用求偏导数与多元函数的极值练习参考解答

1、10用Mathematica求偏导数与多元函数的极值 练习参考解答求下列函数的偏导数。xy z = e(1) z =1 =x2y2u u(xy)z求下列函数的偏导数或导数。设 z 二arctg(xy), y =e(2)3z设 z = xln( xy),求 jx y dz,求 。dx一37二 z- 2xy1rL 2u& z=xlny, x= ,y = v3u 2v,求：z；:u二 zojvx y ：u 设u = f (一,一)，求 丁 ,.u;zo:x设 z = f (x +y,xy,二)，求 yzx,zxx,zxy O求下列方程所确定的隐函数的导数。/for。(2)e/y -2z +ez =0

2、 ,求豆， ：:x-yz = f (x +y +z, xyz),求, :x:x:y-222_=a ,x + y = ax,求dxdzodx求函数f (x, y)=x2+ 5y2 6x+10y+6 的极值。求函数z = x2 - y2,在(x, y) | x2 + y2 4范围内的最大最小值。1求下列函数的偏导数。1xyV z xzz = z=e u=)-(4) u=(xy)x2 y2x x z解(1) In1:=D1/SqrtxA2+yA2,xIn2:= D1/SqrtxA2+yA2,yxOut1=(x y )Out2=-2)23/2(x y )In3:= DEA(x*y),xIn4:= DE

3、A(x*y),xOut3= exyyOut4= exyxIn5:= Dy/x+z/x-x/z,xIn6:= Dy/x+z/x-x/z,yIn7:= Dy/x+z/x-x/z,zy 1 zOut5=x z z TOC o 1-5 h z 1Out6=-x1 xOut7=x zIn8:= D(x*y)Az,xIn9:= D(x*y)Az,xIn10:= D(x*y)Az,zOut8= y(xy)/zzOut9= x(xy)1 zzOut10= (xy)z Logxy2求下列函数的偏导数或导数。(1) 设 z =arctg(xy), y =ex , 求 dz。 dx解 In1:= yx_:EAx;z

4、x_,y_:=ArcTanx*y;Dzx,y,xOut1=(2)设 z = xln(xy),求：3z?z2-xy解 In1:=zx_,y:=x*Logx*yDzx,y,x,2,y;Dzx,y,x,y,3;Simplify%O所空中6(x2 -6x4 x6y4)(1 x2y2)43vz ：:z(3)设 z = x sin y,x =1一-，y =u+3v,求,。uu ;v解 In1:= xu_,v_:=1-v/u;yu_,v_：=u+3v;zx_,y:=xu,vA2*Sinyu,v;Dzx,y,u;Simplify%Dzx,y,v;Simplify%Out1=(u2 -v)(u(u2 -v)Co

5、su 3v 2(u2 v)Sinu 3v)22(u -v)(3(u -v)Cosu 3v -2Sinu 3v)（4）设口 = f（二），求孚,-：u ：:u-Z解 In1:= ux_,y_,z_:=fx/y,y/z;Dux,y,z,xDux,y,z,yDux,y,z,zOut1=xf(1，0) y2yzf(0,1)rx yyf ,y z2zx(5) 设 z = f (x + y, xy,一),求 zx, zxx, z。 y解 In1:= zx_,y_:=fx+y,x*y,x/y;Dzx,y,xDzx,y,x,xDzx,y,x,yOut1=2xy3(1 x2y2)22xy3_L_(1 x2y2)

6、2 1 x2y2三求下列方程所确定的隐函数的导数。(1)x2y 十 3x2y3 -4=0 ,求 dy。dx解 In1:= DxA2*yx+3xA2yxA3-4= =0,x;Solve%,yxOut1=yx -2xyx -6xyx3x2 9x2yx2e-y -2z+ez =0 ,求名，ox .:y解 In1:=DEA(-x*y)-2*zx+EAzx= =0,x;SimplifySolve%,z xDEA(-x*y)-2*zy+EAzy= =0,y;SimplifySolve%,z ye -xy yOut1= zx y-2 - eeyx zyzy2 ez = f(x+y+z,xyz),求名，空,

7、乌。 ；x二y二y解 In1:= Dzx-fx+y+zx,x*y*zx= =0,x;SimplifySolve%,z xDzy-fx+y+zy,x*y*zy= =0,y;SimplifySolve%,z yOut1= zx 一(yzxf (0,1)x y zx, xyzxf (1,0)x y zx, xyzx) /(-1 xyf (0,1)x y zx, xyzx f (1,0)x y zx, xyz x) zy； (xz y f (0,1)x y zy, xyzyf(1，0)x - y zy,xyzy)/(-1 xyf (0,1)x y zy, xyzy f (1,0)x y zy, xyz

8、 y)x2 +y2 +z2 = a2 ,x2 + y2 = ax,求 dy , dz。dx dx解 In1:= DxA2+yxA2+zxA2-a= =0,xA2+yxA2-a*x= =0,x;SimplifySolve%,yxSimplifySolve%,zxx z xz x、Out1= yx ， zx-4 求函数 f (x, y) = x2+5y2 6x+10y+6的极值。解 In1:= Clearx,y,z,a,b,c,d,t;fx_,y_:=xA2+5yA2-6x+10y+6;a=Dfx,y,x,2;b=Dfx,y,x,y;c=Dfx,y,y,2;d=a*c-bA2;t=SloveDfx

9、,y= =0,x,Dfx,y= =0,y,x,y;l=LengthtlFori=1,i0&a10&a10,Print fmin= ,z,d1= =0,Print No Sure,z,d1= =0,PrintNo Out1= x-3,y-1fmin=-85求函数z = x2 - y2,在(x, y) | x2 + y2 M 4范围内的最大最小值。 解 先求z = x2 - y2在圆域内x2 +y2 0,y-0)In2:=xA2+yA2-4/.t1Out2=-4该驻点在圆外，圆内无驻点，故不取极值.下面考虑圆x2 + y2 = 4上的最值.这是在约束条x2 + y2 =4下的条件极值，用Lagrange乘数法求解.In3:= Clearx,y,F,t;Fx_,y_,t_:=fx,y+t(xA2+yA2-4);s=SolveDfx,y,t=0,Dfx,y,t=0,y,DFx,y,t=