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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

Use the table of values of f(x, y) to estimate the values of fx(3, 2), fx(3, 2.2),and fxy(3, 2).

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To determine

To estimate: The values of fx(3,2) , fx(3,2.2) and fxy(3,2) by using the table of values of f(x,y) .

Explanation

Definition used:

If f is a function of two variables, its partial derivatives fx and fy are defined as a limit by,

fx(x,y)=limh0f(x+h,y)f(x,y)hfy(x,y)=limh0f(x,y+h)f(x,y)h

Calculation:

By definition, fx(3,2)=limh0f(3+h,2)f(3,2)h .

Assume h=0.5 and use the given Table to obtain the value of fx(3,2) .

fx(3,2)f(3+0.5,2)f(3,2)0.5f(3.5,2)f(3,2)0.5=22.417.50.5=4.90.5=9.8

Then, assume h=0.5 and use the given Table to obtain the values of fx(3,2) .

fx(3,2)f(30.5,2)f(3,2)0.5f(2.5,2)f(3,2)0.5=10.217.50.5=7.30.5=14.6

Obtain the average of above two values of fx(3,2) ,

fx(3,2)=9.8+14.62=24.42=12.2

Thus, the value of fx(3,2)=12.2 .

The partial derivative fxy(3,2) is defined as,

fxy(3,2)=y[fx(3,2)]=limh0fx(x,y+h)fx(x,y)h=limh0fx(3,2+h)fx(3,2)h

If h=0.2 , then fxy(3,2)=limh0fx(3,2.2)fx(3,2)h . (1)

If h=0.2 , then fxy(3,2)=limh0fx(3,1.8)fx(3,2)h . (2)

By definition, fx(3,2.2)=limh0f(3+h,2.2)f(3,2.2)h .

Assume h=0.5 and use the given Table to obtain the value of fx(3,2.2) .

fx(3,2.2)f(3+0.5,2.2)f(3,2

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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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