The time L (in hours) that a certain drug stays in a person’s system is a function of the quantity q (in milligrams) administered; that is, L = f (q).

Question

The time L (in hours) that a certain drug stays in a person’s system is a function of the quantity q (in milligrams) administered; that is, L = f (q).

1. (a) Write a sentence explaining the practical meaning of f (8) = 12 in terms of time and drug quantity administered. Include units in your answer.
2. (b) Write a sentence explaining the practical meaning of f ′(8) = 0.5 in terms of time and drug quantity administered. Include units in your answer.
3. (c) Use the information given in parts (a) and (b) to approximate how long 11 milligrams of the drug stays in a person’s system. Include units in your answer.

Answer 1

Hi Preshyous J.,

Not really calculus required. The meaning of the first is, if you give 8 mg of drug, it stays in the body for 12 hours. The meaning of the second is, for each 1 mg of drug more (above 8 mg) that you give, the residence time of drug in the body is 0.5 hr more. Then it's pretty easy math to say, if this trend continues with dosage, then for 3 mg more (than 8 mg), it stays in the body for 3*0.5=1.5 hrs more, or 13.5 hr total.

This may be OK math, but it's terrible pharmacology. It implies that 0 mg of drug would stay in the body for 8 hr (an intercept on the y-axis). More likely, the dose vs. time within the effective serum range of the drug (which is quite something else than saying that the drug just disappears from the bloodstream!), is non-linear with respect to dosage, and has an intercept on the x-axis -- a minimum effective dose. And suddenly drops to zero again at a somewhat higher dosage: the patient was fatally overdosed!

--Cheers, --Mr. d.