Calculus with applications

509 Pages·2014·3.86 MB·English

by  Peter D. Lax

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
©SpringerScience+BusinessMediaNewYork1976, 2014
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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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