To solve the problem, we need to calculate the partial derivatives of the wind chill function C(T,w) using the given table and the empirical formula. For the table, we will use the forward, backward, and symmetric difference quotients to approximate the partial derivatives CT(15,25) and Cw(15,25). For the empirical formula, we will compute the partial derivatives analytically.
- Forward Difference Quotient for CT(15,25): Use the values from the table for T=15 and T=25 with w=25.
- Backward Difference Quotient for CT(15,25): Use the values from the table for T=15 and T=10 with w=25.
- Symmetric Difference Quotient for CT(15,25): Average the forward and backward difference quotients.
- Forward Difference Quotient for Cw(15,25): Use the values from the table for w=25 and w=30 with T=15.
- Backward Difference Quotient for Cw(15,25): Use the values from the table for w=25 and w=20 with T=15.
- Symmetric Difference Quotient for Cw(15,25): Average the forward and backward difference quotients.
- Empirical Formula Partial Derivatives: Differentiate the empirical formula with respect to T and w and evaluate at T=15 and w=25.
Using the values from the table, the forward difference quotient is calculated as follows:
CT(15,25)≈25−15C(25,25)−C(15,25)=103−(−4)=107=0.7
The backward difference quotient is calculated using:
CT(15,25)≈15−10C(15,25)−C(10,25)=5−4−(−5)=51=0.2
The symmetric difference quotient is the average of the forward and backward quotients:
CT(15,25)≈20.7+0.2=20.9=0.45
For the wind speed, the forward difference quotient is:
Cw(15,25)≈30−25C(15,30)−C(15,25)=5−5−(−4)=5−1=−0.2
The backward difference quotient is calculated as:
Cw(15,25)≈25−20C(15,25)−C(15,20)=5−4−(−2)=5−2=−0.4
The symmetric difference quotient is:
Cw(15,25)≈2−0.2+(−0.4)=2−0.6=−0.3
Using the empirical formula:
CT(15,25)≈1.2113
Cw(15,25)≈−0.1619
The results for the partial derivatives are:
CT(15,25) (Forward): 0.7
CT(15,25) (Backward): 0.2
CT(15,25) (Symmetric): 0.45
Cw(15,25) (Forward): −0.2
Cw(15,25) (Backward): −0.4
Cw(15,25) (Symmetric): −0.3
Empirical CT(15,25): 1.2113
Empirical Cw(15,25): −0.1619
Thus, the final boxed answers are:
CT(15,25) (Forward)=0.7,CT(15,25) (Backward)=0.2,CT(15,25) (Symmetric)=0.45
Cw(15,25) (Forward)=−0.2,Cw(15,25) (Backward)=−0.4,Cw(15,25) (Symmetric)=−0.3
CT(15,25) (Empirical)=1.2113,Cw(15,25) (Empirical)=−0.1619