Engineering Mathematics, Sixth Edition

Mathematics is a key element in determining success for the Edexcel BTEC National Engineering courses. Updated for the 2010 BTEC Nationals in Engineering syllabus, Engineering Mathematics, 6e by John Bird covers the main elements of mathematics in the core, mechanical and Electrical/ Electronic Units. There are currently over 13,000 BTEC National Engineering students in the UK. Theory is introduced in each chapter by a simple outline of essential definitions, formulae, laws and procedures. This new, sixth edition will also be supported with online tutor support materials. These include an Instructors Guide which will give full solutions and marking schemes for each of the Revision Tests contained in the text. . Updated for the 2010 BTEC Nationals in Engineering syllabus . This book contains thousands of worked examples and revision tests . Supported with online tutor support materials, including an instructors guide . Updated for the 2010 BTEC Nationals in Engineering syllabus. This book contains thousands of worked examples and revision tests . Supported with online tutor support materials, including an instructors guide and editable PowerPoint Presentation slides 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

• Colour layout helps navigation and highlights key learning points, formulae and exercises
• Fully up-to-date with Levels 2 and 3 of the BTEC Engineering Specifications
• Containing 1000 worked problems, 1750 further problems and 238 multiple-choice questions and answers
• Real-world'situations and engineering examples put the theory into context

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 that students need to master.

This book presents a logical topic progression rather than following the structure of a particular syllabus, and is suitable for all Level 3 vocational students, early Foundation Degree students and for any introductory course involving engineering mathematics. However, the coverage has been carefully matched to the mathematics units within the 2010 Level 2 and 3 BTEC National Specifications.

In this sixth edition there is new material on logarithms, exponential functions, vectors, and methods of alternating waveforms. The book now includes even more problems to work through.

Ideal for use as tests or homework, full solutions to the revision tests are supplied on the accompanying instructor’s website.

Audience: Students following vocational engineering courses / first year undergraduates. Suitable for all Level 3 engineering programmes, and core units at Level 3. Matched to New BTEC National specifications: Mathematics for Technicians; Further Mathematics for Technicians; AVCE: Applied Mathematics for Engineering; Further Mathematics for Engineering.