Math 121 Calculus for Biology Spring Semester, 2010 Lab Help	18-Feb-10
	San Diego State University

The first question examines exponential and logarithmic functions. These functions are compared to power functions, x^r, where the power, r, is either an integer or a fraction. The second and third questions in this lab parallels the allometric lecture material and extends your work with Power Functions or Allometric Models. These questions examine the modeling of species biodiversity on islands and the volume of trees. Biological data can often be fit by a power law or allometric model, which implies a relationship of the following form for the data:

y = Ax^r.

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 will need to search for appropriate intervals in Maple that show you the points of intersection. For the exponential function and the power function, you should find the first two points of intersection in a small interval near the origin, such as -2 < x < 2. The third point of intersection requires a larger interval, but should occur before x = 50. The second pair of graphs are a logarithmic function and a fractional power of x. In this case, you should be able to find your first point of intersection for 0< x < 10 . However, the second point of intersection could require a very large value of x. It should be less than x = 10²⁰. You will want to take intervals 0< x < 10^a, increasing the value of a until you see the graphs clearly intersecting.

As you did before, you first create the graphs in Maple, then use the information that you glean from the graphs 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 will have your graphs in Excel, but Maple graphs will be the quickest to find the points of intersection. Be sure to make your Excel graphs satisfy the same standards that we have applied to the graphs in the first 3 Labs. 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)).

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. You are also asked to take the logarithm of the data by simply typing "=ln(x)" where x is the value of the data that you want. This part of the problem uses the linear fit, so after modifying the data to the logarithm of the data, then this problem is just like the linear fit problems that you have done before.

Question 3: This problem is very similar to the one above. The problem addresses the issue of volume of wood from trees of differing height and diameter. You find the best model for volume of wood based on diameter or height, then study linear and allometric models. The allometric model gives you information on how trees change as they increase in volume.

To work this problem, you enter the data into a new Excel worksheet. You highlight the data and create a graph. You click on the data, then select Add Trendline from the menu. Under Trendline, you select either the linear or power law option, depending on what model you are considering, then have Excel put the equation on your graph. This should be very similar to what you have done already in previous labs. Thus, you should not have too much difficulty with these problems. You are also asked to take the logarithm of the data by simply typing "=ln(x)" where x is the value of the data that you want. This part of the problem uses the linear fit, so after modifying the data to the logarithm of the data, then this problem is just like the linear fit problems that you have done before.

In the last part of the problem, you are creating log-log plots with Excel. This is actually very simple to do. You simply make a copy of the graph with the original allometric model. Next you either double click or right click on each of the axes. This gives you a box labeled Format Axis. You select the Scale tab, then check the box for Logarithmic scale. Be sure to answer all the questions asked in these problems.