Functions Modeling Change A Preparation For Calculus, 5th Edition

629 Pages·2015·7.995 MB·English

by  Hughes-Hallett, Gleason, Et. Al. Wiley

FUNCTIONS MODELING CHANGE:
A Preparation for Calculus
FifthEdition
ProducedbytheCalculusConsortiumandinitiallyfundedbyaNationalScienceFoundationGrant.
EricConnally DeborahHughes-Hallett
HarvardUniversityExtension TheUniversityofArizona
AndrewM.Gleason PhilipCheifetz
HarvardUniversity NassauCommunityCollege
AnnDavidian DanielE.Flath
Gen.DouglasMacArthurHS MacalesterCollege
SelinKalaycıoğlu BrigitteLahme
NewYorkUniversity SonomaStateUniversity
PattiFrazerLock GuadalupeI.Lozano
St.LawrenceUniversity TheUniversityofArizona
WilliamG.McCallum JerryMorris
TheUniversityofArizona SonomaStateUniversity
KarenRhea EllenSchmierer
UniversityofMichigan NassauCommunityCollege
PatShure AdamH.Spiegler
UniversityofMichigan LoyolaUniversityChicago
CarlSwenson AaronD.Wootton
SeattleUniversity UniversityofPortland
ElliotJ.Marks
withtheassistanceof
FrankAvenoso DouglasQuinney KatherineYoshiwara
NassauCommunityCollege UniversityofKeele LosAngelesPierceCollege
DedicatedtoBen,Jonah,andIsabel
PUBLISHER LaurieRosatone
ACQUISITIONSEDITOR JoannaDingle
ASSOCIATEEDITOR JacquelineSinacori
MARKETINGMANAGER KimberlyKanakes
SENIORPRODUCTDESIGNER TomKulesa
SENIORCONTENTMANAGER KarolineLuciano
SENIORPRODUCTIONEDITOR KenSantor
FREELANCEPROJECTEDITOR AnneScanlan-Rohrer
EDITORIALPROGRAMASSISTANT CourtneyWelsh
MARKETINGASSISTANT JesseAdler
COVERDESIGNER MadelynLesure
COVERANDCHAPTEROPENINGPHOTO ©PatrickZephyr/PatrickZephyrNaturePhotography
ThisbookwassetinStixbytheConsortiumusingTeX,Mathematica,andthepackageAsTeX,whichwaswritten
byAlexKasman.TheprocesswasmanagedbyElliotMarks.
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PREFACE
Inthe21st century,studentsneedtheabilitytosolveproblems;innovationdependsoncitizenswhocan
thinkcritically.The fiftheditionof FunctionsModelingChange:A Preparationfor Calculusreaffirmsour
efforttorefocustheteachingofmathematicsinabalancedblendofconceptsandprocedures.Ourapproach
encouragesstudents to develop their problem-solvingskills while acquiring the mathematical background
neededtolearncalculusorpursuetheircareers.
FifthEdition:Focus
ThiseditionofPrecalculuscontinuestostressconceptualunderstandingandtheconnectionsamongmathe-
maticalideas.Understandingexponentialchange,forexample,meansbeingabletorelatethesizeofapercent
growthtotheshapeofafunction’sgraph,aswellasassessingthereasonabilityofamodel.
Functionsarethefoundationofmuchofmathematics.Ourapproachencouragesarobustunderstanding
offunctionsintraditionalandnovelcontexts.Skillsareintroducedandreinforcedthroughoutthebook.This
balanceofskillsandunderstandingpromotesstudents’criticalthinking.
FifthEdition:Flexibility
Precalculus courses are taken by a wide range of students and are taught in a wide variety of styles. As
instructors ourselves, we know that the balance we choose dependson the students we have: sometimes a
focusonconceptualunderstandingisbest;sometimesmoreskill-buildingisneeded.
Toenableinstructorstoselectthebalanceappropriatefortheirstudents,thefiftheditionexpandstheop-
tionsavailableforcustomizingmaterial.Forexample,wehaveintegratedexamplesinvolvinglinearinequal-
itiesinourtreatmentoflinearequations,andofferedmoreexamplesonratesofchangeinseveralchapters.
Thesectionsinvolvinglimitnotationandphaseshifthavebeenrestructuredsothatthesetopicscanbeeasily
skipped.TransformationsoffunctionsarenowintroducedinChapter2,whilethefulltreatmentremainsin
Chapter6,providinginstructorswithmorefreedomaboutwhentointroducethemandhowdeeplytocover
thistopic.
OriginofText:TheCalculusConsortiumforHigherEducation
Thisbookistheworkoffacultyatadiverseconsortiumofinstitutions,andwasoriginallygenerouslysup-
portedbytheNationalScienceFoundation.Itrepresentsthefirstconsensusamongsuchadiversegroupof
facultytohaveshapedamainstreamprecalculustext.Bringingtogethertheresultsofresearchandexperience
withtheviewsofmanyusers,thistextisdesignedtobeusedinawiderangeofinstitutions.
GuidingPrinciples:VariedProblemsandtheRuleofFour
Conceptualunderstandingisenhancedwhenstudentsengageinnon-proceduralproblem-solving.Problems
classified into clear-cut types tend to develop proficiency at finding answers, but not necessarily a robust
conceptualunderstanding.Strongproblem-solversfeel capable of makingprogresson problemsthey have
notseenbefore,notjustthoseofknowntype.Consequently,weareguidedbythefollowingprinciples:
• Problemsare variedand often challenging.Many of our problemscannotbe answeredby following a
templateinthetext.
• Textexamplesarediverseandrepresentthenaturalintegrationofskillsandconcepts.
• TheRuleofFour:eachconceptandfunctionisrepresentedsymbolically,numerically,graphically,and
verbally.Thisprinciple,originallyintroducedbytheConsortium,promotesmultiplerepresentations.
v
vi Preface
• Weprovidestudentsandinstructorsquickmeansofassessingcomprehensionbeforemovingonbyusing
thetrue-falseStrengthenYourUnderstandingproblemsattheendofeachchapter.
• Ourcentraltheme,functionsasmodelsofchange,linksthecomponentsofourprecalculuscurriculum.
Algebraisintegratedwhereappropriate.
• Topicsarefewerinnumberthanitiscustomarysotheycanbetreatedingreaterdepth.Thetopicsare
thoseessentialtothestudyofcalculus.
• Problemsinvolvingrealdatagivestudentspracticewithmodeling.
• We include both classic and current applications to prepare students to model with mathematics in a
varietyofcontexts.
• Ourproblemsallowstudentstobecomeproficientintheuseoftechnology,includingsymbolicmanipu-
lators,computers,tablets,andonlinesoftware,asappropriate.
• Ourprecalculusmaterialsallowforabroadrangeofteachingstyles.Theyareflexibleenoughforusein
largelecturehalls,smallclasses,oringrouporlabsettings.
ChangesintheFifthEdition
The fifth edition reflects the many helpful suggestions from users while preserving the focus and guiding
principlesofpreviouseditions.Wehavemadethefollowingchanges:
• Many examples and problems are new or have been rewritten. Data has been updated and new data
introduced.
• Thethreechaptersontrigonometryhavebeenreorganizedandrewritten:
− Trigonometric functions on the circle are introduced both in radians and degrees in Chapter 7,
highlightingthenaturalrelationshipbetweenradiansandtheunitcircleassoonasperiodicfunctions
areintroduced.
− Sinusoidalfunctionsmodelingperiodicphenomenaandsimpletrigonometricequationsarealso
introducedinChapter7,whilemoreinvolvedtrigonometricmodelsandequationsrequiringrelation-
shipssuchasdouble-angleidentitiesaretreatedinChapter9.
− PhaseshifthasbeenmadeanoptionaltopicinChapter7.
− TriangletrigonometryisdiscussedseparatelyinChapter8,aftertrigonometricandsinusoidalfunc-
tionsaretreatedinChapter7.
• Abriefexplorationoflinearinequalitieshasbeenintegratedwiththematerialonsolvinglinearequations
inChapter1.
• VerticalandhorizontalshiftsareintroducedinChapter2,andreferencedinChapter3whenintroducing
thevertexformofaquadraticequation.ShiftsarereviewedatthebeginningofChapter6andconsidered
incombinationwithothertransformations.
• Oddandevenfunctionsareintroducedbylookingatinvarianceofcertainfunctionsunderreflections,
andthusbetterintegratedwiththetransformationfocusofChapter6.
• Theeffectofchangingtheorderoftransformations,thelastsectioninChapter6inthefourthedition,
hasbeenshortenedandincludedintheprevioussection.
• The section on power functions in Chapter 11 has been rewritten to increase the focus on graphical
behaviorandproportionality.
• Examples on average rate of change have been added throughoutthe book, such as for exponential
functions(inChapter5)andperiodicfunctions(inChapters7and9).
• WileyPLUS,theprimaryonlineresourcesuitepairedwiththetextbook,hasbeenupdatedwithimproved
problemsandhints,includingmanynewproblemsfromthefifthedition.
Preface vii
WhatStudentBackgroundisExpected?
Studentsusingthisbookshouldhavesuccessfullycompletedacourseinintermediatealgebraorhighschool
algebraII.Thebookisthought-provokingforwell-preparedstudentswhilestillaccessible tostudentswith
weaker backgrounds.Providingnumerical, graphical,and algebraic approachesbuilds on different student
strengthsandprovidesstudentswithavarietyofwaystomasterthematerial.Multiplerepresentationsgive
studentstoolstopersist,loweringfailurerates.
OurExperiences
Thefirstfoureditionsofthisbookwereusedathundredsofschoolsaroundthecountryinawidevarietyof
settings.Ithasbeenusedsuccessfullyinbothsemesterandquartersystems,inlargelecturesandsmallclasses
aswellasinfull-yearcoursesinsecondaryschools.Ithasalsobeenusedincomputerlabsandsmallgroups,
oftenwiththeintegrationofanumberofdifferenttechnologies.
Content
Thecentralthemeofthisbookisfunctionsasmodelsofchange.Weemphasizethatfunctionscanbegrouped
intofamiliesandthatfunctionscanbeusedasmodels.We explorehowfunctioncharacteristicsconnectto
differencequotientsandratesofchange,naturallypreviewingkeycalculusideas.
Becauselinear,quadratic,exponential,power,andperiodicfunctionsaremostfrequentlyusedtomodel
physicalphenomena,theyareintroducedbeforepolynomialandrationalfunctions.Onceintroduced,afamily
offunctionsiscomparedandcontrastedwithotherfamiliesoffunctions.
Alargenumberoftheexamplesthatstudentsseeinthisprecalculuscoursearereal-worldproblems.By
theendofthecourse,wehopethatstudentswillusefunctionstohelpthemunderstandtheworldinwhich
theylive.Weincludenon-routineproblemstoemphasizethatsuchproblemsarenotonlypartofmathematics,
butinsomesensearethereasonfordoingmathematics.
Technology
Thebookdoesnotrequireanyspecificsoftwareortechnology.Instructorshaveusedthematerialwithgraph-
ingcalculators,graphingsoftware,orscientificcalculators.
Chapter1:LinearFunctionsandChange
Thischapterintroducestheconceptofafunctionaswellasgraphical,tabular,symbolic,andverbalrepresen-
tationsoffunctions,discussingtheadvantagesanddisadvantagesofeachrepresentation.Itintroducesratesof
changeandusesthemtocharacterizelinearfunctions.Examplesonmodelingwithlinearfunctions,including
interpretinglinearinequalities,arediscussed.Asectiononfittingalinearfunctiontodataisincluded.
TheSkillsRefreshersectionforChapter1reviewslinearequations,linearinequalities,andthecoordi-
nateplane.
Chapter2:Functions
Thischapterstudiesfunctionsin moredetail.Itintroducesdomain,range,previewsfunctionshiftsandthe
conceptsofcompositeandinversefunctions,andinvestigatestheideaofconcavityusingratesofchange.A
sectiononpiecewisefunctionsisincluded.
Chapter3:QuadraticFunctions
This chapter introducesthe standard,factored,and vertex formsof a quadraticfunctionand explorestheir
relationshiptographs,includingshifts.Thefamilyofquadraticfunctionsprovidesanopportunitytoseethe
effectofparametersonfunctionalbehavior.
TheSkillsRefreshersectionforChapter3reviewsfactoring,completingthesquare,andquadraticequa-
tions.
viii Preface
Chapter4:ExponentialFunctions
Thischapterintroducesthefamilyofexponentialfunctionsandthenumber𝑒.Itcomparesexponentialand
linearfunctions,solvesexponentialequationsgraphically,andgivesapplicationstocompoundinterest.
TheSkillsRefreshersectionforChapter4reviewsthepropertiesofexponents.
Chapter5:LogarithmicFunctions
Thischapterintroduceslogarithmicfunctionswithbase10andbase𝑒,bothinordertosolveexponentialequa-
tionsandasinversesofexponentialfunctions.Afterdiscussingmanipulationswithlogarithms,thechapter
focusesonmodelingwithexponentialfunctionsandlogarithms.Logarithmicscalesandasectiononlineariz-
ingdataconcludethechapter.
TheSkillsRefreshersectionforChapter5reviewsthepropertiesoflogarithms.
Chapter6:TransformationsofFunctionsandTheirGraphs
Thischapterinvestigatestransformations.Itrevisitsshifts,introducesreflectionsandstretches,andexplores
even and odd symmetry using transformations. The effect of orderingwhen combining transformationsis
explored.
Chapter7:TrigonometryandPeriodicFunctions
Thischapterintroducestrigonometricfunctionsinbothradiansanddegreesasmodelsforperiodicmotion.
Afterexploringgraphsandformulasofsine,cosine,andtangent,generalsinusoidalfunctionsareintroduced.
Thischapterintroducestheinversetrigonometricfunctionstosolvetrigonometricequations.
TheSkillsRefreshersectionforChapter7reviewssineandcosinevaluesofspecialanglesmeasuredin
bothradiansanddegrees.
Chapter8:TriangleTrigonometryandPolarCoordinates
Thischapterdevelopsright-triangletrigonometry,andintroducestheLawofSinesandtheLawofCosines.It
alsodefinespolarcoordinates,explorestheirrelationshiptoCartesiancoordinates,andconsidersthegraphs
ofpolarinequalities.
Note:Chapter8canbeskippedbyinstructorswhoprefernottoemphasizetriangletrigonometry.
Chapter9:TrigonometricIdentities,ModelsandComplexNumbers
Thischapteropenswithadiscussionoftrigonometricequations,andthenexplorestrigonometricidentities
and their rolein trigonometricmodels,such as dampedoscillationsand acoustic beats. The chaptercloses
withcomplexnumbers,includingEuler’sFormulaanddeMoivre’stheorem.
Chapter10:Compositions,Inverses,andCombinationsofFunctions
Thischapterdiscussescombinationsoffunctions.Itinvestigatescompositeandinversefunctions,whichwere
introducedinChapter2,inmoredetail.
Chapter11:PolynomialandRationalFunctions
Thischapterdiscussespowerfunctions,polynomials,andrationalfunctions.Thechapterexploresdominance
and long-run behavior and concludes by comparing polynomial and exponential functions, and by fitting
functionstodata.
TheSkillsRefreshersectionforChapter11reviewsalgebraicfractions.
Chapter12:Vectors
This chapter contains materialon vectors and operationsinvolving vectors. An introductionto matrices is
includedinthelastsection.
Preface ix
Chapter13:SequencesandSeries
Thischapterintroducesarithmeticandgeometricsequencesandseriesandtheirapplications.
Chapter14:ParametricEquationsandConicSections
The concluding chapter looks at parametric equations, implicit functions, hyperbolic functions, and conic
sections:circles,ellipses,andhyperbolas.Thechapterincludesasectiononthegeometricalpropertiesofthe
conicsectionsandtheirapplicationstoorbits.
Note:Chapter14isavailableonlineonly.
SupplementaryMaterials
Thefollowingsupplementarymaterialsareavailableforthefifthedition:
• The Instructor’s Manual contains teaching tips, lesson plans, syllabi, and worksheets. It has been
expanded and revised to include worksheets, identification of technology-orientedproblems, and new
syllabi.(ISBN978-1-119-01383-9)
• ThePrintedTestBankcontainstestquestionsarrangedbysection.
• TheInstructor’sSolutionManualhascompletesolutionstoallproblems.(ISBN978-1-118-94162-1)
• The Student Solution Manual has complete solutionsto half of the odd-numberedproblems.(ISBN
978-1-118-94163-8)
• TheComputerizedTestBank,availableinbothPCandMacintoshformats,allowsinstructorstocreate,
customize, and print a test containing any combinations of questions from a large bank of questions.
Instructorscanalsocustomizethequestionsorcreatetheirown.
• Classroom Activities are posted at the book companion website. These activities were developed to
facilitatein-classgroupworkaswellastointroducenewconceptsandtopracticeskills.Inadditionto
theblankcopiesforeachactivitythatcanbehandedouttothestudents,acopyoftheactivitywithfully
workedoutsolutionsisalsoavailable.
• TheBookCompanionSiteatwww.wiley.com/college/connallycontainsallinstructorsupplements.
• WileyPLUSisapowerfulonlinesuiteofteachingandlearningresourcestightlyintegratedwiththetext.
WileyPLUSenablesinstructorstoassign,deliverandgradeindividuallycustomizedhomeworkassign-
mentsusingexercisesandproblemsfromthetext.Studentsreceiveimmediatefeedbackontheirhome-
workandaccesstofullsolutionstoassignedproblemselectronically.Studentsmayalsoaccesshintsto
theproblems.Inadditiontoonlinehomework,WileyPLUSprovidesstudenttutorials,aninstructorgrade-
book,integratedlinkstotheelectronicversionofthetext,andallofthetext’ssupplementalmaterials.
Formoreinformation,visitwww.wiley.com/college/wileyplusorcontactyourlocalWileyrepresentative
formoredetails.
• Mini-lectureVideoslinkedwithexamplesintheWileyPLUSstudentversionofthetextprovidegreater
detailtothesolutionofexamplesineachsectionofthetext.Thesemayassiststudentsinreadingthetext
priortoclassorinreviewingmaterialafterclass.
• The Faculty Network is a peer-to-peernetwork of academic faculty dedicated to the effective use of
technologyintheclassroom.Thisgroupcanhelpyouapplyinnovativeclassroomtechniques,implement
specificsoftwarepackages,andtailorthetechnologyexperiencetothespecificneedsofeachindividual
class.Visitwww.wherefacultyconnect.comoraskyourWileyrepresentativefordetails.
ConcepTests
ConcepTests, modeledon the pioneeringwork of Harvardphysicist Eric Mazur,are questionsdesignedto
promoteactivelearningduringclass,particularly(butnotexclusively)inlargelectures.Ourevaluationdata
showstudentstaughtwithConcepTestsoutperformedstudentstaughtbytraditionallecturemethods73%ver-
sus17%onconceptualquestions,and63%versus54%oncomputationalproblems.ConcepTestsarrangedby
x Preface
sectionareavailableinPowerPointandClassroomResponseSystem-readyformatsfromyourWileyrepre-
sentative.(ISBN978-1-118-94161-4)
Acknowledgments
We would like to thank the many people who made this book possible. First, we would like to thank the
NationalScienceFoundationfortheirtrustandtheirsupport;weareparticularlygratefultoJimLightbourne
andSpudBradley.
WearealsogratefultoourAdvisoryBoardfortheirguidance:BenitaAlbert,LidaBarrett,SimonBernau,
RobertDavis,LoveniaDeconge-Watson,JohnDossey,RonaldDouglas,EliFromm,BillHaver,DonLewis,
SeymourParter,JohnPrados,andStephenRodi.
Working with Laurie Rosatone, Anne Scanlan-Rohrer, Ken Santor, Shannon Corliss, Joanna Dingle,
KimberlyKanakes,JacquelineSinacori,CourtneyWalsh,andMadelynLesureatJohnWileyisapleasure.
Weappreciatetheirpatienceandimagination.
Manypeoplehavecontributedsignificantlytothistext.Theyinclude:LaurenAkers,FahdAlshammari,
DavidArias,TimBean,CharlotteBonner,BillBossert,BrianBradie,NoahS.Brannen,MikeBrilleslyper,
DonnaBrouillette,MatthewM.Campbell,JoCannon,RayCannon,KennyChing,AnnaChung,PierreCres-
sant,MarciaCrump,LaurieDelitsky,BobDobrow,HelenM.Doerr,IanDowker,CarolynEdmond,Maryann
Faller, Aidan Flanagan, Brendan Fry, Brad Garner, Carrie Garner, John Gerke, Christie Gilliland, Wynne
Guy,DonnieHallstone,David Halstead,LarryHenly,DeanHickerson,Jo EllenHillyer,Bob Hoburg,Phil
Hotchkiss, MikeHuffman,Mac Hyman,Rajini Jesudason,LorenJohnson,David Jones,Scott Kaplowitch,
Nazanin Karimi, Thomas Kershaw, Mary Kilbride, Steve Kinholt, Kandace Kling, Angela Kubena, Rob
LaQuaglia, Barbara Leasher, Kenneth Lemp, Richard Little, David Lovelock, Guadalupe Lozano Terán,
Nicholas Lyktey,Chaimaa Makoudi,Len Malinowski, NancyMarcus, Kate McGivney,GowriMeda, Bob
Megginson,DeborahMoore,EricMotylinski,BillMueller,MichaelNakamaye,KyleNiedzwiecki,Karina
Nielsen, KathrynOswald, IgorPadure,BridgetNeale Paris, Janet Ray,Ritam Ray,Ken Richardson,Halip
Saifi,SharonSanders,MarySchumacher,MikeSeery,MikeSherman,DonnaSherrill,MaxShuchman,Fred
Shure,KanwalSingh,MyraSnell,ReginaSouza,NatashaSpeer,SonyaStanley,MichaelSteuer,JimStone,
PeggyTibbs,JeffTaft,EliasToubassi,JerryUhl,PatWagener,BenjaminWest,DaleWinter,TimothyWise-
carver,andXianbaoXu.
Reportsfromthefollowingreviewersweremosthelpfulinshapingthesecond,third,fourth,andfifthedi-
tions:VictorAkatsa,JeffreyAnderson,BethBorel,LindaBraddy,MichaelBrassington,IngridBrown-Scott,
Linda Casper, Kim Chudnick, Ted Coe, Ray Collings, Joe Coyle, Pam Crawford, Marcia Crump, Monica
Davis, Laurie Delitsky, Phyllis Desormeaux, Helen Doerr, Diane Downie, Peter Dragnev, Patricia Dueck,
Julie Fisher, Jennifer Fowler, Alyne Fulte, David Gillette, Jack Green, Zdenka Guadarrama, Donnie Hall-
stone,JeffHoherz,MajidHosseini,RickHough,AnnHumes,BryanIngham,MaryLouJumonville,Pallavi
Ketkar,WilliamKiele,MileKrajcevski,JohnLaMaster,RichardLane,DebbieLee,PhyllisLeonard,Daphne
MacLean, Diane Mathios, Krista Maxson, Michael McAfee, Vince McGarry, Maria Miles, Laura Moore-
Mueller,EllenMusen,DaveNolan,LindaO’Brien,ChrisParks,ScottPerry,JeffreyS.Powell,AnnePraderas,
MichaelPrice,MaryRack,GregoryRhoads,JoeRody,EmilyRoth,JohnSawka,AmyN.Schachle,Barbara
ShabellDeirdreSmith,ErnieSolheid,ReginaP.Souza,SandySpears,DianaStaats,JonathanStadler,Mary
JaneSterling,DennisStramiello,DebbieStrance,AllisonSutton,JohnThomason,DianeVanNostrand,Jim
Vicich,LindaWagner,NicoleWilliams,JimWinston,VauhnWittman-Grahler,BruceYoshiwara,FangZhao,
andXinyunZhu.SpecialthanksareowedtoFayeRiddleforadministeringtheprojectandtoAlexKasman
forhissoftwaresupport.
EricConnally DeborahHughes-Hallett AndrewM.Gleason PhilipCheifetz
AnnDavidian DanFlath SelinKalaycıoğlu BrigitteLahme
PattiFrazerLock GuadalupeI.Lozano ElliotJ.Marks WilliamG.McCallum
JerryMorris KarenRhea EllenSchmierer PatShure
AdamH.Spiegler CarlSwenson AaronD.Wootton