Tangent plane to a surface at a point

 Goal: Determine the equation of a plane tangent to a surface at a point.

Tangent plane

In a two-dimensional space only one line can be tangent to a curve at a point  However, in a three-dimensional space many lines can be tangent to a curve at a point. A tangent plane is a plane made of all the tangent lines to a curve at a point. When a plane is tangent to a surface at a point, that surface is smooth there meaning there are no corners or discontinuities at that point. A tangent line to the surface at that point  in any direction doesn't have any abrupt changes in slope because the direction changes smoothly.

Definition

Let P₀ = (x₀, y₀, z₀) bee a point at a surface S, and let C be any curve passing by P and lying entirely on S, if the tangent lines to all such curves  C at P₀ lie in the same plane, then the plane is called the tangent plane to S at P₀

Tangent plane

In order for a tangent plane to a surface at a given point to exist, the function that defines the surface has to be differentiable at that point.

Definition

Let S be a surface defined by a differentiable function z = f(x. y) and let P₀ = (x₀, y₀) a point in the domain of. Then, the equation the equation of the tangent plane to S at P₀ is defined by:

Example

Find the equation of the tangent line to the surface defined by the equation:

Solution

Let's calculate f(2, -1):

Let's calculate the partial derivatives with respect to x and y respectively:

Let's calculate the partial derivatives with respect to x and y for the point (2, -1):

LFet's substitute all the elements in the equation of the tangent plane given above:

Figure

Practice

Find the equation of the  tangent plane to the surface defined by the equation: