高数(下)综合练习题.doc

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软件学院高数(下)综合练习题参考答案周世国：软件学院高等数学（下）综合练习参考答案高等数学(下)综合练习(一)一、计算下列各题：(每小题5分,共30分)1.设z=求xzx+yzy. 解：z11x=x+y.x+y)x=x+y.12x；由轮换对称性知：z11y=x+y.2y.所以xzz11x +yy=2x+y.x+yxy=112x+y.x+y)=12.2.设由方程 xz=lnzy 可以确定函数z=z(x,解一：（公式法）（这里x,y,z令F(x,y,z)=xz-lnzy=xz-lnz+lny.则，F111x=z,Fy=y,z.故有1 zFy=z；z=-=-y=z2,z-x+zyFzx1xy+yzz2-z-z2-z所以 dz=zx+zdx+z2xy+yzdy.解二：（一）方程两边同时关于x求导（视z为z=z(x,y)，并视y常数），有：11zz+x-11z1xz1z2.x=zy.xz+z2.x=z，y1周世国：软件学院高等数学（下）综合练习参考答案 故zz； =1xxx+z+2zz1（二）方程两边同时关于x求导（视z为z=z(x,y)，并视y常数），有：x-1z2.zy=1z-1y2.z+1z1xz1y.yz+z2.y=y，y故有1zyx=1=z2.z+xyz+xyz2所以 dz=zx+zdx+z2xy+yzdy.解三：方程两边取全微分，得：dxzz=dlny111zzdx+x-z2dz=z-y2dyy1z+xz2dz1所以，dzy+dx=zdx+z2dy.1xx+zxy+yzz+z23. (yx,x2-y2),其中f(u,v)可微, 求zzx,y.解：zyx=fx+f(x2-y2)y12x =f1-2xx+f2.2x；zy=fyxf(2).y1+2x2-yy =f1+f2.(-2y).yx； 2周世国：软件学院高等数学（下）综合练习参考答案4. 求曲面z=x2+y2-1在点(2,1,4) 处的切平面与法线方程.解：设 F(x,y,z)=x2+y2-z-1，.则，=Fx(M0),Fy(M0),Fz(M0)=2x,2y,-1|(2,1,4)=4,2,-1.所以，切平面的方程为：4(x-2)+2(y-1)-(z-4)=0切平面的法线方程为：x-2y4=-12=z-4-1.n!sinp5. 判定级数 n=12n 的敛散性.解： 记un因为r=n =n6. 求函数z解：（一）解方程组fx(x,y)=3x2-3y=x=0,x=1,fy2y(x=3-3x=0.y=0.或者y=1.P1(0,0),P2(1,1).A=fxx(x,y)=6x,B=fxy(x,y)=-3,C=fyy(x,y)=6y.（1）.在P1(0,0)处，因为A=0,B=-3,C=0,D=AC-B2=-90,B=-3,C=6,D=AC-B2=270, 故f(1,1)=-2为极小值.二、计算下列积分：(每题10分,共70分)3周世国：软件学院高等数学（下）综合练习参考答案 1.I=x2y2s,其中D是由直线y=1,x=2及y=x所围成的闭区域. D解：I=22x122x3x21xdx1y2dy=1x2-1xy|1dx=1(x2-x)dx=23-2|51=6.2.I=(x2+y2)dv,其中W为曲面z=x2+y2与平面z=4所围成的闭区域.W解：本题宜采用“切片法”计算(x2+y2)dxdydz=4()Wdzx20+y2dxdy Dz=4dz2pdqzr2.4r4pz3000rdr=2p040dz=43223|0=3p.如采用柱面坐标系：(x2+y2)dxdydzW=2pdq2422300rdrrp0r.(4-r2)dr=2p4.r44-r66|2rdz=23.I=x+yL3，其中L为圆周x2+y2x2+y2解：I=x+yLx2+y2=x+y3L1=+yds0+0=0. 4. I=l(x2+y)dx+(x+2)dy,其中l为圆周x=上由点(0,0)到(0,2)的一段弧.解一：l可化为x2+(y-x0). l:x=cost,ppy=1+sint,t:-22.(ldx+(x+y2)pdy=22pcost+1+sint(-sint)+cost+(1+sint)2-()().costdt 2p=2-p-cos2t.sint-sint+2sint.costpdt+2p-sin2t+cos2t+cost+sin2tcostdt2-2p =0+220cos2t+cost+sin2tcostdt=212sin2t+sint+13p83sint20=3.4周世国：软件学院高等数学（下）综合练习参考答案 解二：记A(0,2)，记D为l+所围成的区域.由格林公式，得：x+y2x2+y I+=-AOxyD又AO()()dxd=y(1-1)dxdy=0. D(x208+ydx+x+y2dy=0+y2dy=-，故： 23)()()88 I=0-=. 33x+y2x2+y解三：由于=1，故I=(x2+y)dx+(x+y2)dy与路径无关， lxyI=(x2+y)dx+(x+y2)dy=lOA28=0+y2dy=. 03()()()5.I=z2dS,其中为锥面zz=1解：由S:z=x2+y2,zzdS=+yx22=2dxdy. dxdy故222zdS=2x+ydxdy=SDxy()6. I=(8y+1)xdydz+2(1-4yzdxdy,其中为半球面z=.解：若取S10（下侧）.则S与S1一起构成一个封闭曲面.记它们所围成的半球型闭区域为W.在W(8y+1)x)21-y2(-4yz)=+dxdyd zxyzW+S1()=(8y+1)+(-4y)+(-4y)dxdydz=W21dxdydz=p. 3W所以，S=2p2p2p-=-0dxdy=. 3S13Dxy35周世国：软件学院高等数学（下）综合练习参考答案8. 求函数f(x,y)=x2+12xy+y2在闭区域4x2+y225上的最大值与最小值.解：（一） （1）上，问题转化为求f(x,y)=x2+12xy+y2 （2）在条件4x2+y2-25=0下的极值.5由（*）式，可设x=cosq,q0,2p2y=5sinq,代入（2）得：2f=52cosq+12.52coqs.(5sinqinq)2 =254cos2q+75.sin2q+25sin2q =251+cos2q4.2+75.sin25.1-cos2q2 =1258-758cos2q+75sin2q=1258(2q+j).+75865cos(2q+j). (tanj=-18)所以f(x,yy2 在边界上的最大值是1258+75865；最小值是1258-75865.f(0,0)=0与1258+75865及1258-75865知f(x,y)=x2+12xy+y2在闭区域4x2+y225上的最大值与最小值分别1258+75865和1258-75865.9求nxn的和函数,并求级数nn=1n=12n的和.解：（一）记 an=n.(n=1,2,L)6周世国：软件学院高等数学（下）综合练习参考答案 因为r=liman+1na=limn+1nnn=1,所以，R=1.又在端点x=-1(