Engineering Mathematics 1A/1B


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Engineering Mathematics

 

Contents

Preface xii

Section 1 Number and Algebra 1

1 Revision of fractions, decimals

and percentages 3

1.1 Fractions 3

1.2 Ratio and proportion 5

1.3 Decimals 6

1.4 Percentages 9

2 Indices, standard form and engineering

notation 11

2.1 Indices 11

2.2 Worked problems on indices 12

2.3 Further worked problems on indices 13

2.4 Standard form 15

2.5 Worked problems on standard form 15

2.6 Further worked problems on

standard form 16

2.7 Engineering notation and common

prefixes 17

3 Computer numbering systems 19

3.1 Binary numbers 19

3.2 Conversion of binary to decimal 19

3.3 Conversion of decimal to binary 20

3.4 Conversion of decimal to

binary via octal 21

3.5 Hexadecimal numbers 23

4 Calculations and evaluation of formulae 27

4.1 Errors and approximations 27

4.2 Use of calculator 29

4.3 Conversion tables and charts 31

4.4 Evaluation of formulae 32

Revision Test 1 37

5 Algebra 38

5.1 Basic operations 38

5.2 Laws of Indices 40

5.3 Brackets and factorisation 42

5.4 Fundamental laws and

precedence 44

5.5 Direct and inverse

proportionality 46

6 Further algebra 48

6.1 Polynominal division 48

6.2 The factor theorem 50

6.3 The remainder theorem 52

7 Partial fractions 54

7.1 Introduction to partial

fractions 54

7.2 Worked problems on partial

fractions with linear factors 54

7.3 Worked problems on partial

fractions with repeated linear factors 57

7.4 Worked problems on partial

fractions with quadratic factors 58

8 Simple equations 60

8.1 Expressions, equations and

identities 60

8.2 Worked problems on simple

equations 60

8.3 Further worked problems on

simple equations 62

8.4 Practical problems involving

simple equations 64

8.5 Further practical problems

involving simple equations 65

Revision Test 2 67

9 Simultaneous equations 68

9.1 Introduction to simultaneous

equations 68

9.2 Worked problems on

simultaneous equations

in two unknowns 68

9.3 Further worked problems on

simultaneous equations 70

9.4 More difficult worked

problems on simultaneous

equations 72

9.5 Practical problems involving

simultaneous equations 73

10 Transposition of formulae 77

10.1 Introduction to transposition

of formulae 77

vi Contents

10.2 Worked problems on

transposition of formulae 77

10.3 Further worked problems on

transposition of formulae 78

10.4 Harder worked problems on

transposition of formulae 80

11 Quadratic equations 83

11.1 Introduction to quadratic

equations 83

11.2 Solution of quadratic

equations by factorisation 83

11.3 Solution of quadratic

equations by ‘completing

the square’ 85

11.4 Solution of quadratic

equations by formula 87

11.5 Practical problems involving

quadratic equations 88

11.6 The solution of linear and

quadratic equations

simultaneously 90

12 Inequalities 91

12.1 Introduction in inequalities 91

12.2 Simple inequalities 91

12.3 Inequalities involving a modulus 92

12.4 Inequalities involving quotients 93

12.5 Inequalities involving square

functions 94

12.6 Quadratic inequalities 95

13 Logarithms 97

13.1 Introduction to logarithms 97

13.2 Laws of logarithms 97

13.3 Indicial equations 100

13.4 Graphs of logarithmic functions 101

Revision Test 3 102

14 Exponential functions 103

14.1 The exponential function 103

14.2 Evaluating exponential functions 103

14.3 The power series for ex 104

14.4 Graphs of exponential functions 106

14.5 Napierian logarithms 108

14.6 Evaluating Napierian logarithms 108

14.7 Laws of growth and decay 110

15 Number sequences 114

15.1 Arithmetic progressions 114

15.2 Worked problems on

arithmetic progressions 114

15.3 Further worked problems on

arithmetic progressions 115

15.4 Geometric progressions 117

15.5 Worked problems on

geometric progressions 118

15.6 Further worked problems on

geometric progressions 119

15.7 Combinations and

permutations 120

16 The binomial series 122

16.1 Pascal’s triangle 122

16.2 The binomial series 123

16.3 Worked problems on the

binomial series 123

16.4 Further worked problems on

the binomial series 125

16.5 Practical problems involving

the binomial theorem 127

17 Solving equations by iterative methods 130

17.1 Introduction to iterative methods 130

17.2 The Newton–Raphson method 130

17.3 Worked problems on the

Newton–Raphson method 131

Revision Test 4 133

Multiple choice questions on

Chapters 1–17 134

Section 2 Mensuration 139

18 Areas of plane figures 141

18.1 Mensuration 141

18.2 Properties of quadrilaterals 141

18.3 Worked problems on areas of

plane figures 142

18.4 Further worked problems on

areas of plane figures 145

18.5 Worked problems on areas of

composite figures 147

18.6 Areas of similar shapes 148

19 The circle and its properties 150

19.1 Introduction 150

19.2 Properties of circles 150

19.3 Arc length and area of a sector 152

19.4 Worked problems on arc

length and sector of a circle 153

19.5 The equation of a circle 155

Contents vii

20 Volumes and surface areas of

common solids 157

20.1 Volumes and surface areas of

regular solids 157

20.2 Worked problems on volumes

and surface areas of regular solids 157

20.3 Further worked problems on

volumes and surface areas of

regular solids 160

20.4 Volumes and surface areas of

frusta of pyramids and cones 164

20.5 The frustum and zone of

a sphere 167

20.6 Prismoidal rule 170

20.7 Volumes of similar shapes 172

21 Irregular areas and volumes and

mean values of waveforms 174

21.1 Area of irregular figures 174

21.2 Volumes of irregular solids 176

21.3 The mean or average value of

a waveform 177

Revision Test 5 182

Section 3 Trigonometry 185

22 Introduction to trigonometry 187

22.1 Trigonometry 187

22.2 The theorem of Pythagoras 187

22.3 Trigonometric ratios of acute angles 188

22.4 Fractional and surd forms of

trigonometric ratios 190

22.5 Solution of right-angled triangles 191

22.6 Angle of elevation and depression 193

22.7 Evaluating trigonometric

ratios of any angles 195

22.8 Trigonometric approximations

for small angles 197

23 Trigonometric waveforms 199

23.1 Graphs of trigonometric functions 199

23.2 Angles of any magnitude 199

23.3 The production of a sine and

cosine wave 202

23.4 Sine and cosine curves 202

23.5 Sinusoidal form A sin(ωt ±α) 206

23.6 Waveform harmonics 209

24 Cartesian and polar co-ordinates 211

24.1 Introduction 211

24.2 Changing from Cartesian into

polar co-ordinates 211

24.3 Changing from polar into

Cartesian co-ordinates 213

24.4 Use of R→P and P→R

functions on calculators 214

Revision Test 6 215

25 Triangles and some practical

applications 216

25.1 Sine and cosine rules 216

25.2 Area of any triangle 216

25.3 Worked problems on the solution

of triangles and their areas 216

25.4 Further worked problems on

the solution of triangles and

their areas 218

25.5 Practical situations involving

trigonometry 220

25.6 Further practical situations

involving trigonometry 222

26 Trigonometric identities and equations 225

26.1 Trigonometric identities 225

26.2 Worked problems on

trigonometric identities 225

26.3 Trigonometric equations 226

26.4 Worked problems (i) on

trigonometric equations 227

26.5 Worked problems (ii) on

trigonometric equations 228

26.6 Worked problems (iii) on

trigonometric equations 229

26.7 Worked problems (iv) on

trigonometric equations 229

27 Compound angles 231

27.1 Compound angle formulae 231

27.2 Conversion of a sin ωt +b cos ωt

into R sin(ωt +α) 233

27.3 Double angles 236

27.4 Changing products of sines

and cosines into sums or

differences 238

27.5 Changing sums or differences

of sines and cosines into

products 239

Revision Test 7 241

Multiple choice questions on

Chapters 18–27 242

viii Contents

Section 4 Graphs 247

28 Straight line graphs 249

28.1 Introduction to graphs 249

28.2 The straight line graph 249

28.3 Practical problems involving

straight line graphs 255

29 Reduction of non-linear laws to

linear form 261

29.1 Determination of law 261

29.2 Determination of law

involving logarithms 264

30 Graphs with logarithmic scales 269

30.1 Logarithmic scales 269

30.2 Graphs of the form y=axn 269

30.3 Graphs of the form y=abx 272

30.4 Graphs of the form y=aekx 273

31 Graphical solution of equations 276

31.1 Graphical solution of

simultaneous equations 276

31.2 Graphical solution of

quadratic equations 277

31.3 Graphical solution of linear

and quadratic equations

simultaneously 281

31.4 Graphical solution of cubic

equations 282

32 Functions and their curves 284

32.1 Standard curves 284

32.2 Simple transformations 286

32.3 Periodic functions 291

32.4 Continuous and

discontinuous functions 291

32.5 Even and odd functions 291

32.6 Inverse functions 293

Revision Test 8 297

Section 5 Vectors 299

33 Vectors 301

33.1 Introduction 301

33.2 Vector addition 301

33.3 Resolution of vectors 302

33.4 Vector subtraction 305

34 Combination of waveforms 307

34.1 Combination of two periodic

functions 307

34.2 Plotting periodic functions 307

34.3 Determining resultant

phasors by calculation 308

Section 6 Complex Numbers 311

35 Complex numbers 313

35.1 Cartesian complex numbers 313

35.2 The Argand diagram 314

35.3 Addition and subtraction of

complex numbers 314

35.4 Multiplication and division of

complex numbers 315

35.5 Complex equations 317

35.6 The polar form of a complex

number 318

35.7 Multiplication and division in

polar form 320

35.8 Applications of complex

numbers 321

36 De Moivre’s theorem 325

36.1 Introduction 325

36.2 Powers of complex numbers 325

36.3 Roots of complex numbers 326

Revision Test 9 329

Section 7 Statistics 331

37 Presentation of statistical data 333

37.1 Some statistical terminology 333

37.2 Presentation of ungrouped data 334

37.3 Presentation of grouped data 338

38 Measures of central tendency and

dispersion 345

38.1 Measures of central tendency 345

38.2 Mean, median and mode for

discrete data 345

38.3 Mean, median and mode for

grouped data 346

38.4 Standard deviation 348

38.5 Quartiles, deciles and

percentiles 350

39 Probability 352

39.1 Introduction to probability 352

39.2 Laws of probability 353

39.3 Worked problems on

probability 353

39.4 Further worked problems on

probability 355

Contents ix

39.5 Permutations and

combinations 357

Revision Test 10 359

40 The binomial and Poisson distribution 360

40.1 The binomial distribution 360

40.2 The Poisson distribution 363

41 The normal distribution 366

41.1 Introduction to the normal

distribution 366

41.2 Testing for a normal

distribution 371

Revision Test 11 374

Multiple choice questions on

Chapters 28–41 375

Section 8 Differential Calculus 381

42 Introduction to differentiation 383

42.1 Introduction to calculus 383

42.2 Functional notation 383

42.3 The gradient of a curve 384

42.4 Differentiation from first

principles 385

42.5 Differentiation of y=axn by

the general rule 387

42.6 Differentiation of sine and

cosine functions 388

42.7 Differentiation of eax and ln ax 390

43 Methods of differentiation 392

43.1 Differentiation of common

functions 392

43.2 Differentiation of a product 394

43.3 Differentiation of a quotient 395

43.4 Function of a function 397

43.5 Successive differentiation 398

44 Some applications of differentiation 400

44.1 Rates of change 400

44.2 Velocity and acceleration 401

44.3 Turning points 404

44.4 Practical problems involving

maximum and minimum

values 408

44.5 Tangents and normals 411

44.6 Small changes 412

Revision Test 12 415

45 Differentiation of parametric

equations 416

45.1 Introduction to parametric

equations 416

45.2 Some common parametric

equations 416

45.3 Differentiation in parameters 417

45.4 Further worked problems on

differentiation of parametric

equations 418

46 Differentiation of implicit functions 421

46.1 Implicit functions 421

46.2 Differentiating implicit

functions 421

46.3 Differentiating implicit

functions containing

products and quotients 422

46.4 Further implicit

differentiation 423

47 Logarithmic differentiation 426

47.1 Introduction to logarithmic

differentiation 426

47.2 Laws of logarithms 426

47.3 Differentiation of logarithmic

functions 426

47.4 Differentiation of [f (x)]x 429

Revision Test 13 431

Section 9 Integral Calculus 433

48 Standard integration 435

48.1 The process of integration 435

48.2 The general solution of

integrals of the form axn 435

48.3 Standard integrals 436

48.4 Definite integrals 439

49 Integration using algebraic

substitutions 442

49.1 Introduction 442

49.2 Algebraic substitutions 442

49.3 Worked problems on

integration using algebraic

substitutions 442

x Contents

49.4 Further worked problems on

integration using algebraic

substitutions 444

49.5 Change of limits 444

50 Integration using trigonometric

substitutions 447

50.1 Introduction 447

50.2 Worked problems on

integration of sin2 x, cos2 x,

tan2 x and cot2 x 447

50.3 Worked problems on powers

of sines and cosines 449

50.4 Worked problems on integration of

products of sines and cosines 450

50.5 Worked problems on integration

using the sin θ substitution 451

50.6 Worked problems on integration

using the tan θ substitution 453

Revision Test 14 454

51 Integration using partial fractions 455

51.1 Introduction 455

51.2 Worked problems on

integration using partial

fractions with linear factors 455

51.3 Worked problems on integration

using partial fractions with

repeated linear factors 456

51.4 Worked problems on integration

using partial fractions with

quadratic factors 457

52 The t =tan

θ

2

substitution 460

52.1 Introduction 460

52.2 Worked problems on the

t =tan

θ

2

substitution 460

52.3 Further worked problems on

the t =tan

θ

2

substitution 462

53 Integration by parts 464

53.1 Introduction 464

53.2 Worked problems on

integration by parts 464

53.3 Further worked problems on

integration by parts 466

54 Numerical integration 469

54.1 Introduction 469

54.2 The trapezoidal rule 469

54.3 The mid-ordinate rule 471

54.4 Simpson’s rule 473

Revision Test 15 477

55 Areas under and between curves 478

55.1 Area under a curve 478

55.2 Worked problems on the area

under a curve 479

55.3 Further worked problems on

the area under a curve 482

55.4 The area between curves 484

56 Mean and root mean square values 487

56.1 Mean or average values 487

56.2 Root mean square values 489

57 Volumes of solids of revolution 491

57.1 Introduction 491

57.2 Worked problems on volumes

of solids of revolution 492

57.3 Further worked problems on

volumes of solids of

revolution 493

58 Centroids of simple shapes 496

58.1 Centroids 496

58.2 The first moment of area 496

58.3 Centroid of area between a

curve and the x-axis 496

58.4 Centroid of area between a

curve and the y-axis 497

58.5 Worked problems on

centroids of simple shapes 497

58.6 Further worked problems

on centroids of simple shapes 498

58.7 Theorem of Pappus 501

59 Second moments of area 505

59.1 Second moments of area and

radius of gyration 505

59.2 Second moment of area of

regular sections 505

59.3 Parallel axis theorem 506

59.4 Perpendicular axis theorem 506

59.5 Summary of derived results 506

59.6 Worked problems on second

moments of area of regular

sections 507

59.7 Worked problems on second

moments of area of

composite areas 510

Revision Test 16 512

Section 10 Further Number and

Algebra 513

60 Boolean algebra and logic circuits 515

60.1 Boolean algebra and

switching circuits 515

60.2 Simplifying Boolean

expressions 520

60.3 Laws and rules of Boolean

algebra 520

60.4 De Morgan’s laws 522

60.5 Karnaugh maps 523

60.6 Logic circuits 528

60.7 Universal logic gates 532

61 The theory of matrices and

determinants 536

61.1 Matrix notation 536

61.2 Addition, subtraction and

multiplication of matrices 536

61.3 The unit matrix 540

61.4 The determinant of a 2 by 2 matrix 540

61.5 The inverse or reciprocal of a

2 by 2 matrix 541

61.6 The determinant of a 3 by 3 matrix 542

61.7 The inverse or reciprocal of a

3 by 3 matrix 544

62 The solution of simultaneous

equations by matrices and

determinants 546

62.1 Solution of simultaneous

equations by matrices 546

62.2 Solution of simultaneous

equations by determinants 548

62.3 Solution of simultaneous

equations using Cramers rule 552

Revision Test 17 553

Section 11 Differential Equations 555

63 Introduction to differential

equations 557

63.1 Family of curves 557

63.2 Differential equations 558

63.3 The solution of equations of

the form

dy

dx

=f (x) 558

63.4 The solution of equations of

the form

dy

dx

=f (y) 560

63.5 The solution of equations of

the form

dy

dx

=f (x) · f (y) 562

Revision Test 18 565

Multiple choice questions on

Chapters 42–63 566

Answers to multiple choice questions 570

Index 571