A function of the form C(t) = Ate^−kt, where A and k are positive constants, is called a surge function and is sometimes used to model the concentration of drug in a patient’s bloodstream t hours after the drug is administered. Assume t 0.

a. Find C'(t), and determine the time interval where the drug concentration is increasing and where it is decreasing. For what value of t is the concentration maximized? What is the maximum concentration? (Your answers will be in terms of A and k.)

b. Find C"(t), and determine intervals of concavity for the graph of C(t). Find all points of inflection, and explain what is happening to the rate of change of drug concentration at the times that correspond to the inflection points.

c. Sketch the graph of C(t) = te^−kt for k = 0.2, k = 0.5, k = 1.0, and k = 2.0. Describe how the shape of the graph changes as k increases.