Functions modeling change : a preparation for calculus

618 Pages·2007·6.206 MB·English

by  Eric Connally, et al

Openmirrors.com

FUNCTIONS MODELING CHANGE:
A Preparation for Calculus
ThirdEdition
FUNCTIONS MODELING CHANGE:
A Preparation for Calculus
ThirdEdition
ProducedbytheCalculusConsortiumandinitiallyfundedbyaNationalScienceFoundationGrant.
EricConnally DeborahHughes-Hallett
HarvardUniversityExtension UniversityofArizona
AndrewM.Gleason PhilipCheifetz
HarvardUniversity NassauCommunityCollege
AnnDavidian DanielE.Flath
Gen.DouglasMacArthurHS MacalesterCollege
BrigitteLahme PattiFrazerLock
SonomaStateUniversity St.LawrenceUniversity
JerryMorris KarenRhea
SonomaStateUniversity UniversityofMichigan
EllenSchmierer PatShure
NassauCommunityCollege UniversityofMichigan
CarlSwenson KatherineYoshiwara
SeattleUniversity LosAngelesPierceCollege
ElliotJ.Marks
withtheassistanceof
FrankAvenoso
NassauCommunityCollege
(cid:13)
JohnWiley&Sons,Inc.
iv
DedicatedtoMaria,Ben,Jonah,andIsabel
PUBLISHER LaurieRosatone
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COVERPHOTO (cid:13)cScottBerner/PictureQuest
ThisbookwassetinTimesRomanbytheConsortiumusingTeX,Mathematica,andthepackageAsTeX,which
was written by Alex Kasman. It was printed and bound by Von Hoffmann Press. The cover was printed by Von
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ThismaterialisbaseduponworksupportedbytheNational
ScienceFoundationunderGrantNo.DUE-9352905.Opinions
expressedarethoseoftheauthorsandnotnecessarilythose
oftheFoundation.
ISBN-13978-0-471-79302-1
ISBN-100471-79302-7
PrintedintheUnitedStatesofAmerica
10987654321
PREFACE
Mathematics has the extraordinary power to reduce complicated problems to simple rules and proce-
dures.Thereinliesthedangerinteachingmathematics:itispossibletoteachthesubjectasnothingbutthe
rulesandprocedures—therebylosingsightofboththemathematicsandofitspracticalvalue.Thethirdedi-
tionofFunctionsModelingChange:APreparationforCalculuscontinuesourefforttorefocustheteaching
ofmathematicsonconceptsaswellasprocedures.
ACurriculumBalancingSkillsandConceptualUnderstanding
Thesematerialsstressconceptualunderstandingandmultiplewaysofrepresentingmathematicalideas.Our
goal is to provide students with a clear understanding of the ideas of functions as a solid foundation for
subsequentcoursesinmathematicsandotherdisciplines.WhenwedesignedthiscurriculumunderanNSF
grant,westartedwithacleanslate.Wefocusedonthekeyconcepts,emphasizingdepthofunderstanding.
Skillsaredevelopedinthecontextofproblemsandreinforcedinavarietyofsettings,therebyencourag-
ingretention.Thisbalanceofskillsandunderstandingenablesstudentstorealizethepowerofmathematics
inmodeling.
TheCalculusConsortiumforHigherEducation
This book is the work of faculty at a consortium of ten institutions, generously supported by the National
ScienceFoundation.Itrepresentsthefirstconsensusbetweensuchadiversegroupoffacultytohaveshaped
a mainstream precalculus text. Bringing together the results of research and experience with the views of
manyusers,thistextisdesignedtobeusedinawiderangeofinstitutions.
GuidingPrinciples:VariedProblemsandtheRuleofFour
Sincestudentsusuallylearnmostwhentheyareactive,theexercisesinatextareofcentralimportance.In
addition, we have found that multiple representations encourage students to reflect on the meaning of the
material.Consequently,wehavebeenguidedbythefollowingprinciples.
• Ourproblemsarevariedandsomearechallenging.Manycannotbedonebyfollowingatemplateinthe
text.
• The Rule of Four, originally introduced by the consortium, promotes multiple representations. Each
conceptandfunctionisrepresentedsymbolically,numerically,graphically,andverbally.
• Thecomponentsofaprecalculuscurriculumshouldbetiedtogetherbyclearlydefinedthemes.Functions
asmodelsofchangeisourcentraltheme,andalgebraisintegratedwhereappropriate.
• Fewertopicsareintroducedthaniscustomary,buteachtopicistreatedingreaterdepth.Thecoresyllabus
ofprecalculusshouldincludeonlythosetopicsthatareessentialtothestudyofcalculus.
• Problemsinvolvingrealdataareincludedtopreparestudentstousemathematicsinotherfields.
• Tousemathematicseffectively,studentsneedskillinbothsymbolicmanipulationandtheuseoftech-
nology.Theexactproportionsofeachmayvarywidely,dependingonthepreparationofthestudentand
thewishesoftheinstructor.
• Materials for precalculus should allow for a broad range of teaching styles. They should be flexible
enoughtouseinlargelecturehalls,smallclasses,oringrouporlabsettings.
v
vi Preface
TheThirdEdition:IncreasedFlexibility
Thethirdeditionretainsthehallmarksofearliereditionsandprovidesinstructorswithmoreflexibility.
• Composition of functions is introduced in Chapter 2 for instructors who want to introduce this topic
early.However,instructorshavetheflexibilitytodelaythistopicuntilChapter8.
• The former Section 3.4 has been split into two sections, the first on continuous growth and e and the
second on compound interest. This gives instructors the flexibility to discuss the properties of e and
compoundinteresttogetherorseparately.
• Theintroductionofehasbeenrewrittentoincreaseunderstanding.
• LimitnotationhasbeenintroducedinChapters1and3forinstructorswhowishtoincludethis.
• ThematerialontrigonometricidentitiesinChapter7hasbeenexpanded,andnewproblemsadded.
• AnewsectiononthegeometricpropertiesofconicshasbeenaddedtoChapter12.
• Algebra review is integrated throughout the text. More skill building has been emphasized wherever
appropriate.Thealgebraictoolsusedinachapteraresummarizedforeasyreferenceina“Tools”section.
• Dataandproblemshavebeenupdatedandrevisedasappropriate.Manynewproblemshavebeenadded.
• ConcepTests for precalculus are now available for instructors looking for innovative ways to promote
activelearningintheclassroom.FurtherinformationisprovidedundertheSupplementaryMaterialson
pageviii.
WhatStudentBackgroundisExpected?
Studentsusingthisbookshouldhavesuccessfullycompletedacourseinintermediatealgebraorhighschool
algebraII.Thebookisthought-provokingforwell-preparedstudentswhilestillaccessibletostudentswith
weakerbackgrounds.Providingnumericalandgraphicalapproachesaswellasthealgebraicgivesstudents
anotherwayofmasteringthematerial.Thisapproachencouragesstudentstopersist,therebyloweringfailure
rates.
OurExperiences
Previous editions of this book were used by hundreds of schools around the country. In this diverse group
of schools, the first two editions were successfully used with many different types of students in semester
andquartersystems,inlargelecturesandsmallclasses,aswellasinfullyearcoursesinsecondaryschools.
They were used in computer labs, small groups, and traditional settings, and with a number of different
technologies.
Content
The central theme of this course is functions as models of change. We emphasize that functions can be
grouped into families and that functions can be used as models for real-world behavior. Because linear,
exponential,power,andperiodicfunctionsaremorefrequentlyusedtomodelphysicalphenomena,theyare
introducedbeforepolynomialandrationalfunctions.Onceintroduced,afamilyoffunctionsiscomparedand
contrastedwithotherfamiliesoffunctions.
Alargenumberoftheexamplesandproblemsthatstudentsseeinthisprecalculuscoursearegiveninthe
contextofreal-worldproblems.Indeed,wehopethatstudentswillbeabletocreatemathematicalmodelsthat
willhelpthemunderstandtheworldinwhichtheylive.Theinclusionofnon-routineproblemsisintended
toestablishtheideathatsuchproblemsarenotonlypartofmathematics,butinsomesense,arethepointof
mathematics.
Thebookdoesnotrequireanyspecificsoftwareortechnology.Instructorshaveusedthematerialwith
graphingcalculatorsandgraphingsoftware.Anytechnologywiththeabilitytographfunctionswillsuffice.
Preface vii
Chapter1:LinearFunctionsandChange
Thischapterintroducestheconceptofafunctionandthegraphical,tabular,symbolic,andverbalrepresenta-
tionsoffunctions.Theadvantagesanddisadvantagesofeachrepresentationarediscussed.Ratesofchange
are introduced and used to characterize linear functions. A section on fitting a linear function to data is
included.
TheToolssectionforChapter1reviewslinearequationsandthecoordinateplane.
Chapter2:Functions
FunctionnotationisintroducedinChapter1andstudiedinmoredetailinthischapter.Domainandrange,
and the concepts of composite and inverse functions are introduced. The idea of concavity is investigated
usingratesofchange.Thechapterincludessectionsonpiecewiseandquadraticfunctions.
TheToolssectionforChapter2reviewsquadraticequations.
Chapter3:ExponentialFunctions
Thischapterintroducesthefamilyofexponentialfunctionsandthenumbere.Exponentialandlinearfunc-
tionsarecompared,andexponentialequationsaresolvedgraphically.
TheToolssectionforChapter3reviewsthepropertiesofexponents.
Chapter4:LogarithmicFunctions
Logarithmic functions to base 10 and base e are introduced to solve exponential equations and to serve
as inverses of exponential functions. After manipulations with logarithms, Chapter 4 focuses on modeling
withexponentialfunctionsandlogarithms.Logarithmicscalesandasectiononlinearizingdataconcludethe
chapter.
TheToolssectionforChapter4reviewsthepropertiesoflogarithms.
Chapter5:TransformationsofFunctionsandTheirGraphs
Thischapterinvestigatestransformations—shifting,reflecting,andstretching.Theseideasareappliedtothe
familyofquadraticfunctions.
TheToolssectionforChapter5reviewscompletingthesquare.
Chapter6:TrigonometricFunctions
This chapter, which focuses on modeling periodic phenomena, introduces the trigonometric functions: the
sine,cosine,tangent,andbriefly,thesecant,cosecant,andcotangent.Theinversetrigonometricfunctionsare
introduced.
TheToolssectionforChapter6reviewsrighttriangletrigonometry.
Chapter7:Trigonometry
Thischapterfocusesontriangles,identities,andpolarcoordinates.Itincludesasectiononmodelingusing
trigonometricfunctions.Complexnumbersandpolarcoordinatesareintroduced.
Chapter8:Compositions,Inverses,andCombinationsofFunctions
Thischaptercoverscombinationsoffunctions.Compositeandinversefunctions,whichwereintroducedin
Chapter2,areinvestigatedinmoredetailinthischapter.
viii Preface
Chapter9:PolynomialandRationalFunctions
Thischaptercoverspowerfunctions,polynomials,andrationalfunctions.Thechapterconcludesbycompar-
ingseveralfamiliesoffunctions,includingpolynomialandexponentialfunctions,andbyfittingfunctionsto
data.
TheToolssectionforChapter9reviewsalgebraicfractions.
Chapter10:Vectors
This chapter contains material on vectors and operations involving vectors. An introduction to matrices is
includedinthelastsection.
Chapter11:SequencesandSeries
Thischapterintroducesarithmeticandgeometricsequencesandseriesandtheirapplications.
Chapter12:ParametricEquationsandConicSections
Theconcludingchapterlooksatparametricequations,implicitfunctions,thehyperbolicfunctions,andthe
conicsections:circles,ellipses,andhyperbolas.Thechapterincludesasectiononthegeometricalproperties
oftheconicsectionsandtheirapplicationstoorbits.
SupplementaryMaterials
ThefollowingsupplementarymaterialsareavailablefortheThirdEdition:
• Instructor’s Manual contains teaching tips, lesson plans, syllabi, and worksheets. The Instructor’s
Manual has been expanded and revised to include worksheets, identification of technology-oriented
problems,andnewsyllabi.(ISBN0470-10819-3)
• PrintedTestBank containstestquestionsarrangedbysection.(ISBN0470-10821-5)
• Instructor’sSolutionManualwithcompletesolutionstoallproblems.(ISBN0470-10820-7)
• Student Solution Manual with complete solutions to half the odd-numbered problems. (ISBN 0470-
10561-5)
• Student Study Guide includes study tips, learning objectives, practice problems, and solutions. The
topicsaretieddirectlytothebook.(ISBN0470-5-3)
• Getting Started Graphing Calculator Manual instructs students on how to utilize their TI-83 and
TI-84 series calculators with the text. Contains samples, tips, and trouble shooting sections to answer
students’questions.(ISBN0470-10558-5)
• ComputerizedTestBank:AvailableinbothPCandMacintoshformats,theComputerizedTestBank
allowsinstructorstocreate,customize,andprintatestcontaininganycombinationsofquestionsfroma
largebankofquestions.Instructorscanalsocustomizethequestionsorcreatetheirown.(ISBN0470-
10822-3)
• Book Companion SiteThe accompanying website contains all instructor supplements as well as web
quizzesforstudentpractice.
• Wiley PLUS This powerful and highly integrated suite of online teaching and learning resources pro-
vides course management options to instructors an students. Instructors can automate the process of
assigning, delivering, and grading algorithmically generated homework exercises, hints and solutions
while providing students with immediate feedback. Additionally student tutorials, an instructor grade-
book, integrated links to the electronic version of the text and all of the text supplemental materials
areprovided.Formoreinformation,visitwww.wiley.com/college/wileyplusorcontactyourlocalWiley
representativeformoredetails.