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Chapter 1: Problem 6

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).

Short Answer

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

Step by step solution

01

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.
02

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 \( \text{Fahrenheit degrees per milliliter} \).
03

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鈥檚 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.

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Key Concepts

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

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.

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Most popular questions from this chapter

A diver leaps from a 3 meter springboard. His feet leave the board at time \(t=0\), he reaches his maximum height of \(4.5 \mathrm{~m}\) at \(t=1.1\) seconds, and enters the water at \(t=2.45\). Once in the water, the diver coasts to the bottom of the pool (depth \(3.5 \mathrm{~m}\) ), touches bottom at \(t=7\), rests for one second, and then pushes off the bottom. From there he coasts to the surface, and takes his first breath at \(t=13\). a. Let \(s(t)\) denote the function that gives the height of the diver's feet (in meters) above the water at time \(t\). (Note that the "height" of the bottom of the pool is -3.5 meters.) Sketch a carefully labeled graph of \(s(t)\) on the provided axes in Figure 1.1.5. Include scale and units on the vertical axis. Be as detailed as possible. b. Based on your graph in (a), what is the average velocity of the diver between \(t=2.45\) and \(t=7 ?\) Is his average velocity the same on every time interval within [2.45,7]\(?\) c. Let the function \(v(t)\) represent the instantaneous vertical velocity of the diver at time \(t\) (i.e. the speed at which the height function \(s(t)\) is changing; note that velocity in the upward direction is positive, while the velocity of a falling object is negative). Based on your understanding of the diver's behavior, as well as your graph of the position function, sketch a carefully labeled graph of \(v(t)\) on the axes provided in Figure \(1.1 .6 .\) Include scale and units on the vertical axis. Write several sentences that explain how you constructed your graph, discussing when you expect \(v(t)\) to be zero, positive, negative, relatively large, and relatively small. d. Is there a connection between the two graphs that you can describe? What can you say about the velocity graph when the height function is increasing? decreasing? Make as many observations as you can.

Use linear approximation to approximate \(\sqrt{25.3}\) as follows. Let \(f(x)=\sqrt{x}\). The equation of the tangent line to \(f(x)\) at \(x=25\) can be written in the form \(y=m x+b\). Compute \(m\) and \(b\). \(m=\) \(\square\) \(b=\) \(\square\) Using this find the approximation for \(\sqrt{25.3}\). Answer: \(\square\)

The temperature, \(H\), in degrees Celsius, of a cup of coffee placed on the kitchen counter is given by \(H=f(t),\) where \(t\) is in minutes since the coffee was put on the counter. (a) Is \(f^{\prime}(t)\) positive or negative? [Choose: positive | negative] (Be sure that you are able to give a reason for your answer.) (b) What are the units of \(f^{\prime}(35)\) ?\(\square\) heln (units) Suppose that \(\left|f^{\prime}(35)\right|=1.2\) and \(f(35)=52 .\) Fill in the blanks (including units where needed) and select the appropriate terms to complete the following statement about the temperature of the coffee in this case. At \(\square\) minutes after the coffee was put on the counter, its [Choose: derivative | temperature change in temperature] is \(\square\) and will [Choose: increase | decrease] by about \(\square\) in the next 90 seconds.

The velocity of a ball that has been tossed vertically in the air is given by \(v(t)=16-32 t\), where \(v\) is measured in feet per second, and \(t\) is measured in seconds. The ball is in the air from \(t=0\) until \(t=2\). a. When is the ball's velocity greatest? b. Determine the value of \(v^{\prime}(1)\). Justify your thinking. c. What are the units on the value of \(v^{\prime}(1) ?\) What does this value and the corresponding units tell you about the behavior of the ball at time \(t=1 ?\) d. What is the physical meaning of the function \(v^{\prime}(t) ?\)

For each prompt that follows, sketch a possible graph of a function on the interval \(-3<\) \(x<3\) that satisfies the stated properties. a. \(y=f(x)\) such that \(f\) is increasing on \(-3
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