Engineering mathematics PDF
Preview Engineering mathematics
Engineering Mathematics In memory of Elizabeth Engineering Mathematics Sixth edition John Bird ,BSc(Hons), CEng, CSci, CMath, FIET, MIEE, FIIE,FIMA,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 Firstedition 1989 Secondedition 1996 Reprinted 1998(twice),1999 Thirdedition 2001 Fourthedition 2003 Reprinted 2004 Fifthedition 2007 Sixthedition 2010 Copyright©2001,2003,2007,2010,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 AcatalogrecordforthisbookisavailablefromtheLibraryofCongress. ISBN:978-0-08-096562-8 ForinformationonallNewnespublications visitourWebsiteatwww.elsevierdirect.com Typesetby:diacriTech,India PrintedandboundinHongKong,China 10 11 12 13 14 10 9 8 7 6 5 4 3 2 1 Contents Preface xii 7 Partialfractions 55 7.1 Introductiontopartialfractions 55 7.2 Workedproblemsonpartialfractionswith Section 1 Number andAlgebra 1 linearfactors 55 7.3 Workedproblemsonpartialfractionswith 1 Revisionoffractions,decimalsandpercentages 3 repeatedlinearfactors 58 1.1 Fractions 3 7.4 Workedproblemsonpartialfractionswith 1.2 Ratioandproportion 5 quadraticfactors 59 1.3 Decimals 6 1.4 Percentages 9 8 Solvingsimpleequations 61 8.1 Expressions,equationsandidentities 61 8.2 Workedproblemsonsimpleequations 61 2 Indices,standardformandengineeringnotation 11 8.3 Furtherworkedproblemsonsimple 2.1 Indices 11 equations 63 2.2 Workedproblemsonindices 12 8.4 Practicalproblemsinvolvingsimple 2.3 Furtherworkedproblemsonindices 13 equations 65 2.4 Standardform 15 8.5 Furtherpracticalproblemsinvolving 2.5 Workedproblemsonstandardform 15 simpleequations 66 2.6 Furtherworkedproblemsonstandardform 16 2.7 Engineeringnotationandcommonprefixes 17 RevisionTest2 68 3 Binary,octalandhexadecimal 19 9 Solvingsimultaneousequations 69 3.1 Introduction 19 9.1 Introductiontosimultaneousequations 69 3.2 Binarynumbers 19 9.2 Workedproblemsonsimultaneous 3.3 Octalnumbers 22 equationsintwounknowns 69 3.4 Hexadecimalnumbers 24 9.3 Furtherworkedproblemsonsimultaneous equations 71 4 Calculationsandevaluationofformulae 28 9.4 Moredifficultworkedproblemson 4.1 Errorsandapproximations 28 simultaneousequations 73 4.2 Useofcalculator 30 9.5 Practicalproblemsinvolvingsimultaneous 4.3 Conversiontablesandcharts 32 equations 74 4.4 Evaluationofformulae 33 10 Transpositionofformulae 78 10.1 Introductiontotranspositionofformulae 78 RevisionTest1 38 10.2 Workedproblemsontranspositionof formulae 78 5 Algebra 39 10.3 Furtherworkedproblemsontransposition 5.1 Basicoperations 39 offormulae 79 5.2 Lawsofindices 41 10.4 Harderworkedproblemsontransposition 5.3 Bracketsandfactorisation 43 offormulae 81 5.4 Fundamentallawsandprecedence 45 11 Solvingquadraticequations 84 5.5 Directandinverseproportionality 47 11.1 Introductiontoquadraticequations 84 11.2 Solutionofquadraticequationsby 6 Furtheralgebra 49 factorisation 84 6.1 Polynomialdivision 49 11.3 Solutionofquadraticequationsby 6.2 Thefactortheorem 51 ‘completingthesquare’ 86 6.3 Theremaindertheorem 53 11.4 Solutionofquadraticequationsbyformula 88 vi Contents 11.5 Practicalproblemsinvolvingquadratic MultiplechoicequestionsonChapters1–17 138 equations 89 11.6 Thesolutionoflinearandquadratic equationssimultaneously 91 Section 2 AreasandVolumes 143 12 Inequalities 92 12.1 Introductiontoinequalities 92 18 Areasofcommonshapes 145 12.2 Simpleinequalities 92 18.1 Introduction 145 12.3 Inequalitiesinvolvingamodulus 93 18.2 Propertiesofquadrilaterals 145 12.4 Inequalitiesinvolvingquotients 94 18.3 Areasofcommonshapes 146 12.5 Inequalitiesinvolvingsquarefunctions 95 18.4 Workedproblemsonareasofcommon 12.6 Quadraticinequalities 96 shapes 146 13 Logarithms 98 18.5 Furtherworkedproblemsonareasof 13.1 Introductiontologarithms 98 planefigures 149 13.2 Lawsoflogarithms 100 18.6 Workedproblemsonareasofcomposite 13.3 Indicialequations 102 figures 151 13.4 Graphsoflogarithmicfunctions 103 18.7 Areasofsimilarshapes 152 19 Thecircle 154 RevisionTest3 105 19.1 Introduction 154 14 Exponentialfunctions 106 19.2 Propertiesofcircles 154 14.1 Introductiontoexponentialfunctions 106 19.3 Radiansanddegrees 156 14.2 Thepowerseriesforex 107 19.4 Arclengthandareaofcirclesandsectors 157 14.3 Graphsofexponentialfunctions 109 19.5 Workedproblemsonarclengthandarea ofcirclesandsectors 157 14.4 Napierianlogarithms 111 19.6 Theequationofacircle 160 14.5 Lawsofgrowthanddecay 113 15 Numbersequences 117 20 Volumesandsurfaceareasofcommonsolids 162 15.1 Arithmeticprogressions 117 20.1 Introduction 162 15.2 Workedproblemsonarithmetic 20.2 Volumesandsurfaceareasofregular progressions 117 solids 162 15.3 Furtherworkedproblemsonarithmetic 20.3 Workedproblemsonvolumesandsurface progressions 118 areasofregularsolids 163 15.4 Geometricprogressions 120 20.4 Furtherworkedproblemsonvolumesand 15.5 Workedproblemsongeometric surfaceareasofregularsolids 165 progressions 121 20.5 Volumesandsurfaceareasoffrustaof 15.6 Furtherworkedproblemsongeometric pyramidsandcones 170 progressions 122 20.6 Thefrustumandzoneofasphere 173 15.7 Combinationsandpermutations 123 20.7 Prismoidalrule 176 16 Thebinomialseries 125 20.8 Volumesofsimilarshapes 178 16.1 Pascal’striangle 125 21 Irregularareasandvolumesandmeanvalues 16.2 Thebinomialseries 126 ofwaveforms 179 16.3 Workedproblemsonthebinomialseries 126 21.1 Areaofirregularfigures 179 16.4 Furtherworkedproblemsonthebinomial 21.2 Volumesofirregularsolids 181 series 128 21.3 Themeanoraveragevalueofawaveform 182 16.5 Practicalproblemsinvolvingthebinomial theorem 131 RevisionTest5 187 17 Solvingequationsbyiterativemethods 134 17.1 Introductiontoiterativemethods 134 17.2 TheNewton–Raphsonmethod 134 Section 3 Trigonometry 189 17.3 Workedproblemsonthe Newton–Raphsonmethod 135 22 Introductiontotrigonometry 191 22.1 Trigonometry 191 RevisionTest4 137 22.2 ThetheoremofPythagoras 191 vii Contents 22.3 Trigonometricratiosofacuteangles 192 27 Compoundangles 238 22.4 Fractionalandsurdformsoftrigonometric 27.1 Compoundangleformulae 238 ratios 194 27.2 Conversionofasinωt+bcosωt into 22.5 Evaluatingtrigonometricratiosofany Rsin(ωt+α) 240 angles 195 27.3 Doubleangles 244 22.6 Solutionofright-angledtriangles 199 27.4 Changingproductsofsinesandcosines 22.7 Anglesofelevationanddepression 201 intosumsordifferences 245 22.8 Trigonometricapproximationsforsmall 27.5 Changingsumsordifferencesofsinesand angles 203 cosinesintoproducts 246 23 Trigonometricwaveforms 204 RevisionTest7 248 23.1 Graphsoftrigonometricfunctions 204 23.2 Anglesofanymagnitude 204 MultiplechoicequestionsonChapters18–27 249 23.3 Theproductionofasineandcosinewave 207 23.4 Sineandcosinecurves 207 23.5 Sinusoidalform Asin(ωt±α) 211 23.6 Waveformharmonics 214 Section 4 Graphs 255 24 Cartesianandpolarco-ordinates 216 28 Straightlinegraphs 257 24.1 Introduction 216 28.1 Introductiontographs 257 24.2 ChangingfromCartesianintopolar 28.2 Thestraightlinegraph 257 co-ordinates 216 28.3 Practicalproblemsinvolvingstraightline 24.3 ChangingfrompolarintoCartesian graphs 263 co-ordinates 218 24.4 UseofPol/Recfunctionsoncalculators 219 29 Reductionofnon-linearlawstolinearform 269 29.1 Determinationoflaw 269 RevisionTest6 221 29.2 Determinationoflawinvolving logarithms 272 25 Trianglesandsomepracticalapplications 222 25.1 Sineandcosinerules 222 30 Graphswithlogarithmicscales 277 30.1 Logarithmicscales 277 25.2 Areaofanytriangle 222 30.2 Graphsoftheformy=axn 277 25.3 Workedproblemsonthesolutionof trianglesandtheirareas 222 30.3 Graphsoftheformy=abx 280 25.4 Furtherworkedproblemsonthesolution 30.4 Graphsoftheformy=aekx 281 oftrianglesandtheirareas 224 25.5 Practicalsituationsinvolving 31 Graphicalsolutionofequations 284 trigonometry 226 31.1 Graphicalsolutionofsimultaneous 25.6 Furtherpracticalsituationsinvolving equations 284 trigonometry 228 31.2 Graphicalsolutionofquadraticequations 285 31.3 Graphicalsolutionoflinearandquadratic 26 Trigonometricidentitiesandequations 231 equationssimultaneously 289 26.1 Trigonometricidentities 231 31.4 Graphicalsolutionofcubicequations 290 26.2 Workedproblemsontrigonometric identities 231 32 Functionsandtheircurves 292 26.3 Trigonometricequations 232 32.1 Standardcurves 292 26.4 Workedproblems(i)ontrigonometric 32.2 Simpletransformations 294 equations 233 32.3 Periodicfunctions 299 26.5 Workedproblems(ii)ontrigonometric 32.4 Continuousanddiscontinuousfunctions 299 equations 234 32.5 Evenandoddfunctions 299 26.6 Workedproblems(iii)ontrigonometric 32.6 Inversefunctions 301 equations 235 26.7 Workedproblems(iv)ontrigonometric RevisionTest8 305 equations 236 viii Contents 38 Measuresofcentraltendencyanddispersion 367 Section 5 ComplexNumbers 307 38.1 Measuresofcentraltendency 367 38.2 Mean,medianandmodefordiscretedata 367 33 Complexnumbers 309 38.3 Mean,medianandmodeforgroupeddata 368 33.1 Cartesiancomplexnumbers 309 38.4 Standarddeviation 370 33.2 TheArganddiagram 310 38.5 Quartiles,decilesandpercentiles 372 33.3 Additionandsubtractionofcomplex numbers 310 39 Probability 374 33.4 Multiplicationanddivisionofcomplex 39.1 Introductiontoprobability 374 numbers 311 39.2 Lawsofprobability 375 33.5 Complexequations 313 39.3 Workedproblemsonprobability 375 33.6 Thepolarformofacomplexnumber 314 39.4 Furtherworkedproblemsonprobability 377 33.7 Multiplicationanddivisioninpolarform 316 39.5 Permutationsandcombinations 379 33.8 Applicationsofcomplexnumbers 317 RevisionTest10 381 34 DeMoivre’stheorem 321 34.1 Introduction 321 40 ThebinomialandPoissondistributions 382 34.2 Powersofcomplexnumbers 321 40.1 Thebinomialdistribution 382 34.3 Rootsofcomplexnumbers 322 40.2 ThePoissondistribution 385 41 Thenormaldistribution 388 Section 6 Vectors 325 41.1 Introductiontothenormaldistribution 388 41.2 Testingforanormaldistribution 393 35 Vectors 327 35.1 Introduction 327 RevisionTest11 396 35.2 Scalarsandvectors 327 35.3 Drawingavector 327 35.4 Additionofvectorsbydrawing 328 MultiplechoicequestionsonChapters28–41 397 35.5 Resolvingvectorsintohorizontaland verticalcomponents 330 35.6 Additionofvectorsbycalculation 331 Section 8 Differential Calculus 403 35.7 Vectorsubtraction 336 35.8 Relativevelocity 338 42 Introductiontodifferentiation 405 42.1 Introductiontocalculus 405 35.9 i, j,andknotation 339 42.2 Functionalnotation 405 36 Methodsofaddingalternatingwaveforms 341 42.3 Thegradientofacurve 406 36.1 Combinationoftwoperiodicfunctions 341 42.4 Differentiationfromfirstprinciples 407 36.2 Plottingperiodicfunctions 341 42.5 Differentiationofy=axn bythegeneral 36.3 Determiningresultantphasorsbydrawing 343 rule 409 36.4 Determiningresultantphasorsbythesine 42.6 Differentiationofsineandcosinefunctions 411 andcosinerules 344 42.7 Differentiationofeaxandlnax 412 36.5 Determiningresultantphasorsby horizontalandverticalcomponents 346 43 Methodsofdifferentiation 415 36.6 Determiningresultantphasorsbycomplex 43.1 Differentiationofcommonfunctions 415 numbers 348 43.2 Differentiationofaproduct 417 43.3 Differentiationofaquotient 418 RevisionTest9 351 43.4 Functionofafunction 420 43.5 Successivedifferentiation 421 Section 7 Statistics 353 44 Someapplicationsofdifferentiation 423 44.1 Ratesofchange 423 37 Presentationofstatisticaldata 355 44.2 Velocityandacceleration 424 37.1 Somestatisticalterminology 355 44.3 Turningpoints 427 37.2 Presentationofungroupeddata 356 44.4 Practicalproblemsinvolvingmaximum 37.3 Presentationofgroupeddata 360 andminimumvalues 431 ix Contents 44.5 Tangentsandnormals 434 50.4 Workedproblemsonintegrationof 44.6 Smallchanges 435 productsofsinesandcosines 472 50.5 Workedproblemsonintegrationusingthe sinθ substitution 473 RevisionTest12 437 50.6 Workedproblemsonintegrationusingthe tanθ substitution 475 45 Differentiationofparametricequations 438 45.1 Introductiontoparametricequations 438 RevisionTest14 476 45.2 Somecommonparametricequations 438 45.3 Differentiationinparameters 439 51 Integrationusingpartialfractions 477 45.4 Furtherworkedproblemson 51.1 Introduction 477 differentiationofparametricequations 440 51.2 Workedproblemsonintegrationusing 46 Differentiationofimplicitfunctions 443 partialfractionswithlinearfactors 477 46.1 Implicitfunctions 443 51.3 Workedproblemsonintegrationusing 46.2 Differentiatingimplicitfunctions 443 partialfractionswithrepeatedlinearfactors 479 46.3 Differentiatingimplicitfunctions 51.4 Workedproblemsonintegrationusing containingproductsandquotients 444 partialfractionswithquadraticfactors 480 46.4 Furtherimplicitdifferentiation 445 52 Thet=tanθ substitution 482 2 47 Logarithmicdifferentiation 448 52.1 Introduction 482 47.1 Introductiontologarithmicdifferentiation 448 52.2 Workedproblemsonthet =tanθ2 47.2 Lawsoflogarithms 448 substitution 482 47.3 Differentiationoflogarithmicfunctions 448 52.3 Furtherworkedproblemsonthet=tanθ2 substitution 484 47.4 Differentiationoffurtherlogarithmic functions 449 53 Integrationbyparts 486 47.5 Differentiationof[f(x)]x 451 53.1 Introduction 486 53.2 Workedproblemsonintegrationbyparts 486 RevisionTest13 453 53.3 Furtherworkedproblemsonintegration byparts 488 54 Numericalintegration 491 Section 9 IntegralCalculus 455 54.1 Introduction 491 54.2 Thetrapezoidalrule 491 48 Standardintegration 457 54.3 Themid-ordinaterule 493 48.1 Theprocessofintegration 457 54.4 Simpson’srule 495 48.2 Thegeneralsolutionofintegralsofthe formaxn 457 RevisionTest15 499 48.3 Standardintegrals 458 48.4 Definiteintegrals 461 55 Areasunderandbetweencurves 500 55.1 Areaunderacurve 500 49 Integrationusingalgebraicsubstitutions 464 55.2 Workedproblemsontheareaunderacurve 501 49.1 Introduction 464 55.3 Furtherworkedproblemsonthearea 49.2 Algebraicsubstitutions 464 underacurve 504 49.3 Workedproblemsonintegrationusing 55.4 Theareabetweencurves 506 algebraicsubstitutions 464 49.4 Furtherworkedproblemsonintegration 56 Meanandrootmeansquarevalues 509 usingalgebraicsubstitutions 466 56.1 Meanoraveragevalues 509 49.5 Changeoflimits 466 56.2 Rootmeansquarevalues 511 50 Integrationusingtrigonometricsubstitutions 469 57 Volumesofsolidsofrevolution 513 50.1 Introduction 469 57.1 Introduction 513 50.2 Workedproblemsonintegrationof 57.2 Workedproblemsonvolumesofsolidsof sin2x,cos2x,tan2xandcot2x 469 revolution 514 50.3 Workedproblemsonpowersofsinesand 57.3 Furtherworkedproblemsonvolumesof cosines 471 solidsofrevolution 515
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