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Q42E

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Found in: Page 444

### Essential Calculus: Early Transcendentals

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

# 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 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, $$D \ge C{e^{aT}}$$