Examples

Tangent plane to paraboloid

Tangent plane to paraboloid

Problem: Find the equation of the plane tangent to the paraboloid $z=2x^2+y^2$ at $(x,y,z)=(1,1,3)$.
Solution: Set $f(x,y)=2x^2+y^2$. Compute $f_x=4x$, $f_y=2y$. So $f_x(1,1)=4$, $f_y(1,1)=2$, and the linear approximation is given by:

$$L(x,y)=3+4(x-1)+2(y-1).$$

The tangent plane is found by setting $z=L(x,y)$.