数学专业英语8.docx
- 文档编号:11324896
- 上传时间:2023-02-26
- 格式:DOCX
- 页数:11
- 大小:100.78KB
数学专业英语8.docx
《数学专业英语8.docx》由会员分享,可在线阅读,更多相关《数学专业英语8.docx(11页珍藏版)》请在冰豆网上搜索。
数学专业英语8
MathematicalEnglish
Dr.XiaominZhang
Email:
zhangxiaomin@
§2.8TheDerivativeofaFunctionandIts
GeometricInterpretation
TEXTAThederivativeofafunction
Theexampledescribedintheforegoingsectionpointsthewaytotheintroductionoftheconceptofderivative.Webeginwithafunctionfdefinedatleastonsomeopeninterval(a,b)onthex-axis.Thenwechooseafixedpointxinthisintervalandintroducethedifferencequotient
(8.1)
Wherethenumberh,whichmaybepositiveornegative(butnotzero),issuchthatx+halsoliein(a,b).Thenumeratorofthisquotientmeasuresthechangeinthefunctionwhenxchangesfromxtox+h.Thequotientitselfisreferredtoastheaveragerateofthechangeoffintheintervaljoiningxtox+h.
Nowwelethapproachzeroandseewhathappenstothisquotient.Ifthequotientapproachessomedefinitevalueasalimit(whichimpliesthatthelimitisthesamewhetherhapproacheszerothroughpositivevaluesorthroughnegativevalues),thenthislimitiscalledthederivativeoffatxandisdenotedbythesymbolf(x)(readas“fprimeofx”).Thus,theformaldefinitionoff(x)maybestatedasfollows:
DEFINITIONOFDERIVATIVEThederivativef(x)isdefinedbytheequation
(8.2)
providedthelimitexists.Thenumberf(x)isalsocalledtherateofchangeoffatx.
Bycomparing(8.2)with(7.3)intheforegoingsection,weseethattheconceptofinstantaneousvelocityismerelyanexampleoftheconceptofderivative.Thevelocityv(t)isequaltothederivativef(x),wherefisthefunctionwhichmeasureposition.Thisisoftendescribedbysayingthatvelocityistherateofchangeofpositionwithrespecttotime.IntheexampleworkedoutinSection7.2,thepositionfunctionfisdescribedbytheequation
F(t)=144t-16t2,
anditsderivativefisanewfunction(velocity)givenby
(8.3)f(t)=144-32t.
Ingeneral,thelimitprocesswhichproducesf(x)fromf(x)givesusawayofobtaininganewfunctionffromagivenfunctionf.Theprocessiscalleddifferentiation,andfiscalledthefirstderivativeoff.Iff,inturn,isdefinedonanopeninterval,wecantrytocomputeitsfirstderivative,denotedbyfcalledthesecondderivativeoff.similarly,thenthderivativeoff,denotedbyf(n),isdefinedtobethefirstderivativeoff(n-1).Wemaketheconventionthatf(0)=f,thatis,thezerothderivativeisthefunctionitself.
Forrectilinearmotion,thefirstderivativeofvelocity(secondderivativeofposition)iscalledacceleration.Forexample,tocomputetheaccelerationintheexampleofSection7.2,wecanuseEquation(7.2)toformthedifferencequotient
Sincethisquotienthastheconstantvalue-32foreachh0,itslimitash0isalso-32.Thus,theaccelerationinthisproblemisconstantandequalto-32.Thisresulttellsusthatthevelocityisdecreasingattherateof32feetpersecondeverysecond.In9secondsthetotaldecreaseinvelocityis932=288feetpersecond.Thisagreeswiththefactthatduringthe9secondsofmotionthevelocitychangesformv(0)=144tov(9)=-144
TEXTBGeometricinterpretationofthederivateasaslope
Theprocedureusedtodefinethederivativehasageometricinterpretationwhichleadsinanaturalwaytotheideaofatangentlinetoacurve.AportionofthegraphofafunctionfisshowninFigure2-8-1.TwoofitspointsPandQareshownwithrespectivecoordinates(x,f(x))and(x+h,f(x+h)).ConsidertherighttrianglewithhypotenusePQ;itsaltitude,f(x+h)-f(x),representsthedifferenceoftheordinatesofthetwopointsQandP.Therefore,thedifferencequotient
(8.4)
representsthetrigonometrictangentoftheanglethatPQmakeswiththehorizontal.TherealnumbertaniscalledtheslopeofthelinethroughPandQanditprovidesawayofmeasuringthe“steepness”ofthisline.Forexample,itfisalinearfunction,sayf(x)=mx+b,thedifferencequotient(8.4)hasthevaluem,somistheslopeoftheline.
SomeexamplesoflinesofvariousslopesareshowninFigure2-8-2,Forahorizontalline,=0andtheslope,tan,isalso0.Ifliesbetween0and/2,thelineisrisingaswemovefromlefttorightandtheslopeispositive.Ifliesbetween/2and,thelineisfallingaswemovefromlefttorightandtheslopeisnegative.Alineforwhich=/4hasslope1.Asincreasesform0to/2,tanincreaseswithoutbound,andthecorrespondinglinesofslopetanapproachaverticalposition.Sincetan/2isnotdefined,wesaythatverticallineshavenoslope.
Supposenowthatfhasaderivativeatx.Thismeansthatthedifferencequotientapproachesacertainlimitf(x)ashapproaches0.Whenthisisinterpretedgeometricallyittellsusthat,ashgetsnearerto0,thepointPremainsfixed,QmovesalongthecurvetowardP,andthelinethroughPQchangesitsdirectioninsuchawaythatitsslopeapproachesthenumberf(x)asalimit.ForthisreasonitseemsnaturaltodefinetheslopeofthecurveatPtobethenumberf(x).ThelinethroughPhavingthisslopeiscalledthetangentlineatP.
SUPPLEMENTCalculus,derivativeandintegral
Newton,Isaac(1642-1727)Leibniz,Gottfried(1646-1716)
CalculusIngeneral,"a"calculusisanabstracttheorydevelopedinapurelyformalway."The"calculus,moreproperlycalledanalysis(orrealanalysisor,inolderliterature,infinitesimalanalysis)isthebranchofmathematicsstudyingtherateofchangeofquantities(whichcanbeinterpretedasslopesofcurves)andthelength,area,andvolumeofobjects.Thecalculusissometimesdividedintodifferentialandintegralcalculus,concernedwithderivatives
andintegrals
respectively.
Whileideasrelatedtocalculushadbeenknownforsometime(Archimedes'methodofexhaustionwasaformofcalculus),itwasnotuntiltheindependentworkofNewtonandLeibnizthatthemoderneleganttoolsandideasofcalculusweredeveloped.Evenso,manyyearselapseduntilthesubjectwasputonamathematicallyrigorousfootingbymathematicianssuchasWeierstrass.
DerivativeThederivativeofafunctionrepresentsaninfinitesimalchangeinthefunctionwithrespecttowhateverparametersitmayhave.The"simple"derivativeofafunctionfwithrespecttoxisdenotedeitherf(x)or
.Whenderivativesaretakenwithrespecttotime,theyareoftendenotedusingNewton'soverdotnotationforfluxions,
Whenaderivativeistakenntimes,thenotationf(n)(x)or
isused,with
etc.,thecorrespondingfluxionnotation.Whenafunctionf(x,y,…)dependsonmorethanonevariable,apartialderivative
canbeusedtospecifythederivativewithrespecttooneormorevariables.
Thederivativeofafunctionf(x)withrespecttothevariablexisdefinedas
butmayalsobedefinedmoresymmetricallyas
Insertingtheusualmathematicaldisclaimer,inbothcases,thederivativeexistsonlyifthecorrespondinglimitsdo.
Ifthefirstderivativeexists,thesecondderivativemaybesimilarlydefinedas
againassumingthelimitexists,aresultthatcanbegeneralizedtohigher-orderderivatives.
Notethatinorderforthelimittoexist,both
and
mustexistandbeequal,sothefunctionmustbecontinuous.However,continuityisanecessarybutnotsufficientconditionfordifferentiability,suchasWeierstrass’function
.
Sincesomediscontinuousfunctionscanbeintegrated,inasensethereare"more"functionswhichcanbeintegratedthandifferentiated.InalettertoStieltjes,Hermitewrote,"Irecoilwithdismayandhorroratthislamentableplagueoffunctionswhichdonothavederivatives."
Performingnumericaldifferentiationisinmanywaysmoredifficultthannumericalintegration.Thisisbecausewhilenumericalintegrationrequiresonlygoodcontinuitypropertiesofthefunctionbeingintegrated,numericaldifferentiationrequiresmorecomplicatedpropertiessuchasLipschitzclasses.
Thereareanumberofimportantrulesforcomputingderivativesofcertaincombinationsoffunctions.Derivativesofsumsareequaltothesumofderivativessothat
Inaddition,ifcisaconstant,
Theproductrulefordifferentiationstates
wherefdenotesthederivativeoffwithrespecttox.
Thequotientruleforderivativesstatesthat
Otherveryimportantruleforcomputingderivativesisthechainrule,whichstatesthat
ormoregenerally,
where
denotesapartialderivative.
IntegralAnintegralisamathematicalobjectthatcanbeinterpretedasanareaorageneralizationofarea.Integrals,togetherwithderivatives,arethefundamentalobjectsofcalculus.Otherwordsforintegralincludeantiderivativeandprimitive.TheRiemannintegralisthesimplestintegraldefinitionandtheonlyoneusuallyencounteredinphysicsandelementarycalculus.Infact,accordingtoJeffreys,itappearsthatcaseswherethesemethods[i.e.,generalizationsoftheRiemannintegral]areapplicableandRiemann's[definitionoftheintegral]isnotaretoorareinphysicstorepaytheextradifficulty."
TheRiemannintegralofthefunctionf(x)overxfromatobiswritten
Everydefinitionofanintegralisbasedonaparticularmeasure.Forinstance,theRiemannintegralisbasedonJordanmeasure,theLebesgueintegralisbasedonLebesguemeasureandtheLebesgue-StieltjesIntegralisbasedonLebesgue-Stieltjesmeasure.Theprocessofcomputinganintegraliscalledintegration(amorearchaictermforintegrationisquadrature),andtheapproximatecomputationofanintegralistermednumericalintegration.
Therearetwoclassesof(Riemann)integrals:
definiteintegrals,whichhaveupperandlowerlimits,andindefiniteintegrals,suchas
whicharewrittenwithoutlimits.Thefirstfundamentaltheorem
- 配套讲稿:
如PPT文件的首页显示word图标,表示该PPT已包含配套word讲稿。双击word图标可打开word文档。
- 特殊限制:
部分文档作品中含有的国旗、国徽等图片,仅作为作品整体效果示例展示,禁止商用。设计者仅对作品中独创性部分享有著作权。
- 关 键 词:
- 数学 专业 英语