In this lab, the first question examines exponential and logarithmic functions. The last two problems are more allometric models like you did in the previous lab. All of these questions use very similar techniques on the computer that you have previously employed.

Question 1: This problem compares the relative rate of growth of exponential functions to power functions and logarithmic functions to fractional power functions. You will be finding points of intersection for these graphs very much like you did last week, using Maple's fsolve command. You are given two intervals to aid your search for the points of intersection. As you did before, you first create the graph, then use the information that you glean from the graph to help you find the points of intersection (i.e., you restrict the range you search with fsolve for these points of intersection. Your lab report can have your graphs in either Maple or Excel, but in either case, make sure that you clearly label which graph is which. The only new Maple command that you will need is that exp(x) is used to give you e^x (remember that the natural logarithm is given by ln(x)). The key to finding the dominance of one function over the other (the intervals where f(x) < g(x) or vice versa) is simply to find the points of intersection, then observe which function is higher than the other between two successive points of intersection. (They will swap positions on the next interval between points of intersection.) This part of the question is asking you to interpret the graphs of the functions.

Question 2: This problem is very similar to last week. The problem addresses the issue of biodiversity and the amount of land required to maintain a certain level of biological diversity. The model you produce gives a more quantitative answer to how much land is required, using Excel's power law.

Question 3: This Malthusian growth problem is very much like the ones that you might encounter on an exam, so you may want to do most of the work on this problem by hand. You may want to read the Worked examples section of the lecture notes to see how to obtain the annual growth from census taken every 10 years. Be sure to use at least 4 significant figures for your value of r. Once you have your values of r for each country, then you create a table in Excel with the appropriate range of dates and their corresponding populations. I like to label my columns so the labels in A1 is the date, while B1 and C1 are the names of the countries. In A2, you put 1980, while B2 and C2 have the initial population that agrees with the data. A3 will have "= A2 + 1" to update the date by one year. The entry in B3 is given by "= (1 + r)B2" (similar for C3), which updates B3 using the discrete Malthusian growth model. The models are simulated by simply filling down to the desired date. Create your graph using Chart Wizard, highlighting the first 3 columns over the range of dates requested in the question.