Engineering Mathematics Sixth editif'n John Bird, BSc (Hons), CEng, CSci, CMath, FrET, MIEE, FIlE, FrMA, FCollT ELSEVIER AMSTERDAM. BOSTON. HEIDELBERG LONDON. NEW YORK. OXFORD PARIS. SAN DIEGO. SAN FRANCISCO. SINGAPORE. SYDNEY. TOKYO Newnes is an imprint of Elsevier Newnes
Contents Preface xii 7 Partial fractions 55 7.1 Introduction to partial fractions 55 7.2 Worked problems on partial fractions with Section 1 Number and Algebra 1 linear factors 55 7.3 Worked problems on partial fractions with 1 Revision of fractions, decimals and percentages 3 repeated linear factors I 58 1.1 Fractions 3 7.4 Worked problems on partial fractions with 1.2 Ratio and proportion 5 quadratic factors 59 1.3 Decimals 6 1.4 Percentages 9 8 Solving simple equations 61 8.1 Expressions, equations and identities 61 2 Indices, standard form and engineering notation 11 8.2 Worked problems on simple equations 61 2.1 Indices 11 2.2 Worked problems on indices 12 2.3 Further worked problems on indices 13 2.4 Standard form 15 2.5 Worked problems on standard form 15 2.6 Further worked problems on standard form 16 2.7 Engineering notation and common prefixes 17 8.3 Further worked problems on simple equations 63 8.4 Practical problems involving simple equations 65 8.5 Further practical problems involving simple equations 66 Revision Test 2 68 3 Binary, octal and hexadecimal 19 3.1 Introduction 19 9 Solving simultaneous equations 69 9.1 Introduction to simultaneous equations 69 3.2 Binary numbers 19 9.2 Worked problems on simultaneous 3.3 Octal numbers 22 equations in two unknowns 69 3.4 Hexadecimal numbers 24 9.3 Further worked problems on simultaneous equations 71 4 Calculations and evaluation of formulae 28 9.4 More difficult worked problems on 4.1 Errors and approximations 28 simultaneous equations 73 4.2 Use of calculator 30 9.5 Practical problems involving simultaneous 4.3 Conversion tables and charts 32 equations 74 4.4 Evaluation of formulae 33 10 Transposition of formulae 78 Revision Test 1 38 10.1 Introduction to transposition of formulae 78 10.2 Worked problems on transposition of formulae 78 5 Algebra 39 10.3 Further worked problems on transposition 5.1 Basic operations 39 of formulae 79 5.2 Laws of indices 41 10.4 Harder worked problems on transposition 5.3 Brackets and factorisation 43 of formulae 81 5.4 Fundamental laws and precedence 45 11 Solving quadratic equations 84 5.5 Direct and inverse proportionality 47 11.1 Introduction to quadratic equations 84 11.2 Solution of quadratic equations by 6 Further algebra 49 factorisation 84 6.1 Polynomial division 49 11.3 Solution of quadratic equations by 6.2 The factor theorem 51 'completing the square' 86 6.3 The remainder theorem 53 11.4 Solution of quadratic equations by formula 88
vi Contents 11.5 Practical pro blems involving quadratic equations 89 11.6 The solution of linear and quadratic equations simultaneously 91 12 Ineqti~lities 92 12. i Introduction to inequalities 92 12.2 Simple inequalities 92 12.3 Inequalities involving a modulus 93 12.4 Inequalities involving quotients 94 12.5 Inequalities involving square functions 95 12.6 Quadratic inequalities 96 13 Logarithms 98 13.1 Introduction to logarithms 98 13.2 Laws of logarithms 100 13.3 Indicial equations 102 13.4 Graphs of logarithmic functions 103 Revision Test 3 105 14 Exponential functions 106 14.1 Introduction to exponential functions 106 14.2 The power series for ex 107 14.3 Graphs of exponential functions 109 14.4 Napierian logarithms 111 14.5 Laws of growth and decay 113 15 Number sequences 117 15.1 Arithmetic progressions 117 15.2 Worked problems on arithmetic progressions 117 15.3 Further worked problems on arithmetic progressions 118 15.4 Geometric progressions 120 15.5 Worked problems on geometric progressions 121 15.6 Further worked problems on geometric progressions 122 15.7 Combinations and permutations 123 16 The binomial series 125 16.1 Pascal's triangle 125 16.2 The binomial series 126 16.3 Worked problems on the binomial series 126 16.4 Further worked problems on the binomial series 128 16.5 Practical problems involving the binomial theorem 131 17 Solving equations by iterative methods 134 17.1 Introduction to iterative methods 134 17.2 The Newton-Raphsonmethod 134 17.3 Worked problems on the Newton-Raphson method 135 Revision Test 4 137 Section 2 Areas and Volumes 143 18 Areas of common shapes 145 18.1 Introduction 145 18.2 Properties of quadrilaterals 145 18.3 Areas of common shapes 146 18.4 Worked problems on areas of common shapes 146 18.5 Further worked problems on a,reas of plane figures 149 18.6 Worked problems on areas of composite figures 151 18.7 Areas of similar shapes 152 19 The circle 154 19.1 Introduction 154 19.2 Properties of circles 154 19.3 Radians and degrees 156 19.4 Arc length and area of circles and sectors 157 19.5 Worked problems on arc length and area of circles and sectors 157 19.6 The equation of a circle 160 20 Volumes and surface areas of common solids 162 20.1 Introduction 162 20.2 Volumes and surface areas of regular solids 162 20.3 Worked problems on volumes and surface areas ofregularsolids 163 20.4 Further worked problems on volumes and surface areas of regular solids 165 20.5 Volumes and surface areas of frusta of pyramids and cones 170 20.6 The frustum and zone of a sphere 173 20.7 Prismoidal rule 176 20.8 Volumes of similar shapes 178 21 Irregular areas and volumes and mean values of waveforms 179 21.1 Area of irregular figures 179 21.2 Volumes of irregular solids 181 21.3 The mean or average value of a waveform 182 Revision Test 5 187 Section 3 Trigonometry 189 22 Introduction to trigonometry 22.1 Trigonometry 22.2 The theorem of Pythagoras 191 191 191
Contents vii 22.3 Trigonometric ratios of acute angles 192 27 Compound angles 238 22.4 Fractional and surd forms of trigonometric 27.1 Compound angle formulae 238 ratios 194 27.2 Conversion of a sinwt + b cos wt into 22.5 Evaluating trigonometric ratios of any R sin(wt +a) 240 angles'i'" 195 27.3 Double angles 244 22.6 Solutio~ of right-angled triangles 199 22.7 Angles of elevation and depression 201 27.4 Changing products of sines and cosines into sums or differences 245 22.8 Trigonometric approximations for small 27.5 Changing sums or differences of sines and angles 203 cosines into products 246 23 Trigonometric waveforms 204 Revision Test 7 248 23.1 Graphs of trigonometric functions 204 23.2 Angles of any magnitude 204 23.3 The production of a sine and cosine wave 207 Multiple choice questions on Chapters 18-27 249 23.4 Sine and cosine curves 207 23.5 Sinusoidal form Asin(wt ± a) 211 23.6 Waveform harmonics 214 Section 4 Graphs 255 24 Cartesian and polar co-ordinates 216 24.1 Introduction 216 24.2 Changing from Cartesian into polar co-ordinates 216 24.3 Changing from polar into Cartesian co-ordinates 218 24.4 Use of Pol/Rec functions on calculators 219 Revision Test 6 221 28 Straight line graphs 257 28.1 Introduction to graphs 257 28.2 The straight line graph 257 28.3 Practical problems involving straight line graphs 263 29 Reduction of non-linear laws to linear form 269 29.1 Determination oflaw 269 29.2 Determination oflaw involving logarithms 272 25 Triangles and some practical applications 222 25.1 Sine and cosine rules 222 30 Graphs with logarithmic scales. 277 25.2 Area of any triangle 222 30.1 Logarithmic scales 277 25.3 Worked problems on the solution of 30.2 Graphs of the form y = ax n 277 triangles and their areas 222 30.3 Graphs of the form y = ab x 280 25.4 Further worked problems on the solution 30.4 Graphs of the form y = ae kx 281 of triangles and their areas 224 25.5 Practical situations involving 31 Graphical solution of equations 284 trigonometry 226 31.1 Graphical solution of simultaneous 25.6 Further practical situations involving equations 284 trigonometry 228 31.2 Graphical solution of quadratic equations 285 31.3 Graphical solution of linear and quadratic 26 Trigonometric identities and equations 231 equations simultaneously 289 26.1 Trigonometric identities 231 31.4 Graphical solution of cubic equations 290 26.2 Worked problems on trigonometric identities 231 32 Functions and their curves 292 26.3 Trigonometric equations 232 32.1 Standard curves 292 26.4 Worked problems (i) on trigonometric 32.2 Simple transformations 294 equations 233 32.3 Periodic functions 299 26.5 Worked problems (ii) on trigonometric 32.4 Continuous and discontinuous functions 299 equations 234 32.5 Even and odd functions 299 26.6 Worked problems (iii) on trigonometric 32.6 Inverse functions 301 equations 235 26.7 Worked problems (iv) on trigonometric equations 236 Revision Test 8 305
viii Contents Section 5 Complex Numbers 307 33 Complex numbers 309 33.1 Cartesian complex numbers 309 ';~'~ 33.2' The Argand diagram 310 33.3~ Addition and subtraction of complex numbers 310 33.4 Multiplication and division of complex numbers 311 33.5 Complex equations 313 33.6 The polar form of a complex number 314 33.7 Multiplication and division in polar form 316 33.8 Applications of complex numbers 317 34 De Moivre's theorem 321 34.1 Introduction 321 34.2 Powers of complex numbers 321 34.3 Roots of complex numbers 322 Section 6 Vectors 325 35 Vectors 327 35.1 Introduction 327 35.2 Scalars and vectors 327 35.3 Drawing a vector 327 35.4 Addition of vectors by drawing 328 35.5 Resolving vectors into horizontal and vertical components 330 35.6 Addition of vectors by calculation 331 35.7 Vector subtraction 336 35.8 Relative velocity 338 35.9 i, j, and k notation 339 36 Methods of adding alternating waveforms' 341 36.1 Combination of two periodic functions 341 36.2 Plotting periodic functions 341 36.3 Determining resultant phasors by drawing 343 36.4 Determining resultant phasors by the sine and cosine rules 344 36.5 Determining resultant phasors by horizontal and vertical components 346 36.6 Determining resultant phasors by complex numbers 348 Revision Test 9 351 Section 7 Statistics 353 37 Presentation of statistical data 37.1 Some statistical terminology 37.2 Presentation of ungrouped data 37.3 Presentation of grouped data 355 355 356 360 38 Measures of central tendency and dispersion 367 38.1 Measures of central tendency 367 38.2 Mean, median and mode for discrete data 367 38.3 Mean, median and mode for grouped data 368 38.4 Standard deviation 370 38.5 Quartiles, deciles and percentiles 372 39 Probability 374 39.1 Introduction to probability 374 39.2 Laws of probability 375 39.3 Worked problems on probability 375 39.4 Further worked problems on probability 377 39.5 Permutations and combination& 379 Revision Test 10 381 40 The binomial and Poisson distributions 382 40.1 The binomial distribution 382 40.2 The Poisson distribution 385 41 The normal distribution 388 41.1 Introduction to the normal distribution 388 41.2 Testing for a normal distribution 393 Revision Test 11 396 Multiple choice qmestions On Chapters 28-41 397 Section 8 Differential Calculus 403 42 Introduction to differentiation 405 42.1 Introduction to calculus 405 42.2 Functional notation 405 42.3 The gradient of a curve 406 42.4 Differentiation from first principles 407 42.5 Differentiation of y = ax n by the general rule 409 42.6 Differentiation of sine and cosine functions 411 42.7 Differentiation of e ax and In ax 412 43 Methods of differentiation 415 43.1 Differentiation of common functions 415 43.2 Differentiation of a product 417 43.3 Differentiation of a quotient 418 43.4 Function of a function 420 43.5 Successive differentiation 421 44 Some applications of differentiation 423 44.1 Rates of change 423 44.2 Velocity and acceleration 424 44.3 Turning points 427 44.4 Practical problems involving maximum and minimum values 431
Contents ix 44.5 Tangents and nonnals 44.6 Small changes Revision Test 12 437 45 Differentiation~f parametric equations 45.1 Introduction to parametric equations 45.2 Some common parametric equations 45.3 Differentiation in parameters 45.4 Further worked problems on differentiation of parametric equations 434 435 438 438 438 439 440 46 Differentiation of implicit functions 46.1 Implicit functions 443 443 46.2 Differentiating implicit functions 443 46.3 Differentiating implicit functions containing products and quotients 444 46.4 Further implicit differentiation 445 47 Logarithmic differentiation 448 47.1 Introduction to logarithmic differentiation 448 47.2 Laws oflogarithms 448 47.3 Differentiation oflogarithmic functions 448 47.4 Differentiation of further logarithmic functions 449 47.5 Differentiation of [f ex)y 451 Revision Test 13 453 Section 9 Integral Calculus 455 48 Standard integration 457 48.1 The process of integration 457 48.2 The general solution of integrals of the fonn ax" 457 48.3 Standard integrals 458 48.4 Definite integrals 461 49 Integration using algebraic substitutions 464 49.1 Introduction 464 49.2 Algebraic substitutions 464 49.3 Worked problems on integration using algebraic substitutions 464 49.4 Further worked problems on integration using algebraic substitutions 466 49.5 Change of limits 466 50 Integration using trigonometric substitutions 469 50.1 Introduction 469 50.2 Worked problems on integration of sin 2 x, cos 2 x, tan 2 x and cot 2 x 469 50.3 Worked problems on powers of sines and cosines 471 50.4 Worked problems on integration of products of sines and cosines 472 50.5 Worked problems on integration using the sin e substitution 473 50.6 Worked problems on integration using the tan e substitution 475 Revision Test 14 476 51 Integration using partial fractions 477 51.1 futroduction 477 51.2 Worked problems on integration using partial fractions with linear factors 477 51.3 Worked problems on integration using partial fractions with repeated linear factors 479 51.4 Worked problems on integration using partial fractions with quadratic factors 480 52 The t = tan substitution 482 52.1 futroduction 482 52.2 Worked problems on the t = tan ~ substitution 482 52.3 Further worked problems on the t = tan ~ substitution 484 53 Integration by parts 486 53.1 Introduction 486 53.2 Worked problems on integration by parts 486 53.3 Further worked problems on integration by parts 488 54 Numerical integration 491 54.1 Introduction 491 54.2 The trapezoidal rule 491 54.3 The mid-ordinate rule 493 54.4 Simpson's rule 495 Revision Test 15 499 55 Areas under and between curves 500 55.1 Area under a curve 500 55.2 Worked problems on the area under a curve 501 55.3 Further worked problems on the area under a curve 504 55.4 The area between curves 506 56 Mean and root mean square values 509 56.1 Mean or average values 509 56.2 Root mean square values 511 57 Volumes of solids of revolution 513 57.1 Introduction 513 57.2 Worked problems on volumes of solids of revolution 514 57.3 Further worked problems on volumes of solids of revolution 515
x Contents 58 Centroids of simple shapes 518 58.1 Centroids 518 58.2 The first moment of area 518 58.3 Centroid of area between a curve and the ~~-axis 518 58.4 ifcentroid of area between a curve and the y-axis 519 58.5 Worked problems on centroids of simple shapes 519 58.6 Further worked problems on centroids of simple shapes 520 58.7 Theorem of Pappus 523 59 Second moments of area 527 59.1 Second moments of area and radius of gyration 527 59.2 Second moment of area of regular sections 527 59.3 Parallel axis theorem 528 59.4 Perpendicular axis theorem 528 59.5 Summary of derived results 528 59.6 Worked problems on second moments of area of regular sections 529 59.7 Worked problems on second moments of area of composite areas 532 Revision Test 16 534 Section 10 Further Number and Algebra 535 60 Boolean algebra and. logic circuits 537 60.1 Boolean algebra and switching circuits 537 60.2 Simplifying Boolean expressions 542 60.3 Laws and rules of Boolean algebra 542 60.4 De Morgan's laws 544 60.5 Kamaugh maps 545 60.6 Logic circuits 550 60.7 Universal logic gates 554 61 The theory of matrices and determinants 558 61.1 Matrix notation 558 61.2 Addition, subtraction and multiplication of matrices 558 61.3 The unit matrix 562 61.4 The determinant of a 2 by 2 matrix 562 61.5 The inverse or reciprocal of a 2 by 2 matrix 563 61.6 The determinant of a 3 by 3 matrix 564 61.7 The inverse or reciprocal of a 3 by 3 matrix 566 62 The solution of simultaneous equations by matrices and determinants 568 62.1 Solution of simultaneous equations by matrices 568 62.2 Solution of simultaneous equations by determinants 570 62.3 Solution of simultaneous equations using Cramers rule 574 Revision Test 17 575 Section 11 Differential Equations 577 63 Introduction to differential equations 579 63.1 Family of curves 579 63.2 Differential equations 580 63.3 The solution of equations of the form dy dx = f(x) 580 63.4 The solution of equations of the form dy. - = fey) dx 63.5 The solution of equations of the form dy. dx = I(x). fey) Revision Test 18 587 Multiple choice questions on Chapters 42-63 588 582 584 Answers to multiple choice questions 592 Index 593, On the Website 64 Linear correlation 1 64.1 Introduction to linear correlation 64.2 The product-moment formula for determining the linear correlation coefficient 64.3 The significance of a coefficient of correlation 2 64.4 Worked problems on linear correlation 2
Contents xi 65 Linear regression 6 66.3 The sampling distribution of the means 12 65.1 Introduction to linear regression 6 66.4 The estimation of population parameters 65.2 The least-squares regression lines 6 based on a large sample size 15 65.3 Worked problems on linear regression 7 66.5 Estimating the mean of a population based 66 Sampling and l~timation theories 12 66.1 Introd~ction 12 66.2 Sampling distributions 12 on a small sample size 20 Revision Test 19 24
Index Abscissa, 257 Acute angles, 192 Addition of vectors, 328, 331 Adjoint of matrix, 566 Algebra, 39, 49 Algebraic expression, 61 substitution, integration, 464 Amplitude, 209, 212 And-function, 537 And-gate, 550 Angle of depression, 201 elevation, 201 Angles of any magnitude, 204 compound, 238 double, 244 Angular velocity, 212 Approximations, 24 Arbitrary constant of integration, 457 Arc, 154 Arc length of sector, 157 of circle, 157 Area, 145 Area between curves, 506 of triangle, 222 of sector of circle, 157 under a curve, 500 Areas of common shapes, 146 composite figures, 151 irregular figures, 179 similar shapes, 152 Argand diagram, 310 Argument, 314 Arithmetic progression, 117-120 Astroid, 439 Average value, 367, 509 Average value of waveform, 182 Base, 11, 19,98 Binary, 19 to decimal, 19 to hexadecimal, 26 Binomial distribution, 382 expression, 125 series, 126-130 theorem, 126-130 practical problems, 131-133 Blunder, 28 BODMAS, 4, 45 Boolean algebra, 537 laws and rules, 542 Boundary conditions, 580 Boyle's law, 47 Brackets, 43 Calculations, 28 Calculator, 30, 106, 111, 195,219,316 Calculus, 405 Cancelling, 3 Cardioid, 439 Cartesian axes, 257 co-ordinates, 216 complex numbers, 309 form, 309 Centre of gravity, 518 Centroids, 518 Chain rule, 420 Change of limits, integration, 466 Charles's law, 47, 263 Chord, 154 Circle, 146, 154,293 equation of, 160,293 Circumference, 154 Class, 360 interval, 360 limits, 363 Coefficient of proportionality, 47 Cofactor of matrix, 564 Combinational logic networks, 551 Combination of waveforms, 341 Combinations, 123, 380 Common difference, 117 logarithms, 98 ratio, 120 Completing the square, 86, 96 Complex conjugate, 312 equations, 313 numbers, 309 applications of, 317 powers of, 321 roots of, 322 waveforms, 214 Compound angles, 238 Conditional probability, 374 Cone, 163 Continuous data, 355 functions, 299 Conversion of a sin (JJt + b cos (JJt into R sin (M + a), 240 Conversion tables and charts, 32 Co-ordinates, 257 Cosecant, 192 Cosine, 192 curves, 207 rule, 222 wave production, 207 Cotangent, 192 Couple, 546 Cramer's rule, 574 Cubic equations, 290, 292 units, 162 Cuboid, 162 Cumulative frequency curve, 361 distribution, 361, 364 Cycles of log graph paper, 277 Cycloid, 439. Cylinder, 162 Deciles, 372 Decimal fraction, 6 places, 7 system, 19 to binary, 20 via octal, 22 to hexadecimal, 25 Decimals, 6 Definite integrals, 461 Degree of a differential equation, 580 Degrees, 156 De Moivres theorem, 321 De Morgan's laws, 544 Denary number, 19 Denominator, 3 Dependent event, 374 variable, 47 Depression, angle of, 201 Derivatives, 407 Determinant, 562,564 Determinants, to solve simultaneous equations, 570 Determination of law, 263,269 involving logarithms, 272 Diameter, 154 Difference of two squares, 85 Differential calculus, 405 coefficient, 407
594 Index Differential equations, 580 dy solving - = f(x), 580 dx dy = f(x). f(x), 584 dx, dy -<j'" dx =,.y), 582 Differentiation, 405,407 applications of, 423 from first principles, 407 function of a function, 420 in parameters, 439 logarithmic, 448 methods of, 415 ofax", 409 of common functions, 415 of e ax and 1n ax, 412 of [f(x)y, 451 of implicit functions, 443 of parametric equations, 438 of product, 417 of quotient, 418 of sine and cosine functions, 411 successive, 421 Digits, 6 Direct proportion, 6, 47 Discontinuous function, 299 Discrete data, 355,370 Distribution curve, 388 Dividend, 49 Divisor, 49 Double angles, 244 Drawing a vector, 327 Elevation, angle of, 201 Ellipse, 146,293,438 Engineering notation, 17 Equation of a circle, 160 Equations, 61,84 complex, 313 cubic, 290, 292 graphical solution of, 284 indicial, 102 quadratic, 84 simple, 61-65 simultaneous, 69-74 solving by iterative methods, 134 Newton-Raphson,134 trigonometric, 232-237 Errors, 28 Expectation, 374 Explicit function, 443 Exponent, 15,98 Exponential functions, 106 graphs of, 109, 294 series, 107 Extrapolation, 263 Evaluation of formulae, 33 trigonometric ratios, 195 Even function, 299 Factorisation, 43, 84, 96 Factor theorem, 51 False axes, 264 Family of curves, 579 Finite discontinuities, 299 First moment of area, 518, 527 order differential equations, 580 Formula, 33 quadratic, 88 Formulae, evaluation of, 33 transposition of, 78-83 Fractional form of trigonometric ratios, 194 Fractions, 3 partial, 55-60 Frequency, 212, 355 curve, 388 distribution, 360,363 polygon, 361, 363 relative, 355 Frustum of pyramids and cones, 170 sphere, 173 Functional notation, 405, 407 Function of a function rule, 420 Functions and their curves, 292 Fundamental, 214 General solution, 580 Geometric progression, 120-123 Gradient of a curve, 406 straight line graph, 257 Graphical solution of equations, 284 Graphs, 257 of cubic equations, 290,'exponential functions, 109 linear and quadratic equations simultaneously, 289 logarithmic functions, 103 quadratic equations, 285 simultaneous equations, 284 straight lines, 257 trigonometric functions, 204 y = ab x, 280 y = ae kx, 281 y = ax", 277 Graphs with logarithmic scales, 277 Grouped data, 360, 371 Growth and decay laws, 113 Harmonic analysis, 214 Harmonics, 214 H.C.P.,41 Heptagon, 145 Hexadecimal numbers, 19,24 to binary, 27 to decimal, 24 Hexagon, 145, 151 Histogram, 360, 362, 364, 369 of probability, 384, 386 Hooke's law, 47, 264 Horizontal bar chart, 356, 357 component of vector, 330 Hyperbola, 294, 438 rectangular, 294, 438 Hyperbolic logarithms, 98, 111 Hypotenuse, 191 Identity, 61 trigonometric, 231 i, j, k notation, 339 Imaginary part of complex number, 309 Implicit functions, 443 differentiation of, 443 Improper fraction, 3 Independent event, 374 variable, 47 Index, 98 Indices, 11 laws of, 11,41 Indicial equations, 102 Inequalities, 92-97 Integral calculus, 405 definite, 461 indefinite, 461 Integrals, standard, 458 Integration, 405,457 areas under and between curves, 500 by parts, 486 centroids,518 mean values, 509 numerical, 491 r.m.s. values, 511 second moment of area, 527 volumes, 513 Integration using algebraic substitutions, 464 partial fractions, 477 e t = tan - substitution, 482 2 trigonometric substitutions, 469 Intercept, 258 Interpolation, 263 Inverse functions, 301 proportion, 6, 47 trigonometric functions, 302 Inverse matrix, 2 by 2, 563 3 by 3, 566 Invert-gate, 550 Irregular areas and volumes, 179 Iterative methods, 134
Index 595 Kamaugh maps, 545 Lagging angles, 210 Lamina, 518 Laws of algebra, 39 1'" Boo1ean algebra, ~2 growth and decay, 113 indices, 11,41 logarithms, 100,448 precedence, 45 probability, 375 L.e.M.,3 Leading angles, 210 Leibniz notation, 407 Limiting value, 407 Linear and quadratic equations simultaneously, 91 graphically, 289 Logarithmic differentiation, 448 Logarithmic graphs, 277 scales, 277 Logarithms, 98 determination of law, 272 graphs of, 103,294 laws of, 100, 448 Logic circuits, 550 universal, 554 Log-linear graph paper, 280 Log-log graph paper, 277 Lower class boundary value, 360 Mantissa, 15 Matrices, 558 to solve simultaneous equations, 568 Matrix, 558 adjoint, 566 determinant of, 562,564 inverse, 563,566 reciprocal, 563, 566 unit, 562 Maximum value, 285, 427 Maximum and minimum problems, 431 Mean value of waveform, 182 Mean values, 182, 367, 509 Measures of central tendency, 367 for grouped data, 368 Median, 367 Mid-ordinate rule, 179,493 Minimum value, 285, 427 Minor of matrix, 564 Mixed number, 3 Modal value, 367 Modulus, 93, 314 Multiple-choice questions, 138,249, 397,588 Nand-gate, 550 Napierian logarithms, 98, 111 Natural logarithms, 98, 111 Newton-Raphson method, 134-136 Non-terminating decimal, 6 Nor-gate, 551 Normal curve, 388 distribution, 393 testing for, 393 probability paper, 393 standard variate, 388 Normals, 434 Nose-to-tail method, 328 Not-function, 537 Not-gate, 550 Number sequences, 117 Numerator, 3 Numerical integration, 491 Octagon; 145, 151 Octal, 19,22 Odd function, 299 Ogive, 361,364 Ohm's law, 47 Order of magnitude error, 28 Ordinate, 257 Or-function, 537 Or-gate, 550 Pappus' theorem, 523 Parabola, 285, 438 Parallel-axis theorem, 528 Parallelogram, 145, 146 method, 328 Parameter, 438 Parametric equations, 438 Partial fractions, 55-60 integration of, 477-481 Particular solution, 580 Pascal, 264 Pascal's triangle, 125 Pentagon, 145 Percentage component bar chart, 356, 357 relative frequency, 355 Percentages, 9 Percentiles, 372 Perfect square, 85 Period, 209 Periodic function, 208,299 plotting, 341 Periodic time, 212 Permutations, 124, 379 Perpendicular-axis theorem, 528 Phasor,211 Pictograms, 356 Pie diagram, 356, 358 Planimeter, 179 Plotting periodic functions, 341 Points of inflexion, 427 Poisson distribution, 385 Polar co-ordinates, 216 curves, 294 form of complex numbers, 309,314 Pol/Rec function on calculator, 219, 316 Polygon, 145 Polynomial, 49 Polynomial division, 49 Population, 355 Power, 11,98 of complex number, 321 series for ex, 107 Practical problems, binomial theorem, 131-133 maximum and minimum, 431 quadratic equations, 89 simple equations, 65-67 simultaneous equations, 74-77 straight line graphs, 263 trigonometry, 226 Prefixes, 17, 18 Presentation of grouped data, 360 statistical data, 355 ungrouped data, 356 Prism, 162 Prismoidal rule, 176 Probability, 374 laws of, 375 paper, 393 Product rule, differentiation, 417 Proper fraction, 3 Properties of circle, 154 quadrilaterals, 145 Proportionality, 47 Pyramid, 162 Pythagoras' theorem, 191,333 Quadrant, 154 Quadratic equations, 84, 285 by completing the square, 86 by factorisation, 84 by formula, 88 graphically, 285 practical problems, 89 Quadratic graph, 292 inequalities, 96 Quadrilaterals, 145 Quartiles, 372 Quotient rule, differentiation, 418 Quotients, 94
596 Index Radians, 156,212 Radius, 154 Radius of gyration, 527 Radix,19 Rates of cl;j.~ge, 423 Ratio and ~roportion, 5 Real part at complex number, 309 Reciprocal, 11 of a matrix, 563 ratios, 193 Rectangle, 145 Rectangular axes, 257 co-ordinates, 219 hyperbola, 294, 438 prism, 162 Reduction of non-linear to linear form, 269 Relative frequency, 355 velocity, 338 Remainder theorem, 53 Resultant phasor, by complex numbers, 348 drawing, 343 horizontal and vertical components, 346 sine and cosine rules, 344 Rhombus, 145, 146 Right angled triangle, solution of, 199 Root mean square value, 511 Roots of complex number, 322 equation, 84 Rounding off errors, 28 Sample data, 355 Scalar quantity, 327 Secant, 192 Second moments of area, 527 order differential equation, 580 Sector, 146,154 Segment, 154, 192 Semicircle, 146, 154 Semi-interquartile range, 372 Separation of variables, 580 Set, 355 Significant figures, 7 Simple equations, 61-65 practical problems, 65-67 Simpson's rule, 180,495 Simultaneous equations, 69-74 by Cramer's rule, 574 by determinants, 570 by matrices, 568 graphically, 284 practical problems, 74-77 Sine, 192 curves, 207 rule, 222 Sine wave, 182,204 production, 207 Sinusoidal form Asin(wt ± ex), 211 Slope, 257 Small changes, 435 Solid of revolution, 513 Solution of differential equations, 580 right-angled triangles, 199 Space diagram, 338 Sphere, 163 frustum of, 173 Square, 11, 145 root, 11 units, 145 Standard curves, 292 derivatives, 415 deviation, 370 form, 15 integrals, 457, 458 Stationary points, 427 Statistics, 355 terminology, 355 Straight line graphs, 257, 292 practical problems, 263 Subject of formula, 36 Subtraction of vectors, 336 Successive differentiation, 421 Sum to infinity of G.P., 120, 121 Surd form, 194 Surface area of regular solids, 162 frusta of pyramid and cones, 170 Switching circuits, 537 Symbols, 33 Table of partial areas under normal curve, 389, 390 Talley diagram, 360, 361, 363 Tangent, 154, 192,434 e Tan "2 substitution, 482 Terminating decimal: 6 Theorem of Pappus, 523 Pythagoras, 191 Transformations, 294 Transposition of formulae, 78-83 Trapezium, 145, 146 Trapezoidal rule, 179, 491 Triangle, 145 area of, 222 solution of, 222 Trigonometric approximation for small angles, 203 equations, 232-237 identities, 231 inverse function, 302 ratios, 192 substitution, integration, 469 waveforms, 204 Trigonometry, 191 evaluation of trigonometric ratios, 195 I practical situations, 226 right-angled triangles, solution of, 199 Truth tables, 537 Turning points, 285, 427 Two-state device, 537 Ungrouped data, 356 Unit matrix, 562 Universal logic gates, 554 Upper class boundary value, 360 Vector addition, 328,331 quantity, 327 subtraction, 336 Vectors, 327 drawing, 327 resolution by horizontal and vertical 'components, 330, Velocity and acceleration, 424 Velocity, angular, 212 relative, 338 Vertical bar chart, 356,357 component of vector, 330 Volume, 162 Volumes of frusta of pyramids and cones, 170 irregular solids, 181 regular solids, 162 similar shapes, 178 solids of revolution, 513 Waveform harmonics, 214 y-axis intercept, 259 Young's modulus of elasticity, 264 Zone of sphere, 173