Exercise 32

The Heat Index, $I$, (measured in apparent degrees F) is a function of the actual temperature $T$ outside (in degrees F) and the relative humidity $H$ (measured as a percentage). A portion of the table which gives values for this function, $I=I(T,H)$, is reproduced in Table.

1. State the limit definition of the value $I_T(94,75)$. Then, estimate $I_T(94,75)$, and write one complete sentence that carefully explains the meaning of this value, including its units.

2. State the limit definition of the value $I_H(94,75)$. Then, estimate $I_H(94,75)$, and write one complete sentence that carefully explains the meaning of this value, including its units.

3. Suppose you are given that $I_T(92,80) = 3.75$ and $I_H(92,80) = 0.8$. Estimate the values of $I(91,80)$ and $I(92,78)$. Explain how the partial derivatives are relevant to your thinking.

4. On a certain day, at 1 p.m. the temperature is 92 degrees and the relative humidity is 85%. At 3 p.m., the temperature is 96 degrees and the relative humidity 75%. What is the average rate of change of the heat index over this time period, and what are the units on your answer? Write a sentence to explain your thinking.