If a patient takes a dose of D units of a particular drug, the amount of the dosage that remains in the patient’s bloodstream after t days is De^−kt, where k is a positive constant depending on the particular drug.

(a) Show that if the patient takes a dose D every day for an extended period, the amount of drug in the bloodstream approaches R = De^−k / 1 − e^−k .

(b) Show that if the patient takes a dose D once every t days for an extended period, the amount of drug in the bloodstream approaches R = De^−kt/1 − e^−kt.

(c) Suppose that it is considered dangerous to have more than S units of the drug in the bloodstream. What is the minimal time between doses that is safe? D + R ≤ S.