Problem 42 - Essential Calculus Early Transcendentals

Question

After injection of a dose \(D\) of insulin, the concentration of insulin in a patient's system decays exponentially and so it can be written as \(D e^{-a t}\) , where \(t\) represents time in hours and \(a\) is a positive constant. (a) If a dose \(D\) is injected every \(T\) hours, write an expression for the sum of the residual concentrations just before the \((n+1)\) st injection. (b) Determine the limiting pre-injection concentration. (c) If the concentration of insulin must always remain at or above a critical value \(C,\) determine a minimal dosage \(D\) in terms of \(C, a,\) and \(T\)

Step-by-Step Solution

Answer

(a) The sum is \( De^{-aT} \frac{1-e^{-naT}}{1-e^{-aT}} \). (b) Limiting concentration is \( De^{-aT} \frac{1}{1-e^{-aT}} \). (c) Minimal dosage \( D \) is \( C \cdot (1-e^{-aT}) \cdot e^{aT} \).

Step 1: Establish the concentration after each injection

After the first injection, the concentration before the second injection is a residue of the first dose, expressed as \( D e^{-aT} \). After the second injection, the concentration before the third injection is \( D e^{-2aT} \) from the first dose and a full \( D e^{-aT} \) from the second dose. This pattern continues for \( n \) injections.

Step 2: Sum of residual concentrations

The sum of the residual concentrations just before the \((n+1)\)th injection is given by a geometric series:\[\text{Total concentration} = D e^{-aT}(1 + e^{-aT} + e^{-2aT} + \ldots + e^{-(n-1)aT})\].This is a geometric series with the first term \( a_1 = De^{-aT} \) and a common ratio \( r = e^{-aT} \). The sum \( S_n \) of this geometric series is: \[ S_n = a_1 \frac{1-r^n}{1-r} = De^{-aT} \frac{1-e^{-naT}}{1-e^{-aT}} \]

Step 3: Find the limiting pre-injection concentration

The limiting concentration as \( n \to \infty \) can be derived from the formula for the sum of the infinite geometric series. Since \( e^{-aT} < 1 \), \[ \lim_{n \to \infty} S_n = De^{-aT} \frac{1}{1-e^{-aT}}\]. This is the limiting pre-injection concentration.

Step 4: Determine minimal dosage based on critical concentration

To ensure the insulin concentration is always at or above a critical value \( C \), solve the inequality from the limiting concentration:\[ De^{-aT} \frac{1}{1-e^{-aT}} \geq C \].Rearrange to find:\[ D \geq C \cdot (1-e^{-aT}) \cdot e^{aT}\]

Key Concepts

geometric series

Geometric series are a type of sequence in mathematics where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In the context of the exercise, we're dealing with an exponential decay represented as a geometric series. The formula for the sum of the first \( n \) terms of a geometric series with an initial term \( a_1 \) and a common ratio \( r \) is:

• \( S_n = a_1 \frac{1 - r^n}{1 - r} \) if \( r eq 1 \)

In our exercise, the series models the insulin concentration in the patient's body, where each term decreases exponentially over time. Here, the first term \( a_1 \) is \( De^{-aT} \), and the common ratio \( r \) is also \( e^{-aT} \). This means each subsequent dose contributes less to the total concentration than the previous one because of the decay.
This representation is crucial in predicting the insulin levels just before a new dose is administered, ensuring effective management of the patient's condition.

limiting concentration

Limiting concentration refers to the steady-state concentration that the insulin level approaches over a long time period as more injections are made. In this context, as the number of doses \( n \) approaches infinity, the insulin concentration stabilizes despite the exponential decay of each individual dose. Mathematically, we calculate the limiting concentration by evaluating the infinite sum of the geometric series:

• The formula for an infinite geometric series sum is \( S = \frac{a_1}{1 - r} \), assuming that the common ratio \( |r| < 1 \).

In the exercise, \( a_1 \) is \( De^{-aT} \) and \( r \) is \( e^{-aT} \), so the limiting concentration before each injection is:

• \( \lim_{n \to \infty} S_n = De^{-aT} \frac{1}{1 - e^{-aT}} \)

This means that no matter how many doses are administered, the concentration approaches a maximum value determined by the injection parameters \( D \), \( a \), and \( T \). Understanding this concept ensures that the correct dosage and timing are selected to maintain glucose levels safely.

inequalities in calculus

Inequalities in calculus are used to find the boundaries or limits that certain mathematical expressions must adhere to. They are essential in optimizing functions and ensuring results fall within specific constraints. In our exercise, we're tasked with ensuring the insulin concentration is always at or above a critical value \( C \). This is expressed by the inequality:

• \( De^{-aT} \frac{1}{1 - e^{-aT}} \geq C \)

Rearranging this inequality helps determine the minimal dose \( D \) required to meet or exceed this concentration threshold:

• \( D \geq C \cdot (1-e^{-aT}) \cdot e^{aT} \)

This calculation ensures patient safety, providing that their insulin levels remain effective enough to manage their glucose without dropping below a critical level. Such inequalities are common in calculus for ensuring that variables, in this case, concentration, remain within specified limits necessary for successful application.