Part2: Linearization and Tangent Planes

If we let x = p + h, then h = x-p, so that Definition 5.1 can be restated in the following form.    

Definition 5.1a: A function f( x,y) is differentiable at a point ( p,q) if there exists a vector Ñf( p) for which

lim x® p     | f(x) - f( p) -  Ñf( p) ·( x-p) | || x-p || = 0

(4)

When it exists, Ñf( p) is the total derivative of f( x) at p.

The definition of the limit then implies that e > 0, there is a neighborhood of p for which

| f( x) -  [ f( p)+Ñf( p) · (x-p) ] | < e || x-p ||

(5)

for all x in that neighborhood.

Consequently, we define the linearization of  f(x) at p to be

Lp(x) = f( p) + Ñf( p) · (x-p)

which in non-vector form is

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

It follows that

| f( x) - Lp(x)| < e || x-p ||

which means that the graph of L_p( x,y) is a plane that is practically the same as the surface z = f( x,y) near the point ( p,q,f( p,q) ) . We say that z = L_p( 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 = 11 - 2x - 2y

which is shown below.

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The tangent plane is the plane that best approximates a surface in the neighborhood of a point. However, this does not exclude the possibility that a tangent plane may intersect a surface in an infinite number of points.

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:

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LiveGraphics3d Applet