Linear approximations for functions of two variables

 Goal: Use the tangent line to approximate a function of two variables at a point.

Linear approximations

Let's recall that the linear approximation for a function of one variable f(x) at a point x = a  is given by:

Here is the diagram of the linear approximation of a function of one variable:

When we look at the diagram, for values close to x = a., the tangent line is confounded with the curve of the function represented by y =f(x).

The tangent line can be used as an approximation to the function y = f(x) for values of x reasonably close to x = a. 

The concept is the same for a function of two variables. The tangent line is replaced by the tangent plane.

Definition

Given the function  z = f(x, y) with partial derivatives at the point (x₀, yₒ), the linear approximation of at the point (x₀, yₒ) is given by the equation:

We can notice that this equation represents the tangent line to the surface z = f(x,y) at the point  (x₀, yₒ). It holds all the points (f(x,y) , (x,y)) close to (f(x₀, y₀), (x₀,y₀)) for values of (x,y) close to (x₀,y₀).

Exemple

Given the function: 

Approximate the point f(2.1, 2.9) using the point (2, 3) for (x₀, y₀). What is the approximate value of 

f(2.1, 2.9 to 4 decimal places.

Solution

Let's apply the formula of the equation of the linear approximation:

Let's calculate f(x₀, y₀), fₓ(x₀, y₀), fy(x₀, y₀) for x₀ = 2 and y₀ = 3.

Let's substitute all the elements in the equation of the linear approximation above:

L(x, y) = 4 - 2(x - 2) - 3/4(y - 3)

Let's substitute x =2.1 and y = 2.9 into L(x, y):

The approximate value of f(2.1, 2.9) to 4 decimal places is:

This corresponds to an error of approximation of 0.2%.

Practice

Given the function 

Approximate f(4.1, 0.9)  using the point (4, 1) for (x₀, y₀). What is the approximation of f(4.1, 0.9) using 4 decimal places?

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: