Chapter 6
Differentiation of Functions of
More Than One Variable
6.1
Function of More Than One Variable
Definition 29 Let be a set of order pairs of real numbers. If to each ordered pair ( ) in there corresponds a unique real number ( ) then is called a function of and . The set is the domain of and the corresponding set of values for ( ) is the range of
Example 146 Find the domain of each of the following functions. () ( ) = √ 2+2−9 () ( ) = √ 9−2−2−2
Functions of several variables can be combined in the same ways as func-tions of single variables
(± ) ( ) = ( ) ± ( ) ( ) ( ) = ( ) ( ) ( ) = () () ( )6= 0
If is a function of several variables and is a function of a single variable, we can form the composite function
(◦ ) ( ) = ( ( )) Example 147 ( ) = 2+ 2
− 2 + + 2 79
The graph of a function of two variable:
Example 148 What is the range of ( ) =p16− 42− 2 Describe the
graph of 5 2.5 0 -2.5 -5 5 2.5 0 -2.5 -5 0 -25 -50 -75 -100 x y z x y z = p16− 42− 2 2 = 16− 42− 2 Thus 2 4 + 2 16 + 2 16 = 1 0≤ ≤ 4 Example 149 Describe the graph of = −1812+ 12 + 8
−181 2+1 2 + 8 50 250 -25 -50 50 25 0 -25 -50 0 -50 -100 -150 -200 -250 x y z x y z 80
$ Linear Functions:
Linear functions are functions that can be expressed in the form = + +
The graph of such functions are nonvertical planes, not line.
Example 150 Describe the graph of linear function = −2 − 4 + 6
5 2.5 0 -2.5 -5 5 2.5 0 -2.5 -5 25 12.5 0 -12.5 x y z x y z 81
