Question

Drug Concentration The concentration \(C\) of a medication in the bloodstream \(t\) minutes after sublingual (under the tongue) application is given by \(C(t)=\frac{3 t-1}{2 t^{2}+5}, \quad t>0\) (a) Use a graphing utility to graph the function. Estimate when the concentration is greatest. (b) Does this function have a horizontal asymptote? If so, discuss the meaning of the asymptote in terms of the concentration of the medication.

Short answer

Step 1 would provide an estimate for when the maximum concentration occurs, but the actual timing can be confirmed by finding the maximum point using the derivative in Steps 2 and 3. The presence and the meaning of the horizontal asymptote will come out clear after Steps 4 and 5.

Step-by-step solution

Graph the function

Firstly, we need to graph the function \(C(t)=\frac{3 t-1}{2 t^{2}+5}\). The graph will give a visual representation and an estimate of the time when the concentration is at its greatest.

Compute the derivative of the function

Let's compute the derivative of \(C(t)\). This will help us in knowing when the concentration of the medication is increasing or decreasing. The derivative is computed as: \[C'(t)=\frac{[2t^2+5](3)-(3t-1)(4t)}{(2t^2+5)^2}\]Simplify the above expression to find actual derivative function \(C'(t)\).

Identify critical points

Critical points occur at values of \(t\) where either \(C'(t)\) equals zero or does not exist. Solve the equation \(C'(t) = 0\) to find these points. These are the potential positions for which the maximum concentration can occur.

Compute the limit as t approaches infinity

We need to compute the limit of the function \(C(t)\) as \(t\rightarrow\infty\). This would show whether our function has a horizontal asymptote and the position of the possible asymptote. Heuristically, divide both the numerator and denominator by \(t^{2}\) and compute the limit once again to determine whether \(C(t)\) has an horizontal asymptote.

Interpret the result

Based on your calculations for maximum concentration and the asymptote, interpret the result in terms of the concentration of the medication over time.

Key concepts

Graphing Functions

Understanding the visual representation of mathematical functions is crucial for comprehending their behavior. Graphing the function, in this instance, involves plotting the medication concentration over time, with the concentration, denoted as C(t), on the y-axis, and time, t, on the x-axis. By using a graphing utility or software, students can easily observe how the concentration changes as time progresses. The peak of the graph indicates the maximum concentration level of the medication in the bloodstream after application. Graphs serve as valuable tools for predicting and visualizing the outcomes of mathematically expressed situations.

While discussing graphing, it's important to keep scales in mind. Equally spaced intervals on axes ensure accurate representation. Also, carefully analyzing the graph can give insights into the behavior of the concentration, such as how quickly it reaches its peak and how it stabilizes or decreases over time. This understanding is fundamental to applying calculus in practical scenarios.

Calculating Derivatives

The derivative is a powerful tool in calculus, offering the rate of change of a function at any given point. In our context, calculating the derivative of the concentration function, C(t), tells us how the concentration of medication in the bloodstream is changing at each moment in time. By evaluating the derivative, stated as C'(t), we can determine when the rate of change is positive, meaning the concentration is increasing, and when it's negative, which indicates the concentration is decreasing.

The process of finding the derivative, also called differentiation, involves applying rules of calculus, such as the quotient rule, which was used to find C'(t) in the step-by-step solution. Understanding how to carry out this calculation is essential for analyzing the behavior of functions, making it a cornerstone of calculus.

Critical Points in Calculus

A critical point in calculus is where the first derivative of a function is either zero or undefined, indicating a potential maximum or minimum value of the function. For the medication concentration example, identifying the critical points of C(t) helps us find when the concentration reaches its peak. This point is of particular importance in medical applications, as it might correspond to the most effective concentration of medication in the bloodstream.

By setting the derivative equal to zero, C'(t) = 0, and solving for t, we discover these critical points. A further investigation involving the second derivative or a sign analysis can confirm whether it's a maximum or minimum. Understanding critical points can also aid in predicting the behavior of differentiable functions, making it an essential aspect of calculus.

Horizontal Asymptotes

Horizontal asymptotes are horizontal lines that the graph of a function approaches as the input, or x-values, head towards positive or negative infinity. These asymptotes are indicative of the function's end behavior and can represent constraints or limits in real-life situations, such as a maximum possible concentration of a medication that can be reached in the bloodstream, no matter the dosage.

To determine if a horizontal asymptote exists for the given function C(t), we compute the limit of the function as t approaches infinity. If the degree of the polynomial in the denominator is greater than the degree of the numerator, as it is in our example, the horizontal asymptote will be the x-axis or y = 0. The presence of a horizontal asymptote in the context of medication concentration might signify that after a certain time, regardless of the initial dosage, the medication's efficacy will diminish to a negligible or stable level.