Part2: Linearization and Tangent Planes

In single variable calculus, we learned that if f( x) is differentiable at a point p, then

L( x) = f( p) +f'( p) (x-p)

is the linearization of f( x) at x = p. Moreover, the graph of L( x) is the straight line that is tangent to the graph of f( x) at x = p.

Analogously, let us define the linearization of a function f(x,y) at a point ( p,q) to be

L( x,y) = f( p,q) +fx( p,q) (x-p) +fy( p,q) ( y-q)

It follows from definition 4.1 that if f( x,y) is differentiable at ( p,q) , then f( x,y) » L( x,y) for all ( x,y) which are sufficiently close to a point ( p,q) . Moreover, it also follows that the graph of L( x,y) is a plane that practically the same as the surface z = f( x,y) near the point ( p,q,f(p,q) ) . Thus, we say that the graph of L( x,y) is the tangent plane to z = f( x,y) at ( p,q) .       

EXAMPLE 2    Find the tangent plane to f( x,y) = 9-x²-y² when ( x,y) = ( 1,1) .       

Solution: To begin with, f_x( x,y) = -2x and f_y( x,y) = -2y, so that f_x( 1,1) = f_y(1,1) = -2 and

L( x,y) = f( 1,1) +fx( 1,1) (x-1) +fy( 1,1) ( y-1) = 7-2( x-1) -2( y-1)

which simplifies to L( x,y) = 11-2x-2y. Thus, the equation of the tangent plane to f( x,y) at the point ( 1,1,7) is

z = 7-2( x-1) -2( y-1)

which is shown below.

       

The tangent plane is the plane that ''best approximates'' a surface in the neighborhood of a point. However, a tangent plane may intersect a surface in an infinite number of points. Thus, intuitive ''working definitions'' of tangency are not tenable for functions of 2 variables, as illustrated in example 3.       

EXAMPLE 3    Find the linearization of f( x,y) = x³-xy² at the point ( 1,2) .       

Solution: Since f_x = 3x²-y² and f_y = -2xy, we have

fx( 1,2) = 3-4 = -1,        fy( 1,2) = -2·1·2 = -4

Since f( 1,2) = -3, the linearization of f( x,y) =x³-xy² at ( 1,2)  is

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

which simplifies to L( x,y) = 6-x-4y. The graph of L(x,y) is shown versus the graph of f( x,y) below: