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1、北京信息科技大学2010-2011学年第2学期《高等数学》176学时课程期中考试试题(答案)一、填空题(共10分,每小题2分)1.向量的方向余弦2.求过点且平行于直线的直线方程是3.将面的抛物线绕轴旋转一周所成的旋转曲面方程是4.曲线在面的投影曲线是5.设由几何意义知二、解答题(共63分,每小题7分)1.已知三点,求同时垂直于与的单位向量解:由已知得:,┄┄┄┄┄(4分)同时垂直于与的单位向量为┄┄┄┄┄(7分)2.已知求otherstaffoftheCentre.Duringthewar,ZhuwastransferredbacktoJiangxi,andDirecto
2、rofthenewOfficeinJingdezhen,JiangxiCommitteeSecretary.Startingin1939servedasrecorderoftheWestNorthOrganization,SecretaryoftheSpecialCommitteeAfterthevictoryofthelongMarch,hehasbeentheNorthwestOfficeoftheFederationofStateenterprisesMinister,ShenmufuguSARmissions,DirectorofNingxiaCountypart
3、yCommitteeSecretaryandrecorderoftheCountypartyCommitteeSecretary,Ministersand5解:┄┄┄┄(1分)2分2分2分┄┄┄┄(7分)3.已知其中具有二阶连续偏导数,解:┄┄┄┄(3分)┄┄┄┄(7分)4.设由方程所确定,解:方程两边分别对和求偏导数得┄┄┄┄(3.5分)┄┄┄┄(7分)5.求曲面在点处的切平面解:令┄┄┄┄(1分)曲面在点处的法向量为┄┄(4分)所以曲面在点处的切平面方程为即┄┄┄┄(7分)6.问函数在点处沿什么方向的方向导数最大?并求此方向导数的最大值答:函数在点处沿梯度方向的方向导
4、数最大,且otherstaffoftheCentre.Duringthewar,ZhuwastransferredbacktoJiangxi,andDirectorofthenewOfficeinJingdezhen,JiangxiCommitteeSecretary.Startingin1939servedasrecorderoftheWestNorthOrganization,SecretaryoftheSpecialCommitteeAfterthevictoryofthelongMarch,hehasbeentheNorthwestOfficeoftheFeder
5、ationofStateenterprisesMinister,ShenmufuguSARmissions,DirectorofNingxiaCountypartyCommitteeSecretaryandrecorderoftheCountypartyCommitteeSecretary,Ministersand5┄┄┄┄(5分)方向导数的最大值为梯度的模┄┄┄┄(7分)7.解:令解得驻点┄┄┄┄(3分)又┄┄┄┄(4分)又,所以不是极值点┄┄┄┄(5分)又,又┄┄┄┄(6分)所以是极小值点,且┄┄┄┄(7分)8.交换二次积分的积分次序解:积分区域为y—型区域(如图)又
6、积分区域为可表示为┄┄┄┄(3分)┄┄┄┄(7分)9.化为柱面坐标系下的三次积分解:解方程组otherstaffoftheCentre.Duringthewar,ZhuwastransferredbacktoJiangxi,andDirectorofthenewOfficeinJingdezhen,JiangxiCommitteeSecretary.Startingin1939servedasrecorderoftheWestNorthOrganization,SecretaryoftheSpecialCommitteeAfterthevictoryofthelongMa
7、rch,hehasbeentheNorthwestOfficeoftheFederationofStateenterprisesMinister,ShenmufuguSARmissions,DirectorofNingxiaCountypartyCommitteeSecretaryandrecorderoftheCountypartyCommitteeSecretary,Ministersand5于是将Ω投影到xoy面得投影区域(如图)利用柱面坐标Ω可表示为┄┄┄┄(3分)┄┄┄┄(7分)三、计算下列各题(27分)1.及直线(