Precalculus: A Prelude to Calculus [with WileyPlus Access Code]

578 Pages·2016·13.38 MB·English

by  Sheldon Axler

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Precalculus
A Prelude to Calculus
Third edition
with
Student Solutions Manual
Sheldon Axler
San Francisco State University
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LibraryofCongressCataloging-in-PublicationData
Names: Axler,SheldonJay,author.
Title: Precalculus: apreludetocalculus/SheldonAxler.
Description: 3rdedition. |Hoboken,NJ:JohnWiley&Sons,2016. |Includesindex.
Identifiers: LCCN2016033848(print)|LCCN2016039164(ebook)|ISBN9781119330431(looseleaf)|ISBN
9781119334767(pdf)|ISBN9781119321514(epub)
Subjects: LCSH:Precalculus.
Classification: LCCQA331.A982016(print)|LCCQA331(ebook)|DDC512/.13–dc23
LCrecordavailableathttps://lccn.loc.gov/2016033848
ISBN-13: 978-1-119-32151-4
PrintedintheUnitedStatesofAmerica
10987654321
About the Author
SheldonAxler
Sheldon Axler was valedictorian of his high school in Miami, Florida. He
receivedhisABfromPrincetonUniversitywithhighesthonors,followedbyaPhD
inMathematicsfromtheUniversityofCaliforniaatBerkeley.
AsaMooreInstructoratMIT,Axlerreceivedauniversity-wideteachingaward.
Hewasthenanassistantprofessor,associateprofessor,andprofessorintheMath-
ematics Department at Michigan State University, where he received the first J.
SutherlandFrameTeachingAwardandtheDistinguishedFacultyAward.
AxlercametoSanFranciscoStateUniversityasChairoftheMathematicsDepart-
mentin1997. In2002,hebecameDeanoftheCollegeofScience&Engineeringat
SFState,apositionhehelduntilreturningfull-timetomathematicsin2015.
AxlerreceivedtheLesterR.FordAwardforexpositorywritingfromtheMathe-
maticalAssociationofAmericain1996. Inadditiontopublishingnumerousresearch
papers,Axleristheauthoroffivemathematicstextbooks,rangingfromfreshmanto
graduatelevel. HisbookLinearAlgebraDoneRighthasbeenadoptedasatextbook
atover300universities.
Axler has served as Editor-in-Chief of the Mathematical Intelligencer and as As-
sociate Editor of the American Mathematical Monthly. He has been a member of
theCounciloftheAmericanMathematicalSocietyandamemberoftheBoardof
Trustees of the Mathematical Sciences Research Institute. Axler currently serves
on the editorial board of Springer’s series Undergraduate Texts in Mathematics,
GraduateTextsinMathematics,andUniversitext.
TheAmericanMathematicalSocietyhonoredAxlerbyselectinghimasamember
ofitsinauguralgroupofFellowsin2013.
Thediagramonthecovercontainsthecrucialdefinitionsoftrigonometry. AbouttheCover
The1showsthatthetrigonometricfunctionsaredefinedinthecontextofthe
•
unitcircle.
Thearrowshowsthatanglesaremeasuredcounterclockwisefromthepositive
•
horizontalaxis.
The point labeled (cosθ,sinθ) shows that cosθ is the first coordinate of the
•
endpoint of the radius corresponding to the angle θ, and sinθ is the second
coordinate of this endpoint. Because this endpoint is on the unit circle, the
identitycos2θ+ sin2θ = 1followsimmediately.
The equation “slope = tanθ” shows that tanθ is the slope of the radius
•
sinθ
correspondingtotheangle θ;thustanθ = .
cosθ
v

Contents
About the Author v
Preface to the Instructor xv
Acknowledgments xxi
Preface to the Student xxiii
Chapter0 The Real Numbers 1
0.1 The Real Line 2
ConstructionoftheRealLine 2
IsEveryRealNumberRational? 3
Problems 5
0.2 Algebra of the Real Numbers 6
CommutativityandAssociativity 6
TheOrderofAlgebraicOperations 7
TheDistributiveProperty 8
AdditiveInversesandSubtraction 9
MultiplicativeInversesandtheAlgebraofFractions 10
SymbolicCalculators 13
Exercises,Problems,andWorked-outSolutions 15
0.3 Inequalities, Intervals, and Absolute Value 20
PositiveandNegativeNumbers 20
Inequalities 21
Intervals 23
AbsoluteValue 25
Exercises,Problems,andWorked-outSolutions 29
Chapter Summary and Chapter Review Questions 35
Chapter1 Functions and Their Graphs 37
1.1 Functions 38
DefinitionandExamples 38
TheDomainofaFunction 41
TheRangeofaFunction 42
input output
FunctionsviaTables 44
Exercises,Problems,andWorked-outSolutions 45
vii
viii Contents
1.2 The Coordinate Plane and Graphs 50
TheCoordinatePlane 50
TheGraphofaFunction 52
DeterminingtheDomainandRangefromaGraph 54
WhichSetsAreGraphsofFunctions? 56
Exercises,Problems,andWorked-outSolutions 56
1.3 Function Transformations and Graphs 63
VerticalTransformations: Shifting,Stretching,andFlipping 63
HorizontalTransformations: Shifting,Stretching,Flipping 66
CombinationsofVerticalFunctionTransformations 68
EvenFunctions 71
OddFunctions 72
Exercises,Problems,andWorked-outSolutions 73
1.4 Composition of Functions 81
CombiningTwoFunctions 81
DefinitionofComposition 82
DecomposingFunctions 85
ComposingMorethanTwoFunctions 85
FunctionTransformationsasCompositions 86
Exercises,Problems,andWorked-outSolutions 88
1.5 Inverse Functions 93
TheInverseProblem 93
One-to-oneFunctions 94
y TheDefinitionofanInverseFunction 95
TheDomainandRangeofanInverseFunction 97
TheCompositionofaFunctionandItsInverse 98
CommentsAboutNotation 99
Exercises,Problems,andWorked-outSolutions 101
1.6 A Graphical Approach to Inverse Functions 106
TheGraphofanInverseFunction 106
GraphicalInterpretationofOne-to-One 107
x IncreasingandDecreasingFunctions 108
InverseFunctionsviaTables 110
Thegraphsof x2(orange)on[0,2]
anditsinversefunction√x(blue) Exercises,Problems,andWorked-outSolutions 111
on[0,4]aresymmetricaboutthe
Chapter Summary and Chapter Review Questions 115
liney= x.
Chapter2 Linear,Quadratic,Polynomial,andRationalFunctions 119
2.1 Lines and Linear Functions 120
Slope 120
TheEquationofaLine 121
ParallelLines 125
PerpendicularLines 126
Exercises,Problems,andWorked-outSolutions 128