Engineering Mathematics

Cover......Page 2
Copyright......Page 3
Table of Contents......Page 5
Preface......Page 23
Section 1 Number and algebra......Page 26
1 Revision of fractions, decimals and percentages......Page 27
1.1 Fractions......Page 28
1.2 Ratio and proportion......Page 31
1.3 Decimals......Page 32
1.4 Percentages......Page 37
2 Indices, standard form and engineering notation......Page 40
2.1 Indices......Page 41
2.2 Worked problems on indices......Page 42
2.3 Further worked problems on indices......Page 44
2.4 Standard form......Page 46
2.5 Worked problems on standard form......Page 47
2.6 Further worked problems on standard form......Page 49
2.7 Engineering notation and common prefixes......Page 50
2.8 Metric conversions......Page 52
2.9 Metric - US/Imperial Conversions......Page 59
3 Binary, octal and hexadecimal numbers......Page 68
3.1 Introduction......Page 69
3.2 Binary numbers......Page 71
3.3 Octal numbers......Page 76
3.4 Hexadecimal numbers......Page 79
4.1 Errors and approximations......Page 86
4.2 Use of calculator......Page 89
4.3 Conversion tables and charts......Page 92
4.4 Evaluation of formulae......Page 95
Revision Test 1......Page 101
5 Algebra......Page 103
5.1 Basic operations......Page 104
5.2 Laws of indices......Page 106
5.3 Brackets and factorisation......Page 110
5.4 Fundamental laws and precedence......Page 114
5.5 Direct and inverse proportionality......Page 117
6.1 Polynomial division......Page 123
6.2 The factor theorem......Page 127
6.3 The remainder theorem......Page 130
7.1 Introduction to partial fractions......Page 135
7.2 Worked problems on partial fractions with linear factors......Page 136
7.3 Worked problems on partial fractions with repeated linear factors......Page 140
7.4 Worked problems on partial fractions with quadratic factors......Page 142
8 Solving simple equations......Page 146
8.2 Worked problems on simple equations......Page 147
8.3 Further worked problems on simple equations......Page 150
8.4 Practical problems involving simple equations......Page 153
8.5 Further practical problems involving simple equations......Page 155
Revision Test 2......Page 158
9.1 Introduction to transposition of formulae......Page 159
9.2 Worked problems on transposition of formulae......Page 160
9.3 Further worked problems on transposition of formulae......Page 162
9.4 Harder worked problems on transposition of formulae......Page 165
10.1 Introduction to simultaneous equations......Page 170
10.2 Worked problems on simultaneous equations in two unknowns......Page 171
10.3 Further worked problems on simultaneous equations......Page 175
10.4 More difficult worked problems on simultaneous equations......Page 178
10.5 Practical problems involving simultaneous equations......Page 182
11 Solving quadratic equations......Page 190
11.2 Solution of quadratic equations by factorisation......Page 191
11.3 Solution of quadratic equations by ‘completing the square’......Page 195
11.4 Solution of quadratic equations by formula......Page 198
11.5 Practical problems involving quadratic equations......Page 200
11.6 The solution of linear and quadratic equations simultaneously......Page 203
12.1 Introduction in inequalities......Page 205
12.2 Simple inequalities......Page 206
12.3 Inequalities involving a modulus......Page 208
12.4 Inequalities involving quotients......Page 209
12.5 Inequalities involving square functions......Page 211
12.6 Quadratic inequalities......Page 213
13 Logarithms......Page 216
13.1 Introduction to logarithms......Page 217
13.2 Laws of logarithms......Page 220
13.3 Indicial equations......Page 224
13.4 Graphs of logarithmic functions......Page 226
Revision Test 3......Page 228
14.1 Introduction to exponential functions......Page 230
14.2 The power series for e......Page 232
14.3 Graphs of exponential functions......Page 235
14.4 Napierian logarithms......Page 238
14.5 Laws of growth and decay......Page 242
15 Number sequences......Page 249
15.1 Arithmetic progressions......Page 250
15.2 Worked problems on arithmetic progressions......Page 251
15.3 Further worked problems on arithmetic progressions......Page 252
15.4 Geometric progressions......Page 255
15.5 Worked problems on geometric progressions......Page 257
15.6 Further worked problems on geometric progressions......Page 258
15.7 Combinations and permutations......Page 261
16.1 Pascal’s triangle......Page 263
16.2 The binomial series......Page 266
16.3 Worked problems on the binomial series......Page 267
16.4 Further worked problems on the binomial series......Page 269
16.5 Practical problems involving the binomial theorem......Page 272
17.1 Introduction to iterative methods......Page 276
17.2 The Newton–Raphson method......Page 277
17.3 Worked problems on the Newton–Raphson method......Page 278
Revision Test 4......Page 282
Multiple choice questions on Chapters 1–17......Page 284
Section 2 Areas and volumes......Page 290
18 Areas of common shapes......Page 291
18.2 Properties of quadrilaterals......Page 292
18.4 Worked problems on areas of common shapes......Page 295
18.5 Further worked problems on areas of plane figures......Page 301
18.6 Worked problems on areas of composite figures......Page 304
18.7 Areas of similar shapes......Page 307
19 The circle and its properties......Page 309
19.2 Properties of circles......Page 310
19.3 Radians and degrees......Page 313
19.5 Worked problems on arc length and area of circles and sectors......Page 315
19.6 The equation of a circle......Page 320
20 Volumes and surface areas of common solids......Page 324
20.2 Volumes and surface areas of regular solids......Page 325
20.3 Worked problems on volumes and surface areas of regular solids......Page 326
20.4 Further worked problems on volumes and surface areas of regular solids......Page 330
20.5 Volumes and surface areas of frusta of pyramids and cones......Page 339
20.6 The frustum and zone ofa sphere......Page 345
20.7 Prismoidal rule......Page 350
20.8 Volumes of similar shapes......Page 354
21 Irregular areas and volumes and mean values of waveforms......Page 356
21.1 Area of irregular figures......Page 357
21.2 Volumes of irregular solids......Page 362
21.3 The mean or average value of a waveform......Page 363
Revision Test 5......Page 371
Section 3 Trigonometry......Page 374
22 Introduction to trigonometry......Page 375
22.2 The theorem of Pythagoras......Page 376
22.3 Trigonometric ratios of acute angles......Page 379
22.4 Fractional and surd forms of trigonometric ratios......Page 382
22.5 Evaluating trigonometric ratios of any angles......Page 384
22.6 Solution of right-angled triangles......Page 390
22.7 Angle of elevation and depression......Page 393
22.8 Trigonometric approximations for small angles......Page 397
23 Trigonometric waveforms......Page 399
23.1 Graphs of trigonometric functions......Page 400
23.2 Angles of any magnitude......Page 401
23.3 The production of a sine and cosine wave......Page 406
23.4 Sine and cosine curves......Page 407
23.5 Sinusoidal form A sin ( )......Page 415
23.6 Waveform harmonics......Page 420
24.1 Introduction......Page 422
24.2 Changing from Cartesian into polar co-ordinates......Page 423
24.3 Changing from polar into Cartesian co-ordinates......Page 426
24.4 Use of Pol/Rec functions on calculators......Page 430
Revision Test 6......Page 432
25.1 Sine and cosine rules......Page 434
25.2 Area of any triangle......Page 435
25.3 Worked problems on the solution of triangles and their areas......Page 436
25.4 Further worked problems on the solution of triangles and their areas......Page 439
25.6 Further practical situations involving trigonometry......Page 446
26 Trigonometric identities and equations......Page 452
26.1 Trigonometric identities......Page 453
26.2 Worked problems on trigonometric identities......Page 454
26.3 Trigonometric equations......Page 455
26.4 Worked problems (i) on trigonometric equations......Page 456
26.5 Worked problems (ii) on trigonometric equations......Page 459
26.6 Worked problems (iii) on trigonometric equations......Page 461
26.7 Worked problems (iv) on trigonometric equations......Page 462
27.1 Compound angle formulae......Page 465
27.2 Conversion of a sin t b cos t into R sin( t )......Page 469
27.3 Double angles......Page 476
27.4 Changing products of sines and cosines into sums or differences......Page 478
27.5 Changing sums or differences of sines and cosines into products......Page 481
Revision Test 7......Page 483
Multiple choice questions on Chapters 18–27......Page 485
Section 4 Graphs......Page 493
28.1 Introduction to graphs......Page 494
28.2 The straight line graph......Page 495
28.3 Practical problems involving straight line graphs......Page 506
29 Reduction of non-linear laws to linear form......Page 520
29.1 Determination of law......Page 521
29.2 Determination of law involving logarithms......Page 526
30 Graphs with logarithmic scales......Page 536
30.1 Logarithmic scales......Page 537
30.2 Graphs of the form y......Page 538
30.3 Graphs of the form......Page 543
30.4 Graphs of the form y ae......Page 546
31.1 Graphical solution of simultaneous equations......Page 551
31.2 Graphical solution of quadratic equations......Page 554
31.3 Graphical solution of linear and quadratic equations simultaneously......Page 561
31.4 Graphical solution of cubic equations......Page 563
32 Functions and their curves......Page 567
32.1 Standard curves......Page 568
32.2 Simple transformations......Page 577
32.3 Periodic functions......Page 581
32.5 Even and odd functions......Page 582
32.6 Inverse functions......Page 586
Revision Test 8......Page 591
Section 5 Complex numbers......Page 593
33 Complex numbers......Page 594
33.1 Cartesian complex numbers......Page 595
33.3 Addition and subtraction of complex numbers......Page 597
33.4 Multiplication and division of complex numbers......Page 601
33.5 Complex equations......Page 603
33.6 The polar form of a complex number......Page 605
33.7 Multiplication and division in polar form......Page 608
33.8 Applications of complex numbers......Page 610
34.1 Introduction......Page 617
34.2 Powers of complex numbers......Page 618
34.3 Roots of complex numbers......Page 619
Section 6 Vectors......Page 623
35 Vectors......Page 624
35.2 Scalars and vectors......Page 625
35.3 Drawing a vector......Page 626
35.4 Addition of vectors by drawing......Page 627
35.5 Resolving vectors into horizontal and vertical components......Page 632
35.6 Addition of vectors by calculation......Page 634
35.7 Vector subtraction......Page 643
35.8 Relative velocity......Page 647
35.9 , and notation......Page 649
36 Methods of adding alternating waveforms......Page 651
36.2 Plotting periodic functions......Page 652
36.3 Determining resultant phasors by drawing......Page 656
36.4 Determining resultant phasors by the sine and cosine rules......Page 659
36.5 Determining resultant phasors by horizontal and vertical components......Page 661
36.6 Determining resultant phasors by complex numbers......Page 665
Revision Test 9......Page 670
Section 7 Statistics......Page 672
37 Presentation of statistical data......Page 673
37.1 Some statistical terminology......Page 674
37.2 Presentation of ungrouped data......Page 676
37.3 Presentation of grouped data......Page 683
38 Mean, median, mode and standard deviation......Page 695
38.2 Mean, median and mode for discrete data......Page 696
38.3 Mean, median and mode for grouped data......Page 698
38.4 Standard deviation......Page 701
38.5 Quartiles, deciles and percentiles......Page 705
39 Probability......Page 708
39.1 Introduction to probability......Page 709
39.2 Laws of probability......Page 711
39.3 Worked problems on probability......Page 712
39.4 Further worked problems on probability......Page 715
39.5 Permutations and combinations......Page 720
39.6 Bayes’ theorem......Page 722
Revision Test 10......Page 725
40 The binomial and Poisson distributions......Page 727
40.1 The binomial distribution......Page 728
40.2 The Poisson distribution......Page 734
41 The normal distribution......Page 740
41.1 Introduction to the normal distribution......Page 741
41.2 Testing for a normal distribution......Page 749
Revision Test 11......Page 756
42.1 Introduction to linear correlation......Page 758
42.2 The Pearson product-moment formula for determining the linear correlation coefficient......Page 759
42.4 Worked problems on linear correlation......Page 761
43.1 Introduction to linear regression......Page 767
43.2 The least-squares regression lines......Page 768
43.3 Worked problems on linear regression......Page 770
44.1 Introduction......Page 777
44.3 The sampling distribution of the means......Page 778
44.4 The estimation of population parameters based on a large sample size......Page 784
44.5 Estimating the mean of a population based on a small sample size......Page 794
Revision Test 12......Page 801
Multiple choice questions on Chapters 28–44......Page 803
Section 8 Differential calculus......Page 811
45 Introduction to differentiation......Page 812
45.2 Functional notation......Page 813
45.3 The gradient of a curve......Page 814
45.4 Differentiation from first principles......Page 816
45.5 Differentiation of by the general rule......Page 821
45.6 Differentiation of sine and cosine functions......Page 822
45.7 Differentiation of and ln ax......Page 826
46.1 Differentiation of common functions......Page 829
46.2 Differentiation of a product......Page 833
46.3 Differentiation of a quotient......Page 835
46.4 Function of a function......Page 837
46.5 Successive differentiation......Page 840
47 Some applications of differentiation......Page 843
47.1 Rates of change......Page 844
47.2 Velocity and acceleration......Page 846
47.3 Turning points......Page 851
47.4 Practical problems involving maximum and minimum values......Page 857
47.5 Points of inflexion......Page 863
47.6 Tangents and normals......Page 866
47.7 Small changes......Page 869
48 Maclaurin’s series......Page 872
48.1 Introduction......Page 873
48.2 Derivation of Maclaurin’s theorem......Page 874
48.4 Worked problems on Maclaurin’s series......Page 875
Revision Test 13......Page 881
49 Differentiation of parametric equations......Page 883
49.2 Some common parametric equations......Page 884
49.3 Differentiation in parameters......Page 885
49.4 Further worked problems on differentiation of parametric equations......Page 888
50.1 Implicit functions......Page 891
50.2 Differentiating implicit functions......Page 892
50.3 Differentiating implicit functions containing products and quotients......Page 893
50.4 Further implicit differentiation......Page 894
51.1 Introduction to logarithmic differentiation......Page 899
51.3 Differentiation of logarithmic functions......Page 900
51.4 Differentiation of further logarithmic functions......Page 901
51.5 Differentiation of [......Page 904
Revision Test 14......Page 907
Section 9 Integral calculus......Page 908
52.1 The process of integration......Page 909
52.2 The general solution of integrals of the form ax......Page 910
52.3 Standard integrals......Page 911
52.4 Definite integrals......Page 915
53 Integration using algebraic substitutions......Page 918
53.3 Worked problems on integration using algebraic substitutions......Page 919
53.4 Further worked problems on integration using algebraic substitutions......Page 921
53.5 Change of limits......Page 922
54.1 Introduction......Page 925
54.2 Worked problems on integration of x, x, x and x......Page 926
54.3 Worked problems on integration of powers of sines and cosines......Page 928
54.4 Worked problems on integration of products of sines and cosines......Page 930
54.5 Worked problems on integration using the substitution......Page 931
54.6 Worked problems on integration using the substitution......Page 933
Revision Test 15......Page 935
55.1 Introduction......Page 936
55.2 Worked problems on integration using partial fractions with linear factors......Page 937
55.3 Worked problems on integration using partial fractions with repeated linear factors......Page 938
55.4 Worked problems on integration using partial fractions with quadratic factors......Page 940
56.1 Introduction......Page 942
56.2 Worked problems on the substitution......Page 943
56.3 Further worked problems on the substitution......Page 945
57.1 Introduction......Page 948
57.2 Worked problems on integration by parts......Page 949
57.3 Further worked problems on integration by parts......Page 951
58 Numerical integration......Page 956
58.2 The trapezoidal rule......Page 957
58.3 The mid-ordinate rule......Page 961
58.4 Simpson’s rule......Page 963
58.5 Accuracy of numerical integration......Page 968
Revision Test 16......Page 969
59.1 Area under a curve......Page 970
59.2 Worked problems on the area under a curve......Page 973
59.3 Further worked problems on the area under a curve......Page 978
59.4 The area between curves......Page 982
60.1 Mean or average values......Page 986
60.2 Root mean square values......Page 990
61.1 Introduction......Page 993
61.2 Worked problems on volumes of solids of revolution......Page 995
61.3 Further worked problems on volumes of solids of revolution......Page 997
62.1 Centroids......Page 1001
62.3 Centroid of area between a curve and the x-axis......Page 1002
62.4 Centroid of area between a curve and the y-axis......Page 1003
62.5 Worked problems on centroids of simple shapes......Page 1004
62.6 Further worked problems on centroids of simple shapes......Page 1006
62.7 Theorem of Pappus......Page 1009
63 Second moments of area......Page 1015
63.2 Second moment of area of regular sections......Page 1016
63.3 Parallel axis theorem......Page 1017
63.5 Summary of derived results......Page 1019
63.6 Worked problems on second moments of area of regular sections......Page 1020
63.7 Worked problems on second moments of area of composite areas......Page 1026
Revision Test 17......Page 1030
Section 10 Differential equations......Page 1032
64.1 Family of curves......Page 1033
64.2 Differential equations......Page 1035
64.3 The solution of equations of the form......Page 1036
64.4 The solution of equations of the form......Page 1038
64.5 The solution of equations of the form......Page 1042
Revision Test 18......Page 1047
Section 11 Further number and algebra......Page 1048
65 Boolean algebra and logic circuits......Page 1049
65.1 Boolean algebra and switching circuits......Page 1050
65.2 Simplifying Boolean expressions......Page 1059
65.3 Laws and rules of Boolean algebra......Page 1060
65.4 De Morgan’s laws......Page 1063
65.5 Karnaugh maps......Page 1066
65.6 Logic circuits......Page 1076
65.7 Universal logic gates......Page 1084
66 The theory of matrices and determinants......Page 1090
66.2 Addition, subtraction and multiplication of matrices......Page 1091
66.4 The determinant of a 2 by 2 matrix......Page 1096
66.5 The inverse or reciprocal of a 2 by 2 matrix......Page 1097
66.6 The determinant of a 3 by 3 matrix......Page 1098
66.7 The inverse or reciprocal of a 3 by 3 matrix......Page 1100
67.1 Solution of simultaneous equations by matrices......Page 1103
67.2 Solution of simultaneous equations by determinants......Page 1107
67.3 Solution of simultaneous equations using Cramers rule......Page 1113
67.4 Solution of simultaneous equations using the Gaussian elimination method......Page 1114
Revision Test 19......Page 1119
Multiple choice questions on Chapters 45–67......Page 1121
List of essential formulae......Page 1127
Answers to Practice Exercises......Page 1144
Answers to multiple choice questions......Page 1197
Index......Page 1198