英文版 微积分试卷答案 (1)

[第1页共5页]1、(1)sin2limxxx0.(2)d(arctan)x21d1+xx(3)21dsinxx-cot+Cxx(4).2()()xne22nxe.(5)4012dxx26/32、(6)Therightpropositioninthefollowingpropositionsis___A_____.A.Iflim()xafxexistsandlim()xagxdoesnotexistthenlim(()())xafxgxdoesnotexist.B.Iflim()xafx,lim()xagxdobothnotexistthenlim(()())xafxgxdoesnotexist.C.Iflim()xafxexistsandlim()xagxdoesnotexistthenlim()()xafxgxdoesnotexist.D.Iflim()xafxexistsandlim()xagxdoesnotexistthen()lim()xafxgxdoesnotexist.(7)Therightpropositioninthefollowingpropositionsis__B______.A.Iflim()()xafxfathen()faexists.B.Iflim()()xafxfathen()fadoesnotexist.C.If()fadoesnotexistthenlim()()xafxfa.D.If()fadoesnotexistthenthecure()yfxdoesnothavetangentat(,())afa.(8)Therightstatementinthefollowingstatementsis___D_____.A.sinlim1xxxB.1lim(1)xxxeC.11d1xxxCD.5511dd11bbaaxyxy(9)Forcontinuousfunction()fx,theerroneousexpressioninthefollowingexpressionsis____D__.A.d(()d)()dbafxxfbbB.d(()d)()dbafxxfaaC.d(()d)0dbafxxxD.d(()d)()()dbafxxfbfax(10)Therightpropositioninthefollowingpropositionsis__B______.A.If()fxisdiscontinuouson[,]abthen()fxisunboundedon[,]ab.[第2页共5页]B.If()fxisunboundedon[,]abthen()fxisdiscontinuouson[,]ab.C.If()fxisboundedon[,]abthen()fxiscontinuouson[,]ab.D.If()fxhasabsoluteextremevalueson[,]abthen()fxiscontinuouson[,]ab.3、Evaluate2011lim()xxexx201=lim()xxexx01=lim()2xxex01=lim=22xxe（考点课本4.4节洛比达法则，每年都会有一道求极限的解答题，大多数都是用洛比达法则去求解，所以大家要注意4.4节的内容。注意洛比达法则的适用范围。）4．Find0d|xyand(0)yif20()dxxxyytte.20'(()d)'xxxyytte（）1'2()'2()1xxyxyxeyxyxe00(20(0)1)0xdyyedxdx''(2()1)'2()2'()xxyxyxeyxxyxe200()d-(0)0-01xxyyttexye0''02(0)20'(0)=3yyye（）（考察微积分基本定理与微分，书上5.3节）5、Find22arctand(1)xxxx=22221)arctand(1)xxxxxx（22arctanarctan=dd(1)xxxxxx-12311=-arctan+darctan+2xxxxxx22-1221++1=-arctan+darctan1+2xxxxxxxx（）-12211=-arctan+ddarctan1+2xxxxxxxx（）-12211=-arctan+InIn1+arctan22xxxxx-1221=-arctan+Inarctan+C21+xxxxx（凑微分求不定积分，积分是微积分的重点及难点，大家一定要掌握透彻。）6、Giventhat22()1xfxx.[第3页共5页](1)Findtheintervalsonwhich()fxisincreasingordecreasing.22’22212()1xxxxfxx（）（）2221xx（）When’()00fxx’()00fxxTherefore,theincreasingintervalis0，,thedecreasingintervalis0，(2)Findthelocalmaximumandminimumvaluesof()fx’()00fxxThefunctionisincreasingininterval0，,decreasingininterval0，,therefore,thefunctionexistthelocalminimumvalue,itis()0fx(3)Findtheintervalsofconcavityandtheinflectionpoints.'22224222242422181642''()111xxxxxxfxxxx（）（）（）（）（）422464233''()0133xxfxxorxx（）422464233''()000133xxfxxorxx（）331''()''()=334ffTherefore,theconcaveupwardintervalare33，,33，,theconcavedownwardintervalare3-03，,303，andtheinflectionpointsare31-34，,3134，(4)Findtheasymptotelinesofthecure()yfx2221lim=1111+xxxxTherefore,theliney=1isahorizontalasymptote（考点：4.3节，4.5、4.6节。近几年经常会考一道作图题。这种题目应该在注意的点主要包括函数的定义域，对称性，增减区间，极值点，凹凸性，拐点，以及渐近线等。大家参照课本的4.5节进行作图）7、LetRbetheregionboundedbythecurve1yx,andthelineyxand2x.(a)EvaluatetheareaoftheregionR.R=211xdxx2211=In2xx2211=2In21In1223=In22(b)FindthevolumeofthesolidgeneratedbyrevolvingtheRaboutthey-axis.[第4页共5页]V=21212121 4)?4)ydydyy（（12311211443xyyy3311114224114142331283（考点：求面积以及体积，课本6.1、6.2节。这类题目是常考题，较简单。望同学一定要做相应的题目加以巩固。）8、Determinetheproductionlevelthatwillmaximizetheprofitforacompanywithcostanddemandfunctions23()1450360.580.001Cxxxxand()600.01pxx.Solution2()(600.01)600.01Rxxxxx223()()()600.01(1450360.580.001)PxRxCxxxxxx320.0010.57241450xxx'2()0.0031.142PxxxLet'()040020Pxxorxsincex0,thenx=400''()0.0061.14PxxWhenx=400''()0.0064001.141.260Px32(400)0.0014000.5740024400145035350PTherefore,whentheproductionlevelis400thatwillmaximizetheprofit35350（考点：经济函数，课本4.8节。此题型为常考题，属于送分题，大家可以做相应的4.8节的练习加以巩固）9、Statethesecondderivativetesttheoremtestingmaximumandproveit.Suppose''fiscontinuousnearcIf'cf（)=0and''cf（)0,thenhasalocalmaximumatc.Proof:Becausenear''cf（)0candsofisconcavedownwardnearc.Thismeansthatthegraphofliesbelowitshorizontaltangentatcandsohasalocalmaximumatc.10、Showthattheequation3150xxchasatmostonerootintheinterval[第5页共5页][2,2].Supposetheequation3150xxchastworoot1212,andxxxxintheinterval12,[2,2]xx.Let3F()15xxxcthenthereexist12F()F()=0xxUsingtheRolle’sTheoremwecanknowthatthereexistonenumberccansatisfy12F'(c)0(,)cxx2F'()3150[2,2]xxwhenxBut2F'()3150[2,2]xxwhenx,Therefore,thesupposeiswrong.Thentheequation3150xxchasatmostonerootintheinterval[2,2].（考点：罗尔地理。书上4.2节。中值定理的证明题是历年考试证明题的热点，大家一定要吃透该定理，做一定的题目加以巩固）