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Q42E

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Essential Calculus: Early Transcendentals
Found in: Page 444
Essential Calculus: Early Transcendentals

Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

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Short Answer

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 \(D{e^{ - at}}\), 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 \(T\).

a) Expression for the sum of the residual concentrations just before the \((n + 1)\)st injection: \({a_n} = \sum\limits_{k = 1}^n {D{e^{ - akT}}} \).

(b) Limiting pre-injection concentration: \(\sum\limits_{k = 1}^n {(D{e^{ - aT}}).{{({e^{ - aT}})}^{k - 1}}} = \frac{{D{e^{ - aT}}}}{{1 - {e^{ - aT}}}} = \frac{D}{{{e^{aT}} - 1}}\) .

(C) Minimal dosage \(D\) in terms of \(C\) , \(a\), and \(T\):\(D \ge C{e^{aT}}\).

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1 

Finding an expression for the sum of the residual concentrations just before the \((n + 1)\)st injection.

(a)Consider the following situation. After a patient takes a dose of insulin of size \(D\). the amount of insulin in the patient's system afterhours can be expressed as \(D{e^{ - at}}\) for some positive constant \(a\). A dose administered every \(T\) hour.

Let’s find an expression for the amount of insulin in the patient's system right before the \({(n + 1)^{th}}\)injection. Call this amount \({a_n}\).

We'll compute the first few values of \(a\), to see how the terms behave

\({a_1}\) = \(D{e^{ - aT}}\)

Since a dose was administered at the beginning and \(T\) hours have elapsed right before the

second dose

\({a_2} = D{e^{ - a2T}} + D{e^{ - aT}}\)

Since the first dose has now decayed for \(2T\) hours while the second has decayed for \(T\) hours

\({a_3} = D{e^{ - a3T}} + D{e^{ - a2T}} + D{e^{ - aT}}\)

Since the first dose has now decayed for \(3T\), the second for \(2T\) , and the latest for \(T\) hours

So, we see that the general pattern is

\({a_n} = \sum\limits_{k = 1}^n {D{e^{ - akT}}} \)

Step 2 

Determining the limiting pre-injection concentration.

(b) Now let's find the limit to which the insulin concentration stabilizes. Since each term \({a_n}\) represents the concentration at a time indexed by \(n\), the limiting concentration is the limit of \({a_n}\)as \(n\) goes to infinity.

Limiting concentration = \(\mathop {\lim }\limits_{n \to \infty } {a_n}\)\( = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {D{e^{ - akT}}} \),

Note that this is a geometric series, since each term has an increasing factor of \({e^{ - aT}}\). We'll re-express the series to make this more apparent:

\(\mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {D{e^{ - akT}} = } \sum\limits_{k = 1}^n {(D{e^{ - aT}}).{{({e^{ - aT}})}^{k - 1}}} \)

Now we can use the geometric series formula with \(a = D{e^{ - aT}}\), ratio \(r = {e^{ - aT}} < 1\) :

\(\)\(\sum\limits_{k = 1}^n {(D{e^{ - aT}}).{{({e^{ - aT}})}^{k - 1}}} = \frac{{D{e^{ - aT}}}}{{1 - {e^{ - aT}}}} = \frac{D}{{{e^{aT}} - 1}}\).

Step 3

(c) Now, suppose that the concentration always needs to be above some critical level \(C\). Let's find a dosage \(D\)such that the concentration is always above \(C\):

The lowest that the concentration will ever be is \(T\) hours after a dose, when there are no other doses in the system.

Thus, \(D{e^{ - aT}} \ge C\)

Therefore, \(D \ge C{e^{aT}}\)

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