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Chapter 3: Problem 60

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.

Short Answer

Expert verified
\(\frac{dR}{dM} = CM - M^2\).

Step by step solution

01

Understand the Problem

The function we have is given by \( R = M^{2} \left( \frac{C}{2} - \frac{M}{3} \right) \). We need to find the derivative of \( R \) with respect to \( M \), which is denoted as \( \frac{dR}{dM} \). This requires the application of certain differentiation rules.
02

Expand the Function

Before differentiating, let's expand the function \( R \). Distribute \( M^2 \) across the terms inside the parentheses: \[ R = M^2 \cdot \left( \frac{C}{2} - \frac{M}{3} \right) = \frac{CM^2}{2} - \frac{M^3}{3} \].
03

Differentiate Each Term

Differentiate \( R = \frac{CM^2}{2} - \frac{M^3}{3} \) with respect to \( M \). 1. Differentiate the first term: \( \frac{d}{dM}\left( \frac{CM^2}{2} \right) = C \cdot M \). 2. Differentiate the second term: \( \frac{d}{dM}\left( \frac{M^3}{3} \right) = M^2 \).
04

Combine the Derivatives

Combine the results from differentiating each term to get the total derivative: \[ \frac{dR}{dM} = C \cdot M - M^2 \]. Thus, the sensitivity of the body to the medicine, expressed as a function of \( M \), is \( C \cdot M - M^2 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sensitivity Analysis
Sensitivity analysis is crucial in understanding how a small change in one variable can influence the outcome of a model or system. In the context of medicine, it helps us understand how adjustments in the dose of a drug can affect the body's reaction.
The sensitivity of the body to a particular medicine means how the body responds to different doses. For any medicine, determining this sensitivity is important because it tells us the optimal dose needed to achieve the desired therapeutic effect without causing harm.
In this exercise, the sensitivity is represented by the derivative \( \frac{dR}{dM} \), showing how the body's reaction \( R \) changes with respect to the medicine dose \( M \). Understanding this relationship helps doctors and pharmacists in deciding the right dose for patients.
Differentiation
Differentiation is a technique in calculus used to find the rate at which one quantity changes with respect to another.
By calculating the derivative, we obtain a function that tells us the slope of the tangent line to the graph of the original function at any point. In simple terms, it shows how a function is increasing or decreasing as its input changes.
In this exercise, the function \( R = M^2 \left( \frac{C}{2} - \frac{M}{3} \right) \) represents the reaction of the body to medicine. Differentiation helps us find \( \frac{dR}{dM} \), which represents the sensitivity of reaction \( R \) with respect to medicine \( M \).
Through differentiation, we explore how even small increases in dose \( M \) can significantly affect the reaction \( R \), which is a crucial consideration when prescribing medicines.
Medicine Dosage
Medicine dosage is a critical aspect of both pharmaceutical practice and medical treatment.
Determining the right amount of medication is essential to achieve the therapeutic effect without causing adverse effects or toxicity.
Understanding the concept of dosage involves appreciating how the concentration of the drug in the bloodstream (\( M \)) influences the desired effect (\( R \)). In this exploration, we use the equation \( R = M^2 \left( \frac{C}{2} - \frac{M}{3} \right) \) to model this relationship.
Doctors rely on sensitivity analyses and derivatives to guide them in prescribing the most appropriate dosage, ensuring that the body achieves maximum benefit from the medicine while minimizing risks. Effective dosage strategies are paramount to successful treatment outcomes in various medical fields.

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Most popular questions from this chapter

A spherical iron ball \(8 \mathrm{cm}\) in diameter is coated with a layer of ice of uniform thickness. If the ice melts at the rate of \(10 \mathrm{cm}^{3} / \mathrm{min},\) how fast is the thickness of the ice decreasing when it is \(2 \mathrm{cm}\) thick? How fast is the outer surface area of ice decreasing?

In the late 1860 s, Adolf Fick, a professor of physiology in the Faculty of Medicine in W眉rzberg, Germany, developed one of the methods we use today for measuring how much blood your heart pumps in a minute. Your cardiac output as you read this sentence is probably about \(7 \mathrm{L} / \mathrm{min} .\) At rest it is likely to be a bit under \(6 \mathrm{L} / \mathrm{min}\). If you are a trained marathon runner running a marathon, your cardiac output can be as high as \(30 \mathrm{L} / \mathrm{min}\). Your cardiac output can be calculated with the formula $$y=\frac{Q}{D},$$ where \(Q\) is the number of milliliters of \(\mathrm{CO}_{2}\) you exhale in a minute and \(D\) is the difference between the \(\mathrm{CO}_{2}\) concentration \((\mathrm{mL} / \mathrm{L})\) in the blood pumped to the lungs and the \(\mathrm{CO}_{2}\) concentration in the blood returning from the lungs. With \(Q=233 \mathrm{mL} / \mathrm{min}\) and \(D=97-56=41 \mathrm{mL} / \mathrm{L}\), \(y=\frac{233 \mathrm{mL} / \mathrm{min}}{41 \mathrm{mL} / \mathrm{L}} \approx 5.68 \mathrm{L} / \mathrm{min}\), fairly close to the \(6 \mathrm{L} / \mathrm{min}\) that most people have at basal (resting) conditions. (Data courtesy of J. Kenneth Herd, M.D., Quillan College of Medicine, East Tennessee State University.) Suppose that when \(Q=233\) and \(D=41,\) we also know that \(D\) is decreasing at the rate of 2 units a minute but that \(Q\) remains unchanged. What is happening to the cardiac output?

Find \(d y / d t\) $$y=4 \sin (\sqrt{1+\sqrt{t}})$$

Find \(y^{\prime \prime}\) $$y=9 \tan \left(\frac{x}{3}\right)$$

A highway patrol plane flies 3 km above a level, straight road at a steady \(120 \mathrm{km} / \mathrm{h}\). The pilot sees an oncoming car and with radar determines that at the instant the line-of-sight distance from plane to car is \(5 \mathrm{km},\) the line-of-sight distance is decreasing at the rate of \(160 \mathrm{km} / \mathrm{h}\). Find the car's speed along the highway.
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