微積分A學(xué)期期末試題解答

1、2006-2007 學(xué)年微積分A第二學(xué)期期末考試參考答案及評分標(biāo)準(zhǔn)2007年7月10日1LM (1, a,2),Ls k,1, 4 ,n 3,2,1. 332a27,3k240a1,k2. 62z1y.3xx2fx fy2 z1fx1fyxyx yx2xfxyfy. 63 2x,4y, zn1 ,2,1 . 2666uxu2 yz2 ,u2zz2 .xe ,y1 y 2z1 y2u1,u|(0,1,1)2,u|( 0,1,1)2.4x |(0,1,1)y3z3f |(0,1,1)1122126.6n6636324Iy2dxdy1x2dy. 3dxx2 yD01 1(x361. 60 3x)dx

2、2851ln(11)n 14n1n1 / 61ln(11 )n1，.lim4n 1nlim2 分n1n4n 12n而1 發(fā)散，由正項(xiàng)級數(shù)的比較判別法知，級數(shù)n 1 n1ln(11 )發(fā)散. 3分n 14n1n又原級數(shù)為交錯級數(shù)，記un1ln(11) 0，4n 1nlim un0, 易見， un 1 un ，由萊布尼茨判斂法知，原級數(shù)收n斂，從而原級數(shù)條件收斂 . 6 分f3x22y022二、 1x，解得駐點(diǎn)： (0,0),f( ,).2分2 y2x033y2 f6x,2 f2f2.x2xy2,2y在駐點(diǎn) (0,0) 處有：2f2f2f2.A20, By2, C2xxyB2AC 40,所以f (

3、x, y)在（ ，）處不取極值。4 分0 0在駐點(diǎn) ( 2 , 2)處有：33A2 f4,B2f2f2.x2xy2, C2yB2AC40,且A40f (x)在點(diǎn)（2 22 24，）處取得極小值，極小 值f ( , ).3 33 3272 / 6(2,2). .73 32:xoyD : x2y 21.zxx,zyy. . 2x21x2y 21y2I(xy1x2y2 ) 1z 2z 2 dxdy .4xyD( xy1x2y2 )1dxdy.5D1x2y2dxdy. 7D3BA,LBA. GreenI(eyyx2 )dx( xey2sin yxy2 ) dyLBA(eyyx2 )dx( xey2si

4、n yxy2 )dyBA( x2y2 )dxdy1e0dx.4D1d13d242. 7004f ( x) ,f ( x).bn0,a02(2x)dx0,. 20an20(2x)cosnxdx42 (1)n1n0n2k, k1,2,8n2k1, k1,2,4n23 / 6f ( x)2 | x |811) 2cos(2n1) x,x. .7n 1 (2ndz2xdy3xdx(1, 2,1)dx232 dy0dz2dxdx(1,2,1)1,3 ,2. 32Ls 3,5,51,0,5 25,10,5 .6s 1,3,2 25,10,50. 82an 1lim3n n1R 3,. 1liman3nn3

5、n 1(n 1)| x 1| 3,2x4.2（n1時，級數(shù)為）,收斂x1;x4時，級數(shù)為2, 發(fā)散。n 1nn 1 n2,4).4S( x)( x1) n ,n 13n nS (x)(x1)n 11(x1)n 1111. .6n 13n3n 13n 131x 14 x3S( x)S(1)xS (x)dxx1dxln34xx114( x1) nln3,x2,4).8n 13n n4xS : z1, x2y21,Gauss1I(x2y21)dxdydzdxdy.4SS1S1VD:x 2y 2 14 / 61( x2y21) dxdy0dzD z :x 2y2z1dz2dz21)d6000(3. 8VdV2d3 d1r 2 sindr3.3V000xy01. 5zdVzV. 7dVVzdV23 d1r 2 sindr3. 9dr cosV00016z3 39 .(0,0,9).10161616(1)2ddtf (r 2 )r 2 sindr4t2 )r 2 dr .3F (t )00f (r00f ( x) 0,)F (t )(0,)F (t )4 t 2 f (t 2 ).4(2f (0)0,f (0)0,f ( x)f ( x)nn1 F ( 1 )4f ( 1),n 1nn 1 n