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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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BuyFindarrow_forward

Multivariable Calculus

8th Edition
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 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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P-14RECh-14 P-15RECh-14 P-16RECh-14 P-17RECh-14 P-18RECh-14 P-19RECh-14 P-20RECh-14 P-21RECh-14 P-22RECh-14 P-23RECh-14 P-24RECh-14 P-25RECh-14 P-26RECh-14 P-27RECh-14 P-28RECh-14 P-29RECh-14 P-30RECh-14 P-31RECh-14 P-32RECh-14 P-33RECh-14 P-34RECh-14 P-35RECh-14 P-36RECh-14 P-37RECh-14 P-38RECh-14 P-39RECh-14 P-40RECh-14 P-41RECh-14 P-42RECh-14 P-43RECh-14 P-44RECh-14 P-45RECh-14 P-46RECh-14 P-47RECh-14 P-48RECh-14 P-49RECh-14 P-50RECh-14 P-51RECh-14 P-52RECh-14 P-53RECh-14 P-54RECh-14 P-55RECh-14 P-56RECh-14 P-57RECh-14 P-58RECh-14 P-59RECh-14 P-60RECh-14 P-61RECh-14 P-62RECh-14 P-63RECh-14 P-64RECh-14 P-65RECh-14 P-1PCh-14 P-2PCh-14 P-3PCh-14 P-4PCh-14 P-5PCh-14 P-6PCh-14 P-7PCh-14 P-8P

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