Tangent Planes

Definition (Tangent Plane) Suppose is a surface with the equation where has continuous first partial derivatives and . Let be a point on and let be the curve of intersection of with the plane and the intersection of with the plane . The tangent lines and to and , respectively determine a unique plane and this plane actually contains the tangent to every smooth curve that passes through We call this plane the tangent plane to at .  

Proposition (Tangent Plane) The equation of the tangent plane to the surface at the point can be given by  



    Proof. Any plane that passes through has an equation of the form By dividing this equation by and letting and , we can write it in the form If this equation represents the tangent plane at , then its intersection with the plane must be the tangent line with slope . Therefore, Similarly, putting , we get which must represent the tangent line with slope and so   Thus, the equation of the tangent plane is

Example (Tangent Plane) Find the tangent plane to the elliptic paraboloid at the point

    Solution. Let Then and Then the equation of the tangent plane is or

Example (Tangent Plane) Find an equation of the tangent plane to the surface at the point

    Solution. We compute the partial derivatives
    
    
    
Further, at this point we have, and Therefore the equation of the tangent plane is or equivalently

Example (Tangent Plane) Find an equation of the tangent plane to the surface at the point

    Solution. We compute the partial derivatives
    


Further, at this point we have, and Therefore the equation of the tangent plane is or equivalently   

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