]> First Approximation

First Approximation

of a function with many variables.

by Samuel Dagan (Copyright © 2007)


As known from functions with one variable, the tangent line at a point x0 is used, as a first approximation (y1) of the function  y = y(x) , around this point:

y 1 = y 0 + ( dy dx ) 0 ( x x 0 )  

where the index 0, denotes at point x0.

The first approximation of a function of n variables

y=y( x 1 ,...., x n )  

around the point

( x 1 ,...., x n ) 0 =( x 1,0 ,...., x n,0 ) (1)

has the form of:

y 1 = y 0 + k=1 n ( y x k ) 0 ( x k x k,0 ) (2)

since for the point (1)

y 1 = y 0 and y 1 x k = ( y x k ) 0 k=1,...,n } (3)

As seen from (2), the first approximation itself is in general a function of n variables. Care should be taken that the function and the derivatives are continuous at the point (1).

In the case of a function of two variables,

z=z( x,y ) (4)

the first approximation is the tangent plane:

z 1 = z 0 + ( z x ) 0 ( x x 0 )+ ( z y ) 0 ( y y 0 ) (5)

It is called "tangent", since its derivatives are equal to these of the function at the point (x, y, z)0, but is it a plane?

A plane in a three dimensional space is defined as any linear combination of the Cartesian coordinates:

Ax+By+Cz=D (6)

where at least one of the constant factors of the coordinates (A,B,C) does not vanish. Indeed any intersection of (6) with the planes of constant x, y or z, yields a straight line, as can be easily seen. From the definition (6) it follows that (5) is a plane.

A point of a function of many variables with vanishing and continuous derivatives is called a stationary point. As in the case of a function of a single variable, a stationary point indicates that for an infinitesimal deviation from this point, the function remains constant. In the case of two variables (5), the tangent plane at a stationary point is

z= z 0 where    z 0    is   z   at that point (9)

The following example will be used to illustrate the tangent plane.

z=3[ 1 ( x 4 ) 2 ( y 3 ) 2 ] (10)

The section of this function with a constant z plane is

( x 4 ) 2 + ( y 3 ) 2 =1 z c 3 with z c =constant (11)

which is an ellipse for zc<3, a point for zc=3 and non existing for zc>3. Therefore

x=y=0 z=3 }isapointofmaximum (12)

The sections with constant x or y values are parabolas:

z= y 2 3 +3[ 1 ( x c 4 ) 2 ] x c =constant z= 3 16 x 2 +[ 3 y c 2 3 ]    y c =constant } (13)

The shape is similar to a paraboloid of revolution, except that instead of circles there are ellipses, and therefore it is called an elliptic paraboloid.

The tangent plane at point (x,y,z)0, according to (5) is:

z= z 0 3 x 0 8 ( x x 0 ) 2 y 0 3 ( y y 0 ) (14)

For the maximum (12), the tangent plane becomes

z=3 (15)

which corresponds to a stationary point (9).

This example (9) and some display of the tangent planes are shown in Fig. Elliptic paraboloid.