Engineering mathematics

Contents......Page 5
Preface......Page 11
1.1 Fractions......Page 13
1.2 Ratio and proportion......Page 15
1.3 Decimals......Page 16
1.4 Percentages......Page 19
2.2 Worked problems on indices......Page 21
2.3 Further worked problems on indices......Page 23
2.5 Worked problems on standard form......Page 25
2.6 Further worked problems on standard form......Page 26
3.2 Conversion of binary to decimal......Page 28
3.3 Conversion of decimal to binary......Page 29
3.4 Conversion of decimal to binary via octal......Page 30
3.5 Hexadecimal numbers......Page 32
4.1 Errors and approximations......Page 36
4.2 Use of calculator......Page 38
4.3 Conversion tables and charts......Page 40
4.4 Evaluation of formulae......Page 42
Assignment 1......Page 45
5.1 Basic operations......Page 46
5.2 Laws of Indices......Page 48
5.3 Brackets and factorisation......Page 50
5.4 Fundamental laws and precedence......Page 52
5.5 Direct and inverse proportionality......Page 54
6.1 Polynomial division......Page 56
6.2 The factor theorem......Page 58
6.3 The remainder theorem......Page 60
7.2 Worked problems on partial fractions with linear factors......Page 63
7.3 Worked problems on partial fractions with repeated linear factors......Page 66
7.4 Worked problems on partial fractions with quadratic factors......Page 67
8.2 Worked problems on simple equations......Page 69
8.3 Further worked problems on simple equations......Page 71
8.4 Practical problems involving simple equations......Page 73
8.5 Further practical problems involving simple equations......Page 74
Assignment 2......Page 76
9.2 Worked problems on simultaneous equations in two unknowns......Page 77
9.3 Further worked problems on simultaneous equations......Page 79
9.4 More difficult worked problems on simultaneous equations......Page 81
9.5 Practical problems involving simultaneous equations......Page 82
10.2 Worked problems on transposition of formulae......Page 86
10.3 Further worked problems on transposition of formulae......Page 87
10.4 Harder worked problems on transposition of formulae......Page 89
11.2 Solution of quadratic equations by factorisation......Page 92
Multiple choice questions on chapters 44 – 61 ( page 522)......Page 0
11.4 Solution of quadratic equations by formula......Page 96
11.5 Practical problems involving quadratic equations......Page 97
11.6 The solution of linear and quadratic equations simultaneously......Page 99
12.1 Introduction to logarithms 12.2 Laws of logarithms......Page 101
12.3 Indicial equations......Page 104
12.4 Graphs of logarithmic functions......Page 105
Assignment 3......Page 106
13.2 Evaluating exponential functions......Page 107
13.3 The power series for ex......Page 108
13.4 Graphs of exponential functions......Page 110
13.6 Evaluating Napierian logarithms......Page 112
13.7 Laws of growth and decay......Page 114
14.1 Arithmetic progressions 14.2 Worked problems on arithmetic progression......Page 118
14.3 Further worked problems on arithmetic progressions......Page 119
14.4 Geometric progressions......Page 121
14.5 Worked problems on geometric progressions......Page 122
14.6 Further worked problems on geometric progressions......Page 123
14.7 Combinations and permutations......Page 124
15 The binomial series......Page 126
15.3 Worked problems on the binomial series......Page 127
15.4 Further worked problems on the binomial series......Page 129
15.5 Practical problems involving the binomial theorem......Page 132
16.1 Introduction to iterative methods......Page 135
Assignment 4......Page 138
17.2 Properties of quadrilaterals......Page 143
17.3 Worked problems on areas of plane figures......Page 144
17.4 Further worked problems on areas of plane figures......Page 147
17.5 Worked problems on areas of composite figures......Page 149
17.6 Areas of similar shapes......Page 150
18.2 Properties of circles......Page 151
18.3 Arc length and area of a sector......Page 152
18.4 Worked problems on arc length and sector of a circle......Page 153
18.5 The equation of a circle......Page 155
19.1 Volumes and surface areas of regular solids 19.2 Worked problems on volumes and surface areas of regular solids......Page 157
19.3 Further worked problems on volumes and surface areas of regular solids......Page 159
19.4 Volumes and surface areas of frusta of pyramids and cones......Page 163
19.5 The frustum and zone of a sphere......Page 167
19.6 Prismoidal rule......Page 169
19.7 Volumes of similar shapes......Page 171
20.1 Areas of irregular figures......Page 173
20.2 Volumes of irregular solids......Page 175
20.3 The mean or average value of a waveform......Page 176
Assignment 5......Page 181
21.2 The theorem of Pythagoras......Page 183
21.3 Trigonometric ratios of acute angles......Page 184
21.4 Fractional and surd forms of trigonometric ratios......Page 186
21.5 Solution of right-angled triangles......Page 187
21.6 Angles of elevation and depression......Page 188
21.7 Evaluating trigonometric ratios of any angles......Page 190
21.8 Trigonometric approximations for small angles......Page 193
22.2 Angles of any magnitude......Page 194
22.4 Sine and cosine curves......Page 197
22.5 Sinusoidal form A sin.!t......Page 201
22.6 Waveform harmonics......Page 204
23.2 Changing from Cartesian into polar co- ordinates......Page 206
23.3 Changing from polar into Cartesian co- ordinates......Page 208
Assignment 6......Page 210
24.3 Worked problems on the solution of triangles and their areas......Page 211
24.4 Further worked problems on the solution of triangles and their areas......Page 213
24.5 Practical situations involving trigonometry......Page 215
24.6 Further practical situations involving trigonometry......Page 217
25.2 Worked problems on trigonometric identities......Page 220
25.3 Trigonometric equations......Page 221
25.4 Worked problems (i) on trigonometric equations......Page 222
25.5 Worked problems (ii) on trigonometric equations......Page 223
25.7 Worked problems (iv) on trigonometric equations......Page 224
26.1 Compound angle formulae......Page 226
26.2 Conversion of a sin wt+b cos wt into R sin (wt+a)......Page 228
26.3 Double angles......Page 232
26.4 Changing products of sines and cosines into sums or differences......Page 233
26.5 Changing sums or differences of sines and cosines into products......Page 234
Assignment 7......Page 236
27.1 Introduction to graphs 27.2 The straight line graph......Page 243
27.3 Practical problems involving straight line graphs......Page 249
28.1 Determination of law......Page 255
28.2 Determination of law involving logarithms......Page 258
29.2 Graphs of the form y =axn......Page 263
29.3 Graphs of the form y = abx......Page 266
29.4 Graphs of the form y = aekx......Page 267
30.1 Graphical solution of simultaneous equations......Page 270
30.2 Graphical solution of quadratic equations......Page 271
30.3 Graphical solution of linear and quadratic equations simultaneously......Page 275
30.4 Graphical solution of cubic equations......Page 276
31.1 Standard curves......Page 278
31.2 Simple transformations......Page 280
31.5 Even and odd functions......Page 285
31.6 Inverse functions......Page 287
Assignment 8......Page 291
32.2 Vector addition......Page 293
32.3 Resolution of vectors......Page 295
32.4 Vector subtraction......Page 296
33.2 Plotting periodic functions......Page 299
33.3 Determining resultant phasors by calculation......Page 300
34.1 Cartesian complex numbers......Page 303
34.3 Addition and subtraction of complex numbers......Page 304
34.4 Multiplication and division of complex numbers......Page 305
34.5 Complex equations......Page 307
34.6 The polar form of a complex number......Page 308
34.7 Multiplication and division in polar form......Page 310
34.8 Applications of complex numbers......Page 311
35.2 Powers of complex numbers......Page 315
35.3 Roots of complex numbers......Page 316
Assignment 9......Page 318
36.1 Some statistical terminology......Page 319
36.2 Presentation of ungrouped data......Page 320
36.3 Presentation of grouped data......Page 324
37.2 Mean, median and mode for discrete data......Page 331
37.3 Mean, median and mode for grouped data......Page 332
37.4 Standard deviation......Page 334
37.5 Quartiles, deciles and percentiles......Page 336
38.2 Laws of probability......Page 338
38.3 Worked problems on probability......Page 339
38.4 Further worked problems on probability......Page 341
38.5 Permutations and combinations......Page 343
39.1 The binomial distribution......Page 345
39.2 The Poisson distribution......Page 348
Assignment 10......Page 351
40.1 Introduction to the normal distribution......Page 352
40.2 Testing for a normal distribution......Page 356
41.2 The product-moment formula for determining the linear correlation coefficient......Page 359
41.4 Worked problems on linear correlation......Page 360
42.2 The least-squares regression lines......Page 363
42.3 Worked problems on linear regression......Page 364
43.3 The sampling distribution of the means......Page 368
43.4 The estimation of population parameters based on a large sample size......Page 371
43.5 Estimating the mean of a population based on a small sample size......Page 376
Assignment 11......Page 380
44.2 Functional notation......Page 387
44.3 The gradient of a curve......Page 388
44.4 Differentiation from first principles......Page 389
44.5 Differentiation of y = axn by the general rule......Page 391
44.6 Differentiation of sine and cosine functions......Page 392
44.7 Differentiation of eax and ln ax......Page 394
45.1 Differentiation of common functions......Page 396
45.2 Differentiation of a product......Page 398
45.3 Differentiation of a quotient......Page 399
45.4 Function of a function......Page 401
45.5 Successive differentiation......Page 402
46.1 Rates of change......Page 404
46.2 Velocity and acceleration......Page 405
46.3 Turning points......Page 408
46.4 Practical problems involving maximum and minimum values......Page 411
46.5 Tangents and normals......Page 415
46.6 Small changes......Page 416
Assignment 12......Page 418
47.2 The general solution of integrals of the form axn......Page 419
47.3 Standard integrals......Page 420
47.4 Definite integrals......Page 423
48.3 Worked problems on integration using algebraic substitutions......Page 426
48.5 Change of limits......Page 428
49.2 Worked problems on integration of sin2 x, cos2 x, tan2 x and cot2 x......Page 430
49.3 Worked problems on powers of sines and cosines......Page 432
49.4 Worked problems on integration of products of sines and cosines......Page 433
49.5 Worked problems on integration using the sin theta substitution......Page 434
49.6 Worked problems on integration using the tan theta substitution......Page 436
Assignment 13......Page 437
50.2 Worked problems on integration using partial fractions with linear factors......Page 438
50.3 Worked problems on integration using partial fractions with repeated linear factors......Page 439
50.4 Worked problems on integration using partial fractions with quadratic factors......Page 440
51.1 Introduction......Page 442
52.2 Worked problems on integration by parts......Page 446
52.3 Further worked problems on integration by parts......Page 448
53.2 The trapezoidal rule......Page 451
53.3 The mid-ordinate rule......Page 453
53.4 Simpson\'s rule......Page 455
Assignment 14......Page 459
54.1 Area under a curve......Page 460
54.2 Worked problems on the area under a curve......Page 461
54.3 Further worked problems on the area under a curve......Page 464
54.4 The area between curves......Page 466
55.1 Mean or average values......Page 469
55.2 Root mean square values......Page 471
56.2 Worked problems on volumes of solids of revolution......Page 473
56.3 Further worked problems on volumes of solids of revolution......Page 475
57.3 Centroid of area between a curve and the x- axis......Page 478
57.5 Worked problems on centroids of simple shapes......Page 479
57.6 Further worked problems on centroids of simple shapes......Page 480
57.7 Theorem of Pappus......Page 483
58.3 Parallel axis theorem......Page 487
58.6 Worked problems on second moments of area of regular sections......Page 488
58.7 Worked problems on second moments of areas of composite areas......Page 492
Assignment 15......Page 494
59.1 Boolean algebra and switching circuits......Page 495
59.3 Laws and rules of Boolean algebra......Page 500
59.5 Karnaugh maps......Page 503
59.6 Logic circuits......Page 507
59.7 Universal logic gates......Page 512
60.2 Addition, subtraction and multiplication of matrices......Page 516
60.4 The determinant of a 2 by 2 matrix......Page 520
60.5 The inverse or reciprocal of a 2 by 2 matrix......Page 521
60.6 The determinant of a 3 by 3 matrix......Page 522
60.7 The inverse or reciprocal of a 3 by 3 matrix......Page 523
61.1 Solution of simultaneous equations by matrices......Page 526
61.2 Solution of simultaneous equations by determinants......Page 528
61.3 Solution of simultaneous equations using Cramers rule......Page 532
Assignment 16......Page 533
Answers to multiple choice questions......Page 538
Index......Page 539