SMP Further Mathematics Series Extensions of Calculus COLIN GOLDSMITH and DAVID NELSON ~·.' l • •I T~t rflht tit< UltiPtt'l'sily 11/ Omtbthlt~ tQ J"lnt ottd ;t/1 a~J--•r-. MWJTIMW/by VIII in JjJ4. H~ary 71tf Unl"tfsily 1tos ,mt~ "'"''"uou.sly t»fd pw61fshtd Jlita IJU. ' CAMBRIDGE UNIVERS ITY PRESS Cambridge New York Port Chester Melbourne Sydney Contents .. Preface page vn 1 Hyperbolic functions 1 I Introduction I 2 Basic properties 2 3 The inverse hyperbolic functions 7 2 De Moivre's theorem 12 I Introduction 12 2 Polynomial equations 13 3 De Moivre's theorem 16 3 Limits 20 1 Introduction 20 2 Functions n j{n) with domain N 21 ~ 3 Functions x ,..... j{x) with domain !R 26 4 Some standard results 28 5 Continuity and differentiability 31 4 Infinite series 37 1 An example 37 2 Sums of finite series 39 3 Infinite series 40 4 Taylor series 44 5 Derivation of power series by integration 47 6 Applications to the evaluation of functions 48 5 Euler's relation 55 1 Limits of complex sequences 55 2 Series of complex terms 56 3 The exponential function 57 4 Euler's relation 59 5 The link between the circular and hyperbolic functions 61 6 Reduction formulae v 65 1 The basic idea 65 2 Reduction formulae with two parameters 67 ••• 111 iv Contents 7 Second-order differential equations 71 ·1 Foundations 71 2 Second-order equations with constant coefficients 77 3 Simple harmonic motion 79 4 Control mechanisms· 83 5 Forced oscillations 84 6 Formulation of differential equations 86 8 Differential geometry 91 1 Arc length 91 2 Polar coordinates 94 3 Areas of polar curves 96 4 Radius of curvature 100 5 Signs . 104 9 Improper integrals and limits of sequences 108 1 Improper integrals · 108 2 Convergence of sequences 112 3 Proof 116 4 Second-order convergence 117 10 Differential equations and substitution 124 1 Introduction 124 2 Critical damping 126 3 Resonance 127 4 Examples summarised 129 5 Differential equations associated with electrical circuits 130 11 Complex-number geometry 136 1 Lines and circles 136 2 Further loci 139 3 The transformation z 1/z 143 ~ 4 Geometry using complex numbers 146 12 Partial·differentiation I 51 1 Tangent planes I 52 2 Tangents and tangent planes; increment notation 154 3 Function notation and Taylor approximations 156 13 Double integrals 165 1 Volumes 165 2 Polar coordinates 170 3 Applications to probability 172 Contents v 14 transformations of the complex plane 177 Confor~al 1 Transformation diagrams 177 2 Orthogonal families 180 3 The Newton- Raphson process 186 15 Jacobians 192 1 Local distortions 192 2 The Jacobian matrix 194 3 Functions mapping W to !Rn 197 16 Triple integrals and substitution 202 1 Applications to mechanics 202 2 Substitution in double integrals 207 3 Polar coordinates in three dimensions 212 4 Two important applications 215 Answers, hints and comments on the exercises 222 Preface In the preparation of this book, I have made extensive use of a partial draft written by David Nelson. The original SMP book with the same title has also had an important influence. Thanks are also due to many others, pupils as well as teachers, who have provided valuable criticism and constructive ideas. Special mention must be made of Terry Hawkes, who has worked through all the exercise material, and of Sam Boardman, Michael Hall and Donald Miller for their contribution to the development of the book in the early stages. The book interweaves a number of mathematical threads which are more usually developed separately. As the title suggests, a basic knowledge of calculus, such as is included in any British sixth-form mathematics course, is assumed (though this course could be started well before the single-subject calculus is completed). Several of the chapters in the first half of the book develop this work on functions of a single real variable, introducing hyperbolic functions and reduction formulae and then extending techniques for solving differential equations. The second major strand involves complex numbers, with Euler's relation providing useful connections with the two-dimensional calculus. Then the geometry of the complex plane prepares the ground for the third ingredient, the higher-dimension calculus. The treatment of partial differentiation and of double and triple integrals is kept simple, but it is hoped that the concentration on geometrical illustrations will make difficult ideas come alive. The differentiation of functions of a complex variable provides an important link and is interpreted in terms of special transformations of two-dimensional space. The fourth main theme is the one that students find the most difficult: the encouragement of greater rigour and the habit of proof. Two early chapters, on limits and infinite series, set the scene, and opportunities to develop one's expertise occur thereafter. It is believed that, even more than other aspects of mathematics, the philosophy and practice of proof only grows through experi ence. Largely for this reason, the text has been kept fairly brief, while the answers to the exercises are given in considerable detail. There are many occasions when a programmable calculator or computer will be found valuable, not all of them indicated as such explicitly. There are also plenty of opportunities for the curious reader to investigate topics which are only touched on in the text. The book is designed to cover the appropriate section of the SMP Further Mathematics syllabus, and a number of questions from past SMP A-level papers are included. The Oxford and Cambridge Schools Examination Board is •• Vll viii Preface thanked for permission to reproduce these. Much of the material here is more often found in higher education than in school courses, and the book should be found useful also in colleges and universities. Colin Goldsmith 1 Hyperbolic functions 1. INTRODUCTION . We recollect that inverse circular functions ar.e very useful in integration. Thus I 1 d . k v'( -1 + _ x x = sm x 1 2) I ~ x2 1 + dx = tan- x k. 1 We cannot at the moment integrate the similar functions 1I v' (1 + x2 and ) I v' 2 1 (x 1) , and in order to do this we introduce new functions called - hyperbolic functions, which behave in many ways like the circular functions. (The reason for their name is that they are connected with the rectangular x2 - 2 hyperbola y = 1 in much the same way as the circular functions are connected with the circle x2 + y2 = 1.) The hyperbolic .functions were first studied by the Swiss' mathematician Johann Lambert (1728-1777). We first define the functions then, in Exercise 1A, provide the opportunity to develop their simple properties. Definition 1 For all real values of x, sinh x = i(eX- e-x), Example 1 + Simplify sinh a cosh b cosh a sinh b. Solution In exponentials we have + + + ~(ea- e - a) X ~(eh e-h) i(ea e-:-D) X ~(eh- e - h) + + + = !(ea+b ea- b- e-a+b- e-a-b ea+b- ea-b e-a+b- e-a-b) = ~(2ea+h- 2e-a-h) = sinh(a +b). This is reminiscent of the addition formula for sin(a + b). 0 Exercise 1A 1 Express each of the following in terms of exponential functions and simplify: 1 2 1 Hyperbolic functions (a) 2 sinh x cosh x (b) cosh2 x + sinh2 x (c) cosh2 x- sinh2 x (d) cosh a cosh b +sinh a sinh b (e) 2 cosh2 x- I (f) I + 2 sinh2 x Comment on your results. y 2 From the graphs of e" and e-" (Figure 1), sketch the graphs of y = sinh x and y = cosh x. For each of the hyperbolic functions, state (a) whether it is odd or even, (b) whether it is periodic, and (c) the range of the function. Carefully define the inverse functions sinh- 1 x and cosh_, x and state their domains. 3 Find the derivatives of sinh x and cosh x. Are your answers consist 0 ent with the graphs from question X 2? What is the gradient of y = sinh x at the origin? Figure I 2. BASIC PROPERT IES When developing the long list of formulae for sines and cosines, we started with· the four 'addition formulae' and deduced all the others from these. The same can be done for the hyperbolic functions. First, as in Example 1 and Exercise lA, question l (d), we can prove that sinh(a +b) =sinh a cosh b +cosh a sinh b sinh(a- b) =sinh a cosh b- cosh a sinh b cosh(a +b)= cosh a cosh b +sinh a sinh b cosh(a- b)= cosh a cosh b- sinh a sinh b. Replacing both a and b by x, we then obtain sinh 2x = 2 sinh x cosh x cosh 2x = cosh2 x + sinh2 x 1 = cosh2 x - sinh2 x. We combined the formulae cos 28 = cos2 0- sin2 8 and 1 = cos2 0 + sin2 8 to give two further formulae for cos 28, one invoJving cos 8 only, the other involving sin ()only. Similarly, we find that cosh 2x = 2 cosh2 x - 1 = 1 + 2 sinh2 x. Derivatives From the definitions, we obtain 1 Hyperbolic functions 3 j{x) =sinh x => f'(x) =cosh x and g(x) =cosh x => g'(x) = sinh x. Graphs y X Figure 2 . The graph of y = sinh x has rotational symmetry about the origin because sinh x is an odd function. The graph of y = cosh x (an even function) has reflectional symmetry in they-axis. Note that for large positive x, both sinh x ! ex. and cosh x are approximately equal to It can be proved that a thin, flexible cable supported at either end and hanging freely under gravity takes up a shape similar to the graph ofy = cosh x. We call such a curve a catenary (from the Latin catena, a chain). 2.1 Inverse functions Before introducing the important inverse functions for sinh and cosh, it would be sensible to review ideas about the inverse circular functions, which may not be familiar. A calculator will give sin 0.850 = 0.751 and sin - 1 0.751 = 0.850 (to 3 in both cases). SF It also gives sin 3.910 = - 0.695, but sin-.: 1 0.695) = - 0. 768. ( - Many different numbers have the same sine, but the inverse sine button on a