Question

The concentration \(C\) (in milligrams per milliliter) of a drug in a patient's bloodstream \(t\) hours after injection into muscle tissue is modeled by $$ C=\frac{3 t}{27+t^{3}} $$ Use differentials to approximate the change in the concentration when \(t\) changes from \(t=1\) to \(t=1.5\).

Short answer

The approximate change in the concentration when \(t\) changes from \(t=1\) to \(t=1.5\) is 0.005 milligrams per milliliter.

Step-by-step solution

Calculate Differential

To get an approximation of the change, use calculus with differentials. To begin with, differentiate function \(C(t) = \frac{3t}{27 + t^{3}} \) with respect to \(t\). Using quotient rule for differentiation, the derivative \(\frac{dC}{dt}\) is \(\frac{(27 + t^{3}) * 3 - 3t * 3t^{2}}{(27 + t^{3})^{2}} = \frac{81 - 9t^{2}}{(27 + t^{3})^{2}}.\)

Find dC

Now you find the value of \(\frac{dC}{dt}\) when \(t = 1\). Substitute \(t = 1\) into \(\frac{dC}{dt} = \frac{81 - 9t^{2}}{(27 + t^{3})^{2}}\) and find \(\frac{dC}{dt}\). The value comes out to be \(\frac{72}{54^2}\).

Determine Change

To determine the change, compute the change in time \(\delta t\) as \(\delta t = t_{2} - t_{1} = 1.5 - 1 = 0.5.\)

Approximate Change in Concentration

Then, approximate the change in concentration using the given \(\delta t\) and the calculated value of \(\frac{dC}{dt}\). The change in \(C\), denoted by \(\delta C\), can be approximated using \(\delta C \approx \frac{dC}{dt} * \delta t = \frac{72}{54^2} * 0.5\). The value for this comes as approximately 0.005.

Key concepts

Differentiation

Differentiation is a fundamental concept in calculus that measures how a function changes as its input changes. It's like taking a snapshot of a moving object at a precise moment to determine its speed. The derivative of a function is the result of differentiation and it represents the rate of change of the function with respect to a variable. In practical terms, if we have a function that represents the height of a growing plant over time, the derivative will tell us how fast the plant is growing at any given moment.

In our drug concentration model example, the function represents the concentration of a drug over time. By differentiating it, we can determine how the concentration rate changes with time. The mathematical process to find a derivative can involve rules such as the power rule, the product rule, and the quotient rule depending on the structure of the function.

Drug Concentration Model

A drug concentration model is used to describe how a drug disperses and remains active in a patient's system over time. The model is based on factors like metabolism, drug interactions, and how the drug is administered. In our scenario, the drug is injected into muscle tissue, and its concentration in the bloodstream is modeled mathematically.

This model can help health professionals determine the most effective dosage schedule and predict how the drug will behave in the body. Understanding the model allows for precise predictions about the concentration at different times—crucial information for ensuring the drug provides its intended effect without causing harm due to overdose.

Quotient Rule for Differentiation

The quotient rule is a technique in calculus used when differentiating functions that are formed as the ratio of two other functions. If we have a function that is expressed as one function divided by another, we can’t just differentiate the top and the bottom separately. The quotient rule provides a way to handle this situation.

Mathematically, if we have a function \(\frac{u(t)}{v(t)}\), the derivative of this function with respect to \(t\) is given by \(\frac{v(t)\cdot u'(t) - u(t)\cdot v'(t)}{[v(t)]^2}\), where \(u'(t)\) and \(v'(t)\) are the derivatives of \(u(t)\) and \(v(t)\), respectively. In the exercise, the quotient rule is used to find the derivative of the drug concentration with respect to time.

Change Approximation

Change approximation in calculus refers to the technique of estimating how much a function's value will change given a small change in its input value. Differentials, which are small changes in variables, are used along with the derivative to make these estimations. It's like predicting how much the temperature will rise or fall with a slight weather change.

In our example, the differential used is \(\delta t\), which represents a small change in time. By multiplying the derivative of the concentration function by \(\delta t\), we get an approximation of the change in drug concentration. This method is valuable because it often provides a good estimate of change without the need for complex calculations that would involve the original function directly.