Engineering Mathematics

Transcription

1 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

2 Contents Preface xii 7 Partial fractions Introduction to partial fractions Worked problems on partial fractions with Section 1 Number and Algebra 1 linear factors Worked problems on partial fractions with 1 Revision of fractions, decimals and percentages 3 repeated linear factors I Fractions Worked problems on partial fractions with 1.2 Ratio and proportion 5 quadratic factors Decimals Percentages 9 8 Solving simple equations Expressions, equations and identities 61 2 Indices, standard form and engineering notation Worked problems on simple equations Indices Worked problems on indices Further worked problems on indices Standard form Worked problems on standard form Further worked problems on standard form Engineering notation and common prefixes Further worked problems on simple equations Practical problems involving simple equations Further practical problems involving simple equations 66 Revision Test Binary, octal and hexadecimal Introduction 19 9 Solving simultaneous equations Introduction to simultaneous equations Binary numbers Worked problems on simultaneous 3.3 Octal numbers 22 equations in two unknowns Hexadecimal numbers Further worked problems on simultaneous equations 71 4 Calculations and evaluation of formulae More difficult worked problems on 4.1 Errors and approximations 28 simultaneous equations Use of calculator Practical problems involving simultaneous 4.3 Conversion tables and charts 32 equations Evaluation of formulae Transposition of formulae 78 Revision Test Introduction to transposition of formulae Worked problems on transposition of formulae 78 5 Algebra Further worked problems on transposition 5.1 Basic operations 39 of formulae Laws of indices Harder worked problems on transposition 5.3 Brackets and factorisation 43 of formulae Fundamental laws and precedence Solving quadratic equations Direct and inverse proportionality Introduction to quadratic equations Solution of quadratic equations by 6 Further algebra 49 factorisation Polynomial division Solution of quadratic equations by 6.2 The factor theorem 51 'completing the square' The remainder theorem Solution of quadratic equations by formula 88

3 vi Contents 11.5 Practical pro blems involving quadratic equations The solution of linear and quadratic equations simultaneously Ineqti~lities i Introduction to inequalities Simple inequalities Inequalities involving a modulus Inequalities involving quotients Inequalities involving square functions Quadratic inequalities Logarithms Introduction to logarithms Laws of logarithms Indicial equations Graphs of logarithmic functions 103 Revision Test Exponential functions Introduction to exponential functions The power series for ex Graphs of exponential functions Napierian logarithms Laws of growth and decay Number sequences Arithmetic progressions Worked problems on arithmetic progressions Further worked problems on arithmetic progressions Geometric progressions Worked problems on geometric progressions Further worked problems on geometric progressions Combinations and permutations The binomial series Pascal's triangle The binomial series Worked problems on the binomial series Further worked problems on the binomial series Practical problems involving the binomial theorem Solving equations by iterative methods Introduction to iterative methods The Newton-Raphsonmethod Worked problems on the Newton-Raphson method 135 Revision Test Section 2 Areas and Volumes Areas of common shapes Introduction Properties of quadrilaterals Areas of common shapes Worked problems on areas of common shapes Further worked problems on a,reas of plane figures Worked problems on areas of composite figures Areas of similar shapes The circle Introduction Properties of circles Radians and degrees Arc length and area of circles and sectors Worked problems on arc length and area of circles and sectors The equation of a circle Volumes and surface areas of common solids Introduction Volumes and surface areas of regular solids Worked problems on volumes and surface areas ofregularsolids Further worked problems on volumes and surface areas of regular solids Volumes and surface areas of frusta of pyramids and cones The frustum and zone of a sphere Prismoidal rule Volumes of similar shapes Irregular areas and volumes and mean values of waveforms Area of irregular figures Volumes of irregular solids The mean or average value of a waveform 182 Revision Test Section 3 Trigonometry Introduction to trigonometry 22.1 Trigonometry 22.2 The theorem of Pythagoras

4 Contents vii 22.3 Trigonometric ratios of acute angles Compound angles Fractional and surd forms of trigonometric 27.1 Compound angle formulae 238 ratios Conversion of a sinwt + b cos wt into 22.5 Evaluating trigonometric ratios of any R sin(wt +a) 240 angles'i'" Double angles Solutio~ of right-angled triangles Angles of elevation and depression Changing products of sines and cosines into sums or differences Trigonometric approximations for small 27.5 Changing sums or differences of sines and angles 203 cosines into products Trigonometric waveforms 204 Revision Test Graphs of trigonometric functions Angles of any magnitude The production of a sine and cosine wave 207 Multiple choice questions on Chapters Sine and cosine curves Sinusoidal form Asin(wt ± a) Waveform harmonics 214 Section 4 Graphs Cartesian and polar co-ordinates Introduction Changing from Cartesian into polar co-ordinates Changing from polar into Cartesian co-ordinates Use of Pol/Rec functions on calculators 219 Revision Test Straight line graphs Introduction to graphs The straight line graph Practical problems involving straight line graphs Reduction of non-linear laws to linear form Determination oflaw Determination oflaw involving logarithms Triangles and some practical applications Sine and cosine rules Graphs with logarithmic scales Area of any triangle Logarithmic scales Worked problems on the solution of 30.2 Graphs of the form y = ax n 277 triangles and their areas Graphs of the form y = ab x Further worked problems on the solution 30.4 Graphs of the form y = ae kx 281 of triangles and their areas Practical situations involving 31 Graphical solution of equations 284 trigonometry Graphical solution of simultaneous 25.6 Further practical situations involving equations 284 trigonometry Graphical solution of quadratic equations Graphical solution of linear and quadratic 26 Trigonometric identities and equations 231 equations simultaneously Trigonometric identities Graphical solution of cubic equations Worked problems on trigonometric identities Functions and their curves Trigonometric equations Standard curves Worked problems (i) on trigonometric 32.2 Simple transformations 294 equations Periodic functions Worked problems (ii) on trigonometric 32.4 Continuous and discontinuous functions 299 equations Even and odd functions Worked problems (iii) on trigonometric 32.6 Inverse functions 301 equations Worked problems (iv) on trigonometric equations 236 Revision Test 8 305

5 viii Contents Section 5 Complex Numbers Complex numbers Cartesian complex numbers 309 ';~'~ 33.2' The Argand diagram ~ Addition and subtraction of complex numbers Multiplication and division of complex numbers Complex equations The polar form of a complex number Multiplication and division in polar form Applications of complex numbers De Moivre's theorem Introduction Powers of complex numbers Roots of complex numbers 322 Section 6 Vectors Vectors Introduction Scalars and vectors Drawing a vector Addition of vectors by drawing Resolving vectors into horizontal and vertical components Addition of vectors by calculation Vector subtraction Relative velocity i, j, and k notation Methods of adding alternating waveforms' Combination of two periodic functions Plotting periodic functions Determining resultant phasors by drawing Determining resultant phasors by the sine and cosine rules Determining resultant phasors by horizontal and vertical components Determining resultant phasors by complex numbers 348 Revision Test Section 7 Statistics Presentation of statistical data 37.1 Some statistical terminology 37.2 Presentation of ungrouped data 37.3 Presentation of grouped data Measures of central tendency and dispersion Measures of central tendency Mean, median and mode for discrete data Mean, median and mode for grouped data Standard deviation Quartiles, deciles and percentiles Probability Introduction to probability Laws of probability Worked problems on probability Further worked problems on probability Permutations and combination& 379 Revision Test The binomial and Poisson distributions The binomial distribution The Poisson distribution The normal distribution Introduction to the normal distribution Testing for a normal distribution 393 Revision Test Multiple choice qmestions On Chapters Section 8 Differential Calculus Introduction to differentiation Introduction to calculus Functional notation The gradient of a curve Differentiation from first principles Differentiation of y = ax n by the general rule Differentiation of sine and cosine functions Differentiation of e ax and In ax Methods of differentiation Differentiation of common functions Differentiation of a product Differentiation of a quotient Function of a function Successive differentiation Some applications of differentiation Rates of change Velocity and acceleration Turning points Practical problems involving maximum and minimum values 431

6 Contents ix 44.5 Tangents and nonnals 44.6 Small changes Revision Test 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 Differentiation of implicit functions 46.1 Implicit functions Differentiating implicit functions Differentiating implicit functions containing products and quotients Further implicit differentiation Logarithmic differentiation Introduction to logarithmic differentiation Laws oflogarithms Differentiation oflogarithmic functions Differentiation of further logarithmic functions Differentiation of [f ex)y 451 Revision Test Section 9 Integral Calculus Standard integration The process of integration The general solution of integrals of the fonn ax" Standard integrals Definite integrals Integration using algebraic substitutions Introduction Algebraic substitutions Worked problems on integration using algebraic substitutions Further worked problems on integration using algebraic substitutions Change of limits Integration using trigonometric substitutions Introduction Worked problems on integration of sin 2 x, cos 2 x, tan 2 x and cot 2 x Worked problems on powers of sines and cosines Worked problems on integration of products of sines and cosines Worked problems on integration using the sin e substitution Worked problems on integration using the tan e substitution 475 Revision Test Integration using partial fractions futroduction Worked problems on integration using partial fractions with linear factors Worked problems on integration using partial fractions with repeated linear factors Worked problems on integration using partial fractions with quadratic factors The t = tan substitution futroduction Worked problems on the t = tan ~ substitution Further worked problems on the t = tan ~ substitution Integration by parts Introduction Worked problems on integration by parts Further worked problems on integration by parts Numerical integration Introduction The trapezoidal rule The mid-ordinate rule Simpson's rule 495 Revision Test Areas under and between curves Area under a curve Worked problems on the area under a curve Further worked problems on the area under a curve The area between curves Mean and root mean square values Mean or average values Root mean square values Volumes of solids of revolution Introduction Worked problems on volumes of solids of revolution Further worked problems on volumes of solids of revolution 515

7 x Contents 58 Centroids of simple shapes Centroids The first moment of area Centroid of area between a curve and the ~~-axis ifcentroid of area between a curve and the y-axis Worked problems on centroids of simple shapes Further worked problems on centroids of simple shapes Theorem of Pappus Second moments of area Second moments of area and radius of gyration Second moment of area of regular sections Parallel axis theorem Perpendicular axis theorem Summary of derived results Worked problems on second moments of area of regular sections Worked problems on second moments of area of composite areas 532 Revision Test Section 10 Further Number and Algebra Boolean algebra and. logic circuits Boolean algebra and switching circuits Simplifying Boolean expressions Laws and rules of Boolean algebra De Morgan's laws Kamaugh maps Logic circuits Universal logic gates The theory of matrices and determinants Matrix notation Addition, subtraction and multiplication of matrices The unit matrix The determinant of a 2 by 2 matrix The inverse or reciprocal of a 2 by 2 matrix The determinant of a 3 by 3 matrix The inverse or reciprocal of a 3 by 3 matrix The solution of simultaneous equations by matrices and determinants Solution of simultaneous equations by matrices Solution of simultaneous equations by determinants Solution of simultaneous equations using Cramers rule 574 Revision Test Section 11 Differential Equations Introduction to differential equations Family of curves Differential equations The solution of equations of the form dy dx = f(x) 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 Multiple choice questions on Chapters Answers to multiple choice questions 592 Index 593, On the Website 64 Linear correlation 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 Worked problems on linear correlation 2

8 Contents xi 65 Linear regression The sampling distribution of the means Introduction to linear regression The estimation of population parameters 65.2 The least-squares regression lines 6 based on a large sample size Worked problems on linear regression Estimating the mean of a population based 66 Sampling and l~timation theories Introd~ction Sampling distributions 12 on a small sample size 20 Revision Test 19 24

9 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, 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, theorem, practical problems, 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

10 594 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, simultaneous, solving by iterative methods, 134 Newton-Raphson,134 trigonometric, 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, Fractional form of trigonometric ratios, 194 Fractions, 3 partial, 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, 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, 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, trigonometric substitutions, 469 Intercept, 258 Interpolation, 263 Inverse functions, 301 proportion, 6, 47 trigonometric functions, 302 Inverse matrix, 2 by 2, by 3, 566 Invert-gate, 550 Irregular areas and volumes, 179 Iterative methods, 134

11 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, 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, integration of, 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, maximum and minimum, 431 quadratic equations, 89 simple equations, simultaneous equations, 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

12 596 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, practical problems, Simpson's rule, 180,495 Simultaneous equations, by Cramer's rule, 574 by determinants, 570 by matrices, 568 graphically, 284 practical problems, 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, Trapezium, 145, 146 Trapezoidal rule, 179, 491 Triangle, 145 area of, 222 solution of, 222 Trigonometric approximation for small angles, 203 equations, 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