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武大《高等数学》期末考试试题

武大《高等数学》期末考试试题

2000~2001学年第二学期《 高等数学 》期末考试试题(180学时) 专业班级 学号_______________ 姓名

一、 已知一个二阶常系数线性齐次微分方程有相等的实根a ,试写出此微分方程及通解。

(8分)

二、 设幂级数∑∞=?0

)1(n n n x a

在x =3处发散,在x =1处收敛,试求出此幂级数的收敛半径。(8分) 三、 求曲面323

=+xz y x 在点(1,1,1)处的切平面方程和法线方程 。(10分)

四、 设)(,0x f x >为连续可微函数,且2)1(=f ,对0>x 的任一闭曲线L,有0)(43=+∫L dy x xf ydx x ,求)(x f 。

(10分) 五、 设曲线L (起点为A ,终点为B )在极坐标下的方程为36(,2sin πθπθ≤≤=

r ,其中θ=6π

对应起点A ,3

π

θ=对应终点B ,试计算∫+?L xdy ydx 。(10分) 六、 设空间闭区域Ω由曲面222y x a z ??=与平面0=z 围成,其中0>a ,Σ为Ω的

表面外侧,且假定Ω的体积V 已知,计算:

∫∫Σ=+?.)1(2222dxdy xyz z dzdx z xy dydz yz x 。(10分)

七、 函数),(y x z z =由0),(=z y

y x F 所确定,F 具有连续的一阶偏导数,求dz 。

(12分) 八、 计算∫∫∫Ω

+,)(22dxdydz y x 其中Ω是由平面z =2与曲面2222z y x =+所围成的闭区域。(12分)

九、 已知级数

∑∞=1n n U 的部分和arctgn S n =,试写出该级数,并求其和,且判断级数∑∞=1n n tgU

的敛散性。(12分)

十、 设)(x f 连续,证明∫∫∫??=?A

A D dt t A t f dxdy y x f |)|)(()(,其中A 为正常数。D :2||,2||A y A x ≤≤

。(8分)

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