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Chapter 4: Problem 21

A dose, \(D,\) of a drug causes a temperature change, \(T,\) in a patient. For \(C\) a positive constant, \(T\) is given by $$T=\left(\frac{C}{2}-\frac{D}{3}\right) D^{2}$$ (a) What is the rate of change of temperature change with respect to dose? (b) For what doses does the temperature change increase as the dose increases?

Short Answer

Expert verified
(a) The rate of change is \(\frac{dT}{dD} = CD - D^2\). (b) Temperature change increases when \(0 < D < C\).

Step by step solution

01

Identify the function

The temperature change T is given as a function of the dose D: \[ T = \left(\frac{C}{2} - \frac{D}{3} \right) D^2 \]
02

Find the rate of change (derivative)

To find the rate of change of temperature with respect to the dose, we need to differentiate the function T with respect to D. Using the product rule:\[ \frac{dT}{dD} = \frac{d}{dD} \left( \left(\frac{C}{2} - \frac{D}{3} \right) D^2 \right) \]First, identify u and v where \(u = \left(\frac{C}{2} - \frac{D}{3} \right)\) and \(v = D^2\), thus:\[ \frac{du}{dD} = -\frac{1}{3}, \quad \frac{dv}{dD} = 2D \]Applying the product rule:\[ \frac{dT}{dD} = u \cdot \frac{dv}{dD} + v \cdot \frac{du}{dD} = \left(\frac{C}{2} - \frac{D}{3}\right) \cdot 2D + D^2 \cdot \left(-\frac{1}{3}\right) \]Simplifying gives:\[ \frac{dT}{dD} = 2D \left(\frac{C}{2} - \frac{D}{3}\right) - \frac{D^2}{3} = CD - \frac{2D^2}{3} - \frac{D^2}{3} \]\[ \frac{dT}{dD} = CD - D^2 \]
03

Determine when the temperature increases

To find when the temperature change increases as the dose increases, we need \(\frac{dT}{dD} > 0\):\[ CD - D^2 > 0 \]This simplifies to:\[ CD > D^2 \]\[ D(C - D) > 0 \]This inequality implies that 0 < D < C, meaning the temperature change increases for doses between 0 and C.

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Key Concepts

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

Rate of Change
The rate of change is an important concept in calculus that describes how one quantity changes in relation to another. In this context, we are interested in how the temperature change, denoted as \( T \), varies with the drug dose, denoted as \( D \). Understanding this relationship is crucial because it informs us about the effect different doses have on the temperature.In mathematics, the rate of change is often represented by a derivative. It provides a precise measurement of how a function responds to changes in its input variables. Here, by finding the derivative of \( T \) with respect to \( D \), we can quantify exactly how the temperature will change as the dose increments by a small amount.To compute this, we use calculus techniques like the product rule, which helps find the derivative of complex functions.
Temperature Change
In the exercise, temperature change refers to how a patient's body temperature responds to a given dose of a drug. The function provided, \( T = \left(\frac{C}{2} - \frac{D}{3}\right) D^{2} \), models this relationship.Temperature change can be affected by:
  • The dosage of the drug, which influences the function \( T \).
  • The constant \( C \), representing a factor unique to this problem that might relate to a standard temperature effect or patient's baseline.
By analyzing this function, we can predict how temperature is expected to rise or fall as the dose changes. This understanding is critical in medicine to ensure optimal dosing, providing the desired therapeutic effect without adverse reactions.
Product Rule
The product rule is a fundamental rule in calculus used for differentiating products of two functions. In our problem, we use the product rule to find the derivative of the temperature function \( T \) concerning dose \( D \).For a function that can be written as \( u \times v \), where both \( u \) and \( v \) are functions of \( D \), the product rule states that:\[ \frac{d}{dD}(u \cdot v) = u \frac{dv}{dD} + v \frac{du}{dD} \]In this exercise:
  • \( u = \left(\frac{C}{2} - \frac{D}{3}\right) \)
  • \( v = D^{2} \)
  • \( \frac{du}{dD} = -\frac{1}{3} \)
  • \( \frac{dv}{dD} = 2D \)
By applying the product rule, we derive \( \frac{dT}{dD} = CD - D^{2} \), explaining the rate at which temperature changes as a response to varying doses. This technique is an essential tool for studying how different factors interact multiplicatively.

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Most popular questions from this chapter

Give an example of a function \(f\) that makes the statement true, or say why such an example is impossible. Assume that \(f^{\prime \prime}\) exists everywhere. \(f\) is concave down and \(f(x)\) is positive for all \(x\).

Explain what is wrong with the statement. Every function of the form \(f(x)=x^{2}+b x+c,\) where \(b\) and \(c\) are constants, has two zeros.

Explain what is wrong with the statement. An increasing function has no inflection points.

In a \(19^{\text {th }}\) century sea-battle, the number of ships on each side remaining \(t\) hours after the start are given by \(x(t)\) and \(y(t) .\) If the ships are equally equipped, the relation between them is \((x(t))^{2}-(y(t))^{2}=c,\) where \(c\) is a positive constant. The battle ends when one side has no ships remaining. (a) If, at the start of the battle, 50 ships on one side oppose 40 ships on the other, what is the value of \(c ?\) (b) If \(y(3)=16,\) what is \(x(3) ?\) What does this represent in terms of the battle? (c) There is a time \(T\) when \(y(T)=0 .\) What does this \(T\) represent in terms of the battle? (d) At the end of the battle, how many ships remain on the victorious side? (e) At any time during the battle, the rate per hour at which \(y\) loses ships is directly proportional to the number of \(x\) ships, with constant of proportionality k. Write an equation that represents this, Is \(k\) positive or negative? (f) Show that the rate per hour at which \(x\) loses ships is directly proportional to the number of \(y\) ships, with constant of proportionality \(k\) (g) Three hours after the start of the battle, \(x\) is losing ships at the rate of 32 ships per hour. What is \(k ?\) At what rate is \(y\) losing ships at this time?

For any constant \(a\), let \(f(x)=a x-x \ln x\) for \(x>0\) (a) What is the \(x\) -intercept of the graph of \(f(x) ?\) (b) Graph \(f(x)\) for \(a=-1\) and \(a=1\). (c) For what values of \(a\) does \(f(x)\) have a critical point for \(x>0 ?\) Find the coordinates of the critical point and decide if it is a local maximum, a local minimum, or neither.
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