The concentration C(t) in milligrams per cubic centimeter of a particular drug in a patient’s bloodstream is given by:

where t is the number of hours after the drug is taken.

- How many hours after the drug is given will the concentration be maximum ? (Verify there is indeed a maximum at this time.)
- What is the maximum concentration?

- To find the maximum, it is necessary to
find where the function for concentration has a slope or
derivative = 0.
Find

*C*′(*t*) using the quotient rule.Set

*C*′(*t*) = 0 and solve for*t*:Note that time = - 2 hours does not provide a meaningful answer.

Check to see whether a maximum does occur at t = 2 hours by applying the first derivative test and checking a point on either side of

*C*′(2).*C*′(1) = +*C*′(2) = 0*C*′(3) = -This means as the slopes go from + to 0 to - as t goes from 1 to 2 to 3 which maps out a maximum at time 2 h.

- Evaluate the concentration function
*C*(2)=0.02 mg/cc. The maximum concentration is 0.02mg/cc at time 2 hours. Here is a graph of the concentration as a function of time, verifying these results: