Question

Find the equations of the tangent plane and normal line to the surface x + y + z = e^(xyz) at the point (0,0,1).

Answer 1

I think this is what should be done:

\[F_x (x,y,z)= 1-yze^{xyz}\]
\[F_x (0,0,1)= 1\]

\[F_y (x,y,z)= 1-xze^{xyz}\]
\[F_y (0,0,1)= 1\]

\[F_z (x,y,z)= 1-xye^{xyz}\]
\[F_z (0,0,1)= 1\]


Then the tangent plane is:
\[1(x-0)+1(y-0)+1(z-1)=0\]
\[x+y+z=1\]


The normal line is:
\[x=y=z-1\]