Answer to Question #136788 in Calculus for Promise Omiponle

Question

Find the linear approximation to f(x,y,z)=xy/z at the point (2,2,−1):

f(x,y,z)≈

Answer 1

f(x,y,z)=xy/z,

f(2,2,-1)=2*2/(-1)=4,

f^{`</sup><sub>x</sub>= y/z , f<sup>`}_x(2,2,-1)=2/(-1) =-2,

f^{`</sup><sub>y</sub>= x/z , f<sup>`}_y(2,2,-1)=2/(-1) =-2,

f<sup>`</sup><sub>z</sub>= -xy/z<sup>2</sup> , f`_z(2,2,-1)=-2*2/(-1)²=-4,

f(x,y,z) = f(2,2,-1) + f^{`</sup><sub>x</sub>(2,2,-1)*(x-2)+ f<sup>`}_y(2,2,-1)*(y-2)+ f`_z(2,2,-1)*(z+1),

f(x,y,z) = 4+(-2)(x-2)+(-2)(y-2)+(-4)(z+1) = 8-2x-2y-4z,

f(x,y,z)=4-x-y-2z .