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

1. (Mathematica) Sketch the graph of C for 0 £ t £ 48.
2. What do you notice about the slope of the line tangent to the graph of C at the point where C attains its maximum value?
3. Determine C¢(t).
4. How many hours after the drug is taken will the concentration be maximum? What is the maximum concentration?
5. Estimate the average concentration in the bloodstream over the first two days after taking the drug. (Hint: Compute some values of C(t) and find the average of these sample values.)
6. (Mathematica) Later in the semester we will learn how to compute the actual average value. For now, compute this value by issuing the following command:

   N[Integrate[0.2t/(t^2+4t+4),{t,0,48}]]/48

7. Suppose that after receiving a first dose, a patient must take a second dose once the concentration drops to 0.01 mg/cc. When should the second dose be taken? Round your answer to two decimal places.
8. (Mathematica) Assuming a second dose is taken at the recommended time, sketch the graph of concentration vs. time for 0 £ t £ 72.
9. (Extra Credit) Assuming a second dose is taken at the recommended time, estimate the maximum concentration and average concentration over the 72 hours after the first dose is taken.