同济大学课程考核试卷（A卷）20102011学年第一学期命题教师签名：梁进梁进审核教师签名：课号：122008122008课名：高等数学高等数学DD（英语）（英语）考试考查：考试考试此卷选为：期中考试()、期终考试()、重考()试卷年级专业学号姓名任课教师题号一二三四五总分得分（注意：本试卷共（注意：本试卷共55大题，大题，33大张，满分大张，满分100100分考试时间为分考试时间为120120分钟分钟。要。要求写出解题过程，否则不予计分求写出解题过程，否则不予计分）1.Choosearightansweroffourtothefollowingquestions(10marks)1)Forthefollowingconceptsofafunction,_D_isnotrelativetoalimitationA.continuityB.derivativeC.integrationD.variable2)If,abareinthedomainofadecreasingfunction()fx,andab,then_B_A.()()fafbB.()()fafbC.()()fafb=D.()()fafb3)If()fxisaboundedfunctiondefinedona,b,then()fxmustbe_C_A.continuousB.differentiableC.integrableD.increasing4)lim()xafx+exists,then_D_A.lim()()xafxfa=,B.lim()()xafxfa+=C.lim()lim()xaxafxfa+=D.noneisA.B.C.5)If(),()fxgxaredifferentiableina,b,where()()0fxgx,then_C_A.()()()()()()0fbfagbgaB.()()0fxgxC.()()0fxdxgxdxD.()()0bbaafxdxgxdx_andtheregionofthisfunctionis_33(,log2)(log2,)+_.2)Thediscontinuouspointofsin1xxeis_x=0_.3)Theinversefunctionof215yx=+is_2(5)12x+_.4)Suppose()fxisdifferentiable,thenthevalueof()fxatxa=is_()fa_,andtheslopeofthetangentlineof()fxatthispointis_()fa_.5)If()()fxfx=,then(0)f=_0_,andforanyconstanta,thedefiniteintegration()aafxdx=_0_.6)If()xfxe=,then(ln)fxdxx=_|x|+C_.7)If(),(),(),()Fxfxgxhxarecontinuousin(,).()()()gxfxhxwithlim()lim()xaxagxhxL=,()Fxisdecreasing,thenlim()xaFfx=_F(L)_.3.Calculations(30marks)1)2coslim|xxx=02)83sin100lim0.17ln1xxxxex+=03)()sin0lim1ln(1)xxx+=14)()tanxdexdx=21(tan)cosxexx+5)2011xdxdxx=+=16)Supposelnxxyey=+,thendydx=121xeyxxy7)()ln(cos)tanfxxx=+,then222312sincoscosdfxdxxx=+8)()32231445cos()5sin23ttttttteeeedteeeC+=+9)021240(1)1945xxxdx+=10)If(0)0,(1)2,ff=,then()21()2202()4(1)fxfxedxe=4.GraphAnalysisAnalysisfunction432()23fxxxx=+:1)(3marks)Writeoutallrootsof()fxifthereexist;3,0,1xxx=2)(3marks)Writeoutallrelativeextremepointsof()fxifthereexist;333333,0,44xxx+=3)(3marks)Writeoutallinflectionpointsof()fxifthereexist;1313,44xx+=4)(3marks)Writeouttheincreaseanddecreaseintervalsof()fx;Decreasingintervals:333333(,),(0,)44+Increasingintervals:333333(,0),(,)44+5)(3marks)Writeouttheconcaveupandconcavedownintervalsof()fxConcaveup1313(,),(,)44+,concavedown1313(,)44+6)(3marks)Writeouttheinfinitebehaviorsof()fxlim()xfx=+7)(5markes)Sketchthefigureofthefunction5.Applications1)(8marks)Calculatetheareaoftheregionwhichisenclosedbyfunctionscosyx=and2|1yx=.AREA2022cos12(1)4xxdx=+=+2)(11marks)A3-mladderisleaningagainstawall.Ifthetopoftheladderslipsdownthewallataconstantrateof0.3m/s,a)Whatistheratefunctionofthefootmovingawayfromthewall,wheretheindependentvariableofthefunctionisthedistancebetweenthebottomandthewall?b)Whatistheexactfigureoftheratewhenthetopis1mabovetheground.Thedistancetothewallisy,thetopoftheladdertothebottomofthewallish，then223yhm+=,if0.3/dhmsdt=,wehave:a)220.399dyhdhhdtdthh=b)120.30.3|0.106919hdyhdth=3)(8marks)Aclosedrectangularcontainerwithasquarebaseistohaveavolumeof1000cm3.Itcoststwiceasmuchpersquarecentimeterforthetopasitdoesforthesidesandbottom.Findthedimensionsofthecontainerofleastcost.Squareside=x,height=y,thenboxvolume=231000 xycm=Cost:Theboxbottom=2x,sides=4xy,top=22xTotalcostC=22221000400034343xxyxxxxx+=+=+Since332200010006,0,90003CxCxyx=hasleastcost.