tangent plane

The flat plane through a point on a surface that matches the surface's height and both partial-derivative slopes there.

The tangent plane to $z=f(x,y)$ at $(a,b)$ is $z=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)$ — it is just the linearization $L(x,y)$ from the differentiability test, renamed: differentiability was defined precisely by requiring this plane to be a good local approximation. Rearranged to $f_x(a,b)(x-a)+f_y(a,b)(y-b)-(z-f(a,b))=0$ , the vector $\langle f_x(a,b),f_y(a,b),-1\rangle$ is normal to the surface at that point.

Examples

The tangent plane to $z=9-x^2-y^2$ at $(1,2)$ is $z=-2x-4y+14$ .