Problem 147 - Calculus Volume 1

Question

[T] The concentration of antibiotic in the bloodstream \(t\) hours after being injected is given by the function \(C(t)=\frac{2 t^{2}+t}{t^{3}+50},\) where \(C\) is measured in milligrams per liter of blood. a. Find the rate of change of \(C(t)\) . b. Determine the rate of change for \(t=8,12,24,\) and \(36 .\) c. Briefly describe what seems to be occurring as the number of hours increases.

Step-by-Step Solution

Answer

The rate of change decreases over time as the number of hours increases.

Step 1: Understand the Problem

We are given a function \(C(t) = \frac{2t^2 + t}{t^3 + 50}\), which represents the concentration of an antibiotic in the bloodstream \(t\) hours after injection. Our job is to find the rate of change of \(C(t)\) with respect to time \(t\). This is equivalent to finding the derivative \(C'(t)\).

Step 2: Apply the Quotient Rule

The function \(C(t)\) is a quotient of two functions, so we use the quotient rule to find the derivative: \(\frac{d}{dt} \left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}\). Here, \(u = 2t^2 + t\) and \(v = t^3 + 50\). We need to find \(u'\) and \(v'\) first.

Step 3: Differentiate \(u\) and \(v\)

Calculate \(u' = \frac{d}{dt}(2t^2 + t) = 4t + 1\) and \(v' = \frac{d}{dt}(t^3 + 50) = 3t^2\). These are the derivatives needed for the quotient rule.

Step 4: Find the Derivative \(C'(t)\)

Apply the quotient rule: \[C'(t) = \frac{(4t+1)(t^3+50) - (2t^2+t)(3t^2)}{(t^3+50)^2}\]. Expand and simplify the expression in the numerator: \(4t(t^3) + 200t + t^3+50 - 6t^4 - 3t^3\). Simplified, this is \((-2t^4 + t^3 + 200t + 50) / (t^3 + 50)^2\).

Step 5: Evaluate the Derivative at Specific \(t\) Values

Substitute \(t = 8, 12, 24,\) and \(36\) into \(C'(t)\). For each \(t\), calculate: \[C'(8), C'(12), C'(24), C'(36)\]. This requires substituting each value into the expression from Step 4 and simplifying.

Step 6: Interpretation of the Results

As \(t\) increases, the terms \(-2t^4\) and \(t^3\) in the numerator grow very large compared to the constant term \(50\), dominating the behavior. Thus, the rate \(C'(t)\) tends to decrease as \(t\) increases, suggesting that the concentration rate changes at a slower rate.

Key Concepts

Rate of Change

In the context of Calculus, the rate of change refers to how a quantity changes with respect to another variable. Here, we are interested in how the concentration of antibiotic in the bloodstream changes over time. This is captured by the derivative of the concentration function, denoted as \(C'(t)\). The derivative tells us the speed and direction in which the concentration changes.
The concept of rate of change is crucial because it provides insights into whether the concentration is increasing or decreasing at any given moment. A positive derivative means the concentration is increasing, while a negative derivative suggests it's decreasing. By finding \(C'(t)\) for various values of \(t\), we can understand how quickly or slowly the concentration changes at different times.

Quotient Rule

The quotient rule in Calculus is a technique used to differentiate functions that are composed as a ratio of two other functions. It is expressed as:

• Given two functions \(u(t)\) and \(v(t)\), the derivative of their quotient is: \(\frac{d}{dt} \left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}\).

In this exercise, the concentration function \(C(t)\) is a quotient where \(u(t) = 2t^2 + t\) and \(v(t) = t^3 + 50\).
To apply the quotient rule, calculate the derivatives of the numerator and denominator separately. These derivatives are \(u'(t) = 4t + 1\) and \(v'(t) = 3t^2\). Substituting these into the quotient rule formula allows us to find \(C'(t)\), the rate of change of the concentration function. Understanding and accurately applying the quotient rule is essential to solve problems involving fractions of functions.

Differentiation

Differentiation is the mathematical process used to find the derivative of a function. It is a key concept in Calculus that helps us understand how functions behave—particularly with respect to changes in their variables.
In our exercise, differentiation is used to determine \(C'(t)\), the rate at which the antibiotic concentration changes with time. The process involves a series of steps, including:

• Identifying the function to be differentiated, \(C(t) = \frac{2t^2 + t}{t^3 + 50}\).
• Applying the quotient rule to find the derivative.
• Simplifying the resulting expression.

These steps reveal the behavior of the concentration over time, allowing us to make informed predictions about trends and patterns.

Concentration Function

A concentration function describes how the concentration of a substance varies with respect to some variable, often time. In this exercise, the function \(C(t)\) represents the concentration of an antibiotic in the blood over time after an injection.
A concentration function provides a mathematical representation that can help predict future levels of a substance. As time passes, factors such as degradation and distribution can influence the concentration. Understanding the behavior of \(C(t)\) helps in determining how effective a medication remains in the bloodstream and how often doses might need to be repeated.
By differentiating \(C(t)\), we obtained insight into how the rate of change of concentration evolves, indicating that over time, the concentration tends to change more slowly. This is crucial for medical applications where maintaining effective drug levels is important.