Higher Engineering Mathematics

745 Pages·2007·15.57 MB·English

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
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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
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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
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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
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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
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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