Linear Approximation of a Function with two independent variables      

 

Example:  Find the linear approximation of the function, f(x,y) = x2ey near the point (1,0). 

                   Use it to evaluate the function at (1.1, 0.1).

 

 

Recall:  

                               

                    L(x,y)  =  f(a,b) +  ∂f(a,b)/∂x [ x ˗ a ]  +  ∂f(a,b)/∂y [ y ˗ b ] 

 

In this example   a = 1,  b = 0,  x = 1.1, and  y = 0.1.

                                

 

First calculate the partial derivatives.

 

    ∂f/∂x  =  2x ey      and   ∂f/∂y  =  x2ey  

 

Then evaluate them at (1,0).

 

      ∂f/∂x|(1,0)  =  2     and   ∂f/∂|(1,0)  =  1     ,  Also  f(1,0)  =  1

 

The linear approximation becomes

 

L(1.1, 0.1)  =       f(1,0) +  ∂f/∂x |(1,0)  [x - 1]  +  ∂f/∂y |(1,0) [y - 0] 

with  x = 1.1  and  y = 0 .

 

The result is:

 

 

   L(1.1,0.1)  =  1 + 2(1.1 - 1)  +  1(0.1 - 0)  =  1 + 0.3  = 1.3

 

 



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