高等数学竞赛模拟题十四 Microsoft PowerPoint 演示文稿

.)12)...(1(lim.11nnnnnn计算)(十四高等数学竞赛模拟试卷:解;)12)...(1(:1nnnnnnx令nxnnnnnlnln)12ln(...)1ln(ln)];1ln(...)1ln()1[ln(1211nnnnn)]1ln(...)1ln()1[ln(limlim1211nnnnnnnnx10)1ln(dxx101010)1ln()1()1()1ln(dxxxxdx.12ln2.)()(,)(:.2和的函数与一线性函数之可表为一周期为则函数的连续函数是周期为若证明TdttfxFTxfxa:证明.),(])([)(为待定常数kaxkdtktfxFxadsksTfdtktfTxTatTsxa])([])([dsksfTxTa])([dsksfdsksfTxaaTa])([])([dtktfdtktfTaaTxa])([])([dtktfTxa])([令0])([dtktfTaa;)(01TTdttfk).()(])()([)(0101axdttfdtdttftfxFTTxaTT....,cos)cos(:.322121111满足其中证明kknknnkkn:证明);,(,0cos)(,cos)(:22xxxfxxf令:由泰勒公式211!21111111))(())(()()(nkknnkknnkknnkknxfxffxf)(),)(()(11111111nkknnkknnkknnkknxxff),)(()()(111111nkknknkknnkknkfffnknkknknkknnknkknnkkfff111111111))(()()(nknkknknkknnkknfnf1111111)()()(nknkknnkknkknnkknfnf11111111))(()();())(()(11111111nkknnkknkknkknnkknnffnf.)cos()cos()()(11111111nkknnkknnkknnkknff.)()(:.)(,]1,0[)(.421110nMnknknfdxxfMxfxf证明可导在区间设函数:证明nknknnknknknfdxxffdxxfnknk1111110)()()()(1nknknknkdxfxf11)]()([).,(,))((111nknkknknkkLagrangenknkdxxfnknknknkknknknnknknknkdxxMdxxffdxxf11111011)())(()()(nknknknkdxxM11)(.)0()(21121122121nMnknnknkMxMnknk.)12(])([lim,998)(lim,0)(lim,12)(.531212121212xdtduuftxfxfxxfxtxxx求极限且的邻域内为可导函数在设:解21212212123121212)12()(lim4)12(3)(lim)12(])([lim00xduufxduufxxdtduuftxxxxxtx)12(2)(lim41200xxfx1)(lim212)(lim2121200xfxxfxx.19969982..?,,)(.6最大的表面积并求此问部的表面积最大为了使前者夹在定球内的定球面上为常数的球的球心在半径为设半径为raar:解araxyzo222222yxazyxraz;244222arryx;,:244222222arryxyxrazdddxdyrAararrrrryxyxrr244222244222222020)(的表面积24422442220222202)(ararrrrqrrrdr.20),(222arrrar0)2(2)2(2)(23223rrrrrAaa.)(,227323434aaAar最大唯一驻点.lim).2(;)(],,[).1(:,.....2,1),(],,[,10,,)()(],[,)(.7000011xxxxfbaxnxfxbaxkkyxkyfxfbaxbxfannnn使存在唯一的证明设且为常数其中且若:证明];,[).1(bax);0(0)()(xxkxfxxf.],[)(上连续在baxf],[,)()(:baxxxfxF令0)()(,0)()(bbfbFaafaF由零点存在定理;)(],,[000xxfbax使:0是唯一的用反证法证明x.,)(,)(212211xxxxfxxf若.)()(21212121矛盾xxxxkxfxfxx....)()().2(02201010xxkxxkxfxfxxnnnn.lim)(00011xxnxxknnn.1lim:)10(),()()(,0,0)(,1,...,3,2,0)(,),()(.8100000)(0)(00nhnknhxfhxfhxfhxfnkxfnxxxf试证时当且且阶连续导数有在设函数:证明Taylorhxfhxfhxf)()()(000)()(...)()()()([0)1(2)!2(102!2100xfhxfhxfhxfhnnn)])(()()(10)(1)!1(1nnnnhoxfh)])(()()()([10)(1)!1(10nnnnhoxfhxfh))(()()()(10)(1)!1(10nnnnhhoxfhhxfh))(()()()()(10000)(1)!1(1nnnnnhhoxfhxfhxfxfh)()!1())(()()()(10)(1000xfnhhhoxfhxfhxfnnnn)()!1())(()()()(10)(1000xfnhhhoxfhxfhxfnnnn)()!1()((0)()!1()()()(0100)(110)(000limlimlimxfnhhohxfnhxfhxfhxfhnhnnnnnnnhxfhxfhxfhxfn)()()(0)()!1(0000)(lim1000)(00)()(0)()!1(limnnnhxfhxfhxfn200)(00)1()(0)()!1(limnnhnnhxfhxfn!)(0)()!1(0)(0)(0000lim......nhxfhxfnnnnnxfxfnnn1!)()()!1(0)(0)(.lim110nnh