Engineering Mathematics

Description

Now in its eighth edition, Engineering Mathematics is an established textbook that has helped thousands of students to succeed in their exams. John Bird's approach is based on worked examples and interactive problems. Mathematical theories are explained in a straightforward manner, being supported by practical engineering examples and applications in order to ensure that readers can relate theory to practice. The extensive and thorough topic coverage makes this an ideal text for a range of Level 2 and 3 engineering courses. This title is supported by a companion website with resources for both students and lecturers, including lists of essential formulae and multiple choice tests. 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