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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 AnimprintofElsevier LinacreHouse,JordanHill,OxfordOX28DP 30CorporateDrive,Suite400,Burlington,MA01803,USA Firstpublished1993 Secondedition1995 Thirdedition1999 Reprinted2000(twice),2001,2002,2003 Fourthedition2004 Fifthedition2006 Copyright(cid:1)c 2006,JohnBird.PublishedbyElsevierLtd.Allrightsreserved TherightofJohnBirdtobeindentifiedastheauthorofthisworkhasbeen assertedinaccordancewiththeCopyright,DesignsandPatentsAct1998 Nopartofthispublicationmaybereproduced,storedinaretrievalsystemor transmittedinanylausv orbyanymeanselectronic,mechanical,photocopying, recordingorotherwisewithoutthepriorwrittenpermissionofthepublisher PermissionmaybesoughtdirectlyfromElsevier’sScience&Technology RightsDepartmentinOxford,UK:phone(+44)(0)1865843830; fax(+44)(0)1865853333;email:[email protected] youcansubmityourrequestonlinebyvisitingtheElsevierwebsiteat http://elsevier.com/locate/permissions,andselectingObtainingpermission touseElseviermaterial Notice Noresponsibilityisassumedbythepublisherforanyinjuryand/ordamageto personsorpropertyasamatterofproductsliability,negligenceorotherwise,or fromanyuseoroperationofanymethods,products,instructionsorideas containedinthematerialherein.Becauseofrapidadvancesinthemedical sciences,inparticular,independentverificationofdiagnosesanddrugdosages shouldbemade BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary LibraryofCongressCataloging-in-PublicationData AcatalogrecordforthisbookisavailablefromtheLibraryofCongress ISBN13:9-78-0-75-068152-0 ISBN10:0-75-068152-7 ForinformationonallNewnespublications visitourwebsiteatbooks.elsevier.com TypesetbyCharonTecLtd,Chennai,India www.charontec.com PrintedandboundinGreatBritain 0607080910 10987654321 FM-H8152.tex 19/7/2006 18:59 Pagev 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