   Chapter 14.4, Problem 24E

Chapter
Section
Textbook Problem

The wind-chill index W is the perceived temperature when the actual temperature is T and the wind speed is v, so we can write W = f(T, v). The following table of values is an excerpt from Table 1 in Section 14.1. Use the table to find a linear approximation to the wind-chill index function when T is near − 15°C and v is near 50 km/h. Then estimate the wind-chill index when the temperature is −17°C and the wind speed is 55 km/h. To determine

To find: The linear approximation of the wind-chill index function W=f(T,v) at actual temperature near 15C and the wind speed is near 50km/h , use this approximation and approximate the value of f(T,v)=f(17,55) .

Explanation

Given:

The function is, W=f(T,v) where T is actual temperature, v is wind speed.

Calculation:

The linear approximation of the function W=f(T,v) at (15,50) is defined as,

f(15,50)f(15,50)+fT(15,50)(T+15)+fv(15,50)(v50) . (1)

The value of fT(15,50) is defined as, fT(15,50)=limh0f(15+h,50)f(15,50)h

Let h=5 in the above equation,

fT(15,50)=limh0f(15+5,50)f(15,50)5=limh0f(10,50)f(15,50)5=22+295=1.4

Let h=5 and find fT(15,50) ,

fT(15,50)=limh0f(155,50)f(15,50)5=limh0f(20,50)f(15,50)5=27+295=1.2

Compute the average value of, fT(15,50)=1.4 and fT(15,50)=1.2 .

fT(15,50)=1.4+1.22=2.62=1.3

Thus, the value of fT(15,50)=1.3 .

Similarly, the function fv(15,50) is defined as, fv(15,50)=limh0f(15,50+h)f(15,50)h .

Let h=10 and use table to find fv(15,50) .

fv(15,50)=limh0f(15,50+10)f(15,50)10=limh0f(15,60)f(15,50)10=30+2910=0

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