Calculus Differential and integral là bộ sách hiếm về giải tích cao cấp dùng cho sinh viên đại học tham khảo bằng tiếng Anh. Dùng cho giáo sinh sư phạm toán năm nhì. Dùng cho giảng viên Đại học tham khảo thêm.
raws Q A 30 CojpgM COPHRIGHT DSPOSm DIFFERENTIAL AND INTEGRAL CALCULUS WITH EXAMPLES AND APPLICATIONS BY GEORGE OSBORNE, A WALKER PROFESSOR OF MATHEMATICS IN S.B THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY REVISED EDITION BOSTON, D C HEATH & U.S.A CO., 1906 PUBLISHERS ao2> a A UBRARY of CONGRESS Two Copies Received JAN 111907 A Cooyriarht Entry Kioto L It) iSS o^ XXc„ COPY No B Copyright, 1891 By GEORGE A and 1906, OSBORNE PREFACE In the original work, tile author endeavored to prepare a text- book on the Calculus, based on the method of limits, that should be within the capacity of students of average mathematical ability and yet contain all that is essential to a working knowledge of the subject In the revision of the book the same object has been kept in view Most the of text been rewritten, the has demonstrations have been carefully revised, and, for the most part, new examples have been substituted for the of subjects in a There has been some rearrangement old more natural order In the Differential Calculus, illustrations of the " derivative" aave been introduced in Chapter "ion will be found, also, II., and applications of among the examples differentia- in the chapter imme- diately following Chapter VII on Series, is entirely new In the Integral Calculus, immediately after the integration of standard forms, Chapter XXI has been added, containing simple applications of integration In both the Differential and Integral Calculus, examples illustrat- ing applications to Mechanics and Physics will be found, especially in Chapter X of the Differential Calculus, on and in ter has Maxima and Minima, Chapter XXXII of the Integral Calculus been prepared by my The latter chap- colleague, Assistant Professor N It George, Jr The author also acknowledges his special obligation to his col- leagues, Professor H W Tyler and important suggestions and criticisms Professor F S Woods, for CONTENTS DIFFERENTIAL CALCULUS CHAPTER I Functions PAGES AF.T* I -7, Variables and Constants Definition and Classification of Examples Notation of Functions CHAPTER Limit 10 Definition of Limit 11 Notation of Limit 12 Special Limits (arcs 13-15 16 17-21 22 Increment, 1-5 Functions Increment 5-7 II Derivative 8 and chords, the base e) 8-10 Expression for Derivative Derivative Illustration of Derivative .11, Examples Three Meanings of Derivative Continuous Functions Discontinuous Functions CHAPTER 12 13-15 10-21 Examples 22-25 III Differentiation Algebraic Functions Examples Logarithmic and Exponential Functions Examples Trigonometric Functions Examples Inverse Trigonometric Functions Examples Relations between Certain Derivatives Examples 26-39 39-45 45-51 51-57 57-60 CHAPTER IV Successive Differentiation 57, 58 Definition and Notation 60 The nth 60 Leibnitz's Theorem Derivative Examples Examples 61 63-65 65-67 CONTENTS VI CHAPTER V Differentials Infinitesimals PAGES 61-63 Definitions of Differential 64 Formulae for Differentials 65 Infinitesimals 68-70 Examples 71-73 73,74 CHAPTER VI Implicit Functions 66 Examples Differentiation of Implicit Functions CHAPTER Series 75-77 VII Power Series Convergent and Divergent Series Positive and Negative Terms Absolute and Conditional Convergence 69-71 Tests for Convergency Examples Convergence of Power Series Examples 72, 73 Power Series 67,68 CHAPTER 78, 79 85-87 79-85 VIII Expansion of Functions 74-78 Maclaurin's Theorem Examples 88-93 Huyghens's Approximate Length of Arc 80,81 Computation by Series, by Logarithms 82 Computation of -w 83-87 Taylor's Theorem Examples 93 94-96 79 96,97 97-100 89 90-93 94 Rolle's Theorem Remainder 101 Mean Value Theorem 101-104 105 CHAPTER IX Indeterminate Forms 95 96, 97 98-100 Value 106 of Fraction as Limit Evaluation of 106-110 Examples Evaluation of g, 0- oo, oo- ex), 0°, 1", oo° Examples 110-113 INTEGRAL CALCULUS 378 EXAMPLES Find the attraction perpendicular to the wire in Example 1 when the particle is at a distance - above o Ans 21 SiM c [_V9 c- I + ] Find the attraction of a thin, straight, homogeneous wire upon a particle or mass m which is situated I and mass a distance c from one end of the wire and in its line of direction length M Ans of a m at KinM c(c Find the attraction upon a particle of mass of + l) homogeneous circular disk of radius a and at a distance c from the in its axis disk Ans Kirmpl — Vc + a J where p is the density of the disk Find the attraction due to a homogeneous right circular cylinder I and radius a upon a mass m in the axis produced of the cylinder and distant c from one end of length Ans TTKinp [2 ?+Va + c - Va + (c + 2 2 J) ] CHAPTER XXXIII INTEGRALS FOR REFERENCE 277 We give for reference a list of some of the integrals of the preceding chapters / o I J x" dx r n+l = ra-fl —X = log 05 C dx— =-tan I-— J xr -r cr a -f- a." - a r — dx = log # — a J x —a 2a x-\-a — I 2 , EXPONENTIAL INTEGRALS ax Ca x dx = log a j e x dx =e x TRIGONOMETRIC INTEGRALS dx = I j sin x I cos £ dx — cos x = sin x 379 INTEGRAL CALCULUS 380 I tan x dx = log sec x 10 I cot x dx = log sin x 11 a sec x = log (sec x + tan x) dx = logtan(|+|) 12 I cosec x dx = log (cosec x — cot x) = log tan - dx = tan 13 j sec x 14 j cosec 15 I sec x tan x 16 j cosec 17 j sin xd£ 18 cc a? dx =- a? I cot a? = sec x cot x dx = — cosec # sin2a\ /