James Stewart

ISBN: 9781305266643

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 ε1 and ε2 by using Definition 7.

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 ε1 and ε2→0 .as (Δx,Δy)→(0,0) .”

Calculation:

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

Take the partial derivatives with respect to x and y 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)=2x and fy(x,y)=2y .

Compute the value of ε1 and ε2 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

Check out a sample textbook solution.

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Solutions

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Show that the function is differentiable by finding values of ɛ1 and ɛ2 that satisfy Definition 7. 43. f(x, y) = x2 + y2

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Chapter 14 Solutions

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Math
Calculus Multivariable Calculus 8th Edition

Show that the function is differentiable by finding values of ɛ₁ and ɛ₂ that satisfy Definition 7.

43. f(x, y) = x² + y²