   Chapter 14.4, Problem 23E

Chapter
Section
Textbook Problem

Use the table in Example 3 to find a linear approximation to the heat index function when the temperature is near 94°F and the relative humidity is near 80%. Then estimate the heat index when the temperature is 95°F and the relative humidity is 78%. To determine

To find: The linear approximation of the heat index function I=f(T,H) at the temperature is near 94F and the relative humidity is near 80% , use this approximation and approximate the value of f(T,H)=f(95,78) .

Explanation

Given:

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

Calculation:

The linear approximation of the function I=f(T,H) at (94,80) is defined as,

f(94,80)f(94,80)+fT(94,80)(T94)+fH(94,80)(H80) . (1)

Find the value of fT(94,80) as follows,

fT(94,80)=limh0f(94+h,80)f(94,80)h

Let h=2 in fT(94,80) ,

fT(94,80)=limh0f(94+2,80)f(94,80)2=limh0f(96,80)f(94,80)2=13512710=4

Let h=2 in fT(94,80)=limh0f(94+h,80)f(94,80)h ,

fT(94,80)=limh0f(942,80)f(94,80)2=limh0f(92,80)f(94,80)2=1191272=4

Compute the average value of fT(94,80)=4 and fT(94,80)=4 .

fT(94,80)=4+42=82=4

Thus, the value of fT(94,80)=4 .

The value of fH(94,80) is defined as fH(94,80)=limh0f(94,80+h)f(94,80)h

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