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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