Engineering mathematics

Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
Contents......Page 6
Preface......Page 13
Section 1: Number and Algebra......Page 16
1.1 Fractions......Page 18
1.2 Ratio and proportion......Page 20
1.3 Decimals......Page 21
1.4 Percentages......Page 24
2.1 Indices......Page 26
2.2 Worked problems on indices......Page 27
2.3 Further worked problems on indices......Page 28
2.5 Worked problems on standard form......Page 30
2.6 Further worked problems on standard form......Page 31
2.7 Engineering notation and common prefixes......Page 32
3.2 Binary numbers......Page 34
3.3 Octal numbers......Page 37
3.4 Hexadecimal numbers......Page 39
4.1 Errors and approximations......Page 43
4.2 Use of calculator......Page 45
4.3 Conversion tables and charts......Page 47
4.4 Evaluation of formulae......Page 48
Revision Test 1......Page 53
5.1 Basic operations......Page 54
5.2 Laws of indices......Page 56
5.3 Brackets and factorisation......Page 58
5.4 Fundamental laws and precedence......Page 60
5.5 Direct and inverse proportionality......Page 62
6.1 Polynomial division......Page 64
6.2 The factor theorem......Page 66
6.3 The remainder theorem......Page 68
7.2 Worked problems on partial fractions with linear factors......Page 70
7.3 Worked problems on partial fractions with repeated linear factors......Page 73
7.4 Worked problems on partial fractions with quadratic factors......Page 74
8.2 Worked problems on simple equations......Page 76
8.3 Further worked problems on simple equations......Page 78
8.4 Practical problems involving simple equations......Page 80
8.5 Further practical problems involving simple equations......Page 81
Revision Test 2......Page 83
9.2 Worked problems on simultaneous equations in two unknowns......Page 84
9.3 Further worked problems on simultaneous equations......Page 86
9.4 More difficult worked problems on simultaneous equations......Page 88
9.5 Practical problems involving simultaneous equations......Page 89
10.2 Worked problems on transposition of formulae......Page 93
10.3 Further worked problems on transposition of formulae......Page 94
10.4 Harder worked problems on transposition of formulae......Page 96
11.2 Solution of quadratic equations by factorisation......Page 99
11.3 Solution of quadratic equations by ‘completing the square’......Page 101
11.4 Solution of quadratic equations by formula......Page 103
11.5 Practical problems involving quadratic equations......Page 104
11.6 The solution of linear and quadratic equations simultaneously......Page 106
12.2 Simple inequalities......Page 107
12.3 Inequalities involving a modulus......Page 108
12.4 Inequalities involving quotients......Page 109
12.5 Inequalities involving square functions......Page 110
12.6 Quadratic inequalities......Page 111
13.1 Introduction to logarithms......Page 113
13.2 Laws of logarithms......Page 115
13.3 Indicial equations......Page 117
13.4 Graphs of logarithmic functions......Page 118
Revision Test 3......Page 120
14.1 Introduction to exponential functions......Page 121
14.2 The power series for ex......Page 122
14.3 Graphs of exponential functions......Page 124
14.4 Napierian logarithms......Page 126
14.5 Laws of growth and decay......Page 128
15.2 Worked problems on arithmetic progressions......Page 132
15.3 Further worked problems on arithmetic progressions......Page 133
15.4 Geometric progressions......Page 135
15.5 Worked problems on geometric progressions......Page 136
15.6 Further worked problems on geometric progressions......Page 137
15.7 Combinations and permutations......Page 138
16.1 Pascal’s triangle......Page 140
16.3 Worked problems on the binomial series......Page 141
16.4 Further worked problems on the binomial series......Page 143
16.5 Practical problems involving the binomial theorem......Page 146
17.2 The Newton–Raphson method......Page 149
17.3 Worked problems on the Newton–Raphson method......Page 150
Revision Test 4......Page 152
Multiple choice questions on Chapters 1–17......Page 153
Section 2: Areas and Volumes......Page 158
18.2 Properties of quadrilaterals......Page 160
18.4 Worked problems on areas of common shapes......Page 161
18.5 Further worked problems on areas of plane figures......Page 164
18.6 Worked problems on areas of composite figures......Page 166
18.7 Areas of similar shapes......Page 167
19.2 Properties of circles......Page 169
19.3 Radians and degrees......Page 171
19.5 Worked problems on arc length and area of circles and sectors......Page 172
19.6 The equation of a circle......Page 175
20.2 Volumes and surface areas of regular solids......Page 177
20.3 Worked problems on volumes and surface areas of regular solids......Page 178
20.4 Further worked problems on volumes and surface areas of regular solids......Page 180
20.5 Volumes and surface areas of frusta of pyramids and cones......Page 185
20.6 The frustum and zone of a sphere......Page 188
20.7 Prismoidal rule......Page 191
20.8 Volumes of similar shapes......Page 193
21.1 Area of irregular figures......Page 194
21.2 Volumes of irregular solids......Page 196
21.3 The mean or average value of a waveform......Page 197
Revision Test 5......Page 202
Section 3: Trigonometry......Page 204
22.2 The theorem of Pythagoras......Page 206
22.3 Trigonometric ratios of acute angles......Page 207
22.4 Fractional and surd forms of trigonometric ratios......Page 209
22.5 Evaluating trigonometric ratios of any angles......Page 210
22.6 Solution of right-angled triangles......Page 214
22.7 Angle of elevation and depression......Page 216
22.8 Trigonometric approximations for small angles......Page 218
23.2 Angles of any magnitude......Page 219
23.4 Sine and cosine curves......Page 222
23.5 Sinusoidal form Asin(ωt ± α)......Page 226
23.6 Waveform harmonics......Page 229
24.2 Changing from Cartesian into polar co-ordinates......Page 231
24.3 Changing from polar into Cartesian co-ordinates......Page 233
24.4 Use of Pol/Rec functions on calculators......Page 234
Revision Test 6......Page 236
25.3 Worked problems on the solution of triangles and their areas......Page 237
25.4 Further worked problems on the solution of triangles and their areas......Page 239
25.5 Practical situations involving trigonometry......Page 241
25.6 Further practical situations involving trigonometry......Page 243
26.2 Worked problems on trigonometric identities......Page 246
26.3 Trigonometric equations......Page 247
26.4 Worked problems (i) on trigonometric equations......Page 248
26.5 Worked problems (ii) on trigonometric equations......Page 249
26.6 Worked problems (iii) on trigonometric equations......Page 250
26.7 Worked problems (iv) on trigonometric equations......Page 251
27.1 Compound angle formulae......Page 253
27.2 Conversion of a sinωt + bcosωt into Rsin(ωt + α)......Page 255
27.3 Double angles......Page 259
27.4 Changing products of sines and cosines into sums or differences......Page 260
27.5 Changing sums or differences of sines and cosines into products......Page 261
Revision Test 7......Page 263
Multiple choice questions on Chapters 18–27......Page 264
Section 4: Graphs......Page 270
28.2 The straight line graph......Page 272
28.3 Practical problems involving straight line graphs......Page 278
29.1 Determination of law......Page 284
29.2 Determination of law involving logarithms......Page 287
30.2 Graphs of the form y = axn......Page 292
30.3 Graphs of the form y = abx......Page 295
30.4 Graphs of the form y = aekx......Page 296
31.1 Graphical solution of simultaneous equations......Page 299
31.2 Graphical solution of quadratic equations......Page 300
31.3 Graphical solution of linear and quadratic equations simultaneously......Page 304
31.4 Graphical solution of cubic equations......Page 305
32.1 Standard curves......Page 307
32.2 Simple transformations......Page 309
32.5 Even and odd functions......Page 314
32.6 Inverse functions......Page 316
Revision Test 8......Page 320
Section 5: Complex Numbers......Page 322
33.1 Cartesian complex numbers......Page 324
33.3 Addition and subtraction of complex numbers......Page 325
33.4 Multiplication and division of complex numbers......Page 326
33.5 Complex equations......Page 328
33.6 The polar form of a complex number......Page 329
33.7 Multiplication and division in polar form......Page 331
33.8 Applications of complex numbers......Page 332
34.2 Powers of complex numbers......Page 336
34.3 Roots of complex numbers......Page 337
Section 6: Vectors......Page 340
35.3 Drawing a vector......Page 342
35.4 Addition of vectors by drawing......Page 343
35.5 Resolving vectors into horizontal and vertical components......Page 345
35.6 Addition of vectors by calculation......Page 346
35.7 Vector subtraction......Page 351
35.8 Relative velocity......Page 353
35.9 i, j, and k notation......Page 354
36.2 Plotting periodic functions......Page 356
36.3 Determining resultant phasors by drawing......Page 358
36.4 Determining resultant phasors by the sine and cosine rules......Page 359
36.5 Determining resultant phasors by horizontal and vertical components......Page 361
36.6 Determining resultant phasors by complex numbers......Page 363
Revision Test 9......Page 366
Section 7: Statistics......Page 368
37.1 Some statistical terminology......Page 370
37.2 Presentation of ungrouped data......Page 371
37.3 Presentation of grouped data......Page 375
38.2 Mean, median and mode for discrete data......Page 382
38.3 Mean, median and mode for grouped data......Page 383
38.4 Standard deviation......Page 385
38.5 Quartiles, deciles and percentiles......Page 387
39.1 Introduction to probability......Page 389
39.3 Worked problems on probability......Page 390
39.4 Further worked problems on probability......Page 392
39.5 Permutations and combinations......Page 394
Revision Test 10......Page 396
40.1 The binomial distribution......Page 397
40.2 The Poisson distribution......Page 400
41.1 Introduction to the normal distribution......Page 403
41.2 Testing for a normal distribution......Page 408
Revision Test 11......Page 411
Multiple choice questions on Chapters 28–41......Page 412
Section 8: Differential Calculus......Page 418
42.2 Functional notation......Page 420
42.3 The gradient of a curve......Page 421
42.4 Differentiation from first principles......Page 422
42.5 Differentiation of y = axn by the general rule......Page 424
42.6 Differentiation of sine and cosine functions......Page 426
42.7 Differentiation of eax and ln ax......Page 427
43.1 Differentiation of common functions......Page 430
43.2 Differentiation of a product......Page 432
43.3 Differentiation of a quotient......Page 433
43.4 Function of a function......Page 435
43.5 Successive differentiation......Page 436
44.1 Rates of change......Page 438
44.2 Velocity and acceleration......Page 439
44.3 Turning points......Page 442
44.4 Practical problems involving maximum and minimum values......Page 446
44.5 Tangents and normals......Page 449
44.6 Small changes......Page 450
Revision Test 12......Page 452
45.2 Some common parametric equations......Page 453
45.3 Differentiation in parameters......Page 454
45.4 Further worked problems on differentiation of parametric equations......Page 455
46.2 Differentiating implicit functions......Page 458
46.3 Differentiating implicit functions containing products and quotients......Page 459
46.4 Further implicit differentiation......Page 460
47.3 Differentiation of logarithmic functions......Page 463
47.4 Differentiation of further logarithmic functions......Page 464
47.5 Differentiation of [f(x)]x......Page 466
Revision Test 13......Page 468
Section 9: Integral Calculus......Page 470
48.2 The general solution of integrals of the form axn......Page 472
48.3 Standard integrals......Page 473
48.4 Definite integrals......Page 476
49.3 Worked problems on integration using algebraic substitutions......Page 479
49.5 Change of limits......Page 481
50.2 Worked problems on integration of sin2x, cos2x, tan2x and cot2x......Page 484
50.3 Worked problems on powers of sines and cosines......Page 486
50.4 Worked problems on integration of products of sines and cosines......Page 487
50.5 Worked problems on integration using the sinθ substitution......Page 488
50.6 Worked problems on integration using the tanθ substitution......Page 490
Revision Test 14......Page 491
51.2 Worked problems on integration using partial fractions with linear factors......Page 492
51.3 Worked problems on integration using partial fractions with repeated linear factors......Page 494
51.4 Worked problems on integration using partial fractions with quadratic factors......Page 495
52.2 Worked problems on the t = tanθ/2 substitution......Page 497
52.3 Further worked problems on the t = tanθ/2 substitution......Page 499
53.2 Worked problems on integration by parts......Page 501
53.3 Further worked problems on integration by parts......Page 503
54.2 The trapezoidal rule......Page 506
54.3 The mid-ordinate rule......Page 508
54.4 Simpson’s rule......Page 510
Revision Test 15......Page 514
55.1 Area under a curve......Page 515
55.2 Worked problems on the area under a curve......Page 516
55.3 Further worked problems on the area under a curve......Page 519
55.4 The area between curves......Page 521
56.1 Mean or average values......Page 524
56.2 Root mean square values......Page 526
57.1 Introduction......Page 528
57.2 Worked problems on volumes of solids of revolution......Page 529
57.3 Further worked problems on volumes of solids of revolution......Page 530
58.3 Centroid of area between a curve and the x-axis......Page 533
58.5 Worked problems on centroids of simple shapes......Page 534
58.6 Further worked problems on centroids of simple shapes......Page 535
58.7 Theorem of Pappus......Page 538
59.2 Second moment of area of regular sections......Page 542
59.5 Summary of derived results......Page 543
59.6 Worked problems on second moments of area of regular sections......Page 544
59.7 Worked problems on second moments of area of composite areas......Page 547
Revision Test 16......Page 549
Section 10: Further Number and Algebra......Page 550
60.1 Boolean algebra and switching circuits......Page 552
60.3 Laws and rules of Boolean algebra......Page 557
60.4 De Morgan’s laws......Page 559
60.5 Karnaugh maps......Page 560
60.6 Logic circuits......Page 565
60.7 Universal logic gates......Page 569
61.2 Addition, subtraction and multiplication of matrices......Page 573
61.4 The determinant of a 2 by 2 matrix......Page 577
61.5 The inverse or reciprocal of a 2 by 2 matrix......Page 578
61.6 The determinant of a 3 by 3 matrix......Page 579
61.7 The inverse or reciprocal of a 3 by 3 matrix......Page 581
62.1 Solution of simultaneous equations by matrices......Page 583
62.2 Solution of simultaneouse equations by determinants......Page 585
62.3 Solution of simultaneous equations using Cramers rule......Page 589
Revision Test 17......Page 590
Section 11: Differential Equations......Page 592
63.1 Family of curves......Page 594
63.3 The solution of equations of the form dy/dx = f(x)......Page 595
63.4 The solution of equations of the form dy/dx = f(y)......Page 597
63.5 The solution of equations of the form dy/dx = f(x) · f(y)......Page 599
Revision Test 18......Page 602
Multiple choice questions on Chapters 42–63......Page 603
Answers to multiple choice questions......Page 607
Index......Page 608