A TEXT-BOOK OP DIFEEBENTIAL CALCULUS A TEXT-BOOK OF DIFFERENTIAL CALCULUS WITH NUMEROUS WORKED OUT EXAMPLES BY GANESH^^RASAD B.A. (Cantab.), D.Sc. (Allahabad) MEMBER OP THE LONDON MATHEMATICAL. SOCIETY, OF THE DEUTSCHE MATHEMATIKER-VEREINIGUNG, OP THE CTRCOLO MATEMATICO DI PALERMO, ETC. FELLOW OF THE UNIVERSITY OF ALLAHABAD AND PROFESSOR OF MATHEMATICS IN QUEEN'S COLLEGE, BENARE8 LONGMANS, GEEEN, AND CO. 39 PATERNOSTER ROW, LONDON NEW YORK, BOMBAY, AND CALCUTTA 1909 All rights reserved OfV f ^ PREFACE. In this work it has been my aim to lay before students a strictly rigorous and, at the same time, simple exposition of the Differential Calculus and its chief applications. The present volume is intended for beginners and is so designed as to meet the requirements of Part I. of the Cambridge Mathematical Tripos Examination, and of the Examinations for the B.A. and B.Sc. degrees of Indian Universities. The chief characteristics of the present work may be indicated as follows : — (1) The fundamental principles of the Differential Calculus have been based on a purely arith- metical foundation. Thus, the various theorems have been carefully enunciated and their proofs have been made quite independent of geometrical intuition. In this connection, I may specially mention the chapters on Bolle's Theorem and Taylor's Theorem, Maxima and Minima, and Indeter- minate Forms. (2) Almost every article is followed by worked out examples, specially suited for illustrating the article. There are also numerous exercises in every chapter. (3) A special chapter deals with curve-tracing and the im- portant properties of the best- known curves. (4) The order in which the chapters are arranged is intended to enable the beginner to study the simple geometrical applications of the Differential Calculus immediately after he has learnt differen- tiation. (5) The miscellaneous notes A and B are intended r- O t i* f> O vi PREFACE to give the ambitious student a glimpse of the modern researches in the Differential Calculus. This volume is based on my experience in teaching the elements of the Differential Calculus to a large number of pupils. It is, therefore, throughout of an elementary character. But, as certain parts of it may be found difficult by beginners, they have been marked with an asterisk and may be omitted in a first reading. A few words may be said here about Chapter I. In a mathematical book, which professes to be rigorous in its treatment, it is essential that the definitions be carefully worded. This has been done in the present work. It is, however, possible that, for this reason, Chapter I. may be found heavy reading by some students. To them my advice is this : If you do not grasp the full meaning of a definition in a first reading, leave it and after reading Chapters II. and III. come back to Chapter I., and then you will understand the definition better. In writing the present volume, I have derived much help from two books, viz. the excellent little manual, ' Calcolo Differenziale ' of Professor Ernesto Pascal, and Todhunter's 1 Treatise on the Differential Calculus.' For the historical notes in Chapter VIII., as well as for most of the examples on that chapter, I am indebted to Professor Gino Loria's ' Spezielle algebraische und transscendente ebene Kurven.' Of the re- maining examples in this volume, a large number are common to all English text-books on the subject, some are original and the others are taken from the Tripos Examinations of recent years ; in the case of the more important examples belonging to the last category, the sources are cited in the text. To ray friend and former pupil, Mr. Lakshmi Narayan, M. A., Professor of Mathematics at the Central Hindu College, PEE FACE vii Benares, I am much indebted for some valuable suggestions and for assistance in revising the proof-sheets. I am also indebted to my friend Dr. S. C. Bagchi, B.A. (Cantab.), LL.D. (Dublin), Principal of the University Law College of Calcutta, whose criticisms and suggestions, relating to Chapters V., VIII., and XL, have materially enhanced the usefulness of the present volume. G. PBASAD. Benares, July 1909. CONTENTS. [N.B. — The portions marked with an asterisk may be omitted in a first reading.] CHAPTEE I. Definitions. ART. PAGE 1. Variable 1 2. Function 1 3. Limit 2 4. Continuity 4 5. Differential coefficient 5 *Examples on Chapter 1 6 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. " CHAPTER II. Standard Forms. Introductory Four important limits Differential coefficient of x H Differential coefficient of a x Differential coefficient of sin x Differential coefficient of cos x Differential coefficient of tan x Differential coefficient of cot x Differential coefficient of sec x Differential coefficient of cosec x Differential coefficient of vers x Differential coefficient of log a x Differential coefficient of sin -1 x Differential coefficient of cos -1 x Differential coefficient of tan -1 x and cot -1 x Differential coefficient of sec -1 x and cosec -1 x Differential coefficient of vers -1 x Table of results to be committed to memory Examples on Chapter II 8 9 11 12 13 14 14 14 15 15 15 16 17 18 19 19 20 21 22 x CONTENTS CHAPTER in. Fundamental Rules for Differentiation. ART. PAGE 24. Constant 24 25. Product of constant and function 24 26. Sum of two functions 24 27. Product of two functions 25 28. Product of more than two functions 25 29. Quotient of two functions 27 30. Function of a function 28 31. Inverse functions , . . 28 32. The form \4>{x)}W> 29 Examples on Chapter III. 30 CHAPTER IV. Tangents and Normals. 33. Tangent : Definition and Cartesian equation . 34. 35. 36. 37. 38. 39. 40.' 41. 42. 43. 44. 45.' 46. 47. ds v- sv Note on the equation ^r Normal : Definition and Cartesian equation Cartesian subtangent and subnormal .... Polar coordinates. Angle between tangent and radius vector ds /„, . /dr\ 2 \de) r»+ Note on the equation —-= . / Polar subtangent and subnormal Perpendicular from pole on tangent . Pedal equation Inversion. Pedal curves. Polar reciprocals Examples on Chapter IV 35 36 39 40 41 42 43 44 45 46 47 CHAPTER V. Asymptotes. Definition of asymptote 52 General rule for finding asymptotes from Cartesian equation . . 53 Parallel asymptotes 56 Asymptotes by inspection 57 Note on curvilinear asymptotes 58 General rule for finding asymptotes from polar equation ... 59 Note on circular asymptotes 59 Examples on Chapter V 60 CHAPTER VI. Curvature. Centre of curvature : Definition and Cartesian coordinates Radius of curvature : Definition and formulae . 63 66 STENTS xi ART. PAOT 48. Circle and chord of curvature 70 49. Evolute 71 50. Concavity and convexity. Point of inflexion 73 Examples on Chapter VI 75 CHAPTER VII. - Envelopes. 51. Family of curves 78 52. Definition of envelope 78 53. Rule for finding the envelope of a family of straight lines . . 7& 54. General rule 81 Examples on Chapter VII. 82 ♦CHAPTER VIII. Curve Tracing. Properties of Special Curves. 55. Introductory 84 56. Rules for Cartesian equations 84 Note on multiple points 85 57. Rules for polar equations 87 58. Parabola. Ellipse. Hyperbola 89 59. Semi-cubical parabola. Cissoid. Folium 91 60. Lemniscate. Cardioid. Conchoid 92 61. Cycloid. Catenary. Tractrix . . 93 62. Logarithmic spiral. Archimedian spiral. Sine spiral ... 95 Examples on Chapter VIII. . . ■ 96 CHAPTER IX. Successive Differentiation. 63. Definitions 99. 64. Standard results 100 65. Leibnitz's theorem 102 66. Use of partial fractions. Recurrence formulas 103 Examples on Chapter IX 107 CHAPTER X. Rolle's Theorem and Taylor's Theorem. 67. Rolle's theorem. The theorem of the mean value . . . .110' 68. Taylor's development in finite form 112 69. Taylor's theorem. Maclaurin's theorem 115 *Note on contact of curves . 117 Examples on Chapter X H^ xii CONTENTS CHAPTER XI. Maxima and Minima. ART. PA' 70. Definitions 121 71.* Two theorems ". 122 72. General rule for finding maxima and minima 125 Examples on Chapter XI ; 127 CHAPTER XII. Indeterminate Forms. 73. Introductory 128 74. Cauchy's theorem. The fundamental form - 130 75. The Forms ~ , x °o co _ co o°, 1°°, coo 132 CO Note on compound indeterminate forms 135 Note on infinitesimals and infinities 137 Examples on Chapter XII. 138 MISCELLANEOUS NOTES. A. Weierstrass's function 140 B. Rolle's theorem and Taylor's theorem , 143 C. Partial differentiation 146 \J- DIFFERENTIAL CALCULUS. CHAPTEE I. DEFINITIONS. 1. Variable. Let # be a symbol which takes successively every numerical value from a given number a to another given number /3. Then x is called a variable and the totality of the values of x constitutes the domain of x. We will represent the domain of x by the symbol (a, fi). Note. If k be a number, it will be convenient to use the symbol | k \ to denote the absolute value of k, i.e., the value of k without regard to its sign. Thus I -2 I = I 2-1 = 2. 2. Function. By a function of x, denned for a given domain, is understood a quantity which has a single and definite value for every value of x in its domain. We generally denote functions of x by such symbols as f(x), *(*), F(x), f(x). EXAMPLES. 1. # 2 , 2 X , sin x are functions of x whatever be the domain of x. But sin -1 x cannot be a function of x for such a domain as (2, 3). 2. The temperature curve at a certain place is y = 80 + 10 sin x. If the temperatures recorded are all different, the highest and lowest temperatures being 90° and 80° respectively, find the domain of x. 3. For the domain (0, 1), a function may be defined by saying that it is x 11 zero or - according as£C = 0or->#> , n having the values 1 2, 3, etc. n n— n+l 2 »>V DIFFERENTIAL CALCULUS 4. For .the -domain (#, -1)[, & function f(x) may be defined by saying that f(x) is 2 or 3 according as x is rational or irrational. 3. Limit. A and a being both finite, A is said to be the limit of f(x) for #=a if, for any number 8, however small but greater than zero, there exists a corresponding number c>0 such that x having every value such that 0< <8, <€. Note 1. This definition may be expressed in a different form, viz. : A is said to be the limit of f(x) for x = a, if f(x) differs from A by less than any assigned quantity, however small, when x has any value sufficiently near to a. We will use the notation __ f(x) to denote the limit of f(x) for x=a. f(x) is said to be oo, if, for any positive number N however large, there x = a exists a corresponding number e > such that f(x) > N, x having every value such that 0< 05-a <€. f(x) = A, if, for any number 5, however small but greater than zero, there exists a corresponding number N>0 such that U-/(a?) I <8, x having every value greater than N. li Definitions similar to the above hold for the cases when fix)- -oo, x = a lim f(x) = A lita /(*)=oo,etc. X— — QO X = QO Note 2. The notion of limit, on which Differential Calculus is based, is not so unfamiliar to the beginner as he might at first imagine. For, in his algebraic studies he must have become acquainted with this notion in connec- tion with the sums of infinite series. For example, what is meant by saying that 2 is the sum of the series i + 2+ 2 \+23 + • ■ • toinfinit y ? Nothing but this : 2 is the limit to which S,„ the sum to n terms, tends as n is made greater and greater. DEFINITIONS EXAMPLES. 1. **"" x 2 is a 2 . For, take any number £ however small, but Lim greater than zero. Now, if #— a <«, where < 1 ; hence x=a + 6e, x + a=2a + 0e, and, consequently, Therefore, since x + a J <2 + *. x 2 —a 2 =(x- -a)(x+a), x 2 —a 2 such that |/(z,)-/(x 2 )| <5 for every pair x v x 2 satisfying the conditions 0< a?,— a < € , 0< \ x 2 — a\ <€. 5. Prove that im cos is non-existent. Let x, and x 2 be respectively x = x equal to — and ; r- t n being an integer. Then * 2mr (2n + l)?r cos — cos = 2, x x x 2 however large n may be. Hence it follows that in * cos is non-existent. x = x For, if this limit existed, according to the preceding example, it would be possible to find a value of ft so large that cos cos — < 5 even when 5 < 2. 6. Prove that im rt , is non-existent. x = L 2 + e x 4. Continuity. A function f(x) is said to be continuous for x=a, if im J(x) exists, is finite and equals f(a). x=a EXAMPLES. 1. x 1 is continuous for #=a. For, x 2 exists and equals a 2 . x=a ^ 2. A function f(x) is defined by saying that it equals 1 or e~*\ according as x is zero or different from zero. This function is discontinuous for #=0. For, e x ' 1 is zero and is not equal to /(0). 3. If f(x) be continuous for x = 0, prove that /(0) must be zero, when f(x) = x sin - for values of x different from zero. 4. What are the points of discontinuity of the function given in Ex. 3 of Art. 2 ? DEFINITIONS 5 5. Differential Coefficient. By the differential coefficient of a function f{x) for x=a is understood lim /(s)-/(a) ^ lim /(a + ft)-/(a) #=a #--a ' ' 7&=0 /& The differential coefficient of f(x) for #=& will be denoted by the symbol f(a) ; the differential coefficient, considered as a function df(x) of x, will be generally represented by -} orf'(x). ctx Note 1. The process of finding/' (x) is called differentiating f(x). Note 2. For the geometrical meaning of f'(x), see Art. 33, Note 1. Note 3. The beginner should not think that •} ' means the ratio of ax df(x) to dx. To think so would be as wrong as to think that sin x means the product of sin and x. Just as sin is meaningless, so are df(x) and dx. As has been already stated, the symbol *} ' stands for dx lim f(x + h) -f(x) h = Q h This notation was introduced by Leibnitz in 1676. EXAMPLES. 1. Find the differential coefficient of x 6 for x=-l. Here x—1 x—1 Therefore /'(1)= Km /(*)-Al) = lim (a .* + s+l)=3. x=l x—1 X=I 2. Find *&. dx 3. A function f(x) equals zero or x cos -, according zero or different from zero. Prove that/'(0 Here rio)= lim SMSM J v ' x=0 x lim 1 #=0 x But, by Ex. 5, Art. 3, lm * cos - is non-existent. Therefore /'(()) x=0 x J v ' is non-existent. as x is x stan-(l). 6 DIFFEBENTIAL CALCULUS 4. A function f(x) is denned by saying that it equals 0, x or - x according as a: is 0, > or <0. Prove that/'(0) does not exist. j 5. Find the differential coefficient of x 3 for x = 0. 6. If f(x) has the same value whatever x may be, prove that /'(*)-0. Here f'tmX- Hm / ( a? + ft )-/( a? ) = lira 9. 1 v ; fc=0 & *=0 ft ♦Examples on Chapter I. 1. If /(*)»- h ™ tan" 1 ( x \ prove that f(x) is equal to 0, 1 or -1 according as x is 0, > or < 0. 2. Trace the curve y= l[m 3. If /(a?) = , prove that f(x) is or 1 according as a; is or different from 0. 4. Give the graph of lim (l + simro;)' 1 — 1 y a » * , n- co (1 + sinirx) w + 1 5. If <£>(#) s - — — p prove that the limit to which (sin n ! *#) tends when the integer n is made larger and larger, is or 1 according as x is rational or irrational. 6. A function f(x) is denned by saying that it equals 0. or sin (- ) \sin xj according as x is, or is not, a multiple of ». Find the points of discontinuity of /(*). 7. Trace the curve y* lim (simp)*- 1 . n = co 8. A function f(x) is denned by saying that it equals or sin / \ l Sin xj according as x is either zero or a submultiple of , or neither of these. Prove IT that there are an infinite number of points of discontinuity of f(x) between a and |3, where a < < 3. DEFINITIONS 7 9. If f(x) = or according as x is zero or different from zero, trace l-e x the curve y =f{x). 10. Trace the curve li m x 2n sin-- +x 2 find the values of y at the points x ». ± 1, and discuss whether 2/ is continuous at these points. [Math. Tripos, 1901.] . 1 sin ~ 11. A function f(x) is defined by saying that it equals or £ according log x* as x is zero or different from zero. Has f(x) a differential coefficient for x = ? 12. In each of the following cases discuss the question of the existence of the differential coefficient for x = a : — _ _i (i) f(x) = e ( * -a)a whenx^a, /(#) = when x = a. (ii) /(#) = (# -a) 2 cos when x±a, x-a ' /(#) s when sc = a. (iii) /(x) = a, /(ic) = a-cc when x < a. (iv) f(x) = (x — a) cos when #=j=a, /(#) = when x = a. [Calcutta Univ., 1906.] CHAPTEE II. STANDARD FORMS. 6. Introductory. It is the object of the present chapter to investigate and tabulate the results of differentiating the simple elementary functions, viz., x n , a x , sin x, cos x, tan x, cot x, sec x, cosec x, vers x } \og a x, sin -1 x, cos -1 x, tan -1 x, cot -1 x, sec -1 x, cosec -1 x, vers -1 x. It will be seen later on that, by means of certain rules to be given in Chapter III. and a knowledge of the standard forms of the present chapter, most of the ordinary functions can be easily differentiated. Throughout the book we shall always consider the inverse functions to be so denned that — -j. ,./.(*) *«*«*<•>• TSi— •

t + ' ") remains less than a finite quantity as t tends to zero. Therefore lim f n(n-l) w(n-l ) (n-2) , j £ = 0*1 2! 3! * ■' * i ' and, consequently, lim(l + 0' l -l , = 1 =n. i (II) We have to prove that (1 + 1) 1 tends to e when t tends to zero by assuming positive values, as well as when t tends to zero by assuming negative values. Case 1. t remains positive. For each value of t, we can find two integers n, n + 1, such that 1 Thus n n + 1 10 DIFFEEENTIAL CALCULUS and But and \ n) \ n) \ n) % (i + jl y=(i + J lv +1 i . V n + lj \ n + lj 1 | 1 n + 1 Therefore we have from (1) n + 1 (1 \ m 1 + — 1 tends to e as m / the integer ?;i becomes greater and greater. Therefore fl + - J and (1 V+ 1 1 + i both tend to e as the integer n becomes greater and greater ; n+1/ also it is obvious that (1+ ) and ■ both tend to 1. Hence it n+1 i follows from (2) that (1 + t) l tends to e as t tends to zero. Case 2. t remains negative. PuU=-v. Then i (1- = (l + w)' c (l + w) i) where Now when t tends to zero by assuming negative values, v and, consequently, w tend to zero by assuming positive values. Therefore it follows from Case 1, i i that (1 + w)' r tends to the limit e. Hence (1 + 1) l tends to the limit e. (Ill) By the Exponential Theorem, «> s uih 6 . + p ( i *«)' t g(y t . 2 ! 3 ! STANDARD FORMS 11 Therefore But the numerical value of • } f (log«g) 2 *(log g g) 3 12! 3! remains less than a finite quantity as t tends to zero. Therefore lim / (log, a)* t(\og e af \ =Q Ml 21 81 ' ' ' J and, consequently, lim a 1 — ! t = t = log e a. (IV) Describe a circle of unit radius and construct as in the adjoined figure. Then PMt>-l i.e., 1< - (1). But it is obvious that hm rt -i- - 1. t = cos t Hence , = 1 > it follows from (1) that lim t * = s i n $ . m lim sin t -. •**«-o-r" L 8. Differential coefficient of x\ If /(a?) =# n , then f(x + fc) = (a? + fc)* and W* lim (a? + ft)»-s» /w- fc=s0 a : Two cases arise. Case I. x^:0. h\* Fig. 1. (x+hy-x» =xn b ( 1+ 3 "L ^-i . ( 1+ *) ~ 1 In li h h x 12 DIFFERENTIAL CALCULUS Now as h tends to zero, - also tends to zero. Therefore x ■/..or x f lim (l + Q"-l l - x \ t =o r ~ I ■ But by (I) of Art. 7, l im (i + < )»_i Therefore /'(x)=wx"- 1 , i.e., ^=nx"-'. Case II. x = 0. When n<0, /'(0) is non-existent, for/(0) has no meaning. When n > .,,„> lim h»_ lim .„., Hence, if n < 1, /'(0) is oo or non-existent according as h n changes, or does not change, its sign with h ; if n^ 1, /'(0) is 1 or according as n- 1 or > 1. 9. Differential coefficient of a*. If/(#)=a*, then f(x + h)=a x+h , and lim a x+ *— a x /(*) = /i=0 ft lim a' 4 — 1 But by (III) of Art. 7, ~ a •/*=() ~7T- lim a*— 1 . „ fc-0 A =1 ° g ' a - Therefore f'(x)=a x log e a, i.e., ?^-'==a r log, a. STANDARD FORMS 13 EXAMPLES. 1. Write down the differential coefficients of x, x*, x*, ar T , x~*, 2 r , (i) T , e r > e- r , 10*. 2. If f(x) = 2* 7 find f'(x). 7* = 0l h(2x + h) hi' As ft tends to zero, h(2x + ft) tends to zero. Therefore Hm 2 ft(2 *+* ) -l =; lim ff-l, ft = 0~ft(2z + ft) £ = * ~ l0 & 2 - Also evidently Therefore / '(x) = 2*" x 2a; x log e 2. 10. Differential coefficient of sin x. If /(#)=sin x, then f(x + h)= sin (# + /&), and fi(„\— lim sm fo + fe)— sin # = lim^os(^^ ; l= o JJ 2 lim cos (#-{-- ) =cos x But evidently Io, by (IV) of Art. 7, Therefore/'(^) = cos^ i.e.,— ( 4 ?L ^=cosx. dx lim 2__ lim sin t + /i=o"T""~^=o * : 14 DIFFERENTIAL CALCULUS 11. Differential coefficient of cos x. If f(x)=cos x, then f(x + h) = cos (x + h), and ,,, N lim cos (x + h)— cos x o • ( , h\ ■ h h=0 S lim J . / , *\ =— sin a? ; d (cos x) i.e., * , ; =— sin x. dx EXAMPLES. 1. Find the differential coefficient of sin 2x. 2. If f(x)= sin 2 a, find /'(a). , u x lim sin 2 (x + h) — sin 2 x f{x)= h=o — h~ _ lim sin (2x + h ) sin h ~h=0~ h = sin 2x. 12. Differential coefficient of tan x. If f(x) =tan x, then f(x + /i) = tan (# + /z), and r l( x__ lim tan (# + ^) — tan x jw- h=0 - h ___ lim sin (x + fo) cos # — sin # cos (a?4-/t) /i=0 h cos (# + /&) cos a; — ^ m sin h 1 h=0 h cos (x + h) cos x cos 2 # ' d (tan x) „ o i.e., v , / =sec 2 x. dx 13. Differential coefficient of cot x. Proceeding as in the last article, we find that d(cotx) = _ cosec2x dx STANDARD FORMS 14. Differential coefficient of sec x. If/(#)=sec x, then f(x + h)-= sec (x + h), and , H x lim sec (x + h)— sec x _ lim cos #— cos (# + /&) ~~ /*,=() /& cos (# + /&) cos a; h . f h\ 15 __ lim I 2shi2 sin (# + o) I m I /& cos (a: + h) cos a; J lim ~h=0 sin- c L 2 sin (-D cos (a; -f h) cos # sin« cos 2 X ' i.e. d (sec x) _ sin x^ dx cos 2 x* 15. Differential coefficient of cosec x. Proceeding as in the last article, we find that (d cosec) x___ cosx dx sin 2 x* 16. Differential coefficient of vers x. If /(#)=vers x, then /(# + /t)= vers (x + h), and r, ( N __ lim vers (x + h)— vers x J w -/ i= o h __ lim fl— cos (x + h)} — {1— cosx} ~~h=0 h __ lim f cos (x + h)— cos x \ — fe=0l h J d (cos #) __ tZo: sin #; d (vers x) . dx 16 DIFFERENTIAL CALCULUS EXAMPLES. 1 if /(aj) = (tan a) 1 , find /'(*). 9H k lim A/tan (x + ft) — Vta n x lira tan (x + h)- tan a; ~h = Oh{^t &n ( x + h)+ -v/tanz} - 1 ^ (tan a;) . sec 2 x 2x/tm~x d% 2 -/tan a; , 2. If /(z) = e* ec *, find /'(a:). lim e sec (x + W — e Bec * /'(*) ft = ft gsec (.r + ft)- sec x ]_* lim f .... , lim f e 8ec < 1 - | -''»- 8ec:r -l see (a + ft) - sec x 1 ft i ft = 1 sec (x + h) — sec x ' sin a; for and = e sec r x cos 2 x lim e sec(.r + 70-sec:r_l _ \[ m ^-1 h = Q sec (a? + h) — sec a? t = £ lim sec (a? -f ft) — sec x _ d ( sec a;) __ sin x ft = h dx cos 2 x' 17. Differential coefficient of log a x. If/(;r)=log a x, then f (x + h) =log a (x + h), and /'M= lim lo &* (a? + fe)-logqa? ft=0 STANDARD FORMS 17 Now as h tends to zero, also tends to zero. Therefore x SWK) ; }=S{'°^'> ; } ■MSN*} =log a a by (II.) of Art. 7. Therefore /»=- log a e, i.e., d ( log " S«l log a e. 18. Differential coefficient of sin -1 x. If /(.?)= sin -1 x, then f(x + /t)=sin _1 (# -h h), and /•// \ lim sin -1 (# + /&)— sin -1 a; Now take a circle of unit radius and construct as in the adjoined figure. Let PM and QN represent x and (x + h) respectively. Then h, sin -1 x and sin -1 (x + h) are represented by BQ, arc^P and arc AQ respectively. There- fore sin" 1 (x + h) — sin" 1 #__arc AQ— arc AP h BQ arc [PQ—PQ x arc PQ -~BQ- BQ PQ * Hence Fig. 2. ZPOQ But ^5-qpo / POJ2 and * rcP Q= 2 ; i?g~ sec z ^ Vjh; ancl pg ^Tpoq ' also lim lim IJPQQ- 2 lim t b! , =0 zP^=/.P01f, and ^—^^^ sin 2 y (IV) of Art. 7. Therefore = 1 18 DIFFERENTIAL CALCULUS r LFOQ /'(^o sec z.PQB,—^ Si J sin 2 =sec zPOM=^= d (sin -1 x) 1 i.e., -v-.— -/= dx n/1— x 2 Note 1. In order to see that Z P$P tends to L POM, the student has only to note that in the limit z OPQ = - and that, consequently, i PQR tends to Z OPE which is equal to Z POM. Note 2. For another method of finding the differential coefficient of sin -1 x and the other inverse trigonometrical functions, see Ex. 1, Art. 22. 19. Differential coefficient of cos -1 x. If f(x) = cos -1 x, then f(x + h) = cos -1 (x + h), and -,, v lim cos -1 (x + h) — cos -1 x f(x)= h=o *- - In Fig. 2, let ON and OM represent x and (x + h) respectively. Then h cos -1 x and cos -1 (x + h) are represented by NM, arc AQ and arc 4P respec- tively. Therefore cos -1 (x + h) -cos -1 x arc_4.P - arc ^Q h NM arcPQ PP __PQ arcPQ PP PQ = -cosec^PgPx^|-^. Therefore, as in the preceding article, /i*-&{-«*»»S| fl } i.e., STANDAED FOBMS 20. Differential coefficient of tan -1 x and cot -1 x. If /(#)=tan _1 x> then /(#-h^)= tan -1 (x + h), and 19 /'(*) = lim tan"" 1 (x + h)— tan" 1 x h=0 h Now take a circle of unit radius and construct as in the adjoined figure. Let AM and AN represent x and x + h respectively. Then h, tan -1 x and tan -1 (x + h) are represented by MNj arc AP and arc AQ respectively. Therefore tan -1 (x + h) — tan -1 # __arc PQ h ~~MN MN Pq PQ ' But Pq ^P^ MB^OP MB MN MB MN OM MN Fig. 3. Therefore from the similar triangles MBN and ^4 OiV. /.// x lim f ;r T X P(Q v arcPg\ OM.ON Pq 1 PQ J 'OM 2 1 + z 2 ' i.e., Similarly dftan- 1 x) = 1 dx 1 + x 2 dfcot- 1 x) dx 1 + x 2 * 21. Differential coefficient of sec -1 x and cosec -1 x. If f(x) = sec -1 x, then f(x + 7*) = sec -1 (x + 7*), and fr/~\- nm sec -1 (a? + 7*) — sec -1 # In Fig. 2, let — - and — , represent x and x + h respectively. Then h, sec- 1 x . and sec- 1 (x + h) are represented by ^ T - p—* arc AP and arc respectively. T heref ore ON OM 7 c2 20 DIFFERENTIAL CALCULUS ,,. . _ lim arc PQ f( '~h = ~NM~ ON. OM Iim I ON . OM . H«iS . _~« 1 ■at**- pg ' nmJ = om* = 1 PJIf # V^ 2 — j ' ^(sec- 1 x) 1 ace a; J x * _ i Similarly <*(eosec-i z) = !_:. ax x\/ x 2 -l 22. Differential coefficient of vers -1 x. If f(x) = vers" 1 x, then f(x + 7i) = vers -1 (x + h), and -,, N lim vers 1 (x + h) — vers* 1 x /W-».b H* — In Fig. 2, let 1 - O-M" and 1 — ON represent x and # + h respectively. Then h, vers" 1 x and vers _) (a? + h) are represented by NM, arc AP and arc^Q respectively. Therefore .,, v_ lim arc_P9 ' W *«0 2Mf~ _ lim f PQ arc PQ ] fc = \NM' PQ i i.e., PM V2x -x 2 ' cove rs' 1 x) _ 1 dx \/2x-x* EXAMPLES. 1. The differential coefficient of sin -1 # can also be found as follows : — Let u and U represent sin" 1 ^ and sin" 1 (x + h) respectively. Then x=sinu, x + h=sinU; and hence /i=sin U— sinu . U—u U+u =2 sin - cos . A A STANDARD FORMS 21 Therefore d (sin -1 x)__ lim sin -1 (x + h)— sin -1 x dx &=0 h __ lim U— xl 2 sin — — - cos — ^~ U-u = lim „_ 2 ... i_ h=0 . U-u' U+u SU1 ___ cos _ 2 Now, as h tends to zero, U tends to u. Therefore U-u lim 2 1 sm __ cos ._|_ lim 2 111 Therefore ^ (sin -1 #) __ & Vl^c 2 2. Find — ^— - ' by a procedure similar to that given Ex.1. 23. Table of results to be committed to memory. f(x) = x 1l f f'(x) = nx n ~ l . f[x) = a\ f'(x) = a* log e a. /(#)=: sin #, /'(#)= cos a;. /(#)=cos a?, /'(#)= — sin #. /(.r)=tan#, /'(a;) = sec 2 x. /(#)=cot .r, /'(#)= — cosec 2 #. in /(#)=secar, /'(a;)! , sin x cos 2 #' 22 DIFFEBENTIAL CALCULUS /(z)=cosec x, /'(*)»- g^. f(x) = vers x, /'(#)= sin #. /(a?)=loga a, /'(#)=- lo ga 6. f(x)=log e x, /'(*)=-• a; /(x)=sin- 1 a?, / / (^)=- T ==. /(a;) ss cos -1 x, f'(x) = n/1-z 2 ' /(^)=tan- 1 a?, f , ( x )= 1 ~-^ r f(x)=cot~>x, /'(*)« 1+0 a /(#) =sec~ 1 a?, /'(a?) =— 7 _ /(#) = cosec -1 a:, /'(#) = X sj x 2 — l 1 a; \/x*-i f{x) = vers -1 x f f\x) =-7= >S2x-x 2 Examples on Chapter II. Find from first principles the differential coefficient of each of the following functions : — I. x° (degrees). 2. (x + a)". 3. x h + a h . 4. Vx 2 + a 2 . 5. sinh x which is — « — • '6. cosh x which is — — • 7. cV r . 8. x sin a. u 9. e 8in *. 10. sec x 2 . II. log -^. 12. log since. 13. log tan z. 14. sin log x. 15. sin 2 log x) 16. tan -1 x . a 17. log tan -1 x. STANDARD FOBMS 18. Uf(x) = 2at where x = at\ find f'(x). Let (x + h)=a(t + r) 2 . Then M _ lim /(s + fc)-/(s) lim 2a (t + r)-2at "t = a(* + T) 2 -a* 2 _ lim 2t "t = 2(t + t 2 - lim _1_ = 1_ /a" 19. If f(x) = a sin 2 where a = a cos £, find /'(#). 20. If f(x) = a(l - cos t) where (x) 1 (X) -h=0 ft im J 0(x + h)—(p (x)\ =0 1 * J lim ''h = lim l tfe+^zLtfel^ ai >'(x). ft=0 ft 26. Sum of Two Functions. Eule III. Iff(x)^(x)+%(x), thenf'(x)=+'(x) + \{zy For f(x + ft) = (x + ft) + ^(a + ft) . Therefore ,,, s lim fr(a? + ft) + if fo + ft) } - fy(s) + +(x)} J w ft=0 ft _ lim <\>(x + ft) — (x) + +(z)+ .... then f(x)=(x)i|/(x)-f *(x)(x)v|/(x). Fovf(x + h)=(l)(x + h)\p(x + h). Therefore ,,,x_ lim (x) ) h=0 h ='(x). 28. Dividing both sides of the equation f , (x) = 2 (x)4> 3 (x). Then, denoting

( x + h) ^(j)(x)^ (x-\-h) ~~h=0 h\fr(x+h)4>(x) lim m { ^±f=^ } -»<*) { jfe±fafefe) } EXAMPLES, 1. If f(x)=-^-, wehavebyEule V. sin x d(x A ) 4 d(sin x) sin #-V~ # 7 — /'(s) = ^ ^ sin 2 x 4# 3 sin a;— re 4 cos x sin 2 # _ff 3 ( 4 sin x—x cos #) sin 2 x 2. Write down the differential coefficients of 1 x-g a 2 x sinh as a; *+_?*! tana; 1 as + a aj + a a 2 + a; 2 ' cosh a;' e x -l* •/% x xe x sin x' 28 DIFFERENTIAL CALCULUS 3. Find the values of x for which the differential coefficient of ( s-l)(s-6 ) oj-10 is zero. 4. Find the differential coefficient of / gin g-eos g v^. Vsin oj + cos av 30. Function of a function. Kule VI. 1/ /(a?)=^(0 wftere *=i//(a), *fte?z /'fcHfWfc). For, let ^(# + ft)=£ + r, then f(x + h)=(p(t + r). Therefore ' W-fc=0 ft _lim f^ + r )-»ft) r] -fc=0l r ft/ .lirnf »( t + r)-»(fl ^(a? + ft)-J,(s) 1 fc=0l r ft / f lim (/ ^ + r)-0(Q 1 f lim \l,(x + h)—\l,(x) \ "U=0 r J lft=0 ft J = f lim ^ + r)-^)|^ } But as ft tends to zero, r also tends to zero. Therefore lim {t 4-r) — 0(^) == lim 0(£ + r)-- 0(0 ,/,\ ft = r r=0 r V ' U ' Therefore /'(*)=?W'(s)- 31. Inverse functions. If in Rule VI. we put /(#)=#, then, f'(x) being 1, we obtain the result «/(0f(*) = l, or, as it may be written, dx v dt_. at x fi *' This result gives a simple method of differentiating inverse functions. For, let t=f~ 1 (x) be the inverse function of f(t) ; then x=f(t). Therefore dt_d , f-u \* = 1 1 1 db d* 17 V " ^ /'(|) f'{f~\*)Y dt FUNDAMENTAL RULES FOB DIFFERENTIATION 29 32. The Form {(#). Therefore we have by Eules VI. and IV. f^ x)=e ^ = f {x )^ x ) ^{log +(x)} +log f(x) . +'(*)] EXAMPLES. 1. If /(#) = (1 + x n ) m , we have by Eule VI. where £ represents (l-h^'O- Therefore f'(x)=7nt m - 1 xnx n - 1 =mn(l + x n ) m ~ 1 ^ n ' 1 2. If /(#) = (log sin #) 2 , we have by Eule VI. f( r \-d&_ #°_g sin a?) where t represents log sin x. Again, putting v for sin x, we have by Eule VI. d (log sin x)_ jl (log v) d ( sin x) dx dv dx 1 = -> cos X. V Therefore f'(x) = 2t x cos x=2 cot x log sin #. 3. Apply Art. 31 to find the differential coefficient of tan -1 x. Let ^=tan~ 1 x, then #=tan t. Therefore dt d u _i x 1 1 1 — =-=- (tan l x) = -—= — 5-:=*? b- dx dx dx sec 2 £ 1 + x* dt 4. Find the differential coefficient of of. Let t=x log #, then x x =e\ Therefore 30 DIFFERENTIAL CALCULUS d(x x ) t dt x ( d , , i dx\ =af(l + log x). 5. Write down the differential coefficients of (a 2 — x 2 ) e , (ax + b) n , 0-* 3 , a + bx a - bx' \/tan x, sin \/x, cos (# ? ), log cos x, log cosh x, log _, » * an ' (# 2 )- 6. Differentiate the following functions : — 3 s a • — > rA^ 7—^ 2 * log *> el+ *> ^ton v/^ log(x+y^^I), tan log sin #^% a? 4 log tan -1 a;, log sin a 8in r . 7. Apply Art. 31 to find the differential coefficients of sec -1 x and vers -1 x. 8. Differentiate the following functions : — n * / rp\1lX (sin x) x , x x , e e , I - J , (tan o?) 8in * + (sin x) sin r , (sin- 1 a?)*. Examples on Chapter III. fl£ Find /'(a?) in the following cases : — 3. /W= ^g±g^S l 4. fix) = (or + xT. 5. f{x)= A /i + * inx . 6. /(^) = tan 3 x_ tana , + a , V 1-sina; 3 7. /(a:) = sin M (nx n ). 8. /(a*) = sin x sin 2# sin 3x. 9. f(x)=log'i±p a ^. a — b tan a; log * +_^^L? = log (a + 6 tan a:) - log (a - b tan a;) a-b tan a; . ,.,/ v & sec 2 x b sec 2 a; 2ab sec 2 # a + b tan aj a — 6 tan x a? — b 2 tan 2 x 2ab a 2 cos 2 a;-6 2 sin 2 £c' 1 + aj\ * 1 . 10. /(^^log^J^y-^tan- 1 ^ 11. A a? ) = lo g ^ a + f g -^^ IS . -/a + bx + Va-bx 12. /(x) = a; v/aJ 2 !^^- a 2 log (a + *M + a-). FUNDAMENTAL BULES FOB DIFFEBENTIATION 31 13. fix) = tan- 1 e r . log cot x. l 14. fix) = sin-* f + 6cosa; . yv ' /w 6 + acosz U. /W^an-^y^tan-). 16. f(x) = (1 + a/)* sin (m tan- 1 »). 17. /(a;) = log (log x). 18. f(x) = log n a?, where log n means log log log . . . (repeated n times). 19. f[»)=af*\ 20. /(a?) = («*j*. i 21. /(a;) = tan a*. 22. /(a?) = sin- 1 - x. 24. /(a,)=^ S gf + log(^i^). ok //-i i 1 -f- a? 1 , _ 1 + a? + a; 2 ,- x^% 25. /W-kgj^ + .lo gl — _ ?+ ^3tan-i f=55 . I_ , 1 + s^ + s' , J_ . . ^/2 26 -a^ . /"(a;) = — r- log 7= + — n tan- 1 £. •^ 4^2 B l-a;^2 + aj 2 2^2 1- rtW „ , . a? 2 + ax + *> / (x 2 + axY — bx 27. f(x)= log t 1 - — — 7 — =-• 28. /(#) = log (x + ^x 2 -l) + sec- 1 x. 29. /(a;) = (cos a?) cosh * + (cosh x) cos *. so. /w»(i+i)T + « l+ «. vl + aJ 2 -£C 33. /(#) = e cot-1 * log (cosec 2 a; 3 ). o, - x 1 i f (z 4 -l) 2 1 1 * -i /2a3 4 + l\ *■ ^)=24 ^ {ih^ti} + 4^3 tan nr) 35. /W.-jf- tan (^J-j log (— ?" ) 36. /W = ^ + ^^g- 2 Va; + a/2 + a; 2 37. /(*) = -- { ( 7^}. 38. /w.—fctSL (4 etc - Tnen f(x + h) -f(x) rC x K 2 n> 3 w x w 2 w 3 u x u 2 u 3 t, k h h h h + M, U 2 «3 *i h t 3 w x w 2 w 3 m. m 2 m 3 + terms each of which contains two or three of the nine quantities Jc x , l x , m x , h 2 , etc., as factors. Therefore lim k x h h 2 h h h = io x w 2 w 3 u x u 2 u 3 .,. > _ lim f (x + h) - f(x) 1 [X) " h = h lim h = ', i 2 38 DIFFERENTIAL CALCULUS Therefore the required equation is Y-a(l — cos t)= {X-a(* + sin t)} tan l A i.e., Y=Xtan -fa< (1 — cos t) — (tf-f-sin t) tan >, i.e., Z=Xtan- — at tan-. a A 4. Find the equation to the tangent at the point (x, y) on each of the following curves : — (i) xy = a 2 . (ii) y = e - -* 3 . (iii) y = a cosh -. a (iv) y' z = x 3 . (v) x 3 + y* = 3axy. (vi) aj 3 + ic?/ 2 = 2a2/ 2 . (vii) (z 2 + # 2 ) 2 = a 2 (sc 2 -2/ 2 ). (viii) ^-+|- = 1. 5. Prove that - + % = 1 touches the curve a b X y = be~ a at the point where the curve crosses the axis of y. 6. Find the equation to the tangent at the point determined by t on (i) The ellipse x = a cos t \ y = b sin t f ' (ii) The semi-cubical parabola x = a(2 •»- St 2 ) 1 y = 2at 3 J 7. In the curve x i -{-y i =a i , prove that the portion of the tan- gent intercepted between the axes is of constant length. Here|a?-* + |y-»^=0. Therefore §*■— £ Therefore the 8 8 w dx dx or equation to the tangent at the point (x, y) is Y-y=-(X-x)t, i.e., Y^-xX+y^+y*), X i.e., F=-^+ay. x* Therefore the intercepts on the axes of x and y are respectively cfix 1 and a*yK Therefore the length of the portion of the tangent intercepted between the axes = Va*x l +a*yt=a. TANGENTS AND NOBMALS 39 8. In the curve x'"y' l = a m +M , prove that the portion of the tangent inter- cepted between the axes is divided at its point of contact into segments which are in a constant ratio. 34. Normal : Definition and Cartesian Equation. By the normal to a curve at a point is understood the straight line which passes through that point and is at right angles to the tangent at that point. Since, by Art. 33, the equation to the tangent at a point (x, y) is Y-y=(X-x) |, the equation to the normal at the point (x, y) is Y-y=-dy{X-x), dx ;. e .,(X-x) + (Y-y)g=0. EXAMPLES. 1. Find the equation to the normal at the point (x, y) on the ellipseg+g=l. Here ~ +|f ^=0. Therefore ~ y -=-^ x . Therefore the a* b' ax ax a 2 y required equation is (X-x)-^(Y-y)=Q, i.e., X- b ^Y=x(l- b \ a 2 y \ a 2 / 2. Find the equation to the normal at the point (x, y) on each of the curves of Ex. 4, Art. 33. 3. Find the equation to the normal at the point determined by t on the parabola x ~°l A • F y = 2ati 4. If the normal to the curve x% + y% = a§ makes an angle $ with the axis of x, show that its equation is Fcos . 5. In the catenary y = a cosh -, prove that the length of the portion of a the normal intercepted between the curve and the axis of x varies as y 2 . 40 DIFFEBENTIAL CALCULUS 6. In the ellipse ~ + v t = 1, prove that the length of the portion of the normal intercepted between the curve and the axis of x varies inversely as the perpendicular from the origin on the tangent. 35. Cartesian Subtangent and Subnormal. Let P be a given point on a curve. Let PM be tbe ordinate of P, and let the tangent and normal at P meet the axis of x at T and N respectively. Then the length TM is called the subtangent at P, the length MN is called the subnormal at P. Now construct as in the adjoining figure. ~f M~N X Then and *"•«• tanzPTX=g; TM=PM cot ^PTX, i.e., subtangent==^. dx Also L MPN= L PTM ; therefore MN=PM tanzPTX i.e., subnormal=y J . dx Note. The subtangent is measured from T towards the right, and the JL subnormal is measured from M towards the right. If in any curve dy is a dx negative quantity, it indicates that M lies to the left of T and, as in that case yjL is also negative, N lies to the left of M. EXAMPLES. 1. Find the subtangent and subnormal at the point (x, y) on the curve y n =a n ~ ] x Here ny n ~ l ^-=a n ~\ Therefore ax dy__ a n ~ l __ y dx ny n ~ l nx TANGENTS AND NORMALS 41 V Therefore the subtangent =-^-=7z#, and the subnormal is dx ^ dx nx 2. Prove that the subnormal in the parabola y 2 = 4ax is equal to 2a. 3. In the curve y = be a , prove that the subtangent is constant. 4. In the curve x m y n = a m+n , prove that the subtangent varies as the abscissa. 5. In the curve x m+n = a m - n y 2n , prove that the mth power of the subtangent varies as the nth power of the subnormal. 6. Find the subtangent and the subnormal at the point determined by t on the cycloid x = a{t + sin t) 1 y = a(l-cos t) f 36. Polar Coordinates. Angle between Tangent and Radius Vector. Let be a given point and OA a given straight line ; then it is evident that the position of any point P is uniquely determined if the distance OP and the angle POA are known : these are called the polar coordinates of P and are generally denoted by r and 0. is called the pole, OA the initial line, r the radius vector of P, and 6 the vectorial angle of P. Let P be a given point on a curve, and Q any other point on it. Let r, d be the coordinates of P, r-f/3, 0-f-a the coordinates of Q. Then, drawing QM perpendicular to OP produced, we have .QM 'MP QM 'OM-OP tan lQPM=. (r-f /3) sin a ~~(r+/3) cos a— r (r+/3) Bin a /3 cos a 2r sin' 2 a 42 DIFFERENTIAL CALCULUS Now let Q move along the curve and tend to P. Then by definition secant PQ tends to the tangent PL. But as Q tends to P, a tends to zero. Therefore, denoting L LPM by $, tan<2>= lim „tanz QPM a=U , ^ sin a = lim fr + fl is the angle between OP produced and that direction of the tangent at P in which those points on the curve lie whose vectorial angles are greater than the vectorial angle of P. Note 2. .Let the length of the arc A'P measured from some fixed point A' on the curve be denoted by s. Then ds CLd -*3*©* This result may be easily deduced from the result proved in Art. 33, Note 2. For ds _ds dx m dd~dx ' dd~ : - /i+f d y\ z dx = /f dx W( d y\ 2 ? V *\dx) ' dd V \dd) \ddj' But Therefore Hence and, consequently, x - r cos 0, y = r sin 0. d ? = d r cos d-r sine, dd dd d ;{ = d ~ sin 6 + r coze, dd dd TANGENTS AND NORMALS 43 By a procedure similar to that of Art. 33, Note 2, the above value of L dd can be obtained without using the result 5 - v 1 + Vw - J 37. Polar Subtangent and Subnormal. Let P be a given point on a curve. Let the straight line, drawn through the pole at right angles to OP, meet the tangent and normal at N ^ P, at T and N respectively. Then the length OT is called the polar sub- tangent, the length NO is called the polar subnormal. Now construct as in the adjoined figure. Then /_OPT=(p. Therefore OT= OP tan , i.e., polar subtangent=r tan =r 2 -^ t Also L ONP= L OPT ; therefore NO = OP cot , i.e., polar subnormal=r cot ^= Fig. 8. dr de Note. The polar subtangent is measured to the righ from an observer at dd looking towards P. If in any curve r 2 ^ is negative, it indicates that T is to dr the left of the observer. Similar statements hold for the polar subnormal. EXAMPLES. 1. In the logarithmic spiral r=ae ecota , prove that the angle between the tangent and the radius vector is equal to o. Here we have dr dO =G0taxae 6C0 a Therefore = r cot a tan 0=r— = -—=tan o. dr dr dd 44 DIFFEBENTIAL CALCULUS Hence the angle between the tangent and the radius vector is equal to a. 2. Show that in the curve rd=a the polar subtangent is constant. Here dr___ _a__jiP dd~~ 6 2 ~ a Therefore the polar subtangent=r 2 j-= — a. 3. Find the angle

. Therefore 11 2 i j=- 2 cosec 2

• the point where it cuts the curve at right angles being ( — , — -^— A \uTl ci 111' / Now let the straight lines (1) and (2) be identical. Then , 4 o 8 a, 3n 2 \ m-n and -—m 3 = — — ( 1 + — r- i. 27 27n 3 V 2 ) Therefore 3 2 /- 3n 2 \ » 8 =- (1+ - 1, n 3 V 2 / i.e., n 2 = 2 or —1. But the value —1 must be rejected. Therefore ra 2 = n 2 = 2, and the abscissa} in question are f and §. 12. Show that the normal to the parabola y 2 = 4ax touches the curve 27 ay 2 = 4{x-2a)\ 13. Show that the normal to the ellipse - + %- - 1 touches the curve a 2 o 2 (ax) % +(by)*={a 2 -b 2 )*. 14. Show that the normal to the tractrix x - a( cos t + log tan - \ } y = a sin t ) touches the catenary y = a cosh x . a 15. Prove that the equation to the tangent at the point determined by t on the curve ■■#1 \ may be written in the form X P 1 | ♦W +W /(* TANGENTS AND NORMALS 51 16. Chords of the curve x 3 + y 3 - ax 2 are drawn subtending a right angle at the origin, prove that the tangents at their extremities intersect on the conic 5a? 2 + Sxy - 2ax - 3ay + a 2 = 0. [Math. Tripos, 1907.] 17. If the tangents at P, Q, B to the cardioid r = a(l + cos 9) be parallel, then of the quantities \/UP, \ZOQ] 0 y n +a n _ lil y n - 1 x+ . .. . + a 0>n x n ) + K-i ) o2/ n " 1 + ^- 2 ,i2/ n " 2 ^+ • ■ • +ct 0>n . 1 x n - 1 ) the a's being all constants, put y=mx + c in the expression on the left side of (1), equate to zero the coefficients of x n and x n ~ l } and solve for m and c the two equations thus obtained. Then two cases arise.* I Case I. a n> is not zero. Tnere are, in general, n sets of values (m lt c { ) } (w 2 , c 2 ), . . . , (m n , c„) satisfying the two equations : thus all the asymptotes are of the form y=mx + c, their equations being I2/=w 1 t + c 1 , y=m 2 x + c 2 , . . . , y=m n x + c n . 54 DIFFERENTIAL CALCULUS Case II. a n> is zero. There are, in general, n—1 sets of values (ra^ C,), . . . , (m n _ x , c n _ x ) satisfying the two equations : thus there are only (n—1) asymptotes of the form y=mx + c, their equations being y=m l x + c u . . . ;y=m n -iz + c n - 1 ; the remaining asymptote is of the form x=d. In order to obtain it, put x=d in the expression on the left side of (1), equate to zero the coefficient of y n ~ 1 ) and solve for d the simple equation thus obtained; the required asymptote is x=d lt d v being the root of this equation. Note 1. The student is advised to verify the general rule, given above, by means of the definition of Art. 41. Note 2. If a curve is such that its Cartesian equation is not of the form (1), then, in order to find the asymptotes, it will be necessary either to use the method indicated in Ex. 12, given in the set of examples on the present chapter, or to apply the result of Art. 45 after transforming the equation into polar coordinates. EXAMPLES. 1. Find the asymptotes of the curve y z — yx 2 + y 2 + x 2 — a 2 = 0. Put y=mx + c in the expression on the left side of the above equation. Then we have (mx+c) 3 —x 2 (mx + c) + (mx + c) 2 +x 2 — a 2 , i.e., x 3 (m 3 —m)+x 2 (3m 2 c— c+m 2 + l) + etc. Equating to zero the coefficients of x 3 and x 2 , we obtain the two equations 3m 2 c—c + m 2 - which are satisfied by the three sets of values c- Therefore all the asymptotes are of the form y=mx + c, their equations being y»i, y=x—l, y=— x— 1. 2. Find the asymptotes of the curve xy 2 —x 2 y=a 2 (x f y) + b 3 . n°— m=0~l 2 + l=0J =01 m=ll ra=-l\ =1/' c=-ir ow-ljf' ASYMPTOTES 55 Putting y=mx + c in xy 2 — x*y— a 2 (x+y)— & 3 , we have x(mx + c) 2 — x 2 (mx +■ c) — a 2 (# + mx + c) — 6 3 , i.<3., x s (m 2 — m)+x 2 (2mc— c)+ etc. Equating to zero the coefficients of # 3 and # 2 , we have the two equations m 2 —m=Q 1 2mc — c=0 J which are satisfied by the two sets of values m=0 "I m=l 1 c=0J' c=0J* Thus there are only two asymptotes of the form y=mx + c, their equations being 2/=0, y=x. As, in the equation to the curve, the coefficient of y 3 is zero, the remaining asymptote is of the form x=d. Putting x=d in xy 2 — x 2 y — a 2 (x + y) — b 3 , we have dy 2 —d 2 y— etc. Equating to zero the coefficient of £/ 2 , we have d=Q. Thus the remaining asymptote is x=0. The required asymptotes are therefore y=0, y=x } x=0. 3. Find the asymptotes of the curves— (i) x 2 y-xy 2 + xy + y 2 + x-y = Q. (ii) xy(x 2 - y 2 ) + a 2 y 2 + b 2 x 2 = a 2 b 2 . (iii) y 3 -6y 2 x + llyx 2 -6x 3 + x + y = 0. 4. Find the real asymptotes of the curves— (i) x 3 + y 3 = Saxy. (ii x(x 2 + y 2 )=a(y 2 -3x 2 ). (iii) y 3 = (x-a) 2 (x-b). 5. Prove that the asymptotes of the curve ax 3 - (lx 2 y + my 3 ) + hx + ky + b = Q pass through the origin. 6. Prove that the asymptotes of the curve ax 2 y + bxy 2 + a'x 2 + b'y 2 + a"x + b"y = are always real, and find their equations. 56 DIFFERENTIAL CALCULUS 43. Parallel Asymptotes. In connexion with the general rule given in Art. 42, it should be noted that, in some curves, the two equations, obtained by equating to zero the coefficients of x n and x n ~ 1 J are satisfied by one or more values of m and any value of c. Thus if n be such a value of ra, according to the definition given in Art. 41, y—jjx + c should be called an asym- ptote whatever c may be. But it is customary to call such straight lines of the system y=fxx + c asymptotes as satisfy the following condition : c is stick that the greatest number of the points of inter- section of any secant S tend to points infinitely distant from the origin, when S is made to tend to y=/jx + c. If y n be absent from the equation of the curve, a statement, similar to that given above, would apply to the straight lines of the system x=d. The method of finding parallel asymptotes is made clear by the two examples worked below. 1. Find the asymptotes of the curve (x 2 - y 2 ) 2 - 2a 2 (x 2 + y 2 ) + xa 3 = a\ Putting y = mx + c in (x 2 - y 2 ) 1 - 2a 2 (x 2 + y 2 ) + xa 3 - a\ we have j (1 - m z )x 2 — 2mcx - c 2 } - — 2a 2 { (m 2 + l)x 2 + 2mcx + c 2 } + xa 3 — a\ i.e., (1 - m 2 ) 2 x ' - 4wc(l - m 2 )x 3 + {4 w 2 c 2 - 2c 2 (l - m 2 ) - 2a 2 (l + m 2 ) ) x 2 + etc. Equating to zero the coefficients of x A and x 3 , we have the two equations (l-m 2 ) 2 = 0) 4mc(l-w 2 ) = 0J ' which are satisfied by the values ra = l, m= —1, whatever c may be. Now choose c so that the coefficient of x 2 is zero for m=l. Then 4c 2 — 4a ? = 0, i.e., c= ±a. Thus, of the straight lines of the system y - x + c, only y = x±a are asymptotes. For, corresponding to each of them, three points of intersection of S tend to points infinitely distant from the origin. Similarly, of the straight lines of the system y = —x + c, only y= —x±a are asymptotes. The required asymptotes are therefore y = x±a, y= -x±a. 2. Find the asymptotes of the curve x 2 y 2 = a 2 (x 2 + y 2 ). ASYMPTOTES 57 Putting y-mx + c in x 2 y 2 -a 2 {x 2 + y 2 ), we have m 2 x k + 2mcx 3 + { c 2 — a 2 (l + ?;&*) [ cc 2 — 2ma 2 cx — a 2 c 2 . Equating to zero the coefficients of x* and x 3 , we have the two equations ra 2 = j 2mc = 0J' which are satisfied by m = 0, whatever c may be. Now choose c so that the coefficient of x 2 is zero for m - 0. Then we have c 2 — a 2 = 0, i.e., c= ±a. Thus, of the straight lines of the system y = c, only 2/ = ± a are asymptotes. Also y x is absent from the equation of the curve. Putting x = d in x 2 y 2 -a 2 {x 2 + y 2 ), we have 2/ 2 (d 2 -a 2 )-a 2 d 2 . The coefficient of y 3 is zero whatever d may be. Now choose d so that the coefficient of y 2 is zero. Then we have d 2 - a 2 = 0, i.e., d = ± a. Thus, of the straight lines of the system x = d, only cc = ± a are asymptotes. The required asymptotes are therefore y = ± a, £c = ± a. 3. Find the asymptotes of the curves — (i) y 3 - 5xy 2 + Sx 2 y - ix 3 - 3y 2 + dxy -6x 2 + 2y-2x = l. (ii) x 3 y — 2x 2 y 2 + xy 3 = a 2 x 2 + b 2 y 2 . 4. Find the asymptotes of the curves— (i) a 2 y 2 = (b-y) 2 (x 2 + y 2 ). (ii) 2/ 2 (z 4 -a 4 )=a 6 . 44. Asymptotes by Inspection. By the following rules, which are easily deducible from Arts. 42 and 43, sometimes the equations of some or all of the asymptotes of a curve can be written down from an inspection of the equation of the curve : — Kule I. If y n be absent from the equation of a curve of the wth degree, the asymptotes parallel to the y — axis are obtained by equating to zero the coefficient of the highest power of y in the equa- tion; similarly, if x n be absent, the asymptotes parallel to the x—axis are obtained by equating to zero the coefficient of the highest power of x. Kule II. Let the symbol F n represent a product of n different linear factors. Then if the equation of a curve of the nth. degree be of the form F n + P=0, where P is of a degree not higher than n — 2 all the asymptotes are given by the equation F n =Q. 58 DIFFERENTIAL CALCULUS Note 1. If it be possible to express the equation of a curve in the form y = Ax + B +-+-. + . . . , x x 2 A, B, C, etc., being constants, then the straight line y=Ax+B is evidently an asymptote to the curve. This method of finding asymptotes is sometimes useful. In order to determine on which side of the curve the asymptote lies, we have only to consider the sign of C. For example, it is clear that, in the first quadrant the curve will approach the asymptote from above or below according as C is positive or negative. Note 2. Curvilinear asymptotes. t Let there be two curves which con- tinually approach each other so that for a common abscissa the limit of the difference of the ordinates is zero, or for a common ordinate the limit of the difference of the abscissae is zero when that common abscissa or common ordinate tends to become infinite ; then each curve is said to be a curvilinear asymptote of the other. In particular, if one of the curves is a parabola, it is said to be a parabolic asymptote of the other. For an example of a parabolic asymptote, see Ex. 19, p. 62. EXAMPLES. 1. Write down the equations of the asymptotes of the curve x 2 y 2 =a 2 (x 2 + y 2 ). By Kule I. the required equations are x 1 — a 2 =0, i.e., x=±a and ?/ 2 — a 2 =0, i.e., y=±a. 2. Write down the equations of the asymptotes of the curve xy 2 —x 2 y=a 2 (x + y) + b*. By Eule II. the asymptotes are given by xy 2 —x 2 y=0. There- fore the required equations are x=0, ?/=0, y=x. 3. Write down the equations of the asymptotes of the curves — (i) y' 2 (x-l) {x-2)=x 2 + B. (ii) xy(x' 2 - y' 1 ) + a 2 y' 1 + b 2 x' 2 = d*b 2 . (hi) xy(x 2 — y' 2 ) (x 2 — \.y -) + 2xy(x' i — y-) + x* + y' 2 = 1 . 4. Write down the equations of the real asymptotes of the curves — (i) xy 3 + x 3 y = a 4 . (ii) xy' 2 = -la' 2 ('2a-x). (iii) x' 2 y' 2 = a' 2 (x' 2 — y' 2 ). f A fuller treatment wil^be given in Vol. II. ASYMPTOTES 59 45*. General Rule for finding Asymptotes from Polar Equation. Let the equation of a curve be — =/(ti) ; then if a be a root of the equation /(0)=O, rsin(e-a)= f/ ^ ) .... (1) is an asymptote to the curve. For, the vectorial angles of the points of intersection of the secant r sin (0— -fi)=a with the curve are the roots of the equation 0(0)=a/(0)— sin (0-/3)=O. Now when the secant is made to tend to (1), /3 and a must tend to a and ,..- ■ respectively, and both <£(«) and (fl)=0 tend to «, and consequently the corresponding radii vectores tend to oo. Note 1. The first part of the above sentence will become quite clear to the reader if he notes that, according to a general theorem which he will study in Chapter X., (e)='(a) tend to zero, the equation $(6) = tends to become equiva- lent to the equation (0-a) 2 P = O, and, consequently, two of its roots tend to a. Note 2. Circular asymptotes. Let the polar equation of a curve be such that r tends to a limit, c say, when 6 tends to become infinite. Then the circle r = c is said to be a circular asymptote of the curve ; for evidently the curve approaches nearer and nearer to the circle as 9 is made larger and larger. EXAMPLES. 1, Find the asymptotes of the curve r sin 2d=a. Here /(0)= sm , and therefore the roots of the equation ;0)=O are 0, *", etc. Also/ / (0)= 2 -^^. Therefore the required 2k a asymptotes are * 60 DIFFERENTIAL CALCULUS a V rsin0=? >"HM r sin i.e., r cos 6= - ; A i.e., r sin 0=—?; A i.e., r cos 0=- 2' 2. Find the asymptote of the cissoid <% sin 2 cos 3. Find the asymptotes of the curve r sin w0 = a. 4. Prove that r sin = a is the only asymptote of the conchoid r-a cosec 6 + b. Examples on Chapter V. 1. Show that the asymptotes of the curve (x 2 -y 2 ) 2 = 2(x 2 + y 2 ) form a square. 2. Show that the asymptotes of the curve x 2 y 2 - a 2 (x 2 + if) - a?(x + y) + a A = form a square, through two of whose angular points the curve passes. 3. Find the asymptotes of the following curves :— (i) xy 2 = a*(a + x). (ii) x 2 y 2 - y* + 2ay 3 - a 2 x 2 = 0. (iii) (x + y) 2 (x + 2y) (x + 3y) - 2a(x* + y*) - 2a 2 (x + 2y) (x + y) = 0. 4. Find the asymptotes of the following curves :— (i) r sin 20 = a cos 30. (ii) r = a(sec + cos 0). (iii) r n sin nd = a n . 5. Prove that every curve of the 7ith degree has in general n asymptotes, real or imaginary. ASYMPTOTES 61 6. Prove that, in general, the points, in which the asymptotes of a curve of the nth degree cut it at a finite distance from the origin, lie on a curve of the (n — 2)th degree. 7. Find the asymptotes of the cubic ax 2 y + bxy 2 + a^x 2 + b x y 2 + a.p + b. 2 y = 0, and show that the three points, in which the asymptotes meet the curve at a finite distance from the origin, lie on the straight line b-{a 2 a 2 + a?b)x + a 2 (b% + ab x 2 )y + a l b l (a' 2 b l + a x b 2 ) = 0. 8. Show that the four asymptotes of the curve (x 2 - y 2 )(y 2 - ix 2 ) + 6x 3 - 5x 2 y - dxy 2 + 2i/-x 2 + Sxy-l = cut the curve in eight points which lie on the circle x 2 +.y 2 = l. 9. Prove that, in general, the distance of a point in any branch of a curve from the corresponding asymptote diminishes indefinitely as its distance from the origin is increased indefinitely. 10. If a right line be drawn through the point (a, 0) parallel to the asymptote of the cubic (x — a) 3 — x 2 y = 0, prove that the portion of the line intercepted by the axes is bisected by the curve. 11. If from the origin a right line be drawn parallel to any of the asym- ptotes of the cubic y(ax 2 + 2hxy + by 2 + 2gx + 2fy + c)—x 3 = 0, show that the portion of this line intercepted between the origin and the line gx+fy + c = is bisected by the curve. x 2 v 2 12. Find the asymptotes of the hyperbola ~ — ^ = 1, using the following definition : An asymptote to a curve is the limiting position of tlie tangent when the point of contact moves to an infinite distance from tlie origin. The tangent at the point (x, y) is r -4*Hg> Therefore the straight line Y=mX + c is an asymptote, where m and c are respectively the limits to which -f- and y — x~~ tend as x 2 -\-y 2 tends to <». dx dx But here dy _b 2 x _ b x and dx a 2 y a V x 2 — a? dy b 2 x 2 ab ii x-—y - — t dx ' a l y V x 2 -a 2 62 DIFFERENTIAL CALCULUS Therefore m and c are respectively ± - and 0. Thus the required asymptotes are F=±-X a 13. Show that the straight line x = a is an asymptote to the curve y-- x-a if (a) and '(a) are finite. 14. Find the asymptotes of the following curves : — (i) y - tan x. (ii) y = e~ x *. 15. Apply the definition of Ex. 12 to establish the general rule given in Art. 42. 16. Show that there is an infinite series of parallel asymptotes to the curve 0sin and show that their distances from the pole are in Harmonical Progression. 17. Prove that the curve y-x . — 2 lies above its oblique asymptote in the first quadrant. 18. In connexion with the curve y(x 2 - 8a 2 ) + 2a(Sx 2 - lax + 8a 2 ) = 0, find the equations of the asymptotes, and determine on which side of each symptote the curve lies at infinity. 19. Show that the curve y = ?-^. has a parabolic asymptote ay = x 2 . /y»3 rt3 /yi2 yy2 ■ = . Therefore, for the same value of x, the difference between ax ax the ordinates of y a x ~ a an cl ay = x 2 ax a 2 is , which tends to zero when x tends to become infinite. Hence, by Art. 44, Note 2, the two curves are asymptotic to each other ; and, since the second curve is a parabola, it is a parabolic asymptote of the first. 20. Show that the curve, whose equation is of the form y = A n x» + A n _ 1 x»-i+ . . . +^+f»+§+ • • • • has a curvilinear asymptote y = A n x n + A n _ x x n - 1 + . . . + A Q ; the ^4's and the jB's being all constants and n being greater than 1. 21. Show that the curve a 3 + aby — axy = has a parabolic asymptote x 2 + bx + b 2 = ay. [Suggestion : Expand r in descending powers of x. n ] ax — ao 22. Find the circular asymptote of the curve r = . — -. 1 + CHAPTEE VI. CUKVATUEE. 46. Centre of Curvature : Definition and Cartesian Coordinates. Let P be a given point on a curve and Q any other point on it. Then, if M denote the point of intersection of the normals at P and Q, by the centre of curvature of the curve at P is understood the limiting position M , on the normal at P, to which M tends as Q, travelling along the curve, tends to P. Let x, y be the coordinates of P,, x + h, y+k the coordinates of Q. Then, representing -=2 by (x), the normals at P and Q are F IG< 10. respectively and (X-x-h) + {Y-y-k)(t>(x + h)=0. Therefore, denoting by a, fi the coordinates of M, we have ( a -x) + (P-y)(x)=0. . . . (1), ( n - x -h) + (i3-y-k)(p(x + h)=0. . . (2). Hence we have, by subtraction, A_ ___ h + k(x + h ) y ~~(x + h)-( x + h ) — '=(» 2 « where \J/ = tan "~ 1 3f and s denotes the length of the arc measured from a fixed dx point A on the curve to the point (x, y), i.e., P. The formula (d) maybe easily deduced from the formula (a). For, ds\ 2 _ \dxj \dx/ dty) (d$y~ ( " 35 ^ 2 . Here ~^«— a sin and -— £=— a cos 6. Therefore dd dd 2 »}' = a 6 {(l + cos6) 2 + sin 2 6}^ {a 2 (l + cos 0) 2 + 2a 2 sin 2 0+a 2 cos 0(1 + cos 0)} s Therefore p=f(2ar)*. 70 DIFFERENTIAL CALCULUS 4. Find the radius of curvature at the point (x, y) on each of the following curves : — (i) x 2 = ±ay. (ii) xy = a 2 . (iii) a 2 y = x*. (iv) y = a log sec -. (v) 2 r 2 (vi ) i^+y-f. V ' p 2 a 2 b 2 7. Find the radius of curvature at the point (r, 0) on each of the following curves : — (i) r - a cos 0. (ii) r = - — - — -. 7 v ' 1 + cos a (iii) r n - a n cos nO. (iv) r n sinh n0 = a n . M a Vr 2 ^ 2 ^.f ,.v 1 cos 2 0^ sin 2 a (V) = COS *-. (VI) — = = h TO * v ' a a v ' r 2 a 2 6 2 8. In the conic ax 2 + 2hxy + by 2 = l, prove that the radius of curvature varies inversely as the cube of the central perpendicular on the tangent. 48. Circle and Chord of Curvature. Let P be a given point on a curve, and M the centre of curva- ture of the curve at P. Then the circle whose centre is M and radius M () P is called the circle of curvature of the curve at P ; and by the chord of curvature of the curve at P in a given direction is understood that chord of the circle of curvature which passes through P in the given direction. EXAMPLES. 1. In the curve y=a log sec -, prove that the chord of curva- ture parallel to the axis of y is of constant length. CURVATURE 71 It is clear that the required chord PQ=2 P cos /_M Q PQ. But, since -^=tan - and, consequently, -r-l=- sec 2 -, dx a * Jr dx 2 a a P 2 = b0; if _ ^=0, in general P is a point of inflexion on the curve. Note. For proofs of the above statements, as well as for a discussion of the question of contact, see the Note at the end of Chapter X. EXAMPLES. 1. Prove that the parabola y 2 =4:ax is everywhere concave with regard to the axis of x. Here -^= — , and consequently -^-1= — ^-- Therefore w r ? ax y ax 2 y 3 ax 2 4a 2 = 2~ which is less than zero at every point on the curve. Therefore the curve is everywhere concave with regard to the axis of x. 2. Show that the points of inflexion on the cubic y a 2 +x 2 are given by x=0 and x=+a*/3. Here j*» ffi*HS? i and consequently dx (a 2 + x 2 ) 2 ' * J ffiy __ 2a *x(3a 2 -x 2 ) dx 2 (a*+x 2 Y * Therefore the points of inflexion are given by x{3a 2 -x 2 )=0, i.e., #=0, x^+a^S. 3. Show that the curve y-er x * has points of inflexion where x = ± — -• 4 Find the point of inflexion on the curve x 3 -Sbx 2 + a 2 y = 0. 5. Prove that, in general, the point (r, 6) is a point of inflexion, if \dd) dd* 6. In the curve rd m = a, prove that there is a point of inflexion when 0= ^m(l-m). CUBVATUBE 75 Examples on Chapter VI. 1. Find the coordinates of the centre of curvature of the catenary y = a cosh -, and show that the radius of curvature is equal in length to the a portion of the normal intercepted between the curve and the axis of x. 2. In the equiangular spiral r = ae m \ prove that the centre of curvature is the point where the perpendicular to the radius vector through the pole intersects the normal. 3. Prove that the distance between the pole and the centre of curvature corresponding to any point on the curve r n = a n cos nB ja*»+(n*-l)r**} h (n + l)r n ~ l 4. Prove that [Math. Tripos, 1907.] 1 dt dt> dt dt 2 J at the point determined by t on the curve % = (t)\ 5. If />, p' be the radii of curvature at the extremities of two conjugate diameters of an ellipse, prove that ( P U p' 1 ) (abf = a*+b\ 6. Prove that the radius of curvature at any point of the curve x = a +bt +ct 2 -» y^a' + Vt + c'Pj varies as sec 3 , where is the angle the tangent at the point makes with the line ex + c'y = 0. 7. If a curve be given by the equations 2x= Vp + 2t+ ^t 2 -2t, %= «'t 2 + 2t- ^t 2 -2U find the radius of curvature in terms of t. 8. In the curve whose intrinsic equation is s = a log sec ^, prove that p oc tan ij/. [A relation between the arc s and the angle of contingence 4/ in a curve is called an intrinsic equation of the curve.] 76 DIFFERENTIAL CALCULUS 9. A tangent TPU is drawn to the curve ©•♦(0'-> at any point P, and intersects the axes of x and y in T and U respectively. Prove that the radius of curvature at P is (r^TJ .th of that of a conic having the axes of coordinates for principal axes and touching TU at P. 10. In the curve p =f{r), prove that the chord of curvature through the pole is numerically equal to Wpr • 11. At the point on the Archimedian spiral r = ad, at which the tangent makes half a right angle with the radius vector, the chords of curvature along and perpendicular to the radius vector are both equal to fa. 12. If p and p' be the radii of curvature at corresponding points of the curve y =f(x) and its evolute, prove that {p'\ 2 _ Ldx\dx 2 ) 1 \dx) \dx\dx 2 )] \p) /b and <(l + n 2 )b. If a lies between these limits, prove that the points of inflexion lie on a circle, and show how to find its radius. CHAPTEK VII. ENVELOPES. 51. Family of Curves. — Consider the equation x cos a +y sin a=a. This represents a straight line for every value of a. Thus, giving successively all values to a, we obtain an infinite number of straight lines which are equidistant from the origin : these >-*^ straight lines may therefore be said to constitute, a family, and ~ the quantity a whose different values serve to distinguish the individual members of the family may be called the 'parameter of the family. In general, denoting by the symbol F(x, y, a) an expression containing x, y, and o, the curves corresponding to the equation F(x, y, ci)=0 may be said to constitute a family of which the parameter is a. 52. Definition of Envelope. By the envelope of the family of curves corresponding to F(x,y, a)=0, where a is the parameter of the family, is understood the curve which touches every member of the family, and which, at each point, is touched by some member of the family. For an illustration of this definition, see the figure given in the third of the following examples. Note. The student is advised to make figures for the other examples. EXAMPLES. 1. Write down the equation of the family of straight lines parallel to y~x. 2. What is the family of curves corresponding to (i) (x a) 2 + y 2 - a 2 , parameter a ; (ii) (x - a) 2 + y 2 = a 2 , parameter o ; (iii) y 2 b Aa(x - a), parameter a ; x 2 v 2 (iv) - + -J 2 = 1, parameter a ? a 2 -a 2 ENVELOPES 79 3. Prove that the circle x 2 + y 2 =a 2 is the envelope of the family of straight lines corresponding to x cos a +y sin a=a, a being the parameter of the family. The straight line x cos a + y sin a =a touches the circle, the point of contact being (a cos a, a sin a). Thus the circle touches every member of the family, and is touched at each point by one member of the family. Therefore by definition the circle is the envelope of the family. 4. Prove that the curve y 2 = a 2 is the en- velope of the family of circles corresponding to (x — a) 2 + 2/ 2 = a 2 , a being the parameter of Fig. 11. the family. 5. Prove that the curve y 2 -x 2 = is the envelope of the family of parabolas corresponding to 2/ 2 = 4a(cc — a), a being the parameter of the family. 6. Explain why the family of circles corresponding to (x — a) 2 + y 2 zza 2 has no envelope, a being the parameter of the family. 53. Rule for finding the Envelope of a Family of Straight Lines. The equation of the envelope of the family of straight lines corresponding to F(x, y, a)=y—ax—f(a) = 0, a being the parameter of the family, is obtained by eliminating a between d F(x, y, «)=0 and ~F(x } y, a)=0. da F(x,y,a)=0, i.e., y-ax-f(a)=0 . (1), £F(x,y, a)=0, i.e., x+f'(a)=0 . (2), For, from and we have *=-/'(«). 2/=/(«)-«/'(«) ; and consequently the curve whose equation is obtained by eliminating a between (1) and (2) is the same as the curve I 80 DIFFEBENTIAL CALCULUS Now the tangent at the point determined by a on this curve is the straight line ie.,F-/(«) + a/'(«)=g{X+/'(«)}i '~dx da i.e., F=«X+/(a). Hence the family of straight lines corresponding to F(x,y, a)=0 is the same as the family of tangents to the curve whose equation is obtained by eliminating a between Fix, y, a)=0 and §-F(x, y, a)=0. da EXAMPLES. 1. Find the envelope of the family of straight lines correspond- ing to x cos a + y sin a =a. Eliminating a between x cos a +y sin a— a=0 and — x sin a + y cos a=0, we have x 2 +y 2 =a 2 , which is the required envelope. 2. Find the envelopes of the families of straight lines corresponding to (i) x + y sin a = a cos a, parameter a. (ii) JUL — hL =a 2 -6 2 , parameter o. cos o sin a (iii) y = mx + - , parameter m. m (iv) y = wo; — 2am — am 3 , parameter m. (v) x cos 30 + 2/ sin 30 = a (cos 20)*, parameter 0. (vi) 2/ cos - x sin = a - a sin log tan (?- + £ Y parameter 0. ENVELOPES 81 2 2 2 3. Prove that £2-n + yz-n = ofc-n is the envelope of the family of straight lines corresponding to x cos w 6 + y sin n 6 = a, 6 being the parameter of the family. 4. Find the envelope of the straight line, the product of whose intercepts on the axes of coordinates is a 2 . Here the family corresponds to - + -^=1 .... (1), a a 2 /a v a being the parameter. Therefore, eliminating a between (1) and a 2 a 2 U ' we have 4:xy=a 2 . Therefore the required envelope is the a 2 rectangular hyperbola xy-=—. 5. Find the envelope of a straight line of fixed length a which moves with its extremities on two straight lines at right angles to each other. 6. Find the envelopes of straight lines at right angles to the radii vectores of the following curves drawn through their extremities : — (i) r = ae dcota . (ii) r = a + 6cos0. (iii) r n = a n cos n9. 54. General Rule. In general, the rule given in Art. 53 is true even when F(x, y, a)=0 represents a curve of any hind. Note. A rigorous proof of the above statement is beyond the scope of the present volume. Such a proof, as well as a discussion of the exceptional cases, will be given in Vol. II. EXAMPLES. 1. Find the envelope of the family of circles corresponding to (#-— a) 2 + 2/ 2 =a 2 , a being the parameter of the family. Here F(x, y, a)=(x—a) 2 +y 2 —a 2 . Therefore fF(x,y,«) = -2(x-a). a/3, 0). (vii) The point ( a, - ) lies on the curve. Therefore it follows from the foregoing statements that the form of the curve is that shown in the adjoined figure. Trace the following curves : — 4. c&ifi - 5. 6. ay 2 = x(a 2 —x 2 ). y = (x-l)(x-2)(x-3). 7. x(x* + y 2 ) = a(x 2 -y 2 ). 8. x 2 y 2 = a 2 (x 2 + y 2 ). 9. y 2 =x*. X -^- 9 x — a 10. aj 4 = a 2 (» 2 -2/ 2 ). v 9 ' Fig. 17. 57. Rules for Polar Equations. The rules of Art 56 are also applicable to the case of polar equations, certain points being kept in view For example, there is symmetry with respect to the initial line if the equation remains unaltered by changing the sign of 6. Again, r being essentially positive, the curve is non-existent in the region bounded by the lines 0=a, 0=/3, if r is negative or unreal when 6 lies between a and /3. EXAMPLES. 1. Trace the curve r=a(l + cos 6) [Cardioid]. The following statements hold true in this case : — (i) There is symmetry with respect to the initial line, (ii) The curve is non-existent outside the circle r=2a. (hi) There is no asymptote. 88 DIFFEBENTIAL CALCULUS (iv) The curve passes through the origin, its tangent there being the initial line, (v) The curve crosses the initial line at (2a, 0), the tangent there being perpendicular to the initial line. (vi) The curve has no point of inflexion. (vii) The points (|, §), («,f)> and ( ^, —J lie on the curve. Therefore it follows from the foregoing statements that the form of the curve is that shown in the adjoined figure. Fig. 18.— Cardioid. 2. Trace the curve r=a sec 6±b, a being less than b [Con- choid of Nicomedes]. The following statements hold true in this case : — (i) There is symmetry with respect to the initial line. (ii) Denoting cos -1 j- by d, the curve is non-existent in the region bounded by the lines 0=^> 0=7r— u as well as in the region bounded by 0=?r-|-a, 0= 3tt 2' (iii) There is only one asymptote, viz., r cos 0= a. (iv) The curve passes through the origin, its tangents there being 0=«, 6=2*— a. (v) The curve crosses the initial line at (a + b, 0) and (b— a, ?r), the tangent at each of these points being perpendicular to the initial line. (vi) The curve has two points of inflexion given by sec 3 0— ! sec 0- sec a =0. (vii) The points (2b, a) and 1 2a + 6, ~j lie on the curve. CURVE TRACING, PROPERTIES OF SPECIAL CURVES 89 Therefore it follows from the foregoing statements that the form of the curve is that shown in the adjoined figure. Trace the following curves : — 3. r = a sin 20. r 2 = a 2 coa 30. 4. 5. 6. 7. 8. r = a(sec + cos I r = a6 cos 0. r = a cos 30. . 30 r = a cot -— . 58. Parabola. Ellipse. Hyperbola. I. Parabola, (a) The simplest cartesian equa- tion is ?/ 2 =4a#; the sim- plest polar equation is == 2a T 1-cos 0' where (a, 0) is the pole and the # — axis is the initial line ; the pedal equation 2a of r= 1— cos d is p z =ar. Fig. 19. — Conchoid of Nicomedes. (/3) y 2 = &ax is the locus of a point whose distance from (a, 0) is equal to its perpendicular distance from x + a=0: the point (a, 0) is the focus of the parabola and the straight line x + a=0 is the directrix. (y) A characteristic property of y 2 = 4tax is the constancy of its sub- normal. (8) The evolute of y 2 — &ax is 27ay 2 =±(x-2a)\ , II. Ellipse, (a) The simplest car- *-*-ll the Fig. 20.— Parabola. tesian equation is -i+ T , a 2 b V simplest polar equation is r= + ^ , wne re (>/a*^5* 0) is the 90 DIFFEBENTIAL CALCULUS pole, the x— axis is the initial line, and e= ; the pedal a qnl qj2 equation of 2 + ^ 2 =l is l = j. 1 r*_ p 2 a 3 6 s a 2 b r (ft) -j+?5=sl is the locus of a point whose distance from a 1 b l (±.ae, 0) is equal to e times its perpendicular distance from Y foci of the ellipse and the straight x+ ct =0: the points (±ae, 0) are the e lines #±-=0 are the directrices, e (y) A characteristic property of lis the constancy of the sum a 2 b 2 Fig. 21 Ellipse. of the distances of (x, y) from (ae, 0) and {—ae, 0). (8) The evolute of §+|»^l is {axf + (by)*=:(a 2 -b 2 )K (See Ex. 1, Art. 49.) III. Hyperbola. («) The simplest cartesian equation is V x 2 y 2 — — -^— v 2 =l; the simplest polar equation is r= 2 where a ° 1+e cos 0> Wa?~+b 2 , 0) is the pole, the x— axis is the initial line, and s/a 2 + b 2 , a e= ; the pedal equa- x II tion of — «/ =l is a 2 2 4+ r ' 2 • 6» a 2 6* 1 = 1 _p 2 a a 2 6 2 of a point whose distance from (±ae, 0) is equal to e times iti Fig. 22. — Hyperbola. (/3) ^"""^ = 1 is tne l° cus CUBVE TBACING, PBOPEBTIES OF SPECIAL CUBVES 91 perpendicular distance , from x T =0 : the points (±ae, 0) are the foci of the hyperbola and the straight lines x±~ =0 are the directrices. x II (y) A characteristic property of — 2—7,2=1 * s the constancy of the difference of the distances of (#, y) from (ae, 0) and (— ae, 0). (3) The evolute of ^-fi=i is (aaOM%) 8 =(a 2 + 6 2 )l [°/- the equation of the evolute of ^2 + p=l- 59. Semi-cubical Parabola. Cissoid. Folium. I. Semi-cubical Parabola, (a) The simplest cartesian equa- tion is ay 2 =x 3 ; the simplest polar equation is r= ^-- , where (0, 0) is cos 3 d the pole and the x— axis is the initial line. (/3) The semi-cubical parabola is the evolute of the common parabola 2/2= 16a/ + 8a\ 27 \ 27/ Fig. 23. — Semi-cubical Parabola. II. Cissoid. (ci) The simplest cartesian equation is . X" y ~2a-x'' the simplest polar equation is r= — -, where (0, 0) is the pole and the x— axis is the initial line. (i(3) The. cissoid is the pedal of the parabola y 2 =—8ax. For the form of the curve, see Fig. 16 given in Art. 56. III. Folium. The simplest cartesian equation is x 3 -f y 3 = Saxy ; , , , ... 3a sin 6 cos 6 , /A AN . the simplest polar equation is r= . Q .. ?— :, where (U, U; is r r u sin 3 6 + cos 3 6 the pole and the a;—- axis is the initial line. 92 DIFFEBENTIAL CALCULUS Fig. 24.— Folium of Descartes. Note. The semi-cubical parabola is also called Neil's parabola after William Neil (1637-1670), who found the length of any arc of this curve in 1657. Diodes, the discoverer of the cissoid, was a Greek mathematician who flourished some time between 250 and 100 b.c The curve x s + y 3 = Zaxy is called Folium of Descartes after Bene Descartes (1596-1650), the French mathematician and philosopher, who was one of the first to discuss its form. 60. Lemniscate. Gardioid. Conchoid. I. Lemniscate. (a) The simplest cartesian equation is (^+2/ 2 ) 2 =a 2 (0J 2 -2/ 2 ); the simplest polar equation is r 2 =a 2 cos 2d, where (0, 0) is the pole and the a;— axis is the initial line ; the pedal equation is r 3 =a 2 p. (fi) The lemniscate is the pedal of the rectangular hyperbola r 2 cos 20=a 2 . (See Ex. 2, Art. 40, n being —2.) (y) A characteristic property of the lemniscate is the constancy of X the product of the distances of (x t y) from (±^,0). Fig. 25.— Lemniscate of Bernoulli. II. Cardioid. (a) The simplest cartesian equation is (x 2 +y 2 -ax) 2 =a 2 (x 2 + y 2 ); the simplest polar equation is r=a(l + cos 6), where (0, 0) is the pole and the re— axis is the initial line ; the pedal equation is r 3 =2ap 8 . (/3) The cardioid is the pedal of the circle r=2a cos 6, and the inverse of the parabola r= 1 + cos tf (See Ex. 2, Art. 40, n being 1.) (y) The evolute of the cardioid is the cardioid r= Q (l— cos 0), o CURVE TRACING, PROPERTIES OF SPECIAL CURVES 93 the pole in this equation being taken as the point ( «?> 0) and the initial line as the x — axis. [Use the result of Ex. 3, Art. 47.] For the form of the curve, see Fig. 18 given in Art. 57. III. Conchoid, («) The simplest cartesian equation is {x.-afix^+y^bV \ the simplest polar equation is r=— ° ! — -±b, where (0, 0) is the pole and the x— axis is the initial line. (l3) The conchoid may be geometrically constructed as follows : Take any point S on the straight line x=a, and let P and p be two points on the line joining (0, 0) to S such that the distances SP and Sp are both equal to b ; then the locus of P and p is the conchoid. For the form of the curve when a=r sin a. (/3) The logarithmic spiral is called the equiangular spiral be- cause of the equality of the angle, between the tangent and the radius vector, to the constant angle a. (y) The logarithmic spiral has for its pedal, inverse, polar reci- FlG - 29.— Logarithmic Spiral. procal and evolute logarithmic spirals which are equal to it. (For evolute, see Ex. 13, Examples on Chapter VI.) II. Archimedian Spiral, (a) The simplest equation is r=aO ; r 4 the pedal equation is p 2 —-^ §« (/3) The Archimedian spiral is the path described by a point which moves from the pole with uniform velocity along a straight line while the straight line rotates with uniform velocity about the pole, the ratio of the velocities being a. .... . . , „ . . Fig. 30. — Archimedian Spiral. (y) A characteristic property of the Archimedian spiral is the constancy of its polar subnormal. (See Ex. 4, Art. 37.) 96 DIFFERENTIAL CALCULUS III. Sine Spiral, (a) The simplest equation is. r n =a n cos nO ; the pedal equation is pa n =r n+1 . (/3) A characteristic property of the sine spiral is the equality of the angle, between the tangent and the radius vector, to ^-\-nd. (y) For the values —1, +1, —2, + 2, — \ and +^ of n, the sine spiral is respectively a straight line, a circle, a rectangular hyperbola, a lemniscate, a parabola, and a cardioid. Note. The spiral r = ad is called after its discoverer Archimedes (287-212 B.C.), the famous Greek mathematician of Syracuse. The discovery of this spiral is wrongly ascribed by some writers to Conon. Examples on Chapter VIII. 1. In the* cissoid % = — — 2 1 1 t{i + t 2 )> prove that the points a, j8, y are collinear if a +0+7=0 and that the points a, 0, 7, 5 are concyclic ifa + j8 + 7 + 5 = 0. 2. Prove that the polar reciprocal of the cissoid y 2 = with respect to 2a -x (0, 0) is a semi-cubical parabola. 3. Prove that in the cissoid x 2 = —-^ — the subtangent is equal to the 2a-y portion of the tangent intercepted between the axes. 4. In the folium Bat \ Sat 2 f ' prove that the points a, 0, y are collinear if ajSy + 1 = 0. 5. From a point P on the curve x* + y 3 = daxy two tangents PQ, PQ' dis- tinct from the tangent at P are drawn to the curve. Show that QQ' touches the rectangular hyperbola ±xy = 9a 2 . Let the parameters of P, Q and Q' be respectively o, o, and a 2 . Then it follows from Ex. 4 that Now the equation to I is v- "i 2 = « 2 2 3aa, 2 "l + «i 3 _ 1 a Sao. 2 Saa* "l + o, 8 X 3aa, "l + a, 8 3aa 2 l + V" 3aa, which simplifies to y + xa l * — 3aa 2 = 0. Therefore the envelope of QQ' is 4xy = 9a 2 . CURVE TRACING, PROPERTIES OF SPECIAL CURVES 97 6. Trace the strophoid a(x 2 — y l ) - x(x 2 + y 2 ) and prove that it is the pedal of the parabola if-^ax with respect to the point ( — a, 0). a 3 7. Trace the witch of Agnesi y = , -. x l + a 2 a 8. Trace the trisectrix of Maclaurin r = a sec -. 3 9. Trace the limacon r = a cos — 6 and show that for a = 26 the curve is a 30" cos 2 Note that 2 cos 0—1 = trisectrix. 10. In the cardioid cos - 2 a(l-t 2 ) ) y- (1 + t 2 ) 2 ) prove that the points a, #, y are collinear if a&y(a f j8 4- 7) + « 2 + £ 2 + y 2 + &y + ya + aj3 = 0. 11. From any point on the circle r + a cos = 0, tangents are drawn to the cardioid r = a(l + cos 0) ; prove that the points of contact are collinear. 12. In the cardioid r = a(l + cos 0), prove that s 2 «-9/> 2 = 16a" 2 , s being taken to be zero at the point (2a, 0). ChY / d <$\ 2 Here -rz - - a sin 0. Therefore, remembering that ( 3^ ) stands for /dr\2 ds r 2 +(^j , it follows that ^ = 2a cos g. . Therefore s must differ from 4a sin g only by a constant ; and this constant is zero, for s vanishes with 0. a Thus s = 4a sin -. 2 Now f = i*^^) } _ **»' cos2 » 1 v^/ ^ 2 f Therefore s 2 + 9r = 16a-. 13. In the lemniscate r 2 = a 2 cos 20, prove that the angle made by the normal with the initial line is three times the vectorial angle. (x\ n /v\ n - j + (r) =1> prove that 1 f dx dy . dx dy. i 15. In the cycloid x = a(t- sin £) ] 2/ = a(l— cos £) I ' prove that p 2 + s 2 = 16a 2 , s being taken to be zero at the point determined by t = ir. H 98 DIFFEBENTIAL CALCULUS x 16. In the catenary y = a cosh .-, prove that y 2 = s 2 + a 2 = ap and s - a tan \|/, where s is taken to be zero at the point (0, a) and ty denotes the angle made by the tangent with the x - axis. 17. Prove that the sum of the curvatures at the points of contact of per- pendicular tangents to a catenary is constant. 18. Prove that the catenary y = a cosh is the caustic of reflexion of a the logarithmic curve x = a log U for rays perpendicular to the x — axis. [The a caustic of reflexion is the envelope of the reflected rays.] 19. In the tractrix cos t + log tan ~ J x = al y = a sin t prove that s = a log -, s being taken to be zero at the cusp. y 20. Prove that in the curve, whose intrinsic equation is s = a log cosec if*, the product of the radius of curvature and the normal is constant, the normal being terminated by the asymptote to the curve. [Math. Tripos, 1897.] 21. In the polar tractrix = + sin -1 -, prove that the portion of the r a tangent, intercepted between the curve and the straight line drawn through the pole perpendicular to the radius vector, is of constant length a. 22. Establish the following construction for a logarithmic spiral having its pole at : — Two points M, N on the spiral being known, construct on ON the triangle ONP similar to the triangle OMN; then P is a third point on the spiral ; similarly a fourth point and then a fifth point, etc., can be determined. 23. In the logarithmic spiral r = ae ecota , prove that s = p tan a, s 2 = r 2 + S t 2 and p 1 - r 2 + £ n 2 , where s is taken to be zero at the pole and S t and S n denote respectively the subtangent and the subnormal. 24. Prove that the pedal of the involute of a circle is an Archimedian spiral. 25. Prove that the sine spiral a r >» = (?<*)" cos 910 2 is the locus of a point the continued product of whose distances from the vertices of the regular n-agon, having its centre at the pole, is constant. CHAPTEK IX. SUCCESSIVE DIFFEEENTIATION. 63. Definitions. The second differential coefficient oif(x) means df(x) the differential coefficient of J v ', and is represented by the symbol dx 2 ' In like manner the third differential coefficient of f(x) means the d 2 f(x) . d 3 f(x) differential coefficient of i \ > an ^ is represented by *V ' ; and ax ax so on. Thus, if y=f(x), the successive differential coefficients of y are % 2' % : • ■• ' ;& • • • ; these are also denoted by y 1>yfJ Ifo ••• ». If* '•'•-•' • Note. The value of y n for # = & is generally denoted by the symbol (y n )x=a d n f(x) or, briefly, (y ) • It should be also noted that Jy ' is frequently denoted by the symbol f in) (x), its value for x = a being denoted by f (n) (a). 1. The first, second, and third differential coefficients of x 3 are respectively 3# 2 , 6#, and 6 ; all the higher differential coefficients are equal to zero. 2. If y = log (cos a?), find y 3 . Here y x — — tan a?. Therefore 2/ 2 = — sec 2 a?, and y <*(-«*>' *) = _2 tan ^ sec2 x= _2*n* V6 dx cos 3 a? 3. If y=a sin a? + b cos a, prove that — -^ + ^=0. h2 EXAMPLES. 100 DIFFERENTIAL CALCULUS Here y v =a cos x— b sin x. Therefore y 2 — —a sin x — b cos x=—y. 4. Find the second differential coefficient of each of the following functions : — (i) 4a: 3 + 4x + 2. (ii) tan x + sec x. (iii) sin(z 2 ). (iv) tan-' Vl + X "'~ 1 + t&n' 1 -fe-. sc 1 — ar 5 Ii y = x 3 log £c, prove that 2/ 4 = -. 6 If i/ = sin (ra sin* 1 #), prove that 7. If ?y = a and, in general, dx..( aI ) =aI ( l0 S a )"- III. If 7/=sin a;, then yy—cos x=sinlx+ ), ^M*+i)h^M-i)H4 + f)- and, in general, £>in X )= S in( X + -). IV. Proceeding as in the case of sin x, we find that £ B (C0SX) = C0s(x + 5?). V. If y=e ax sin bx } then y i =ae ax sin bx + be nx cos bx=e ar (a sin &r + & cos bx). SUCCESSIVE DIFFEREN'J/ATICN V f 101 Let a=r cos 0, b=r sin , so that r 2 =a 2 +b 2 and tan 4,=-; a then we get 2/,=re a *{cos sin foe -f sin f cos 6a?} =re ax sin (bx + ). Again, y i =sref ts {a sin (for-f ) + & cos (&# + )} = rV*{cos 9 sin (&# + <£) + sin cos (bx + )} By repeating this process it is easily seen that, in general, An n (e ax sin bx)=r w e a * sin (bx+n). VI. Proceeding as in the case of e ax sin bx, we find that An ^ t (e ax cos bx)=i n e ax cos (bx+n). EXAMPLES. 1. If y=-— r-, find y u . * (b + cx) m ' yn tt,™ n . —mac n (— l) 2 ra(ra + l)ac 2 -, • 1 Here 2/l= (F+7x)-' y *= -(H^cr* • and ' m general - y — ( ~~ l) w ra(ra-H ) ._. • (m+n— l)ac n Vn (b + cx) m+>l 2. If 2/= sin # sin 2a;, find y„. Here ?/ = i(cos #— cos 3#). Therefore __1 / ^(cosjz^^c^sJ&e) 1 ** 2 1 (ii) tog( M + n. (iii) cos 3 x. (iv) e ax sin 3 bx. (v) sin 2x sin 3a? sin 4a?. (vi) e T sin a; sin 2 2x. 5. If 2/ = "* sin bx, prove that ^_ 2a # + (a 2 +6 2 )2/ = 0. dx 2 dx 6. If # = sin mx + cos mcc, prove that y =m n {l + ( — l) n sin 2mx\K 65. Leibnitz's Theorem. / and assume that Leibnitz's theorem holds for any particular value of n, say m, so that y m =fn

and /stand for x 2 and e ax respectively. Now i =2x ) 2 = 2, and 3 , etc., are all zero. Also f r ~a r e ax . Therefore by Leibnitz's theorem we have f ! (^ 2 ^)=aV u x aj a + M 1 a n " 1 e a * x 2x+ n C 2 a n - 2 x n ~ 2 x 2 SUCCESSIVE DIFFEBENTIATION 103 2. Find the nth differential coefficient of each of the following functions :— (i) x* sin ax. (ii) e ax { a' 2 x 2 — 2nax + n(n + 1) J . (iii) x m e ax . (iv) (a + bx) m log (a + bx). 3. If y = a;"- 1 log a?, prove that y n = ^ n ~ ^ \ n x 4. Prove that feC^)-{'--(' + 5)*«-(- + T)} + ^ where P and Q stand for a; ?l -n(n-l)ic H - 2 + w(^-l)(n-2)(w-3)a;' 1 * 4 - . . . and nx n ~ l —n(n — l)(n — 2)x n ~ 3 + . . . respectively. 5. Prove that f - (««#) = «"/-£- + a Vy. 66. Use of Partial Fractions. Recurrence Formulae. I. Use of Partial Fractions. In the case of a fractional expression whose numerator and denominator are both rational integral algebraic expressions, the nth. differential coefficient may be obtained in a convenient form by first resolving the fraction into partial fractions ; and if the denominator contains some imaginary linear factors, the differential coefficient may be reduced to real form by aid of Demoivres theorem, as in Ex. 4. II. Recurrence Formulce. Sometimes the value of the ?ith differential coefficient for a particular value of the variable is most easily obtained by first obtaining a recurrence formula, as in Ex. 6. Note. If, in his course of Algebra, the student has not become familiar with the resolution of a rational fraction into partial fractions, he is specially advised to follow carefully the various steps in the solutions of Examples 1 and 2 given below. These solutions are intended to make clear the practical points that are necessary for his present purposes. It should be noted that the partial fractions, corresponding to a factor (x — a) r in the denominator of the original fraction, are always assumed to be r A r-i A i -A±- For a full discussion of the subject (X~ay (aj-a)*-!' ' ' " (a -a)* (x-a)- of partial fractions, see some good book on Algebra, or Ch. III., Vol. I., of the author's book on ' Integral Calculus.' EXAMPLES. 1. ' Find the nth differential coefficient of ax 2 + bx+c (a?— a)(a?-/3)(«— y)' 104 DIFFEBENTIAL CALCULUS Assume that ax 2 + bx+c _ A B C {x—u)(x—ft)(x—y) x — a x — ft x — y where A, B and C are constants whose values are to be determined. On clearing of fractions, ax 2 + bx + c=A(x-ft)(x—y) + B(x— u)(x-y) + C(x— a)(x— ft). Therefore, putting x=-a in the above identity, we have A qa> + ba + C imilarl B= f* + W + C . mAC= ay' + by + e («-«(«-y) J (tf-«)(0-y)' (y-«)(r-0) Therefore the required differential coefficient is (-!)■»! A ■ (-!)*» 1 B (-l)"w ! C (x-a)»+ l + (OS- /3)" +1 + (x-y)** 1 -r_i\M I / «" 2 + &«+ c ^ 'l(a-/3)(n-7)"(^-«)" + l a/3 2 + fy3+c 1 ffly 2 + 6y+c y 2 + 6y+c 1_ 1 - a )(y-,3) (Z— y)"»J" (tf_„)(/3- y ) (aj-/3)«« (y- a )(y-ll) ( X -yf 2. Find the nth differential coefficient of (x-a)\x-ft)' Assume that A +7 _A. B (x—a) 2 (x—ft) X — a (x — a) 2 X—jV where A u A 2 and B are constants whose values are to be deter- mined. On clearing of fractions, x 2 =AXx-«)(x-ft) + AXx-ft) + B(x-a) 2 . Therefore, equating the coefficients of like powers of x in the above identity } we have A^B^l, AXa + ft)-A 2 + 2aB=0, and A l ap-A 2 ft+Ba 2 =0. Hence, solving the above system of simultaneous equations for A u A 2 and B, we have A a (a — 2/3) A a 2 n -n ft 2 Therefore the required differential coefficient is ( -!)«» 14 , (-l)»(n + l) !_4 2 ( - 1)"^ ! B (z-a) w+1 (a— n) M " (#-/3)' Ml SUCCESSIVE DIFFERENTIATION 105 ' l (a-/3) 2 (iC-a) n+1 ^ («-/?) (*— a)"« (a-/3) 2 (z-/3)" +1 J 3. Find the nth differential coefficient of each of the following functions :— (1) 7=uTi m fiCf (a>-l)(a>-2)(aj-3) v ' {x-l)*{x + 2) 4. If y= prove that a 2 + # 2 _ (— l) w wl sin w+1 sin(?i + l )^ where 0= tan -1 -. Here t/ == ,== - ■ — ( 1 . (x + ia) (x — ia) 2ia \x —ia x + ia) ' (x + «a) (4j — ia) 2ia \ Therefore t jbS&L!/ L_ i 1 Ua 2ia l(a— ia) w+1 (x + ia) n+1 ) Now let r and

. Therefore, since by Demoivre's theorem (cos d>— i sin ^)~ n_1 =cos (w + l)/> + i sin (n + l)(f> and (cos + i sin — isin (w-hl), (-1)**! . , Liw (XT* = (""^ ! sin" +1 sin (n+l)-/>. (1 — 2x cos a + cc 2 ) 2 15. If y = tan ^ , prove that ** 1 — a: cos a' F 106 DIFFEBENTIAL CALCULUS 6. If y=e a » in ~ lx , prove that (l—x*)y n+2 —(2n + l)xy n + l --(n? + a 2 )y H =0, and hence find the value of y n when x is zero. Here y 1 =e aBl,l ~ 1 *x ._ — , v 1— x 2 i.e., (l—x 2 )yi 2 =a 2 y 2 ; whence, differentiating and dividing by 2y i} we have (1— x 2 )y 2 — xy v — a 2 y=0. Therefore ^{(1-^)^-^0-^=0 . . (i). But by Leibnitz's theorem dx 7 J{l-x 2 )y 2 }=(l-x 2 )y n+2 -2nxy n+1 --n(n--l)y n and d Jxy x )=xy n + x +ny n . Hence (1) becomes (l-x 2 )y n+2 -(2n + l)xy n+1 -(ri 2 + a 2 )y n =0. Putting x=0 in the above equation, we have the recurrence formula (^) =(rc 2 +a 2 )(2/„)o • • • (2). Now (y ] ) () =a and (y i ) a =a 2 . Therefore the formula (2) gives (2/2m t i)o= K 2 + (2ra-l) 2 } (y 2m -!) = {a 2 + (2m-l) 2 } {a 2 + (2m-3) 2 j (y. im _ a ) = {a 2 + (2m-l) 2 }{a 2 + (2w-3) 2 } . . . (a 2 + l)a, and (&»)o= {« 2 + (2to-2) 2 } (2/ 2m _ 2 ) = {a 2 + (2m- 2) 2 } {a 2 + (2m - 4) 2 } (^ 4 ) = {a 2 + (2w-2) 2 } {a 8 + (2w-4) 2 } . . . (a 2 + 2> 2 . Thus (2/„) is a(a* + l s )(o* + 3 2 ) . . . (a 2 + (w-2) 2 } SUCCESSIVE DIFFERENTIATION 107 or a2(a 2 + 2 2 )(a 2 + 4 2 ) . . . ( a 2 + ( w _2) 2 } according as n is odd or even. 7. If y ^ sin -1 x, prove that m Mg % (2/n + 2)o = W%n)o and hence find (y n ) . 8. If 2/ = sin (m sin -1 a;), prove that (2/n+2)o = (™ 2 -W 2 )(?/n)o and hence find (yj . Examples on Chapter IX. 1. If x 2 = 2y, prove that the sixth differential coefficient of sin y with respect to x is 15(4i/ 2 — 1) cos y — 2y(4y 2 -4:5) sin y. 2. If 2/ = log \ / i + ft^ + a 2 + tan" 1 £^-| prove that da;*' 8^2ai (1+a 4 ) 1 3. The first, second, third and fourth differential coefficients of y with respect to x are denoted by t, a, b, c respectively, and those of x with respect to y by t, a, j3, 7. Show that 3ac-5fr 2 _ 3a7-5£ 2 t* r* ' 4. If Y=sX, Z — tX, and all the letters denote functions of x, then X Y Z X x Y x Z x \ =X* X 2 Y 2 Z 2 I [Math. Tripos, 1906.] s 2 t, 5. If 2/ = tan- , {(e r + l)/(^-l)J i , prove that s=i{ i+i2 (i))( i+4 (tn- [Math. Tripos, 1907.] d» 6. Prove that ?-U"x m ) = (-\ dx n \xj n\n—m ftm a\ <*—(e** x »). dx mX ' 7. Show that ^(^ = l=^(lo gaJ -5lY dsc*\ sb / z ,l+1 V 1 rj 8. Show that *{«-H.+i)-V+^< ..(^ + m)-i| »w! f»»O "C, v 'm!U M+1 (« + l) n+1 (x + 2) n+i - . . . + (-!)» "^ }. (x+m)*+ l J 108 DIFFERENTIAL CALCULUS 9. If y - , prove that a 2 + sc 2 fry = (-l)»nlBm»+'»ooB(n + lte where = tan _, a dx n a n+l x 10. If y = X l~~ X + l and tan

-a)» +l 3a 2 5±» ' (cc 2 + aic + a 2 ) * where (# 2 ), prove that U = M% + *(» - 1)(2*)"- V.., 18. Prove that d>l t(l jn^k ^ 1 *" 11 '* . 5 . . . (2»-l) Binnf d£" ll ; . * <^> where = cos -1 a;. SUCCESSIVE DIFFERENTIATION 109 19. If y = w , 2 2! (2a/x) w + (-l)^(n+_p-l)(n+_p-2) . . . (n+l)n(n-\) . . . (n-p) n _ p + . . . 20. Prove that (6 )_e \2vW o I 1 i, r!(n-r-l)!(2a^KJ [Math. Tripos, 1886.] CHAPTEK X. rolle's theorem and taylor's theorem. 67. Rolle's Theorem. The Theorem of the Mean Value. I. Holies Theorem, Let f(x) be continuous for every value of x in the domain (a, b) and let f(a)=f(b) ; further, let f'{x) be existent for every value of x within the domain (a, b). Then there must be at least one value c of x, intermediate between a and b, such thatf f (c)=:0. In the case when f(x) is repre- sentable graphically, the truth of -^ w ^- x Eolle's theorem becomes obvious Fig. 31. from the figure of the curve y=.f(x) ; for, the ordinates of the points, whose abscissae are a and b, being equal, there is at least one inter- mediate point at which the tangent is parallel to the x— axis. For a rigorous proof of Eolle's theorem, see Note B. II. The Theorem of the Mean Value. Let f(x) be continuous for every value of x in the domain (a, b) ; further, let f'(x) be existent for every value of x within the domain (a, b). Then there must be at least one value c of x, intermediate between a and b } such that f(b)-f(a) = (b-a)f'(c) . . . (1). For consider the function 1> ( x )=f(b)-f( x )-M2 f a {a \b-x). Now (p(a) and '(#) is existent, its value being o—a ROLLE'S THEOREM AND TAYLOR'S THEOREM 111 Therefore (j>{x) satisfies all the conditions of Rolle's theorem. Therefore b — a vanishes for at least one value c of x intermediate between a and b. Hence m-A«) =(*-«)/'(«)• Note 1. Rolle's theorem has been called by some writers the fundamental theorem of Differential Calculus, because of the many important theorems which are based on it. It is named after the French mathematician Michel Rolle (1652-1719), who first attempted a rigorous proof of it. Note 2. Putting h for b — a, we get (1) in the form f(a + h)=f(a) + M'(a + 0h), where <6< 1. EXAMPLES. 1. (- tan" 1) give the graphs of the functions and hence show that Rolle's For the domain ■nx -. lim f2x , _i x cos ¥ and * ob tan T\ theorem is applicable to the first function but not to the second. Since the second function equals 0, x or — x according as x is 0, or < 0, the required graphs are A GB and DOE respectively. Further, it " is obvious from the figure that, while there is a point, viz., 0, on ACB where the tangent is parallel to the x— axis, there is no such point on DOE. Thus it is shown that, in the domain ( — 1, 1), Rolle's theorem is appli- cable to the first function but not to the second. 2. In the case when f(x) is represen table graphically, give the geometrical interpretation of the theorem of the mean value. Let ACB be the graph of f(x) in the domain (a, b), the points whose abscissae are a, c and b being A, C and B respectively. Then, constructing as in the adjoined figure, we have OTM Fig. 33. MN AB 112 DIFFERENTIAL CALCULUS Now /'(c)=tan L GTX. Therefore the following is the geo- metrical interpretation of the theorem of the mean value x — If the graph ACB of f(x) is a continuous curve having everywhere a tangent, then there must be at least one 'point, intermediate between A and B, at which the tangent is parallel to the chord AB. 3. In each of the following cases give the graph of the function and decide whether the theorem of the mean value is applicable : — (i) f(x)=x* t a= -1, 6 = 1. (ii) /(&)=&*, <*=-l, 6 = 1. (iii) f(x)=x\ a= -1, 6 = 1. (iv) /(*)= Hm (l+jm«)"_rl |fl= _ 1,6 = 1. v ' JK ' n = oo(l +sin irx) n + l 4. For the function Ax 2 + Bx + C, prove that c = ^— in the theorem of the mean value. 68. Taylor's Development in Finite Form. Let f(x) and its first (n—1) differential coefficients be continuous for every value of x in the domain (a, b) ; further, let f {n) (x), the nth differential coefficient of f(x), be existent for every value of x within the domain (a, b). Then there must be at least one number 0, intermediate between and 1, such that f(b)=f(aj * (b- a)f '(a) + (b "* )2 f "(a) +;.;.. +( b ~ a )>>(a) f . . . + ( J ) n " a i ) !V f(n ->) .(b-a)» + v n! 'f«>(a + eb-a) (1). This is Taylor's theorem with the remainder in Lagrange's form, the remainder being ^^'f^Xa + eb-a); the equation (1) is called Taylor's formula or Taylor's development in finite form, The above theorem may be proved in the following manner : — Consider the function ROLLE'S THEOBEM AND TAYLORS THEOREM 113 _ Sb-x) n K (2 ^ where it is a constant given by the equation m+ . . . +|: + (/>-«)" A - K§ . ( 3 ), f(b) = f(a) + tb~-a)f(a)+ . . + ^=^/-» t.6., (a) = 0. Now ^(a) and <^(&) are zero; and (x)), vanishes for at least one value c of x intermediate between a and b. Hence, denoting c by a + 8(b— a), we have K=f (n) (a + 0b—a) ; and, substituting this value of .ST in (3), we get (1). Note 1. Putting h for b — a and writing x for a, we get Taylor's formula in the form f(x + h)=f(x) + hf(x) + |*f"(x)+. .. +£f» + . . . +(n h ^ i 1 )! f<"- 1 '(x) + ^f> + eh). Note 2. Putting a = and writing x for b, we get, as a particular case of (1), f(x) = f(0) + xf'(0) + ^f"(0)+. . . V!^ + --- + (Sr!^ 1,(0) v^ >x) - 114 DIFFERENTIAL CALCULUS EXAMPLES. 1. Find the remainder after n terms in the expansion of sin (a + h) in powers of h. Here f{x) — sin x and, consequently, all the conditions of Taylor's formula are satisfied. Also f kn \x) — sin f x -f * ) . Therefore the remainder required is JTf*(a + Bh) . fc w Bin(g + efc + ^) w- ! n ! 2. Prove that tan" 1 x =x-^+°£--~+ . o o 7 ^2m-i # 2m sin I 2mf| + tan- 1 ftrj j + ( -1 ) W * 2^1 + 2m(l + 6V 2 )" 1 Take /(#)=tan~ l x. Then /(#) satisfies all the conditions of Taylor's formula. Also by Ex. 4, Art. 66, fw^^in— 1) I shW l+tan" 1 #j (1+S*)t and consequently f (n) (0) = (n—l) ! sin ^. A Therefore /W=/(0)+^/'(0)+^/"(0)+ . . . +£/««»+ • • • +( jg^ /<•»-<(>) =»-£+. .. +(-1)-^:;+... „. 2 »-i x™ sin ( 2to ( I + tan- 1 ftz ) } + (-1)- 1 £—. ; + Ln V ?iiJ l u — '- 1 ■ ( 1\m-l ± I I Vj V ; 2m- 1 2m(l + (a) + ... to infinity for every value of h within the domain (0, H). For a discussion of the conditions of Taylor's theorem, see Note B. II. Maclaurin s Theorem. Let f(x) be a function defined for a given domain (0, X) of x. Then, when certain conditions are satisfied, f(x)«f(0)+xf(0)+||f"(0)+. . . +*>>((>) + ... to infinity for every value of x ivithin the domain. Maclaurin's theorem is a particular case of Taylor's theorem, for it is obtained from Taylor's theorem by putting &=0 and writing x and X for h and H respectively. Note. Taylor's theorem is called after the English jurist and mathematician Brook Taylor (1685-1731). EXAMPLES. 1. Apply Taylor's theorem to obtain the Binomial Expansion (a + h) m =a m + ma m - 1 h^ m ^~ 1 \c m - 2 h 2 A ! + I m ( m ~ 1 ) • • • ( m ~~ n + 1 > )a m - n ti l | n\ Here f(x)=x m and consequently f in \a)=m(m—l) . . . (m— n + l)a m ' n . Therefore the general term in the series for (a t- h) m is *L ;wf fl ) = fft (^ 1 ) • • • ( m -rc + l) m _ Mfelt . nV ■' n\ and thus we get the required expansion. 12 116 DIFFERENTIAL CALCULUS 2. Apply Maclaurin's theorem to prove that •v.2 ~4 Here /(^)=<3 8in *, .-./(0)=1; /' (a?) = e sin ■ cos a?, /' (0) = 1 ; f ,f (x)=e sinx (cos 2 x— sin x),f'(Q) = l ; f n, (x)=e sinx cos #(cos 2 a;— sin #) — <3 sin *(2 sin # cos # + eos #) = _ e -*(3 + sin*) s *^, .../'"(0)=0; /<'>(a;) = — ie sin * cos #(3-f sin a;) sin 2x —\e*™ x {2 cos 2#(3 + sin x) + cos x sin 2#} , /./<«(0)=-3. Therefore ^*=/(0)+«/<(0) + ~/ ,/ (0) +f|/ ,,; (0)+^(0)+ . . . = l+* + „___ . . . 3. Apply Maclaurin's theorem to find the coefficient of x n in the expansion of e a 8in ~ x . HGYef(x)=e aBin ~ lx , and consequently by Ex. 6, Art. 66,/ (n) (0) is a(a 2 + l 2 )(a 2 + 3 2 ) . . . {a 2 + (n-2) 2 } or a> 2 + 2 2 )(a 2 + 4 2 ) . . . (a 2 4>-2) 2 } according as n is odd or even. Therefore the coefficient required is q(a 2 + l 2 )(a 2 + 3 2 ) . . . (^ + (n~2) 2 } ft! or a 2 ( q 2 + 2 2 ) ( a 2 + 4 2 ) . . . {a 2 + (n-2 ) 2 } n! according as n is odd or even. 4. Apply Taylor's theorem to prove that tan" 1 (a + h) = tan -1 a + (h sin - (/l 7/ )2 sin2^, + . . . +( _i r i( fc !^Binn*+ ... to infinity, where a = cot (x), y=^(x) are said to have a contact of the nth order at the point x=-a ; if ^(a)-^(a)=f(a)-+ , (a)=*>)-*"(a)= . . . =™(a)-V n \a)=0 and f"+v(a)-V n+1) {a)=^Q. It follows from the above definition that the following results hold true : — (I) Two curves, having a contact of the nth order at the point x=a, cross each other, or do not cross each other, at that point according as n is even or odd. For, denoting by y L and y 2 the ordinates of the two curves for x=a + h, we have, by Taylor's theorem and by the above definition, y L —y 2j i.e., (l>(a + h)—\l,(a + h) + (S)iU < ' ,+2,(a) - r+2,(a) } + --- Hence, when h is sufficiently small, yi—y> has the same sign as Tjn+i r "] . < (n+1) (a) — ty n+1) (a) > ; and, consequently, y L — y 2 changes (n+1) ! I J sign, or does not change sign, with h according as n is even or odd. 118 DIFFEBENTIAL CALCULUS [To obtain the criterion for a point of inflexion, first note that, if y = \\t(x) represent the tangent to y- {n+1) (a) for sufficiently small values of h. Now by definition (Art. 50) the tangent crosses the curve at a point of inflexion. Hence x - a is a point of inflexion, if - V) - u^K r \ J W • • • ( n -l)l J W n\ ^ where K is a constant given by the equation «-iA-T' (ro-1) . . . (ot-to + 1) (1 — n_1 where w n stands for * ~ * . ~~ #"• (w — 1) ! (1 — \ n ~ 1 1 must remain less than a finite number A, however large n may be. Therefore [ R ni X ) < A \ m ( 1 + 6x ) m ~ l x J W n • But it is well known that, as | x \ < 1, w n tends to zero when n tends to infinity. Hence R n tends to zero when n tends to infinity. 3. Prove that a statement, exactly similar to that in Ex. 2, holds true in the case of the expansion of tan -1 x. 4. In the case of each of the following functions, investigate the behaviour of the remainder after n terms as n tends to infinity : — (i) a v . (ii) sin x. (iii) log(l + cc). 5. If f(x) is finite, continuous, and has a continuous differential coefficient f'(x) for all values of x from to 1, prove without the use of geometrical repre- sentation that/(l) -/(0) =/'(0), where 6 is some proper fraction. 120 DIFFERENTIAL CALCULUS Deduce the relation T \ 71 I stating the further assumptions necessary. [Math. Tripos, 1903.] 6. Prove that under certain conditions to be specified f(x)=f(o)+xf(o)+f l r(o)+ . ■ . + ( -™jyr/"" 1) ( ) + f,/ < " w ' where < 6 < 1. Expand f(x) = (1 — #) n+ * in the above form, and prove that as x tends to unity the value of 6 in f^ n) (Bx) tends to the limit J K } (2;i+l) 2 [Math. Tripos, 1906.] 7. In the equation f(x + h)=f(x) + hf'(x) + £f"(x) + . . . +J ! /w(x i -e/ i ) ) show that the limiting value of as h is indefinitely diminished is . n + 1 8. Apply Maclaurin's theorem to prove that _5_ =1 -| + B i g_B,f.; + B|- . . . , where B, = |, B 2 = ^, B a = ±, B 4 = ±. [The numbers J5„ J5 2 , etc., are called Bernoulli's Numbers.] 9. Prove that xx x 2 x* x 6 10. Prove that if 2/ = sin log (x 2 + 2x + 1), Hence or otherwise expand 2/ in ascending powers of x as far as x 6 . [Math. Tripos, 1904.] 11. If A be the chord of any circular arc, and B that of half the arc ; then Q T> A the length of the arc is very nearly equal to ., • [This is Huyghens' approximation.] 12. Determine the parabola y - a + bx + ex- so that it shall have a contact* of the second order with the curve y = x* at the point (1, 1). CHAPTEE XI. MAXIMA AND MINIMA. 70. Definitions. A function f(x), defined for a domain (a, b), is said to have a maximum for a value c of x } within the domain, if there exists a quantity e >0 such that f(x)-f(c) <0 for every value of x satisfying the condition 0< x—c 0, taking the place of /(a?) —/(c) <0. Note. The student is advised to make himself familiar with these defi- nitions by studying carefully the three examples worked below. The beginner should guard himself against confusing the words maximum and minimum, as defined above, with the words greatest and least. A maximum of a function is not necessarily the greatest possible value the function can have ; likewise, a minimum is not necessarily the least possible value. Cf. Ex. 6 given below. EXAMPLES. 1. The function sin x has a maximum for x=^. For, let c= 7r , 2 4 say. Then sin x— sin- < A for every value of x satisfying the con- dition 0< # — 7 0 for every value of x satisfying the con- — - dition 0< x—0 0 for every value of x satisfying the con- dition Fig. 36. 0< I x-0 See the adjoined figure. <*. 4. Show, by a graph, that the function x 3 has neither a maximum nor a minimum. 5. Show, by a graph, that the function ax — x°- has a maximum for x= . 6. Give the graph of the function +cos x for the domain (0, 4tt), and hence find its maxima and minima within that domain. 71.* Two Theorems. Theorem I. If a function f(x), defi?ied for a domain (a, b), has either a maximum or a minimum for a value c of x within the domain; then f '(c) must be zero, if it exists.^ For, suppose f(x) has a maximum for x = c. Then, by Art. 70, there exists a quantity e > such that /(x)-/(c)<.0 for every value of x satisfying the condition 0< x-c <€. Therefore according as jfl5b/@< or >0 . X-C < (x - c) < € or > (x — c) > - 6. (i) f For a case in which /'(c) is non-existent, see Fig. 36 where c = 0. MAXIMA AND MINIMA 123 Now let the symbols (x) and 0. For, under the conditions, by Art. 68, where O<0<1. Also, f w (x) being a continuous function, there exists a quantity e > such that f° l) (x) has the same sign as f {n) (c) for every value of x satisfying the condition 0< <£. Therefore, for such values of x, f(x) —f{c) has the same sign as (x— c) n f in \c). Hence, if n is odd, f(x)—f(c) changes sign with (x—c) and, consequently, f(x) has neither a maximum nor a minimum for x=-c\ if n is even, f(x)—f(c) has the same sign as f {n) (c) and, consequently, f(x) has a maximum or a minimum for #=c according as / (n) (c)<0or >0. 124 DIFFERENTIAL CALCULUS EXAMPLES. 1. Prove that sin x has a maximum for #= and a minimum A for x=— -. 2 — ■» Here /(a?) = sin a;, and, consequently, f(x) = cos a; and /"(#) =— sin #. Therefore /'(5) = ^(-l) =0 ^"(l) <0and ^(-l) >0 - Hence, by Theorem II., sin g has a maximum for x=- and a A minimum for x=— s . 2 2. Prove that # 5 — 5# 4 + 5a^ — 1 has a maximum for x=l ; a minimum for #=3 ; neither for x=0. Here f(x)=x*—5x 4 +-5# 3 — 1, and, consequently, /'(#)= 5(z 4 -4# 3 + 3a; 2 ), /''(#)=10(2z 3 -6z 2 + 3;r), and Therefore /'''(z) = 10(6z 2 --12a; + 3). /'(0)=/'(l)=/'(3)=0, /"(0)=0, / /, (1)<0, / / '(3)>0 and /'"(0) f 0. Hence, by Theorem II., we have the required result. i 3. Prove that x x has a maximum for x=e. i Here f(x)=x x , and, consequently, /'(*)=/(*) x±(l-log«) and x* f"(x) = f'(x)x±(l-logx)+f(x)xl 3 (21ogx-3). X X 1 Therefore /'(e)=0 and/ /; (e) <0. Hence, by Theorem II., 05" has a maximum for x=e. 4. Prove each of the following statements : — (i) x s has neither a maximum nor a minimum for x = 0. (ii) (x -l)(x- 2) 2 has a maximum for x = £ and a minimum for x = 2. MAXIMA AND MINIMA 125 (iii) — has a maximum for x = 2 and a minimum for x - 14. x-S (iv) sin x(l + cos x) has a maximum for x = ~. 5. If a 3 + 2/ 3 - 3az?/ = and a > 0, prove that y has a maximum for x = a ^2. 6. Prove that the minimum value of the radius vector of the curve — + — = 1 is a + b. x 2 tf 72. General Rule for finding Maxima and Minima. The following rule, which is applicable to most of the ordinary functions follows immediately from the two theorems of Art. 71 : To find the values of x for which a function f(x) has a maximum or a minimum, first solve the equation f'(x)—0 and let c { , c 2t . . . > c r , . . . be its roots. Then f{x) has a maximum or a minimum for x=c r according as f "(c) <0 or > 0. Note. The student will sometimes come across a practical problem in which the quantity, whose maximum or minimum is sought, is given in terms of two variables ; these variables being connected with each other by a given relation. In such a case the quantity should be first made a function of a single variable with the help of the given relation ; and then the general rule should be applied to this function. See the solution of Ex. 2 given below. EXAMPLES. 1. Find the values of x for which x(l—x)(l—x 2 ) has a maxi- mum or a minimum. Here f(x)=x(l— x)(l— x 2 ), and, consequently, f(x) = (l-x)(l-x-±x 2 ). Therefore the roots of the equation f'(x)=0 are -. -lWlT j -l-\/l7 1, 3— and — -^— . Now f"(x)=12x 2 -6x-2. Therefore /"(D>0, /«(=i^) <0 and /«(=i^S)>a Hence x(l—x)(l-x 2 ) has a minimum for x~l, and for -1-V17. . , -1+^17 x= ~ ; a maximum for #=- . o 8 126 DIFFERENTIAL CALCULUS 2. An open tank of assigned volume has a square base and vertical sides. If the inner surface is the least possible, what is the ratio of the depth to the width ? Let V' denote the assigned volume, a the width, and ax the depth ; further, let S denote the inner surface. Then S=a 2 + 4:a 9 x and V =a*x. Therefore S=[ ) (l + 4#), which is to be a minimum. [Note this step.] Now ^=F f (— |ar*+#ar*). Therefore the equation ~=0 U.X (XX has only one finite root, viz., #=| ; and this value of x makes S a minimum, for which is greater than zero when x~\. Hence the required ratio is 1 : 2. 3. In the case of each of the following functions, find the values of a for which the function has a maximum or a minimum : — (i) 2a 3 - 15a 2 + 36a + 2. (ii) x{a + x) 2 (a-x)\ ,..« 3a 2 -a 2 ,. x (v) sin x cos 3 x. (vi) . w 1 + cot x 4. A Norman window consists of a rectangle surmounted by a semi-circle. Given the perimeter, required the height and breadth of the window when the quantity of light admitted is a maximum. 5. In a submarine telegraph cable the speed of signalling varies as x 2 log -, where x is the ratio of the radius of the core to that of the covering : show that the speed is greatest when this ratio is 1 : \/c7 6. What is the ratio of the height to the radius of an open cylindrical can , of given volume, when the surface is a minimum ? 7. Examine each of the following functions for maxima and minima : — (i) rf(i+i)(«-aj«, (ii) g=g [X + O) , (iii) 5x e + 12a 5 - 15a; 1 - 40z 3 + 15a 2 4 60a + 17. (iv) 4 cos x + cos 2a. MAXIMA AND MINIMA 127 8. Prove that x 2 sin 2 - has an infinite number of maxima and minima in x every domain containing 0. Examples on Chapter XI. 1. Investigate the maxima and minima of each of the following functions : — (i) sin x cos 2x. (ii) sin nx sin" x. (iii) e x sin x. , (iv) cos x + cos 2x + cos 3x. 2. Find the maximum and minimum of x 2 + y 2 where ' ax' 2 + 2hxy + by 2 = 1. 3. Find the maximum cone of given slant height. 4. Show that the semi-vertical angle of the cone of given surface and maximum volume is sin -1 1. 5. Show that the shortest normal chord of the parabola y 2 - 4ax is Ga */$[ 6. The portion of the tangent to an ellipse intercepted between the axes is a minimum : find its length. 7. From a fixed point A on the circumference of a circle of radius a, the perpendicular A Y is let fall on the tangent at P. Prove that the greatest 3 a/r area ^PFcan have is a 2 . 8 8. Find the height of the cone of least volume which can be circumscribed about a sphere of given radius a. 9. Find the area and position of the maximum triangle, having a given angle, which can be inscribed in a given circle, and prove that the area cannot have a minimum value. 10. Show that the total surface of a cylinder inscribed in a right circular cone cannot have a maximum value if the semi-angle of the cone exceeds tan _1 |. 11. Find the area of the largest rectangle that can be inscribed in an ellipse. 12. P, Q are two points on a circle whose centre is C and radius a. Prove that the maximum value of the radius of the circle inscribed in the triangle CPQ is ^*/iovT- 22 - 13. If the sum of the edges of a rectangular parallelopiped is Z, and the m of the areas of the faces is „, show that when the excess of the volume 2o of the parallelopiped over that of a cube, whose edge is its smallest edge, is as • great as poss other edges. sum of the areas of the faces is - , show that when the excess of the volume 2)0 cube, whc • great as possible, the smallest edge must be - ; and find the lengths of the 20 [Math. Tripos, 1903.] 128 DIFFERENTIAL CALCULUS 14. The circle of curvature at a point P of a parabola meets the parabola again at Q, and its centre is C. Prove that the area of the triangle CPQ is a maximum when CP makes with the axis an angle whose tangent is — -. [Math. Tripos, 1901.] 15. Prove that f(x) is stationary for values of x which are the roots of the equation /'(#) = 0; and determine the conditions necessary to distinguish the values of f(x), corresponding to the roots of f'(x) = 0, as maxima, minima, or neither maxima nor minima. If A, A' are the areas of the two maximum isosceles triangles which can be described with their vertices at the pole and their base angles on the cardioid r = a(l + cos 0), prove that 256 A A' = 25a 4 ^5. [Math. Tripos, 1907]. CHAPTEK XII. INDETERMINATE FORMS. 73 Introductory. Letf(x)=pP\, where lim 0(^)and lim Mx) Y(x) x=a x=a are both zero. Then __ f(x) cannot be taken to be equal to X——CC lhn ,/ x — fW . . x — a ; for, this quotient is of the form - and is, consequently, lim meaningless. The form - is ordinarily called an indeterminate form; the other simple indeterminate forms are — ,0x 00,00—00, 00 0°, I 00 and ooo. It is the object of the present chapter to show how, in general, we may determine the limit of a function, when the ordinary pro- cedure, indicated in the Note to Art. 6, fails on account of the appearance of one or more of the above-mentioned indeterminate forms. The form - may be considered as fundamental ; for. every problem corresponding to any other simple indeterminate form may be reduced to a problem corresponding to the form -. Note. It should be noted that in particular cases it is possible to find the limit, corresponding to an indeterminate form, by common algebraical or trigonometrical transformations. For example, the various differential co- efficients, which were investigated in Chapter II. and which all correspond to the form -, were obtained by means of such transformations. 180 DIFFERENTIAL CALCULUS 74. Cauchy's Theorem. The Fundamental Form -. I. Cauchy's Theorem. Let (j)(x) and yj/(x) be continuous for every value of x in the domain (a, b) ; further, for every value of x within the domain {a, b), let '(x) and ^'(x) be existent and \p'(x) finite and =f=0. Then »(b)-4>(a)^4/(o) a (n) (a) and v// n) («) be existent and i// n) (a) finite and 4=0. Then lim 4>(x) = V" ) (a) X = avKx) V' 7, (a)' For, by Cauchy's theorem, 4,(x) Hx)-t(«) f( Ci ) *"(<*) '•• r- 1} (^ i)' INDETERMINATE FORMS 131 Ci being intermediate between x and a, c 2 between c { and a, etc. Therefore, as (,i_1) (a) and t// n_1) (a) are both zero by supposition, U*) * ( * _1> ((x) V n) («)' Note. Cauehy's theorem is named after the French mathematician A. L. Cauchy (1789-1857). EXAMPLES. 1. Find lim i l0 ^. x—\ x—1 Here (#)= log x, 4>(x)=x—l; /. f'(*)=4 f (1) = 1, Hence Um . ^«1. 2. Find #=1 #— 1 lim x— sin -1 re (x)—x— sin -1 x, i/,(#)=sin 3 x=l(S sin x- sin 3#) ; v(/(a;)=l(cos #-cos 3a;), if/(°) = °> (i-£ 2 )* ^'(a?)«=|(3 sin 3#-sin x), \l>"(0)=0, *'"(*)=-{ —J + ^ },«/"(0)=-l, l (l-# 2 ) (l-# 2 )* J i|/"(a;)=f(9 cos 3^-cos a?), f "(0)Hfc -□- lim #— sin * a; , Hence — r^ = — e- x=0 sm d a; 3. Evaluate the following : — ,.v lim x — sin x cos a; /..x lim cosh x — cos a; W a: = ^i~ * ' z = a sin a; ..... lim a r -fc* ,. , lim ^-e 8ina: (ill) „ . (lV) n ; . x = x x = 0x — sin a; , x lim a; 2 ' — x , . x lim or tan # (v) ^ . (vi) . v a; = 1 1 - £ + log a? x - / e r _ j\$ 4. If #(#) = or jc 2 sin ■ according as a; is zero or different from zero, find lim , oo-oo, 0°, 1°°, co°. I. The Form . Let „__ (#) and -\(x) be both infinite, OO X — (aj X Cb and let ttw and 77-T satisfy the conditions of Cauchys theorem lim d)(x) in a domain containing a within it ; further, let __ '3j\ and n , [ be both existent. Then x—a \p'(x) lim 4>(x) lim 4>'(x) x=a vi/(x) =: x=a V(x)' INDETERMINATE FORMS 133 For, by Cauchy's theorem, J_ j/(c) »fa) {x) {f(c)}* where c is intermediate between a and x. Therefore lim ' (x) + ty \x) f lim _j/(a;) 1 hence X= lim *M #=a y(x) Case 3. A is infinite. It follows from Case 2 that , i.e., lim ;£(#) __ lim \^'{x) , #=a (xy x=a may be determined by first writing ty(x)^(x) X — CL \ J fundamental form—. III. The Form oo-oo. Let llm (x) and lim \L(x) be both x=a x=a infinite and have the same sign. Then, ^ < (p(x) - \^(x) > may be determined by first writing (x)— -\^{x) in the form J L H °c) ip(x) J_ 1 \P(x) ' (x) and then applying the rule corresponding to the fundamental form q. IV. The Form 0°. Let lim q>(x) and lim Mx) be both zero. x=a x=a Then, x=a{+ {x) ^+ {x) } determined by first writing \^(x) k>£ and then applying the rule corresponding to the funda- may be determined by first writing \^(x) log (x) in the form m log (x) mental form ^ ; hence .*£.{«*-} is known, for it is equal to lim v) logoff) )>. lim r x=a< \^(x) e I V. The Forms 1°°, oo°. The forms 1°° and oo° are treated exactly as the form 0°. INDETERMINATE FORMS 135 Note. Compound indeterminate forms. If a function f(x) be the prod' t of two or more functions, the limit of each of which corresponds to a simple indeterminate form for the same value of x ; then the limit of f(x) may be determined by finding the limits of each of its factors separately, provided that the product of these limits is itself not an indeterminate form. It is evident that similar remarks apply to other compound indeterminate forms. For an illustration of the procedure indicated above, see the solution of Ex. 6. EXAMPLES. 1. Find lim lQ g fr'> . x=0 log cot 2 x This limit corresponds to the form — ; for, in l log (x 2 ) and oo #=0 lm ^ log cot 2 x are both infinite. Therefore lim log (x % ) _ Km dx ^ log ( ^ )} #=0 log cot 2 x x=0 d n ,o x v (log cot 2 x) 2 __ lim x ' -~r=0 2 ~7~' cosec' x cot x lim —sin x> cos x "- ^=0" x which corresponds to the form- . Therefore the required limit= — rt — - — = — 1. * #=0 1 2. Find lim , (1-s) tan -jf. This limit corresponds to the form x oo. Therefore lim .. , 7tx lim 1— x ^(Mton-j- a . =1 — — (which corresponds to the form _^j cot - lim -1 X = l v o irX -- cosec 2 ~ 2 - 136 DIFFERENTIAL CALCULUS 3. Find lim ft ft— rW. o;=0 Vc 2 sin 2 xj This limit corresponds to the form oo-oo. Therefore lim / 1 _ 1 \ _ lim sin 2 x— x 2 #=0 \x 2 sin 2 x) x=0 x 2 sin 2 x (which corresponds to the form -J lim sin 2x— 2x x=0 2x sin 2 x-\-x 2 sin 2x lim 2 cos 2a; — 2 o;=0 2 sin 2 x + ix sin 2o; + 2a; 2 cos 2a; _ lim —4 sin 2x x=0 6 sin 2x + 12x cos 2a;— 4a; 2 sin 2x _ lim —8 cos 2x x=0 24 cos 2x— 32a; sin 2x- Sx 2 cos 2a; JL — 3* 4. Find ln \x 2 *. x=0 This limit corresponds to the form 0°. Therefore, im x log (x 2 ) being 0, the required limit =c°=l. 5. Find h ™ (cos x) coi *\ a;=0 This limit corresponds to the form 1*. Therefore, since _ lim —ta n a; x=0 2 tan a; sec 2 x =-h the required limit =e _i . 6. Find lim ■ /(i + ^^.^-^V). Lim /-, . v ., ^(1+^=1; lim o 3 lim e* 3 lim dx a;=0 *«0T a==0 d/l\ a; 2 da; \a; 2 / lim / L INDETERMINATE FORM 137 and, finally, lim e x — e~ x __ lim #=01og (1 + x) x=0 d dx w-^ {log(l + *)} % 1 S) (1+a,)(,B ' + *"" ) =2. Therefore the required limit =1 xco x2=oo. 7. Evaluate the following : — (i) lim 1U S 1 V~aJ f x = a CQt2 *x a (iii) im t (sec ~ . log x). » = 1 V 2x h ) (v) lim /l 1 \ # = \# 2 a; tan x)' (vii) lim 2 sin x x = (ix) ajs=0 (cos WWJ)* (xi) lim /IN 2 *'** (viii) Vm . (sin as)"*". a; = M (xii) aj-0 ( oosha 0*"- lim x = (cot #) : Note. Infinitesimals and Infinities. The following definitions are not of much importance to the beginner ; but we give them, because it is sometimes convenient to use them in the applications of Differential Calculus : — (I) By an infinitesimal is understood a variable quantity having zero for its limit ; by an infinity is understood a variable quantity having + oo or -oo for its limit. (II) Let 0. Taking x as a standard, we say that x* is an infinitesimal of order n, if x > 0. f(x) In general, any function f(x) is called an infinitesimal of orders; if JK ' tends to a limit, which is finite and * 0, as x tends to zero by assuming positive values. Definitions similar to those, given in (II) and (III), hold for infinities. Examples on Chapter XII. 1. Evaluate the following by algebraical or trigonometrical transforma- tions : — /.v lim cos 2 wx /..x lim 1 — x + lo g x U x = le**-2ex U x = 1 1 - V2x^7 2 ' ,..., lim ^2x — x x — ^x ,. x Urn (log xfi+jl — x 2 )* x = l X- : V3P. ' ® = 1 nx?{z-l) 2. Evaluate the following : — i i m lim /sinh x\x 2 .... lim / coshcc — l\x 2 W x=0 (— ) ' (11) **o ( 2 • — 5T— ) • log COS X /...x lim / tan a ?\g«. /. x lim sin' 2 # £C — U \ X / (v) # m (sin x) Un *{— - — — l. v ' x = -?_ v f \2sin2ic/ x = *, x log cos - • •> X Z sin 2 2 • v lim f /2 + cos 2a; — sin x / ir — 2x \ 2 i ^ _ * 1 ■ \/ a; sin 2a; + x cos a; \2 sin 2x) )' 3. Prove that lim • *" = <). 4. Find lim 2*sin a -. a: = oo 2 X 5. Prove that in the equation (x + h)-$(x) ~yp'(x + eii)' the limiting value of when h tends to zero is £ ; and that when <£>(.!•) = sin x and i//(ic)=cos x, the value of is always \. 6. If ^ = b <£(#) and = f(flj) be both zero, prove that , _ {0(a)***} will always be 1 provided that either lim ^ or lim *(*)*'(*) is finite. x = a $(x) x-a ty(x)'(x) INDETERMINATE FORMS 139 7. If hm (*) = land hm ^(x) = «>, prove that x = a x = a jti^ri-* where m = (x — a) ^ (x) . ic = a Apply this to find the limits of (** -?)* and (* ^J"*^ tends to zero. 2sinh£— 2£\x 2 w hen a; 3 8. If im (£) =oo, prove, by Cauchy's theorem, that lim (cc + l) Hence prove that x = TO {(*) } * = ^ = ^ T ^ ' • 9. Find _ PL when ?/ = — — and 6 = cos -1 (1—x). 6=z0dx 2 * sine v ' 10. In the curve y = 15 ( - . Y prove that the radius of curvature at \sin 2 a; x 1 ) the point (0, 5) is \. 11. Define an infinitesimal. If a; and /(#) are infinitesimals, what is meant by the statement that f(x) is of order m with respect to x ? Find the orders of log (1 + x) and e r -x — cos £ with respect to #. [Math. Tripos, 1906.] 12. If a side of a regular polygon be an infinitesimal of the first order in comparison with the radius of its inscribed circle, prove that the difference between the perimeter of the polygon and the circumference of the circle is an infinitesimal of the second order. MISCELLANEOUS NOTES. Note A. Weierstrass's Function. Letf(x)= 2 a H cos (b'Vx), where a is a positive constant less than n=0 1, b is an odd integer, and ab>l + £. Thenf(x) is Weierstrass's A function^ and possesses the property that, although continuous for every value of x, it has no differential coefficient for any value of x. I. To prove that f(x) is continuous for every value of x. Let x be a given value of x and 8 an arbitrarily small quantity greater than zero ; further, choose a value m of n such that a m Then 71=00 h . 2 a n <-,i.e., . ~-< v n=m O L-—a O f( x )—f( x o)= 2 a n cos (b n 7rx)+ 2 a 11 cos (b n wx) H=0 |a|| n=m— 1 n=ao — 2 a H cos (b n 7rx )— 2 a" cos (b"rrx ) n=0 n=m n=m-l S = 2 a n {cos(b n Trx)-cos(b n 7rx )}+2t): . . »=0 (1)- where Now cos hence <1. (ftVaO-cos (^ )=2sin^rf sin h% <**+*\ A A COS (&"7T^) — cos (b n 7rx ) "^° ) ) . Since C OS (& n 7rff,)--COS (& n ffa?o ) ==; _o • b n ir(Xi +% ) sin 2 #i— #o = —b n 7r sin sin ft w ir(si -s it ) & w 7r(a? T +a?o ) 2_ 2 Z? n 7r(or 1 — ff ) ~ 2 and since sin ^(a?!— a? ) b n 7c(xi—x ) ~2~ <1, " = ^" 1 n f cos ( 6 w wa?i)— cos (Wa? ) 1 ?i=0 1 Xy—Xq J <;i -1 (a6)"<-vp »=o ab— 1 Second, consider "aV~ ( ^ (6— ^,)-cos (6'^^ ) 1 n=0 I OSi—Xq 1 \Jcos(6^ )-coB(^a?o) 1 I < ;i"W< 7r( ^ ) COS (fr^'Trff])— COS (6 ;t+w 7ra? ) Since b is an odd integer, COS (fr l+ ™7r#i) = COS {&"(a w — 1)tt} = -( — lY m and COS (6 n+w 7T^ ) = COS (^ w « m T + & w 7ra? w+1 ) = (-l) a m COS (& n ^ wtl )' Hence 7^ | COS (^"VflQ-COS (^^^ q) 1 »=0 l iCl — # J n=0 I 1 + ^m+i J Now all the terms of the above series are positive ; and the first term <§, since cos (^x m+1 ) > and J < (1 +# m+1 ) < }• Therefore "7 *+m J cos ( ^"VaQ-cos (fr t+m 7nr ) 1 £* 1 ^^~~ - )=(-ir(airx§, (4), where )?>1. MISCELLANEOUS NOTES 143 From (3) and (4), we have /o — . \ I Vi <1. Similarly it can be proved that # 2 — #o , \3 ab—lj Now - + ~ L — and - + ^~~ are both greater than zero, for 3 ab— 1 3 a6— 1 l>-^r- Therefore /feWfe) and &dbfisi have always opposite signs ; and their numerical values increase beyond all limit as m is made greater and greater. Hence f'(x ) is non- existent; for, if f'(x ) existed, both .felr^fe) and .feHfe) fl/]_ Xq X>2 — Xq would have tended to one and the same limit. Example. /(#)= 2 ^ is a particular case of Weier- 1 13 strass's function. For, taking a— - and b =13, we have ab=-~ A A o which is greater than 1-f ~k • Note B. Rolle's Theorem and Taylor's Theorem. [This Note is supplementary to Arts. 67 and 69.] 1. Proof of Rolle's Theorem. Since f(x) is continuous for every value of x in the domain (a, b), it follows from a theorem of Weierstrasst that there exist at least two values x u x 2 of x such that /fc r )>/(*),/fo)* according as x is a or different from a. Then it can be easily shown that / (r) (a)=0 for every value of r. Therefore r-o r ! _ l while/(a + 7&)=e _/t ' which is not 0. II. A sufficient condition which is not always necessary. It is sufficient for the validity of Taylor s theorem that there exist a finite constant A such that, for every value of r, r\x) ft) «\ 146 DIFFERENTIAL CALCULUS p having every positive integral value and 6 having every value satisfying the condition < < 1. This condition was first given by Professor A. Pringsheim of Munich in 1893. It was also obtained independently by Professor E. Pascal of Pavia. The proof of the necessity and sufficiency of this condition is beyond the scope of the present volume. Note C. Partial Differentiation. 1. Introductory. As a knowledge of partial differentiation is necessary for understanding even the elementary applications of Differential Calculus to Physics, we propose to define and exemplify partial differentiation in this note. A fuller treatment of this subject will be given in Vol. II. 2. Definitions. Let/(#, y) denote a quantity which has a single and definite value for every pair x, y satisfying the conditions a © v > e x + e y e x + e y J 3. If /(#, y)=ax + by, find — ( ^ J, i.e., ^L Here -^/=& ; ex \cy/ Ixey ty hence £sjrl {b)s * Q ' 4. If /(#, y)=e my cos m#, find — ( * J, i.e., -^ 2 . [Answer : ( X \@X / ( X —m 2 e my cos mx.] 5. If w=(t>(x + ay) + ty(x—ay), prove that - 2 =a 2 — 2 . 1 £2/ £2/ £2/ 6. If f(x, y, ^)=v ^+^+^ ' prove that fo« + ^ + ^~ ' l2 ANSWERS TO THE EXAMPLES. (<4 4. |, fc J, etc. 2. 2a\ 5. cxd. Chaptee I. Art. 2. Art. 4. Art. 5. Page 6. 6. The points are given by a?=?^7^, where n is any integer. 10. For a?=l, 2/ is 1 and continuous; for #= — 1, # is and discontinuous. 11. No. 12. The differential coefficient is existent in the first and the second cases, and non-existent in the third and the fourth. Chapter II. Art. 9. 1. 1, bx\ \x~\ -7z" 8 , -£ar% 2* log, 2,- (^) l log, 2, e\ 2e 2 *, 10* log, 10. Art. 11. 1. 2 cos 2x. Page 22. 1 P. 2. nix + aY 1 . 3. |ar*. 4. _JL_ . 5. cosh x. 6. sinh x. 7. . 8. sin # + £ cos x. 2^/x 9. e Bini cos x. 10. 2# sec x 2 tan a; 2 . _2 1-z 2, ANSWERS TO THE EXAMPLES 149 12. cot x. 13, 2 cosec 2x. 14. cos log re. x ° 15. sin log x 2 . X 16. a 17 * a 2 +a; 2 (1 + rc*) tan -1 a;' 19. -coW. 20. tan|. Chapter III. Art. 26. 3. l-f2 r log 2, cos x- -10 sin x, 2x, + e r , X 3 -f sin a; 1 3 cos* re' >/i— x z' 4. x l0 -l ~ 1 1 • x + 1 ' Art. 28. 3. re n ~V(w + re), re 2 cos re + 2rc sin re, log (ex), x 3 (x sec 2 re + 4 tan re), 2 r (- + log 2 . log A sin x tan" 1 rc + rc cos x tan" 1 ^+^?-? \# / 1+nr' e x (x- log re + 2rc log re + re), _ °. ~ . sec x + —==. + x l log x . sec x tan a?. 'As/X sf x 4. (l-_rc)(l--re-4rc 2 ). Art. 29. 2. 1 2a a 2 ( a 2 -rc 2 ) 1 e *( l-rc)-l (rr+a) 2 ' (z+a) 2 ' (a 2 +re 2 ) 2 ' cosftV (^-l) 2 ' e x ( 2x— 1) — \ x sec 2 a; —tan re rs + 1 + ffcotrs 3. 4, 16. 4. f x I **-~(P+M***> I . I (sin re + cos re) 2 J (s Art. 32. 5. -12x(a 2 -a; 2 ) s ( *»(«:+&)-*, -2^-", J^L , ^~» 2vtan« 2v^« -2r sin ** -tan r sinh x 2ab 2x 2x S mx, tanz,-^, -j—-, __, 6 - (« 2 Sr (JS? (Sfp- »"-••***.*».-* 150 DIFFERENTIAL CALCULUS _2x__ Jjp sec 2 s/x 1 . _ n _u^2\2 e > A ,— n — =^ 9 /-o — n> sec J log sin e^ . cot &**. (± + x ) 4ls/x s/t&n^x V# 2 — 1 — x A e^ . jVi, 4; log tan- 1 ^ + ( 1 + a; 2) tan -i^ cot a sin * . a 8i »* cos *. log a. 8. (sin a?)* (log sin x + x cot #), af* . x 1l ~ l log (#£*), e 6 * . e x , (/y>\ tlX PIT - I log — , (tan #) 8in x (sec x H- cos x log tan x) -h (sin #) 8in ■ cos x log (e sin x), (sin" 1 #)*( - _ . — — + log sin" 1 x ) . \V 1— x 2 sin -1 x I Page 30. L ^^^\{™-~n)x + mb--na}. \X-T 0) o s/a(s/x— s/a) 2s/xs/a-\-x{s/a-\-sJx) 2 a; 3 \ \/a 4 —xv 4. ^(aH^i 5. — - 1 . n 1— sin # 6. tan 4 x. 7. w 3 # n_1 sin M_1 (nx n ) cos (wo;"). 8. i(cos 2x + 2 cos 4a;— 3 cos 6x). 10 - »• u 35? 12 - 2 ^'- 13. log V"cotl;_ 2 tan" 1 g^ 14 __ y/6 2 — a 2 cosh a; sin 2a; &+acos# # 15. v^ 2 -^ 2 . 16. w(l + V cos { (m-1) tan" 1 £ ) . 2(a + &cos#) I J 17. -,-— . 18, \ x log a? re log x log 2 ^ . . . log" x # 19. «j^ . x x I - + loga;4-(log£) 2 1. 20. (x x j . x log (ez 2 ). ANSWEBS TO THE EXAMPLES 151 21. _?^ log a. a" x 2 nn cos x oq # sk 1 " 1 x A'sina? vl — sin # -s/1— x 2 24. £2l* 25. -iU 26. X (1-3*)* 1-z 6 ' 1 + z 4 27 3# + a 23 # + * N/(^ 2 + a^) 2 — bx xs/x 2 —! ■ 29. (cos x) COBhr ( — cosh x tan # + sinh x log cos #) + (cosh x) COBX ( cosaj -Srf- siniclo § cosha; )- 31. -/(«) { Wan «•' . log r^+ (1+ °Xl+2z) } ■ »■ Tib- 3s - -n'l^^ 15 ^ 3 }- 37 1 39 Qr-l)*(72a 2 + 152a-149 ). (l + x 2n )^(l+ x 2n^- x 2 I ' 10(^ + 2)1(4^-3)1 40. f(x) - logsin# + cot#.log#— x=+a. Art. 44. 3. (i) #=±1, a=l, a>=2. (ii) ?/=0, y«a?, 2/=— sp, x=0. (iii) ?/=0, y=±», y=±ix, x=0. 4. (i) y=0, x=0. (ii) #=0. (iii) y=±<*> 154 DIFFERENTIAL CALCULUS Art. 45. 2. r cos 6-=a. 3. r sin Id ^1= sec nnr. where wis \ n J n any integer. Page 60. 3. (i) #=a, x=0. (ii) y=±a, y=±.x + a. (iii) 2/ = — x + a, y=—x + 2a, x + 2^ + 14^=0, # + 32/= 13a. 4. (i) rsine=|e=|. (ii) r cos 0=a. (iii) 0= — , where m is any integer. 7. a* + &i=0 f ay + a 1 ==0 l aa? + ^=^±*i^. ab 9-m 4- 1 14. (i) x= — i^~ w > where m is any integer. (ii) y=0. 18. 2/=— 6a, x=2^2a f #= — 2s/ 2a. Above the first in the first quadrant and below it in the fourth ; to the left of the second in the first quadrant and to its right in the fourth ; to the right of the third in the second quadrant and to its left in the third. 22. rasa. Chapter VI. Art. 46. a =a cos t (cos 2 £ + 3 sin 2 t), /3 =a sin t (sin 2 t + 3 cos 2 t). Art. 47. (i) m* „ fe^T „ (^M. (iv) <,,cc*. (v) 3(«x,,).. (vi) -* cos 1-71* Art. 54. 3. (i) y 2 ~x 2 . (ii) #±?/=:±a. (iii) #2/=±*| 4. *> + »»= g)'. 5. (i) r 2 =a 2 cos 2 tf + 6 2 sin 2 tf. (ii) x + a=0. (iii) r» +1 =a n+1 cos i rc-f 1 Page 82. 1. A conic having the two points as foci. 4. (x 2 + y 2 ) 2 =±a 2 (x 2 -y 2 ). 7. {ax)* + {byY={a 2 -b 2 )K 8. p 2 2 _w^-w¥ Chapter IX. Art. 63. cos X (1— sin x) 2 ' (iii) 2 cos (# 2 )- 4:x 2 sin (x 2 ). 4. (i) 24z. (ii) (iii) 2 ( (iv) - (i+z 2 ) 2 ANSWERS TO THE EXAMPLES 157 Art. 64. 4 m (-in bc-ad)*- 1 n l m (_i)»-i a *(w— 1) ! * U (cx + d) n + l ' K } {ax + by (iii) \ { 3" cos (s*+^) + 3 cos [*+y) } " (iv) e ~ | 3(a 2 + 6 2 )* sin (te+* tan" 1 £) -(a 2 + 9& 2 )* sin fete* » tan" 1 -\ j . (v) |{«m ( a?+ ^) + JMin(ar+^) + 5* sin (5* + ^)- 9" sin ($z + y) } ' (vi) -*J22 +1 sin (x+t)+1& sin (3x+ntonr l 3\ — (26)^ sin (5x + n tan" 1 5] j. Art. 65. 2. (i) a) 3 a" sin (as 4- n *\ + 3w a 2 a""* 1 sin (ax + ^p*j 3w(n-l)s a*" 2 sin(a#+ ~M + w(n— l)(w— 2)a»- s (ii) a" +2 s 2 e ar . (iii) «~ { a»x«> + i«r»H + *(*=: lH^pl)a"-V»^ + w(w--l)(M-2)?^(m-l)(m-2) aW _3^ w _3 + 1 (iv) m(m— 1) . . . (w— ?i + l)(a + ^) w_w 6"< log (a + Z>s) + __ W C L _ 1 ! n C<> 4 2 ! "C 3 m — n + 1 (in— n + l)(m— 7i + 2) (w-?i + l)(w-?i-f2)(m-w + 3) xsin 158 DIFFERENTIAL CALCULUS Art. 66. 3 - » (-^^{(^F'WI * 9 V i VlpTP 1 (z-2)" +1 ^>-3) w+1 J rw f-iv»*t./ n+1 + 6 1 — + 4 x 1 7. or l 2 . 3 2 . 5 2 . . . (w— 2) 2 according as n is even or odd. 8. or m(l 2 — ra 2 )(3 2 — m 2 ) . . . {(w — 2) 2 — ra 2 } according as w is even or odd. Page 107. 15. m(m 2 -l 2 )(m 2 -3 2 ) . . . { m 2 -(w-2) 2 } or m 2 (m 2 —- 2 2 )(m 2 — 4 2 ) . . . {m 2 — (n— 2) 2 } according as n is odd or even. 16. or (—1) 2 m(m— l)(m— 2) . . . (ra— ?z + l) according as n is even or odd. Chapter X. Art. 67. 3. (i) Applicable, (iv) Not applicable. (ii) Not applicable. Art. 69. (iii) Applicable. e. o) *-*;<- /y.2r-l • • • +(- i )- i 2 :_i+ • . . . (ii) l+ax+^jfe* + . tun .,1« , _ L 1.3^ L n\ \ a) ,1. 3... (2r-3) a; 2 ;" 1 + . . . 2.4... (2r-2) 2r-l (vi) 2 {fl + 4T 4 - AN8WEES TO THE EXAMPLES 159 . r A mx m(m 2 -l 2 ) 3 , ^K-l 2 )(w 2 -3 2 ) 5 ■ r lV-i ^( ^-l 2 )(^ 2 -3 2 ) . . . (m 2 -(2r-3 ) 2 } 2M / \ i , ,i 2 9, m(ra 2 — V) 8 , m 2 (ra 2 — 2 2 ) , , (v) l + w#+~ ^ + of + — Sn x + ■ • • , m(m 2 -i 2 )(m 2 --3 2 ) . . . |m 2 -(2r- 3) 2 } ;_, ■ (2r-l)l ~ , m'(ro'-2») . . . {m 2 -(2r-2)} 2 2r , + " (2r)! "* + 2 2 . 4 2 ,. ITT 2 2 .4 2 .6 2 . . . (2r-2) 2 1 + ~(2r)7~ ' * "/ Page 119. 4. (i) The remainder tends to 0, whatever x may be. (ii) The remainder tends to 0, whatever x may be. (iii) The remainder tends to 0, when x satisfies the condi- tion —1 /2. 6. Sum of the semi-axes. 8. 4a. 9. Maximum area=4a 2 sin a cos 3 «, where a is the radius of the circle and 2a the given angle. 11. Half the area of the rectangle circumscribing the ellipse. 13. Each is -4- 10 ANSWEES TO THE EXAMPLES 161 Chapter XII. Art. 74. 8. (i) 2 3 (ii) 1 ( lii) log |. (iv) 1. (v) -2. (vi) 1. 4. 0. Art. 75. 7. (i) 0. (ii) 1. (m) -. 7T (iv) i (v) (ix) i 3* e 2 . (vi) W t eK Page (vii) 1. (xi) 1. 138. (viii) (xii) 1. 1. 1. (i) 7T 2 2e' (ii) -1. (iii) ?• (iv) 1. 2. (i) eK (ii) 6tV, (iii) el. (iv) 4. M Om+n' (vi) -h 4. a. 7. The required limits are e* and e^ o respe stiv< 9. 4 IK* n. There quired or ders are 1 and 2 respe ctiv PRINTED BY srornswooDE and co. ltd., London COLCHESTER AND ETON 14 DAY USE RETURN TO DESK FROM WHICH BORROWED LOAN DEPT. This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. 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