   Chapter 14.4, Problem 43E

Chapter
Section
Textbook Problem

Show that the function is differentiable by finding values of ɛ1 and ɛ2 that satisfy Definition 7.43. f(x, y) = x2 + y2

To determine

To show: The function f(x,y)=x2+y2 is differentiable by obtain the values of ε1andε2 by using Definition 7.

Explanation

Given:

The function is, f(x,y)=x2+y2 .

Definition 7:

“If z=f(x,y) , then f is differentiable at (a,b) if Δz can be expressed in the form

Δz=fx(a,b)Δx+fy(a,b)Δy+ε1Δx+ε2Δy , where ε1andε20 .as (Δx,Δy)(0,0) .”

Calculation:

Consider the function z=f(x,y)=x2+y2 .

Take the partial derivatives with respect to xandy in the function z=f(x,y)=x2+y2 .

fx(x2+y2)=2x+0=2xfy(x2+y2)=0+2y=2y

Thus, the value of fx(x,y)=2xandfy(x,y)=2y .

Compute the value of ε1andε2 by using Definition 7 as follows.

Δz=fx(x,y)Δx+fy(x,y)Δy+ε1Δx+ε2Δy (1)

The value of Δz can be expressed as follows

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