After injection of a dose D of insulin, the concentration of...

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 $T$

Transcript

00:04 In this question, we are given that the concentration of insulin in a patient system is is decaying exponentially and is written as and is written as d, e to the par, minus a, t.

00:46 So this is d, e to the par minus a t, where a is given as the positive constant, positive constant, and t is given the time in our.

01:04 Time in r.

01:08 All right.

01:10 So now, we have to find out, we have to find out.

01:15 If a dose, if a dose is injected every t r's, injected every t r's, this dose is represented by t.

01:35 So, then we have to write an expression for the sum of, then to write an expression for the sum of an expression for the sum of residual concentrations, concentration, before n plus 1th injection.

02:02 All right.

02:03 So let us consider after the nth dose.

02:13 So this clearly means before the n plus 1th dose, before n plus 1th dose or injection.

02:26 All right.

02:28 So after the nth dose, it is now.

02:32 Consider after the first.

02:33 Dose the concentration after first dose is d to the par minus 80 so the value for and after second dose the concentration will be d to the power minus 280 and so on after nth dose it is d to the power minus n a t so on as after nth dose, the sum of the residual concentrations, the sum of residual concentration will follow a geometric series as the sum is the sum at time capital t, capital t r.

03:49 So that will be equals to.

03:51 D to the power minus a t to the power minus 2a capital t and so on d to the power minus n a t.

04:02 All right.

04:04 So observe that...

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