Problem 82 - College Algebra

Question

For the following exercises, use the given rational function to answer the question. The concentration \(C\) of a drug in a patient's bloodstream \(t\) hours after injection in given by \(C(t)=\frac{2 t}{3+t^{2}} .\) What happens to the concentration of the drug as \(t\) increases?

Step-by-Step Solution

Answer

As time increases, the drug concentration approaches zero.

Step 1: Understand the Problem

We are given a rational function \(C(t) = \frac{2t}{3 + t^2}\) that describes the concentration of a drug over time. We need to determine what happens to \(C(t)\) as \(t\) approaches infinity.

Step 2: Identify Behavior at Infinity

To find out what happens as \(t\) increases, specifically as \(t\) approaches infinity, we need to analyze the behavior of the function \(C(t) = \frac{2t}{3 + t^2}\). As \(t\) gets larger and larger, the \(t^2\) term in the denominator \(3 + t^2\) will dominate the constant term, making the denominator approximately equal to \(t^2\).

Step 3: Simplify the Expression

When \(t\) is very large, \(3 + t^2 \approx t^2\). Substitute this approximation in the function to simplify: \(C(t) \approx \frac{2t}{t^2} = \frac{2}{t}\).

Step 4: Evaluate the Limit as t Approaches Infinity

Evaluate the limit of the simplified function as \(t\) approaches infinity: \[\lim_{t \to \infty} \frac{2}{t} = 0.\]

Step 5: Interpret the Result

The limit calculated in the previous step indicates that as time continues indefinitely, the drug concentration \(C(t)\) in the bloodstream approaches zero.

Key Concepts

Asymptotic Behavior

The asymptotic behavior of a function describes how it behaves as the input gets very large or very small. In our example function, \(C(t) = \frac{2t}{3 + t^2}\), we are interested in what happens as \(t\) becomes very large. When analyzing asymptotic behavior, we often simplify the function to understand what happens when terms become negligible.
For rational functions, the terms with higher powers generally dominate as \(t\) increases. Here, the \(t^2\) in the denominator dominates the constant 3, leading to an approximation \(C(t) \approx \frac{2t}{t^2} = \frac{2}{t}\).
This insight into the function's behavior helps reveal that as \(t\) grows very large, the concentration \(C(t)\) decreases towards zero. This is how asymptotic analysis simplifies complexity to show trends.

Limits

Understanding limits is crucial when approaching values like infinity. A limit describes the value a function approaches as the input draws closer to a particular point. Here, we found that as \(t\) approaches infinity, our function simplifies to \(\lim_{t \to \infty} \frac{2}{t} = 0\).
Limits are instrumental in analyzing rational functions because they help in predicting long-term behavior. They effectively tell you where the function is headed, which, in this case, is a concentration of zero.
When evaluating limits, especially for rational functions, we can use simplifications to reveal unchanged behavior trends. By focusing on dominant terms, we clarify the function's tendencies as the input becomes extremely large or small.

Function Analysis

Function analysis involves dissecting functions to understand their properties and behaviors across different inputs. Rational functions, like \(C(t) = \frac{2t}{3 + t^2}\), can be broken down by evaluating the numerator and denominator separately to anticipate changes as time evolves.
In our analysis, we noticed that the denominator grows much faster than the numerator as \(t\) increases owing to the \(t^2\) term. By substituting and simplifying, we can predict that \(C(t)\) approaches zero with time. Critical to function analysis is recognizing that while individual terms inform behavior, their interplay provides the complete picture.
This analytical method doesn't only apply to time-related scenarios. It offers powerful insights into how outputs respond over expansive inputs, honing our ability to judge end-behavior without complex calculations.