Problem 6 - Applied Calculus

Question

A drug is injected into a patient's blood vessel. The function \(c=f(x, t)\) represents the concentration of the drug at a distance \(x\) mm in the direction of the blood flow measured from the point of injection and at time \(t\) sec. onds since the injection. What are the units of the following partial derivatives? What are their practical interpretations? What do you expect their signs to be? (a) \(\partial c / \partial x\) (b) \(\partial c / \partial t\)

Step-by-Step Solution

Answer

(a) Units: mg/mL/mm; likely negative. (b) Units: mg/mL/s; likely negative.

Step 1: Understand the Context

Understand that we have a function \( c = f(x, t) \) that represents the concentration of a drug. Here, \( x \) is the distance in millimeters from the injection point in the direction of blood flow, and \( t \) is the time in seconds since the injection.

Step 2: Identifying Units: \( \frac{\partial c}{\partial x} \)

\( \frac{\partial c}{\partial x} \) is the rate of change of concentration with respect to distance. The units of \( c \) are concentration (perhaps mg/mL), and the units of \( x \) are millimeters (mm). Therefore, the units of \( \frac{\partial c}{\partial x} \) are concentration per millimeter (e.g., mg/mL/mm).

Step 3: Interpreting \( \frac{\partial c}{\partial x} \)

This derivative represents how the concentration changes per unit distance. A negative \( \frac{\partial c}{\partial x} \) would indicate that concentration decreases with distance, which is likely due to the drug diffusing away from the injection site.

Step 4: Identifying Units: \( \frac{\partial c}{\partial t} \)

\( \frac{\partial c}{\partial t} \) is the rate of change of concentration with respect to time. The units of \( c \) are concentration (e.g., mg/mL), and the units of \( t \) are seconds. Therefore, the units of \( \frac{\partial c}{\partial t} \) are concentration per second (mg/mL/s).

Step 5: Interpreting \( \frac{\partial c}{\partial t} \)

This derivative represents how the concentration changes over time. A negative \( \frac{\partial c}{\partial t} \) suggests that the concentration of the drug is decreasing with time, possibly due to metabolism or elimination from the bloodstream.

Step 6: Expectation of Signs

For \( \frac{\partial c}{\partial x} \), we expect a negative sign if the drug concentration is decreasing with distance due to diffusion. For \( \frac{\partial c}{\partial t} \), a negative sign is expected if the concentration decreases over time, as the drug is metabolized or cleared from the bloodstream.

Key Concepts

Partial Derivatives

Partial derivatives are a way to measure how a function changes as one of its variables changes, while keeping the other variables constant. In applied calculus, they are especially useful in multi-variable functions like the one given in the exercise.
When we talk about \(\partial c / \partial x\), we are measuring how the concentration of a drug (\