Drug Concentration After a certain drug is injected into a...

Drug Concentration After a certain drug is injected into a patient, the concentration $c$ of the drug in the bloodstream is monitored. At time $t \geq 0$ (in minutes since the injection) the concentration (in mg/L) is given by $$c(t)=\frac{30 t}{t^{2}+2}$$ (a) Draw a graph of the drug concentration. (b) What eventually happens to the concentration of drug in the bloodstream?

Drug Concentration After a certain drug is injected into a patient, the concentration $c$ of the drug in the bloodstream is monitored. At time $t \geq 0$ (in minutes since the injection) the concentration (in mg/L) is given by
$$c(t)=\frac{30 t}{t^{2}+2}$$
(a) Draw a graph of the drug concentration.
(b) What eventually happens to the concentration of drug in the bloodstream?

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Key Concepts

Graphing Functions

Graphing functions involves plotting points based on the input-output pairs of a mathematical function. It helps in visualizing the behavior of a function over time, identifying key features such as maximum or minimum values, intercepts, and overall trends, which is crucial in understanding how the drug concentration changes after injection.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. Understanding rational functions is important because they often have features such as vertical and horizontal asymptotes, and their behavior at extremes (such as large values of the independent variable) helps predict long-term trends.

Asymptotic Behavior and Limits

Asymptotic behavior refers to how a function behaves as the independent variable approaches a particular value or infinity. In the context of the drug concentration function, analyzing limits and asymptotes allows us to determine what happens to the concentration as time progresses indefinitely, which is essential for understanding the eventual decrease in drug concentration.

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