Find an equation of the tangent plane to the given surface at the specified point.

\(z=2(x-1)^2+6(y+3)^2+4,(3,-2,18)\)

\(\Rightarrow z_x=4(x-1)\)

\(=4x-4\),

\(z_y=12(y+3)\)

\(=12y+36\)

At the point (3,-2,18):

\(z_x=4x-4\)

\(=4(3)-4\)

\(=12-4\)

\(=8\)

\(z_y=12y+36\)

\(=-12(-2)+36\)

\(=-24+36\)

\(=12\)

Equation of the tangent plane is,

\(z-z_1=z_x(x-x_1)+z_y(y-y_1)\)

\(z-18=8(x-3)+12(y-(-2))\)

\(z-18=8x+12y\)

\(z=8x+12y+18\)

\(z=2(x-1)^2+6(y+3)^2+4,(3,-2,18)\)

\(\Rightarrow z_x=4(x-1)\)

\(=4x-4\),

\(z_y=12(y+3)\)

\(=12y+36\)

At the point (3,-2,18):

\(z_x=4x-4\)

\(=4(3)-4\)

\(=12-4\)

\(=8\)

\(z_y=12y+36\)

\(=-12(-2)+36\)

\(=-24+36\)

\(=12\)

Equation of the tangent plane is,

\(z-z_1=z_x(x-x_1)+z_y(y-y_1)\)

\(z-18=8(x-3)+12(y-(-2))\)

\(z-18=8x+12y\)

\(z=8x+12y+18\)