Calculus with Applications

509 Pages·2013·6.782 MB·English

by  Lax, Peter D;Terrell, Maria Shea

Undergraduate Texts in Mathematics
Peter D. Lax
Maria Shea Terrell
Calculus With
Applications
Second Edition
Undergraduate Texts in Mathematics
Undergraduate Texts in Mathematics
SeriesEditors:
SheldonAxler
SanFranciscoStateUniversity,SanFrancisco,CA,USA
KennethRibet
UniversityofCalifornia,Berkeley,CA,USA
AdvisoryBoard:
ColinAdams,WilliamsCollege,Williamstown,MA,USA
AlejandroAdem,UniversityofBritishColumbia,Vancouver,BC,Canada
RuthCharney,BrandeisUniversity,Waltham,MA,USA
IreneM.Gamba,TheUniversityofTexasatAustin,Austin,TX,USA
RogerE.Howe,YaleUniversity,NewHaven,CT,USA
DavidJerison,MassachusettsInstituteofTechnology,Cambridge,MA,USA
JeffreyC.Lagarias,UniversityofMichigan,AnnArbor,MI,USA
JillPipher,BrownUniversity,Providence,RI,USA
FadilSantosa,UniversityofMinnesota,Minneapolis,MN,USA
AmieWilkinson,UniversityofChicago,Chicago,IL,USA
Undergraduate Texts in Mathematics are generally aimed at third- and fourth-
year undergraduate mathematics students at North American universities. These
texts strive to provide students and teachers with new perspectives and novel
approaches.Thebooksincludemotivationthatguidesthereadertoanappreciation
ofinterrelationsamongdifferentaspectsofthesubject.Theyfeatureexamplesthat
illustratekeyconceptsaswellasexercisesthatstrengthenunderstanding.
Forfurthervolumes:
http://www.springer.com/series/666
Peter D. Lax Maria Shea Terrell
(cid:129)
Calculus With Applications
Second Edition
123
PeterD.Lax MariaSheaTerrell
CourantInstituteofMathematicalSciences DepartmentofMathematics
NewYorkUniversity CornellUniversity
NewYork,NY,USA Ithaca,NY,USA
ISSN0172-6056
ISBN978-1-4614-7945-1 ISBN978-1-4614-7946-8(eBook)
DOI10.1007/978-1-4614-7946-8
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2013946572
MathematicsSubjectClassification:00-01
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Preface
Ourpurposein writinga calculustexthasbeentohelpstudentslearnatfirsthand
thatmathematicsis thelanguagein whichscientific ideascan bepreciselyformu-
lated,thatscienceis a sourceofmathematicalideasthatprofoundlyshapethe de-
velopment of mathematics, and that mathematics can furnish brilliant answers to
importantscientificproblems.ThisbookisathoroughrevisionofthetextCalculus
withApplicationsandComputingbyLax,Burstein,andLax.Theoriginaltextwas
predicatedonanumberofinnovativeideas,anditincludedsomenewandnontradi-
tionalmaterial.Thisrevisioniswritteninthesamespirit.Itisfairtoaskwhatnew
subjectmatterornewideascouldpossiblybeintroducedintosooldatopicascalcu-
lus.Theansweristhatscienceandmathematicsaregrowingbyleapsandboundson
theresearchfrontier,sowhatweteachinhighschool,college,andgraduateschool
must not be allowed to fall too far behind. As mathematiciansand educators, our
goalmustbetosimplifytheteachingofoldtopicstomakeroomfornewones.
To achieve that goal, we present the language of mathematics as natural and
comprehensible,alanguagestudentscanlearntouse.Throughoutthetextweoffer
proofsofalltheimportanttheoremstohelpstudentsunderstandtheirmeaning;our
aimistofosterunderstanding,not“rigor.”Wehavegreatlyincreasedthenumberof
workedexamplesandhomeworkproblems.Wehavemadesomesignificantchanges
in the organizationofthe material;the familiar transcendentalfunctionsare intro-
duced before the derivative and the integral. The word “computing” was dropped
from the title because today, in contrast to 1976, it is generally agreed that com-
putingisanintegralpartofcalculusandthatitposesinterestingchallenges.These
are illustrated in this text in Sects.4.4, 5.3, and 10.4, and by all of Chap.8. But
themathematicsthatenablesustodiscussissuesthatariseincomputingwhenwe
roundoffinputsorapproximateafunctionbyasequenceoffunctions,i.e.,uniform
continuityanduniformconvergence,remains.Wehaveworkedhardinthisrevision
toshowthatuniformconvergenceandcontinuityaremorenaturalandusefulthan
pointwiseconvergenceandcontinuity.Theinitialfeedbackfromstudentswhohave
usedthetextisthatthey“getit.”
Thistextisintendedforatwo-semestercourseinthecalculusofasinglevariable.
Onlyknowledgeofhigh-schoolprecalculusisexpected.
v
vi Preface
Chapter 1 discusses numbers,approximatingnumbers,and limits of sequences
ofnumbers.Chapter2 presentsthebasic factsaboutcontinuousfunctionsandde-
scribestheclassicalfunctions:polynomials,trigonometricfunctions,exponentials,
and logarithms.It introduceslimits of sequencesof functions,in particularpower
series.
In Chapter3, the derivative is defined and the basic rules of differentiationare
presented.Thederivativesofpolynomials,theexponentialfunction,thelogarithm,
andtrigonometricfunctionsarecalculated.Chapter4describesthebasictheoryof
differentiation,higherderivatives,TaylorpolynomialsandTaylor’stheorem,andap-
proximatingderivativesbydifferencequotients.Chapter5describeshowthederiva-
tiveentersthelawsofscience,mainlyphysics,andhowcalculusisusedtodeduce
consequencesoftheselaws.
Chapter 6 introduces,throughexamplesof distance,mass, and area,the notion
oftheintegral,andtheapproximateintegralsleadingtoitsdefinition.Therelation
betweendifferentiationandintegrationisprovedandillustrated.InChapter7,inte-
grationbypartsandchangeofvariableinintegralsarepresented,andtheintegralof
theuniformlimitofasequenceoffunctionsisshowntobethelimitoftheintegrals
of the sequence of functions. Chapter 8 is about the approximation of integrals;
Simpson’s rule is derived and compared with other numerical approximations of
integrals.
Chapter 9 shows how many of the concepts of calculus can be extended to
complex-valuedfunctions of a real variable. It also introduces the exponential of
complexnumbers.Chapter10appliescalculustothedifferentialequationsgovern-
ingvibratingstrings,changingpopulations,andchemicalreactions.Italsoincludes
averybriefintroductiontoEuler’smethod.Chapter11isaboutthetheoryofprob-
ability,formulatedinthelanguageofcalculus.
ThematerialinthisbookhasbeenusedsuccessfullyatCornellinaone-semester
calculus II course for students interested in majoring in mathematics or science.
The students typically have credit for one semester of calculus from high school.
Chapters 1, 2, and 4 have been used to present sequences and series of numbers,
power series, Taylor polynomials, and Taylor’s theorem. Chapters 6–8 have been
usedtopresentthedefiniteintegral,applicationofintegrationtovolumesandaccu-
mulationproblems,methodsof integration,and approximationof integrals. There
hasbeenadequatetimeleftinthetermthentopresentChapter9,oncomplexnum-
bersandfunctions,andtoseehowcomplexfunctionsandcalculusareusedtomodel
vibrationsinthefirstsectionofChapter10.
Wearegratefultothemanycolleaguesandstudentsinthemathematicalcommu-
nitywhohavesupportedoureffortstowritethisbook.Thefirsteditionofthisbook
was written in collaborationwith SamuelBurstein. We thank him for allowing us
todrawonhiswork.WewishtothankJohnGuckenheimerforhisencouragement
andadviceonthisproject.We thankMattGuay,JohnMeluso,andWyattDeviau,
who while they were undergraduatesat Cornell, carefully read early drafts of the
manuscript,and whose perceptivecommentshelped us keep our studentaudience
inmind.WealsowishtothankPatriciaMcGrath,ateacheratMaloneyHighSchool
in Meriden, Connecticut, for her thoughtful review and suggestions, and Thomas
Preface vii
KernandChenxiWu,graduatestudentsatCornellwhoassistedinteachingcalcu-
lus II with earlier drafts of the text, for their help in writing solutions to some of
thehomeworkproblems.ManythanksgotothestudentsatCornellwhousedearly
draftsofthisbookinfall2011and2012.Thankyouallforinspiringustoworkon
thisproject,andtomakeitbetter.
This current edition would have been impossible without the support of Bob
Terrell,Maria’shusbandandlong-timemathematicsteacheratCornell.FromTEX-
ingthemanuscripttomakingthefigures,tosuggestingchangesandimprovements,
ateverystepalongthewayweoweBobmorethanwecansay.
PeterLaxthankshiscolleaguesattheCourantInstitute,withwhomhehasdis-
cussedover50yearsthechallengeofteachingcalculus.
NewYork,NY PeterLax
Ithaca,NY MariaTerrell

Contents
1 NumbersandLimits............................................ 1
1.1 Inequalities................................................ 1
1.1a RulesforInequalities................................. 3
1.1b TheTriangleInequality ............................... 4
1.1c TheArithmetic–GeometricMeanInequality.............. 5
1.2 NumbersandtheLeastUpperBoundTheorem.................. 11
1.2a NumbersasInfiniteDecimals.......................... 11
1.2b TheLeastUpperBoundTheorem ...................... 13
1.2c Rounding........................................... 16
1.3 SequencesandTheirLimi√ts.................................. 19
1.3a Approximationof 2 ................................ 23
1.3b SequencesandSeries................................. 24
1.3c NestedIntervals ..................................... 36
1.3d CauchySequences ................................... 37
1.4 TheNumbere ............................................. 44
2 FunctionsandContinuity ....................................... 51
2.1 TheNotionofaFunction .................................... 51
2.1a BoundedFunctions................................... 54
2.1b ArithmeticofFunctions............................... 55
2.2 Continuity ................................................ 59
2.2a ContinuityataPointUsingLimits...................... 61
2.2b ContinuityonanInterval.............................. 64
2.2c ExtremeandIntermediateValueTheorems............... 66
2.3 CompositionandInversesofFunctions ........................ 71
2.3a Composition ........................................ 71
2.3b InverseFunctions .................................... 74
2.4 SineandCosine............................................ 81
2.5 ExponentialFunction ....................................... 86
2.5a RadioactiveDecay ................................... 86
2.5b BacterialGrowth .................................... 87
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