Question

How do you find on what time interval is the concentration of the drug increasing if suppose a certain drug is administered to a patient, with the percent of concentration in the bloodstream t hr later given by `K(t)= 8t / (t^2 + 1)`?

Answer 1

The concentration is increasing on `0 ≤ t < 1`

Explanation:

You will need to start by differentiating.

`K'(t) = (8(t^2 + 1) - 8t(2t))/(t^2 + 1)^2`

`K'(t) = (8t^2 + 8 - 16t^2)/(t^2 + 1)^2`

`K'(t) = (8 - 8t^2)/(t^2 + 1)`

This will have a critical number when `K'(t) = 0`.

`0 = 8 - 8t^2`

`8t^2 = 8`

`t = +-1`

But since `t > 0`, only `t =1` is acceptable.

You will notice that whenever `t > 1`, the derivative turns negative, therefore the amount of drug in the patients system is increasing for the first hour, with `0 ≤ t < 1`.

Hopefully this helps!