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Higher Engineering Mathematics

745 Pages · 2007 · 15.57 MB · English

  • Higher Engineering Mathematics

    FM-H8152.tex 19/7/2006 18:59 Pagei


    HIGHER ENGINEERING MATHEMATICS FM-H8152.tex 19/7/2006 18:59 Pageii


    InmemoryofElizabeth Higher Engineering Mathematics


    Fifth Edition


    John Bird,


    BSc(Hons),CMath,FIMA,FIET,CEng,MIEE,CSci,FCollP,FIIE


    AMSTERDAM (cid:127) BOSTON (cid:127) HEIDELBERG (cid:127) LONDON (cid:127) NEW YORK (cid:127) OXFORD


    PARIS (cid:127) SAN DIEGO (cid:127) SAN FRANCISCO (cid:127) SINGAPORE (cid:127) SYDNEY (cid:127) TOKYO


    Newnes is an imprint of Elsevier FM-H8152.tex 19/7/2006 18:59 Pageiv


    Newnes


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    Firstpublished1993


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    Thirdedition1999


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    Copyright(cid:1)c 2006,JohnBird.PublishedbyElsevierLtd.Allrightsreserved


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    Contents


    Preface xv 5 Hyperbolicfunctions 41


    Syllabusguidance xvii 5.1 Introductiontohyperbolicfunctions 41


    5.2 Graphsofhyperbolicfunctions 43


    SectionA: Number andAlgebra 1 5.3 Hyperbolicidentities 44


    5.4 Solvingequationsinvolving


    1 Algebra 1 hyperbolicfunctions 47


    5.5 Seriesexpansionsforcoshxand


    1.1 Introduction 1 sinhx 48


    1.2 Revisionofbasiclaws 1


    1.3 Revisionofequations 3 Assignment1 50


    1.4 Polynomialdivision 6


    1.5 Thefactortheorem 8 6 Arithmeticandgeometricprogressions 51


    1.6 Theremaindertheorem 10


    6.1 Arithmeticprogressions 51


    2 Inequalities 12 6.2 Workedproblemsonarithmetic


    progressions 51


    2.1 Introductiontoinequalities 12 6.3 Furtherworkedproblemson


    2.2 Simpleinequalities 12 arithmeticprogressions 52


    2.3 Inequalitiesinvolvingamodulus 13 6.4 Geometricprogressions 54


    2.4 Inequalitiesinvolvingquotients 14 6.5 Workedproblemsongeometric


    2.5 Inequalitiesinvolvingsquare progressions 55


    functions 15 6.6 Furtherworkedproblemson


    2.6 Quadraticinequalities 16 geometricprogressions 56


    3 Partialfractions 18 7 Thebinomialseries 58


    3.1 Introductiontopartialfractions 18 7.1 Pascal’striangle 58


    3.2 Workedproblemsonpartialfractions 7.2 Thebinomialseries 59


    withlinearfactors 18 7.3 Workedproblemsonthebinomial


    3.3 Workedproblemsonpartialfractions series 59


    withrepeatedlinearfactors 21 7.4 Furtherworkedproblemsonthe


    3.4 Workedproblemsonpartialfractions binomialseries 61


    withquadraticfactors 22 7.5 Practicalproblemsinvolvingthe


    binomialtheorem 64


    4 Logarithmsandexponentialfunctions 24


    8 Maclaurin’sseries 67


    4.1 Introductiontologarithms 24


    4.2 Lawsoflogarithms 24 8.1 Introduction 67


    4.3 Indicialequations 26 8.2 DerivationofMaclaurin’stheorem 67


    4.4 Graphsoflogarithmicfunctions 27 8.3 ConditionsofMaclaurin’sseries 67


    4.5 Theexponentialfunction 28 8.4 WorkedproblemsonMaclaurin’s


    4.6 Thepowerseriesforex 29 series 68


    4.7 Graphsofexponentialfunctions 31 8.5 Numericalintegrationusing


    4.8 Napierianlogarithms 33 Maclaurin’sseries 71


    4.9 Lawsofgrowthanddecay 35 8.6 Limitingvalues 72


    4.10 Reductionofexponentiallawsto


    linearform 38 Assignment2 75 FM-H8152.tex 19/7/2006 18:59 Pagevi


    vi CONTENTS


    9 Solvingequationsbyiterativemethods 76 13 Cartesianandpolarco-ordinates 133


    9.1 Introductiontoiterativemethods 76 13.1 Introduction 133


    9.2 Thebisectionmethod 76 13.2 ChangingfromCartesianintopolar


    9.3 Analgebraicmethodofsuccessive co-ordinates 133


    approximations 80 13.3 ChangingfrompolarintoCartesian


    9.4 TheNewton-Raphsonmethod 83 co-ordinates 135


    13.4 UseofR→PandP→Rfunctions


    oncalculators 136


    10 Computernumberingsystems 86


    10.1 Binarynumbers 86 14 Thecircleanditsproperties 137


    10.2 Conversionofbinarytodenary 86


    14.1 Introduction 137


    10.3 Conversionofdenarytobinary 87


    14.2 Propertiesofcircles 137


    10.4 Conversionofdenarytobinary


    14.3 Arclengthandareaofasector 138


    viaoctal 88


    14.4 Workedproblemsonarclengthand


    10.5 Hexadecimalnumbers 90


    sectorofacircle 139


    14.5 Theequationofacircle 140


    11 Booleanalgebraandlogiccircuits 94 14.6 Linearandangularvelocity 142


    14.7 Centripetalforce 144


    11.1 Booleanalgebraandswitching


    circuits 94 Assignment4 146


    11.2 SimplifyingBooleanexpressions 99


    11.3 LawsandrulesofBooleanalgebra 99 15 Trigonometricwaveforms 148


    11.4 DeMorgan’slaws 101


    15.1 Graphsoftrigonometricfunctions 148


    11.5 Karnaughmaps 102


    15.2 Anglesofanymagnitude 148


    11.6 Logiccircuits 106


    15.3 Theproductionofasineand


    11.7 Universallogicgates 110


    cosinewave 151


    15.4 Sineandcosinecurves 152


    Assignment3 114


    15.5 SinusoidalformAsin(ωt±α) 157


    15.6 Harmonicsynthesiswithcomplex


    waveforms 160


    Section B: Geometry and


    trigonometry 115


    16 Trigonometricidentitiesandequations 166


    12 Introductiontotrigonometry 115 16.1 Trigonometricidentities 166


    16.2 Workedproblemsontrigonometric


    12.1 Trigonometry 115 identities 166


    12.2 ThetheoremofPythagoras 115 16.3 Trigonometricequations 167


    12.3 Trigonometricratiosofacute 16.4 Workedproblems(i)on


    angles 116 trigonometricequations 168


    12.4 Solutionofright-angledtriangles 118 16.5 Workedproblems(ii)on


    12.5 Anglesofelevationanddepression 119 trigonometricequations 169


    12.6 Evaluatingtrigonometricratios 121 16.6 Workedproblems(iii)on


    12.7 Sineandcosinerules 124 trigonometricequations 170


    12.8 Areaofanytriangle 125 16.7 Workedproblems(iv)on


    12.9 Workedproblemsonthesolution trigonometricequations 171


    oftrianglesandfindingtheirareas 125


    12.10 Furtherworkedproblemson


    17 Therelationshipbetweentrigonometricand


    solvingtrianglesandfinding


    hyperbolicfunctions 173


    theirareas 126


    12.11 Practicalsituationsinvolving 17.1 Therelationshipbetween


    trigonometry 128 trigonometricandhyperbolic


    12.12 Furtherpracticalsituations functions 173


    involvingtrigonometry 130 17.2 Hyperbolicidentities 174 FM-H8152.tex 19/7/2006 18:59 Pagevii


    CONTENTS vii


    18 Compoundangles 176 22.3 Vectorproducts 241


    22.4 Vectorequationofaline 245


    18.1 Compoundangleformulae 176


    18.2 Conversionofasinωt+bcosωt


    Assignment6 247


    intoRsin(ωt+α) 178


    18.3 Doubleangles 182


    18.4 Changingproductsofsinesand


    Section E: Complex numbers 249


    cosinesintosumsordifferences 183


    18.5 Changingsumsordifferencesof


    23 Complexnumbers 249


    sinesandcosinesintoproducts 184


    18.6 Powerwaveformsina.c.circuits 185 23.1 Cartesiancomplexnumbers 249


    23.2 TheArganddiagram 250


    Assignment5 189 23.3 Additionandsubtractionofcomplex


    numbers 250


    23.4 Multiplicationanddivisionof


    Section C: Graphs 191


    complexnumbers 251


    23.5 Complexequations 253


    19 Functionsandtheircurves 191 23.6 Thepolarformofacomplex


    number 254


    19.1 Standardcurves 191


    23.7 Multiplicationanddivisioninpolar


    19.2 Simpletransformations 194


    form 256


    19.3 Periodicfunctions 199


    23.8 Applicationsofcomplexnumbers 257


    19.4 Continuousanddiscontinuous


    functions 199


    19.5 Evenandoddfunctions 199 24 DeMoivre’stheorem 261


    19.6 Inversefunctions 201


    24.1 Introduction 261


    19.7 Asymptotes 203


    24.2 Powersofcomplexnumbers 261


    19.8 Briefguidetocurvesketching 209


    24.3 Rootsofcomplexnumbers 262


    19.9 Workedproblemsoncurve


    24.4 Theexponentialformofacomplex


    sketching 210


    number 264


    20 Irregularareas,volumesandmeanvaluesof


    Section F: Matrices and


    waveforms 216


    Determinants 267


    20.1 Areasofirregularfigures 216


    20.2 Volumesofirregularsolids 218


    25 Thetheoryofmatricesand


    20.3 Themeanoraveragevalueof


    determinants 267


    awaveform 219


    25.1 Matrixnotation 267


    Section D:Vector geometry 225 25.2 Addition,subtractionand


    multiplicationofmatrices 267


    21 Vectors,phasorsandthecombinationof 25.3 Theunitmatrix 271


    waveforms 225 25.4 Thedeterminantofa2by2matrix 271


    25.5 Theinverseorreciprocalofa2by


    21.1 Introduction 225 2matrix 272


    21.2 Vectoraddition 225 25.6 Thedeterminantofa3by3matrix 273


    21.3 Resolutionofvectors 227 25.7 Theinverseorreciprocalofa3by


    21.4 Vectorsubtraction 229 3matrix 274


    21.5 Relativevelocity 231


    21.6 Combinationoftwoperiodic 26 Thesolutionofsimultaneousequationsby


    functions 232 matricesanddeterminants 277


    26.1 Solutionofsimultaneousequations


    22 Scalarandvectorproducts 237


    bymatrices 277


    22.1 Theunittriad 237 26.2 Solutionofsimultaneousequations


    22.2 Thescalarproductoftwovectors 238 bydeterminants 279 FM-H8152.tex 19/7/2006 18:59 Pageviii


    viii CONTENTS


    26.3 Solutionofsimultaneousequations 31.3 Differentiationoflogarithmic


    usingCramersrule 283 functions 324


    26.4 Solutionofsimultaneousequations 31.4 Differentiationof[f(x)]x 327


    usingtheGaussianelimination


    method 284 Assignment8 329


    Assignment7 286 32 Differentiationofhyperbolicfunctions 330


    32.1 Standarddifferentialcoefficientsof


    SectionG:Differentialcalculus 287 hyperbolicfunctions 330


    32.2 Furtherworkedproblemson


    27 Methodsofdifferentiation 287 differentiationofhyperbolic


    functions 331


    27.1 Thegradientofacurve 287


    27.2 Differentiationfromfirstprinciples 288


    33 Differentiationofinversetrigonometricand


    27.3 Differentiationofcommon


    hyperbolicfunctions 332


    functions 288


    27.4 Differentiationofaproduct 292 33.1 Inversefunctions 332


    27.5 Differentiationofaquotient 293 33.2 Differentiationofinverse


    27.6 Functionofafunction 295 trigonometricfunctions 332


    27.7 Successivedifferentiation 296 33.3 Logarithmicformsoftheinverse


    hyperbolicfunctions 337


    28 Someapplicationsofdifferentiation 298 33.4 Differentiationofinversehyperbolic


    functions 338


    28.1 Ratesofchange 298


    28.2 Velocityandacceleration 299


    34 Partialdifferentiation 343


    28.3 Turningpoints 302


    28.4 Practicalproblemsinvolving 34.1 Introductiontopartial


    maximumandminimumvalues 306 derivaties 343


    28.5 Tangentsandnormals 310 34.2 Firstorderpartialderivatives 343


    28.6 Smallchanges 311 34.3 Secondorderpartialderivatives 346


    29 Differentiationofparametric


    35 Totaldifferential,ratesofchangeand


    equations 314


    smallchanges 349


    29.1 Introductiontoparametric


    35.1 Totaldifferential 349


    equations 314


    35.2 Ratesofchange 350


    29.2 Somecommonparametric


    35.3 Smallchanges 352


    equations 314


    29.3 Differentiationinparameters 314


    36 Maxima,minimaandsaddlepointsfor


    29.4 Furtherworkedproblemson


    functionsoftwovariables 355


    differentiationofparametric


    equations 316


    36.1 Functionsoftwoindependent


    variables 355


    30 Differentiationofimplicitfunctions 319


    36.2 Maxima,minimaandsaddlepoints 355


    36.3 Proceduretodeterminemaxima,


    30.1 Implicitfunctions 319


    minimaandsaddlepointsfor


    30.2 Differentiatingimplicitfunctions 319


    functionsoftwovariables 356


    30.3 Differentiatingimplicitfunctions


    36.4 Workedproblemsonmaxima,


    containingproductsandquotients 320


    minimaandsaddlepointsfor


    30.4 Furtherimplicitdifferentiation 321


    functionsoftwovariables 357


    36.5 Furtherworkedproblemson


    31 Logarithmicdifferentiation 324


    maxima,minimaandsaddlepoints


    31.1 Introductiontologarithmic forfunctionsoftwovariables 359


    differentiation 324


    31.2 Lawsoflogarithms 324 Assignment9 365 FM-H8152.tex 19/7/2006 18:59 Pageix


    CONTENTS ix


    Section H: Integral calculus 367 41 Integrationusingpartialfractions 408


    41.1 Introduction 408


    37 Standardintegration 367


    41.2 Workedproblemsonintegrationusing


    37.1 Theprocessofintegration 367 partialfractionswithlinearfactors 408


    37.2 Thegeneralsolutionofintegralsof 41.3 Workedproblemsonintegration


    theformaxn 367 usingpartialfractionswithrepeated


    37.3 Standardintegrals 367 linearfactors 409


    37.4 Definiteintegrals 371 41.4 Workedproblemsonintegration


    usingpartialfractionswithquadratic


    factors 410


    38 Someapplicationsofintegration 374


    38.1 Introduction 374 42 Thet=tanθ substitution 413


    38.2 Areasunderandbetweencurves 374 2


    38.3 Meanandr.m.s.values 376 42.1 Introduction 413


    θ


    38.4 Volumesofsolidsofrevolution 377 42.2 Workedproblemsonthet=tan


    38.5 Centroids 378 2


    substitution 413


    38.6 TheoremofPappus 380


    42.3 Furtherworkedproblemsonthe


    38.7 Secondmomentsofareaofregular


    θ


    sections 382 t= tan substitution 415


    2


    39 Integrationusingalgebraic Assignment11 417


    substitutions 391


    39.1 Introduction 391


    43 Integrationbyparts 418


    39.2 Algebraicsubstitutions 391


    39.3 Workedproblemsonintegration 43.1 Introduction 418


    usingalgebraicsubstitutions 391 43.2 Workedproblemsonintegration


    39.4 Furtherworkedproblemson byparts 418


    integrationusingalgebraic 43.3 Furtherworkedproblemson


    substitutions 393 integrationbyparts 420


    39.5 Changeoflimits 393


    Assignment10 396 44 Reductionformulae 424


    44.1 Introduction 424


    44.2 Usingreductionform(cid:1)ulaefor


    40 Integrationusingtrigonometricand integralsoftheform xnexdx 424


    hyperbolicsubstitutions 397 44.3 Usingreductionform(cid:1)ulaefor


    i(cid:1)ntegralsoftheform xncosxdxand


    40.1 Introduction 397


    xn sinxdx 425


    40.2 Workedproblemsonintegrationof


    44.4 Usingreductionform(cid:1)ulaefor


    sin2x,cos2x,tan2xandcot2x 397 (cid:1)integralsoftheform sinnxdxand


    40.3 Workedproblemsonpowersof


    cosnxdx 427


    sinesandcosines 399


    44.5 Furtherreductionformulae 430


    40.4 Workedproblemsonintegrationof


    productsofsinesandcosines 400


    40.5 Workedproblemsonintegration


    45 Numericalintegration 433


    usingthesinθ substitution 401


    40.6 Workedproblemsonintegration 45.1 Introduction 433


    usingtanθ substitution 403 45.2 Thetrapezoidalrule 433


    40.7 Workedproblemsonintegration 45.3 Themid-ordinaterule 435


    usingthesinhθ substitution 403 45.4 Simpson’srule 437


    40.8 Workedproblemsonintegration


    usingthecoshθ substitution 405 Assignment12 441


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