M112 Lab 1 The Surge Function and Nicotine

We are going to investigate the surge function in this lab, and see how it applies to the concentration of a drug in the blood system. In this lab, x represents time, y represents the concentration of the drug in the blood system, and a and b are called parameters or constants. One of the goals of the lab is to understand how the parameters affect the shape of the graph of the surge function. Then, you will be able to fit the model to data that represents drug concentration levels.

1. Graph the function for various values of a and b. Use the range settings , and . Start with the values and . First keep the value of b fixed and vary a. To do this, choose new a values that are close to 1, such as 0.5, 2, 3 . Observe the graph as you vary that parameter, and explain what happens as to the graph as the parameter gets larger or smaller. Then reverse the process, switching the roles of the parameter that is fixed and the one that is varied. Thus, you will put a back to 1, and choose new b values, such as 0.04, 0.06, 0.08, 0.12. Describe what each parameter does to the shape of the graph, paying particular attention to the maximum point on the graph. Discuss what this means in terms of nicotine levels.

2. Using what you learned from part 1, fit the surge function model to the data in Figure 4.97 on page 226 of your book. You will have to estimate the coordinates of the data points as best you can. One thing that you should notice is that the first data point is about so that the initial concentration of nicotine in the blood is 3 mg/ml. In order to fit the surge function to this data, add to the surge function model, so that it now becomes . Now vary the parameters until you get what you consider to be your best fit to the data. Describe briefly how you used your knowledge from part 1 to get a fit to the data. Could you get a good fit to all of the data? Why or why not? What compromises did you make?

3. For the model that you came up with in part 2, determine where the function is increasing, and where it is decreasing. The maximum/minimum feature of your calculator is of great help here. Explain what this means in terms of nicotine levels.

4. For the model that you came up with in part 2, estimate where the function is concave up, and where it is concave down. Explain what this means in terms of nicotine levels.

5. For the model from part 2, estimate the derivative of the surge function at the points . You may use the derivative feature of your calculator. Give units for each, and explain what this means in terms of nicotine levels.

6. Sketch a graph of the derivative function, and explain to the reader what it represents, and how it relates to nicotine levels. Use the values from part 5 to help in sketching the graph.

7. Predict how long it will take for the nicotine level to go back down to within 5% of the starting level.

Write up:

Write a paper that incorporates all of the above ideas. Graphs and tables should be completely explained to the reader of the paper (do not assume that I am the reader). Address everything in parts 1-7, but do not label Part 1, Part 2, and so one. Include an introduction and conclusion.