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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

For what values of the number r is the function f ( x , y , z ) = { ( x + y + z ) r x 2 + y 2 + z 2 if  ( x , y , z ) ( 0 , 0 , 0 ) 0 if ( x , y , z ) = ( 0 , 0 , 0 )

continuous on R3?

To determine

To find: The value of r for which the function f(x,y)={(x+y+z)rx2+y2+z2if(x,y,z)00if(x,y,z)=0 is continuous on 3 .

Explanation

Definition used:

A function f of two variables is continuous at (a,b) if lim(x,y)(a,b)f(x,y)=f(a,b) .

Result used:

“The inequality (a+b+c)24a2+4b2+4c2 .”

Calculation:

The given function is, f(x,y)={(x+y+z)rx2+y2+z2if(x,y,z)00if(x,y,z)=0 .

Condition 1:

If the function is continuous on 3 then by the definition of continuous,

lim(x,y,z)(0,0,0)f(x,y,z)=f(0,0,0)=0

Thus, the value of r should satisfy the condition 1.

From the given result,

(x+y+z)24x2+4y2+4z2(x+y+z)24(x2+y2+z2)

Consider the rational fraction (x+y+z)rx2+y2+z2 ,

The domain of this rational (x+y+z)rx2+y2+z2 is {(x,y,z)|(x,y,z)(0,0,0)} .

Consider (x+y+z)r , then its abosute value is, |(x+y+z)r|=(|(x+y+z)2|)r2 .

Now by the result the above equation can be expressed as,

(|(x+y+z)2|)r2=[(x+y+z)2]r2[(x+y+z)2]r2[4(x2+y2+z2)]r2[(x+y+z)2]r2(4)r2(x2+y2+z2)r2[(x+y+z)2]r22r(x2+y2+z2)r2

Compute the value of |f(x,y,z)0| for (x,y,z)(0,0,0) as follows,

|f(x,y,z)0|=|(x+y+z)rx2+y2+z2|=|(x+y+z)r|x2+y2+z2

Substitute the respective values in the above equation,

|f(x,y,z)0|=|(x+y+z)r|x2+y2+z22r(x2+y2+z2)r2x2+y2+z22r(x2+y2+z2)r2(x2+y2+z2)12r(x2+y2+z2)r21

The value of |f(x,y,z)0|=2r(x2+y2+z2)r21

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Chapter 14 Solutions

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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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