1 Functions of several variables

We know that f ( x ) is used to represent a function of one variable: the input variable is x and the output is the value f ( x ) . Here x is the independent variable and y = f ( x ) is the dependent variable .

Suppose we consider a function with two independent input variables x and y , for example

f ( x , y ) = x + 2 y + 3.

If we specify values for x and y then we have a single value f ( x , y ) . For example, if x = 3 and y = 1 then f ( x , y ) = 3 + 2 + 3 = 8 . We write f ( 3 , 1 ) = 8 .

Task!

Find the values of f ( 2 , 1 ) , f ( 1 , 3 ) and f ( 0 , 0 ) for the following functions.

  1. f ( x , y ) = x 2 + y 2 + 1
  2. f ( x , y ) = 2 x + x y + y 3
  1. f ( 2 , 1 ) = 2 2 + 1 2 + 1 = 6 ; f ( 1 , 3 ) = ( 1 ) 2 + ( 3 ) 2 + 1 = 11 ; f ( 0 , 0 ) = 1
  2. f ( 2 , 1 ) = 4 + 2 + 1 = 7 ; f ( 1 , 3 ) = 2 + 3 27 = 26 ; f ( 0 , 0 ) = 0

In a similar way we can define a function of three independent variables. Let these variables be x , y and u and the function f ( x , y , u ) .

Example 1

Given f ( x , y , u ) = x 2 + y u + 2 , find f ( 0 , 1 , 0 ) , f ( 1 , 1 , 2 ) .

Solution

f ( 0 , 1 , 0 ) = 0 2 + 1 × 0 + 2 = 2 ; f ( 1 , 1 , 2 ) = 1 2 + 2 = 1

Task!
  1. Find f ( 2 , 1 , 1 ) for f ( x , y , u ) = x y + y u + u x .
  2. Evaluate f ( x , y , u , t ) = x 2 y 2 u 2 2 t when x = 1 , y = 2 , u = 3 , t = 1.
  1. f ( 2 , 1 , 1 ) = 2 × ( 1 ) + ( 1 ) × 1 + 1 × 2 = 1
  2. f ( 1 , 2 , 3 , 1 ) = 1 2 ( 2 ) 2 3 2 2 × 1 = 14 (this is a function of 4 independent variables).