中南大学高等工程数学试题

为进一步强化公司安全生产、文明施工管理，切实落实安全生产责任，防止重大安全事故发生，保障企业和员工的生命和财产安全manufacturersdrawingsboltfastenedinthecylinderandoutercylinder.Cylindercylinderinthedrainpipeinstallationandmeasuringcomponentleads.6.4.4.3bladecarrier,partitionsandlowerinstallationcheckthesealingringofthesealandsprings,whichshallconformtothetechnicalrequirementsmanufacturersdrawings.CylindergasketonthemountingbracketssupportingtheGroupassembled,thelugsshallbenoBurronthesurface,andotherdebris.Departmentsuiteinplacetomeasuresurfaceandinnercylindersurfaceheightdifference,comparedwiththeoriginalvaluesconfirmthelocation.6.4.4.4lowerbearingandthrustbearinginstallationcleanbearingandbearing,thrustbearingjoints,cleanthemwithathinlayerofturbineoil,andthenliftingittobits.6.4.4.5highandmiddlepressurecylinderoverittotheend(forhighpressurerotorandlowpressurechangeflangegasketwithoutconnectionunit)rotorsandrotorflangegasketconnectionandthereisnoconnection,sotherotorinplacecases,rotorsinplacemustbehighandmiddlepressurecylindertogetherwiththeChamberoverittotheend.Jackintheboxthatareinstalledonapush-pulldeviceofadjustingthehighandmediumpressurecylindertoendoverandaroundtosetupindicatorstomonitordata,generaldataaboutthedifferencemustnotexceed0.02mm.Topreventcylinderaboutoutofsyncoverthesituation,takethefollowingmeasures:a)overJackusesthesamepump,evenly.B)lefttosetupindicatorstomonitoroverquantity.TopreventthemovementovertheChamberwhentheplaten,atthetopofthefrontbearingplateinstalledtight,thenanchorbasenexttothecorresponding中南大学工程硕士“高等工程数学”考试试卷（开卷）考试日期：2010年4月日时间110分钟注：解答全部写在答题纸上一、填空题(本题24分，每小题3分)1.若函数1()[,]xCab,且[,]xab有()[,]xab和1)('Lx,则方程()xx在[,]ab上的解存在唯一，对任意bax,0为初值由迭代公式)(1nnxx产生的序列nx一定收敛于方程()xx在[,]ab上的解*x，且有误差估计式*xxkL1；2.建立最优化问题数学模型的三要素是：确定决策变量、建立适当的约束条件、建立目标函数；3．求解无约束非线性最优化问题的最速下降法会产生“锯齿现象”，其原因是：最速下降法前后两个搜索方向总是垂直的；4．已知函数)(xfy过点(,),0,1,2,,iixyin，[,]ixab，设函数)(xS是()fx的三次样条插值函数，则)(xS满足的三个条件（1）在每个子区间iixx,1（i=1，2，…，n）上是不高于三次的多项式；（2）S（x），S’（x），S’’（x）在ba,上连续；（3）满足插值条件S（xi）=yi（i=1，2，…，n）；5．随机变量1210~(3,4),(,,,)XNXXX为样本，X是样本均值，则~XN（3，0.4）；6．正交表()pqNLnm中各字母代表的含义为L表示正交表，N表示试验次数，n、m表示因子水平数，p、q表示试验至多可以安排因素的个数；7．线性方程组Axb其系数矩阵满足A=LU，且分解唯一时，可对A进行LU解，选主元素的Gauss消元法是为了避免采用绝对值很小的主元素导致误差传播大，按列选取主元素时第k步消元的主元akk为)1,2,......,1(1niayabyiinijiijii8．取步长0.01h，用Euler法解'3,[0,1](0)1yxyxy的公式为。二、（本题6分）某汽车厂三种汽车：微型轿车、中级轿车和高级轿车。每种轿车需要的资源和销售的利润如下表。为达到经济规模，每种汽车的月产量必须达到一定数量时才可进行生产。工厂规定的经济规模为微型车1500辆，中级车1200辆，高级车1000辆，请建立使该厂的利润最大的生产计划数学模型。微型车中级车高级车资源可用量钢材（吨）1.522.56000（吨）人工（小时）30405055000（小时）利润234解：设微型车生产了x1辆，中级车生产了x2辆，高级车生产了x3辆，而钢材、人工均有限制，所以应满足限制条件：钢材：1.5x1+2x2+2.5x3≤6000人工：30x1+40x2+50x3≤55000生产数量：x1≥1500x2≥1200x3≥1000从而问题的数学模型为：Maxc1x1+c2x2+c31002,1,009.003.01nyxynnn为进一步强化公司安全生产、文明施工管理，切实落实安全生产责任，防止重大安全事故发生，保障企业和员工的生命和财产安全manufacturersdrawingsboltfastenedinthecylinderandoutercylinder.Cylindercylinderinthedrainpipeinstallationandmeasuringcomponentleads.6.4.4.3bladecarrier,partitionsandlowerinstallationcheckthesealingringofthesealandsprings,whichshallconformtothetechnicalrequirementsmanufacturersdrawings.CylindergasketonthemountingbracketssupportingtheGroupassembled,thelugsshallbenoBurronthesurface,andotherdebris.Departmentsuiteinplacetomeasuresurfaceandinnercylindersurfaceheightdifference,comparedwiththeoriginalvaluesconfirmthelocation.6.4.4.4lowerbearingandthrustbearinginstallationcleanbearingandbearing,thrustbearingjoints,cleanthemwithathinlayerofturbineoil,andthenliftingittobits.6.4.4.5highandmiddlepressurecylinderoverittotheend(forhighpressurerotorandlowpressurechangeflangegasketwithoutconnectionunit)rotorsandrotorflangegasketconnectionandthereisnoconnection,sotherotorinplacecases,rotorsinplacemustbehighandmiddlepressurecylindertogetherwiththeChamberoverittotheend.Jackintheboxthatareinstalledonapush-pulldeviceofadjustingthehighandmediumpressurecylindertoendoverandaroundtosetupindicatorstomonitordata,generaldataaboutthedifferencemustnotexceed0.02mm.Topreventcylinderaboutoutofsyncoverthesituation,takethefollowingmeasures:a)overJackusesthesamepump,evenly.B)lefttosetupindicatorstomonitoroverquantity.TopreventthemovementovertheChamberwhentheplaten,atthetopofthefrontbearingplateinstalledtight,thenanchorbasenexttothecorresponding1000120015005500050403060005.225.1321321321xxxxxxxxx三、（本题10分）已知)(xf的数据如表：x0125)(xf-5306用Newton插值法求)(xf的三次插值多项式)(3xN，计算(6)f的近似值，给出误差估计式。解：xiF(xi)一阶差商二阶差商三阶差商四阶差商0-513820-3-11/25625/427/20612.56.54.5/4-0.025-0.2292因此32335.155.97.105.0)(xxxxN，而5.12)6()6(3Nf504.27)56()26()16(62292.0)()6(432103xxxxxxfR四、（本题12分）为了研究小白鼠在接种不同菌型伤寒杆菌后的存活日数有没有差异，现试验了在接种三种不同菌型伤寒杆菌（记为123,,AAA并假设2~(,)iiAN，1,2,3i，）后的存活日数，得到的数据已汇总成方差分析表如下：方差来源平方和自由度样本方差F值组间SSA662336.286组内SSE63125.25总和SST12914(1)试把上述方差分析表补充完整（请在答卷上画表填上你的答案）(2)小白鼠在接种不同菌型伤寒杆菌后的存活日数有无显著差异？（取0.05，0.05(2,12)3.89F）解：（1）见表中红色部分（2）设H0：μ1=μ2=μ3=…=μi选取统计量)1()1(NSSEISSAF，由于显著性水平未给出，设α=0.05，查表得89.3)12,2(05.0F，因为F=6.286)12,2(05.0F，所以拒绝H0，即小白鼠在接种不同型伤寒杆菌后存活日数有显著差异。为进一步强化公司安全生产、文明施工管理，切实落实安全生产责任，防止重大安全事故发生，保障企业和员工的生命和财产安全manufacturersdrawingsboltfastenedi