上海財(cái)經(jīng)大學(xué)英語高數(shù) 課件

Chapter2

DerivativesupdownreturnendChapter212.6

Implicitdifferentiation1)Explicitfunction:Thefunctionwhichcanbedescribedbyexpressingonevariableexplicitlyintermsofanothervariable(othervariables)aregenerallycalledexplicitfunction---forexample,y=xtanx,ory=[1+x2+x3]1/2,

oringeneraly=f(x).2)Implicitfunction:Thefunctionswhicharedefinedimplicitlybyarelationbetweenvariables--xandy--aregenerallycalledimplicitfunctions---suchasx2+y2=4,or7sin(xy)=x2+y3

or,ingeneralF(x,y)=0.Ify=f(x)satisfiesF(x,f(x))=0onanintervalI,wesayf(x)isafunctiondefinedonIimplicitlybyF(x,y)=0,orimplicitfunctiondefinedbyF(x,y)=0.updownreturnend2.6Implicitdifferentiation1)23)DerivativesofimplicitfunctionSupposey=f(x)isanimplicitfunctiondefinedbysin(xy)=x2+y3.Thensin[xf(x)]=x2+[f(x)]3.Fromtheequation,wecanfindthederivativeoff(x)eventhoughwehavenotgottentheexpressionoff(x).Fortunatelyitisnotnecessarytosolvetheequationforyintermsofxtofindthederivative.Wewillusethemethodcalledimplicitdifferentiationtofindthederivative.Differentiatingbothsidesoftheequation,weobtainthat[f(x)+xf'(x)]cos[xf(x)]=2x+3[f(x)]2f'

(x).Thenupdownreturnend3)Derivativesofimplicitfun3Example(a)Ifx3+y3=27,find

(b)

Findtheequationofthetangenttothecurve

x3+y3=28atpoint(1,3).updownreturnendExample(a)Ifx3+y3=27,fi4Example(a)If

x3+y3=6xy,findy'.

(b)

FindtheequationofthetangenttothefoliumofDescartes

x3+y3=6xyatpoint(3,3).updownreturnendExample(a)Ifx3+y3=6xy,f5Orthogonal:

Twocurvesarecalled

Orthogonal,

ifateachpointofintersectiontheirtangentlinesareperpendicular.Iftwofamiliesofcurvessatisfythateverycurveinonefamilyisorthogonaltoeverycurveintheanotherfamily,thenwesaythetwofamiliesofcurvesareorthogonaltrajectoresofeachother.Example

Theequationsxy=c(c0)representsafamilyofhyperbolas.AndtheTheequationsx2-y2=k(k0)representsanotherfamilyofhyperbolaswithasymptotesy=x.Thenthetwofamiliesofcurvesaretrajectoresofeachother.updownreturnendOrthogonal:Twocurvesarecal6Derivative

f'(x)ofdifferentiablefunctionf(x)isalsoafunction.Iff'(x)isdifferentiable,thenwehave[f'

(x)]'.Wewilldenoteitbyf'

'

(x),i.e.,f'

'(x)=[f'

(x)]'.Thenewfunctionf

'

'(x)iscalledthesecondderivativeoff(x).Ify=f(x),wealsocanuseothernotations:

Similarlyf'

'

'(x)=[f''(x)]'iscalledthethirdderivativeoff(x),and2.7

HigherderivativesupdownreturnendDerivativef'(x)ofdifferent7Andwecandefinef'

'

'

'(x)=[f'

'

'(x)]'.Fromnowoninsteadofusingf'

'

'

'(x)weusef(4)(x)torepresentf

'

'

'

'(x).Ingeneral,wedefine

f(n)(x)=[f(n-1)(x)]',whichiscalledthenthderivativeoff(x).Wealsoliketousethefollowingnotations,ify=f(x),Example

Ify=x4-3x2+6x+9,findy

',y

'',y

''',y(4).updownreturnendExample

Iff(x)=,findf(n)(x).

Example

Iff(x)=sinx,g(x)=cosx,findf(n)(x)andg(n)(x).

Example

Findy'',ifx4+y3=x-y.Andwecandefinef'''82.8

Relatedrates(omitted)2.8Relatedrates(omitted)92.9

Differentials,LinearandQuadraticApproximationsDefinition:

Letx=x-x0,f(x)=f(x)-f(x0).IfthereexistsaconstantA(x0)whichisindependentofxandxsuchthatf(x)=A(x0)x+B(x,x0)where

B(x,x0)satisfies.ThenAxiscalleddifferentialoff(x)atx0.GenerallyAxisdenotedbydf(x)|x=x0

=A(x0)x.Replacingx0byx,thedifferentialisdenotedbydf(x)anddf(x)=A(x)x.updownreturnend2.9Differentials,Linearand10Proof:Fromthedefinition,Corollary:Ifthedifferentialoff(x)isdf(x)=A(x)x,thenf(x)isdifferentiableandA(x)=f

'(x).Corollary:(a)Iff(x)=x,thendx=df(x)=x.

(b)Iff(x)isdifferentiable,thendifferentialoff(x)existsanddf(x)=f'(x)dx.updownreturnendProof:Fromthedefinition,Cor11Example(a)

Finddy,ify=x3+5x4.

(b)Findthevalueofdywhenx=2anddx=0.1.Solution:Geometricmeaningofdifferentialoff(x),df(x)=QS

f(x)=RSxoxyPtSRQdx=xdyy=f(x)Asx=dxisverysmall,

y=dy,i.e.,f(t)-f(x)

f'(x)t.updownreturnendExample(a)Finddy,ify=x12ExampleUsedifferentialstofindanapproximate(65)1/3.Fromdefinitionofthedifferential,wecaneasilygetIff(x)isdifferentiableatx=a,andxisveryclosedtoa,thenf(x)

f(a)+f'(a)(x-a).TheapproximationiscalledLinearapproximationortangentlineapproximationoff(x)ata.AndfunctionL(x)=f(a)+f'(a)(x-a)iscalledthelinearizationoff(x)ata.updownreturnendExampleUsedifferentialstof13Example

Findthelinearizationofthefunctionf(x)=(x+3)1/2andapproximationsthenumbers(3.98)1/2and(4.05)1/2.updownreturnendExampleFindthelinearizati14Quadraticapproximationtof(x)nearx=a:Supposef(x)isafunctionwhichthesecondderivativef

''(a)exists.P(x)=A+Bx+Cx2istheparabolawhichsatisfiesP(a)=f(a),P'(a)=f'(a),andP''(a)=f

''(a).Asxisveryclosedtoa,theP(x)iscalledQuadraticapproximationtof(x)neara.Corolary:SupposeP(x)=A+Bx+Cx2istheQuadraticapproximationtof(x)neara.Then

P(x)=f(a)+f'(a)(x-a)+f''(a)(x-a)2/

2.IfP(x)isthequadraticapproximationtof(x)nearx=a,thenasxisveryclosedtoa,P(x)f(x).Thatis

f(x)

f(a)+f'(a)(x-a)+f''(a)(x-a)1/2/2.updownreturnendQuadraticapproximationtof(x15Example

Findthequadraticapproximationtof(x)=cosxnear0.updownreturnendExampleFindthequadratic16Example

Find

thequadraticapproximationtof(x)=(x+3)1/2nearx=1.updownreturnendExampleFindthequadraticapp17Themethodistogiveawaytogetaapproximationtoarootofanequation.2.10

Newton’smethod(tobeomitted)Supposef(x)isdefinedon[a,b],f'(x)doesnotvalue0.Letx0[a,b],f(a)f(b)<0.Andx1=x0-,x2=x1-.Keepingrepeatingtheprocess(xn=xn-1-),weobtainasequenceofapproximationsx1,x2,...,xn,......If,thenristherootoftheequationf(x)=0.updownreturnendThemethodistogiveawayto18Example

Startingwithx1=2,findthethirdapproximationx3totherootoftheequationx3-2x-5=0.updownreturnendExampleStartingwithx1=2,f192.1

DerivativesWedefinedtheslopeofthetangenttoacurvewithequationy=f(x)atthepointx=atobeGenerallywegivethefollowingdefinition:updownreturnend2.1DerivativesWedefinedthe20Definition:Thederivative

ofafunctionfatanumbera,denotedbyf′(a),isifthislimitexists.Thenwehave:updownreturnendDefinition:Thederivativeof21Example

Findthederivativeofthefunctiony=x2-8x+9ata.Geometricinterpretation:Thederivativeofthefunctiony=f(x)ataistheslopeoftangentlinetoy=f(x)at(a,f(a)).Thelineisthrough(a,f(a)).Soiff′(a)exists,theequationofthetangentlinetothecurvey=f(x)at(a,f(a))isy-f(a)=f′(a)(x-a).updownreturnendExampleFindthederivativeof22Example

Findtheequationofthetangentlineofthefunctiony=x2-3x+5atx=1.Inthedefinitionifwereplaceabyx,thenweobtainanewfunctionf′(x)whichisdeducedfromf(x).updownreturnendExampleFindtheequationof23Example

Iff(x)=(x-1)1/2,findthederivativeoff.Statethedomainoff′(x).ExampleFindthederivativeoffif

1-x

f(x)=2+xOthernotations:Ify=f(x),thentheothernotationsarethatf′(x)=y′====Df(x)=Dxf(x).updownreturnendExampleIff(x)=(x-1)1/2,find24ThesymbolDandd/dxarecalleddifferentialoperators.Wealsousethenotations:DefinitionAfunctionfiscalleddifferentiableataiff′(a)exists.Itisdifferentiableonanopeninterval(a,b)[or(a,+)or(-,b)]ifitisdifferentiableateverynumberintheinterval.Example

Whereisthefunctionf(x)=|x|isdifferentiable?updownreturnendThesymbolDandd/dxarecall25Theorem:

Iff(x)isdifferentiableata,thenf(x)iscontinuousata.(Theconverseisfalse)(3)thepointsatwhichthecurvehasaverticaltangentline,suchas,f(x)=x1/3,atx=0.(1)thepointsatwhichgraphofthefunctionfhas“corners”,suchasf(x)=|x|atx=0;(2)thepointsatwhichthefunctionisnotcontinuous,suchas,thefunction,definedasf(x)=2xforx1,and3xforx<1,atx=1;ThereareseveralcasesafunctionfailstobedifferentiableupdownreturnendTheorem:Iff(x)isdifferenti262.2

Differentiation1).TheoremIffisaconstantfunction,f(x)=c,thenf′(x)=(c)′=0,i.e.,=0.updownreturnend2.2Differentiation1).Theorem272).ThepowerruleIff(x)=xn,wherenisapositiveinteger,thenf′(x)=nxn-1,xn=nxn-1.Example

Iff(x)=x100,findf′(x).updownreturnend2).ThepowerruleIff(x)=xn283)TheoremSupposecisaconstantandf′(x)andg′(x)exist.ThenExampleIff(x)=x50+x100,findf′(x).(c)(f(x)-g(x))′existsand(f(x)-g(x)′=f′(x)-g′(x).(b)(f(x)+g(x))′existsand(f(x)+g(x)′=f′(x)+g′(x);(a)(cf(x))′existsand(cf(x))′=cf′(x);updownreturnend3)TheoremSupposecisaconst294)ProductruleSupposef′(x)andg′(x)exist.Then

f(x)g(x)isdifferentiableand[f(x)g(x)]′=f′(x)g(x)+f(x)g′(x).Example

Iff(x)=(2x5)(3x10),findf′(x).updownreturnend4)ProductruleSupposef′(x)304)QuotientruleSupposef′(x)andg′(x)existandg(x)0,thenf(x)/g(x)isdifferentiableand[f(x)/g(x)]′=[f′(x)g(x)-f(x)g′(x)]/[g(x)]2.Example

Iff(x)=,findf′(x).x2+2x-5x3-6updownreturnend4)QuotientruleSupposef′(x312).Thepowerrule(generalversion)Iff(x)=xn,wherenisanyrealnumber,thenf′(x)=nxn-1,,i.e.,xn=nxn-1.Example

Iff(x)=x,findf′(x).Ifg(x)=x1/2,g′(x)=?updownreturnendExample

Differentiatethefunctionf(t)=(1-t)t1/3.Tableofdifferentiationformulas(inpaper119)2).Thepowerrule(generalve322.3

RateofchangeintheEconomicsSuppose

C(x)isthetotalcostthatacompanyCx=C(x2)-C(x1)

x2-x1=C(x1+x)-C(x1)

xaverageofchangeofthecostistheadditionalcostisC=C(x2)-C(x1),andthenumberofitemsproducedincreasedfromx1tox2,ThefunctionCiscalledacostfunction.Iftheincursinproducingxunitsofcertaincommodity.updownreturnend2.3RateofchangeintheEcon33Thelimitofthisquantityasx0,iscalledthemarginalcostbyeconomist.Marginalcost=Takingx=1andnlarge(sothatxissmallcomparedton),wehaveC'(n)C(n+1)-C(n).Thusthemarginalcostofproducingnisapproximatelyequaltothecostofproducingonemoreunit[the(n+1)stunit].updownreturnendThelimitofthisquantityas342.4

Derivativesoftrigonometricfunctions(1)Theorem

Proof: