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Higher Engineering Mathematics In memory of Elizabeth Higher Engineering Mathematics Sixth Edition John Bird ,BSc(Hons), CMath, CEng, CSci, FIMA,FIET, MIEE, FIIE,FCollT AMSTERDAM•BOSTON•HEIDELBERG•LONDON•NEWYORK•OXFORD PARIS•SANDIEGO•SANFRANCISCO•SINGAPORE•SYDNEY•TOKYO NewnesisanimprintofElsevier NewnesisanimprintofElsevier TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UK 30CorporateDrive,Suite400,Burlington,MA01803,USA Firstedition2010 Copyright©2010,JohnBird,PublishedbyElsevierLtd.Allrightsreserved. TherightofJohnBirdtobeidentifiedastheauthorofthisworkhasbeenassertedinaccordancewith theCopyright,DesignsandPatentsAct1988. Nopartofthispublicationmaybereproduced,storedinaretrievalsystemortransmittedinanyform orbyanymeanselectronic,mechanical,photocopying,recordingorotherwisewithoutthepriorwritten permissionofthepublisher. PermissionsmaybesoughtdirectlyfromElsevier’sScience&TechnologyRightsDepartmentinOxford, UK:phone(+44)(0)1865843830;fax(+44)(0)1865853333;email:[email protected]. AlternativelyyoucansubmityourrequestonlinebyvisitingtheElsevierwebsiteat http://elsevier.com/locate/permissions,andselectingObtainingpermissiontouseElseviermaterial. Notice Noresponsibilityisassumedbythepublisherforanyinjuryand/ordamagetopersonsorpropertyasamatter ofproductsliability,negligenceorotherwise,orfromanyuseoroperationofanymethods,products, instructionsorideascontainedinthematerialherein.Becauseofrapidadvancesinthemedicalsciences, inparticular,independentverificationofdiagnosesanddrugdosagesshouldbemade. BritishLibraryCataloguing-in-PublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary. LibraryofCongressCataloging-in-PublicationData AcataloguerecordforthisbookisavailablefromtheLibraryofCongress. ISBN:978-1-85-617767-2 ForinformationonallNewnespublications visitourWebsiteatwww.elsevierdirect.com Typesetby:diacriTech,India PrintedandboundinChina 10 11 12 13 14 15 10 9 8 7 6 5 4 3 2 1 Contents Preface xiii 6 Arithmeticandgeometricprogressions 51 6.1 Arithmeticprogressions 51 Syllabusguidance xv 6.2 Workedproblemsonarithmetic 1 Algebra 1 progressions 51 1.1 Introduction 1 6.3 Furtherworkedproblemsonarithmetic 1.2 Revisionofbasiclaws 1 progressions 52 1.3 Revisionofequations 3 6.4 Geometricprogressions 54 1.4 Polynomialdivision 6 6.5 Workedproblemsongeometric 1.5 Thefactortheorem 8 progressions 55 1.6 Theremaindertheorem 10 6.6 Furtherworkedproblemsongeometric progressions 56 2 Partialfractions 13 7 Thebinomialseries 58 2.1 Introductiontopartialfractions 13 7.1 Pascal’striangle 58 2.2 Workedproblemsonpartialfractionswith 7.2 Thebinomialseries 59 linearfactors 13 7.3 Workedproblemsonthebinomialseries 59 2.3 Workedproblemsonpartialfractionswith 7.4 Furtherworkedproblemsonthebinomial repeatedlinearfactors 16 series 62 2.4 Workedproblemsonpartialfractionswith 7.5 Practicalproblemsinvolvingthebinomial quadraticfactors 17 theorem 64 3 Logarithms 20 RevisionTest2 67 3.1 Introductiontologarithms 20 3.2 Lawsoflogarithms 22 8 Maclaurin’sseries 68 3.3 Indicialequations 24 8.1 Introduction 68 3.4 Graphsoflogarithmicfunctions 25 8.2 DerivationofMaclaurin’stheorem 68 8.3 ConditionsofMaclaurin’sseries 69 4 Exponentialfunctions 27 8.4 WorkedproblemsonMaclaurin’sseries 69 4.1 Introductiontoexponentialfunctions 27 8.5 NumericalintegrationusingMaclaurin’s 4.2 Thepowerseriesforex 28 series 73 4.3 Graphsofexponentialfunctions 29 8.6 Limitingvalues 74 4.4 Napierianlogarithms 31 9 Solvingequationsbyiterativemethods 77 4.5 Lawsofgrowthanddecay 34 9.1 Introductiontoiterativemethods 77 4.6 Reductionofexponentiallawsto 9.2 Thebisectionmethod 77 linearform 37 9.3 Analgebraicmethodofsuccessive approximations 81 RevisionTest1 40 9.4 TheNewton-Raphsonmethod 84 10 Binary,octalandhexadecimal 87 5 Hyperbolicfunctions 41 10.1 Introduction 87 5.1 Introductiontohyperbolicfunctions 41 10.2 Binarynumbers 87 5.2 Graphsofhyperbolicfunctions 43 10.3 Octalnumbers 90 5.3 Hyperbolicidentities 45 10.4 Hexadecimalnumbers 92 5.4 Solvingequationsinvolvinghyperbolic functions 47 RevisionTest3 96 5.5 Seriesexpansionsforcoshxandsinhx 49 vi Contents 11 Introductiontotrigonometry 97 15.5 Workedproblems(ii)ontrigonometric 11.1 Trigonometry 97 equations 156 11.2 ThetheoremofPythagoras 97 15.6 Workedproblems(iii)ontrigonometric 11.3 Trigonometricratiosofacuteangles 98 equations 157 11.4 Evaluatingtrigonometricratios 100 15.7 Workedproblems(iv)ontrigonometric 11.5 Solutionofright-angledtriangles 105 equations 157 11.6 Anglesofelevationanddepression 106 11.7 Sineandcosinerules 108 16 Therelationshipbetweentrigonometricand hyperbolicfunctions 159 11.8 Areaofanytriangle 108 16.1 Therelationshipbetweentrigonometric 11.9 Workedproblemsonthesolutionof andhyperbolicfunctions 159 trianglesandfindingtheirareas 109 16.2 Hyperbolicidentities 160 11.10 Furtherworkedproblemsonsolving trianglesandfindingtheirareas 110 17 Compoundangles 163 11.11 Practicalsituationsinvolving 17.1 Compoundangleformulae 163 trigonometry 111 17.2 Conversionofasinωt+bcosωt into 11.12 Furtherpracticalsituationsinvolving Rsin(ωt+α) 165 trigonometry 113 17.3 Doubleangles 169 12 Cartesianandpolarco-ordinates 117 17.4 Changingproductsofsinesandcosines 12.1 Introduction 117 intosumsordifferences 170 12.2 ChangingfromCartesianintopolar 17.5 Changingsumsordifferencesofsinesand co-ordinates 117 cosinesintoproducts 171 12.3 ChangingfrompolarintoCartesian 17.6 Powerwaveformsina.c.circuits 173 co-ordinates 119 12.4 UseofPol/Recfunctionsoncalculators 120 RevisionTest5 177 13 Thecircleanditsproperties 122 13.1 Introduction 122 18 Functionsandtheircurves 178 13.2 Propertiesofcircles 122 18.1 Standardcurves 178 13.3 Radiansanddegrees 123 18.2 Simpletransformations 181 13.4 Arclengthandareaofcirclesandsectors 124 18.3 Periodicfunctions 186 13.5 Theequationofacircle 127 18.4 Continuousanddiscontinuousfunctions 186 13.6 Linearandangularvelocity 129 18.5 Evenandoddfunctions 186 13.7 Centripetalforce 130 18.6 Inversefunctions 188 18.7 Asymptotes 190 18.8 Briefguidetocurvesketching 196 RevisionTest4 133 18.9 Workedproblemsoncurvesketching 197 14 Trigonometricwaveforms 134 19 Irregularareas,volumesandmeanvaluesof 14.1 Graphsoftrigonometricfunctions 134 waveforms 203 14.2 Anglesofanymagnitude 135 19.1 Areasofirregularfigures 203 14.3 Theproductionofasineandcosinewave 137 19.2 Volumesofirregularsolids 205 14.4 Sineandcosinecurves 138 19.3 Themeanoraveragevalueofawaveform 206 14.5 Sinusoidalform Asin(ωt±α) 143 14.6 Harmonicsynthesiswithcomplex RevisionTest6 212 waveforms 146 15 Trigonometricidentitiesandequations 152 20 Complexnumbers 213 15.1 Trigonometricidentities 152 20.1 Cartesiancomplexnumbers 213 15.2 Workedproblemsontrigonometric 20.2 TheArganddiagram 214 identities 152 20.3 Additionandsubtractionofcomplex 15.3 Trigonometricequations 154 numbers 214 15.4 Workedproblems(i)ontrigonometric 20.4 Multiplicationanddivisionofcomplex equations 154 numbers 216 vii Contents 20.5 Complexequations 217 25.4 Determiningresultantphasorsbythesine 20.6 Thepolarformofacomplexnumber 218 andcosinerules 268 20.7 Multiplicationanddivisioninpolarform 220 25.5 Determiningresultantphasorsby 20.8 Applicationsofcomplexnumbers 221 horizontalandverticalcomponents 270 25.6 Determiningresultantphasorsbycomplex numbers 272 21 DeMoivre’stheorem 225 21.1 Introduction 225 26 Scalarandvectorproducts 275 21.2 Powersofcomplexnumbers 225 26.1 Theunittriad 275 21.3 Rootsofcomplexnumbers 226 26.2 Thescalarproductoftwovectors 276 21.4 Theexponentialformofacomplex 26.3 Vectorproducts 280 number 228 26.4 Vectorequationofaline 283 22 Thetheoryofmatricesanddeterminants 231 RevisionTest8 286 22.1 Matrixnotation 231 22.2 Addition,subtractionandmultiplication ofmatrices 231 27 Methodsofdifferentiation 287 22.3 Theunitmatrix 235 27.1 Introductiontocalculus 287 22.4 Thedeterminantofa2by2matrix 235 27.2 Thegradientofacurve 287 22.5 Theinverseorreciprocalofa2by2matrix 236 27.3 Differentiationfromfirstprinciples 288 22.6 Thedeterminantofa3by3matrix 237 27.4 Differentiationofcommonfunctions 289 22.7 Theinverseorreciprocalofa3by3matrix 239 27.5 Differentiationofaproduct 292 27.6 Differentiationofaquotient 293 23 Thesolutionofsimultaneousequationsby 27.7 Functionofafunction 295 matricesanddeterminants 241 27.8 Successivedifferentiation 296 23.1 Solutionofsimultaneousequationsby matrices 241 28 Someapplicationsofdifferentiation 299 23.2 Solutionofsimultaneousequationsby 28.1 Ratesofchange 299 determinants 243 28.2 Velocityandacceleration 300 23.3 Solutionofsimultaneousequationsusing 28.3 Turningpoints 303 Cramersrule 247 28.4 Practicalproblemsinvolvingmaximum 23.4 Solutionofsimultaneousequationsusing andminimumvalues 307 theGaussianeliminationmethod 248 28.5 Tangentsandnormals 311 28.6 Smallchanges 312 RevisionTest7 250 29 Differentiationofparametricequations 315 29.1 Introductiontoparametricequations 315 24 Vectors 251 29.2 Somecommonparametricequations 315 24.1 Introduction 251 29.3 Differentiationinparameters 315 24.2 Scalarsandvectors 251 29.4 Furtherworkedproblemson 24.3 Drawingavector 251 differentiationofparametricequations 318 24.4 Additionofvectorsbydrawing 252 24.5 Resolvingvectorsintohorizontaland 30 Differentiationofimplicitfunctions 320 verticalcomponents 254 30.1 Implicitfunctions 320 24.6 Additionofvectorsbycalculation 255 30.2 Differentiatingimplicitfunctions 320 24.7 Vectorsubtraction 260 30.3 Differentiatingimplicitfunctions 24.8 Relativevelocity 262 containingproductsandquotients 321 24.9 i,j andknotation 263 30.4 Furtherimplicitdifferentiation 322 25 Methodsofaddingalternatingwaveforms 265 31 Logarithmicdifferentiation 325 25.1 Combinationoftwoperiodicfunctions 265 31.1 Introductiontologarithmicdifferentiation 325 25.2 Plottingperiodicfunctions 265 31.2 Lawsoflogarithms 325 25.3 Determiningresultantphasorsbydrawing 267 31.3 Differentiationoflogarithmicfunctions 325 viii Contents 31.4 Differentiationoffurtherlogarithmic 38 Someapplicationsofintegration 375 functions 326 38.1 Introduction 375 31.5 Differentiationof[f(x)]x 328 38.2 Areasunderandbetweencurves 375 38.3 Meanandr.m.s.values 377 RevisionTest9 330 38.4 Volumesofsolidsofrevolution 378 38.5 Centroids 380 38.6 TheoremofPappus 381 32 Differentiationofhyperbolicfunctions 331 32.1 Standarddifferentialcoefficientsof 38.7 Secondmomentsofareaofregular hyperbolicfunctions 331 sections 383 32.2 Furtherworkedproblemson 39 Integrationusingalgebraicsubstitutions 392 differentiationofhyperbolicfunctions 332 39.1 Introduction 392 33 Differentiationofinversetrigonometricand 39.2 Algebraicsubstitutions 392 hyperbolicfunctions 334 39.3 Workedproblemsonintegrationusing 33.1 Inversefunctions 334 algebraicsubstitutions 392 33.2 Differentiationofinversetrigonometric 39.4 Furtherworkedproblemsonintegration functions 334 usingalgebraicsubstitutions 394 33.3 Logarithmicformsoftheinverse 39.5 Changeoflimits 395 hyperbolicfunctions 339 33.4 Differentiationofinversehyperbolic RevisionTest11 397 functions 341 34 Partialdifferentiation 345 40 Integrationusingtrigonometricandhyperbolic 34.1 Introductiontopartialderivatives 345 substitutions 398 34.2 Firstorderpartialderivatives 345 40.1 Introduction 398 34.3 Secondorderpartialderivatives 348 40.2 Workedproblemsonintegrationofsin2x, cos2x,tan2x andcot2x 398 35 Totaldifferential,ratesofchangeandsmall 40.3 Workedproblemsonpowersofsinesand changes 351 cosines 400 35.1 Totaldifferential 351 40.4 Workedproblemsonintegrationof 35.2 Ratesofchange 352 productsofsinesandcosines 401 35.3 Smallchanges 354 40.5 Workedproblemsonintegrationusingthe sinθ substitution 402 36 Maxima,minimaandsaddlepointsforfunctions 40.6 Workedproblemsonintegrationusing oftwovariables 357 tanθ substitution 404 36.1 Functionsoftwoindependentvariables 357 40.7 Workedproblemsonintegrationusingthe 36.2 Maxima,minimaandsaddlepoints 358 sinhθ substitution 404 36.3 Proceduretodeterminemaxima,minima 40.8 Workedproblemsonintegrationusingthe andsaddlepointsforfunctionsoftwo coshθ substitution 406 variables 359 36.4 Workedproblemsonmaxima,minima 41 Integrationusingpartialfractions 409 andsaddlepointsforfunctionsoftwo 41.1 Introduction 409 variables 359 41.2 Workedproblemsonintegrationusing 36.5 Furtherworkedproblemsonmaxima, partialfractionswithlinearfactors 409 minimaandsaddlepointsforfunctionsof 41.3 Workedproblemsonintegrationusing twovariables 361 partialfractionswithrepeatedlinear factors 411 RevisionTest10 367 41.4 Workedproblemsonintegrationusing partialfractionswithquadraticfactors 412 37 Standardintegration 368 37.1 Theprocessofintegration 368 θ 42 Thet=tan substitution 414 37.2 Thegeneralsolutionofintegralsofthe 2 formaxn 368 42.1 Introduction θ 414 37.3 Standardintegrals 369 42.2 Workedproblemsonthet=tan 2 37.4 Definiteintegrals 372 substitution 415 ix Contents θ 42.3 Furtherworkedproblemsonthet= tan 48 Linearfirstorderdifferentialequations 456 2 48.1 Introduction 456 substitution 416 48.2 Proceduretosolvedifferentialequations dy RevisionTest12 419 oftheform +Py=Q 457 dx 48.3 Workedproblemsonlinearfirstorder differentialequations 457 43 Integrationbyparts 420 48.4 Furtherworkedproblemsonlinearfirst 43.1 Introduction 420 orderdifferentialequations 458 43.2 Workedproblemsonintegrationbyparts 420 43.3 Furtherworkedproblemsonintegration 49 Numericalmethodsforfirstorderdifferential byparts 422 equations 461 49.1 Introduction 461 49.2 Euler’smethod 461 44 Reductionformulae 426 44.1 Introduction 426 49.3 WorkedproblemsonEuler’smethod 462 44.2 Usingred(cid:2)uctionformulaeforintegralsof 49.4 AnimprovedEulermethod 466 theform xnexdx 426 49.5 TheRunge-Kuttamethod 471 44.3 Usingred(cid:2)uctionformulae(cid:2)forintegralsof theform xncosxdxand xn sinxdx 427 RevisionTest14 476 44.4 Usingred(cid:2)uctionformulae(cid:2)forintegralsof theform sinnxdxand cosnxdx 429 50 Secondorderdifferentialequationsoftheform 44.5 Furtherreductionformulae 432 d2y dy a +b +cy=0 477 dx2 dx 45 Numericalintegration 435 50.1 Introduction 477 45.1 Introduction 435 50.2 Proceduretosolvedifferentialequations 45.2 Thetrapezoidalrule 435 d2y dy 45.3 Themid-ordinaterule 437 oftheforma +b +cy=0 478 dx2 dx 45.4 Simpson’srule 439 50.3 Workedproblemsondifferentialequations d2y dy oftheforma +b +cy=0 478 RevisionTest13 443 dx2 dx 50.4 Furtherworkedproblemsonpractical differentialequationsoftheform d2y dy 46 Solutionoffirstorderdifferentialequationsby a +b +cy=0 480 separationofvariables 444 dx2 dx 46.1 Familyofcurves 444 51 Secondorderdifferentialequationsoftheform 46.2 Differentialequations 445 d2y dy 46.3 Thesolutionofequationsoftheform a +b +cy=f(x) 483 dy dx2 dx = f(x) 445 dx 51.1 Complementaryfunctionandparticular 46.4 Thesolutionofequationsoftheform integral 483 dy = f(y) 447 51.2 Proceduretosolvedifferentialequations dx d2y dy 46.5 Thesolutionofequationsoftheform oftheforma +b +cy=f(x) 483 dy dx2 dx = f(x)· f(y) 449 51.3 Workedproblemsondifferentialequations dx d2y dy oftheforma +b +cy=f(x) dx2 dx 47 Homogeneousfirstorderdifferentialequations 452 where f(x)isaconstantorpolynomial 484 47.1 Introduction 452 51.4 Workedproblemsondifferentialequations 47.2 Proceduretosolvedifferentialequations d2y dy dy oftheforma +b +cy=f(x) oftheform P =Q 452 dx2 dx dx where f(x)isanexponentialfunction 486 47.3 Workedproblemsonhomogeneousfirst 51.5 Workedproblemsondifferentialequations orderdifferentialequations 452 d2y dy 47.4 Furtherworkedproblemsonhomogeneous oftheformadx2+bdx+cy=f(x) firstorderdifferentialequations 454 where f(x)isasineorcosinefunction 488