Math 1215 First Project - Due October 5, 2015 The Excel parts of this project need to be done in Excel, Open Oce or any other spreadsheet software and printed and attached to the rest of your work. The rest of the project may be typed or hand-written, in a neat and legible way (messy work can cost you up to 5 points.) The rst page of the project needs to contain the name(s) and student number(s) of the authors of the project clearly written on the top. Include the question numbers with your answers. Projects consisting of more than one page need to be stapled! Everything is due in hard copy in class or in your professor's oce in the Chase Building by 5 PM on October 3, 2012. This project may be done alone or in pairs (at most two). Students are responsible for choosing their own partner. Names of all the group members and banner numbers must appear clearly at the top of the cover page. If more than two authors are listed, the project will not be graded. Medication in the Bloodstream In class we studied a model for the level of medication in the bloodstream of a patient who receives a xed daily dose of medication, that raises the concentration of the medication in the bloodstream by 1 mg/liter. 1 mt+1 = mt + 1 with m0 = 0. 3 The equation m0 = 0 means that on day 0 (before the treatment starts) there is no medication in the blood. The '+1' represents the dosage and the 1 is the fraction of medication left in the 3 2 bloodstream by the patient's body tissue each day (which means the other 3 is absorbed out). The fraction of the medication that is absorbed out of the blood into the tissue can be dierent for dierent people. Below we explore what would happen if the patient absorbed more of the medication. 1. Assume that the patient absorbs 75% of the medication that is present in the bloodstream (so 75% of what is in the blood stream on day t is absorbed out of the bloodstream into the tissue by the next day, t + 1). The daily dose is still 1 mg/liter, and the patient begins with no medication in the blood on day 0. Give the new discrete dynamical system that models this situation (including the initial condition). Note: For the remaining questions in this section assume the discrete dynamical system is given by mt+1 = .2mt + 1 with m0 = 0 . Note: This is not the solution to question 1 2. Write an excel document that calculates the medication levels in the bloodstream for the rst two weeks of taking this medication (i.e., for 14 days). What do you expect to happen with the amount of medication in the bloodstream if the patient continues to take this dose of medication for a longer period of time? 3. Find the equilibrium point(s) of this system. 4. Draw a cobwebbing diagram that determines whether this equilibrium is stable or unstable. Does this answer agree with what you would have expected from your Excel table? Why? 1 For questions 5-6, we change the daily dose. For a daily dose of d mg/liter the system is given by: mt+1 = .2mt + d , with m0 = 0 . 5. What would the equilibrium concentration be in this case? Give your answer as a function of d. 6. Suppose that a doctor wants to give the patient a daily dose such that eventually the concentration of medication will be 4 mg/liter. What should that dose be? 2