多元函数微分历试题

多元函数微8．设二元函数zxlny,则2z解析:zx, 多元函数微8．设二元函数zxlny,则2z解析:zx, yy15．设zuv,而u2xyvx求和15.解zzuzv2vuv1uvlnu vzzuzvu v又当x1,y0时，u2,v2ln2,19zf(xy2ln2,19zf(xyxylnx(x0x1)，D(xy2xe1y1（1）求全微分dz（2）求f(xDzxy1yxy1lnxxy1(1ylnx),zxyln2dzzdxzdyxy1(1ylnx)dxxyln2（2）f(x,e1ln2De1(xy2(x1)dx(11e2ln23ee2x2229，设二元函数zln(xy(x1)dx(11e2ln23ee2x2229，设二元函数zln(xy2 16．已知三元函数f(uvw)fvfw0。若二元函数zz(x,y是由三f(xyyzz算z所确定的隐函 16．解：F(x,y,z)f(xy,yz,zx)f(u,v,w)其中uxyvyzwz则Fxfufw,Fyfufv,Fzfvfw fu Fffywuv , fv fv故zzfufwfufvfvfw 217．求二元函数zt dt的全微故zzfufwfufvfvfw 217．求二元函数zt dt的全微分dz02偏导。22 yy17x所dzzdxzdyyex2y2dxxex2y2dyex2y2yex2(2x2y)ex2y2(12x2y22则zf(4x2y2在点(1,2)解：设u4x2y2，则zfzf(u)u8xf(u),8f(0)8解：设u4x2y2，则zfzf(u)u8xf(u),8f(0)812zf(u)u2yf(u),4f(0)412dxdy4dx217．已知二元函数zx(2y1)，x17．解：z2x2(2y1)x124x(2y1)x12x2(2y1)x1ln(2y242ln故17．已知二元函数z(3xy)2y求偏导及。17．解法一：设u3xy,v17．已知二元函数z(3xy)2y求偏导及。17．解法一：设u3xy,v2y则zzzuzvvuv13uvlnu u v3vuv16y(3xy)2；zzuzu vvuv11uvlnu2uvln2y(3xy)2y12(3xy)2yln(3x2ln(3x2(3xy)23x解法二：lnz2yln(3xy设F(xyz)2yln(3xylnz，62F(x,y,z),F(x,y,z)2yln(3x设F(xyz)2yln(3xylnz，62F(x,y,z),F(x,y,z)2yln(3xy)x3xy3xF(xyz1zzzFx(x,y,Fz(x,y,(3xy)2y63x6y(3xy)2F(x,y,2 yz2ln(3xy)3xyF(x,y,z(3xy)2y2ln(3xy)23xy4x39．若二元函数z(y0)22 =＿＿＿＿ y2),yy5f(xyy2),yy5f(xyy，f(0，0y)f解：f(0,0)=ysin(202(0lim limsin(17．已知隐函数zf(xy)由方xy2z31所确定，求z和z17．解：设F(xyzxzxy2z31，Fzxz1y2,Fy2xy,FxzlnxFy2zxFzxz1y2,Fy2xy,FxzlnxFy2zx 所以，Flnx3zxzxFy Flnx3zxzf(x,xy， 5f(xyxyx2A．2yC.2xB.解：f(xyxyx2y2xyxf(x,y)x2f(x,y)9．已知二元函数zf(xy)的全微2dzy2dx＿，则解：由dzy2dx2xydy知z216．设隐函数zf(xy)由方程xyz3xz所确定，及。16.【解析】设216．设隐函数zf(xy)由方程xyz3xz所确定，及。16.【解析】设F(xyzxyz3xzFxyxy1z，Fyxylnx，Fz3z2所Fy xzF .x3z2 3z216.设方程exy2zez0zz(xy)，求zzx16．【解析】设F(xyzexy2zez则yexy,Fxexy,F2ezyexe.,29.设uexyexe.,29.设uexcosy,vexsinyuv。 ,(x,y)(0,5f(xy)(x,y)(0,fx(0A.等于C.等于B.等于-D存xy17f(xyxy)xf(x,f(x,x的值17f(xyarctanxy1yf(x,y)1y,x2yf(x,y)1x . f(x,f(x,x的值17f(xyarctanxy1yf(x,y)1y,x2yf(x,y)1x . x22yf(x,y)f(x,yxx故x2yx2y10.设uln(x2y2z2则全微分du 4．设zln(xy，则dz）A.1dx1B.1dx1xyyxdxD.ydx217．设函数zxarctanxy。1xdxD.ydx217．设函数zxarctanxy。1x1(xy，2z2x(x2y2)x2(x2y2(x2y