Problem 6 - Active Calculus

Question

The temperature change \(T\) (in Fahrenheit degrees), in a patient, that is generated by a dose \(q\) (in milliliters), of a drug, is given by the function \(T=f(q)\). a. What does it mean to say \(f(50)=0.75 ?\) Write a complete sentence to explain, using correct units. b. A person's sensitivity, \(s,\) to the drug is defined by the function \(s(q)=f^{\prime}(q) .\) What are the units of sensitivity? c. Suppose that \(f^{\prime}(50)=-0.02 .\) Write a complete sentence to explain the meaning of this value. Include in your response the information given in (a).

Step-by-Step Solution

Answer

At a 50 ml dose, the temperature increases by 0.75°F. Sensitivity has units of Fahrenheit degrees per milliliter. A 50 ml dose decreases temperature at a rate of 0.02°F per ml.

Step 1: Understanding the Function Value - Step 1

The expression \( f(50) = 0.75 \) means that when the dose of the drug, \( q \), is 50 milliliters, the patient's temperature change \( T \) is 0.75 Fahrenheit degrees. In other words, administering a dose of 50 milliliters of the drug results in an increase of 0.75 degrees Fahrenheit in the patient's temperature.

Step 2: Sensitivity Units - Step 2

To determine the units of sensitivity \( s(q) \), which is defined as \( f'(q) \), note that \( f'(q) \) represents the rate of change of temperature with respect to the dose. Since \( T \) is measured in Fahrenheit degrees and \( q \) is measured in milliliters, the units of \( f'(q) \) (and hence sensitivity \( s(q) \)) are \( \text{Fahrenheit degrees per milliliter} \).

Step 3: Interpreting the Derivative Value - Step 3

The expression \( f'(50) = -0.02 \) indicates that at a dose of 50 milliliters, the patient's temperature decreases at a rate of 0.02 Fahrenheit degrees per milliliter. This means that increasing the dose from 50 milliliters by one milliliter will decrease the patient's temperature by 0.02 degrees Fahrenheit. Combining this with the information given in (a), even though the patient’s temperature increases to 0.75 degrees Fahrenheit at a 50 milliliter dose, any further increase in the dose will cause the temperature to start decreasing.

Key Concepts

Function Interpretation

In this problem, we encountered the function \(T = f(q)\), which describes the change in a patient's temperature \(T\) (measured in Fahrenheit degrees) depending on the dose \(q\) (measured in milliliters) of a drug. When we say \(f(50) = 0.75\), it means that administering a dose of 50 milliliters of the drug causes the patient's temperature to increase by 0.75 degrees Fahrenheit. This interpretation is crucial because it tells us precisely how a particular dose affects temperature, which can be vital for determining the correct dose for patients.

Sensitivity Analysis

Sensitivity in this context is defined by the function \(s(q) = f'(q)\). Sensitivity analysis is used to determine how sensitive the patient's temperature is to changes in the drug dose. The unit of sensitivity is derived from the derivative \(f'(q)\).

Since \(f(q)\) represents temperature in Fahrenheit degrees and \(q\) represents the dose in milliliters, the derivative \(f'(q)\) tells us the rate of change in temperature per unit change in the dose. Therefore, the unit of sensitivity \(s(q)\) is Fahrenheit degrees per milliliter. This means we are looking at how much a small change in the drug dose will impact the patient's temperature.

Knowing the sensitivity can help doctors make better decisions about adjusting drug doses to achieve the desired temperature change in patients.

Derivative Interpretation

The derivative \(f'(q)\) shows us how the temperature change rate varies with the dose. When \(f'(50) = -0.02\), it means that at a dose of 50 milliliters, the patient's temperature is decreasing at a rate of 0.02 Fahrenheit degrees per milliliter. This is a negative value, indicating a decrease in temperature.

Combining this with our previous information that \(f(50) = 0.75\), we understand that while a 50 milliliter dose initially increases the patient's temperature by 0.75 degrees Fahrenheit, any increase in dose beyond 50 milliliters will start to decrease the temperature. This highlights the importance of not just knowing the effect of a specific dose but also understanding how small changes in dose can impact the patient's temperature.

By interpreting derivatives, healthcare professionals can better manage doses to achieve the optimal therapeutic effect, avoiding potential adverse effects from incorrect dosing.