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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

At the beginning of this section we discussed the function I = f(T, H), where I is the heat index, T is the temperature, and H is the relative humidity. Use Table 1 to estimate fT(92, 60) and fH(92, 60). What are the practical interpretations of these values?

Table 1 Heat index I as a function of temperature and humidity

images

To determine

To estimate: The values of fT(92,60) and fH(92,60) from the Table 1, interpret the values.

Explanation

Given:

The function I=f(T,H) , where I is the heat index, T is the temperature and H is the relative humidity.

Definition used:

“If f is a function of two variables, its partial derivatives are the functions fx and fy defined by

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

Calculation:

Obtain fT(92,60) by definition.

fT(92,60)=limh0f(92+h,60)f(92,60)h

Assume h=2 and obtain the value of fT(92,60) .

fT(92,60)f(92+2,60)f(92,60)2f(94,60)f(92,60)2=1111052=62=3

Similarly, assume h=2 and use Table 1 to obtain the values of fT(92,60) .

fT(92,60)f(922,60)f(92,60)2f(90,60)f(92,60)2=1001052=52=2.5

Obtain the average of above two values of fT(92,60) ,

fT(92,60)=3+522=6+522=1122=114

Thus, the value of fT(92,60)=2

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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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