testdaily-15門(mén)ap?？伎颇縪pen微積分

)) VerticalandHorizontalAlinex=kisaverticalasymptoteofthecurvey=f(x),if:limf(x)=±ooorlimf(x)= Alineisahorizontalasymptoteofthecurvey=f(x),if:limf(x)=Aorlimf(x)= X?-Supposethatf(x)iscontinuouson[a,b],andNisanumberbetweenf(a)andf(b),thenthereisacE[a,b]suchthatf(c)=N.Iff(x)1)continuouson[a,2)differentiableon(a,ThenatleastexistsonenumberE(ab)satisfythefollowingf(b)-f(互

b-導(dǎo)數(shù)計(jì)算基本初等函數(shù)導(dǎo)數(shù)表(詳見(jiàn)【方法詳解】

[f(x)±g(x)]=f(x)±g(x)=[f(x)g(x)]=f(x)g(x)+f(x)g l

f(x)g(x)-f(x)g ,g(x)-=I復(fù)合函數(shù)(compositefunction)，y=f[g(x)]fgchain令u=g(x)，則y=f(u),dy=dy·du=外層求導(dǎo)·內(nèi)層求 du例:x2y2r2，r 2x+2ydx=0

dx=-切線與割 TangentandSecantTangentline1)(2)求切點(diǎn)SecantLine(1)兩點(diǎn)直接求方程Position,Velocity,andPosition: v(t)=x(t)=dX

Acceleration:a(t)=v(t)=dvOra(t)=x(t)=d fDisplacement:ft2v Distance:t2|v|f

a(t)>Speed vaRelatedKey:L(x)=f(a)+f(a)×(x-IncreasingorLocal(Relative)ConcaveUpandPointof(具體步驟見(jiàn)【方法詳解】TD制作Riemann LeftRiemannRightRiemannMidpointRiemannInterpretingofDefinite 牛萊公f(x)dx=F(x)+積分運(yùn)算與積分表（見(jiàn)【方法積分方CalculusCalculusLongDivisionandCompletingtheIntegrationByudv=uv vPartial(x-a)(x-b)(x-(TD反常積SeparationofRewriteinthedifferentialIntegratebothsidesofthe

dy=M(x)fdy=fExponentialOTheExponentialgrowthanddecaymodel,dy=ky,withinitialconditiony=O

whent=hassolutionsoftheformy=LogisticTheLogisticdifferentialequation,dP=kP(1- KPk增長(zhǎng)速度最快。(TDAveragebf(x)dx=f(c)(b-a求面定截面體bv a旋轉(zhuǎn)體弧 L 1+ )2 Larn-1=a+ar+ar2+ar3+If|r|<1,theseriesconverges;If|r|＿1,theseriesdiverges

=lim

1-1-rCalculusCalculusHarmonicSeriesandp-交錯(cuò)RatioTestforTaylorIfafunctionf(x)hasderivativesofallordersatx=a,thentheTaylorseiresforf(x)x=af(x)=

n!(x-=f(a)+f(a)(x-a)+fll(a)(x-a)2+…+fn(a)(x-a)n+ Word(TD求極當(dāng)f(x)在xc處連續(xù)時(shí)，limf(xx趨向的數(shù)帶入式子即可。x=c0xx=c無(wú)定義，不連續(xù)。第一步：看是否可以因式分解，上下同除消掉為0部分。第二步：看是否可以利用(x+y)(x–y)=x2-y2消除掉為0部分。Supposearationalf(x)=Pn(X)(Q(x)-=IlimPn(X)=

a。XnlX

n-

Xl

-=I0,

-=I0)

a(m=bX?ooQm(X)X?oob。XmlXl+…+bm-lXl ±oo(n>0(n<

= 1 lim(1+

=e?lim(1+x)X=Limit

limOandlimoo Iff(c)=g(c)=o(f(c)=g(c)=oo),f(x)andg'(x)exist,andg'(c)-=I0, =X?c X?c limf(x)與limf(x)與 x?x圖removableCalculus圖jump圖discontinuitiesduetoverticalLocal（Relative）Iff'(a)equalszeroorDNE(Doesnotexist),and??′(??)changesitssignatx=a,wesayhasalocalextremevalueatx=CalculusGlobal（Absolute） SecondConcaveUpandPointof1.積分

PreciseDefinition:WesaylimfxL

LimitatInfinity:Wesay

fxLifforeveryc0thereisao0suchthat canmakefxasclosetoLaswewantbywhenever0xaothenfxLc. takingxlargeenoughandpositive.“Working”Definition:Wesaylimfx

Thereisasimilardefinitionfor

fxifwecanmakefxasclosetoLaswe exceptwerequirexlargeandbytakingxsufficientlyclosetoa(oneithersideofa)withoutlettingxa.

imit:Wesaylim

Righthandlimit:limfxL.This canmakefxarbitrarilylarge(andthesamedefinitionasthelimitexceptitrequiresxa.

bytakingxsufficientlyclosetoa(oneithersideofa)withoutlettingxa.ThereisasimilardefinitionforfxLefthandlimit:

fxL.Thishas

exceptwemakefxarbitrarilylargesamedefinitionasthelimitexceptitxa

Relationshipbetweenthelimfx

x

x

x

fxLlimfx

fx

fxlimfxDoesNotxAssumelimfxandlimg

bothexistandcisanynumber providedlimgxlimcfxclimf providedlimgx limfxgxlimfx

xagxlimfx

limgxlimfx

limfxgxlimfxlimg

nlimf nnlimf xa BasicLimitEvaluationsatNote:sgna1ifa0andsgna1ifa0limex limex 5.neven:limxnx x limlnx limlnx 6.nodd:limxn&limxnx

x xIfr0thenlim 7.neven:limaxn·bxcsgn

xr0andxrisrealfornegative 8.nodd

limaxn·bxcsgnathenlimb

9.nodd:limaxn·cxdsgnax Continuous

EvaluationL’Hospital’sIffxiscontinuousatathenlimfxfa Iflimfx0orlimfxxag xag ContinuousFunctionsand limfxlimfxaisanumber,orfxiscontinuousatbandlimgxb xag xaglimfgxflimgxf Polynomialsat Factorand

pxandqxarepolynomials.Top limx24x12limx2x x xx

士qx

factorlargestpowerofxinqx x ofbothpxandqxthen 2 2

x2

34 3 2 Rationalize 2 x5x x

x5 3 3 3 3

lim3

Piecewise

x25ifx

x2x2813 x

limgx

gxx93 xx93 xComputetwoonesided

gx

x25CombineRational

gxlim13xlim1

Onesidedlimitsaredifferentsolimgx h0hx x h0 xxh doesn’texist.Ifthetwoonesidedlimits1 beenequalthenlimgxwouldhave lim h0hxxhh0xx

andhadthesameSomeContinuousPartiallistofcontinuousfunctionsandthevaluesofxforwhichtheyarePolynomialsforallRationalfunction,exceptforx’sthat

cosxandsinxforalltanxandsecxdivisionbynn(nodd)forall x·nn

冗冗3 (neven)forallx0 2 exforall cotxandcscxlnxforx0 x·,2冗,冗0,冗,2冗Suppose

IntermediateValuefxiscontinuouson[a,b]andletMbeanynumberbetweenfaandfbThenthereexistsanumbercsuchthatacbandfcM Ifyfxthenthederivativeisdefinedtobefxlim xh xh Ifyfxthenallofthefollowing Ifyfxallofthefollowingareequivalentnotationsforthe notationsforderivativeevaluatedatxafxydfdydfxDfx

fa

dx

dx

DfaInterpretationoftheIfyfx faistheinstantaneousratemfaistheslopeofthe changeoffxatxalinetoyfxatxaandthe 3.Iffxisthepositionofanobjectatequationofthetangentlineatxais timexthenfaisthevelocityofgivenbyyfafaxa. theobjectatxa.Iffx

BasicPropertiesandgxaredifferentiablefunctions(thederivativeexists),candnareanyreal cfcf

c fgfxg 6.dxnnxn1–Power fgf

fgfgfgfg

–Product dfgxfgxgg g

–Quotient ThisistheChaindxdsinxcos

Commondcscxcscxcotxdcotxcsc2

daxaxlnadex1dcosxsin dsin1x dlnx1,x1dtanxsec2x

dcos1x 11

dlnx1

xdsecxsecxtan

d

x

x

x

1

ChainRuleThechainruleappliedtosomespecific dfxnnfxn1f 5.dcosfx」fxsinfx efx

fxef 6.dtanf fxsec2fxd

fxf 7.dsecf(x)f(x)secf(x)tanf(f d2dsinfx f f tan1fx f 2 1fxHigherOrderTheSecondDerivativeisdenoted ThenthDerivativeisdenotedfxf2xd2

andisdefinedas

andisdefinedfxfx,i.e.thederivativeofthederivative,fx.

fnxfn1x,i.e.thederivativeofthe(n-1)stderivative,fn1x.ImplicitFindyife2x9yx3y2siny11x.Rememberyyxhere,soproducts/quotientsofxandywillusetheproduct/quotientruleandderivativesofywillusethechainrule.The“trick”istodifferentiateasnormalandeverytimeyoudifferentiateayyoutackonay(fromthechainrule).Afterdifferentiatingsolvefory.e2x9y29y3x2y22x3yycosyy2e2x9y 2x9 2

y

11

2x9

3x29y 3xy2xyycosyy2x3y9e2x9ycosyy112e2x9y3x2

Critical

Increasing/Decreasing–ConcaveUp/Concavexcisacriticalpointoffxprovided1.fc0or2.fcdoesn’tIffx0forallxinanintervalIfxisincreasingontheintervalIffx0forallxinanintervalIfxisdecreasingontheintervalIffx0forallxinanintervalIfxisconstantontheinterval

ConcaveUp/ConcaveIffx0forallxinanintervalIfxisconcaveupontheintervalIffx0forallxinanintervalIfxisconcavedownontheintervalInflectionxcisainflectionpointoffxiftheconchangesatxc.Absolutexcisan umoff

Relative(local)xcisarelative(or umiffcfxforallxin fxiffcfxforallxnearxcisanabsoluteminimumoff xcisarelative(orlocal)minimumfxiffcfxforallxneariffcfxforallxin Fermat’sIffxhasarelative(orlocal)extremaxc,thenxcisacriticalpointoffxExtremeValueIffxiscontinuousonthecloseda,bthenthereexistnumberscanddso

1stDerivativeIfxcisacriticalpointof thenxcarel.max.offxiffx0totheleftofxcandfx0totherightofxc.arel.min.offxiffx0totheleftofxcandfx0totherightofxc.notarelativeextremaoffxiffxac,db, fcistheabs.max. thesamesignonbothsidesofxca,b,3.fdistheabs.min.ina,b 2ndDerivativeIfxcisacriticalpointoffxsuchFindingAbsoluteTofindtheabsoluteextremaofthefunctionfxontheintervala,busethefollowingprocess.Findallcriticalpointsoffxina,b

fc0thenxisa umoffxiffc0isarelativeminimumoffxiffc0maybearelativeum,EvaluatefxatallpointsfoundinStep minimum,orneitheriffc0EvaluatefaandfbIdentifytheabs.max.(largestfunctionvalue)andtheabs.min.(smallestfunctionvalue)fromtheevaluationsinSteps2&3.

FindingRelativeExtremaand/orClassifyCriticalPointsFindallcriticalpointsoffxUsethe1stderivativetestorthe2ndderivativetestoneachcriticalpoint.MeanValueIffxiscontinuousontheclosedintervala,banddifferentiableontheopenintervala,thenthereisanumberacbsuchthatfcfbfabNewton’sIfxisthenthguessfortheroot/solutionoffx0then(n+1)stguessis xfxnnprovidedfxn

fxnRelatedEx.Twopeopleare50ftapartwhenonestartswalkingnorth.Theangle8changesat0.01rad/min.Atwhatrateisthebetweenthemchangingwhen 0.5WeEx.Twopeopleare50ftapartwhenonestartswalkingnorth.Theangle8changesat0.01rad/min.Atwhatrateisthebetweenthemchangingwhen 0.5Wehave8 0.01rad/min.andwanttox.Wecanusevarioustrigfcnsbuteasiest sec8tan88x We 0.05soplugin8andsec0.5tan Remembertohavecalculatorinradians!tiesandsolvefortheEx.Ex.A15footladderisrestingagainstawall.Thebottomisinitially10ftawayandispushedtowardsthewallat1ft/sec.How4isthetopmovingafter12xisnegativebecausexisdecreasing.UsingPythagoreanTheoremanddifferentiating,x2y2 2xx2yy After12secwehave 10147soy 15272 176.Pluginandsolvefory.7 14176 074Sketchpictureifneeded,writedownequationtobeoptimizedandconstraint.Solveconstraintforoneofthetwovariablesandplugintoequation.Findcriticalpointsofequationinrangeofvariablesandverifythattheyaremin/maxasneeded.Ex.We’reenclosingarectangularfieldwith500ftoffencematerialandonesideofthefieldEx.We’reenclosingarectangularfieldwith500ftoffencematerialandonesideofthefieldisabuilding.Determinedimensionsthat izetheenclosedarea.ize xysubjecttoconstraintx2y intoarea. 5002y y5002y500y2Differentiateandfindcritical 5004y By2ndderiv.testthisisarel.max.andsoistheanswerwe’reafter.Finally,findx. 500 Thedimensionsarethen250xEx.Determinepoint(s)on x21thatclosesttoMinimizedx02y22andeconstraintis x21.Solveconstraintx2andplugintothey1 x2yy1y y23yDifferentiateandfindcriticalf 2y y2Bythe2ndderivativetestthisisarel.min.andsoallweneedto sfindxvalue(s).3212 The2pointsare and2 22 2DefiniteIntegral:Supposefxis Anti-Derivative:Ananti-derivativeoffxona,b.Dividea,bintonsubintervals isafunction,Fx,suchthatFxfxwidthxandchoosex*fromeach

fxdxFx

fxdxlimfx

x whereFxisananti-derivativeoffxFundamentalTheoremofPartI:Iffxiscontinuousona,b VariantsofPartI

xftdtisalsocontinuousona,

du

uxft x andgx

dxxftdt x d ftdtvxfvx dxv PartII:fxiscontinuousona,b,Fx u ft xfu(ananti-derivativeoffx(i.e.Fx fxdxbthenafxdxFbFa

dxv fxgxdx afxgxdxafxdxagxa

cfxdxcfxdx,cisa acfxdxcafxdx,cisa afxdx afxdxaft afxdxbfx

afxdx

fxIffxgxonaxbthenafxdxbgxbIffx0onaxbthenafxdxbIfmfxMonaxbthenmba fxdxMbkdxkx

Commonsinuducosusinuducosu

xndx1xn1c,nxndx1xn1c,n

du tanaa a2 aa1ax

dx1lnaxb

secutanudusecu 21

du

ulnuduulnuueudueu

cscucotuducscucsc2uducotuStandardIntegrationNotethatatmanyschoolsallbuttheSubstitutionRuletendtobetaughtinaCalculusII guSubstitution:Thesubstitutionugxwillconvertbfgxgxdxgbfu g 5xcos 22 31215xcos dx 5xcos 22 31215xcos dx 2 381cosuu du3xdxxdx32213x1u131::x2u23sin sin8sin585313IntegrationbyParts:udvuvv andbudvuvbbvdu.Chooseuanddv integralandcomputedubydifferentiatinguandcomputevusingvdvEx.Ex.xexu dv du vxexdxxexexdxxexexEx.3lnx5uln dv du1dxvx53lnxdxxln dxxlnx553 535ln53ln3Productsand(some)QuotientsofTrigForsinnxcosmxdxwehavethefollowing Fortannxsecmxdxwehavethefollowingnodd.Strip1sineoutandconvertresttocosinesusingsin2x1cos2x,thenusethesubstitutionucosx.modd.Strip1cosineoutandconvertresttosinesusingcos2x1sin2x,thenusethesubstitutionusinx.nandmbothodd.Useeither1.ornandmbotheven.Usedoubleangleand/orhalfangleformulastoreducetheintegralintoaformthatcanbe

nodd.Strip1tangentand1secantoutandconverttheresttosecantsusingtan2xsec2x1,thenusetheusecxmeven.Strip2secantsoutandconvertresttotangentsusingsec2x1tan2x,thenusethesubstitutionutanx.noddandmeven.Useeither1.ornevenandmodd.Eachintegralwillbedealtwithdifferently.TrigFormulas:sin2x2sinxcosx,cos2x121cos2x,sin2x121Ex.Ex.tan3xsec5xtan3xsec5xdxtan2xsec4xtanxsecsec2x1sec4xtanxsecu2 usec1sec7x1sec5x75sincos35sincos35dxsinxsincos34dx(sin2sincos3(1cosx)sincos32ucos2du12uu41sec2x2lncosx1cos2x22TrigSubstitutions:Iftheintegralcontainsthefollowingrootusethegivensubstitutionandformulatoconvertintoanintegralinvolvingtrigfunctions.a2b2b2x2a2b2 xasin8 xasec8 xatan8cos28a2b2b2x2a2b2 x249

94sin282cos89

2cos8d8

12 sin 3 x2 dx2cos83 49x2 44sin28 4cos282 x2x.Becausewehaveanintegralwe’llassumepositiveanddropabsolutevaluebars.Ifwehadadefiniteintegralwe’dneedtocompute8’sandremoveabsolutevaluebarsbasedonthatand,

12csc2d812cot8UseRightTriangleTrigtogobacktox’s.2substitutionwehavesin83x2x

ifx

Fromthisweseethatcot849x2. ifxInthiscasewe 49x22cos8

x249

dx

449x

2 PartialFractions:IfintegratingPxdxwherethedegreeofPxissmallerthanthe QQx.Factordenominatorascompleyaspossibleandfindthepartialfraction positionoftherationalexpression.Integratethepartialfraction position(P.F.D.).Foreachfactorinthedenominatorwegetterm(s)inthe positionaccordingtothefollowingtable.FactorFactorinQxaxTermin FactorinQ axTerminA1 axaxaxax2bxaxAxB2kax2bxkA1xB1ax2bxcAkxax2bxcSetcoefficientsequaltogetasystemandsolvetogetconstants.AB CB 4ACA B C27x213xABx2CBx4A(x1)(xlandcollectlikexABxCA(x24)(BxC)( )2(x1)(xSetnumer7x2Hereispartialfractionform22x231lnx484lnx22x x x2dx(x1)(x2(x1)(x24)27x7x2Ex.ternatemethodthatsometimesworkstofindconstants.Startwithsettingnumeratorsequalinpreviousexample:7x213xAx24BxCx1.Chosenicevaluesofxandplugin.Forexampleifx1weget205AwhichgivesA4.Thiswon’talwayswork ApplicationsofNetArea:afxdxrepresentsthenetareabetweenfxandx-axiswithareaabovex-axispositiveandareabelowx-axisAreaBetweenCurves:Teralformulasforthetwomaincasesforeach yfxAaupperfunctionlowerfunctiondx&xfyAcrightfunctionleftfunctiondyIfthecurvesintersectthentheareaofeachportionmustbefoundindividually.Herearesomesketchesofacouplepossiblesituationsandformulasforacoupleofpossiblecases. adAbfxgx Aad

cfygy

Aafxgxdxcgxfx VolumesVolumesofRevolution:ThetwomainformulasareVAxdxandVAydy.Hereissomegeneralinformationaboutea ethodofcomputingandsomeexamples.A冗outerradius2innerradius2Limits:x/yofright/botringtox/yofleft/topringHorz.Axisusefx, Vert.Axisusefy,gx,Axand gy,AyandA2冗radiuswidth/Limits:x/yofinnercyl.tox/yofoutercyl.Horz.Axisusefy Vert.Axisusefxgy,Ayandgx,AxandEx.Axis:ya Ex.Axis:ya Ex.Axis:ya Ex.Axis:yaouterouterradius:afinnerradius:agouterradius:aginnerradius:afradius:awidth:fygyradius:awidth:fygyTheseareonlyafewcasesforhorizontalaxisofrotation.Ifaxisofrotationisthex-axisuseya0casewitha0.Forverticalaxisofrotation(xa0andxa0)interchangexytogetappropriateWork:IfaforceofFxmovesan AverageFunctionValue:Theaverage ainaxb,theworkdoneisWbFx offxonaxbisfavgbaa

fxa ArcLengthSurfaceArea:NotethatthisisoftenaCalcIItopic.Thethreebasic 1222Lads SAa2冗yds(rotateaboutx-axis) SAa1222ds dxifyfx,ax ds dtifxft,ygt,at12r 2ds dyifxfy,ay ds d8ifr12r 2Withsurfaceareayoumayhavetosubstituteinforthexorydependingonyourchoiceofdstomatchthedifferentialintheds.Withparametricandpolaryouwillalwaysneedtosubstitute.ImproperAnimproperintegralisanintegralwithoneormoreinfiniimitsand/ordiscontinuousintegrands.Integraliscalledconvergentifthelimitexistsandhasafinitevalueanddivergentifthelimitdoesn’texistorhasinfinitevalue.ThisistypicallyaCalcIItopic.Infini afxdxlimafx fxdxlimtfx fxdxfxdxcf rovidedBOTHintegralsare 1.Discont.ata:

fxdx

fx 2.Discont.atb:

fxdxlim

fx ta

3.Discontinuityatacb:afxdxafxdxcf rovidedbothare Ifafxdxconv.thenagxdxconv. 2.Ifagxdxdivg.thenafxdxdivg.Usefulfact:Ifa0then 1dxconvergesifp1anddivergesfor