Engineering Mathematics Programmes and Problems

K.A. Stroud formerly Principal Lecturer Department of Mathematics, Coventry University

F O U R T H E D I T I O N

MACMILLAN

Contents

Preface to the first edition xiii

Preface to the second edition xv

Preface to the third edition xvi

Preface to the fourth edition xvii

Hints on using the book xviii

Symbols used in the text xix

Useful background information xx

PART I Foundation Topics 1

Programme F.1 Number Systems 3

, Types of numbers 4 , Approximation - 'rounding off; significant flgures 4 . Number Systems: denary, binary, octal, duodecimal, hexadecimal 6 • Change of base; use of octals as intermediate conversion 15

Programme F.2 Arithmetic and Algebra 23

• Laws of arithmetic and algebra; rules of precedence 24 • Evaluation and transposition of formulae 27 • Multiplication and division of algebraic polynomials 31 • Factorisation; common factors; by grouping; quadratic functions 34

Programme F.3 Polynomials Evaluation and Factorisation 47

• Function notation 48 . Evaluation of a polynomial by 'nesting' 49 - Remainder theorem; factor theorem 50 • Factorisation of cubic and quartic expressions having linear factors 55

Programme F.4 Indices and Logarithms 61

• Rules of indices; Standard form; preferred Standard form 62 • Rules of logarithms 67 • Common logarithms; change of base 69 • Natural (Napierian) logarithms 71 ' Indicial equations 73

vi Contents

Programme F.5 Linear and Simultaneous Linear Equations 79

'Equations and identities 80 «Linear equations; simplification and Solution 80 s Simultaneous linear equations with two and three unknowns;

Solution by Substitution and by equating coefficients 83

Programme F.6 Polynomial Equations 93

, Quadratic equations; Solution by factors, completing the Square; by formula 94

• Solution of cubic equations having at least one linear factor 99 , Solution of quartic equations having at least two linear factors 103

Programme F. 7 Series 109

Sequences and series; finite and infinite series; notation 110 Arithmetic series; arithmetic means; sum of n terms 111 Geometrie series; geometric means; sum of n terms 116 Powers of first n natural numbers 118 Binomial series 122

• Convergent and divergent series; limiting values 126 • Exponential series 129

Programme F.8 Partial Fractions 135

Introduction 136 Rules of partial fractions; linear factors; repeated factors;

quadratic factors 138

Programme F.9 Differentiation 153

Slope of a straight line graph 154 Slope of a curve at a point; graphical and algebraic determination 156 Differential coefficients; Standard differential coefficients 159 Differentiation of polynomial funetions 162 Second differential coefficients 164 Differentiation of produets and quotients of two funetions 168 Functions of a funetion; the chain rule 173

Programme F.10 Integration 181

Integration the reverse of differentiation 182 Constant of Integration; indefinite integrals 182 Standard integrals 183 Functions of a linear funetion of x 185 Integration of polynomial funetions 187 Integration by partial fractions 189 Areas under curves; definite integrals 192 Integration as a summation 196

Contents vii

> PART II 205

Programme 1 Complex Numbers, Part 1 207

Introduction: The symbol j ; powers of j ; complex numbers 208 Multiplication of complex numbers 212 Equal complex numbers 217 Graphical representation of a complex number 219 Graphical addition of complex numbers 222 Polar form of a complex number 223 Exponential form of a complex number 228 Test exercise - 1 232 Further problems - 1 233

Programme 2 Complex Numbers, Part 2 235

Introduction 236 Loci problems 256 Test exercise - II 260 Further problems - II 261

Programme 3 Hyperbolic Functions 263

Introduction 264 Graphs of hyperbolic functions 267 Evaluation of hyperbolic functions 273 Inverse hyperbolic functions 274 Log. form of the inverse hyperbolic functions 277 Hyperbolic identities 279 Trig. identities and hyperbolic identities 282 Relationship between trigonometric & hyperbolic functions 283 Test exercise - III 286 Further problems - III 286

Programme 4 Determinants 289

Determinants 290 Determinants of the third order 297 Evaluation of a third-order determinant 298 Simultaneous equations in three unknowns 301 Consistency of a set of equations 310 Properties of determinants 314 Test exercise - IV 319 Further problems - IV 319

Programme 5 Matrices 323

Definitions; order; types of matrices 324 Operations 327

viii Contents

Transpose and inverse of a Square matrix 332 Solutions of sets of linear equations 341 Gaussian elimination method 343 Eigenvalues and eigenvectors 346 Revision summary 351 Test exercise - V 353 Further problems - V 354

Programme 6 Vectors 357

Introduction: scalar and vector quantities 358 Vector representation 359 Two equal vectors 359 Types of vectors 360 Addition of vectors 360 Components of a given vector 364 Components of a vector in terms of unit vectors 368 Vectors in space 370 Direction cosines 372 Scalar product of two vectors 373 Vector product of two vectors 375 Angle between two vectors 378 Direction ratios 380 Summary 381 Test exercise - VI 382 Further problems - VI 382

Programme 7 Differentiation 385

Standard differential coefficients 386 Functions of a function 388 Logarithmic differentiation 394 Implicit functions 398 Parametric equations 400 Test exercise - VII 403 Further problems - VII 404

Programme 8 Differentiation Applications, Part 1 407

Equation of a straight line 408 Centre of curvature 425 Test exercise - VIII 428 Further problems - VIII 429

Programme 9 Differentiation Applications, Part 2 431

Inverse trigonometrical functions 432 Differentiations of inverse trig. functions 435 Differential coefficient of inverse hyperbolic functions 438 Maximum and minimum values (turning points) 443 Test exercise - IX 454 Further problems - IX 454

Contents ix

Programme 10 Partial Differentiation, Part 1 457

Partial differentiation 458 Small increments 472 Test exercise - X 478 Further problems - X 478

Programme 11 Partial Differentiation, Part 2 481

Partial differentiation 482 Rates of change problems 485 Change of variables 493 Test exercise - XI 495 Further problems - XI 496

Programme 12 Curves and Curve Fitting 499

Standard curves 500 Asymptotes 510 Systematic curve sketching 514 Curve fitting 520 Method of least Squares 526 Test exercise - XII 534 Further problems - XII 534

Programme 13 Series, Part 1 537

Sequences and series 538 Arithmetic and geometric means 540 Series of powers of natural numbers 545 Infinite series: limiting values 548 Convergent and divergent series 551 Tests for convergence: absolute convergence 553 Test exercise - XIII 562 Further problems - XIII 562

Programme 14 Series, Part 2 565

Power series, Maclaurin's series 566 Standard series 574 The binomial series 575 Approximate values 579 Limiting values 580 Test exercise - XIV 590 Further problems - XIV 590

Programme 15 Integration, Part 1 593

Introduction 594 Standard integrals 594 Functions of a linear function 597 Integrals of the form Jf(x). f'(x)dr etc. 600 Integration of products - Integration by parts 604

x Contents

Integration by partial fractions 609 Integration of trigonometrical functions 615 Test exercise - XV 619 Further problems - XV 619

Programme 16 Integration, Part 2 621

Test exercise - XVI 649

Further problems - XVI 649

Programme 17 Reduction Formulae 651

Test exercise - XVII 662

Further problems - XVII 663

Programme 18 Integration Applications, Part 1 665

Parametric equations 675 Mean values 677 R.M.S. values 679 Revision summary 681 Test exercise - XVIII 682 Further problems - XVIII 682

Programme 19 Integration Applications, Part 2 685 Introduction 686 Volumes of solids of revolution 686 Centroid of a plane figure 691 Centre of gravity of a solid of revolution 695 Lengths of curves 696 Lengths of curves - parametric equations 698 Surfaces of revolution 700 Surfaces of revolution - parametric equations 702 Rules of Pappus 704 Revision summary 704 Test exercise - XIX 706 Further problems - XIX 706

Programme 20 Integration Applications, Part 3 709

Moments of inertia 710 Radius of gyration 714 Parallel axes theorem 718 Perpendicular axes theorem 722 Useful Standard results 724 Second moment of area 727 Composite figures 731 Centres of pressure 731 Depth of centre of pressure 735 Test exercise - XX 739 Further problems - XX 740

Contents xi

Programme 21 Approximate Integration

Introduction Approximate Integration Method 1 - by series Method 2 - by Simpson's rule Proof of Simpson's rule Test exercise - XXI Further problems - XXI

Programme 22 Polar Coordinates System

Introduction to polar coordinates Polar curves Standard polar curves Test exercise - XXII Further problems - XXII

Programme 23 Multiple Integrals

Summation in two directions Double integrals: triple integrals Applications Alternative notation Determination of volumes by multiple integrals Test exercise - XXIII Further problems - XXIII

Programme 24 First-Order Differential Equations

Introduction Formation of differential equations Solution of differential equations Method 1 - by direct integration Method 2 - by separating the variables Method 3 - homogeneous equations: by substituting y-vx Method 4 - linear equations: use of integrating factor Test exercise - XXIV Further problems - XXIV

Programme 25 Second-Order Differential Equations

Test exercise - XXV Further problems - XXV

Programme 26 Operator D Methods

The Operator D Inverse Operator 1/D Solution of differential equations by Operator D methods Special cases

743

744 745 746 750 759 760 761

763

764 766 768 784 785

787

788 791 795 800 806 810 810

813

814 815 818 818 819 825 831 848 848

853

877 877

879

880 883 893 905

xii Contents

Test exercise - XXVI 913

Further problems - XXVI 913

Programme 27 Statistics 915

Discrete and continuous data 916 Grouped data; class boundaries and class interval 919 Frequency and relative frequency; histograms 926 Central tendency - mean, mode and median 928 Coding 930 Dispersion - ränge, variance and Standard deviation 938 Frequency polygons and frequency curves 942 Normal distribution curve 942 Standardised normal curve 945 Test exercise - XXVII 949 Further problems - XXVII 950

Programme 28 Probability 953

Empirical and classical probability 955 Addition and multiplication laws of probability 963 Discrete and continuous probability distributions 971 Mean and Standard deviation of a probability distribution 981 Binomial and Poisson distributions 986 Normal distribution curve, Standard normal curve, areas

under the Standard normal curve 988 Test exercise - XXVIII 995 Further problems - XXVIII 996

Answers - Part I (Programmes F. 1 to F. 10) 999

Answers - Part II (Programmes 1 to 28) 1007

Index 1029