Problem 72 - University Calculus: Early Transcendentals

Question

The reaction of the body to a dose of medicine can sometimes be represented by an equation of the form $$R=M^{2}\left(\frac{C}{2}-\frac{M}{3}\right)$$ where \(C\) is a positive constant and \(M\) is the amount of medicine absorbed in the blood. If the reaction is a change in blood pressure, \(R\) is measured in millimeters of mercury. If the reaction is a change in temperature, \(R\) is measured in degrees, and so on. Find \(d R / d M .\) This derivative, as a function of \(M,\) is called the sensitivity of the body to the medicine. In Section \(4.5,\) we will see how to find the amount of medicine to which the body is most sensitive.

Step-by-Step Solution

Answer

The derivative \( \frac{dR}{dM} = CM - M^2 \) measures the sensitivity of the reaction to medicine.

Step 1: Write down the given equation

The equation given to represent the reaction of the body as a function of the amount of medicine absorbed is: \( R = M^2 \left( \frac{C}{2} - \frac{M}{3} \right) \).

Step 2: Expand the equation

Expand the expression inside the parentheses: \[ R = M^2 \cdot \frac{C}{2} - M^2 \cdot \frac{M}{3} \]which simplifies to:\[ R = \frac{CM^2}{2} - \frac{M^3}{3} \].

Step 3: Differentiate with respect to M

Find the derivative of \( R \) with respect to \( M \). Differentiate each term separately:\[ \frac{dR}{dM} = \frac{d}{dM} \left( \frac{CM^2}{2} \right) - \frac{d}{dM} \left( \frac{M^3}{3} \right) \].The first term's derivative is: \( C \cdot M \), since \( \frac{d}{dM}(M^2) = 2M \) and you multiply this by constant \( C/2 \).The second term's derivative is: \( M^2 \), since \( \frac{d}{dM}(M^3) = 3M^2 \) and divide by 3.Thus: \[ \frac{dR}{dM} = C \cdot M - M^2 \].

Step 4: Finalize the derivative

Combine the terms from Step 3 to write the final expression for the derivative:\[ \frac{dR}{dM} = CM - M^2 \].

Key Concepts

Derivative

In calculus, the derivative is a fundamental concept that measures how a function changes as its input changes. The derivative is, in essence, the "rate of change" of the function. For our exercise, the goal was to find the derivative of the function \( R = M^2 \left( \frac{C}{2} - \frac{M}{3} \right) \) with respect to the variable \( M \). This involves:

• Expanding the function to simplify the differentiation process.
• Applying differentiation rules, such as the power rule, to find the rate of change.

The power rule is paramount here, saying if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \). Using this, we differentiate each term of the expanded expression \( R = \frac{CM^2}{2} - \frac{M^3}{3} \). The derivative \( \frac{dR}{dM} = CM - M^2 \) shows how sensitive \( R \) is to changes in \( M \).

Sensitivity Analysis

Sensitivity analysis is a method used to predict how different values of an independent variable affect a particular dependent variable under a given set of assumptions. Essentially, it tells us how the output changes in response to changes in input. In this exercise, after finding the derivative \( \frac{dR}{dM} = CM - M^2 \), we observe how sensitive the body's reaction \( R \) is to the amount of medicine \( M \).

• \( CM \) suggests that the reaction is directly proportional to \( M \) initially.
• \( - M^2 \) indicates diminishing sensitivity as \( M \) increases significantly, balancing the initial boost provided by \( CM \).

This derivative function describes the sensitivity and helps in identifying optimal medicine levels. In Section 4.5, as stated, we'll explore how to determine the exact amount \( M \) that maximizes sensitivity.

Mathematical Modeling

Mathematical modeling involves creating abstract models using mathematical language to describe real-world systems. In this exercise, the equation \( R = M^2 \left( \frac{C}{2} - \frac{M}{3} \right) \) models how the body's reaction, such as blood pressure or temperature, changes with varying medicine doses.Key aspects of this modeling include:

• Utilizing equations to reflect biological processes, like drug absorption.
• Incorporating constants like \( C \) to account for various biological conditions or attributes.
• Predicting outcomes or reactions based on variable inputs, exemplified through finding the sensitivity \( \frac{dR}{dM} \).

Mathematical models serve as powerful tools in assessing hypothetical scenarios, optimizing medical doses, and studying complex biochemical reactions in a structured, quantitative manner.