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Basic Engineering Mathematics

377 Pages · 2013 · 3.35 MB · English

  • Basic Engineering Mathematics

    Basic Engineering Mathematics In memory of Elizabeth Basic Engineering Mathematics


    Fifth 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


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    Firstedition 1999


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    Thirdedition 2002


    Fourthedition 2005


    Fifthedition 2010


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    10 11 12 13 14 10 9 8 7 6 5 4 3 2 1 Contents


    Preface ix 6.3 Directproportion 42


    Acknowledgements x 6.4 Inverseproportion 45


    Instructor’sManual xi 7 Powers,rootsandlawsofindices 47


    7.1 Introduction 47


    1 Basicarithmetic 1


    7.2 Powersandroots 47


    1.1 Introduction 1


    7.3 Lawsofindices 48


    1.2 Revisionofadditionandsubtraction 1


    1.3 Revisionofmultiplicationanddivision 3 8 Units,prefixesandengineeringnotation 53


    1.4 Highestcommonfactorsandlowest 8.1 Introduction 53


    commonmultiples 5 8.2 SIunits 53


    1.5 Orderofprecedenceandbrackets 6 8.3 Commonprefixes 53


    8.4 Standardform 56


    2 Fractions 9


    8.5 Engineeringnotation 57


    2.1 Introduction 9


    2.2 Addingandsubtractingfractions 10


    RevisionTest3 60


    2.3 Multiplicationanddivisionoffractions 12


    2.4 Orderofprecedencewithfractions 13


    9 Basicalgebra 61


    9.1 Introduction 61


    RevisionTest1 15


    9.2 Basicoperations 61


    9.3 Lawsofindices 64


    3 Decimals 16


    3.1 Introduction 16 10 Furtheralgebra 68


    3.2 Convertingdecimalstofractionsand 10.1 Introduction 68


    vice-versa 16 10.2 Brackets 68


    3.3 Significantfiguresanddecimalplaces 17 10.3 Factorization 69


    3.4 Addingandsubtractingdecimalnumbers 18 10.4 Lawsofprecedence 71


    3.5 Multiplyinganddividingdecimalnumbers 19


    11 Solvingsimpleequations 73


    4 Usingacalculator 22 11.1 Introduction 73


    4.1 Introduction 22 11.2 Solvingequations 73


    4.2 Adding,subtracting,multiplyingand 11.3 Practicalproblemsinvolvingsimple


    dividing 22 equations 77


    4.3 Furthercalculatorfunctions 23


    4.4 Evaluationofformulae 28 RevisionTest4 82


    5 Percentages 33


    12 Transposingformulae 83


    5.1 Introduction 33


    12.1 Introduction 83


    5.2 Percentagecalculations 33


    12.2 Transposingformulae 83


    5.3 Furtherpercentagecalculations 35


    12.3 Furthertransposingofformulae 85


    5.4 Morepercentagecalculations 36


    12.4 Moredifficulttransposingofformulae 87


    RevisionTest2 39


    13 Solvingsimultaneousequations 90


    13.1 Introduction 90


    6 Ratioandproportion 40 13.2 Solvingsimultaneousequationsintwo


    6.1 Introduction 40 unknowns 90


    6.2 Ratios 40 13.3 Furthersolvingofsimultaneousequations 92 vi


    Contents


    13.4 Solvingmoredifficultsimultaneous 19.2 Graphicalsolutionofquadraticequations 156


    equations 94 19.3 Graphicalsolutionoflinearandquadratic


    13.5 Practicalproblemsinvolvingsimultaneous equationssimultaneously 160


    equations 96 19.4 Graphicalsolutionofcubicequations 161


    13.6 Solvingsimultaneousequationsinthree


    unknowns 99


    RevisionTest7 163


    RevisionTest5 101


    20 Anglesandtriangles 165


    20.1 Introduction 165


    14 Solvingquadraticequations 102


    20.2 Angularmeasurement 165


    14.1 Introduction 102


    20.3 Triangles 171


    14.2 Solutionofquadraticequationsby


    20.4 Congruenttriangles 175


    factorization 102


    20.5 Similartriangles 176


    14.3 Solutionofquadraticequationsby


    20.6 Constructionoftriangles 179


    ‘completingthesquare’ 105


    14.4 Solutionofquadraticequationsby


    21 Introductiontotrigonometry 181


    formula 106


    21.1 Introduction 181


    14.5 Practicalproblemsinvolvingquadratic


    21.2 ThetheoremofPythagoras 181


    equations 108


    21.3 Sines,cosinesandtangents 183


    14.6 Solutionoflinearandquadraticequations


    21.4 Evaluatingtrigonometricratiosofacute


    simultaneously 110


    angles 185


    15 Logarithms 111 21.5 Solvingright-angledtriangles 188


    15.1 Introductiontologarithms 111 21.6 Anglesofelevationanddepression 191


    15.2 Lawsoflogarithms 113


    15.3 Indicialequations 115 RevisionTest8 193


    15.4 Graphsoflogarithmicfunctions 116


    16 Exponentialfunctions 118 22 Trigonometricwaveforms 195


    16.1 Introductiontoexponentialfunctions 118 22.1 Graphsoftrigonometricfunctions 195


    16.2 Thepowerseriesforex 119 22.2 Anglesofanymagnitude 196


    16.3 Graphsofexponentialfunctions 120 22.3 Theproductionofsineandcosinewaves 198


    16.4 Napierianlogarithms 122 22.4 Terminologyinvolvedwithsineand


    16.5 Lawsofgrowthanddecay 125 cosinewaves 199


    22.5 Sinusoidalform: Asin(ωt±α) 202


    RevisionTest6 129


    23 Non-right-angledtrianglesandsomepractical


    applications 205


    17 Straightlinegraphs 130


    23.1 Thesineandcosinerules 205


    17.1 Introductiontographs 130


    23.2 Areaofanytriangle 205


    17.2 Axes,scalesandco-ordinates 130


    23.3 Workedproblemsonthesolutionof


    17.3 Straightlinegraphs 132


    trianglesandtheirareas 206


    17.4 Gradients,interceptsandequations


    23.4 Furtherworkedproblemsonthesolution


    ofgraphs 134


    oftrianglesandtheirareas 207


    17.5 Practicalproblemsinvolvingstraightline


    23.5 Practicalsituationsinvolvingtrigonometry 209


    graphs 141


    23.6 Furtherpracticalsituationsinvolving


    18 Graphsreducingnon-linearlawstolinearform 147 trigonometry 211


    18.1 Introduction 147


    24 Cartesianandpolarco-ordinates 214


    18.2 Determinationoflaw 147


    24.1 Introduction 214


    18.3 Revisionoflawsoflogarithms 150


    24.2 ChangingfromCartesiantopolar


    18.4 Determinationoflawinvolvinglogarithms 150


    co-ordinates 214


    19 Graphicalsolutionofequations 155 24.3 ChangingfrompolartoCartesian


    19.1 Graphicalsolutionofsimultaneous co-ordinates 216


    equations 155 24.4 UseofPol/Recfunctionsoncalculators 217 vii


    Contents


    RevisionTest9 218 30.4 Determiningresultantphasorsbythesine


    andcosinerules 281


    30.5 Determiningresultantphasorsby


    25 Areasofcommonshapes 219 horizontalandverticalcomponents 283


    25.1 Introduction 219


    25.2 Commonshapes 219


    RevisionTest12 286


    25.3 Areasofcommonshapes 221


    25.4 Areasofsimilarshapes 229


    31 Presentationofstatisticaldata 288


    26 Thecircle 230 31.1 Somestatisticalterminology 288


    26.1 Introduction 230 31.2 Presentationofungroupeddata 289


    26.2 Propertiesofcircles 230 31.3 Presentationofgroupeddata 292


    26.3 Radiansanddegrees 232


    26.4 Arclengthandareaofcirclesandsectors 233 32 Mean,median,modeandstandarddeviation 299


    26.5 Theequationofacircle 236 32.1 Measuresofcentraltendency 299


    32.2 Mean,medianandmodefordiscretedata 299


    32.3 Mean,medianandmodeforgroupeddata 300


    RevisionTest10 238


    32.4 Standarddeviation 302


    32.5 Quartiles,decilesandpercentiles 303


    27 Volumesofcommonsolids 240


    27.1 Introduction 240 33 Probability 306


    27.2 Volumesandsurfaceareasofcommon 33.1 Introductiontoprobability 306


    shapes 240 33.2 Lawsofprobability 307


    27.3 Summaryofvolumesandsurfaceareasof


    commonsolids 247 RevisionTest13 312


    27.4 Morecomplexvolumesandsurfaceareas 247


    27.5 Volumesandsurfaceareasoffrustaof


    pyramidsandcones 252 34 Introductiontodifferentiation 313


    34.1 Introductiontocalculus 313


    27.6 Volumesofsimilarshapes 256


    34.2 Functionalnotation 313


    28 Irregularareasandvolumes,andmeanvalues 257 34.3 Thegradientofacurve 314


    28.1 Areasofirregularfigures 257 34.4 Differentiationfromfirstprinciples 315


    28.2 Volumesofirregularsolids 259 34.5 Differentiationofy=axn bythe


    28.3 Meanoraveragevaluesofwaveforms 260 generalrule 315


    34.6 Differentiationofsineandcosinefunctions 318


    34.7 Differentiationofeaxandlnax 320


    RevisionTest11 264


    34.8 Summaryofstandardderivatives 321


    34.9 Successivedifferentiation 322


    29 Vectors 266 34.10 Ratesofchange 323


    29.1 Introduction 266


    29.2 Scalarsandvectors 266 35 Introductiontointegration 325


    29.3 Drawingavector 266 35.1 Theprocessofintegration 325


    29.4 Additionofvectorsbydrawing 267 35.2 Thegeneralsolutionofintegralsofthe


    29.5 Resolvingvectorsintohorizontaland formaxn 325


    verticalcomponents 269 35.3 Standardintegrals 326


    29.6 Additionofvectorsbycalculation 270 35.4 Definiteintegrals 328


    29.7 Vectorsubtraction 274 35.5 Theareaunderacurve 330


    29.8 Relativevelocity 276


    29.9 i, j andknotation 277 RevisionTest14 335


    30 Methodsofaddingalternatingwaveforms 278


    Listofformulae 336


    30.1 Combiningtwoperiodicfunctions 278


    30.2 Plottingperiodicfunctions 278 Answerstopracticeexercises 340


    30.3 Determiningresultantphasorsbydrawing 280 Index 356 viii


    Contents


    Website Chapters


    (Goto:http://www.booksite.elsevier.com/newnes/bird)


    Preface iv 38.3 Inequalitiesinvolvingamodulus 20


    36 Numbersequences 1 38.4 Inequalitiesinvolvingquotients 21


    36.1 Simplesequences 1 38.5 Inequalitiesinvolvingsquarefunctions 22


    36.2 Then’thtermofaseries 1 38.6 Quadraticinequalities 23


    36.3 Arithmeticprogressions 2


    39 Graphswithlogarithmicscales 25


    36.4 Geometricprogressions 5


    39.1 Logarithmicscalesandlogarithmic


    37 Binary,octalandhexadecimal 9 graphpaper 25


    37.1 Introduction 9 39.2 Graphsoftheformy=axn 25


    37.2 Binarynumbers 9 39.3 Graphsoftheformy=abx 28


    37.3 Octalnumbers 12 39.4 Graphsoftheformy=aekx 29


    37.4 Hexadecimalnumbers 15


    38 Inequalities 19 RevisionTest15 32


    38.1 Introductiontoinequalities 19


    38.2 Simpleinequalities 19 Answerstopracticeexercises 33 Preface


    Basic Engineering Mathematics 5th Edition intro- asproblemsolvingisextensivelyusedtoestablishand


    ducesandthenconsolidatesbasicmathematicalprinci- exemplifythetheory.Itisintendedthatreaderswillgain


    plesandpromotesawarenessofmathematicalconcepts realunderstandingthroughseeingproblemssolvedand


    forstudentsneedingabroadbaseforfurthervocational thensolvingsimilarproblemsthemselves.


    studies. This textbook contains some 750 worked problems,


    Inthisfifthedition,newmaterialhasbeenaddedtomany followed by over 1550 further problems (all with


    of the chapters, particularly some of the earlier chap- answersattheendofthebook)containedwithinsome


    ters,togetherwithextrapracticalproblemsinterspersed 161 Practice Exercises; each Practice Exercise fol-


    throughout the text. The extent of this fifth edition lows on directlyfrom the relevant section of work. In


    is such that four chapters from the previous edition addition,376linediagramsenhanceunderstandingof


    have been removed and placed on the easily available the theory. Where at all possible,the problemsmirror


    website http://www.booksite.elsevier.com/newnes/bird. potentialpractical situationsfoundinengineering and


    The chapters removed to the website are ‘Number science.


    sequences’,‘Binary,octalandhexadecimal’,‘Inequali- Placed at regular intervals throughout the text are


    ties’and‘Graphswithlogarithmicscales’. 14 Revision Tests (plus another for the website


    Thetextisrelevantto: chapters) to check understanding. For example, Revi-


    sion Test 1 covers material contained in Chapters 1


    • ‘Mathematics for Engineering Technicians’ for


    and 2, Revision Test 2 covers the material contained


    BTEC FirstNQF Level 2 –Chapters1–12,16–18,


    in Chapters 3–5, and so on. These Revision Tests do


    20,21,23and25–27areneededforthismodule.


    not have answers given since it is envisaged that lec-


    • Themandatory‘MathematicsforTechnicians’for


    turers/instructors could set the tests for students to


    BTECNationalCertificateandNationalDiplomain


    attempt as part of their course structure. Lecturers/in-


    Engineering,NQFLevel3–Chapters7–10,14–17,


    structors may obtain a complimentary set of solu-


    19, 20–23, 25–27, 31, 32, 34 and 35 are needed


    tionsoftheRevisionTestsinan Instructor’sManual,


    and,inaddition,Chapters1–6,11and12arehelpful


    available from the publishers via the internet – see


    revisionforthismodule.


    http://www.booksite.elsevier.com/newnes/bird.


    • Basic mathematics for a wide range of introduc-


    Attheendofthebookalistofrelevantformulaecon-


    tory/access/foundationmathematicscourses.


    tained within the text is included for convenience of


    • GCSErevisionandforsimilarmathematicscourses


    reference.


    inEnglish-speakingcountriesworldwide.


    Theprincipleof learningbyexampleisattheheartof


    BasicEngineeringMathematics5thEditionprovidesa BasicEngineeringMathematics5thEdition.


    leadintoEngineeringMathematics6thEdition.


    Each topic considered in the text is presented in a


    JOHNBIRD


    way that assumes in the reader little previous know-


    RoyalNavalSchoolofMarineEngineering


    ledge of that topic. Each chapter begins with a brief


    HMSSultan,formerlyUniversityofPortsmouth


    outlineof essential theory, definitions, formulae, laws


    andHighburyCollege,Portsmouth


    andprocedures;however,thesearekepttoaminimum


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