Problem 13 - Active Calculus Multivariable

Question

The Heat Index, \(I,\) (measured in apparent degrees \(F)\) is a function of the actual temperature \(T\) outside (in degrees \(\mathrm{F}\) ) 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 \(10.2 .10 .\) $$\begin{array}{ccccc}\hline T \downarrow \backslash H \rightarrow & 70 & 75 & 80 & 85 \\ \hline 90 & 106 & 109 & 112 & 115 \\\\\hline 92 & 112 & 115 & 119 & 123 \\\\\hline 94 & 118 & 122 & 127 & 132 \\\\\hline 96 & 125 & 130 & 135 & 141 \\\\\hline\end{array}$$ a. 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. b. 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. c. 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. d. 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.

Step-by-Step Solution

Answer

In summary, we estimated the partial derivatives \(I_T(94,75) = 4\) and \(I_H(94,75) = 1\), meaning the Heat Index increases at a rate of 4 apparent degrees \(F\) per actual degree \(F\) of temperature increase and 1 apparent degree \(F\) per \(1\%\) increase in relative humidity at the given point. Using given partial derivatives, we estimated \(I(91,80) \approx 115.25\) and \(I(92,78) \approx 117.4\). Lastly, we found the average rate of change of the Heat Index between 1 p.m. and 3 p.m. to be 3.5 apparent degrees \(F\) per hour.

Step 1: State the limit definition of \(I_{T}(94,75)\)

To estimate \(I_{T}(94,75),\) we'll use the limit definition: \(I_T(94,75)=\lim_{\Delta T \to 0} \frac{I(94+\Delta T, 75) - I(94, 75)}{\Delta T}.\)

Step 2: Estimate \(I_{T}(94,75)\) using table values

As we cannot actually take the limit since we don't have an equation for the function, we can estimate the value \(I_{T}(94,75)\) by taking the difference in Heat Indices and dividing by the difference in temperatures for a small change in temperature: \(I_{T}(94,75) \approx \frac{I(96, 75) - I(94, 75)}{96-94} = \frac{130 - 122}{2} = 4.\)

Step 3: Interpret the meaning of the partial derivative

The estimated value of \(I_T(94,75) = 4\) means that when the temperature is \(94^\circ F\) and the relative humidity is \(75\%\), the Heat Index is increasing at a rate of 4 apparent degrees \(F\) per actual degree \(F\) of temperature increase. b. Estimate the partial derivative \(I_H(94,75)\).

Step 1: State the limit definition of \(I_{H}(94,75)\)

To estimate \(I_{H}(94,75),\) we'll use the limit definition: \(I_H(94,75)=\lim_{\Delta H \to 0} \frac{I(94, 75+\Delta H) - I(94, 75)}{\Delta H}.\)

Step 2: Estimate \(I_{H}(94,75)\) using table values

As before, let's use table values to estimate \(I_{H}(94,75)\): \(I_{H}(94,75) \approx \frac{I(94, 80) - I(94, 75)}{80-75} = \frac{127 - 122}{5} = 1.\)

Step 3: Interpret the meaning of the partial derivative

The estimated value of \(I_H(94,75) = 1\) means that when the temperature is \(94^\circ F\) and the relative humidity is \(75\%\), the Heat Index is increasing at a rate of 1 apparent degree \(F\) per \(1\%\) increase in relative humidity. c. Estimate the values of \(I(91,80)\) and \(I(92,78)\).

Step 1: Estimate \(I(91,80)\) using given partial derivatives

As we don't have the exact location in the table, we can use the given partial derivatives \(I_T(92,80)\) and \(I_H(92,80)\) to estimate nearby values. \(I(91,80) \approx I(92,80) - I_T(92,80) \cdot (92-91) = 119 - 3.75 = 115.25.\)

Step 2: Estimate \(I(92,78)\) using given partial derivatives

Using the same approach, let's estimate \(I(92,78)\). \(I(92,78) \approx I(92,80) - I_H(92,80) \cdot (80-78) = 119 - 2 \cdot 0.8 = 117.4.\) d. Find the average rate of change of Heat Index between 1 p.m. and 3 p.m.

Step 1: Calculate the difference in Heat Index

At 1 p.m., \(I(92,85)=123\) and at 3 p.m., \(I(96,75)=130\). The difference in Heat Index is \(\Delta I = 130 - 123 = 7\).

Step 2: Find the average rate of change

Since the time difference is 2 hours, the average rate of change of the Heat Index over this time period is: \(\frac{\Delta I}{\Delta t} = \frac{7\text{ apparent degrees }F}{2 \text{ hours}} = 3.5\). The average rate of change of the Heat Index over this time period is 3.5 apparent degrees \(F\) per hour. This means that the sensation of the Heat Index increased at an average rate of 3.5 apparent degrees \(F\) every hour between 1 p.m. and 3 p.m.

Key Concepts

Rate of Change

The concept of "Rate of Change" is fundamental in understanding how a function behaves as its input variables change. In the context of the Heat Index function, the rate of change measures how quickly the Heat Index value changes as either temperature or humidity changes. In mathematical terms, this is captured by partial derivatives.

• Partial Derivative with Respect to Temperature: The partial derivative \( I_T(94, 75) = 4 \) indicates that as the temperature increases by 1 degree Fahrenheit, while keeping humidity constant at 75%, the Heat Index increases by approximately 4 apparent degrees Fahrenheit.
• Partial Derivative with Respect to Humidity: Similarly, \( I_H(94, 75) = 1 \) means that as humidity increases by 1%, while keeping temperature constant at 94°F, the Heat Index increases by 1 apparent degree Fahrenheit.

Understanding these rates of change helps in predicting the sensation of heat under varying weather conditions.

Limit Definition

The limit definition of a derivative helps in formally understanding how a function changes as its input variables change infinitesimally. It provides a precise tool to measure the instantaneous rate of change of a function.For partial derivatives of the Heat Index \(I(T, H)\), the limit definitions are utilized as follows:

• Temperature: \( I_T(94, 75) = \lim_{\Delta T \to 0} \frac{I(94+\Delta T, 75) - I(94, 75)}{\Delta T} \) is used to describe how \(I\) changes as \(T\) changes slightly.
• Humidity: \( I_H(94, 75) = \lim_{\Delta H \to 0} \frac{I(94, 75+\Delta H) - I(94, 75)}{\Delta H} \) captures how \(I\) is affected by slight changes in \(H\).

By employing these limit definitions, we translate real-world changes in temperature and humidity into mathematical expressions of change.

Estimated Values

In practical scenarios, estimating the values of partial derivatives is often more feasible than calculating them exactly, especially when only discrete data points are available, like in the Heat Index table provided.To estimate the partial derivative \( I_T(94, 75) \):

• Look at the table values for temperatures close to 94°F with constant humidity at 75%.
• Use the nearest values: \( I(96, 75) \) and \( I(94, 75) \).
• Calculate the rate of change: \( \frac{130 - 122}{96-94} = 4 \).

For \( I_H(94, 75) \):

• Consider humidity levels close to 75% while keeping temperature at 94°F.
• Using \( I(94, 80) \) and \( I(94, 75) \), find \( \frac{127 - 122}{80-75} = 1 \).

These estimates provide crucial insights into how the Heat Index changes with temperature and humidity.

Heat Index Function

The Heat Index is a complex function that relates actual temperature and relative humidity to a perceived measure of heat discomfort. It reflects how hot the weather feels, rather than the actual air temperature, because of the humidity's effect on the body's ability to cool itself through evaporation of sweat.Key points about the Heat Index Function:

• It is defined for specific combinations of temperature \(T\) and humidity \(H\).
• The values \(I(T, H)\) in the table show how the Heat Index changes with variations in \(T\) and \(H\).
• Higher humidity makes the Heat Index higher because moisture in the air reduces evaporation, causing the body to retain heat.

This function is invaluable for weather forecasting and public health advisories, informing us about how weather will actually feel.