Engineering Mathematics, Fifth Edition

John Bird's approach to mathematics, based on numerous worked examples supported by problems, is ideal for students of a wide range of abilities. Theory is kept to a minimum, with the emphasis firmly placed on problem-solving skills, making this a thoroughly practical introduction to the mathematics engineering students need to master.The book presents a logical topic progression, rather than following the structure of a particular syllabus and is suitable for all Level 3 vocational students and first year undergraduates in Engineering. However, coverage has been carefully matched to the mathematics units within the 2007 BTEC National specifications.In this fifth edition, new material on inequalities and differentiation of parametric equations, implicit and logarithmic functions as well as an introduction to differential equations has been added. The book now also includes two new revision tests and even more problems for students to work through.Additional chapters on linear correlation, linear regression and sampling and estimation theories can be downloaded for free from http://books.elsevier.com/companions/9780750685559Support material for tutors is available as a free download at http://textbooks.elsevier.com:Instructor's manual with full solutions and suggested marking scheme for all 18 revision tests in the bookSolutions manual with worked solutions for about 1,250 of the further problems in the bookElectronic files for all illustrations in the book * New colour layout helps navigation and highlights key learning points, formulae and exercises* Over 1,000 worked examples and 2,000 questions, all with answers* Fully up to date with the 2007 BTEC National specification* Free lecturer support material available via textbooks.elsevier.com Front Cover......Page 1 Engineering Mathematics, Fifth edition......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 Conversion of binary to decimal......Page 34 3.3 Conversion of decimal to binary......Page 35 3.4 Conversion of decimal to binary via octal......Page 36 3.5 Hexadecimal numbers......Page 38 4.1 Errors and approximations......Page 42 4.2 Use of calculator......Page 44 4.3 Conversion tables and charts......Page 46 4.4 Evaluation of formulae......Page 47 Revision Test 1......Page 52 5.1 Basic operations......Page 53 5.2 Laws of Indices......Page 55 5.3 Brackets and factorisation......Page 57 5.4 Fundamental laws and precedence......Page 59 5.5 Direct and inverse proportionality......Page 61 6.1 Polynominal division......Page 63 6.2 The factor theorem......Page 65 6.3 The remainder theorem......Page 67 7.2 Worked problems on partial fractions with linear factors......Page 69 7.3 Worked problems on partial fractions with repeated linear factors......Page 72 7.4 Worked problems on partial fractions with quadratic factors......Page 73 8.2 Worked problems on simple equations......Page 75 8.3 Further worked problems on simple equations......Page 77 8.4 Practical problems involving simple equations......Page 79 8.5 Further practical problems involving simple equations......Page 80 Revision Test 2......Page 82 9.2 Worked problems on simultaneous equations in two unknowns......Page 83 9.3 Further worked problems on simultaneous equations......Page 85 9.4 More difficult worked problems on simultaneous equations......Page 87 9.5 Practical problems involving simultaneous equations......Page 88 10.2 Worked problems on transposition of formulae......Page 92 10.3 Further worked problems on transposition of formulae......Page 93 10.4 Harder worked problems on transposition of formulae......Page 95 11.2 Solution of quadratic equations by factorisation......Page 98 11.3 Solution of quadratic equations by ‘completing the square’......Page 100 11.4 Solution of quadratic equations by formula......Page 102 11.5 Practical problems involving quadratic equations......Page 103 11.6 The solution of linear and quadratic equations simultaneously......Page 105 12.2 Simple inequalities......Page 106 12.3 Inequalities involving a modulus......Page 107 12.4 Inequalities involving quotients......Page 108 12.5 Inequalities involving square functions......Page 109 12.6 Quadratic inequalities......Page 110 13.2 Laws of logarithms......Page 112 13.3 Indicial equations......Page 115 13.4 Graphs of logarithmic functions......Page 116 Revision Test 3......Page 117 14.2 Evaluating exponential functions......Page 118 14.3 The power series for e[sup(x)]......Page 119 14.4 Graphs of exponential functions......Page 121 14.6 Evaluating Napierian logarithms......Page 123 14.7 Laws of growth and decay......Page 125 15.2 Worked problems on arithmetic progressions......Page 129 15.3 Further worked problems on arithmetic progressions......Page 130 15.4 Geometric progressions......Page 132 15.5 Worked problems on geometric progressions......Page 133 15.6 Further worked problems on geometric progressions......Page 134 15.7 Combinations and permutations......Page 135 16.1 Pascal's triangle......Page 137 16.3 Worked problems on the binomial series......Page 138 16.4 Further worked problems on the binomial series......Page 140 16.5 Practical problems involving the binomial theorem......Page 142 17.2 The Newton–Raphson method......Page 145 17.3 Worked problems on the Newton–Raphson method......Page 146 Revision Test 4......Page 148 Multiple choice questions on Chapters 1–17......Page 149 Section 2 Mensuration......Page 154 18.2 Properties of quadrilaterals......Page 156 18.3 Worked problems on areas of plane figures......Page 157 18.4 Further worked problems on areas of plane figures......Page 160 18.5 Worked problems on areas of composite figures......Page 162 18.6 Areas of similar shapes......Page 163 19.2 Properties of circles......Page 165 19.3 Arc length and area of a sector......Page 167 19.4 Worked problems on arc length and sector of a circle......Page 168 19.5 The equation of a circle......Page 170 20.2 Worked problems on volumes and surface areas of regular solids......Page 172 20.3 Further worked problems on volumes and surface areas of regular solids......Page 175 20.4 Volumes and surface areas of frusta of pyramids and cones......Page 179 20.5 The frustum and zone of a sphere......Page 182 20.6 Prismoidal rule......Page 185 20.7 Volumes of similar shapes......Page 187 21.1 Area of irregular figures......Page 189 21.2 Volumes of irregular solids......Page 191 21.3 The mean or average value of a waveform......Page 192 Revision Test 5......Page 197 Section 3 Trigonometry......Page 200 22.2 The theorem of Pythagoras......Page 202 22.3 Trigonometric ratios of acute angles......Page 203 22.4 Fractional and surd forms of trigonometric ratios......Page 205 22.5 Solution of right-angled triangles......Page 206 22.6 Angle of elevation and depression......Page 208 22.7 Evaluating trigonometric ratios of any angles......Page 210 22.8 Trigonometric approximations for small angles......Page 212 23.2 Angles of any magnitude......Page 214 23.4 Sine and cosine curves......Page 217 23.5 Sinusoidal form A sin(ωt ± α)......Page 221 23.6 Waveform harmonics......Page 224 24.2 Changing from Cartesian into polar co-ordinates......Page 226 24.3 Changing from polar into Cartesian co-ordinates......Page 228 24.4 Use of R → P and P → R functions on calculators......Page 229 Revision Test 6......Page 230 25.3 Worked problems on the solution of triangles and their areas......Page 231 25.4 Further worked problems on the solution of triangles and their areas......Page 233 25.5 Practical situations involving trigonometry......Page 235 25.6 Further practical situations involving trigonometry......Page 237 26.2 Worked problems on trigonometric identities......Page 240 26.3 Trigonometric equations......Page 241 26.4 Worked problems (i) on trigonometric equations......Page 242 26.5 Worked problems (ii) on trigonometric equations......Page 243 26.7 Worked problems (iv) on trigonometric equations......Page 244 27.1 Compound angle formulae......Page 246 27.2 Conversion of a sin ωt + b cos ωt into R sin(ωt + α)......Page 248 27.3 Double angles......Page 251 27.4 Changing products of sines and cosines into sums or differences......Page 253 27.5 Changing sums or differences of sines and cosines into products......Page 254 Revision Test 7......Page 256 Multiple choice questions on Chapters 18–27......Page 257 Section 4 Graphs......Page 262 28.2 The straight line graph......Page 264 28.3 Practical problems involving straight line graphs......Page 270 29.1 Determination of law......Page 276 29.2 Determination of law involving logarithms......Page 279 30.2 Graphs of the form y = ax[sup(n)]......Page 284 30.3 Graphs of the form y = ab[sup(x)]......Page 287 30.4 Graphs of the form y = ae[sup(kx)]......Page 288 31.1 Graphical solution of simultaneous equations......Page 291 31.2 Graphical solution of quadratic equations......Page 292 31.3 Graphical solution of linear and quadratic equations simultaneously......Page 296 31.4 Graphical solution of cubic equations......Page 297 32.1 Standard curves......Page 299 32.2 Simple transformations......Page 301 32.5 Even and odd functions......Page 306 32.6 Inverse functions......Page 308 Revision Test 8......Page 312 Section 5 Vectors......Page 314 33.2 Vector addition......Page 316 33.3 Resolution of vectors......Page 317 33.4 Vector subtraction......Page 320 34.2 Plotting periodic functions......Page 322 34.3 Determining resultant phasors by calculation......Page 323 Section 6 Complex Numbers......Page 326 35.1 Cartesian complex numbers......Page 328 35.3 Addition and subtraction of complex numbers......Page 329 35.4 Multiplication and division of complex numbers......Page 330 35.5 Complex equations......Page 332 35.6 The polar form of a complex number......Page 333 35.7 Multiplication and division in polar form......Page 335 35.8 Applications of complex numbers......Page 336 36.2 Powers of complex numbers......Page 340 36.3 Roots of complex numbers......Page 341 Revision Test 9......Page 344 Section 7 Statistics......Page 346 37.1 Some statistical terminology......Page 348 37.2 Presentation of ungrouped data......Page 349 37.3 Presentation of grouped data......Page 353 38.2 Mean, median and mode for discrete data......Page 360 38.3 Mean, median and mode for grouped data......Page 361 38.4 Standard deviation......Page 363 38.5 Quartiles, deciles and percentiles......Page 365 39.1 Introduction to probability......Page 367 39.3 Worked problems on probability......Page 368 39.4 Further worked problems on probability......Page 370 39.5 Permutations and combinations......Page 372 Revision Test 10......Page 374 40.1 The binomial distribution......Page 375 40.2 The Poisson distribution......Page 378 41.1 Introduction to the normal distribution......Page 381 41.2 Testing for a normal distribution......Page 386 Revision Test 11......Page 389 Multiple choice questions on Chapters 28–41......Page 390 Section 8 Differential Calculus......Page 396 42.2 Functional notation......Page 398 42.3 The gradient of a curve......Page 399 42.4 Differentiation from first principles......Page 400 42.5 Differentiation of y = ax[sup(n)] by the general rule......Page 402 42.6 Differentiation of sine and cosine functions......Page 403 42.7 Differentiation of e[sup(ax)] and ln ax......Page 405 43.1 Differentiation of common functions......Page 407 43.2 Differentiation of a product......Page 409 43.3 Differentiation of a quotient......Page 410 43.4 Function of a function......Page 412 43.5 Successive differentiation......Page 413 44.1 Rates of change......Page 415 44.2 Velocity and acceleration......Page 416 44.3 Turning points......Page 419 44.4 Practical problems involving maximum and minimum values......Page 423 44.5 Tangents and normals......Page 426 44.6 Small changes......Page 427 Revision Test 12......Page 430 45.2 Some common parametric equations......Page 431 45.3 Differentiation in parameters......Page 432 45.4 Further worked problems on differentiation of parametric equations......Page 433 46.2 Differentiating implicit functions......Page 436 46.3 Differentiating implicit functions containing products and quotients......Page 437 46.4 Further implicit differentiation......Page 438 47.3 Differentiation of logarithmic functions......Page 441 47.4 Differentiation of [f(x)][sup(x)]......Page 444 Revision Test 13......Page 446 Section 9 Integral Calculus......Page 448 48.2 The general solution of integrals of the form ax[sup(n)]......Page 450 48.3 Standard integrals......Page 451 48.4 Definite integrals......Page 454 49.3 Worked problems on integration using algebraic substitutions......Page 457 49.5 Change of limits......Page 459 50.2 Worked problems on integration of sin[sup(2)]x, cos[sup(2)]x, tan[sup(2)]x and cot[sup(2)]x......Page 462 50.3 Worked problems on powers of sines and cosines......Page 464 50.4 Worked problems on integration of products of sines and cosines......Page 465 50.5 Worked problems on integration using the sin θ substitution......Page 466 50.6 Worked problems on integration using the tan θ substitution......Page 468 Revision Test 14......Page 469 51.2 Worked problems on integration using partial fractions with linear factors......Page 470 51.3 Worked problems on integration using partial fractions with repeated linear factors......Page 471 51.4 Worked problems on integration using partial fractions with quadratic factors......Page 472 52.2 Worked problems on the t = tan θ/2 substitution......Page 475 52.3 Further worked problems on the t = tan θ/2 substitution......Page 477 53.2 Worked problems on integration by parts......Page 479 53.3 Further worked problems on integration by parts......Page 481 54.2 The trapezoidal rule......Page 484 54.3 The mid-ordinate rule......Page 486 54.4 Simpson's rule......Page 488 Revision Test 15......Page 492 55.1 Area under a curve......Page 493 55.2 Worked problems on the area under a curve......Page 494 55.3 Further worked problems on the area under a curve......Page 497 55.4 The area between curves......Page 499 56.1 Mean or average values......Page 502 56.2 Root mean square values......Page 504 57.1 Introduction......Page 506 57.2 Worked problems on volumes of solids of revolution......Page 507 57.3 Further worked problems on volumes of solids of revolution......Page 508 58.3 Centroid of area between a curve and the x-axis......Page 511 58.5 Worked problems on centroids of simple shapes......Page 512 58.6 Further worked problems on centroids of simple shapes......Page 513 58.7 Theorem of Pappus......Page 516 59.2 Second moment of area of regular sections......Page 520 59.5 Summary of derived results......Page 521 59.6 Worked problems on second moments of area of regular sections......Page 522 59.7 Worked problems on second moments of area of composite areas......Page 525 Revision Test 16......Page 527 Section 10 Further Number and Algebra......Page 528 60.1 Boolean algebra and switching circuits......Page 530 60.3 Laws and rules of Boolean algebra......Page 535 60.4 De Morgan's laws......Page 537 60.5 Karnaugh maps......Page 538 60.6 Logic circuits......Page 543 60.7 Universal logic gates......Page 547 61.2 Addition, subtraction and multiplication of matrices......Page 551 61.4 The determinant of a 2 by 2 matrix......Page 555 61.5 The inverse or reciprocal of a 2 by 2 matrix......Page 556 61.6 The determinant of a 3 by 3 matrix......Page 557 61.7 The inverse or reciprocal of a 3 by 3 matrix......Page 559 62.1 Solution of simultaneous equations by matrices......Page 561 62.2 Solution of simultaneous equations by determinants......Page 563 62.3 Solution of simultaneous equations using Cramers rule......Page 567 Revision Test 17......Page 568 Section 11 Differential Equations......Page 570 63.1 Family of curves......Page 572 63.3 The solution of equations of the form dy/dx = f(x)......Page 573 63.4 The solution of equations of the form dy/dx = f(y)......Page 575 63.5 The solution of equations of the form dy/dx = f(x) · f(y)......Page 577 Revision Test 18......Page 580 Multiple choice questions on Chapters 42–63......Page 581 Answers to multiple choice questions......Page 585 C......Page 586 F......Page 587 M......Page 588 Q......Page 589 V......Page 590 Z......Page 591