Introductory analysis: a deeper view of calculus

219 Pages·2001·1.32 MB·english

by  Richard J. Bagby

INTRODUCTORY
ANALYSIS
A Deeper View of Calculus
Richard J. Bagby
DepartmentofMathematicalSciences
NewMexicoStateUniversity
LasCruces,NewMexico
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CONTENTS
ACKNOWLEDGMENTS ix
PREFACE xi
I
THE REAL NUMBER SYSTEM
1. FamiliarNumberSystems 1
2. Intervals 6
3. SupremaandInfima 11
R
4. ExactArithmeticin 17
5. TopicsforFurtherStudy 22
II
CONTINUOUS FUNCTIONS
1. FunctionsinMathematics 23
2. ContinuityofNumericalFunctions 28
v
vi CONTENTS
3. TheIntermediateValueTheorem 33
4. MoreWaystoFormContinuousFunctions 36
5. ExtremeValues 40
III
LIMITS
1. SequencesandLimits 46
2. LimitsandRemovingDiscontinuities 49
∞
3. LimitsInvolving 53
IV
THE DERIVATIVE
1. Differentiability 57
2. CombiningDifferentiableFunctions 62
3. MeanValues 66
4. SecondDerivativesandApproximations 72
5. HigherDerivatives 75
6. InverseFunctions 79
7. ImplicitFunctionsandImplicitDifferentiation 84
V
THE RIEMANN INTEGRAL
1. AreasandRiemannSums 93
2. SimplifyingtheConditionsforIntegrability 98
3. RecognizingIntegrability 102
4. FunctionsDefinedbyIntegrals 107
5. TheFundamentalTheoremofCalculus 112
6. TopicsforFurtherStudy 115
VI
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
1. ExponentsandLogarithms 116
2. AlgebraicLawsasDefinitions 119
CONTENTS vii
3. TheNaturalLogarithm 124
4. TheNaturalExponentialFunction 127
5. AnImportantLimit 129
VII
CURVES AND ARC LENGTH
1. TheConceptofArcLength 132
2. ArcLengthandIntegration 139
3. ArcLengthasaParameter 143
4. TheArctangentandArcsineFunctions 147
5. TheFundamentalTrigonometricLimit 150
VIII
SEQUENCES AND SERIES OF FUNCTIONS
1. FunctionsDefinedbyLimits 153
2. ContinuityandUniformConvergence 160
3. IntegralsandDerivatives 164
4. Taylor’sTheorem 168
5. PowerSeries 172
6. TopicsforFurtherStudy 177
IX
ADDITIONAL COMPUTATIONAL METHODS
1. L’Hoˆpital’sRule 179
2. Newton’sMethod 184
3. Simpson’sRule 187
4. TheSubstitutionRuleforIntegrals 191
REFERENCES 197
INDEX 198
ACKNOWLEDGMENTS
I
would like to thank many persons for the support and assistance that
I have received while writing this book. Without the support of my
departmentImightneverhavebegun,andthefeedbackIhavereceived
from my students and from reviewers has been invaluable. I would espe-
cially like to thank Professors William Beckner of University of Texas at
Austin,JungH.TsaiofSUNYatGeneseoandCharlesWatersofMankato
State University for their useful comments. Most of all I would like to
thank my wife, Susan; she has provided both encouragement and impor-
tanttechnicalassistance.
ix
PREFACE
I
ntroductoryrealanalysiscanbeanexcitingcourse;itisthegatewayto
animpressivepanoramaofhighermathematics. Butforalltoomany
students,theexcitementtakestheformofanxietyoreventerror;they
are overwhelmed. For many, their study of mathematics ends one course
soonerthantheyexpected,andformanyothers,thedoorwaysthatshould
have been opened now seem rigidly barred. It shouldn’t have to be that
way,andthisbookisofferedasaremedy.
GOALS FOR INTRODUCTORY ANALYSIS
The goals of first courses in real analysis are often too ambitious. Stu-
dents are expected to solidify their understanding of calculus, adopt an
abstractpointofviewthatgeneralizesmostoftheconcepts,recognizehow
explicit examples fit into the general theory and determine whether they
satisfy appropriate hypotheses, and not only learn definitions, theorems,
andproofsbutalsolearnhowtoconstructvalidproofsandrelevantexam-
plestodemonstratetheneed forthehypotheses. Abstractpropertiessuch
as countability, compactness and connectedness must be mastered. The
xi
xii PREFACE
studentswhoareuptosuchachallengeemergereadytotakeontheworld
ofmathematics.
A large number of students in these courses have much more modest
immediateneeds. Manyareonlyinterestedinlearningenoughmathemat-
ics to be a good high-school teacher instead of to prepare for high-level
mathematics. Others seek an increased level of mathematical maturity,
but something less than a quantum leap is desired. What they need is a
new understanding of calculus as a mathematical theory — how to study
it in terms of assumptions and consequences, and then check whether the
neededassumptionsareactuallysatisfiedinspecificcases. Withoutsuchan
understanding,calculusandrealanalysisseemalmostunrelatedinspiteof
thevocabularytheyshare,andthisiswhysomanygoodcalculusstudents
are overwhelmed by the demands of higher mathematics. Calculus stu-
dentscometoexpectregularitybutanalysisstudentsmustlearntoexpect
irregularity; real analysis sometimes shows that incomprehensible levels
of pathology are not only possible but theoretically ubiquitous. In calcu-
lus courses, students spend most of their energy using finite procedures
to find solutions, while analysis addresses questions of existence when
there may not even be a finite algorithm for recognizing a solution, let
aloneforproducingone. Theobstacletostudyingmathematicsatthenext
levelisn’tjusttheinherentdifficultyoflearningdefinitions,theorems,and
proofs;itisoftenthelackofanadequatemodelforinterpretingtheabstract
conceptsinvolved. Thisiswhymoststudentsneedadifferentunderstand-
ing of calculus before taking on the abstract ideas of real analysis. For
some students, such as prospective high-school teachers, the next step in
mathematicalmaturitymaynotevenbenecessary.
Thebookiswrittenwiththefutureteacherofcalculusinmind,butitis
alsodesignedtoserveasabridgebetweenatraditionalcalculussequence
andlatercoursesinrealornumericalanalysis. Itprovidesaviewofcalculus
thatisnowmissingfromcalculusbooks,andisn’tlikelytoappearanytime
soon. Itdealswithderivationsandjustificationsinsteadofcalculationsand
illustrations, with examples showing the need for hypotheses as well as
casesinwhichtheyaresatisfied. Definitionsofbasicconceptsareempha-
sized heavily, so that the classical theorems of calculus emerge as logical
consequencesofthedefinitions,andnotjustasreasonableassertionsbased
on observations. The goal is to make this knowledge accessible without
dilutingit. Theapproachistoprovideclearandcompleteexplanationsof
thefundamentalconcepts,avoidingtopicsthatdon’tcontributetoreaching
ourobjectives.
PREFACE xiii
APPROACH
Tokeepthetreatmentsbriefyetcomprehensible, familiarargumentshave
been re-examined, and a surprisingly large number of the traditional con-
ceptsofanalysishaveprovedtobelessthanessential. Forexample,open
andclosedintervalsareneededbutopenandclosedsetsarenot,sequences
are needed but subsequences are not, and limits are needed but methods
for finding limits are not. Another key to simplifying the development is
to start from an appropriate level. Not surprisingly, completeness of the
realnumbersisintroducedasanaxiominsteadofatheorem,buttheaxiom
takes the form of the nested interval principle instead of the existence of
supremaorlimits. Thisapproachbringsthepowerofsequencesandtheir
limitsintoplaywithouttheneedforafineunderstandingofthedifference
between convergence and divergence. Suprema and infima become more
understandable, because the proof of their existence explains what their
definition really means. By emphasizing the definition of continuity in-
stead of limits of sequences, we obtain remarkably simple derivations of
the fundamental properties of functions that are continuous on a closed
interval:
existenceofintermediatevalues
existenceofextremevalues
uniformcontinuity.
Moreover,thesefundamentalresultscomeearlyenoughthatthereisplenty
of time to develop their consequences, such as the mean value theorem,
the inverse function theorem, and the Riemann integrability of continu-
ous functions, and then make use of these ideas to study the elementary
transcendentalfunctions. Atthisstagewecanbeginmainstreamrealanal-
ysis topics: continuity, derivatives, and integrals of functions defined by
sequencesandseries.
Thecoverageofthetopicsstudiedisdesignedtoexplaintheconcepts,
not just to prove the theorems efficiently. As definitions are given they
are explained, and when they seem unduly complicated the need for the
complexityisexplained. Insteadofthedefinition-theorem-proofformat
often used in sophisticated mathematical expositions, we try to see how
the definitions evolve to make further developments possible. The rigor
is present, but the formality is avoided as much as possible. In general,
proofsaregivenintheirentiretyratherinoutlineform;thereaderisn’tleft
withasequenceofexercisestocompletethem.
xiv PREFACE
Exercises at the end of each section are designed to provide greater
familiarity with the topics treated. Some clarify the arguments used in
the text by having the reader develop parallel ones. Others ask the reader
to determine how simple examples fit into the general theory, or give
examples that highlight the relevance of various conditions. Still others
address peripheral topics that the reader might find interesting, but that
werenotnecessaryforthedevelopmentofthebasictheory. Generallythe
exercisesarenotrepetitive;theintentisnottoprovidepracticeforworking
exercises of any particular type, and so there are few worked examples to
follow. Computational skill is usually important in calculus courses but
thatisnottheissuehere; theskillstobelearnedaremoreinthenatureof
makingappropriateassumptionsandworkingouttheirconsequences,and
determiningwhethervariousconditionsaresatisfied. Suchskillsaremuch
harder to develop, but well worth the effort. They make it possible to do
mathematics.
ORGANIZATION AND COVERAGE
Thefirstsevenchapterstreatthefundamentalconceptsofcalculusinarig-
orousmanner; they forma solid core fora one-semestercourse. The first
chapter introduces the concepts we need for working in the real number
system, and the second develops the remarkable properties of continuous
functionsthatmakearigorousdevelopmentofcalculuspossible. Chapter
3 is a deliberately brief introduction to limits, so that the fundamentals of
differentiation and integration can be reached as quickly as possible. It
shows little more than how continuity allows us to work with quantities
given as limits. The fourth chapter studies differentiability; it includes a
developmentoftheimplicitfunctiontheorem,aresultthatisnotoftenpre-
sentedatthislevel. Chapter5developsthetheoryoftheRiemannintegral,
establishingtheequivalenceofRiemann’sdefinitionwithmoreconvenient
onesandtreatingthefundamentaltheoremevenwhentheintegrandfailsto
beaderivative. Thesixthchapterstudieslogarithmsandexponentsfroman
axiomaticpoint of viewthat leads naturally to formulasforthem, and the
seventhstudiesarclengthgeometricallybeforeexaminingtheconnections
betweenarclengthandcalculus.
Buildingonthisfoundation,Chapter8getsintomainstreamrealanal-
ysis,withadeepertreatmentoflimitssothatwecanworkwithsequences
andseriesoffunctionsandinvestigatequestionsofcontinuity,differentia-
bility, and integrability. It includes the construction of a function that is
continuouseverywherebutnowheredifferentiableormonotonic,showing
that calculus deals with functions much more complicated than we can