Engineering Mathematics, Fourth Edition

Engineering Mathematics is a comprehensive textbook for vocational courses and foundation modules at degree level. John Bird's approach, based on numerous worked examples supported by problems, is ideal for students of a wide range of abilities, and can be worked through at the student's own pace. Theory is kept to a minimum, placing a firm emphasis on problem-solving skills, and making this a thoroughly practical introduction to the core mathematics needed for engineering studies and practice. The book presents a logical topic progression, rather than following the structure of a particular syllabus. However, coverage has been carefully matched to the two mathematics units within the new BTEC National specifications, and AVCE specifications. New sections on Boolean algebra, logic circuits matrices and determinants have been added to ensure full syllabus match. Includes: 900 worked examples, 1700 further problems, 234 multiple choice questions (answers provided), and 16 assessment papers - ideal for use as tests or homework. These are the only problems where answers are not provided in the book. Full worked solutions are available to lecturers only as a free download from http://textbooks.elsevier.com * A complete foundation mathematics course for engineering students * Student-friendly learn-through-examples method appeals to engineers * Includes 850 worked examples, 1500 problems (answers provided), 200 multiple choice questions, and 15 assessment papers 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