SDSU

Math 121 Calculus for Biology
Spring Semester, 2007
Lab Help

06-Mar-06

San Diego State University


Laboratory Help Page for Lab 6

The first question in this lab examines the Improved Malthusian Nonautonomus growth model.
The second lab problem studies an allometric modeling for the heitgh and weight of girls.
Question 1: Malthusian Growth and Nonautonomous Growth Models (F4).
This problem asks you to repeat what we have done in the notes with the U. S. census data to census data from another country. You will be producing similar graphs (see the graph titled "Growth Rate for U. S." and the following graph called "Discrete Growth Models for U. S."), which should give you a guideline. Be sure to answer all of the questions carefully.

Use the power of Excel to find your growth rates. Suppose that once again you have entered the dates in Column A (starting in A2) and the population values in Column B (starting in B2). I would suggest that you copy all but the last date again into another column, say Column G. (I like to have different computations away from the main table of data. Note that A2 = G2 are the same dates.) In Column H, you start in H2 and enter "=B3/B2-1" to let Excel compute the growth rate associated with the first date. You simply fill down from here to obtain all the growth rates for your country. At the bottom of the calculations in Column H you can use the Excel AVERAGE command to compute the average growth rate for the country. (I find it easiest to simply put this average value in Column I and fill down, so that I can highlight columns G, H, and I to produce the graph like the one in the notes.) Use trendline on the growth rate data to find the best straight line. In order to get good results, it is very important that after you use trendline to find the best straight line through the growth data, you double click (or right click) on the equation (be careful here), then when the Format menu pops up, you can choose "number" of the 3 folders. Under number, you select "number" again, which should give you a menu to select the way you want your numbers displayed. You should use the scientific notation option. Its this equation that will determine the behavior of your nonautonomous model, so you need 6 significant figures for an accurate model.

Now you return to the Columns A and B in your data set. In Column C, you repeat the procedure you did in Question 1 above using the average value just computed for your growth rate r. In Column D, you will produce the nonautonomous model using the equation you just found from trendline. Say trendline found the equation k(t) = 3.148 - 0.002347 t, then after the first entry in D2 (which is the same as B2 and C2), in D3, you enter "=(1 + 3.148 - 0.002347*A2)*D2." You fill down from here to complete the nonautonomous model. You may want to use my discrete template with the nonautonomous model as a guide, making the appropriate changes. From there you simply need to create tables and graphs of the models and answer the questions posed in this Lab question.


Question 2: Weight and Height of Girls (I2).
The first part of this problem gives you more experience with allometric modeling using heights and weights of girls. Part c is related to the previous problem giving you more intuition on rate of gain of weight, which is again a derivative. The rate of gain of weight is similar to the computations we did for the lecture notes.