscribed in a given sphere. With the notation of Ex. 1, whence /S = 4 -n-yx = 4 ttx^R- — or, S' = R-x- - x\ f {x) = 2 R^x - 4 x" = 0, 7? - X = , and 2 a; = i2V2 = altitude. V2 6. Find the cylinder of greatest volume which can be in- scribed in a given ellipsoid. With the notation of Ex. 1, using the equation of the ellipse, F = 2 irf-x = 2 TT-' x(a' - x^) , a- V = a-x - x\ f {x) = a- - 3 x-2 = 0, whence x = — - , and 2 a; = ^^^ = altitude. ^J' V3 V3 7. Find the cone of greatest volume which can be inscribe in a given sphere. With the notation of the figure, V= y?/'^'i ^ o but f =2 Rx - x", hence V= ^ (2 Rx" - x"), V' = 2Rx^- x\ /' (x) = 4 Rx - 3 .'B- = 0, or x = ^R = altitude. \ 8. Find the cone of maximum convex surface which can be inscribed in a given sphere. Fig. 19. >S' = 7r7/V;c-+y-=7rV2 /2a;-.'C-V27e^=7rV4i2V-2i2a;«, S'=2Rx'- x\ f{x) = 4Rx-3x^ = 0, and x = ^R= altitude. 104 THE DIFFEUP^NTIAL CALCULUS. 9. Find the cone of greatest volume which can be inscribed in a given paraboloid, the vertex of the cone being on the axis in the base of the paraboloid. V=-y-{a-x). Altitude =" (Fig. 18.) o ^ 10. Find the cylinder of greatest volume which can be in- scribed in a given right cone. Let b = radius of base, and a = altitude of the cone, and x, y, those of the cylinder. Then V=TTX^y. From similar triangles, h: a::x: (a—y), or y = -{b-x). b h Hence V=~x-{h- .x) , V = bx" - x\ F'g- 20. b f'{x) = 2 bx — o X- — 0, whence .x = 1 6, and y z= -{b — x\ =. =z altitude. 11. Find the cylinder of greatest convex surface Avhich can be inscribed in a given right cone, Ans. Altitude = ^ altitude of cone. 12. Of all right cones of given convex surface determine that one whose volume is greatest. If a; = altitude, y = radius of base, V=^y-x. By condi- o tion, TT?/ V^ + y' = in, a constant. Diif erentiating first, from V'=y'X we have f'{x) = 2yx^ -\- ?/- — 0. From iry^x-y^ = m, dx ^ = ^ Hence f (x) = — =^ 1- ?/- = 0, whence dx x' + 2y' -^ ^ ^ x' + 2f- '' ' X = 2/V2, or the altitude = V2 x radius of base. ^ 13. Of all cones whose slant heights are equal find that which has the greatest volume. A71S. The tangent of the semi-vertical angle = V2. API'LICATIONS OF SUCCESSIVE DIFFERENTIATION. 105 14. From a given quantity of material a cylindrical vessel with circular base and open top is to be made so as to have a maximum content. Find the relation between the radius and altitude. Let .X' = altitude, ?/ = radius. Then V=Try-x, V' = y-x, f'{x) = 2yx-^ -{-y- = 0. By condition 2iryx + -n-y- = m, a con- dx stant; whence -^= '- — , and therefore ' ax x + y f\^)^-^j + f=^^,ovx = y. 15. A square is cut from each corner of a rectangular piece of pasteboard whose sides are a and 6. Find the side of the square that the remainder may form a box of maximum content. a-\-h — Va^ — cib + b'' Ans. The side = 6 16. Prove that in an ellipse referred to its -centre and axes, the product of the co-ordinates of a point on the curve is a maximum when the co-ordinates are in the ratio of the axes. 17. A vertical flagstaff consists of two pieces, the upper being a and the lower b feet long. Find the distance from the foot of the staff at which the visual angle subtended by the upper segment is a maximum. With the notation of the figure, tan 6 = — ' — , tan a = -, X x ,.,./] X tan — tan a tan <^ = tan (0 — a) = 1 + tan tan a a+b b -; whence x = ^b{a -\-b) ^ ab + Jr 106 THE DIFFERENTIAL CALCULUS. 18. Find the least triangle which can be circumscribed about a given ellipse having one side parallel to the trans- verse axis. Ans. Altitude = 36, base = 2 a V3. 19. Find the parabola of maximum area which can be cut from a given right cone, knowing the area of a parabola to be |3/Q.QP(Fig. 22). Let AB = 2b, AC=a, QB = x. Then MQ = ^AQ • QB = ^/{2b-x)x, and AB: AC .:QB:QP,QP=^^^^=—. "^ ^ "^ AB 2b 2aa; Hence A= I MQ'QP: 3 2b ^{2b-x)x, A' = 2 bx^ - X*, f'{x) = 6 bx- - 4 x'' whence x = QB = | b 0; Hence QP = — 2b = fa, or the area of the parabola whose axis is f the slant lieight of the cone is a maximum. 20. Assuming that the work of driving a steamer through the water varies as the cube of her speed, find her most economical rate per hour against a current running c miles per hour. V = speed of steamer in miles per hour. av^ = work per hour, a being constant, V — c = actual distance advanced per hour. Let Then and Hence v — c = work per mile of actual advance. Ans. v = ^c. \ 21. The sides of a triangle are a, x, y, subject to the con- dition X + y = m, a constant. Prove that the triangle of maximum area is isosceles, and that x = y The area of a triangle in terms of its sides = A = Vs(s —a) (.s — x){s — y) in which s = ^ sum of the sides. By condition, 2s=x-^y + a = m + a, whence s ; Hence APPLTCATIOXS OF SUCCESSIVE DIFITERENTIATION. 107 A = ^ Vm-— a^ Vet" — m-+ 4 mx — 4 x^, A'= mx — or, f{x) = m — 2x = 0, and x = — ' 7th From x + y = m, y=-' 1/22. Find the point P of least illumination on the line joining two lights A and B, the intensity at a unit's distance of A being b, and that of B being c, knowing that the intensity varies inversely as the square of the distance. If the distance between A and B is a, and AP=x, the I) c ab illumination at P= --\ — ; whence x = — • x- {a —xy ji _l_ gi 23. Kequired the height of a light directly above the centre of a circle whose radius is i? when the perimeter is most illu- minated, knowing that the illumination varies directly as the sine of the angle of incidence, and inversely as the square of the distance. Let X = height. X R Then the illumination = , .-. x = — -- {x'-R')'^ V2 24. The base of a prism is a given regular polygon whose tea is A and perimeter P. The prism is surmounted by a regular pyramid whose base coincides with the head of the prism. Find the inclination of the faces of the pyramid to the axis, in order that the whole solid may have a given volume C with the least possible surface. Let a; = height of prism. Then its volume is Ax and sur- face Px. Let a — perpendicular from centre of polygon on one side. mi Aa cot 6 • ,-, T J, ,1 1 1 Pa cosec 6 Then is the volume oi the pyramid, and its surface. By condition ., .a cot 6\ ri ^ -~\ "^ ^^t ji[x -\ — 1 = O, . •. a; — <»— ~ • 108 THE DIFFERENTIAL CALCULUS. * The surface which is to be a minimum is P(x-\ Y or, substituting the value of x, ^ a cot 6 + ^a cosec 0. Hence /'(d) = -cosec2^-" cot ^cosec^=0, .-. 2 cosec d= Scot d, or cos d = f , and = cos~' |. 25. Prove that the minimum tangent which can be drawn to an ellipse is divided at the point of tangency into segments , which are equal to the semi-axes. a- b^ If (x, y) is the point of tangency, — , — are the intercepts of X y the tangent, and the length of the tangent \ X- y^ \x- a- — XT a- h^ Hence the function to be examined is -7, + -^ a;- a- — XT From /'(.'r) = we find x^oyl— — , and from the equa- \a-\-h tion of the ellipse, yz=h-J\ If P is the point of tan- gency, and T the point where the tangent meets X, \"^\x ) ^a + 6^l,(„ + j)j; \ a + 6 and in like manner the other segment may be shown to be a. 26. In the straight line bisecting the angle ^ of a triangle ABC, a point P is taken. Prove that the difference of the angles APB, APC is a maximum when AP is a mean proportional between AB and AC. Let AC= a, AB = b, PAB = m, AP= x. Draw PE and PF perpendicular to the sides. Then the function is ^ ''•g- 23. APPLICATIONS OF SUCCESSIVE DIFFERENTIATION. 109 APC - APB = EPC - FPB = tan-i ^ - tan-^^ EP FP ^ih — xcosm , , .a — X cos m — fan— i X sin m = tan" tan a.* sm m whence /' (..) = ^ ^- — a^ + a^ — 2 ax cos m ar' + 6- — 2 bx cos ^m from which we find x = -y/ab- = 0, 27. A paraboloid of revolution whose axis is vertical con- tains a quantity of water into which is sunk a given sphere, the quantity of water being just sufficient to cover the sphere. Find the form of the paraboloid such that the quantity of water may be a minimum, knowing the volume of the paraboloid to be one-half that of the circumscribing cylinder. Let R = radius of sphere, z = OH, the height of the water when the sphere is sunk, and Fig. 24. 2/2 = Ix (1) be the equation of the parabola, in which I is the unknown parameter. The equa- Y tion of the circle is f-^(x-ocy=R', or, since OC — z — R, y- + {x-z-l-Ry=E\ Combining (1) and (2), we have lx + (x-z+ Ry = R-, (2) whence ,^2(z - R)-l ±VAR^ + 1^ -^zl + ^Rl^ 2 But the circle is tangent to the parabola, and there can be but one value for x, and hence 110 THE DIFFERENTIAL CALCULUS. "'-~Ti (3) The vol. of water = vol. of paraboloid - vol. of sphere since (1) gives HS' = Iz. Substituting the value of z from (3) the function becomes or Y>-(i + 2R)\ I whence /'(/) = gives I = ?^ R, which determines the form of the paraboloid. CHAPTER IV. FUNCTIONS OF TTVO OR MORE VARIABLES. 87. A partial differential of a function of two or more vari- ables is its differential on the hypothesis that only one of the variables changes. Thus, if w = sin x log y + zar, and x only is supposed to change, y and z being regarded constant, the par- tial differential of u with respect to x is cos.r log ydx + 2 zxdx ; and the partial differentials of u with respect to y and z are dy and x'dz, respectively. 88. Notation. To distinguish the partial differentials, the variable with respect to which the function is diiferentiated is written as a subscript, thus : d^u — (cos X log y + 2 zx) dx, d^u — dy, d^u = x^dz, which are read ' the a;- differential of w,' etc. d tt -J- is evidently the rate of the function so far as its rate depends upon the rate of x. 89. A partial derivative, or partial differential coefficient, is the ratio of a partial differential to the differential of the variable lohicli is supposed to vary. Thus, in the above case, the partial derivatives of u with respect to x, y, and z, respectively, are du , _ du sin a; du , -y- = cos x log y + 2zx, -j- = ? -^ =xr, dx °^ dy y dz the subscripts being omitted as the denominators indicate the variable with respect to which the differentiation is performed. Ill 112 THE DIFFERENTIAL CALCULUS. A partial derivative is the ratio of the rate of the function to that of the variable supposed to vary so far as the rate of the former depends upon that of the latter, and the function is an increasing or a decreasing function of any one of its variables according as the corresponding partial derivative is positive or negative (Art. 22). Since the first derivative is the factor by which the differen- tial of the variable is multiplied to obtain the differential of the function, the partial differentials are also represented by the notation clu , dw - du ^ -7- clx, -T- ay, -I- az, etc., dx ' dy •^' cZz ' ' which are equivalent to d^xi, d^u, d.,u, etc. Examples. Find the partial differentials of ; 1. u = (or' + f)^. ^dx = ^'^^^ , ^-^dy = y^y . ^^^ {x^ + f)^ "^y {:>? + f)^ o • -\X dii J dx du , xdy 2. u = sin -• — dx = — , — dy= ^ y dx ^y^ — x- dy y^y^-Qi? 3. u = y". — dx = zy" log ydx, —dy=^ xzy"~^dy, dx dy — dz = xy" log ydz. dz Find the partial derivatives of : 4. u = s\\\{xy). ~— — ycos,{xy), — = a;cos(a;y). 5. u = 2/°'°*. — = ?/^'"'= log y cos X, dx du ■ .:„__, sin a; — = sm X • v^'"" ' = dy ^eoversx n T / , \ da du 1 6. « = log(aj4-?/)- -r=-r = dx dy x-\-y FUNCTIONS OF VARIABLES. 113 90. The total differential of a function is its differential ob- tained on the hypothesis that all its variables change. Since the total differential of u =f{x, y, z, etc.) can contain only the first powers of dx, dy, dz, etc., it will be of the form du=f{x, y, z, QtQ.) dx+f_{x, y, z, etc.)dy-^f{x, y, z, etc.)d2 + etc., in which /i(x, y, z, Qtc.),f{x, y, z, etc.), etc., represent the col- lected coefficients of dx, dy, etc. But if all the variables except X be regarded constant, dy = dz = etc. = 0, and all the terms vanish except the first, which is the partial differential with respect to x ; and if all the variables except y be regarded con- stant, all the terms vanish except the second, which is the par- tial differential with respect to y ; and so on. Thus all the terms of the second member will be obtained by differentiating u in succession as if all the variables but one were constant. Hence the total differential of a function is the sum of its partial differentials. Illustration. Let u = 3 ax^y + zrV. du = d^u + dyU + d^u = (6 axy -\- 2rV) dx + 3 aardy -f 3 z^e'dz. (1) Equation (1) is evidently true whether the variables be dependent or independent. If, however, the variables are dependent, the total differential may be expressed in terms of any one of them. Thus if y = bx, 2 = sin x, whence dy = bdx, dz = cos xdx, (1) becomes du = (9 abay^ -f- e'' sin" x — 3e' sin^ x cos x) dx ; or the same result might be obtained by substituting the values of y and z in the original function, giving u = 3abx" -f- e" sin^ x, whence, differentiating, we have (2), as before. If Equation (1) be divided by dt, we have the rate of the 114 THE DIFFERENTIAL CALCULUS. function in terms of the rates of its variables ; and if the vari- ables are dependent, the rate of the function may be expressed in terms of that of any one of its variables assumed as the independent variable. Thus, dividing (2) by dt, we have the rate of u in terms of the rate of x. Examples. Find the total diiferentials of : 1. u = axry + by^'x. du (3 aa^y + by^) dx + {ax^ + 3 by^x) dy. o i. -i^ J ydx — xdy 2. u = tan -• du = ^ •- • y ar-f-?/2 ydx 3. u = log x". dit = H log xdy. 4. u = r cos 6. du = cos 6dr — r sin Odr. xii , a?dy 4- y-dx o. u = — '^ — da = — ^ — x + y {x-i-yy 6. ?t = e'a". ' du = cO'e'dx + e^a" log ady. 7. u = tan~^ (xy) . du = ^ — — — -^• 8. w = x^. du = y3^~^dx + a;" log xdy. 9. ?6 = a'' + e~*'2; + sin v. du = a"' log adx — ze^Hly + e~''d2: -f-cos udy. ,x — y , ydic — a;d?y 10. « = tan-i — ~- da = '^—;r-. — ^^ x-{-y X' + y^ 91. The total derivative, or total differential coefficient, of a function is the ratio of the total differential to the differential of the independent variable. Thus, if M =f{x, y, z), we have for the total differential, by Art. 90, du , du , du , ^^ = d^^'^' + d2,^^ + d^^"5 (1) FUNCTIONS OF VARIABLES. 115 and if x be the independent variable, rdtt"|_ Idxj- du du dy du dz 1 ^H (2) dx dy dx dz dx ^ ' The student will observe that the du's are not the same in the above formulae. In the first member of (1) d?t is the total dili'erential of u, while in the second member du is a partial differential, the notation serving to distinguish the partial differentials from each other and from the total differential. To cancel the equal factors from the terms of the second member would be to destroy the means of distinguishing the dw's, no two of which are the same. In fact, (1) is du = d^u -f- d^n + d^xi. In forming (2) from (1) the first member becomes a total derivative, the bracket being used to distinguish it from the partial derivative — in the second member. It is further to dx be observed that while (1) is true whether the variables be dependent or independent, (2) has no significance unless the (Jv dz variables are dependent ; for -~, — , cannot be evaluated unless CttV CtJb y = cf>{x),z = ip{x). The total derivative is evidently the ratio of the rate of the function, on the hypothesis that all its variables change to the rate of the independent variable. 92. The total derivative with respect to any independent variable may be formed in like manner by dividing the total differential, which is always the sum of the partial differen- tials, by the differential of the independent variable, under- standing that the independent variable is connected with the others by auxiliary relations. Thus, given w =f(x, y, z) and x = i(tv),y =o(w), z = 3(w), to form the total derivative with respect to w, we have du , du , du , 116 THE DIFFERENTIAL CALCULUS. whence Fdul \_dwj du dx du dy du dz dx dw dy dw dz dtv 111 this case the bracket is not necessary, but it is usual to enclose the total derivative in brackets whatever the indepen- dent variable. Examples. 1. Given u = 2axy -^logx, and a; = siny, to form the total derivative of u with respect to y. du ^ du ^ du = ^r-dx -\--r- dy, du _dy_ dx du dx du dx dy dy From the given function Ave find the partial derivatives du dx dx 1 ^" and from x = sin y, — = cos y. Substituting these values, du dy = (2 ay -{- -j cos y -\-2ax = 2a(ycosy -{• smy)-\- cot?/. The same result would be obtained by first substituting the value of X in the function and then differentiating. 2. u — y--}-z*-i-zy, y = smx, z = cosx. du dx = cos2a/'(l — sin2a;). 3. M 1 ^ — V tan '■ — r-^, x = e'', y = e^ x + y' du dz 2e-' By substituting the values of x and y in u and then dif- ferentiating, the student may compare the two processes. In this case the use of the formula is more expeditious. FUNCTIONS OF VARIABLES. 117 4. It = tan ^-, xr-\-y-= R-. y fdul _ 1 (hf ~ X .- • 2; „ o. M = Sin -, 2 = 6*, y = XT. [du~\ e' e 6. u = yz, y = e', z 4ar^+12ar'-24a; + 24. rdH~\ \dx\ ex" 93. Implieit function of two variables. Let f{x, y) = 0. Representing the function by u we have u=f{x, y) = 0. Since the only possible values of the variables are those which render the function zero, u is constant, and hence its differential is zero. Therefore , du J , da, rt du = — dx-\ dy = 0, dx dy whence du dy_ dx dx ~d^' dy (1) or the first derivative of an implicit function is the negative ratio of its partial derivatives. The above depends solely upon the fact that du is constant ; hence (1) is true when f{x, y) = a, where a is any constant. dy dx Thus, f - 2px = 0, -2^=^- Again, i^+a;^=. = — — =—-. These results might of course be ob- dx 2y y tained directly by the ordinary process of differentiation, but it is often useful to employ the value of the derivative in terms of the partial derivatives as given in (1). 118 THE DIFFERENTIAL CALCULUS. Examples, Form by the above method the derivatives of : 1. u = 3 ascry — 2 ay-x = c. 6 axy —2aff_y{2y — 6 x) '3ax' — 4:ayx ~ x('dx — Ay) o 1 1 n ^y yx\ogy-y^ 2. u = x losf y — y losr a; = 0. :j~ = — t :3* o J ^ ty dx yx log x — ar du dy dx~ dx du dy 3. M = y -f ar — 3 mxy = 0. 4. II = ?/e"* — a-r™ = 0, 5. u = sin (x?/) + tan (.ry) = a. 6. M = o?/'' — x^y — ax^ = 0. dy _ my — d? dx y^ — mx dy my dx x{l + ny) ^= -I. dx X dy _ 3 a^?/ + 3 ax^ dx 3 ay^ — x" 94. Evaluation of the first derivative of an implicit function. Let f{x, y) = 0, in which x is the equicrescent variable. du Then dy^_dx dx du dy may be a function of both x and y and assume the illusory form - for particular values of x and y. In such a case we may eliminate y from -^ by means of f{x, y) = 0, and then proceed as in Art. 74 ; or, since y is a function of x, we may apply the process of Art. 74 directly, without eliminating y, forming the successive derivatives of the numerator and denominator with respect to x until a pair is found whose ratio does not become - for the particular values of the variar bles. Thus, if u = y^ — xry — x* = 0, FUNCTIONS OF VARIABLES. 119 dy dx du dx _ '2xy -{•■ia? du~ 3y^ — x^ dy which for x = y = i) becomes -. By Art. 74, dy _2xy 4-4ar^' dx~ 3y^ — X- 2y + 2.^ + 12^ «^|-2^ 0,0 ^dy ^ d'y ^dy dx dx- dx .dy' G-^ + ef^-2 d^ 'd^ 4^ dx 0,0 ,dl dx- ,df ^dy .dy dy Hence 6-^ — 2^^ = 4^, or V^ = 0, and ± 1. dx^ dx dx dx Having seen that the true value of an expression which assumes the form - for a particular value of the variable is its limit (Art. 74), it vi^ould seem, since a quantity can have but one limit (Art. 64), that -^ in the dx above example could have but one value. That it may have several values will appear in Art. 114. dy Examples. 1. If u = x*-{-2axhj— ay'^= 0, shoAv that ~ = 0, or ± V2 when x — y = 0. dy _ 4 .r^ -f 4 axy dx~ 2 aaf — 3 ay' 12 a:? + 4 ay + 4 ax dy dx 0,0 bay-j Aax CM . 4 dti , dij , d'y 24a; + 4a / + 4a ,- + 4aa;-T4, dx dx dx- dy' 6a^-f 6a2/^-4a 'da^ 0,0 ^dy dx Jo,o <:-^« w»-l(S-)=»- 120 THE DIFFERENTIAL CALCULUS. 2. If u = X* - af + 2 axf + 3 ax'y = 0, show that ^ = 0, 3, or — 1, when x = y = 0. 3. u = x^ - a?xy + f = 0. Prove that g = 0, or a% when x = y = 0. 4. u = X' + ax^y - af = 0. Prove that ^ = 0, or ± 1, when X=y=:0. 5. u = ahf - 2 abxhj - .r' = 0. Prove that 5^ = ± 0, when ,, ax ' x = y = (). CHAPTER V PLANE CURVES. CURVATURE. 95. A curve is concave npivard at any of its points when its tangent at that point lies below the curve, arid is convex ujncard when its tangent lies above the curve. When a curve is concave upward, its slope increases with cc; hence if y=zf{x) be its equation, f\x)-\--^ is an increas- ing function. But the first derivative of an increasing function is positive, and the first derivative of /'(a;) is f"{'x) = -~- Hence /"(a;) is positive tvhen the curve is concave upivard. If the curve is convex upward, its slope decreases as x increases ; f'(x) is a decreas- ing function, and its derivative is therefore negative. Hence f"{x) is negative when the curve is convex upivard. 96. A point at which, as x increases, the curvature changes from concave to convex upward, or vice versa, is called a point of inflexion. At a point of inflexion the tangent evidently cuts the curve. Since on one side the curve is convex and on the other con- cave upward, the analytic condition for a point of inflexion is 121 Fig. 27. X 122 THE DIFFERENTIAL CALCULUS. (Art. 95) a change of sign mf"{x). Hence all values of x cor- responding to such points are roots of the equations /"(cc) = 0, f"{x) = cc. These roots are critical val- ues, and do not correspond to points of inflexion iiidess accompanied by a change of sign in f"(x). In approaching a point of inflexion f'{x) is increasing (or decreasing), and after passing this point is decreasing (or increasing) ; hence /' (x) is either a maximum or a minimum at a point of inflexion. Examples. Examine the following curves for curvature and points of inflexion : 1. a; = log y, the logarithmic curve. f"(^x) = y, which is always positive, since negative numbers have no logarithms. The curve is therefore always concave upward and has no point of inflexion. 2. y'-+ x^= R% the circle. 7?- f"(x)= -, which is negative when y is positive, and y3 positive when y is negative ; hence the curve is convex upward above, and concave upward below, X. f"(x) has two signs, but does not change sign for increasing values of x, and there is no point of inflexion. o. xy = m, the hyperbola. 2 m /"(a;) = — ^, which has the sign of x. The curve is there- fore concave upward in the first, and convex upward in the third, angle. f"(x) changes sign at a; = 0; but when a; = 0, y = 00, the curve being discontinuous, and there is no point of inflexion. 4. cry = 4a^(2a — y), the witch. f"(x) = 2y A?!nA^ . Points of inflexion at a; = ± — • (.r-+4a-)- V3 CURVATURE. 123 5. ay-z=oc?, the semi-cubical parabola. G. y = sin x, tlie sinusoid. 7. X = lo^y. A point of inflexion at x = S, where the curva- ture changes from convex to concave upward. 8. y^ = arx, the cubical parabola. 2 a* y " (.1-) = — ^:^, which is positive when ?/ is negative, and y negative when y is positive ; hence the curve is concave up- Avard in the third, and convex upward in the first, angle. f"(x) changes sign at ?/=0, whence x = 0, passing through infinity, and the origin is a point of inflexion. 9. y(a*-b') = x{x-ay-xh\ A point of inflexion when x = la. 10. y = d'-\-x^ 11. y = tan X. 97. Kate of curvature. A plane curve may be defined as the locus of a point which always moves along a straight line while the line always turns around the point. Since the direction of motion is always that of the line, the line is the tangent to the curve. Were the line to remain fixed, the locus would be a straight line, that is, if the tangent does not turn about the moving point there is no curvature ; hence, if <^ be the angle which the tangent makes with any fixed line as X, the curvature will depend upon the change of ^. Since in the circle equal arcs subtend equal angles at the centre, the normal, and therefore the tangent, turns through the same angle for every unit of path describetl by the generating point, and the curvature of the circle is therefore constant whatever the unit by which it is measured. 124 THE DIFFERENTIAL CALCULUS. It is evident that if, in passing a second time through any point of a given curve, the velocity of the generating point be m times what it was before, the rate of turning of the tangent at that point will also be m times its former rate ; or that the ratio of the rate of turning of the tangent to the velocity in the curve is constant. Hence dcfi di _(l ds ~ ds dt is a constant for the same point, whatever the velocity. This expression is evidently the rate of turning of the tangent per unit of length of the curve, and may be taken as a measure of the curvature. This measure is independent of t, as it should be, for the curvature is a geometric proj)erty of the curve independent of the time of its description. dd> Since the rate -j- is the amount by which <^ would change for a unit's length of path, were its rate to remain through this distance what it was at its beginning, the curvature at any point of a plane curve is that of a circle which has a common tangent with the curve at the point considered. This circle is called the circle of curvature, and its radius the radius of cur- vature. 98. To express — ^ in terms of the coordinates of the generating ds ' point. From Art. 25 we have ds=-y/da?-\-dy'^, and, reckoning <^ from the axis of X, tan <^ = — ", whence sec^ d(f> = — —, dx dx dry d^y d'-y dx dx dx or o^ = sec^<^ 1 + tan''<^ -, ..dy'' dx'^ CURVATURE. 125 d-y ^y Hence -^ = 7 T^^ =7 TTTT* i^) To find therefore the curvature of a plane curve y=f{x), differentiate its equation twice and substitute in (1) the values of the first and second derivatives. 99. Curvature of the circle. in o , ., ry, dy X X d-y R , . , 1^ rom X--\- y-= li-,_± = = -:::::: ; — ^ = -, which f^x y Vi2--a^ ^^ 2/ will be ± as ?/ is :f . Hence m dct> _ f _ 1 ds A a^\| R' or the curvature of a circle is the reciprocal of its radius. CoR. 1. Since — =1 when i?= 1, the unit of curvature is ds seen to be the curvature of the circle whose radius is unity.- Cor. 2. The curvatures of any two circles are inversely as their radii. 100. Radius of curvature. Since the curvature of any plane curve at a given point is that of its circle of curvature at that point, and the curvature of this circle is measured by the re- ciprocal of its radius, we have, if p be this radius, p ds^ or p='^' ^ ^"^ dx" 126 y THE DIFFERENTIAL CALCULUS. If we take the positive value of the radical, the radius of curvature will be ± as — ^ d^ is ± ; that is, according as the curve is concave upward or downward at the point considered. The sign of p may thus serve to determine the direction of curvature. CoR. 1. Since — ^ = at a point of inflexion, the radius of dar curvature at a point of inflexion is infinite, and the curvature zero. CoR. 2. Since the circle of curvature at any point has a tangent in common in the curve, the radius of curvature is a normal to the curve. 101. Coordinates of the centre of curvature. Let C be the centre of curvature of the curve MN at any point P, and a, /? the co- i' ordinates of C Then a = OD=OB-DB=x dy /t) sin (3 = DC = BP -\- SC = y + peostji dx ds Substituting the values of ds = -Vdx- + dy- and p = — \ \dxj dry a — X — - dx dx^ dx \dx daP (1) CURVATURE. 127 102. Maximum or minimum curvature. Since the curvature is measured by -> it will be a maximum or minimum when p is a minimum or maximum. It is further evident that if a curve is symmetrical with reference to the normal in the vicin- ity of the point of contact, the curvature, if not constant, will be a maximum or a minimum at that point. 103. In the vicinity of a point of maximum or minimum cur- vature, the circle of curvature lies wholly on one side of the curve ; at all other points it intersects the curve. For at a point of maxi- mum curvature the rate of turning of the tangent is greater than immediately before or after, while the rate of turning of the tangent to the circle of curvature remains constantly what it was at the point of contact ; hence the circle lies within the curve at this point. For a like reason, at a point of minimum curvature, the circle lies without the curve. At all other points of the locus (except when it is a straight line) its cur- vature is continually increasing (or decreasing) while that of the circle remains the same ; on one side, therefore, the curva- ture is less and on the other greater than that of the circle, and hence the curve crosses the circle. Thus, the circle of curvature lies without the ellipse at the extremities of the conjugate axis, within at the extremities of the transverse axis, and at all other points cuts the ellipse. Examples. 1. The parabola. _ „ ^ dy p cPy From y^ = 2px, -/=-» -A = ^ ^ dx y dx^ Hence pi P = 1 +(&"'■ dx if+p')^ cPy daf 1 + dy dx d^ djr = Zx+p>, 128 THE DIFFERENTIAL CALCULUS. At the vertex y = x = 0, p=p, a = p, ^ = 0, or the radius of curvature is one-half tlie parameter and the centre of curvature on the axis twice as far from the vertex as the focus is. We observe also that p is least when y=0, or the curvature at the vertex is a maximum. 2. The ellipse. From ay + 6^x^=a^&S ^ = _ ?>^, ^ = _ _&!.. dx a'y da? ci^if Hence p = (^V + ^-^. At the extremities of the conjugate axis, a; = 0, y = ±b, a? At the extremities of the transverse axis, y = 0, x= ±a, &2 If a=h = R, p = Ri the radius of the circle. 3. The cycloid. From :r=rvers-'^-V2^^^"=:?, dy ^sj^ry-f^ ^=_I. r dx y da? y^ Hence p = 2-\/2ry, or the radius of curvature is twice the corresponding normal (Art. 48, Ex. 9). At the highest point, y = 2r, p = 4?'; at the vertex, y = 0, p = 0. 4. The logarithmic curve. From x = log^, -^ = ^, — ^ = -^ ; hence p = v^^ 'rV ) . dx m dar mr my If a = e, p = ^^ — ^ -^ , and if a; = 0, whence y = 1, p = 2 V2, y the radius of curvature of the Naperian curve at the point where it crosses Y. EVOLUTES AND ENVELOPES. 129 5. The hypocycloid. From. x^+y^ = a^, -^ = — -J~, — ^ = --i-^; hence p = 3^/axy. When either x or y is zero, p = 0. 6. The cubical parabola. From If = a-x, p = -^^-^^ — - — ^• 7. The semi-cubical parabola. 17 2 3 (4a + 9a;)^ l From aif = ar, p = -^ ^x^. 6a 8. The catenary. From 2/ = ^(e« + e"«), p=-|'- 9. The cissoid. i rom ?/ = , p = —5^ ^ , which is zero when 2a — X 'S(2a — xy x = 0, and infinity when x = 2 a. EVOLUTES AND ENVELOPES. 104. The locus of the centre of curvature of a given curve is called the evolute of the curve. TJie given curve is called the involute. 105. Equation of the evolute. Let yz=f{x) be the equation of the involute. The coordi- nates of its centre of curvature are (Art. 101), a = X 1 + \dxj dx ^ \dx P = y + d?y ' ^ ^ ■ ^ da? da? By substituting in these the values of the derivatives ob- tained from y = f{x) , we obtain the values of a and /8 in terms 130 THE DIFFERENTIAL CALCULUS. of X and y. Eliminating x and y between these results and y =f{x), the resulting equation between a and /8 will be the equation of the evolute. Examples. 1. Eind the equation of the evolute of the parabola. From Art. 103, Ex. 1, Ave have a='6x+p, I3 = -K Avhence a—p ~3~"' y = — ^-ip^. Substituting these values of x and y in 7/2= 2px, we have The form of the evolute is shown in the figure. 2. Find the equation of the evolute to the ellipse whence x= y = - b* b'l3 Substituting these in ah/-\- b^x^= a-b', we find and the form of the curve is shown in the figure. 3. The evolute of the cycloid is an equal cycloid. Froma; = rvers-i-^-V2ry-2/^, ^^^^ry-f^ ^ = _21. r dx y dx^ y^ Hence x = a- 2 V- 2ry8 - p\ 2/ = — /8. Substituting these in the equation of the cycloid, a = r vers-'/'- ^V V- 2r;8 - ^. (1) EVOLUTES AND ENVELOPES. If the given cycloid be referred to the axes XiOiY^, 131 0,N =x = CD + QP = MP -{-QP= MP + VMQ • QD which is of the same form as (1). Hence the evokite is an equal cycloid, being its highest point. 4. Show that the evolute of a circle is a point, the centre of the circle. The usefulness of the above method of finding the equation of the evolute is limited by the difficulties of elimination, although the method is general. 5. To find the evolute of the hypocycloid. dy _ y^ d^y _ 1 a' From x^ -\-y^ = a^, we have dx ,.J dx^ 3 »i_™t Hence a = x + 3 x^y^, I3 = y -\- oy^x^. To eliminate x and y we proceed as follows : a+&=x + 3 x^y^ + 3y*r^ + y = (a:^ + fy ; hence (a+ B)^ — x^ + i/^. Similarly, (a — $)* = x^ — y^. Hence, (a+ /S)* + (a-)3)*=: 2.r*, yx' 132 THE DIFFERENTIAL CALCULUS. and [(a + ;8)^ + (a - 3)^]2 + [(a + &)^ - (a - 0)^^ 6. If C is the centre of an ellipse, CG the X-intercept of the normal at P, and the centre of curvature corresponding to P, prove that the area of the triangle COG is a maximum when the distance of P from the conjugate axis is one-fourth the transverse axis. 106. Envelopes. The equation of a locus is a relation be- tween X, y, and one or more constants, upon which latter the position or form of the locus depends. Thus, the constants m and b fix the position of the straight line y = mx -f- b ; the con- stants a and b determine the form of the ellipse a^y--\-b-af=a^b^ ; while the constants of the general equation of a conic deter- mine both its position and form. The constants in y =f{x) are called parameters. It follows that if different values be assigned to one of the parameters of the equation y =f(x), the resulting eqiiations will represent a series or system of curves differing from each other in form, or position, or both. Thus, {x — my + y"^ = R"^ is the equation of a circle whose centre is on X, and if different values be assigned to m, we shall obtain a series of equal circles whose centres are on X. The curve which is tangent to all curves of the system obtained by the coyitinuous variation of any one of the parameters in y = f{x) is called the envelope of the system. The constant thus supposed to vary is called the variable parameter. Thus, in the case of the above circle {x — my + y^ = R^, m being the variable parameter, the envelope of the system is evidently the two tangents to the circle, in any of its positions, which lie parallel to X. E VOLUTES AND ENVELOPES. 133 Denoting the variable parameter by a, the general equation of the system may be represented by f{x, y, a) = 0. Fig. 32. 107. Equation of the envelope. Let SB be the envelope of any system of curves, and Q the point at which the envelope is tangent to any one curve of the system MN. Let u=f{x,y,a) = (1) be the general equation of the system, a being the variable parameter, and P, {x, y), any point on MN. Were jTfiV fixed, that is, a constant, the direc- tion of P's motion would be determined by du dy dx dx~ du' dy But, if MN is not fixed, a is variable, and , du , , du 7 , dii ^ r> du = -^ dx -\- -;- dy -\ da =0, dx dy da whence du du da dy dx da dx dx du dy Now when P coincides with Q, these values of -^ are equal, dx since MN and SR have at Q a common tangent. Hence at Q — — = 0, which will be satisfied if da = 0, that is, if the da dx partial derivative du _ ,. da (2) 134 THE DIFFERENTIAL CALCULUS. The coordinates of any point Q of the envelope must there- fore satisfy (1) and (2). Hence, to determine the equation of the envelope of any system, combine the general equation of the system with the eqtiation formed by jilacing the partial derivative ivith respect to the variable parameter equal to zero^ eliminating the parameter. Examples. 1. Find the envelope of {x — m)--\-y-=R-, m being the variable parameter. XI = (x — my+ y-— E^= 0, — = — 2 (ic — m) =0, or x = m. dm Substituting this value of m in {x — my+y-— B^= we have y=±B, two straight lines parallel to X 2. Find the envelope of the hypothenuse of a right-angled triangle of constant area. Let OAB = c be the constant area, and OA — a. Then, since OB- OA c, 0B = 2c or + y 2( = 0, (1) Hence the equation of ^5 is a^'2c ^' a _ 2 ex a- a in which a is the variable parameter. Hence die 4 ex , 2 c da a° a' whence a = 2x. Substituting this value in (1), we have xy = -, the equation of an hyperbola referred to its asymptotes. 3. Find the envelope of an ellipse whose eccentricity so varies that its area remains constant ; knowing the area of an ellipse to be irab. + ^=0, E VOLUTES AND E^'VELOPES. 135 We have -n-ab = m, a constant, whence a6 = — = c, a constant. TT As a and 6 are both variable, we eliminate either parameter, as b, from a-i/^+6V=a^6- by means of the condition ab=c, and thus obtain u = a* if -\- crxr — o?(? = ; whence — =4 ay — 2 ac^ = 0, or a^=;^ — ■,, which in a^ + c^a^— aV= 0, gives xy=y- Since 2y- 2 the axes are rectangular, the hyperbola is equilateral, as also in Ex. 2. 4. A line of fixed length moves with its extremities in two rectangular axes. Find its envelope. Let AB (Fig. 33) be the line. Its equation is - + ^ = 1, or u=bx -{- ay — ab = 0, (1) and by condition, a^ + b^=AB'=P, (2) I being constant. Proceeding as before, we should eliminate one of the parameters from (1) by means of (2) and then form the partial derivative. But it will be found more expeditious to differentiate first and eliminate afterwards! We have from (1), since 6 is a function of a, m dU ' db , , , rt . . T /x /ov since ;y- = — r from (2) . Substituting in succession the values of X and y from (1) in (3), we find a'y-\-bhj-b' = 0, (4) - a'x - b^x + a? = 0. (5) Substituting from (2) the value of a^ in (4) and of b^ in (5), 6« a" or b^ = yh\ a^ = x¥, which in (2) give cc' + t/"^ = l^, the equation of the hypocycloid. 136 THE DIFFERENTIAL CALCULUS. 5. Find the envelope of y = mx + b, m being the variable parameter. 6. From a point A on the axis of X distant a from the origin lines are drawn. Find the envelope of the perpendic- ulars drawn to these lines at their intersections with Y. A line through Ais y = m{x — a), and its intersection with Y is (0, —ma). The perpendicular to y = m(x — a) at 1 X (0, — ma) is y •\- ma = x. Hence u = y -{- ma -f — = 0, in m m which m is the variable parameter. dm m- ' ^5 Substituting this value of m in y + ma + — = 0, we have y- = 4 ax, a parabola. 7. Find the envelope of a series of equal circles whose centres lie in the circumference of a given circle. Let Xi^ -f 2/i^ = ^1^ be the fixed circle. Then (x - Xi)- -\- {y - yi)- = R^ is the movable circle. Ans. 3? -\-y'^ = (i?i ± liy, two concentric circles Avhose radii are ^i + ^ and R^ — R. 8. Find the envelope of x cos 39 + y&\\\'Sd = a (cos 2 ^) ^, 6 being the variable parameter. 3 xcos3^ + t/sm3e = a(cos2e)^, (1) whence — = — a;siii3e+ r/cos3tf + «(cos2e)^sin2e= 0, Old 1 or a;sin3fl— ?/cos3e= asin2e(cos2^)^. (2) Squaring (1) and (2) and adding, x2 + J/2 ::::, a2[(cos 2 d)^ + COS 2 ^(siu 2 «)2] = oP' cos2 fl. (3) EVOLUTES AND ENVELOPES. 137 Dividing (2) by (1), • o /. o /I tan 3 fl — - % sm 3d — y cos 6 6 x xcos3e + y sin -id y 1 + tan Se- - X = tan 2 9, whence ?' = tan ^. Hence from (3), X „ , 2 o 1 — tan2 e „2^^ — y^ x^ + y^= a^ — =a ^ — — ^, 1 + tan'^ fl x2 + j/2 or {x? + j^2)2 _ a2(j;2 _ j^2)^ the lemniscate. 108. The evolute is the envelope of the normals to the involute. I Let (if', y') be any point P' of the involute, (a, ft) the cor- responding point Q of the evolute, and (j> the angle made by the normal or radius of curvature p = FQ with X. Then for >SQ and ^ SF we have a— a;'=pcos<^, p—y'=psm. (!) o As (x', y') moves along the involute, (a, fi) moves along the evolute, or a, ^3, y' are functions of x' Hence, differentiating (1), da = dx'-\- cos dp — p sin d(f>, dft = d7j'-\- sin dp + p cos d(f>. But, Art. 26, dx' = sin (f>ds, dy'= — cos <^fZi-, 1 d or, since - = -^-j da;'= p sin , dy'= — p cos (f)dcf). Substituting these in (2), we have da = cos ct>dp, dp = sin dp, (3) whence -^ = tan ^. da (2) 138 THE DIFFERENTIAL CALCULUS. But — is the slope of the tangent to the evolute at Q, and da tan <^ is the slope of the normal to the involute at P'. Hence the normal to the involute is tangent to the evohite, and the evolute is tlie envelope of the normals to the invobite. 109. The difference between any two radii of curvature to the involute is equal to the arc of the evolute ivhich they intercept. For, from Art. 108, Eq. 3, da = cos (jidp, dft = sin cf>dp. Hence, squaring and adding, da-+d/3'=dp'; or, if s' be the arc of the evolute (Art. 25), ds'z=±dp; or the rates of change of s' and p are equal. 110. The two preceding properties afford the following mechanical construction of the involute when the evolute is given. Let ES be any curve. Then, if a pattern of RS be made, and a string, one end of which is fixed at S, p,/_ ^'3- 35. be wrapped around the pattern SQB, as the string is unwound from the pattern the free end will describe the \i/ curve MN which will be the involute of MS. Any point of the string will trace the arc of an involute as the string un- winds from the evolute ; hence, while a curve has but one evolute, namely, the locus of its centre of curvature, the evo- lute has an infinite number of involutes. 111. Orders of contact. Let y=f{x), y=(fi(x), be the equations of two curves re- ferred to the same axes and having a common point at xz= a. E VOLUTES AMD ENVELOPES. 139 Then /(a) = {a). Let ^ be a very small increment of a, the ordinates corresponding to a; = a + ^ being /(a + 7i), {a -\- h). By Taylor's formula, /(a + /0= /(a)+/'(a)/i +/"(«) 1+ /'"(a) | .••, <^(a + /0=<^(a) + <^'(a)/i + <^"(«),T;+<^"'(a),|'-, or, by subti-action, f{a+h) - «/,(a + /0 = [/'(a) - <^'(«)]'^+ [/"(«) - <^"(«)] ,| + [/"'(a)-<^"'(a)]f^ + -,(l) which is the difference between corresponding ordinates of the curves on one or the other side of their common ordinate according as h is positive or negative. It thus appears from (1) that two curves are nearer on each side of their common point as the second member is smaller, that is, as the succes- sive derivatives in order are equal each to each when x = a. If /'(a) = '(a), the curves are tangent at a; = a and are said to have contact of the first order. If, also, /"(«) = <^"(«)) the curves are said to have contact of the second order ; and so on. ■• Cob. 1. Since, if the curves have a common point, we must have /(a) = ^(a), contact of the nth order imposes n + 1 condi- tions. Cor. 2. If contact is of an odd order, the first term of (1) which does not vanish contains an even power of h, and the difference between the ordinates has the same sign whether h be positive or negative. Hence one curve lies above or below the other on both sides of the common ordinate, or curves whose order of contact is odd do not intersect. If contact is of an even order, the first term of (1) which does not vanish contains an odd power of h, and the difference between the ordinates changes sign with h. Hence if one curve lies above the other on one side of the common ordinate, it lies below it on the other side, or curves whose order of contact is even intersect. 140 THE DIFFERENTIAL CALCULUS. Cor. 3. Since the number of independent conditions which can be imposed upon a curve is the same as the number of arbitrary constants in its equation, the highest possible order of contact between two curves whose general equations contain n and m arbitrary constants is w — 1, n being less than m. Examples. 1. What is the highest possible order of con- tact of an ellipse and parabola ? The general equation of the conies contains five arbitrary con- stants, and therefore the ellipse has a possible fourth order of contact with other curves. But for the parabola e = 1, the number of arbitrary constants is four, and its highest possible order of contact is the third. Hence the ellipse and parabola cannot have contact with each other above the third order. 2. Prove that in general the highest possible order of con- tact of a straight line is the first, that is, tangency ; and of the circle, the second. 3. Prove that at a point of inflexion the straight line has contact of the second order, and intersects the curve. At a point of inflexion the second derivative of y =f(x), the equation of the curve, is zero (Art. 96). Also, from y = mx -{- b, the second derivative is zero. Hence the line and the curve have contact of the second order. Hence, also, the tangent intersects the curve (Art. Ill, Cor. 2). 4. Prove that at a point of maximum or minimum curvature the circle of curvature has contact of the third order. At such a point the circle does not intersect the curve (Art. 103), hence its contact must be of an odd, and therefore of the third, order (Art. Ill, Cor. 2). SINGULAR POINTS. 112. Points of a curve presenting some peculiarity, inde- pendent of the position of the axes, are called singular points. Such are points of inflexion, already considered (Art. 103). SINGULAR POINTS, 141 Fig. 36 Fig. 37. <■ Multiple points. A multiple point is one common to two or more branches of a curve, and is double, triple, etc., as it lies on two, three, etc., branches. If the branches pass through the point, as in Figs. 36 and 37, P is called a mul- tiple point of intersection or osculation, according as the branches have different tangents or a common tangent. Thus, in Fig. 36, P is a triple multiple point of intersection ; and in Fig. 37, P is a double multiple point of osculation. If the branches meet at the common point but do not pass through it, as in Figs. 38 and 39, P is called a salient point or a cusp point, according as the branches have difterent tangents or a common tan- gent. Cusp points are of the first or second species according as the branches lie on opposite sides or on the same side of the common tangent. 113. A conjugate, or isolated point, is one whose coordinates satisfy the equa- tion of the curve, although no branch of the curve in the plane of the axes passes through it ; as P, Fig. 40. A stop point is one at which a single branch of a curve terminates. Fig. 40. 114. Determination of singular points by inspection. Ascertain if possible, by inspection of the equation, whether for any value of one of the variables, as x, y has a single value. Let a; = a be the value of x which gives a single value h for y. Then the point (a, h) is to be examined. If, for values of x both a little less and a little greater than a, y has more than one real value, the branches pass through 142 THE DIFFERENTIAL CALCULUS. (a, b), which is therefore a multiple point of intersection or osculation. If, for values of x a little greater (or less) than a, y has more than one real value, but is imaginary when x is a little less (or greater) than a, the branches meet but do not pass through (a, 6), which is therefore a salient or cusp point. If y is imaginary for values of x both a little less and a little greater than a, {a, b) is a conjugate point. To determine whether the branches have the same tangent or different tangents at (a, b), we observe that, since (a, b) is common to several branches, '^ must at that point have sev- dx eral values, and the branches will have different tangents or a common tangent according as these values are different or equal. It is evident that -^, as derived from/(a^, y) = 0, may have more thail one limit when f(^x,y) = 0, has multiple points. Thus if POP' is the locus of f{x, y) = 0, -^, being entirely general, applies to both the branch OP and the branch OP', and its value at is the limit of ^ or of ^ X x' according as P or P' approaches a-'o Fig. 41. It is thus a general ex- pression for the limits of different ratios, and these limits may or may not be the same. Examples. 1. Prove that y-= af{l — ar') has a double mul- tiple point of intersection at the origin. Values of x, whether positive or negative, give in general two values of y ; but when x = 0, y has the single value 0. Hence the branches pass through the orgin. dy_ l-2a; dx = ±1; Fig. 42. Vl - x'_ there are therefore two tangents at the origin, making angles of 45° and 135° with X, and the branches intersect. SINGULAR POINTS. 143 2. Prove that ah/—2ah^y — a'^= has a point of osculation at the origin. Solving / for y we have ^ 2/ = ^(6±Vaa; + 6-). If X is positive and very small, the radical >6; hence one value of y is positive and the other nega- tive. If X is negative and very small, both values of y are positive, since the radical is then less than h. li x = 0, y — 0. Hence the branches pass through the origin and lie in the second angle on the left of Y", and in the first and fourth on the right of Y. 20 x" + 4 aby + A ahx^^ dy 5x*-\-4abxy 0,0 Fig. 43. dx dx 2 a^y — 2 abx^_ dy 2a''^y dx — 4 abx 5^ dx whence ^ = ± 0, or the axis of X is a common tangent at the dx origin. Hence there is a double point of osculation at the origin, and for one branch the origin is a point of inflexion. 3. Prove that y^=2x'^-\-x^ has a mul- tiple point of intersection at the origin, the tangent having the slopes ± V2. 4. Prove that y^= „^ has a double ar—x^ multiple point of osculation at the origin. y has in general two real values with opposite signs, whether x be positive or negative, and is zero when a; = ; hence the branches pass through the origin. Fig. 45. 144 THE DIFFERENTIAL CALCULUS. dy _ 2 g- — cr ' = ±0. Hence the axis of X is a common tangent at the origin, 5. Prove that y- = x'^ — af has a double multiple point of osculation at the origin. The locus of Ex. 4 is represented by the dotted line of Fig. 45, and that of Ex. 5 by the full line. 6. Prove that the cissoid has a cusp of the first species at the origin. y- = If X is positive, y has two values with oppo- site signs ; if cc = 0, y = 0; if cc is negative, y is imaginary. Hence branches in the first and fourth angles meet at the origin, but do not pass through it, and the origin is either a salient point or a cusp of the first species. dy _ J 3 a — X ^^ (2a-x)'0> = ±0, or the branches have the axis of X for a common tangent. 7. Prove that ay^ = x^ has a cusp point of the first species at the origin. 8. Prove that (y — x^y = .r' has a cusp point of the second species at the origin. y = a^ ± x'^. If X is negative, y is imagi- nary ; ii X = 0, y = ; if a; is positive and small, y has two positive values. He^ce two branches, both in the first angle in the vicinity of the origin, meet at the origin but do not pass through it. Hence X is a common tangent, and the origin is a cusp of the second species. V SINGULAR POINTS. 145 9. Show that y^=x(a-\-xy has a conjugate point at (—a, 0). y = y/x(a + x) has two values when x is positive, but is imaginary for all negative values of x except x= —a, when y = 0. 10. Prove that the conchoid has a multiple point of inter- section, a cusp of the first species, or a conjugate point, at (0, — b), according as a > &, a = &, a < 5. y-{-b x^y^ = {y •{■by{a? — y^), whence x — ± va^ — y^- If a > 6, values of 2/ a little less or greater than — h give two values of x, and cc = when y= — h. Hence the branches pass through (0, — 6). If a = b, X is imaginary if y is negative and numerically greater than 6 ; is when y= — b; and has two values when y is negative and numerically less than b. Hence the branches meet at (0, — b), but do not pass through it. If a < &, all negative values of y numerically greater than a, except y = — b, render x imaginary. dy fx dx —a^y + a^y — 27f + ba^ — 3by'' -&'2/J.,o 2y.% + f - 2xy + {cr-x'-6y--6by- b') ^ 0,0 b- (a'-.)| ^=± dx Va' - &- If a> b, there are two tangents who slopes are ± /«.2 . Va If a = b, the slope becomes 00, and F is a common tangent. it a < 0, -p IS imaginary. 146 THE DIFFEKENTIAL CALCULUS. 11. Show that y = ictan~^- has a salient point at the origin. If a; = 0, y = 0; whether x be positive or negative, y is positive. The curve therefore lies above X, and branches in the first and second angles meet at the origin. When x is positive, dy = tau~'- dx X = "" = 1.5708, l+a^_ the slope of the branch in the first angle. When x is negative, ^^ = tan-' dx + = tan (- X ) = - ^ = - 1.5708, 1 + X- the slope of the branch in the second angle. 12. Prove that y = x log x has a stop point at the origin. The curve lies to the right of Y, for y negative numbers have no logarithms, and X cannot be negative. When x is positive, y has one real value. When o- x = 0, y = x log X = 11 logic 1 X X •*'"_ Fig. 48. -;r. =0. Hence the curve consists of a single branch terminating at the origin. ASYMPTOTES. 115. A rectilinear asymptote to a curve is a straight line which the curve continually approaches but never reaches ; or it may be defined as the limiting position of the tangent as the point of contact recedes indefinitely from the origin. If the curve has no infinite branch, it can have no asymptote. asy:mptotes. 147 116. Asymptotes parallel to the axes. If PQ is an asymptote parallel to X, and at a distance b from it, then as x increases without limit, y approaches the finite limit b, and y = b is the equation of PQ. So, also, if SR is an asymptote parallel to Y, and at a distance a from it, then as y increases without limit, x approaches the p_ < i .- finite limit a, and x = a is the equation otSR. To determine, therefore, whether f{x, y) = has asymptotes parallel to the axes, observe whether either variable approaches a finite value as a limit, that is, as the other increases indefi- nitely. If such be the case, there is an asymptote parallel to the axis corresponding to the variable which increases indefi- nitely, at a distance from it equal to the corresponding finite limit of the other variable. Examples. 1. Show that x=2R is an asymptote to the cissoid. y- — —— , in which y approaches ± oo as a; approaches 2 R — X 2R. Hence x = 2R is an asymptote to both branches. 2. Show that y = is an asymptote to the conchoid. x = ± --i-Z— Va- — y\ in which, whether y be positive or neg- ative, X approaches ± co as y approaches 0. Hence y = 0, or the axis of X, is an asymptote to both the branch above and that below X. 3. Examine y = tan x for asymptotes. 4. Show that r/ = is an asymptote to the witch x^y = 4:R'-{2R-y). 148 THE DIFFERENTIAL CALCULUS. 5. a-y — x-y = a^. y = — -• As X approaches ±cc, y approaches 0. Hence y = 0, or the axis of X, is an asymptote to two branches. Also, y approaches oo as x approaches ± a. Hence x = a and x = — a are asymptotes. 6. a-x = y{x — aY y a'x As x approaches ± oo, {x-ay y approaches ± 0. Also y approaches CO as cc approaches a. Hence the axis of X and x = a are asymptotes. 7. xy — ay — bx = 0. y = , X = — - — The asymptotes are x = a, y = b. X — a y — b 8. Show that y = is an asymptote to x = log y. 9. Examine aPy^ = a-{x^ — y^) for asymptotes. Ans. y— ±a. 10. Examine y{a- -\- x-) = a^{a — a;) for asymptotes. Ans. w = 0. b^ 11. Examine y = a-\ for asymptotes. {x — cy Ans. y = a, x = c. 12. Examine the locus of Ex. 4, Art. 114, for asymptotes. 117. Asymptotes oblique to the axes. The equation of a tangent to a plane curve being dy' y-y'=dx'^^~^'^' if we make in this equation y = 0, the corresponding value of ASYMPTOTES. 149 X will be the intercept of the tangent on X Kepresenting this intercept by X, we have X=x'-y'^- (1) dy' In like manner, making x = 0, the intercept on Y is found to be Y=y'--'%' (2) from which the accents may be omitted if we understand (x, y) is the point of tangency. Now the asymptote is the limiting position of the tangent, that is, the position which the tangent approaches as the point of contact recedes indefi- dy nitely ; hence its slope is the limit of -^, and its intercepts are the limits of (1) and (2), as the point of contact recedes in- definitely from the origin. The position of the asymptote when oblique to the axes will therefore be known when the limits of X and Y are known, and if these limits be designated by Xi and Yj, the equation of the asymptote is Xj Yr If either Xi or Yi is zero, the asymptote passes through the origin, and its direction is determined by finding the limit of dv — as the point of contact recedes indefinitely from the origin. If both Xi and Ti are infinity, there is no asymptote. If one is infinity and the other finite or zero, the asymptote is parallel to or coincides with the axis on which the intercept is infinite. dy It is usually most expeditious to find first the limit of ^• If this is neither nor co, the asymptote is oblique, and its position is made known by either Xi or Yi; if the limit of dv ~ is zero, there will be an asymptote parallel to the axis of X if Fi is finite ; if the limit of ^ is oo, there will be an asymptote parallel to the axis of Y if Xi is finite. 150 THE DIFFERENTIAL CALCULUS. Examples. Examiue the following curves for asymptotes. 1, The parabola, y^ = 22)x. The curve has infinite branches in the first and fourth angles. -^ = - = ± ; hence if there are asymptotes, they are paral- lei to the axis of X. Yi = y — x there are no asymptotes. '_)», ±0 = ± X ; hence 2. The hyperbola, ary^ — 6V = — a-6l The curve has an infinite branch in each angle. dx X.= b'x , b L , b- ay a ^ y^ dx~\ ( = ± = ±0. Hence the diagonals of the rectangle on the axes are asymp- totes to the curve in each angle. 3. x = log y. Since y approaches as a; approaches — x, the axis of X is an asymptote (Art. 142). Otherwise, dy dx y = ; hence if there is an asymptote to the branch in the second angle, it is parallel to X. 1 1 - log y yi = 2/-2/logy]o = 0, or the axis of X is the asymptote. For the branch in the first angle, a; = oo when y = x. Hence dy -f- = y = 00 ; that is, the asymptote is perpendicular to the dx J^ axis of X, if one exists. Xi = a; — 1]^ = oo, or there is no asymptote to the branch in the first angle. •CtfRVE TRACING. 151 4. y^ = x^(a — x). When X > a, y is negative, and there is an infinite branch in the fourth angle. When x is negative, y is positive, and there is also an infinite branch in the second angle. X,= 2a-3a; Y,= Hence the asymptote is common to both branches, and its equation is y = — x-{--' (See Fig. 53.) o 5. Prove that y = x + 2 is an asymptote to y' = 6 a;^ -j- x^. 6. Prove that r/= — ic is an asymptote to y = a^ — ar"'. CURVE TRACING. 118. The foregoing principles are sufficient for the deter- mination of the forms and singularities of many curves, but a knowledge of the general theory of curves is necessary in order to trace curves with facility from their equations. l-x" 1. y = 1 + ar^ 2a;(a^-3) (3). (2), {l+x'Y Since y has but one value for any value of x, its sign being that of x, and is when y = 0, the curve passes from the third to the first angle through the origin, and has infinite branches in these angles. As x approaches ± cc, y approaches 0, and the axis of X i's therefore an asymptote to both branches. f'{x) changes sign at x = ±l, and these values render f"{x) negative and positive respectively, giving a maximum ordinate in the first, and a minimum ordinate in the third, angle. f"{x) changes sign at Fig. 52. 152 THE DIFFERENTIAL CALCULUS. x=±y/S and a; = 0, giving three points of inflexion. The slope of the curve at the origin is 45°, for /'(x) = l when x = 0. 2. y' = aa^ — of' (1), /"(■v) = /'(•«) = 2a -3a; 2 a" 9.x^(a-a;)^ 3x^{a (3). xy (2), If X is negative, y is positive, and there is an infinite branch in the second angle. f"{x) is negative when x is negative, hence this branch is convex upward. If X is positive, y is positive till x = a, when y = 0, the curve having a branch which crosses X at x = a from the first into the fourth angle. Since y = when a; = 0, the branches meet at the origin, Avhich is a cusp point of the first species, /'(a;) becoming co for x = 0, and Y being the common tan- gent. f"{x) changes sign from — to + at a; = a, which is therefore a point of inflexion, the curve being convex upward in the first angle and concave in the fourth. The slope at x = a is 00, since f (x) = cc when x=a. f'(x) changes sign at x = ^a from + to — , hence x = ^a renders y a maximum. It has been shown in Art. 145, Ex. 4, that y = — a? -f - is an o asymptote to both branches. 3. 2/ = e"«(l), f'{x) = ±-^ (2), f"(x) = -^^ (3). x'e' x'^e" From (1) we observe that y is positive whatever the value of X, or the curve lies above X. Let X be negative. Then y = e^, which increases as x de- creases, becoming ac when a; = ; and decreases as x increases, CURVE TRACING. 153 Fig. 54. y r ^,/"'^ X becoming 1 when x = v:. Hence y = 1, and the axis of Y, are asymptotes in the second angle. Also, when x is negative, f"{x) is positive, and this branch is concave upward. Let X be positive. Then y= — , which increases with x, becoming 1 when a; = x ; or, y = 1 is also an asymptote to the branch in the first angle. Since y = when a; = 0, and y = '-c when x is negative and very small, the origin is a stop point. f'{x) cannot change sign, lience there are neither maxima nor minima ordinates. f"{x) changes at a;= ^ from -f- to — , a point of inflexion at which the cnrvature changes from concave to convex upward. f'i^) = —. = X cc. Placing z = -, whence z = x when are'-' ,_ ^ = 0, /'(a-) = -; orisrin. e' = 0, and X is a tangent at the 4. y = x\ogx (1), /'(a-) = l + logx (2), /"(^) = ^ (3). The curve lies to the right of Y, since x cannot be negative. As the logarithm of a proper fraction is negative, y is negative till a; = 1 , when y = 0. When x>l, y is positive. As f"{x) cannot change sign, ^'' the curve is concave upward. f'{x) = gives log x = — l, or x = ('~^ = ', which ^\ renders y a minimum. When x = 0, f'(^x) = — cc, or the axis of 1' is a tan- gent. When x = l, f'{x)=:l, or the curve crosses X at an angle of 4.5°. Fig. 55. ^^ dx X fly 1 4- log X = T =0C, Y, = y-x^ = -x ax 154 THE DIFFERENTIAL CALCULUS. hence there are no asymptotes. The origin is a stop point (Art. 114, Ex. 11). o. {y - xf = aA or 2/ = r^ ± x^ (1), f\x) = x{2 ± |V^) (2), f"{x) = 2±-'^-Vx (3). See Ex. 7, Art. 114, and Fig. 40. G. r = «^;c(l), /'(,•) =i^, (2), oy- 7. (»r = r'(l), /'(x-) = |^(2), 2 ay ./■"(x-) = f^, (3). 4 a^ir 8. ?/-" = 2ar^ + ^-' (1)- 4 + 3a; .r(x)=± ^+3a- ^3^^ 4(2 + x)^ 2 V2 + X (2), Erom (1), ?/= ± a;V2 + a;, from which we see that the curve is symmetrical to X, passes through the origin, and has x = ~2, x = oo for its limits along X. f'(x) = ± V2 when x=0, hence the origin is a multiple point of inter- section. The tangent at x=—2 is perpen- dicular to X, since f'{x) = oo for x = — 2. f"{x) has two signs, but does not change sign except for x = — |, which is not a point of the curve, since the limit of re is — 2 ; hence there are no points of inflexion. f'(x) changes sign at a; = — |, where there is a maximum and a minimum ordinate. Fig. 56. fi( X _ f^.y_4a5 + 3ar" 4 -f 6 a; , dy z — , whence -^ ()dy ax dx or the asymptote, if there be one, is perpendicular to X. Xi = 4-f3a; cc, and there is no asymptote. CURVE TRACING. 155 9. y' = x* + x^ (1), /'(x) = ^(4-h5a.)= ± i^jtl^ (2), ^,^^^^^8 + 24^+15^ (3). ■ From (1) , y= ± a^ Vl + x, whence we see the curve is symmetrical to X, passes through the origin, and that its limits along X are x = — l and X = X. When x = 0, /' (a?) = ± ; hence the origin is a point of osculation, X being a tangent to both branches. From f"{x) = 0, we find x= ""^^"^^^^ 15 ; the lower sign is impossible since a; = — 1 is a limit, and the upper sign gives points of inflexion. f'{x) = 0, gives x = and a;= — 4, which correspond to maxima and minima ordinates. There are no asymptotes. 10. y- = x^ — X*, ov y=± xl^l — x (1) 3-4a; \x) = = ± 2y /"(^)=± ;a^_12a;4-3 -X) (3). (2), Fig. 58. 4(l-a;)Va;(l-a;) From (1) the curve is seen to be symmetrical with respect to X; and as x cannot be negative and /'(») = when x = 0, the origin is a cusp of the first species. Since x cannot be greater than 1, the curve lies between the limits and 1 along X. There is a maxi- mum and a minimum ordinate at a^ = f , and a; = ' ~"^' corresponds to points of inflexion. When 4 a; = l, /'(x) = x. 11. a-y-x'y^aJK 13. a/ - a^ + &ar' = 0. 12. 4a; = y{x - 2)^ " 14. x"' - ay + 1 = 0. 156 THE DIFFERENTIAL CALCULUS. 15. f = a^(l _ :^Y (Fig. 59). 18. y- = x* - .^■« (Fig. 45). 16. f = x\l - :^Y (Fig. 60) . 19. f + x' _ «« = 0. 17. dY + b'^x^ = d^b^. Fig. 59. Fig. 60. POLAR CURVES. 119. Subtangent and subnormal. Let P be any point of MM ', PT the tangent, PN the normal, the pole, and OX the polar axis. Through the pole draw NT perpendicular to the radius vector to the point of contact, OP, meeting the tangent and the normal at T and N. Then OT is the subtangent and OiVthe subnormal. 120. Lengths of the subtangent and tangent. From the right triangle OPT, 0T= OP tan OPT= rtan OPT. tan a — tan But tan OPT = tan (a - ^) = -r — tan ax 1 + tan a tan 6 dy cos 6 — dx sin 6 1-f-^tan^ (^os 6 + dy sin 6 dx But x = rcos6, y = rsin$, whence dx = cos ^dr - r sin ^d^, (7y = sin Odr + r cos Odd. Making these substitutions, we find tan OPT=r—- hence Subtangent = OT=i" ode dr POLAR CURVES. 157 and Tangent = V OP' -\-OT-=r^ \l + r" ~ Cor. ds = Vdx^ -\- dy^ (Art. 25) = Vrfr' + iHG^. 121. Lengths of the subnormal and normal. 0N= OP tan OPN= root OPT= tan OPT Hence (Art. 120), Subnormal = 0A^=^^-', dO and Normal = PN = ^ / ?•' + -^'• Examples. 1. The lemnisoate r^ = a^ cos 2 B. dr _ _ci- sin 2 6 dd~ r Hence Subt = r- -- = ~ = - r cot 2 ^, dr a' sin 2 ^ Tangent = VylX + r^'^ = --^^, \ di^ Va* - r* Subn = ^"=-^^!-?i2l^, Normal = ^ r' + ^^ = — • \ (7^ T 2. The logarithmic spiral /•=€<*. — = a* losj:a. Hence Subt = , Subn = r log a. log a dO 1 In this spiral tn.\\OPT=r — = , a constant. Hence dr log a the tangent makes a constant angle with the radius vector to the point of contact. For this reason this spiral is often called 158 THE DIFFERENTIAL CALCULUS. the equiangular spiral. In the Naperian logarithmic spii-al, log e = l, and the subnormal is equal to the radius vector. 3. The spiral of Archimedes r = a$. — = a — subn, a constant. 1^ — = a€F ■■ dO dr 4. The cardioide r = (i{'l -f cos^). dr subt. dd z= — a sin 6 = subn. Subt. = — a sin ^ 5. Prove that in the curve r=a sin 6 the radius vector makes equal angles with the tangent and polar axis. tan OPT = r— = a sin — — dr a cos 6 = tan 6. 6. The circle r = 2R cos 0. Subt = 2 Root e cos 6, subn = -2EsmO, Tangent = 2R cos 6 cosec 6, normal =2R. 7. Prove that the subtangent of the reciprocal spiral is con- stant. 122. Curvature of polar curves, A curve at any of its points is said to be convex or concave towards the pole according as its tangent does or does not lie on the same side of the curve as the pole. Let fall from the pole the perpendicular OD=p upon the tangent. If the curve is concave to the pole, 2^ is an increasing function of r, r =f{0) being the equation of the curve. Therefore -^ is positive. dr ^ If the curve is convex to the pole, p is a decreasing function of r, and ~^ is nega- , . dr tive. Fig. 62. /p POLAR CURVES. 159 Hence the curve is concave or convex towards the pole ac- cording as — is positive or negative, and at a point of inflex- dr ion -t- must change sign dr dp dr To find p, we have (Fig. 63), NP being the normal, OD:NP::OT: NT. But, Arts. 120 and 121, NP=Jl^+—; 0T=')^'-^, \ de^ dr- NT=NO+OT=^ - '- + r' ^^^. dO dr Hence p = V^+*^ de" To examine a polar curve for points of inflexion, substitute — from the equation of the curve, r =f(0), in the above value of p, and see if, for any value of r, -^ changes sign. dr Examples. 1. Prove that the logarithmic spiral is always concave to the pole. „ dr , '/•- r r=a'', .-. -3^=rloga, p = dO lr^4.^ Vl+log^a \ d&' Hence ^ — aV2, — is negative. Hence ?-=aV2 indicates a point of inflexion dr at which the curvature changes from concave to convex towards the pole. 3. Prove that the parabola r = — ^ is always concave to the pole. 123. Radius of curvature. From Art. 119, we have P = )^^h d^ dx" (1) in which a; is equicrescent, and the problem is to transform (1) into its equivalent in terms of r and 6 when 6 is equi- crescent. Therefore (Art. 58, Ex. 7), P = + -r \d6J dS" (2) Examples. Find the radius of curvature of : 1. The lemniscate, ?" = a^cos26. dr a^ sin 2 ^ d6 r Va* - r' r dV r* -f a* d^ f^ Hence P = , 'dr POLAK CURVES. 2. The cardioide, r = a{l—GosO). (Id — = acoiiO = a — r. Hence p= 2V2 ar. 3. The spiral of Archimedes, r = aO. ^ 2a- + ^^ 2 + 0' 161 4. The reciprocal spiral, r = -• or u* 5. The lituus, r = — =• Ve ^~2a- Aa*-r* 6. The logarithmic spiral, r = a*. p = a«(l+log2a)^. If a = e, p = V2e*=rV2, or the radius of curvature is V2 X the radius vector. 124. Asymptotes. Since the asymptote is the limiting position of the tangent as the point of contact recedes indefinitely from the pole, if a polar curve has an asymp- tote, r must be infinite for some finite value of 6, and for such value of 6 the subtangent d6 r^— must be finite. dr Let a be the value of B which renders r in- finity. To construct the asymptote make ^jr AOP=a, draw through a perpendicular to 162 THE DIFFERENTIAL CALCULUS. OP, and make OT=t^~ ilr . Then TQ, parallel to OP, is the asymptote. For the point of contact being infinitely dis- tant from 0, the radius vector and asymptote are parallel. If = cc, there is no asymptote. dr Examples. 1. Examine the hyperbola for asymptotes. p , dr ewsin^ ■, ul6 p r = — , whence — = ^- — and r — = — ± ecos^ — 1 dQ {eQ,o^B — \y dr esin^ Now r = cc when cos = -== — ' Hence, if there be an asymptote, it is parallel to the diagonal of the rectangle on the axes. Again, e=Vl- cos^ e = Ve^' - 1 .dO hence r^ — = — 4- — = ctVe^ —1 = 6. dr e sin 6 There is therefore an asymptote. To construct it, draw OP parallel to the diagonal on the axes (or make AOP= cos"^- ), and make OT=h. Then TQ, parallel to OP, is the asymp- tote. Since OC = ^ = — = ae, C is the centre, and sma Ve- — 1 the asymptote coincides with the diagonal. Also, as cos 9 = cos {— 0), there is another asymptote below the axis, and similarly situ- ated. 2. Prove that the parabola r = =- has no asymptote. 3. Prove that the lituus r = — - has the polar axis for an asymptote. ^^ POLAR CURVES. 168 4. Prove that the spiral r = - has an asymptote parallel to 6 the polar axis at a distance a from it. ^ 1,. ■ .. (r'sinS^ ,. , , o. r^xamme r = tor asymptotes. cos 6 (). Examine (r — a) siu d =h for asymptotes. 7. Examine the conchoid r = h cosec 6 + a for asymptotes. r = c» when ^ = 0. ?- — = -^ — ! '-- =b. dr b cos d Jo Hence the asymptote is parallel to the polar axis, and at a distance from it equal to b. 125. Tracing of polar curves. Write the ecpiation of the curve f(r, $) = in the form r=f(6), when possible, and assign such values to 9 as will render it easy to determine those of r. This will usually be sufficient to determine the general form of the locus. For maxima or minima values t)f r, — must change sign. The do locus may then be examined for curvature, points of inflexion, and asymptotes. ExAMPLKs. 1. '/•=asin2^. — = 2«cos2^. dd r = when ^ = and increases with 6 till 6 = -, when — 4 dd changes sign from + to — . Hence 6 = - renders r = a a maximum. From 6 =" - to 6 = - r decreases from a to 0, and the curve is a loop in the first angle. When $ passes -, r becomes negative, in- 2 (If, creasing numerically till = ^Tr, when — dd changes sign from — to +, Hence ^ = ^tt 164 THE DIFFERENTIAL CALCULUS. renders r = — a a minimum. From ^ = f tt to 6 = ir, r is still negative, but decreases numerically from a to 0, giving another loop in the fourth angle. When passes ir, r becomes positive, and as sin 2 6 passes through all its values while 6 varies from to tt, equal loops will be traced for values of 6 between ir and 2 tt. The maximum and minimum values of r are derived from -^=0; namely, Itt, Itt, frr, and In. cW Xo value of 6 renders r = x ; hence the curve has no asymp- tote. 2. r = a sin 3 6. The curve is shown in the figure. From this example and Ex. 1 it may be inferred that in all equations of the form r = a sin ?i^, the curve consists of n, or of 2w, loops, according as n is an odd or an even integer. 3. r = asin|(Fig. 68). 4. r=a(l-tan^) (Fig. 69). 5. r = a cos 2 6. 6. r = a + sin \ 6. 7. r=a + .sinf ^ (Fig. 70). 8. r = a + tan 2 6. 9. y2 = a-(tan2^-l). Prove that there are two asymptotes perpendicular to the polar axis at dis- tances ± a from the pole. Part IL THE INTEGRAL CALCULUS. CHAPTER VI. TYPE INTEGRABLE FORMS. 126. Integral and Integration. If f{x) be any function, and f'(x)dx its differential, then f{x) is called the integral of f\x)dx. Hence, any function is the integral of its differential. The process of finding the function from its differential is called integration. As an operation it is the inverse of differ- entiation, and having seen I. Differentiation to be the process of finding the ratio of the raies of change of the function and its variable, we may define II. Integration to be the process of finding the function when the ratio of its rate of change to that of its variable is given. 127. Symbol of integration. The symbol of integration is I , read ' the integral of.' Thus, if y = ar\ then dy = 3 a^dx, and i dy = y = | 3 ay'dx = ■3?, d and | , as symbols of inverse operations, neutralizing each other. The test of the result of any integration is differentiation ; that is, I ?>:i?dx — ar* because d(ar') = ?>v?dx. 167 168 THE 1NT?:(IHAL CALCULUS. 128. Constant of integ^ration. It is evident that functions which have the same rate, and therefore the same differential, may differ from each other by any constant, but only by a constant. Thus in the function y =mx + &, which for different values of b represents a series of parallel straight lines, the rate of y will be the same whatever the value of b, or cly = mdx for all values of b. Hence any given differential is the differential of an infinite number of functions which differ from each other by a constant, and if the differential only is known, the function cannot be deter- mined. Therefore mdx = mx -f- C, /« in which C is an undetermined constant. Otherwise : since the differential of a constant is zero, if a function contains any constant term, this term will not appear in its differential ; hence a constant C must be added to every integral to represent this term. This constant is called the constant of integration. It will be shown, in the application of integration to definite prob- lems, that it may either be eliminated or that its value may be determined from the conditions. 129. The integral of the sum of any number of terms is the sum of the integrals of the terms. This is an obvious consequence of the proposition (Art. 16) that the differential of the sum of any number of terms is the sum of their differentials. Or, formally, as | and d neutralize each other, d{x — y + z) = x — y-\-z + C, and i dx— j dy + i dz — x — y + z + C; /" TYPE INTEGRABLE FORMS. 169 hence | d(x — y-\-z)= i dx + i dy -\- i dz. Both d and | are, therefore, distributive symbols. 130. If the differential has a constant factor, its integral icill have the same constant factoi: For dlAf{x)^= Adlf{x)l (1) and fcJ[AtX^) ] = Af{^) ; (2) but (2) is the integral of (1). Since a constant factor in the differential also appears in the integral, siich a factor may he loritten before or after the integral sign, at pleasure. Thus, d{ax) = adx, and i adx = a I dx = ax. 131. Type integrable forms. Since a function is the integral of its differential, from d{ax"*) = max'^'^dx, we have I max"'~\]x = ax"*, /__, , ax'" ax"" ^dx = m Putting m — 1 = n, we have in general, /ax^'dx = — ^ a;»+' + C. 71 + 1 71 + 1 Reversing the fundamental processes of differentiation, we obtain thus the twenty following forms : 1. faa;"dr = — ^a;"+i + C'. J 7? + 1 2. r^=loga:+C. J X 170 THE INTEGRAL CALCULUS. 3. I a' log adx = a' + C. 4. I e'dx = e' -\-C. 5. I cos xrix = sin x + C. 6. I — sin xdx = oos a; + C. 7. I sec^ a;c7a; = tan x + C. 8. I — eoseo^ ifdi' = cot x-\-C. 9. I sec X tan a-r/iv = see a; + C. 10. I — cosec X cot a;c/.t* = cosec x -\- C. 11. I sin a^a; = vers x-\-C. 12. j — cos xdx = covers .r + C. 1.3. r ^^' =sin-^T + <7. 14. f -J£— = cos-'.a-4-<7. J Vl - ar' 15. r^^ = tan-ia; + C. 16. r_^^ = eot-^T-HC. 17. r ^^g = sec-^ X + g *^ xVa*^ — 1 18. I ^1::::^:-^ = cosee^ a; + C ^ a; Var^ — 1 TYPE INTEGKABLE FORMS. 171 19. ^ V'^x-x' ^^^^ = covers"^ x -\- C. V2 X — ar 20 132. Remarks on the type forms. The processes of the Integral Calculus consist chiefly in the reduction of differentials to the above forms. When this reduction has been effected, the integral is seen at once by inspection. This being the case, it is evidently indispensable that the student should be thoroughly familiar with the type forms, so as to be able to recognize them at sight. The fol- lowing suggestions will facilitate their recognition and appli- cation. Form 1. Wlienever a differential can be resolved into three factors, viz. : a constant factor, a variable factor vnth any constant exponent except — 1, and a differential factor which is the differ- ential of the variable factor ivithojtt its exponent, then its integral is the product of the constant factor into the variable factor with its exponent increased by 1, divided by the new exponent. For • Ca-x-. dx = -^^ x"+' + C. J n + l Form 2. When the exponent of the variable factor is — 1, the differential falls under the second form f ^ = loga- + C, in which the numerator is the differential of the denominator. Hence, tvhenever the numerator of a fraction is the differential of its denominator, the integral of the fraction is the Naperian logarithm of its denominator. Forms 3 axd 4. These forms are a' ' log adx = a" + C, f" 172 THE INTEGRAL CALCULUS. and Ce' • dx = e'-\-C, in which the differential facto)' must be the logarithm of the base into the differential of the exponent. Forms 5-12. In eacli of these forms the differential factor dx must be the differential of the arc. Forms 13-20. The conditions to which the differential must conform should in each case be carefully noted. Thus, from r-^?^ = tan^..+ J 1 + .t"^ a we see that the first term of the denominator must be 1, and the numerator the differential of the square root of the second term of the denominator. Examples. 1 . | ocp'dx = I 5 • ar' • dr = |ar + C. (Form 1.) 2. Cmx"'dx=-^^^ x'"--^C. J 1 — m ,y Cadx C -2,1 ^^ I ri 3. I — — = I ax hJx = — •-—+ C. J X a' 'Ix- . r2dx 'i , ri '•j35 = -^+^- - rdx^^2Vx-\-a 6. Cfax" -^^ + Vx\ f7.T = V - — + 2 J ^ c. (Art. 12^.) 7. Cb{a+bxydx= C{a + bxy ■ bdx=l(a+hxy + C. (Form 1.) ,^ 8. f ^^^^^' = ^(4 + ar^)-^3ar'f?.^^ = 2(4 + :^-)^ +a ^ ^ (4 + .r')* ^ 9. fm (3 ao^ + 5 a^) ^ (6 ax 4- 25 x*)dx = f w (3 ax' + 5 x") ^ -j- C. TYPE INTEGRABLE FORMS. 173 r?J^ = log(x^+l)^C. (Form 2.) J X- + 1 10 + In logarithmic integrals it is customary to write the constant of integration C = log c. Hence log(a~' + 1) +C= log(cB2 ^ 1) + logc = log[c(ar' + 1)]. . 11. r_^ = log[c(a;±a)]. J X ±a J 2 2 + 3a; + ar' 2^2 + 3a; -far' . =log[c(2 + 3ar + cr')^]. 14. ri-±^2i^da; = logrc(a; + sin.'«)]. .■' X 4- sin X dx ^^- f^x =fhk^ = ^"^^^"^ ^> + ^^^ ^ = ^""^^^ ^"^^-l- 16. flOlog'a; — = flog^a;-fa (Form 1.) J X " I 17. f m log»a; — = ^^ log»+'a- + 0. c/ a; n -|- 1 18. J ae'^clx = | e" • adx = p"' + C (Form 4. ) \j 19. r3 log aa'Vda; = fa'' • log a 3 a^dx = a^ -f C. (Form 3. ) 20. r e"'" Vos a-da; = e"'" "^ + C. 21. I sin X cos xfix* = \ sin- a* + €'. (Form 1.) 22. f- 2 sin 2 .rda- = f- sin 2 x • 2 dx = cos 2 a; + C. 174 THE INTEGRAL CALCULUS. 23. J 4 sin'^ x cos xdx = sin* x + C. 24. I 4 sec* a; tan xdx = I 4 • sec'* x • sec x tan ccdx = sec* x + C 25. I ^ tan* a; sec' xdx = J^ ta n' x-\-C. 26. I 6 tan ar* sec^ a:'' • .T^dr = I 2 • tan x^ sec" a;* • 3 a:?dx ='tan2ar^ + C. 27. C-^^^ = sin-'x' + C. J Vl - a;* oo /^2sin"^^a^a; /^o • -i <^^ / • -i xo ■ /-» 28. I — = I 2 • sm ^a; =(sin ■'a;)-+(7. (Form 1.) 29. I — - = I — = vers^oa; +C. -' VlOa; - 25a^' •' V2(5a;) - (5.^;)- V2(5a;)-(5.'c)-' dx + 4a; + 5 J l+(a;-2)' 30. C-—-^ = f- ~ = tan-^a; - 2) + C. Jx' + ^x + o Jl+(x-2V ^ ' 31. Ce'^e'dx = ("^+C. TiTiTnVTF.NTARY TRANSFORMATIONS. No general method exists for the reduction of differentials to type forms. Much therefore depends upon the ingenuity and insight of the student. In addition to the specific trans- formations applicable to certain differentials of definite forms, given in the next chapter, the following elementary transforma- tions should constantly be borne in mind. 133. By the introduction of a constant factor. When the differential is under a type form so far «.s- the variable is con- cerned, it may frequently be reduced exactly to such form by multiplying and dividing by a constant factor. This reduction ELEMENTARY TRANSFORMATIONS. 175 depends upon the fact that a constant factor may be written before or after the integral sign. Illustrations. Fo'rm 1. | {3aa:r-\- 2xy - {3ax -\-l)dx. Were the differential factor (6ax-\-2)dx, it would be the exact differential of the variable factor without its exponent. Hence, multiplying and dividing by 2, CiSax" + 2xy(Sax + l)rlx = J- C(3ax- + 2xy{6ax -f 2)dx = j^{3ax' + 2xy-]-C. When the proper factor is not readily seen by inspection, we may determine it as follows. Suppose the differential to be (2a;2-|-a-5)'f(6a;? _|_2ic^)dir, and ^ = required constant fac- tor. Then A must satisfy the condition d(2x^ + «*) = (^Ax^ + 2Ax*)dx, or {3x^--\-5x*)dx = {^Ax^ + 2 Ax*) dx ; and as this condition must be fulfilled for all values of x, the coefficients of like powers in the two members must be equal, or ^A = 3, 2A = 5, from either of which we find -<4 = f. In- troducing this factor, iC(2x^ + x^)'(3x^'-\-5a^)dx = -^{2x^ + x^)'i+a Again, suppose the given differential to be (2a^ + 7x)^(5a^-\-3)dx. Then we must have d(2ar' + 7x) = (6a^ -f 7)dx = {oAx? + 3A)dx; whence 6 = 5^, and 1 = 3 A, or J. = |, ^=^. As these values are not the same, there is no constant factor, and the integration cannot be effected by Form 1. 176 thp: integral calculus. Form 2. C^+M J Qx + x'' dx. ox-\-x'* Were the numerator 6 + 4 ar\ it would be the exact differen- tial of the denominator. Hence, introducing the factor 2, I a , ^ ^^ = ^\ i, , , = i^og(6x + x*) + logc = \og[c{(^x+x'y-]. If the constant factor is not readily seen, it may be deter- mined from the condition that the numerator must be the exact differential of the denominator. FoBMS 3 AND 4. The constant is determined from the con- dition that the differential factor must be the product of the logarithm of the base into the differential of the exponent. Thus, to integrate a^dx, the factor to be introduced is 2 log a, and Ca^dx = — — fa^ ■loga2dx = —^ — a^ + C. J 2 log a J 2 log a Forms 5 to 12. The required constant is readily seen from the fact that the differential factor is the differential of the arc. Thus I cos 2xdx = Y I cos 2x-2dx = ^sm2x-{- C. Forms 13 to 20. In the case of the circular differentials the constant must be determined separately for each form. /dx — , we observe that so far as the variable is concerned it has the type form I — • To transform it, we must make the first term under the radi- cal 1, and the numerator the exact differential of the square root of the second term under the radical. We proceed, there- fore, as follows : ELEMENTARY TRANSFORMATIONS. 177 1 h - dx -, _ -dx \ a- \ a- -x" = - sin ^-x 4- C/- 6 a (14-) r ^^ =-cos~'-a;+a 1 1 6 = 1 tan 1 - a; + C. cib a (16') f- -:r^— = \ cot-^i ^ x- + a J a- -\- b-xr ab a h (17') f ^^"' =^ f_Jf==l f ^'^'' 1 , ^ = - sec~* - x-\- C. a a . , r dx 1 , b „ (18') I , = - cosec-i ~x + a 1 6 dx' ^ _ - da; -V — a; .; XT \\ — X 7,x \ a a- \ a a^ 1 -lb , rt = - vers ^ - X + C • 6 a do; 1 ,b J ax L 1 " ^ - = V covers-^ x + C. V2«6x-6V & a 178 THE INTEGRAL CALCULUS. These eight forms are known as the subordinate, or auxiliary, circular forms. It is better to transform each special case directly; thus 4 V3 C Vffte r Adx ^4 r (ix ^4V3 r J 'S-\- i)x- 3 J 1 + ix"^ 3 -y/BJ 1 ^ tan-V|a; + a V15 If the subordinate forms are memorized, or at hand for refer- ence, we have 4:dx _ dx 3 4- oar' f-ffar' whence, by comparison with (15'), a^ = I, //' = ^, and hence 1 . , ^ .. 4 -T tan ' ~ x-\- C = —r— ah a -\/lh tan ^ ~x-\- C= -7== tan ' Vfic + C Examples. y 1. J|(.r' + l)'^a^da;=|4.J(af+l)WdT=^\(af + l)^ + C. 4. rV2a;^-3a:^ + l(.t'' - f a;)da; = tV(2cc* - 3.t2 + 1)2 + C. 5. Which of the following can be integrated by introducing a constant factor ? {o^ 4- Sar' + a;- + 5)^(2 x' + ^ar' + ^a;)da;. (3a^-2x)'(3a;-l)da;. 3.'r— 1 , 4-f6a;- , ■dx. ■ ~ Ax. (l-aj + .r^)- (4a;-3ar')^ ELEMENTARY TRANSFORMATIONS. 179 ^ a,v^ — bx^- "^^ ax" — hx- = log [c (aar* — 6.x'- ) *]. 7. f- ^^^•^' =log[c(12ar^ + 7)^'n. J 412aT^ + 7 ^'- ^ -1- / J J a — o.r* 1 (a — 6ar')".» 9. f ^^"'^•^' =logrc(10ar'4-16)^]. + 10 /' sin xdx _ , a + 6 cos a; + 6 cos X " / , t V r 11. Which of the following can be integrated by the intro- duction of a constant factor? odx 1— V■^'7 x'"~^dx Ax—'S^x-, -dx. ax. 8-6ar^ ^_j^ x^ + l .^_J 2 . I (("dx = a"' + 0. J a log a 3. (a^'xHx = —1— a-' + G. J 3 log a 4. I nifi^^dx — #?«e'-' 4-C. 5. Te'" ' sin .xdit = - e™' ' + C {x), and the sum of the numerators will be equal to f{x). This sum will be a polynomial of the (h — l)th degree, and since the equation formed by placing it equal to f{x) must be true for all values of x, the coefficients of like powers of x must be separately equal. We shall therefore have n equations of condition from which to find the values of the n constants A, B, '•• N. Hence the resolution can always be effected. fix) 2°. The integration of ^^ ' dx is thus made to depend upon jA dcx* that of a series of fractions of the same form, namely /Adx ■ x—a = A log (x — a). Hence the integration is always x — a possible. Examples. 1. ^^ + ^ dec = 5.T -f 15 -I- -l^^^:!?-^-- by division. The factors of x? — '6x-\-2 are x — 1 and .t — 2 ; hence 35a; -29 ^ A B ^ ^(o;- 2) + -B(a;- 1) a^_3a:-|-2 x-1 a;-2 x'-'6x-^2 ' 186 whence THE INTEGRAL CALCULtTS. Sox - 29 = A{x - 2) + B(x - 1) = {A-\-B)x-2A-B. Equating the coefficients of like powers, 3o = A + B, 29 = 2A + B; therefore A = — 6, B = 41, and Jx^-:^x-\-2 J .; J x-l J x-'J =f a^+laiK-f) log(a;— 1) +411og(a;-2) + C = l^ + i5a; + iog (;"-^)V a (x — ly 2- ^^Z^f'^S ^^x. The roots of ar- -7.^2 + 36 = are G, or — ( ar + 36 3, and - 2. Hence 2a^-3x + 5^^^ B ar»-7a^ + 36 x-6 x-'S x-\-2' and 2x2 - 3 cc + - ^ ^^^ _ 3^ ^^ ^ o) ^ ^^3, _ 6) (a; + 2) • +C(a;-6)(a;-3). Instead of proceeding as in Ex. 1, the vahies of the con- stants are readily found by assuming some value for x, since the equation is to be true for all values of x. Thus, making x equal to — 2, 3, and 6, in succession, we find C= \^, B = — 1|-, A = ^, and r 2x^- J a^- —3x+5 7a;2 + 36 dx = lo -6)^^ (a; + 2) a;-3)T^ 3. r4^da. = log j(-^::^c. REDUCTION BY PARTIAL FRACTIONS. 187 6. r^^=iog,p(^-^). Ja^-9 ^\ x + 3 x + h '•/. '•/. J^^^dx=.\og[c(x-^)Hx + 2fy 10 a^ -{-x^ — 4:X — 4 3a^-l dx = log ■^ (x-^i)ix-2y ix+2y . I -^ dic = log [o(.r + 1) (.r - l)x] = log [c(.'r' - a;)]. "^ ^~^ (Form 2.) 140. Case 2. TAe factors real and eqiml f(x) 1°. Let -^^ ^ be the fraction, <^(a;) being resolvable into n fi>(x) real and equal factors x — a, a; — a, •••. Then, to such set of n equal factors there corresponds a set of n partial fractions, A B N (a; — a)"' (a: — a)" ^ x — a stants, or '^-^ = , in M-^hich A, B, "• N are con- + B irn+- N {x) (x — a)" (cc — a)"~" x — a Reducing the second number to a common denominator, this denominator will be equal to {x), and the sum of the numerators will be a polynomial of the (n — l)th degree equal to f(x) . The latter equation is to be an identical one true for all values of x ; hence, equating separately the coefficients of the like powers of x, we have n equations of condition from which to find the values of the n constants A, B, ••• N. The resolution is, therefore, always possible. When the factors of (a;) are not all equal, the two cases can be combined. Thus JM. A {x-2y{x-3y{x-4:) ix-2) D 7-,+ B + ■ C + -.+ {x-2y ' x-2 E F {x-Sy a; -3 x-4: 188 THE INTEGRAL CALCULUS. 2°. The integration of -^ ^ dx is thus made to depend upon (x) that of a series of fractions of the form If n = 1, (a: -a)" rAdx^ = A log {x - a). If n is other than 1, J X — a r^Mx_ ^ ^ r^^ _ . . „^^ ^ _A^ .^ _ .!.„ J {x-ay J ' \-n ' Hence the integration is always possible. x — \ Examples. 1. -da;. Placing a;-l ^ A B ^ ^ + -5(^ + 1) {x + \y {x-ir\f x^-\ (a; + l)- ' and equating the numerators, we have a: — 1 = ^ -|- Bx + B. Placing the coefficients of like powers equal, we obtain B=\, A = — 2; whence , , 1.2 ^^= , . -,,0 + I — — r=""-r+log(a; + l)+a {x + \y J {x + iy J x-{-i x-\-i 2 (a;^ + o)dx • {x-iy{x + 2){x + l) This is a combination of Cases 1 and 2, three only of the factors being equal. Hence we assume ^ + 5 _ A B C (ar-l)''(a; + 2)(x + l) (a;-!)'' (a;-!)- a; - 1 a; + 2 X + 1' whence, reducing to a common denominator, and equating the numerators, »* + 5 = ^(a; + 2) (a; + 1) + B{x - 1) (a; + 2) (a; + 1) ^C{x-iy{x + 2){x + l) + D{x-iy(x + l) + E{x-iy{x + 2), (1) REDUCTIOX BY PARTIAL FRACTIONS. 189 = (C + D + E)x*-{-{B-{-C-2D-E)x^ + {A-\-2B-3C-SE)jf -^{3A-B- C + 2D + 5E)x + 2A-2B + 2C-D-2E. (2) In (1), make x = — l, x = — 2, x = 1, in succession, and we have directly E = — ^, D=^, A=l. Equating the coeffi- cients of a;* in (2), C-\- D + E = 1, whence, having D and E, C= If. Equating also the absolute terms, o = 2A-2B + 2C-D-2E, whence B = — ^. Therefore r (x* + o)dx _ r dx _ 1 r dx 35 r dx J {x-lY{x-\-2){x-\-l)~J (x-iy GJ {x-iy 36J x-1 7 r dx _ 3 r dx ^ _ 1 1 1 QJ x-\-2 4 J X- + 1 2(a; - 1)2 "•" 6 a; - 1 + Mlog(^'-l)+ilog(^-f2)-flog(a^ + l) + C. 3. C ^^-'^ dx = — + log{x-3y-\-C. J {x-3y x-3 ^^ ^ J {x-iy{x-2) x-1 ^x-1 r {2x-5)dx ^ L_ + iilog^+l4-C. J af'_|_5r' + 7a; + 3 2(a--M) ^ ^x-\-3 '■ / (.-2r;+3)- =-2T,G4-2+^3)+^^-^^^^- r ix + 2)dx ^U_l 3_\ ^^-1 J {x-iy(x+i) ^\x-i (x-iyy^ ^x+i^ J {x-3y {x-3y 141. When the factors of <}>(x) are imaginary, the above processes will lead to the logarithms of imaginary quantities. 190 THE INTEGRAL CALCULUS, To avoid such we resolve {x) into quadratic, instead of sim- ple, factors, as follows : The general form of an imaginary quantity being a-|-6V — 1, that of an imaginary factor will be aj — (a + ftV — 1). But for every such factor there must be another, a; — (a — 6V— 1), since (x) is real. Therefore, for every pair of imaginary factors, ft>{x) will have a quadratic factor of the form [a; _ (a + &V^][.x - (a - 6 V^)]= {x - a)- + &'• Cash 3. The factors imagiyiary and unequal. f(x\ 1°. Let -'^ ' be the fraction, ^(x) being resolvable into p {x) unequal quadratic factors (x — a)^ +• ^^ (^* — ^Y + d^, etc. Then, to every such quadratic factor there corresponds a par- tial fraction ^t_^ -, "^ -, etc., in which A, B, (^x-af + h'' {x-cy + d^ C, D, etc., are constants, or f{x) ^ A + Bx C+Dx M+ Nx {x) ~ (^x-ay + b' {x-cy + d' {x-my + n^' For, in reducing the second member to the common denomi- nator (x), any numerator, as A-\-Bx, will be multiplied by p — 1 factors of the form (a; — a)- + 6', and the sum of the numerators [=/(ic)] will therefore be a polynomial of the [2(p — 1) + l]th degree. We shall therefore have 2(p_l) + 2 = 2p equations of condition from which to find the values of the 2p constants A, B, '•• N. and the resolution is always possible. 2°. The integration of -'^ ^ dx is thus made to depend upon (x) that of a series of fractions of the form (A-^-Bx)dx _ {A + Ba)dx B(x — a)dx {x - ay + b^~ {X- ay + b- {x - ay + b^' REDUCTION BY PAKTFAL FRACTIONS. 191 r{A + Ba)dx ^ A±Ba^^^_, x-a .^^.^ ^33 ^^^,.. ^ r B{x-a)dx B^^^^^_ , ,^^ Hence the integration is always possible. Examples. 1. -^- ~ / a;^ — 4 a; + 5 The factors of ic^ — 4a; + o are x — {2± V— 1), and their product is (a; — 2)-+l, or a = 2, 6=1 in the form (x — a)^-f- 6*. Assuming — — ^^^^ = — — - — , we have A= —4, B=l. ^a^^ix + r> (a; -2)2 + 1' ' Hence /(x — 4:)dx A-i-Ba. .x — a.B-, r-, \'> , ^,i-^ x^— Ax-\- i^ b b 2 = - 2 tan-H.r - 2) + ^log [(x - 2)^ + 1] + C. f, (a;^ + a:^ + a; + 1 ) da; (.T-l)*(.T-' + 2) Assume -^^±^i±i^±l- = -^A- + -A_ + £±^, (a;-l)2(a^ + 2) {;x-\y x-\ x'-\-2 whence A = ^, 5 = Y-, ^' = |» D= — h ^"'^ J (ar' + ar^ + a; + l)da; ^4 r da; 10 r dx {x-iy{x' + 2) "sJ (a;-l)- 9 J a;-l 5 r dx _ 1 r xdx 9Ja:2_^2 9Ja;2 + 2 4 1 , IOt , 1, s= log (a; —1) 3a;-l 9 ^ ^ 4. _A^ tan-i _^ - -^ log (ar^ + 2) -f C. 9V2 V2 - 18 „ ar^da; _ ^da; 5da; (C + Dx)dx (a; + l)(a;-l)(a-2 + 2) x + 1 a; - 1 a;- + 2 192 THE INTEGRAL CALCULUS. Then a.^ = A(x-l) (a^ + 2) + B(x + 1) (ar' + 2) + (C+Dx)(x-\-l){x-l) (!) = (A + B-\-D)a-f' + (B-A + C)ar + {2A + 2B-D)x + 2B-2A-C. (2) From (1), when x = \, x = — l, in succession, we have B = \, A = -\. From (2), B-A + C=l, or C = f ; and ^ + 5 + 7) = 0, or Z) = 0. Therefore r ^da; ^ _ 1 r dx 1 r dx 2 r x J {x-{-l){x- 1) (x^ + 2) qJ x + 1 6J x-1 sJ ar'+2 11 a; — 1 , V2 , -1 X , ^ = ilog— — +— -tan '-— + C. a; -(- 1 3 V2 da; J (x-l L-tan-'-^+a 3V2 V2 + ^tan-*a;+C. 142. Case 4. The factors imaginary and equal. 1°. Let {x) be the fraction, ^'^ ^ being resolvable into p {x) equal quadratic factors (x — a)^+ 6^, (a; — a)^+ b^, etc. Then, to such set of factors there corresponds a set of p partial fractions, A + Bx C + Dx M+Nx \_{x - ay + 6-]''' [(a- - ay + Wy-'' '" {X- ay + W' in which A, B, ••• N are constants, or f{x) ^ A + Bx C+Dx M+Nx 4,{x) {{x-ay + Wy {{x-ay + U'Y ''^"' Xx-ay + i/ EEDUCTIOX BY PARTIAL FRACTIONS. 193 Reducing the second member to the common denominator {x), the sum of the numerators [ = /(ic)] will be a poly- nomial of the [2(i> - 1) + l]th, or (2iJ — l)th, degree. This equation will therefore furnish 2p equations of condition, from which the values of the 2p constants can always be determined. ^The resolution is, therefore, always possible. 2°. The integration of -^^^ dx is thus made to depend upon {x) that of the general form — ^^ — — ^— — , in tvhich jy is integral. \_{x — a)^-f 6^]'' If 2^ = Ij tlie integration has been shown to be possible in Art. 141. If p is other than 1, place x — a = z, whence x = z + a, dx= dz. Then / {A-\-Bx)dx / {A -\- Bz + Ba)dz ^ r Bzdz r {A + Ba)dz {z^ + l^y "J (z^ +&')".' (z' + ft^)" 2(p-l)(22-f-62)» be shown in Art is always possible when p is integral. and it will be shown in Art. 147, that the integration of dz (z* + by Examples. ^ r{7?-ifx'-\-2)dx ^ r {A + Bx)dx r{C-\-Dx)dx whence af -\- .i- -\- 2 = A + Bx + (C + Dx) (ar + 2), from which we find ^ = 0, B= -2, C=D = 1. Hence r(x^-{-jr + 2)dx ^ r -2xdx r dx f J {ii^ + 2Y J {x" + 2y J x' + 2 J a xdx x2 + 2 1 +_i_tan~^-i^+ilog(x^'+2)+C. 194 THE INTEGRAL CALCULUS. ,, r (x*+2x^—2jf—2x+5) dx ^ A+Bx C+Dx E 'J (ar' + l)=^(a;-2) ' {x^+\y x^'+l x-2' whence ^1 = — 4, .5 = 0, C = 2. D = 0, E =1, and we have r —-idx r 2dx r dx J (or^ + 1)- J x^ + 1 J X - 2 = 2 tan-^ X 4- log (a; - 2) - 4 f — — 3 ri^-x + l)dx ^^ (x + iy _^tan-'.>; + 4. p^ + -»^ + x-' + .^- ^,^ o J r.r + 2)Hx' 4- .S)^ 2(a^ + 2) a 10 {x' + 2y{x' + l^y 2(a^ + 2) x' + 'S + ^log(r'+2)-91og(ar+3) + a BY RATIONALIZATION. Since rational algebraic polynomials and rational fractions can always be integrated, an irrational differential may be inte- grated if it can be rationalized. The rationalization is effected by substituting for the variable of the given differential a new variable of which it is a function. Of these substitutions the following are the most important : 143. When the only function of x affected with fractional exponents is a linear one, in which case it will be either of the p p form X* or {ax + 6)', assume x = z" or ax + b = z", n being the least common multiple of tlie denominators of the fractional exponents. For, if x=z" or ax-}-b = z", the values of x, ax-\-b, dx, and the surds of the given differential will be rational functions of z. Examples. 1. | dx. x^- r Here n = 12, and x = z'^. Hence x^ = z^, .f' = z; .c- = z'', dx = 12z^hlz ; REDUCTION BY NATIONALIZATION. 195 . .. r ^^--^' dx = f^^ 12 z'klz = 12 Hz'^' -z')dz 3. r ^^' = 2 .r ■ -f- 3 x^ + <> x'^ + 6 log (x^ - 1 ) + C. '^ .T- — X^ ^ r cj-xydx _ r (2-x)^-dx J 3-x ~Jl+{2-x)' Assume 2 — x = z-. Then, (2 — x)^ = z, dx = — 2zdz, and r(2.n^^= rZL2^ = _2 ffl L_U J 3~x J 1+z- J \ l-\-z-J = -2(2 + fot 'z)+C = - 2(cot- ^/iT^ 4- ViT^-) + C. J {2r-yY- / dx = 3[(a;-f- l)i + 2(;c+l)*+21og((a;4-l)*-l)] + C. 7. ^:>?{l + x)^dx = 2(\■Vx)\{\^-xy-■\{\^-x)^\-\^C. 144. When the only surd of the given differential is of the form Va + &x ± ic^, rationalization is effected as follows : I. When the sign of y? is jjositive, place Va + hx + x^ = z — x. Then a-\-hx = z- — 2zx; , z^—a , 2(z- + bz -i-a)dz whence j: = , dx = —^ ' —-^ — , b + 2z {b-{-2zy ' 196 THE INTEGRAL CALCULUS. and Va + hx + x" = z-x = ^'' + ^^ + ^ . b+2z The given differential will then be a rational function of z, since x, dx, and Va + bx-\-oif are rational functions of z. II. Wkeyi the sign ofxr is negative, place Va ■i-bx — 2if= ■y/{x — a) {ft — x) = (x — a)z, in which a and /8 are the roots of a^ — bx — a = 0. Then ^ — x = (x — a)z-; Avhence x= ^2^1 ^ ^^^ = — T^TXTvr-' and Va + to - ar' = (a; - a)z = ^^ ~ ")^ . Z- + 1 The given differential will then be a rational function of z, since x, dx, and ■\fa + bx^^^ are rational functions of z. dx Examples. / Vl + a; + ^2 Assume Vl +a; + a^ = 2! — a;. z'-\ Then 1+22 whence d.r = ^(^l+^+H^. 0^ + 2zy Hence Vl + a; + a^ = 2 - a; = ?-±^-:t_l. l + 2z C '^^ f-^^ = log(l4-22) + C = log (1 + 2 a; -I- 2 Vl + a; + a«) -H C. 2. J*^-^— ^ = log {2Vx'-x-l + 2x - 1) +C. »/ ar ',._i-A/9^a.ri'^ :+V2x-{-x' REDUCTION BY RATIONALIZATION. 197 4. f ^ = f-^ = iog(r+x)+a Or, by the above method, f ^^ - = log(l + a; + Vl+2a; + x-2) _^ ^« = log2(l+a:)+C'. Prove that C'=C-log2. 5. r-,^ = iog(i + x + V^T^) + a V2 4- a; - a^ The roots of x*^ — a; — 2 = are 2 and — 1. Hence x" -x-2 = {x -2){x + l), and V2 + X - a.-2 = V(x- + 1) (2 - a;) = {x + l)z. Squaring, we find x — "^^ — - ; whence dx= , , ..., , ^2-irX-xr = {x + l)z=^—-' {z-->riy z^-\-l Hence f— =^^== f- -^ = 2 cot ^z + C J V2 + a;-a;2 -^ ^ +^ = 2cot-\/2zi^ + a ^ix + l r. f ^^ =2cot-'J^ »/ -x/2 - rr - ar^ \a; — a V2 - a; - ar' \a; + 2 f ^^ =2cot-\/I +a 8. i ""^ ^2cot-^^:^ + a V4a;-3-ar' \a;-3 9. f ^^- =2cot-\/ ^^-^-^ +a -^ V2-2a;-ar^ VS + l+a; 198 THE INTEGRAL CALCULUS. 145. Binomial Differentials. Every binomial differential may be reduced to the form of {a + bx"ydx, in which p may be any fraction, but m and n are integral and n positive. For, let x^{ax^-\-bxfydx be the binomial, and let k < t. Then a;*(a.r* + bx^ydx = x^x^'fa + 6-Yda; = x-''+^*(a + bx^'^ydx; in which t — k may be fractional, but is positive, and h -\-pk is fractional or integral, positive or negative. That is, the bino- mial is of the form X ''(a + bx fydx. Put x = z^, and this becomes z^'\a + bz+<"ydfz^^-'dz = dfz^'^+'V~\a + bz^^'ydz, in v'hich ± cf-\- df— 1 and de are integral, and the latter posi- tive. Hence writing m for the former and n for the latter, we have dfz'^{a + bz"ydz, which is of the required form. As p may be fractional, represent it by -, s We are now to show that a binomial differential of the form r af (a4-6a;'')'dic, in which m and n are integral and n positive, may be rationalized, and therefore integrated : I. When is a whole number or zero, by assuming a + bx" = z". II. Wheyi 1- - is a whole number or zero, by assuming a -f bx" = z'x". To prove that the rationalization is effected when the above conditions are satisfied : REDUCTION BY RATIONALIZATION. 199 I. Let a + bx" = z\ Then . = (£^)i ..= (--^)", <^=i(-_^y-.-... and (tt + te")'' = 2;''. Hence \ J nb\ h J =ni>' \-ir) ^"^ which is rational when — '^^^- is a whole number or zero. n II. Let a + bx" = z'x". Then x = f )", .f"' = dx = -^(^LS;''-^^dz, n\z'-b) {z'-by ' and (a 4- bx")' = z'x' = z''( — - — V Hence a;'" ( a -f- bx" ) 'd.t^ « \i: J_a_. >^ . ^/^_^\^-' _^±_ az f — bj \z' — bj n\z'—bj {z' — b)- = --a" 'z'^M—±—]« ' dz, n yz' — bj which is rational when — — — h - is a whole number or zero. r . n s r When - is a positive integer, the factor {a + bx")' may be ex- panded and integrated directly. Examples. 1. C — ^^^ = Cx'(a-\-baf)-idx. ^' {a-\-b^)^ ^ Here *A±i = 2. 200 THE INTEGRAL CALCULUS. Assume therefore a -|- 6ar = 2- ; whence (a + bx-) - = z^, Hence r x'dx _ r/ z^ - a ^ {a + bx'y^ ^\ ^ . b^{z^-ay- zdz b^z'-ay- ^1 2a + bx^ ^ ^Wa + b^ 3. Cx(l + xf^dx= ^(1 + x)^{rix- 2) + C. 4. r — ^L^!:L_ = ^^(a 4. bx:'y^dx. ^ {a + bx-y~ ^ Here !'i±i +!=_!. n s Assume therefore a -\- 6ar' = z'-'or ; whence a^ Hence (a + bx')-' = ^(-^^—\\ dx=- /x-^rfx- _ r a (i-zdz ( z^ — 6 X1 1 (tt + 6.r2)'~ Jz^-?, (22_5)| V a y ar' rdz a , ^i a .t'' , ^ {a-i-bxy- 5. r — ^^^ — liog^ — +e. / REDUCTION BY PARTS, 201 dx 1 a + 2 bar 7f(a + bx") ^ «" x{a + bar) ^' 7. f ^^=^ = -l±Mvr^=i>^'+a 8. Cx\l + 2 a^) ^dr = ( 1 + 2 af') ^ '^^^- + C. BY PARTS. 146. Let 7c and ^• be any functions of x. Then d(^tiv) = udv -\- vdu. Transposing and integrating, I iidc = nv — I vdu. This formula is known as the formula for integration by parts. It evidently makes the integration of udv to depend upon that of vdu. To apply it, the given differential must be resolved into factors m and dv such that dv and vdu shall be integrable. The following are the most important applications of this formula. 147. Binomial differentials. Formulae of reduction. It has been shown that every binomial differential may be reduced to the form x"'(a + bx^ydx, in which p is any fraction, but m and n are integral and n positive. I. Let u = x"'~"+\ dv = {a -\- bx")"x''-^dx. Then du = (m — n -\- l)x"*~"dx, v = ^^ trr * nb{p + l) Substituting these in | udv = uv — i vdu, I af(a + bx")Pdx = -^^ — ' — —^ — - 'm'-n-\-\ r „_„ . hx'^y^'dx. 202 THE INTEGRAL CALCULUS. But I ^-»(^ci + bx")P^kl.r = i .T'" "(a + bx"y{a -\- hx'')dx Hence or, solving for j af (a + bx"ydx, Cx'^ia-^bx^ydx x'"~"+\a 4- bx"y^^ — a{m — n-^1) Cx"'"{a + bx^ydx ~ TT — ; r~r\ ' ('^^ b{np-\- m 4-1) a formula which makes the integration of the given binomial to depend upon that of another in which the exponent of the variable without the parenthesis is diminished by that of the variable within. Illustratiox. I — = i3if(l—x^)~^dx. We apply (A) to this differential because its integration would thereby be made to depend upon that of x{l — x^y-dx, which comes under Form 1. Substituting therefore in (-4) m = 3, n — 2, p = — ^,a = l, 6 = — 1, Ave have ^ x'(l-a^)^-2Cx(l-a^)~^dx J x'{l - x'yhlx = 4, : — • = _ ^ic2(l _ a;2) 2 _|_ I r^il - x'ykx REDUCTION BY PARTS. 203 If ?»p -j- m 4-1 = 0, the formula fails ; hut in this case m + 1 n and the differential may be rationalized and integrate4 by Art. 145. II. ( af (a + hx^ydx = Cx"'{a + hx")"~\a + I)af)dx = a Cx"'{a + bx")" \1x + b Cx"^"{a + bx''y-'^dx. (1) Applying (A) to the last integral of (1), we obtain Cxr^"(a + bx"y~^dx 3r-^\a + ba^'Y — a{m -f 1) j a-"'(a + bx^y^Hx b{np -i-m +1) Avhich, substituted in (1), gives I (ir{a + bx!^ydx yf'+^a -f 6a?")'' -f- anp Cx'^ia + bx^y-Hx (B) vp + m -\- 1 a formula which makes the integration of the given binomial to depend upon that of another in which the exponent of the parenthesis is diminished by 1. Illustration. | (a^-far)^dx. The application of (B) to this differential makes the integration depend upon that of ^ ^ -, which can be rationalized and integrated by Art. 144. Va" -I- x-2 Substituting, therefore, in (B) m = 0, n = 2. p = ^, a = a*, 6 = 1, we have 204 THE INTEGRAL CALCITLUS. dx fia' + x'ydx •^ Va^ + x" Writing Va^ -(- ar = 2 — a-, we find r^x__ ^ rdz^iQgz+C = log (.T + V^N=^) -f C. Hence C(a^ + ar') ^. dic = ^ x(a- + ar') ^ + ^ log (ic + V^^H^) + C. If rjj9 + m 4- 1 = 0, the formula fails, but Art. 145 applies as before. III. In {A) let m = m-\-n. Then Cxr+"{a + bx''ydx x^+^a + bx''y+^— a{m + 1) j a;'"(a + 6a;")''da; 6(np + m + n + 1) whence I a;"'(a + bx^ydx a formula which makes the integration of the given binomial to depend upon that of another in which the exponent of the variable without the parenthesis is increased by that of the variable within. lLT.USTRATIOJf r ^ = Cx %x' - l)-^rfa;. By ap- ;egration is made to depend upon that of , which is a known form. Hence, making m = — 3, plying (C) the integration is made to depend upon that of dx xy/oi? — 1 n = 2, /) = — ^, a = — 1, 6 = 1, in (C), we have /: dx REDUCTION BY PARTS. 205 If m = — 1, the formula fails ; but in this case = 0, and Art. 145 applies. lY. In {B) let 2^=p-\-l. Then \^{a-\-hx''y^Hx — . d , np -\- n + m -\- 1 whence fee" (a + bx"ydx —x"'+\a-\-bx"y+'^+ (wp+w-l-m+l) Cor(a-{-bx^y+^dx an{p-{-l) AD) a formula which makes the integration of the given binomial to depend upon that of another in which the exponent of the parenthesis is increased by 1. r dx C Illustration. I— ;r5= I {\-\-^y^dx. By applying ./ (1 + xry J (D) twice, we see the integration will be made to depend upon that of ? which is a known form. Hence, making wi = 0, 1 +aH H = 1, 2> = — 3, a = 6 = 1, in (D), we have ""4(1+0^)^ ' 4 ^^^ - x{i-\-x^y'- 3 f(i-\-x'y'-dx (1+^'" =^4 206 THE INTEGRAL CALCULUS. Applying {D) to the last integral, we have m = « = <> i; = - 2, a = 6 = 1, and ' ' f(^+^rM.^Z^^^^lfnzSj^±^ Hx ^' ^+htiin \v.+a 2(1 + ^) Hence Examples. '^ {of — Qi?y- ^ 2, a yd -\- OCT J - -* ^- J (a-'- ar') kx = ^(a^ - o-^^ + ^^' (a^ ^- o;^)* ** 8 ,3 a''. , a; ^ + — -sin->-+C. Appl'y (5) twice. EEDUCTION BY PARTS. 207 8. Cx'{l-af)klx= - ^(^ ~ ^) ' + ^(^ - ^^y +^sin-'x + C. J ■ 4 8 Apply (yA) and (B) in succession. o C 3?dx C 3 N-'^ 9. I — = I .T- (2 a — .t) - ax *^ V2 ox — ar' '^ = ^^ (2 aa; — ar) - H rers^ - + C. 2 ^ ^2 a Apply iyA) twice. 10. f '^ = -^^'-^ + c. Apply (C). 11. f ^ = -^^^^^^-4-^ log -fC. -'ar'(l-.r2)^ S-r^ Vl - a:'^ + 1 12. f ^ = ? +J^tan-'^ + C. J(o; + ar^)2 2aXa^ + iK') 2 a'' a 13. Show that (^) will reduce the following to known forms : , if m is even and positive ; also if m is odd and Va^ — a^ positive. x'^dx , if m is even and positive. /~~i~t — « Va'^ + ar ±? ^(o? ± a^) », if m is odd and positive. What if m is odd and negative ? 14. r(r2 -^)^dx = \x{i~ - x-') ^ -h ^ r^ sin^^ ~+C. 15. r y'^ =- ^y^+^Ky+^^) v2^^j7^^ v' -s/lry — f ^> -l-fr'vers-i^ + C. 208 , THE INTEGRAL CALCULUS. 148. Logarithmic differentials of the form (jc*** (log x)ndx may be integrated by parts when n is a positive integer, by placing af*dx = dv, (log xY=u, in the formula I udv = vv — I vdu ; every application of the formula reducing the exponent of the logarithm by unity and thus finally making the integration depend upon | x'^dx. Examples. 1. \ x- {\o^ xydx. Let a?dx = dv, (loga;)-= n. Then v = -, du = 2 log x — , and 3 X Jtidv = (log xY I ar^log xdx. Placing ^dx = dv, u = log x, whence v = —, du = — , 3 X J udv = - log a.- — - I x'dx = ~ \ogx — ^ + C. Hence ("^^(log x)Hx = - [ (log x)-- f log a; -(- f ] + C. 2. I log xdx = X (log x — \)-\-C. 3. Jar'(loga;)^rf.r = ^'[(loga;)^'-iloga; + i] + C. 149. Exponential differentials of the form x^^tF^dx may be integrated by parts when n is a positive integer, by placing a;"= XL, e'"dx = dv, in I udv = tiv — | vdu, every application of the formula reducing the exponent of x" by unity, and thus finally making the integration depend upon | e'"dx. REDUCTION BY PARTS. 209 Examples. 1. | are^'xdx. Let e'"dx = dv, x'=u\ then v = — , dn = 2 xdx, and a /udv = I e'^xdx. Placing e'"dx = dv, x = m, whence v = — , die = dx, a erxdx =^^ - - I e-fte = ^ - £- + C. a aJ a or Hence (W^d.c = — ^3?^-^ + ^ V C. J a \ a o?j »/ \ cos'' a; + 3 sin a; — G cos a; ) -}- C. 151. Circular differentials of the forms / (a?) sin^ icrio?, /(£r) cos ^ xrfic, etc., (7i iohichf{x) is an algebraic function. Assuming dv=f{x)dx, the formula for integration by parts will make the integration depend upon an algebraic form. Examples. 1. I sin ^Trfa*. If = sin~' X, dv = dx, da = — ' v = x. Then /sin^^ xdx = a; siu"^ -^ ~ I — ^ = ^' ^^^'^ ^' + (1 — •'»^) " + 2. ftan-i a;da; = x tau"^ a? — | log ( 1 + a;-) + C. 218 THE INTEGRAL CALCULUS. 0. j ar'cos-^ xdx= ~ cos ^ x - ^^ ^ ^^ (a'-+ 2) + 0. 4. I xQOii-^xdx=^x^cos~^x — ^x(l — af')- +|^sin^*a; + C. BY SUBSTITUTION. This method has been already employed in the rationaliza- tion of irrational differentials (Arts. 143-4), and consists in substituting for the variable of the given differential a new- variable of which it is a function. 152. Trigonometric functions of the form sin" x cos***xdx. 1. Let sin a; = 2. Then sin" X = z", cos" x = (l— z') '', dx = {l— z') ~^dz. Hence I sin"a;cos'"a;da' = | z"(l — z'-) - dz, or in like manner, writing cos x = z, I sin" x cos"'xdx = I —z"'(l — z-) - dz. The given differential may then be integrated whenever the above binomials can be integrated. Examples. 1. \sm*xdx. s,mx = z, dx=:-^~ = — — — . »/ cos X -y/i _ ^ fsin* xdx = C—^^ = _ ^ (^2 + 3 ) Vr^^ + 3 sin-i ^^c J J Vl - ^2 4 - * ^ (Ex. 2, Art. 147) = _ c^ (si,^3^ _,_ 3 sin x) + f .^• + C. 2. rsin^.d.= r-^=-fi^+ii>8yr^+c (Ex. 3, Art. 147.) = - ^' (sin* a; + A sin^a- + f) + C. REDUCTION BY SUBSTITUTION. 219 3. i sin* X cos^ xdx = i z^(l —z-)-dz Avhen sin a; = 2. fz\l - z')^dz = - ^(^ -''')' + g( ^ - ^') + 1 sin-»z + a (Ex. 8, Art. 147.) -rx r • 9 i J sinit'cos^a? , sin a; cos a; , , , ^ Hence I sin-'a; cos^a;aic = 1 f- ^ .r + C. II. When either m or n is odd, we may integrate directly by treating the factor whose exponent is odd as in Art. 150, 1,, (a). 4. I sin''.rcos^a;da;= 1 (1 — cos^x)cos^a;sina;fte = — ^ COS'^ X-\-\ COS'' X + C. 5. I COS"' X sin'' xdx = ^ sin'' ^ — ^ sin^ x + i sin^ a' + C. 6. I cos^ X sin^ xdx = — i cos' x -\- ^ cos" a; + C 7. I sin a; cos^ xdia; = — ^ cos^a; + C. 8. I cos X sin'ajfia; = ^ sin® a; + C, form 1 applying when n or m is 1. 9. r_,_J?!? = ClA^^ = log tan x+a (Ex. 12, Art. 150.) J sin a; cos a; . ' sin 2 .r 10. f-_^^_=r?iB!^±^dx = tana.-cota.+a J sin^ajcos^a; ./ sin^ajcos^a; 11. I sin^a; cos'^xda; = \ sm*x — ^ sin® a; + C. 153. Many differentials may be integrated by substitution, but no general rule can be given, and the method is best exhibited by examples, of which a few are added. 220 THE INTEGRAL CALCULUS. 1- I —r-, ^ =1 I -^-r, : when x' = z. J x{ci^ + ar*) ,)J z{a^ + z) By Art. 139, Case 1, J z{a' dz 1 T z + 2!) a? (f-\-z Hence i ''^ — - = — -log— -^^ — ^, + C. J a;(a' + .r') '.\a^ (r + .r* 2. r___^^_— = _ log 2 + a; + 2 V ar' + a; + 1 _ »^ a;Vl +x-\- X- ^ Put a; = - ; the differential may then be integrated by Art. 144, I. y Put 1 + .-C = 2. Then f_i±^e=^rfa. = Y^^+2 r^'-2 f^^'Y (1) J (1-f-a;)- e\ J z- J z J ^ ^ Placing M = e* ?a\(i dv = z -dz, and applying the formula for integration by parts to I — , Ave have J r 2- z J z Substituting this value in (1), we have the above result. 4. r^Vl+loga; = i (1 + logx) ' + C. Let 1 + loga; = z. f- C x^dx 1 , a; Vl —x^,^ ^ . o, I — 3—-—- = — A- cos"^^ X h C\ Let x = cos 2. 6. r d^ _2&^Q^ ft + to a + 25a; ^ Put X' = -, whence x^{a-\-bxy- (az + by REDUCTION BY SERIES. 221 In the latter let az + b = y, and it becomes ~ ^^~ ^ dy. a^ y- 7. x^(a-2c^)^dx. Put x^=a-z\ BY SERIES. 154. When the given differential can be expanded into a converging series, its integral may be found by integrating each term of the series. The integral thus obtained will be in the form of a series, and therefore integration by series affords a method of developing a function where the development of the derivative is known. EXAMPLKS. 1. Cy/x'^^\ix= CV'x(l-x^)^dx = C^~x(l -- -~ - ^'-Adx J ^ 2 8 If) ^ = ^x^ -Ix'^- ^^ cc y -^x^ ••' -{- C. 2. C-^dx= Cn+x + x''-{-^a^ + ^x*'..)dx J cos a; ./ = x- -f- - + - + - + ■'"- ••• + C*. 2 3 10 See Ex. 18, Art. 72. ^. Develop log (1 + a;) . log (1 + X) = f-^ = C(l + x)-'dx J 1 + x J = C(l-x-\-x^-x'---)dx 2 3 4 4. Develop sin~^a?. sin-'a; = f /^^ = C(l -j- ^x^ + |^' + H^' + ■■•)dx •^ Vl — X- "^ = x + \x' + ^x^ + jf^^x' ... + C. 222 THE INTEGRAL CALCULUS. Eemark. The process of integration is the inverse of that of differentiation ; but it does not follow that, because we can differentiate every integral, we can integrate every differential. Suppose, for example, the given function be a;" ; its differential is nx"-^dx. Now, in order that the differential of x" should assume the form -, we must have n — 1 = — 1, or « = ; in X which case x" = 1, which has no differential. That is, the algebraic function x" cannot give rise to a differential of the form -^ ; nor can any other known function except log x. It X is evident, therefore, that, before the invention of logarithms and the investigation of their properties, the operation indi- cated by I — would have been impossible. The transcenden- tal functions sin"'.'*-, tan^^a.', etc., whose differentials dx . . ^1-^ ^, etc., are algebraic functions, are further illustrations of l + or the fact that the integration of algebraic differentials may in- volve transcendental, or higher, fvmctions. The integration, therefore, of such forms as do not arise by the differentiation of the known functions cannot be effected until new functions corresponding to these forms have been invented. MlSCELLAXKOUS EXAMPLES. Integrate -, 1 — x" , J- 1 + 2.1; cos^a; , 1. dx. 5. —\ dx. 1 — x cosa; sina; + arcos^x ,^ xdx ^ _^dx_ Va^-x* • (l — xy • '7T~, — Ta" T. xt&w^xdx. {1-^xy 4. ^(^±1 dx. 8- bdx ■y^y. _ 1 ' Vc"^ — a^ — 2 abx — b^oi^ MISCELLANEOUS EXAMPLES. 223 Q • ■"'J- ; oTT. "*^' V3-(ix-<^ar (« + ^^ + «^) 1 + ^ f^f, sin Ticcdaj 11. '^^ a* — x* X* — a' 7ji — COS r?a; fiinxcos'x sec^xdx 26. 7 ■> 1 7rt — n tan .^• 12. ^ ~^ dx. x^^.r' + l nx"hlx 27. da; Va'"-a;"-"' on cos xda; a-daj «^ + sin- a; 14. — X' X* — X- — J, 15. _^^. ^^ (a;-a)c?x 29. e' "dx. 30. (.T-a)- + (.T + a)"'* 16 ^'^~ - — die. o. {x-a)dx ;»3 + 6a;2 + 8a; 'J- (x- «)- ± (a; + a)- 17. — on tZ.X- a^cos^aj + fe^sin^tc ■^^- x(x + \)- 18. _^:i!^. 33 ^^• 19. - '^'-^ ^ dx. 34. L±idda;. ^2mx — ^ 1__x~- 20. ^ + ^^ c?y. 35. xWT^^dx. ' (^ -\- x' 36. a^^i + x'dx. fy^ tndx aT^* ^^- e^'sinSajdoj. „„ wia;da; 38. , 22. ^^:^- a^^Vl + o;^ 224 THE INTEGRAL CALCULUS. aUOCESSIVE INTEGRATION. 155. Successive differentials obtained on the hypothesis that the variable is equicrescent are readily integrated by the preceding methods, the differential of the variable being con- stant. Examples. 1. Given cPi/ = 10 x"rlx-, to find ?/, — ^ = lOar^rfiK: integrating, -^ dx ' * "" (Ix = 10a^dx; integrating, !lf = J^r' + C". dy = -y- r 'd.c + CVZ.r ; integrating, ?/ = |a;^ -|-C".c -fC". d^y 2. Given —4, = f'os x, to find y. dx^ d^y ^ dry . , ^, —4 = oos xdx : . •. — ^„ = sin a* + C . doc- dx- ^ = sin xdx + C 'dx ■ . : v- = - oos x+C 'x +C". dx dx dy = — cos xdx + C 'xdx + C"dx • .-. y =-sina;+^'^^--fO"x-+C"". d^y 3. Given —4, = 0, to find y. dar 2 = 0;.-. g = C". dy = C'dx; ,.y = C'x+C". 4. Given d*y = sin xdx*, to find y. 5. Given d-s = — gdf, to find s. 6. Given — ^ = , to find y. dx" x" ^ THE CONSTANT OF INTEGRATION. 225 THE CONSTANT OF INTEGRATION. 156. All the integrals thus far obtained contain the inde- terminate constant C, and are called indefinite integrals. Integrals from which the constant has been eliminated, or for which its value has been determined, are called definite integrals. 157. Definite integrals. The two methods of disposing of the constant of integration G are best explained by an illus- tration of the processes. Let it be required to find the plane area OM'N^ between the parabola 0M\ the ordinate M'N\ and the axis of X. This area may be regarded as generated by the motion of the ordinate PD from left to right. If this area be repre- sented by z, dz will represent what its change would be in any interval of time, dt, if its rate of increase remained uniformly the same during that interval. But if the rate of z becomes constant at any instant, that is, at any value PD of y, its increase for any interval dt will be represented by PQRD = PD X DR = ydx ; DR = dx being the corresponding differential of x. Hence dz = ydx, and =jyda (1) y 5 value dx = "- dy from the equation of parabola y- = 2pic, y Substituting the value dx = "- dy from the equation of the dz = jdy (2) and z=:^i\/dy=^f-^a (3) First Method. Evidently the area generated cannot be defi- nitely expressed until we assume some initial position of PD 226 THE INTEGRAL CALCULUS. as an origin from which to estimate it. If we reckon the area from the ordinate through the focus F, then « = when 7)' y=FP'=p, and (3) gives C=—\^, and the definite integral is o which gives the area, estimated from FP', to any position of y as M'N' when y' = M'N' is substituted for y. If we reckon the area from 0, then 2; = when y = 0, and (3) gives C = 0, the definite integral being op which gives the area, estimated from 0, to any position M'N' of y, when y' = M'N' is substituted for y. Hence the value of C may be found whenever we know the value of the function for a particular value of the variable ; and it is evident that this will be the case in all problems like the above, in which the origin from which the magnitude is to be estimated may be arbitrarily chosen. Second Method. If we substitute any value of y, as y"= M"N", in (3), z" = I- 4- C Sp is the area generated while the ordinate is moving to the posi- tion M"N". Substituting ?/' = M'N', z' = l- \-C 3p is the area generated while the ordinate is moving to the position M'N'. Hence is the area generated in moving from M'N' to M"N", and is independent of any initial position of the ordinate. In other THE CONSTANT OF INTEGRATION. 227 words, the area is increasing at the rate — = ^ -^, and the dt p dt area generated at that rate while y passes from the value y' to the value y" is found by substituting these values in (3) and taking the difference of the results. In this way C is eliminated, the process being called integration between limits. The symbol for the integral between the limits y' and y" is Xy" (f)(y)dy, y" being the superior and y' the inferior limit; and it indicates that in the integi-al of (ji(y)dy, y" and y' are to be substituted for the variable in succession, and the latter result subtracted from the former. It is to be observed that the two methods are essentially the same, for in the first the inferior limit is assumed in determining the value of C, and the superior limit is the value subsequently assigned to the variable in the definite integral. The constants introduced in successive integration are readily determined from the conditions of the problem if the latter is a determinate one. Thus, suppose a body starts from rest with a constant acceler- ation m in a right line. Taking the axis of X coincident with the rectilinear path, we have (Art. 59), d'x — n-= m. dt' Multiplying by dt and integrating, 'I^ =: V = mt + a (1) dt Reckoning t from the instant the body starts, we have, by condition, v = when t = 0', hence C=0, and ^ = v=:mt. (2) dt ^ ' Integrating again, x = '^ + C\ (3) 228 THE INTEGRAL CALCTLUS. Reckoning x from the initial position of the body, x = when t = 0; hence C" = 0, and mt- , . . x=^. (4) Eliminating t between (li) and (4), we have for the equa- tions of motion, , mt- /n V = mt, X = -— , V = V ^ mx, from which we may find the position of the body at any time, and its velocity at any time or at any point of the path. Had the body an initial velocity Vq when x=t=0, we should have had from (1), C=Vo, and therefore ~ = v = mt-\- Vo ; at whence, integrating again, in which C' = 0, since x = when ^ = 0. The equations of motion in this case would be v = mt + Vo, X = -— + tV, V- = Vq- + 2 mx. And, in general, the equations of motion can be found when- ever the position and velocity of the body at any instant is known. CHAPTER VIII. GEOMETRICAL APPLICATIONS. 158. Determination of the equations of curves. 1. To find the equation of the curve ichose normal is constant. Let R = length of normal. Then (Art. 27, Ex. 21), or .c = ±fy{Ii' - r)-^ dy = ip {H' - f) - + C. (1) la this, as in all like cases, the fact that the position of the origin of coordinates is arbitrary enables us to determine C. Thus if we assume that the origin is so chosen that y = R when x = 0, then, from (1), C = 0. Hence ic = q: V^^ — y^ or, squaring, a^ -\-y^ = R^; the curve being a circle, and the constant of integration being determined upon the condition that the origin is at the centre. 2. To find the curve ichose snbtangent is constant. doc 7/ — = m ; hence x = log„ y + C, or a; = log„ y if x = when dy y = 1- See Ex. 8, Art. 30. 3. To find the curve whose subnormal is constant. dv y-^=P' Hence y- — 2j)x if .t = when y = 0. 229 230 THE INTEGRAL CALCULUS. 4. To find the curve tohose subnormal is always equal to the abscissa ofthej^oint of contact. An equilateral hyperbola. 5. To find the curve ichose tangent is constant. V\ 1+ — I —a: whence dx = x ^^ — ~y ) c^,/ Taking the negative sign, that is, the case in which y is a decreasing function of x, ^^_CW-y-y C f g- y\ ^ y ^\y(a'-y')^ {a?-,/)^) = a log —^ — '-^ - (a- - r ) -" + C. (Ex. 5, Art. 145. ) Assuming the origin so that x = when j- y = a, we have C = 0. The curve is called the tractrix, and is shown in the figure. Fig. 72. 6. Find the curve whose polar subtangent is x constant. dB i^ — = a (Art. 120). The reciprocal spiral. dr 7. Find the curve whose x>olar subnormal is constant. 159. Rectification of plane curves. The ])rocess of finding the length of a curve is called rectification. I. To rectify f{x, y) = 0. ¥'rom Art. 25, ds = Vdaf -\- dy^ ; hence ^s= CVda^ + dy\ (1) II. To rectify f{r, 0) = 0. From \rt. 120, ds=VdJ^+i^d¥; hence s = C^di^ + r^d^. (2) GEOMETRICAL APPLICATIONS. 231 By substituting the value of dx, oi of dij, from the equation of the curve in (1), s may be expressed in terms of a single variable and its value found when the integration is possible. If the curve is given by its polar equation, the second form of s is in like manner expressed in terms of a single variable. Examples. Rectify the following curves : 1. The semi-cubical parabola y^ = ax\ dy = ^^axdx; hence .s = \ ("(4 + i) ax) ^-dx = -~ (4 + ax) ' + C. Estimating the length from the vertex, « = when a; = ; .-. C = - — , and .s = -^ [ (4 + \) ax) ' - 8], '11 a '27 a which is the length of the curve from the vertex to any point whose abscissa is x. 2. The cycloid x = r vers"' " , — V2 i^y — y'^. y 2ry hence s = Vl' r ( (2 r - y)~'-dy = - 2 V^ v-(2 r -y)^ + C. Estimating from the origin, s — O when y = (); whence C=4r, and s = — 2 V2 r (2 r — y) - + 4 r]^=2. = 4 r. Hence the entire length of one branch is 8 r. 3. The parabola y-= 'Jpx. « = 1 C(j/ + f)^.dy = ^V^n^+^log{y + ^fTp') + a pJ Jj) 2 (See Art. 147, II., the illustrative example.) 232 THE INTEGRAL CALCULUS. Estimating from the vertex, 4. The catenary 7/ = - (e'' -j- e '^). Estimating the arc from the point for which x = 0, z z C =:,(«'-« 0- 5. The hypocycloid a;' -f-//^ = a''. Ans. 6a. 6. Determine tlie length of the tractrix. From Ex. 5, Art. 158, ^ dx- = — ^^^ dy'^. y- Hence s= I Vd-ir' + dy- = — | ?(/^ = — a kig y -}- C, taking the negative sign as s is a decreasing function of y. (Fig. 72.) Estimating the arc from T, s = () when y = a; hence C=aloga-, and s = alog-. y 7. Determine the length of the ellipse. Using the central form of the equation in terms of the eccentricity, 2/^ = (l-.^)(a^_a-), c^^^^iLziir)^; (r — X- hence s = CVdx" + dy' = fx rr ^'f da; = f — ^^- (a- - e'.v') ^- ; J J \ a'-x- J Va- - x" and for the length of the entire curve. GEOMETRICAL APPLICATIONS. 233 3eV 1 = 41 — =^zz=r I « „ Jo v;?3^V -'« 2.4a-^ 2. 4 • 6 a' (Ex. 25, Art. 72.) ~ Jo Vrt* - y? "'«^'' Va'-.T^ 2 a'' Jo Vo^^ „ .. e^ 3e* 32 27ra(l--- — - 22 2^2 2^ . 42 . 6- The second and third of the above integrals are given in Art. 147, Exs. 1 and 2. 8. The logarithmic spiral r = a*, a being the basis and m the modulus of the logarithmic system. dr = -de ; s = C(^^ + aA^^dO = (1 + m')^ r + cT= Vl + m J \m- ) Jo the length from the point for which r = 1 to the pole. The corresponding arc of the Naperian spiral = V2. 9. The spiral of Archimedes, r = aQ. s = a C^T+J'dO = i C{a'^ ?-') ^dr nr ^ r(a^-^,^r _^a r + ^a^ + r- ^ (Art. 147, 11.) 2a 2 a when the arc is estimated from the pole. This is also the length of the arc of the parabola y^ = 2ax from the vertex to y = r (Ex. 3) ; hence this spiral is often called the parabolic spiral. 10. r = a(l + cos^). S = C(di^ -f- ^-^dO'-) ^ = C^2d'{l-\-cosd)de = r^4a2cos''^c7^ = 2a('cosfdd = 4a sin^ + C. 234 THE INTEGRAL CALCULUS. Estimating the area from the point for which = 0, we a have C=0, and s = 4asin — 2 The curve is a cardioide, the polar axis being the axis of symmetry, and its entire length is 8 a. 160. Quadrature of plane areas. I. The plane area included between y =f{x) and the axis of X is given (Art. 157) by 2 =jydx. (1) In like naanner z' = j xdy gives the area between the curve and Y. dz If the curve crosses X, y, and therefore — , becomes nega- dx tive, z being a decreasing function of x ; hence areas below X must be considered as negative. II. By the area of a polar curve is meant the area swept over by its radius vector. Thus OPQ is the area of MN between the limits P and Q. Eepresenting the area by z, its change would evidently become uniform at any value of r= OP if at this value the generating point moved uniformly in the circular arc PP'. Hence if d$ = j)p', dz = area OPP' = ^OPx PP' = ir- rdO, or 2 = 1 Cj-^rie. (2) The process of finding the area is called ftuadrature. Examples. 1. Determine the area of the parabola y'^=2px. dx = tdy; hence z= \ ydx = i yhly = -^ + (7. GEOMETRICAL APPLICATIONS. 235 Estimating the area from the vertex, z = when y = ; hence C=0, and 2; = ~- = |a^, or two-thirds the circumscribing rectangle. 2. Determine the area between y = s^in x and X. z = i sin xrlx = — cos x 3. Show that the area between the witch x-y = 4'/--(2/- — y) and its asymptote is 47r)-. ST^dx fydx=f = 4?-^tan~' — a^+4?" 'Jr = 27rr2-(-27r»-2) = 47rr^. 4. Show that the area between X and the hyperbola xy = 1 from X = 1 to X = .t' is log a;'. 5. Find the area of one branch of the cycloid. ' Cyrix = C- y'^y =Cr{2r-y)-^dy «/ ./ -v/'> n-tl 7/2 »/ V2 ry — ?/ = — -^—^ (2 ?•?/ — ?/) - -I vers ' •- = f7rr^; 2 ■ :^ ?• hence the whole area is .StD"'. See Ex. 9, Art. 147. 6. Find the area of the circle x- + y^ = ?-. fydx=f(^^-x^)kx = ^(^-^ + ^^si.'^^^^^ hence the whole area is tt?*^. See Ex. 14, Art. 147. 7. Prove that the area of the ellipse a^y- -f b^a^ = a'b^ is irab. 8. Show that the area between the cycloid y x = 2 vers~^ | — V4 y — y'^ c and the parabola y-= -a; is |7r. The curves intersect at the origin and x = 2ir. 286 THE INTEGRAL CALCULUS. 9. Find the area of y'^ = a;* + ar' on the left of Y (see Fig. 57). fydx = Cf{l + X) 2dr = 2 Cz\z' - Ifdz See Ex. 7, Art. 143. 10. Show that the area of the loop a^y* = a-x* — af is ^o.-. 11. Show that the area of y(af -f a^) = c^(a — x) from a; = to a; = a is c^(^\og2 — -y x^ 12. Prove that the area between the cissoid y^ = and "^a — x its asymptote is Stto^. See Ex. 9, Art. 147. 13. Prove that the area of both loops of y'^ = xr{l — x^y is |-. See Fig. 59. 14. Prove that the area between X and y = 4x — a:^ from a; = — 2 to a; = + 2 is 8. 15. The spiral of Archimedes, r = aO. z = i Circle = ~ Cme = -&' = ^ r'e, C being zero if the area is estimated from ^ = 0. For ^ = 2 TT, z= ^ irr^, or the area of the first spire is ^ that of the measuring circle. When 6 = 4:7r, z = ^Tr'i'^, or the area of the second spire is firr^ — |7rr^ = 27r7-^, the first spire having been traced twice. 16. Prove that the area of r — e^ is one-fourth the square described on the radius vector. 17. Find the area of the lemniscate 7-^ = a^ cos 2 d. Ans. a*. 18. Prove that the area of the cardioide 7- = a(l+cosd) is f iral GEOMETRICAL APPLICATIONS. 237 19. Prove that the area of the three loops of r = a sin 3 (Fig. 67) is ^Tra'. 20. Find the area of the four loops of r= asin2^ (Fig. 66). 161. Volumes and surfaces of revolution. Let the curve ON, whose equation is y =f{x), revolve about X as an axis of revolution. The plane area OQR will generate a solid revolution whose surface will be gener- ated by OQ. A plane section PP perpendicu- lar to X will cut from this solid a circle whose centre is D and radius is PD = y. The volume of the solid may be regarded as generated by this variable circle moving with its centre on X. The rate of every point of this generating area is — ; hence the rate of increase of the dt , . dV odx volume V is — = Trir — , or dt ^ dt V= i iryHx = TT I y^c^a (1) The surface S of the solid may be regarded as generated by the circumference of the circle. The rate of every point of this generating circumference is — ; hence the rate of increase of the. surface is — = 2Try—, or dt dt S= C2 tryds = 2 TT Cy^dx" + dy\ (2) Examples. 1. Find the volume of the paraboloid of revo- lution. V=ir \ ifdx = Tr I 2pxdx = irpsc^ -\-C. Estimating the vol- ume from the vertex, F= when x = 0; hence C = 0, and V = irpx^ = irpx 2^ = \ -rrfx, or one half the volume of the circumscribing cylinder. 238 THE INTEGRAL CALCULUS. 2. Find the volume of the prolate speroid. Jr'a J2 — (a? — oi?\dx = ^TTh-a. Hence the whole volume " a- = |7r6-(2a), or two-thirds the circumscribing cylinder. If .3. Find the volume of the oblate spheroid. Here F=7r Cx'dy = ^7ra^{2b). 4. Find the volume generated by the revolution oiy— — x-\-h about X: " /r I y'hlx = ^ Trb'-a. 5. Find the volume generated by the revolution of the cycloid about X /TTi/dx = I -n-y^ — y y ~ '^ I .^'(-^ ''y ~ V'Y^y ^ V2 ry — y' ^ 2 y- + r>r(y + 3 r) /.^ t, = — TT —^ — f^—^ i V2 ry — y' y + #7r?'" vers' ' " -f- C or the whole volume = 5 ttV. See Ex. 15, Art. 147. 6. Find the volume generated by the revolution of the witch about its asymptote, x^y = 4?-^(2r — ?/) ; TT CyHx = TT f— Mr!_^ dx = 64 /tt f ^^— = 64 rV (^——4 7- + -^ tan-i — ^ \^8?'2(47-2 + ar') IBr* 2ry See Ex. 12, Art. 147, = 47rV. 7. Show that the volume generated by the revolution of a.5 _|_yt _ ^t about the axis of X is y^^Tra^ GEOMETRICAL APPLICATIONS. 239 8. Find the surface of the paraboloid of revohition. S = 27r JyVdx' + clf'=2^JyyJy^^ + lrh, = ^^^y'+p'y^+C. Estimating the surface from the vertex, S = when y = 0; whence C= — --x>•^ and S = -^[(y- +p'^)^ —p'^'l. 3p 3p^^ ' -• 9. Find the surface of the sphere. .S = 2 TT (^'^/-J^ + 1 rfx- = 2 TT rV sf-^—- dx=27r Crdx = 4 TTV. 10. Find the surface generated by the revokition of x^ -\-y^=a^ about X. Ans. ^-tra?. 11. Find the surface generated by the revolution of the cycloid about its base. S = 2ir ryy\~^^-^ + \dy = 2n-y/2^' ry{2r -yyhy Jo ^2ry — y^ Jo = -2W27-(|(4r+^)(2r-i/)2)]2'=^2^r^. Hence the whole surface = -%^ irr-. See Ex. 5, 143. 12. Prove that the surface generated by the revolution of one branch of the tractrix about X is 27ra^. See Ex. 5, Art. 158. 13. Prove the area of the surface of the prolate spheroid is tto.-] — sin 'e. 1 14 DAY USE RETURN TO DESK FROM WHICH BORROWED LOAN DEPT. RENEWALS ONLY — TEL. NO. 642-3405 *■' This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. MAn 1 M: MAR MAY 18 1984 2Af_ \7W s&^M- LD 21-IOOto-! 6 7,-Sm7'8 LD21A-60wi-3,'70 (N5382sl0)476-A-32 General Library University of California Berkeley 5 3Z ^ ^^^.