高数~求切平面方程 高数 切平面方程
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\u628a\u66f2\u9762\u65b9\u7a0b\u5199\u6210F\uff08x\uff0cy\uff0cz\uff09=0\uff0c
\u5219Fx\u3001Fy\u3001Fz\u5728P0\u7684\u503c\u5c31\u662f\u5207\u5e73\u9762\u6cd5\u5411\u91cf\u7684\u4e09\u4e2a\u5750\u6807\uff0c
\u5b83\u4eec\u4e0e1\u30014\u30016\u6210\u6bd4\u4f8b\u2605
\u53c8\u5207\u70b9\u5728\u66f2\u9762\u4e0a\u2605\u2605
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\u6c42\u51faz\u5bf9x\u3001y\u7684\u504f\u5bfc\u6570\uff0cz(x)=(-y/x^2)/(1+y^2/x^2)\uff0cz(y)=(1/x)/(1+y^2/x^2)\uff0c\u628a\u70b9M0\u5750\u6807\u4ee3\u5165\uff0c\u5f97\u5e73\u9762\u7684\u6cd5\u5411\u91cf=\uff08z(x)\uff0cz(y)\uff0c-1\uff09=\uff08-1/2,1/2,-1\uff09\uff0c\u6240\u4ee5\u5207\u5e73\u9762\u65b9\u7a0b\u4e3a(-1/2)(x-1)+(1/2)(y-1)+(-1)(z-\u03c0/4)=0
设G(x,y,z)=F(x-y+z,xy-yz+zx)求偏导数:Gx=Fu*1+Fv*(y+z),Gy=Fu*(-1)+Fv*(x-z),Gz=Fu*1+Fv*(x-y)
代入x=2,y=1,z=-1,Fu=2,Fv=-3,得Gx=2,Gy=-11,Gz=-1
所以切平面的法向量是(2,-11,-1),切平面方程是2(x-2)-11(y-1)-(z+1)=0,答案是 D
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