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CALCULUS forBiologyandMedicine

UniversityofMinnesota

UniversityofCalifornia—LosAngeles

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LibraryofCongressCataloging-in-PublicationData

Names:Neuhauser,Claudia,1962-author. | Roper,MarcusL.,author.

Title:Calculusforbiologyandmedicine.

Description:Fourthedition/ClaudiaNeuhauser,UniversityofMinnesota, MarcusL.Roper,UniversityofCalifornia—LosAngeles. | Boston: Pearson,[2018] | Includesbibliographicalreferencesandindex.

Identifiers:LCCN2017036101 | ISBN9780134070049(hardcover)

Subjects:LCSH:Biomathematics–Textbooks. | Medicine–Mathematics–Textbooks. Classification:LCCQH323.5.N462018 | DDC570.1/51–dc23LCrecordavailable athttps://lccn.loc.gov/2017036101

117

ISBN13:978-0-13-407004-9

ISBN10:0-13-407004-6

ThroughoutthisTableofContentsweuseanasterisk(*)fortopicsthatarenotdirectlyusedinthelatersectionsofthetext.Thatis, youcanstudytheentirebookwithoutstudyingthesetopics.Someofthesetopicsgodeeperintorigorousdefinitionsoflimitsand bounds(forexample,3.6or10.2).Othersexploreextensionsofthemajormathematicalideas,likestabilityinrecurrenceequations (in5.7)ormodelsusingsystemsofrecurrenceequations(in9.3.3and10.9).Athirdclassofasteriskedtopicsprovidesmoremodeling examplesusingthemathematicaltoolsdevelopedinthetext(forexample,2.3.4,5.9,9.4,and11.5).Instructorsmaydecidefor themselveswhichofthesetopicstocover.

Preface viii

1 PreviewandReview 1

j 1.1 PrecalculusSkillsDiagnosticTest 1

j 1.2 Preliminaries 4

1.2.1TheRealNumbers4

1.2.2LinesinthePlane7

1.2.3EquationoftheCircle9

1.2.4Trigonometry9

1.2.5ExponentialsandLogarithms11

1.2.6ComplexNumbersandQuadraticEquations13

j 1.3 ElementaryFunctions 18

1.3.1WhatIsaFunction?18

1.3.2PolynomialFunctions21

1.3.3RationalFunctions23

1.3.4PowerFunctions24

1.3.5ExponentialFunctions25

1.3.6InverseFunctions28

1.3.7LogarithmicFunctions30

1.3.8TrigonometricFunctions33

j 1.4 Graphing 40

1.4.1GraphingandBasicTransformations ofFunctions40

1.4.2TheLogarithmicScale42

1.4.3TransformationsintoLinearFunctions44

1.4.4*FromaVerbalDescription toaGraph49

KeyTerms 58 ReviewProblems 58

2 Discrete-TimeModels, Sequences,andDifference Equations 62

j 2.1 ExponentialGrowthandDecay 62

2.1.1ModelingPopulationGrowthinDiscreteTime62

2.1.2RecurrenceEquations64

2.1.3VisualizingRecurrenceEquations65

j 2.2 Sequences 68

2.2.1WhatAreSequences?68

2.2.2*UsingSpreadsheetstoCalculateaRecursive Sequence71

2.2.3Limits71

2.2.4RecurrenceEquations75

2.2.5Using NotationtoRepresentSums ofSequences78

j 2.3 ModelingwithRecurrenceEquations 81

2.3.1Density-DependentPopulationGrowth81

2.3.2Density-DependentPopulationGrowth:The Beverton–HoltModel84

2.3.3TheDiscreteLogisticEquation85

2.3.4*ModelingDrugAbsorption88

KeyTerms 98 ReviewProblems 98

3 LimitsandContinuity 101

j 3.1 Limits 101

3.1.1ANon-RigorousDiscussion ofLimits102

3.1.2PitfallsofFindingLimits106

3.1.3LimitLaws108

j 3.2 Continuity 112

3.2.1WhatIsContinuity?112

3.2.2CombinationsofContinuousFunctions115

j 3.3 LimitsatInﬁnity 120

j 3.4 TrigonometricLimitsandtheSandwich Theorem 124

3.4.1GeometricArgumentfor TrigonometricLimits124

3.4.2*TheSandwichTheorem126

j 3.5 PropertiesofContinuousFunctions 129

3.5.1TheIntermediate-ValueTheoremandThe BisectionMethod129

3.5.2*UsingaSpreadsheettoImplementthe BisectionMethod132

3.5.3AFinalRemarkonContinuousFunctions134

j 3.6 *AFormalDeﬁnitionofLimits 134

KeyTerms 139 ReviewProblems 139

4 Differentiation 142

j 4.1 FormalDeﬁnitionoftheDerivative 143

j 4.2 PropertiesoftheDerivative 148

4.2.1InterpretingtheDerivative148

4.2.2DifferentiabilityandContinuity150

j 4.3 ThePowerRule,theBasicRulesof Differentiation,andtheDerivativesof Polynomials 154

j 4.4 TheProductandQuotientRules,andthe DerivativesofRationalandPower Functions 160

4.4.1TheProductRule160

4.4.2TheQuotientRule162

j 4.5 TheChainRule 168

4.5.1TheChainRule168

4.5.2ProofoftheChainRule172

j 4.6 ImplicitFunctionsandImplicit Differentiation 174

4.6.1ImplicitDifferentiation174

4.6.2RelatedRates177

j 4.7 HigherDerivatives 180

j 4.8 DerivativesofTrigonometric Functions 184

j 4.9 DerivativesofExponentialFunctions 188

j 4.10 DerivativesofInverseFunctions, LogarithmicFunctions,andtheInverse TangentFunction 194

4.10.1DerivativesofInverseFunctions194

4.10.2TheDerivativeoftheLogarithmic Function199

4.10.3*LogarithmicDifferentiation201

j 4.11 LinearApproximationandError Propagation 204 KeyTerms 211 ReviewProblems 211

5 ApplicationsofDifferentiation 213

j 5.1 ExtremaandtheMean-Value Theorem 213

5.1.1TheExtreme-ValueTheorem213

5.1.2LocalExtrema215

5.1.3TheMean-ValueTheorem219

j 5.2 MonotonicityandConcavity 225

5.2.1Monotonicity226

5.2.2Concavity228

j 5.3 ExtremaandInﬂectionPoints 234

5.3.1Extrema234

5.3.2InflectionPoints240

j 5.4 Optimization 242

j 5.5 L’Hˆopital’sRule 253

j 5.6 GraphingandAsymptotes 260

j 5.7 *RecurrenceEquations:Stability 271

5.7.1ExponentialGrowth271

5.7.2Stability:GeneralCase272

5.7.3PopulationGrowthModels275

j 5.8 *NumericalMethods:The Newton–RaphsonMethod 279

j 5.9 *ModelingBiologicalSystemsUsing DifferentialEquations 285

5.9.1ModelingPopulationGrowth285

5.9.2InterpretingtheMathematicalModel287

5.9.3PassageofDrugsThroughthe HumanBody289

j 5.10 Antiderivatives 294 KeyTerms 301 ReviewProblems 302

6 Integration 306

j 6.1 TheDeﬁniteIntegral 306

6.1.1TheAreaProblem306

6.1.2TheGeneralTheoryofRiemannIntegrals308

6.1.3PropertiesoftheRiemannIntegral314

6.1.4*OrderPropertiesoftheRiemannIntegral316

j 6.2 TheFundamentalTheoremof Calculus 322

6.2.1TheFundamentalTheoremofCalculus (PartI)322

6.2.2*Leibniz’sRuleandaRigorousProof oftheFundamentalTheoremof Calculus323

6.2.3AntiderivativesandIndefiniteIntegrals326

6.2.4TheFundamentalTheoremofCalculus (PartII)329

j 6.3 ApplicationsofIntegration 334

6.3.1CumulativeChange334

6.3.2AverageValues336

6.3.3*TheMeanValueTheorem338

6.3.4*Areas340

6.3.5*TheVolumeofaSolid343

6.3.6*RectificationofCurves346

KeyTerms 352 ReviewProblems 352

7

j 8.2 EquilibriaandTheirStability 441

8.2.1EquilibriumPoints442

8.2.2GraphicalApproachtoFinding Equilibria442

8.2.3StabilityofEquilibriumPoints443

8.2.4SketchingSolutionsUsingtheVector FieldPlot448

8.2.5BehaviorNearanEquilibrium450

355

IntegrationTechniquesand ComputationalMethods

j 7.1 TheSubstitutionRule 355

7.1.1IndefiniteIntegrals355

7.1.2DefiniteIntegrals360

j 7.2 IntegrationbyPartsandPracticing Integration 365

7.2.1IntegrationbyParts365

7.2.2PracticingIntegration370

j 7.3 RationalFunctionsandPartial Fractions 374

7.3.1ProperRationalFunctions374

7.3.2Partial-FractionDecomposition375

7.3.3RepeatedLinearFactors379

7.3.4*IrreducibleQuadraticFactors380

7.3.5Summary385

j 7.4 *ImproperIntegrals 388

7.4.1Type1:UnboundedIntervals388

7.4.2Type2:UnboundedIntegrand392

7.4.3AComparisonResultforImproper Integrals395

j 7.5 NumericalIntegration 398

7.5.1TheMidpointRule398

7.5.2TheTrapezoidalRule401

7.5.3UsingaSpreadsheetforNumerical Integration402

7.5.4*EstimatingErrorinaNumericalIntegration406

j 7.6 *TheTaylorApproximation 409

7.6.1TaylorPolynomials409

7.6.2TheTaylorPolynomialabout x = a 414

7.6.3HowAccurateIstheApproximation?415

j 7.7 *TablesofIntegrals 420

KeyTerms 424

ReviewProblems 424

8 DifferentialEquations 427

j 8.1 SolvingSeparableDifferential Equations 428

8.1.1Pure-TimeDifferentialEquations429

8.1.2AutonomousDifferentialEquations430

8.1.3GeneralSeparableEquations436

j 8.3 DifferentialEquationModels 455

8.3.1CompartmentModels455

8.3.2AnEcologicalModel456

8.3.3ModelingaChemicalReaction457

8.3.4TheEvolutionofCooperation459

8.3.5EpidemicModel463

j 8.4 IntegratingFactorsandTwo-Compartment Models 471

8.4.1IntegratingFactors471

8.4.2Two-CompartmentModels475 KeyTerms 484 ReviewProblems 484

9 LinearAlgebraandAnalytic Geometry 487

j 9.1 LinearSystems 487

9.1.1GraphicalSolution488

9.1.2SolvingEquationsUsingElimination491

9.1.3SolvingSystemsofLinearEquations492

9.1.4RepresentingSystemsofEquations UsingMatrices496

j 9.2 Matrices 501

9.2.1MatrixOperations501

9.2.2MatrixMultiplication503

9.2.3InverseMatrices506

9.2.4*ComputingInverseMatrices513

j 9.3 LinearMaps,Eigenvectors,and Eigenvalues 518

9.3.1GraphicalRepresentation519

9.3.2EigenvaluesandEigenvectors523

9.3.3*IteratedMaps531

j 9.4 *DemographicModeling 535

9.4.1ModelingwithLeslieMatrices535

9.4.2StableAgeDistributionsin DemographicModels540

j 9.5 AnalyticGeometry 547

9.5.1PointsandVectorsinHigher Dimensions547

9.5.2TheDotProduct551

9.5.3ParametricEquationsofLines555

KeyTerms 558 ReviewProblems 559

10 MultivariableCalculus 561

j 10.1 FunctionsofTwoorMoreIndependent Variables 563

10.1.1DefiningaFunctionofTwoor MoreVariables563

10.1.2TheGraphofaFunctionofTwoIndependent Variables–SurfacePlot565

10.1.3HeatMaps566

10.1.4ContourPlots568

j 10.2 *LimitsandContinuity 575

10.2.1InformalDefinitionofLimits575

10.2.2Continuity578

10.2.3FormalDefinitionofLimits579

j 10.3 PartialDerivatives 582

10.3.1FunctionsofTwoVariables582

10.3.2FunctionsofMoreThan TwoVariables586

10.3.3Higher-OrderPartialDerivatives586

j 10.4 TangentPlanes,Differentiability, andLinearization 589

10.4.1FunctionsofTwoVariables589

10.4.2Vector-ValuedFunctions594

j 10.5 *TheChainRuleandImplicit Differentiation 599

10.5.1TheChainRuleforFunctionsof TwoVariables599

10.5.2ImplicitDifferentiation601

j 10.6 *DirectionalDerivativesandGradient Vectors 604

10.6.1DerivingtheDirectional Derivative604

10.6.2PropertiesoftheGradientVector608

j 10.7 *MaximizationandMinimization ofFunctions 610

10.7.1LocalMaximaandMinima610

10.7.2GlobalExtrema617

10.7.3ExtremawithConstraints621

10.7.4Least-SquaresDataFitting626

j 10.8 *Diffusion 635

j 10.9 *SystemsofRecurrenceEquations 640

10.9.1ABiologicalExample640

10.9.2EquilibriaandStabilityinSystemsofLinear RecurrenceEquations641

10.9.3EquilibriaandStabilityofNonlinearSystems ofRecurrenceEquations643 KeyTerms 650 ReviewProblems 650

11 SystemsofDifferential Equations 653

j 11.1 LinearSystems:Theory 655

11.1.1TheVectorField655

11.1.2SolvingLinearSystems657

11.1.3EquilibriaandStability664

11.1.4SystemswithComplexConjugate Eigenvalues666

11.1.5SummaryoftheTheoryofLinearSystems671

j 11.2 LinearSystems:Applications 677

11.2.1Two-CompartmentModels677

11.2.2AMathematicalModelforLove682

11.2.3*TheHarmonicOscillator684

j 11.3 NonlinearAutonomousSystems: Theory 688

11.3.1AnalyticalApproach688

11.3.2GraphicalApproachfor2×2Systems694

j 11.4 NonlinearSystems:Lotka–Volterra ModelforInterspeciﬁcInteractions 698

11.4.1Competition698

11.4.2APredator–PreyModel704

j 11.5 *MoreMathematicalModels 708

11.5.1TheCommunityMatrix709

11.5.2NeuronActivity711

11.5.3EnzymaticReactions713

11.5.4MicrobialGrowthinaChemostat716

11.5.5AModelforEpidemics718

KeyTerms 730 ReviewProblems 730

12 ProbabilityandStatistics

j 12.1 Counting 734

12.1.1TheMultiplicationPrinciple734

12.1.2Permutations735

12.1.3Combinations737

734

12.1.4CombiningtheCountingPrinciples738

j 12.2 WhatIsProbability? 742

12.2.1BasicDefinitions742

12.2.2EquallyLikelyOutcomes746

j 12.3 ConditionalProbabilityand Independence 752

12.3.1ConditionalProbability753

12.3.2TheLawofTotalProbability754

12.3.3Independence755

12.3.4TheBayesFormula758

j 12.4 DiscreteRandomVariablesandDiscrete Distributions 763

12.4.1DiscreteDistributions763

12.4.2MeanandVariance766

12.4.3TheBinomialDistribution774

12.4.4TheMultinomialDistribution778

12.4.5GeometricDistribution779

12.4.6ThePoissonDistribution783

j 12.5 ContinuousDistributions 793

12.5.1DensityFunctions793

12.5.2TheNormalDistribution799

12.5.3TheUniformDistribution805

12.5.4TheExponentialDistribution807

12.5.5ThePoissonProcess811

12.5.6Aging812

j 12.6 LimitTheorems 819

12.6.1TheLawofLargeNumbers819

12.6.2TheCentralLimitTheorem823

j 12.7 StatisticalTools 828

12.7.1DescribingUnivariateData828

12.7.2EstimatingParameters833

12.7.3LinearRegression842

KeyTerms 848

ReviewProblems 849

AppendixA FrequentlyUsedSymbols 851

AppendixB TableoftheStandardNormalDistribution 852

AnswerstoOdd-NumberedProblems A1

References R1

Index I1

Preface

Thegoalof CalculusforBiologyandMedicine hasremainedconstantfromitsinception:

Toshowstudentshowcalculusisusedtoanalyzephenomenainnaturewithoutcompromisingtherigorofthepresentationofcalculusprinciples.

Theresultofthisgoalisacalculustextthathasplentifullifeandhealthsciencesapplicationsandthatprovidesstudentswiththeknowledgeandskillsnecessarytoanalyzeandinterpretmathematicalmodelsofadiversearrayofphenomenaintheliving world.Sincethistextiswrittenforcollegefreshmen,theexampleswerechosensothat noformaltraininginbiologyisneeded.

Therigorofthetextpreparesstudentswellformoreadvancedcoursesinmathematicsandstatistics.Ourhopeisthatstudentswillfindcalculusconceptseasierto understandandmoreinterestingiftheyarerelatedtotheirmajorandcareeraspirations.Whilethetableofcontentsresemblesthatofatraditionalcalculustext,the contentdoesnot:Abstractcalculusconceptsareintroducedinabiologicalcontext, andstudentslearnhowtotransferandapplytheseconceptstobiologicalsituations.

NewtoThisEdition

Modeling –The4thEditionplacesmuchmoreemphasisonmodelingbiological situations.Studentsareinstructedintheprocessesofmodelingreal-worldsituations andgivenmanyopportunitiestopracticethesetechniquesinproblems.

Applications –Theapplicationsinthetexthavebeengreatlyexpandedinnumber. Newapplicationsincludepopulationgenetics,pharmacology,andtheevolutionof microbialcooperation.Manyoftheseapplicationsareadaptedfrompublishedstudiesandothercurrentsources.

Technology –The4thEditionnowincludesclearstudentinstructionsonusing spreadsheetstonumericallysolveequations,visualizedata,andmodelbiological processes.Thismaterialisclearlylabeledsothatinstructorswhoprefernottouseit caneasilyomitit,orassigntechnologysectionsasoptionalreadingstotheirstudents.

Approach –Thelevelofrigorofthetexthasbeenmaintained,butwehavemade adjustmentstohowsometopicsareintroducedtomakethepresentationaccessible tostudents,betterbridgingthegapbetweenwhatstudentsalreadyknowandwhat theyareattemptingtolearn.Wehavealsostreamlinedormadeoptionalmaterial thatisnotusefulforlifesciencesstudents(formaldiscussionoflimits,continuityin multivariatefunctions,etc.).Thismaterialismaintainedsothatinstructorsmaycontinuetoteachit,andstudentswhomaytransferoutoflifesciencescalculuscourses intophysicalsciencesandengineeringcalculuswillstillfindthematerialthatiscoveredinphysicalcalculus;butthemaincurrentofthebookisthroughtopicsthatare directlyneededforlifesciences.

Writing –Everyattempthasbeenmadeintheneweditiontouselanguagethat willenablestudentstobetterusethetextasanindependentlearningresource.In somesectionsthisrequiredlengtheningexplanationsthatwereoverlyterseinorder tomakethemmoreaccessible;inothers,topicsareintroducedinformallyusingexamplesdirectlytakenfromlifesciencestomotivatethemoreformalmathematical materialthatwasthestrengthofthepreviouseditions.

Prerequisites –WeaddedaPrecalculusSkillsDiagnosticTestatthebeginningofthe texttohelpstudentsgaugewhetherreviewofprecalculustopicsisneeded.Answers tothequizareprovidedinthebackofthebookalongwithtipsonwhattoreview inChapter1ifrefreshersareneeded.

Design –Thebookhasbeenredesignedinfullcolortohelpstudentsbetteruseit andtohelpmotivatestudentsastheyputinthehardworktolearnthemathematics.

Figures –Manyfigureswererevisedtotakeadvantageofthenewfull-colordesign. Mostnotably,the3-dimensionalfigureswerere-renderedusingthelatestsoftware. Seethefigureattheleftforanexample.

BiologyNotes –New“BioInfo”notesprovideoptionalbackgroundinformationto supportthenarrativeandexercises.

“HelpText”withinExamples –Weaddedtext(inbluetype)withinexamplesto explainthemathematicalprinciple(s)appliedinthestepsofthesolution.Thistext helpsstudentsunderstandthesolutionandemphasizesthateachstepinamathematicalargumentiscarefullyjustified.

Figure6.45 Thesolidofrotationfor Example14canbemadeupof washer-likeelements.

TopicCoverage –Basedonfeedbackfromreviewers,somenewtopicshavebeen addedtothetext.Themostsignificantamongthesearethefollowing:

Manynewmathematicalmodels,includingmodelsformicrobialcooperation,evolution,andepidemiology,andmulticompartmentmodelsinpharmacology.

Expandedapplicationsforoptimizationmethods.

Toolsforfittingmodelstorealdata.

Expandeddiscussionofmethodsforvisualizingmultivariatefunctions.

MyLabTM MathOnlineHomework –Last,butnotleast,thetextnowhasonline homeworkwithinMyLabMath.TheMyLabMathcoursecontainshundredsof algorithmicallygeneratedexercisesthatprovidestudentswithinstantfeedback,optionallearningaidsformanyexercises,andthecompleteeBook.SeebelowforadditionalfeaturesofMyLabMathforthistext.

FeaturesoftheText

Adistinguishingfeatureofthistextisthebiologicalexamplesandexercises,which arenotably real (manyfrompublishedstudiesorothercurrentsources), relevant,and varied.TheReferencessectionatthebackofthistextcontainsanexhaustivelistof thesourcesweused.

ExamplesandExplanations Eachtopicisinspiredbybiologicalexamples.These motivatingintroductionsarefollowedbyathoroughdiscussionoutsidethelifesciencecontexttoenablestudentstobecomefamiliarwithboththemeaningandthe mechanicsofthemathematicaltopic.Finally,biologicalexamplesarepresentedto teachstudentshowtoapplythematerialinalifesciencecontext.Examplesinthe textarecompletelyworkedout,andthestepsinthesolutionsareexplainedinblue texttotherightofeachstep.

Exercises Calculuscannotbelearnedbywatchingsomeonedoit.Becauseofthis, CalculusforBiologyandMedicine providesstudentswithskill-basedexercisesaswell aswordproblems.Wordproblemsareanintegralpartofteachingcalculusinalifesciencecontext.Thewordproblemscontainedinthetextareup-to-dateandareadapted fromeitherstandardbiologytextsororiginalresearch.Theexercisesandwordproblemsareattheendofeachsectionandareorganizedbysubsectiontohelpstudents refertospecificsubsectionsofcontentwhilecompletinghomework.Thisalsoaids instructorsinassigninghomeworkproblems.

Technology CalculusforBiologyandMedicine assumestheavailabilityofgraphing calculators.Thisallowsstudentstodevelopamuchbettervisualunderstandingofthe conceptsincalculus.Beyondthis,nospecialsoftwareisrequired.

ReﬂectionsandOutlook

Likemanyschoolsnow,bothUCLAandUniversityofMinnesotaofferlifescience studentstheirowncalculustrack.Otheruniversitiesareincreasinglyadoptingseparatecalculustrackstodealwiththedifferentneedsoflifesciencesmajorsandphysicalscience/engineeringmajors.Therearemanywaystodesigncurriculaforthese courses,andfacultyventuringintotherecommendationsofferedbyreportsonnew

needsforlifescienceseducation(suchas Bio2010:TransformingUndergraduateEducationforFutureResearchBiologists fromtheNationalResearchCouncilandthe NationalAcademies,or ScientificFoundationsforFuturePhysicians fromtheAssociationoftheAmericanMedicalCollegesandtheHowardHughesMedicalInstitute) maybeoverwhelmedbytheamountofquantitativetrainingthatisnowexpectedfor lifesciencestudentsandbyhowitgoesfarbeyondwhatstudentscanbeprepared forwithasingleyearofcalculus.Inthisfourtheditionofthetextbookwehavefocusedonretainingthestrengthsofthethirdedition,includinggivingstudentsaccess totherigorousfoundationsofmathematicalideasthatwillenablethemtotakefurtherclassesinmaththatareincreasinglynecessaryforquantitativemindedbiologists. However,wehaverewrittenmuchofthematerialwithaneyetoeliminatingbarrierstostudy(e.g.,byavoidingusingexpressionswithmultipleunknownconstants inthem).

Wearealsoverymindfulofthefutureneedsofstudentstohandlethelargedata streamscreatedbynewinnovationsinomics,personalmedicine,andremotesensing. Muchofthemathunderlyingthesenewareasisoutsidewhatcanbecoveredinthis book,butChapters9and12layfoundationsforstudentswhowillgoontostudybioinformatics.Additionally,wehavebroughtdata(anddatafitting)increasinglyintothe book,especiallyinsupportofthenewmathematicalmodelingtopicswehaveintroduced.Studyofalgorithmsissupportedbyexplicitdirectionsonusingspreadsheetsto implementthealgorithms.Anyspreadsheetsoftwarecanbeused,butwehavefound GoogleSheetstobeespeciallyeffectiveintheclassroom,sinceitallowsspreadsheets tobesimultaneouslysharedandeditedacrossdozensofcomputers.

ChapterSummary

Chapter1 Thischapterreviewsprecalculustools,includingfunctionsandmethods forgraphingdata.Manystudentswillhavestudiedthismaterialintheirprecalculus classes,sosummariesarekeptbrief.Section1.1includesadiagnostictestthatstudents cantake(eitherbyitself,orinconjunctionwithMyLabMath)toreviewtheirknowledgeofthesetopics.Thebasictoolsfromalgebraandtrigonometryaresummarizedin Section1.2.Section1.3thendescribesthefunctionsthatstudentsneedtobefamiliar withforthisbook,includingexponentialandlogarithmicfunctions.Section1.4focuses ongraphing,includinglog-logandsemi-logplotsandtranslatingverbaldescriptions ofbiologicalphenomenaintographs.

Chapter2 Thischaptercoversrecurrenceequations(ordiscretetimemodels)and sequences.Importantly,weusethischaptertointroduce -notationforsummingseries.Thisnotationisusedthroughoutthetext.Wealsousethischaptertointroduce mathematicalmodeling,includingtheassumptionsthatarebuiltintomodels,parsingverbaldescriptions,andcomparingmodelsagainstdata.Ourexampleshereare drawnfrompopulationgrowthandphysiologicalmodelingofhowdrugspassthrough apatient’sbody.

Chapter3 Limitsandcontinuityarekeyconceptsforunderstandingtheconceptualpartsofcalculus.Visualintuitionisemphasizedbeforethetheoryisdiscussed. Weshowhowthebisectionmethodcanbeusedasapracticaltoolforsolvingequations.Theformaldefinitionoflimitsisgivenattheendofthechapterinanoptional section.

Chapter4 Westartwithanintuitiveandvisualdescriptionofthederivativebeforegivingaformaldefinition.Then,beforewegointothemechanicsofdifferentiation,wedescribeinterpretationsofthederivativeindifferentcontexts(including chemicalreactions),buildingstudents’intuitionfurther.Differentiationrulesarediscussedandbrokenintoreadilydigestiblechunkstogivestudentstimetoacquaint themselveswiththem.Errorpropagationanddifferentialequationsarethemain applications.

Chapter5 Thischapterpresentsbiologicalandmoretraditionalapplicationsofdifferentiation.Wemaintainthe3rdEdition’sapproachthatderivesresultsonfunctional extremarigorouslyfromtheMeanValueTheorem.Butwealsoexplaintoskeptical studentswhycalculus-basedtoolsforanalyzingfunctionsanddrawingtheirgraphsare stillrelevantwhencomputersallowfunctionstobesoreadilyplotted.Wehavealsoenlargedthenumberofapplicationsforoptimization,includingmodelsfromphysiology andpopulationgenetics.Wealsoaddedanewsectionondifferentialequation-based models(againfocusingonpopulationgrowthandthepassageofdrugsthroughthe body),sothatstudentsencounterthesevitalapplicationsbeforetheymeetintegration.Finally,weintroduceantiderivatives,inanticipationofstudyingintegrationin Chapter6.Analysisofrecurrenceequationsiscoveredinanoptionalsection.

Chapter6 Integrationismotivatedgeometrically.Wealsodescribethedefinition oftheintegralviaRiemannsumsinawaythathasbeengreatlysimplifiedfromthe 3rdEdition.Inparticular,studentscanstudythismaterialwithoutneedingtoknow -notationandwithoutthefullformalismofpartitions.Inourexperience,thismakes thisdifficulttopicmucheasierforstudentswithoutanoverallcompromiseonthelevel ofrigorintheirunderstanding.Thefundamentaltheoremofcalculusanditsconsequencesarediscussedindepth.Wediscussapplicationsforintegration,buthavereducedtheamountofrequiredmaterialinthischaptertofocusonlyonapplications thataredirectlyrelevanttolifesciences,suchascalculatingthemeanofafunction anditscumulativechange.

Chapter7 Thischaptercontainsintegrationtechniques,focusingonthetechniques thatareimmediatelynecessaryforsolvingdifferentialequations,includingintegration byparts,bysubstitution,andatailoredintroductiontothemethodofpartialfractions.MaterialonTaylorpolynomialsandonusingtablesofintegrals(atechnique nowlargelymaderedundantbycomputeralgebrapackages)iscoveredinoptional sectionsattheendofthechapter.

Chapter8 Thischapterprovidesanintroductiontodifferentialequations,covering separableequationsandlinearfirstorderequations.Thetreatmentisnotcomplete, butitwillequipstudentswithbothanalyticalandgraphicalskillstoanalyzedifferentialequations.Thechaptershowcasesandinterpretsmathematicalmodelsfrommany areasofbiology,includingmicrobialcooperation,ecology,andepidemiology.Additionofintegratingfactorsallowsustodiscusstwo-compartmentmodels,whichare widelyusedforstudyingthemovementofdrugsthroughthebody.

Chapter9 Linearmodels,andthematrixmethodsneededtosolvethem,arecentral tomodernmethodsinbioinformatics.Thematerialinthischapterintroducesstudents tothebasicconceptsneededtostudymultivariatefunctionsinChapters10and11. However,althoughthetreatmentofeigenvaluesandeigenvectorsemphasizestheir importancetomodelsofchange(bothrecurrenceequationsandsystemsofdifferentialequations),matrixmathisintroducedinawaydesignedtoprovidestudentswith afirmplatformforfurtherstudyinlinearalgebraforbioinformaticsapplications.

Chapter10 Thisisanintroductiontomultidimensionalcalculus.Sincestudentsoftenstrugglewiththetransitionfromunivariatefunctionstomultivariatefunctions, wehaveexpandedtheintroductorymaterialtobuildupstudentintuitionmoregradually,withmoreexamples(includingpracticalexampleslikeheatindex)andlarger discussionofhowfunctionscanbevisualized.Themainmathematicaltopicsarepartialderivativesandlinearizationofvector-valuedfunctions.Wecoveratlengthfinding extremaoffunctions(includingunderconstraints).Althoughthistopicisnotneeded forChapter11,optimizationhasmanyuses(andwehighlightitsapplicationtoleast squaresestimationoffittingparameters),andwefindthissectionworthyofclasstime. Thefinalsectionsprovideoptionalmaterialonsystemsofrecurrenceequationsand onpartialdifferentialequationmodels.

Chapter11 Bothgraphicalandanalyticaltoolsaredevelopedtoenablestudents toanalyzesystemsofdifferentialequations.Thematerialisdividedintolinearand

nonlinearsystems.Understandingthestabilityoflinearsystemsintermsofvectorfields,eigenvectors,andeigenvalueshelpsstudentstomasterthemoredifficult analysisofnonlinearsystems.Theoryisexplainedbeforeapplicationsaregiven.Extensiveexamples(withaccompanyingproblems)showcaseapplicationsofthesetools inecology,epidemiology,andphysiology.

Chapter12 Thischapterintroducessomefundamentalprobabilisticandstatisticaltools,takingstudentsfromcounting(i.e.,combinatorial)approachestoprobability,throughimportantdistributionsthatarisewhenmodelingstochasticprocesses. Throughoutstudentsareintroducedtofundamentalideasforworkingwithdata:estimatingprobabilitydistributionsfromhistograms,fittinglinearmodels,andcalculating andinterpretingsummarystatistics.

HowtoUseThisBook

Bydesignthisbookcontainsmorematerialthancanbecoveredinoneyear.Theintent istoallowforschoolsandinstructorstohavemoreflexibilityinthechoiceofmaterial covered.Topicslabeledwithanasterisk(*)intheTableofContentsmaybeomitted attheinstructor’sdiscretion.Theyincludesectionsgoingmorerigorouslyintothe definitionsoflimitsandcontinuityaswellasmanyofthemodelingandapplications sections.

Thebook’scontentcanbearrangedtosupportanylengthofcourse,fromone quartertothreesemesters.Chapter1isprecalculusmaterial.Studentsshouldhave beenexposedtothismaterialbeforestartingtheirfirstcourseincalculus.Thissaid, wefindithighlyusefulforstudentstoself-studythismaterialbeforestartingthe class,whichtheycandomoreeasilyusingthenewPrecalculusSkillsDiagnosticTest. Additionally,weoftencoverinclassthematerialinSection1.4(inparticularon howtographdatausinglogarithmicandsemi-logarithmicaxes).Sections2.1and2.2 givestudentsaminimalintroductiontosequences,series,and -notation.However, westronglyrecommendSection2.3asanintroductiontoderiving,solving,andinterpretingmathematicalmodels,beforestudentsmeetmodelsagaininthecalculus context.

Chapters3and4mustbecoveredinthatorderbeforeanyoftheothersections arecovered.InadditiontoChapters3–4,thefollowingsectionscanbechosen:

Onesemester—integrationemphasis 5.1–5.6,5.10,6.1–6.3(without6.3.4and6.3.5)

Onesemester—differentialequationemphasis 5.1–5.6,5.9–5.10,6.1,6.2,8.2,8.3(withoutsolvinganyofthedifferentialequations)

Onesemester—probabilityemphasis Chapter3(except3.6),Chapter4(without 4.11),5.1–5.4,5.10,6.1,6.2,7.1,7.2.1,12.1–12.5(without12.5.5),12.6(iftimepermits)

Twoquarters 5.1–5.6,5.8,5.10,6.1–6.2,6.3.1and6.3.2,Chapter7,Chapter8

Twosemestersorthreequarters 5.1–5.6,5.10,6.1–6.2,6.3.1and6.3.2,Chapters7,8, and9(without9.2.4or9.4),10.1,10.3,10.4,11.1–11.4

Fourquartersorthreesemesters Allsectionsthatarenotlabeledoptional(with*); optionalsectionsshouldbechosenastimepermits

MyLabMathOnlineCourse (accesscoderequired)

Usedbyover3millionstudentsayear,MyLabMathistheworld’sleadingonline programforteachingandlearningmathematics.MyLabMathdeliversassessment, tutorials,andmultimediaresourcesthatprovideengagingandpersonalizedexperiencesforeachstudent,solearningcanhappeninanyenvironment.Forthefirsttime, instructorsteachingwith CalculusforBiologyandMedicine canassigntext-specific onlinehomeworkandotherresourcestostudentsoutsideoftheclassroom.

TolearnmoreabouthowMyLabMathcombinesprovenlearningapplications withpowerfulassessment,visit pearson.com/mylab/math orcontactyourPearson representative.

Preparedness

Oneofthebiggestchallengesincalculuscoursesismakingsurestudentsareadequatelypreparedwiththeprerequisiteskillsneededtosuccessfullycompletetheir coursework.MyLabMathsupportsstudentswithprecalculuscontentandjust-intimeremediation.Instructorscancreatequizzestoassessnecessaryprerequisiteskills, thenautomaticallyassignpersonalizedremediationforanygapsinskillsthatare identified.

BuildingUnderstanding

MyLabMath’sonlinehomeworkoffersstudentsimmediatefeedbackandtutorial assistancethathelpsthembuildunderstandingofkeyconcepts.

Exerciseswithimmediatefeedback—theassignableexercisesforthistextregeneratealgorithmicallytogivestudentsunlimitedopportunityforpracticeandmastery. MyLabMathprovideshelpfulfeedbackwhenstudentsenterincorrectanswersand includesoptionallearningaidsincludingHelpMeSolveThis,ViewanExample,and aneText.

SetupandSolveExercises askstudentstofirstdescribehowtheywillsetupand approachtheproblem.Thisreinforcesstudents’conceptualunderstandingofthe processtheyareapplyingandpromoteslong-termretentionoftheskill.

AdditionalConceptualQuestions focusondeeper,theoreticalunderstandingof thekeyconceptsincalculus.ThesequestionswerewrittenbyfacultyatCornell UniversityunderaNationalScienceFoundationgrantandarealsoassignable throughLearningCatalytics.

InteractiveFigures havebeenaddedtosupportteachingandlearning.Thefigures aredesignedtobeusedinlectureaswellasbystudentsindependently.Theyare editableusingthefreelyavailableGeoGebrasoftware.

LearningCatalyticsTM isastudentresponsetoolthatusesstudents’smartphones, tablets,orlaptopstoengagetheminmoreinteractivetasksandthinkingduring lecture.LearningCatalyticsfostersstudentengagementandpeer-to-peerlearningwithreal-timeanalytics.LearningCatalyticsisavailabletoallMyLabMath users.

CompleteeText isavailabletostudentsthroughtheirMyLabMathcoursesforthe lifetimeoftheedition,givingstudentsunlimitedaccesstotheeTextwithinany courseusingthateditionofthetextbook.

Mathematica manualandprojects, Maple manualandprojects,and TIGraphing Calculator manualutilizethemostcurrentversionsofMapleandMathematica,as wellastheTI-84PlusandTI-89.Eachprovidesdetailedguidanceforintegratingthe softwarepackageorgraphingcalculatorthroughoutthecourse,includingsyntaxand commands.

Accessibility andachievementgohandinhand.MyLabMathiscompatiblewith theJAWSscreenreader,andenablesmultiple-choiceandfree-responseproblemtypestobereadandinteractedwithviakeyboardcontrolsandmathnotationinput.MyLabMathalsoworkswithscreenenlargers,includingZoomText, MAGic,andSuperNova.Moreinformationisavailableat www.pearson.com/mylab/ math/accessibility.

InstructorSupport

Comprehensivegradebook withenhancedreportingfunctionalityallowsyouto efficientlymanageyourcourse.ThegradebookmeetsallFERPArequirements. ReportingDashboardprovidesinsighttoview,analyze,andreportlearningoutcomes.Studentperformancedataispresentedattheclass,section,andprogram levelsinanaccessible,visualmannersoyou’llhavetheinformationyouneedto keepyourstudentsontrack.

ItemAnalysistracksclass-wideunderstandingofparticularexercisessoyoucan refineyourclasslecturesoradjustthecourse/departmentsyllabus.Just-in-time teachinghasneverbeeneasier!

TrainingandSupport –MyLabMathcomesfromanexperiencedpartnerwith educationalexpertiseandaneyeonthefuture.Whetheryouarejustgetting startedwithMyLabMath,orhaveaquestionalongtheway,we’rehereto helpyoulearnaboutourtechnologiesandhowtoincorporatethemintoyour course.TolearnmoreabouthowMyLabMathhelpsstudentssucceed,visit www.pearson.com/mylab/math/support orcontactyourPearsonrepresentative.

Supplements

Instructor’sSolutionsManual(downloadonly) Providesfullyworked-outsolutions toeverytextbookexercise,includingtheChapterReviewproblems.Availablefor downloadonlinewithinMyLabMath.

Student’sSolutionsManual Providesfullyworked-outsolutionstotheoddnumberedexercisesinthesectionsandChapterReviews.Availableinprint(ISBN-13: 978-013-412269-4)ordownloadablewithinMyLabMath.

Acknowledgments

Thisbookwouldnothavebeenpossiblewithoutthehelpofnumerouspeople.The facultymembersbelowcontributedtheirexpertisetothisproject.Wearemostgratefultothem.

NandiniBhattacharya, UniversityofCaliforniaSantaCruz* AdenaCalden, UniversityofMassachusettsAmherst*

Youn-ShaChan, UniversityofHoustonDowntown

AlbertoCorso, UniversityofKentucky*

HaoGao, UniversityofCaliforniaLosAngeles*

GuillermoGoldsztein, GeorgiaInstituteofTechnology*

YvetteHester, TexasA&MUniversity*

PeterHoward, TexasA&MUniversity

YangKuang, ArizonaStateUniversity*

GlennLahodny,Jr., TexasA&MUniversity*

GlennLedder, UniversityofNebraskaLincoln*

PetrLisonek, SimonFraserUniversity

LawrenceMarx, UniversityofCaliforniaDavis

EdwardMigliore, UniversityofCaliforniaSantaCruz*

DouglasNorton, VillanovaUniversity*

HeatherRamsey, TexasA&MUniversity*

PatrickShipman, ColoradoStateUniversity*

Jeong-MiYoon, UniversityofHoustonDowntown

*Reviewerswhocontributedtothepreparationofthisedition

ThankstostudentswehavetaughtattheUniversityofMinnesota,theUniversityof CaliforniaLosAngeles,andtheUniversityofCaliforniaDavisfortheirconstructive criticismandenthusiasm.WeoweaspecialthankstoGeorgeLobell,formermathematicseditoratPrenticeHall,whomadethisbookpossibleinthefirstplace,and tothosewhohelpedshepherdthis4thEdition:JeffWeidenaar,PattyBergin,Jenn Snyder,MarielleGuiney,KristinaEvans,EmilyOckay,andotherstaffatPearson; KellyRicciatAptara;RonWeickartatNetworkGraphics;RogerLipsett(answers andsolutions);PaulAnagnostopoulos(typist);andPaulLorczakandAlbertoCorso (accuracychecking).

ThanksalsototheNationalScienceFoundation,whichhassupportednew approachestoteachingcalculusforlifesciencesatUCLAundergrantnumber DMS-1351860.

Marcuswouldliketoexpresshisloveandgratitudetohiswife,Dr.MechelHenry, MD,forhersupportduringthisbook-writingjourney,andtohischildren,Eliotand Charlotte.Youhavegivenmesomanywonderfulreasonsformissingdeadlineson thisbook!

ClaudiaNeuhauser neuha001@umn.edu UniversityofMinnesota

MarcusRoper mroper@math.ucla.edu UniversityofCaliforniaLosAngeles

1

PreviewandReview

Thechapterbeginswithadiagnostictestonprecalculusskills.Sections1.2and1.3serveasa reviewofalgebra,trigonometry,andprecalculus,materialneededtomasterthetopicscovered inthisbook.Section1.4reviewsgraphingfunctionsandintroducestheimportantconceptof plottingdataorfunctionsontransformedaxestodeterminehowtwovariablesarerelated.Section1.4alsoincludesasubsectiononvisualizingverbaldescriptionsofbiologicalphenomena.

ABriefOverviewofCalculus

Calculushastwomainingredients:differentiationandintegration.Differentiationallowsustocalculatehowquicklyafunctionischanging(forexample,therateatwhich apopulationoforganismsisgrowing).Integrationallowsustocalculatetheareaundercurves.Althoughknowingtheareaunderacurvemaynotseemveryimportant, wewilllearninChapter6thatintegrationistheoppositeofdifferentiation(thatis,integratingtherateofchangeofafunctiongetsusbacktotheoriginalfunction).Often thefirststeptofindingafunctionistofindanequationforitsderivative;manyphenomenainbiologycanbemodeledusing differentialequations,equationsthatgovern thederivativeorrateofchangeofafunction.Forexample,inChapter5wewilllearn howtofindtherateofchangeofthenumberofcellsgrowinginaflask.Integration enablestosolvethesedifferentialequationsandfindafunctiontocalculatethenumberoforganismsintheflaskatanygiventime.Inadditiontodevelopingthetheoryof differentialandintegralcalculus,youwillbeintroducedtomanyexamplesofdifferentialequationstodescribebiologicalphenomenaincludingpopulationgrowth,the speedofchemicalreactions,thefiringofneurons,andthespreadofinvasivespecies intonewhabitats.

Theuseofquantitativereasoningisbecomingincreasinglymoreimportantin biology—forinstance,inmodelinginteractionsamongspeciesinacommunity,describingtheactivitiesofneurons,explaininggeneticdiversityinpopulations,andpredictingtheimpactofglobalwarmingonvegetation.Today,calculus(Chapters2–11) andprobabilityandstatistics(Chapter12)areamongthemostimportantquantitative toolsofabiologist.

Section1.1PrecalculusSkillsDiagnosticTest

Inthischapterwereviewthefollowingimportantprecalculustopics:

1. Algebra

2. Trigonometry

3. Visualizingfunctionsanddata

4. Translatingworddescriptionsofbiologicalphenomenainto sketches.

Werecommendthatifyouareself-studyingthischapteryou startbytakingadiagnostictest,whichwillshowyouwhich

areasyoumayneedtoreviewbeforemovingontothecalculus material.Theanswerstothesequestionsareinthebackofthis book;theanswerkeyalsotellsyouwhichsubsectiontoreview ifyouareunabletoansweraquestion.

1. Writetheequationofastraightline

(a) withslope1and y-interceptat y = 5.

(b) withslope2andpassingthroughthepoint(x, y) = (2, 3).

2. Toconvertbetweenthetemperaturemeasuredindegrees Fahrenheit(◦ F)anddegreesCelsius(◦ C)weusethefollowing formula:

y = 5 9 (x 32),

where x isthetemperaturegivenindegreesFahrenheitand y is thetemperaturegivenindegreesCelsius.

(a) IfthetemperatureinLosAngelesinFebruaryis80◦ F,what isthetemperatureindegreesCelsius?

(b) IfthetemperatureinRochesterinFebruaryis 10◦ C,what isthetemperatureindegreesFahrenheit?

3. Describeinwordsthesetofpoints(x, y)satisfyingtheequation:

(x + 1)2 + (y 5)2 = 9.

4.(a) Converttheangle θ = π 7 fromradianstodegrees.

(b) Findallsolutionsoftheequationsin x =− √3 2 .

(d) Findallsolutionsfor x ∈ [0,π ]oftheequationcos3x = 1 √2

5.(a) Simplifythefollowingexpressions:

(i) 23

2 2 3 (ii) 2 2 3 × 4

(b) If4x = 1 2 ,find x

(d) Simplifylog10 3x + log10 5x.

(e) Solvefor x:ln(x2 ) + ln x = 2

(f) Ifln x = 3,calculatelog10 x

6.(a) Findthe(complex)rootsofthequadraticequation x2 + x + 1 = 0,simplifyingyouranswerasmuchaspossible.

(b) Evaluate(1 + i) × (2 i)andsimplifyyouranswer.

7.(a) Determinetherangesofthefollowingfunctions:

(i) f (x) = x2 , x ∈ [ 1, 1]. (ii) f (x) = x2 , x ∈ R

(b) If f (x) = √x and g(x) = (x + 1)2 find:

(i) f (3). (ii) ( f ◦ g)(x). (iii) (g ◦ f )(4).

8.(a) Forlargevaluesof x,whichofthefollowingpolynomials willreturnthelargestvalue?

(i) p1 (x) = x (ii) p2 (x) = 1 2 x2 (iii) p3 (x) = 1 3 x3

(b) Thesteadyflowoffluidinapipewithcircularcross-section obeys Poiseuille’sequation.Thefastestflowoccursatthecenter ofthepipeandthereisnoflowwherethefluidtouchesthewalls. If r isthedistancefromthecenterofthepipeand a istheradius ofthepipe,thenthevelocity u varieswith r accordingto

u(r) = u0 1 r2 a2 , where u0 isthemaximumvelocityinthepipe(seeFigure1.1).

(i) Whatisthedegreeof u(r)asapolynomialin r?

(ii) Whatisthedomainofthisfunction?

(iii) Whatistherangeofthisfunction?

(iv) u(r)decreaseswithdistancefromthecenterline.Atwhat distancedoes u(r)decreaseto 1 2 ofitsmaximumvalue?

r 5 a

Figure1.1 DiagramofthegeometryofacircularpipeforQuestion 8.Atdistance r fromthecenterof thepipetheflowis u(r).

9. Metabolismofaparticulardruginthebodyisdescribedby Michaelis-Menten kinetics.Iftheconcentrationis c,therateof metabolization(r,ortheamountremovedfromthebloodinone hour)isgivenbytheformula

r(c) = c c + 10

(a) Forwhichofthefollowingconcentrationsistherateofmetabolizationlargest?

(i) c = 1 (ii) c = 2 (iii) or c = 3

(b) Suppose c = 5initially,thenthepatienttakesapillandmore drugisabsorbed;theconcentrationof c doublesto10.Doesthe rateofmetabolization

(i) double? (ii) morethandouble? (iii) lessthandouble?

(d) If c = 10initially,isthereanyconcentrationincreasethat wouldgettherateofmetabolizationtodouble?

10. Species-areacurvesareusedtopredicthowtheamountof diversity(numberofspecies)inaparticularhabitatdecreasesif thehabitatshrinks.Atypicalspecies-arearelationshipbetween N ,thenumberofspecies,and A,theareaofthehabitat,is:

N = kAz

where k and z arepositiveconstants.Assume z = 1 5 toanswer thefollowingquestions.

(a) Iftheinitialareais A 0 ,calculatehowmuchthehabitatarea mustshrinkforthenumberofspeciestobehalved.

(b) Howmuchmustitshrinktoreducethenumberofspeciesto one-thirdofitsstartingvalue?

11. Thesize N ofapopulationofcellscanbemodeledbyan exponentiallaw:

N = N0 ert

where t isthetimeelapsedsincethepopulationgrowthbegan and N0 and r arepositiveconstants.

(a) Calculate N0 and r giventhefollowingpopulation-sizedata. Therewereinitially1000cellsinthepopulation(i.e., N (0) = 1000).At t = 2thereare1000morecellsinthepopulation(i.e., N (2) = 2000).

(b) Usingtheformulaandtheparametersfrompart(a),calculatehowmuchtimemustelapsebefore1000morecellsare addedtothepopulation(i.e.,forwhat t does N (t ) = 3000)?

12. Calculatetheinversesofthefollowingfunctions(i.e.,find f 1 (x)suchthat( f 1 ◦ f )(x) = x).

(a) f (x) = x2 + 1for x ≥ 0.

(b) f (x) = 2ln(x + 1)for x > 1. (c) f (x) = x5

13.(a) Combinethetermsinthefollowingexpressionsintoa singlelogarithm:

(i) ln(x) + ln(x2 + 1) (ii) log(x1/3 ) log((x + 1)1/3 )

(iii) 2 + log2 (x)

(b) Usethechangeofbaseformulatoturnthelogarithmsbelow intonaturallogs:

(i) log2 7 (ii) log6 (iii) logx 2

14.(a) Givetheperiodandamplitudeofthefollowingtrigonometricfunctions:

(i) f (x) =−2sin x (ii) f (x) = 2cos3x (iii) f (x) = 3cos π x 2

(b) Findtherangeandmaximumdomainof:

(i) f (x) = tan x (ii) f (x) = cos x

15. Thefunction f (x)isdrawninFigure1.2.Sketchthegraphsof:

(a) f (x + 2) (b) f (x) + 1 (c) f (x) (d) f ( x) (e) f ( x 2 )

2. anexponentialdependence y = kax forsomeconstants k and a.Inthiscase,givethevalueof a

(a) Dependenceofnumberoflanguagesspoken(D)onthearea (A)oftheglobethatissampled.

(b) FrequencyofearthquakesinSouthernCalifornia(N )with shakingmagnitudelargerthan m

(d) NumberofHIVvirusesin1mlofblood(N )againsttime(t ) forapatientreceivingHIVtreatment.

Figure1.4 Question 17(a).Numberof languagesspoken, D asafunctionofarea, A.Adaptedfrom Gomesetal.(1999)

Figure1.2 Plotofthefunction f (x) forQuestion15.

16. Thetypicalweights(inkilograms)offivepopulardogbreeds areshownonthelogarithmicnumberlineshowninFigure1.3.

(a) Whatisthetypicalmassofthefollowingbreeds:

(i) Chihuahua (ii) Labradorretriever (iii) St.Bernard

(b) HowmuchheavierisanadultDalmatianthanapuppy?

Dalmatian (puppy) Jack Russell Terrier

Retriever

(adult)

Bernard

Figure1.3 Weightsofpopulardogbreedsin kilograms,plottedonalogarithmicnumber lineforQuestion16.

17. EachofthegraphsinFigures1.4through1.7showshowone quantity(plottedonthe y-axis)varieswithasecondquantity (plottedonthe x-axis).Ineachcasestatewhetherthegraph shows

1. apowerlawdependence y = kxa forsomeconstants k and a. Inthiscase,givethevalueof a. or

Figure1.5 Question 17(b).Numberof earthquakes(N )with magnitudelargerthan m,inoneyearasa functionof m.Graph adaptedfrom Rundle etal.(2003)

Figure1.6 Question 17(c).Bacterial populationsize(N )as afunctionoftime(t ). Dataadaptedfrom Balabanetal.(2004).

Figure1.7 Question 17(d).NumberofHIV viruses(N )asa functionoftime(t ). Dataadaptedfrom Ho etal.(2004)

Figure1.8 Question18.Concentrationof anADHDdrug(c)inapatient’sblood, asafunctionoftime(t ).

18. YouarestudyinghowmedicationthatisusedtotreatAttentionDeficitHyperactivityDisorder(ADHD)ismetabolized.Ahealthypatient(patient1)takesadoseofthedrug at8a.m.andyoumeasuretheconcentrationintheirblood plasmaathourlyintervals.Youobtainthedatashownin Figure1.8.

Threeotherpatientsalsoreceivethedrug,butfollowslightly differentdoseregimens.Yourtaskistoidentifywhichcurvein

1.2Preliminaries

Figure1.9 Question18.Identifythe curvecorrespondingtoeachpatient.

Figure1.9describeswhichpatientbasedontheworddescriptions below.

(a) PatientAreceivesthesamedoseaspatient1butat10:00 ratherthan8:00a.m.

(b) PatientBreceivesanextendedreleaseformofthedrug at8:00a.m.,whichtakeslongertoenterthebloodstream.

Thissectionreviewssomeoftheconceptsandtechniquesfromalgebraandtrigonometrythatarefrequentlyusedincalculus.Theproblemsattheendofthesectionwill helpyoureacquaintyourselfwiththismaterial.

1.2.1TheRealNumbers

The realnumbers canmosteasilybevisualizedonthe real-numberline (seeFig4321 0 1 2 3 4 x ab

Figure1.10 Thereal-numberline. ure1.10),onwhichnumbersareorderedsothatif a < b,then a istotheleftof b. Sets(collections)ofrealnumbersaretypicallydenotedbythecapitalletters A, B, C , etc.Todescribetheset A,wewrite

A ={x :condition}

where“condition”tellsuswhichnumbersareintheset A.Themostimportantsetsin calculusare intervals.Weusethefollowingnotations:If a < b,then the open interval(a, b) ={x : a < x < b} and the closed interval[a, b] ={x : a ≤ x ≤ b}

Wealsouse half-open intervals:

[a, b) ={x : a ≤ x < b} and(a, b] ={x : a < x ≤ b}

Unbounded intervalsaresetsoftheform {x : x > a}.Herearethepossiblecases:

[a, ∞) ={x : x ≥ a} (−∞, a] ={x : x ≤ a} (a, ∞) ={x : x > a} (−∞, a) ={x : x < a}

Thesymbols“∞”and“−∞”mean“plusinfinity”and“minusinfinity,”respectively. Thesesymbolsare not realnumbers,butareusedmerelyfornotationalconvenience. Thereal-numberline,denotedby R,doesnothaveendpoints,andwecanwrite R inthefollowingequivalentforms: R ={x : −∞ < x < ∞}= (−∞, ∞)

EXAMPLE1

Thelocationofthenumber0onthereal-numberlineiscalledthe origin,andwe canmeasurethedistanceofthenumber x totheorigin.Forinstance, 5is5unitsto theleftoftheorigin.Aconvenientnotationformeasuringdistancesfromtheorigin onthereal-numberlineistheabsolutevalueofarealnumber.

Deﬁnition The absolutevalue ofarealnumber a,denotedby |a|,is

|a|= a if a ≥ 0 a if a < 0

Forexample, |− 7|=−( 7) = 7.Wecanuseabsolutevaluestofindthedistance betweenanytwonumbers x1 and x2 asfollows:

distancebetween x1 and x2 =|x1 x2 |

Notethat |x1 x2 |=|x2 x1 |.Sothedistancebetween x1 and x2 is,asyouwould expect,thesameasthedistancebetween x2 and x1 .Tofindthedistancebetween 2 and4,wecompute |− 2 4|=|− 6|= 6,or |4 ( 2)|=|4 + 2|= 6.

Wewillfrequentlyneedtosolveequationscontainingabsolutevalues,forwhich thefollowingpropertyisuseful:

PropertyofAbsoluteValueEquations Let b ≥ 0.Then |a|= b isequivalentto a =±b.

Solve |x 4|= 2.

Solution ApplyingthePropertyofAbsoluteValueEquations,weobtain: x 4 =±2,(i.e., x = 4 ± 2).Soeither x = 4 2 = 2or x = 4 + 2 = 6.Thesolutions,illustrated graphicallyinFigure1.11,aretherefore x = 6and x = 2.Thepointsofintersectionof y =|x 4| and y = 2areat x = 6and x = 2.Solving |x 4|= 2canalsobeinterpreted asfindingthetwonumbersthathavedistance2from4.

Figure1.11 Thegraphof y =|x 4| and y = 2.Thepointsofintersection areat x = 6and x = 2.

Whenthereareabsolutevaluesignsonbothsidesoftheequationtherecanbe ± onbothsidesoftheequationasillustratedinthenextexample.

EXAMPLE2 Solve | 3 2 x 1|=| 1 2 x + 1|

Solution ApplyingthePropertyofAbsoluteValueEquationsruletobothsidesoftheequation wecanreplacetheabsolutevaluesignsby ± onbothsides: ± 3 2 x 1 =± 1 2 x + 1

Nowthe ± signsareindependentofeachother.Wecanchoose + ontheleftsideand ontherightside.Soitseemsthattherearefourpossibilitiestoconsider:(+, +),

Figure1.12 Thegraphsof y =| 3 2 x 1| and y =| 1 2 x + 1|.The pointsofintersectionareat x = 0 and x = 2.

(+, ),( , +),( , ),where(+, )denoteschoosing + signontherightsideand signontheleftside.However,someofthepossibilitiesgiveequivalentanswers;for example:

+, +)gives:

Ifwemultiplybothsidesbytheterm 1wecanturn(1.2)into(1.1),sothetwoequationsareequivalent.

Withthisinmindweneedonlyconsideronlythechoicesofsign:(+, +)and (+, ):

Taking(+, +)gives

2 x 1 = 1 2 x + 1 ( , )givesthesameequation. x = 2. Subtract 1 2 x frombothsides,add1tobothsides. Taking(+, )gives

2x = 0 Add 1 2 x tobothsides,add1tobothsides. x = 0

AgraphicalsolutionofthisexampleisshowninFigure1.12.

ReturningtoExample1,wherewefoundthetwopointswhosedistancefrom4 wasequalto2,wecanalsotrytofindthosepointswhosedistancefrom4islessthan (orgreaterthan)2.Thisamountstosolvinginequalitieswithabsolutevalues.Looking backatFigure1.11,weseethatthesetof x-valueswhosedistancefrom4islessthan 2(i.e., |x 4| < 2)istheinterval(2, 6).Similarly,thesetof x-valueswhosedistance from4isgreaterthan2(i.e., |x 4| > 2)istheunionofthetwointervals(−∞, 2)and (6, ∞),or(−∞, 2) ∪ (6, ∞).

Tosolveabsolute-valueinequalities,thefollowingtwopropertiesareuseful:

UsingtheAbsoluteValueinInequalities Let b > 0.Then

1. |a| < b isequivalentto

Solution (a) Werewrite |2x 5| < 3as 3 < 2x 5 < 3 2 < 2x < 8

Add5toallthreeparts 1 < x < 4

Divideallthreepartsby2

Thesolutionisthereforetheset {x :1 < x < 4}.Inintervalnotation,thesolutioncan bewrittenastheopeninterval(1, 4).

(b) Tosolve |4 3x| ≥ 2,wegothroughthefollowingsteps:

Either: 4 3x ≥ 2or4 3x ≤−2 3x ≥−2 3x ≤−6 x ≤ 2 3 x ≥ 2

Whenbothsidesof inequalityaredividedor multipliedbyanegative number,theinequality mustbereversed.

Thesolutionistheset {x : x ≤ 2 3 or x ≥ 2},or,inintervalnotation,(−∞, 2 3 ] ∪ [2, ∞).

EXAMPLE3

m 1 y x

b Figure1.14 Theslope x-intercept formoftheequationforastraight linerequirestheslope, m,and y-intercept, b.Thetriangleshows thatforeveryunitthelinetravelsin the x-directionitgoesup m unitsin the y-direction.

1.2.2LinesinthePlane

Wewillfrequentlyencountersituationsinwhichtherelationshipbetweenquantities canbedescribedbya linearequation.Forexample,themaximumbitestrengthof spottedhyenasincreaseswithageofthehyena.Specifically,if x istheageofthehyena inmonthsand y isthemaximumbiteforce(innewtons)thatthehyenaiscapableof exerting,thenasshownby BinderandVanValkenburgh(2000):

y = 166.0 + 12.7x (1.3)

Equation(1.3)isanexampleofa linearequation,andwesaythat x and y satisfya linearequation.

Thegraphofalinearequationisastraightline.Theequationofthestraightline canbewrittenusinganyofthreedifferentforms:

1. The standard formofalinearequationisgivenby

Ax + By + C = 0

where A, B,and C areconstants, A and B arenotbothequalto0,and x and y arethetwovariables.

2. Ifthetwopoints(x1 , y1 )and(x2 , y2 )lieonastraightline,thenthe slope ofthe lineis

m = y2 y1 x2 x1

(SeeFigure1.13.)Twopoints(oronepointandtheslope)aresufficienttodeterminetheequationofastraightline.

Ifyouaregivenonepointandtheslope,provided m isfiniteyoucanusethe point–slope formofastraightlinetowriteitsequation,givenby

y y0 = m(x x0 )

where m istheslopeand(x0 , y0 )isapointontheline.Ifyouaregiventwopoints, firstcomputetheslopeandthenuseoneofthepointsandtheslopetofindthe equationofthestraightlineinpoint–slopeform.

3. Lastly,the slope–intercept formis:

y = mx + b

where m istheslopeand b isthe y-intercept,whichisthepointofintersectionof thelinewiththe y-axis;the y-intercepthascoordinates(0, b).(SeeFigure1.14.)

Deﬁnition FormsofLinearEquations

Ax + By + C = 0(StandardForm)

y y0 = m(x x0 )(Point–SlopeForm)

y = mx + b (Slope–InterceptForm)

EXAMPLE4 Determine,inslope–interceptform,theequationofthelinepassingthrough( 2, 1) and(3, 1 2 ).

Solution Theslopeofthelineis

Usingthepoint–slopeform:

y 1 =− 3 10 (x

or,inslope–interceptform,

Wecouldhaveusedtheotherpoint,(3, 1 2 ),andobtainedthesameresult.

Figure1.13 Theslopeofastraight line.