Math 121 Calculus for Biology
Spring Semester, 2013
Lab Help

09-Apr-13

San Diego State University


Laboratory Help Page for Lab 9

This lab continues the study of applications of the derivative. Again you will find Maple useful to help solve these problems. The first problem has you fitting data on radioactive materials used in biomedical imaging. The second question considers classic data from a growing culture of yeast. The third question examines the logistic and Gompertz growth models for tumor cells.

Problem 1: This problem examines two radioactive isotopes that are used in medical imaging. The first isotope simply decays exponentially and is fit with Excel's Trendline with the exponential fit. The second isotope's data are fit using the difference of two decaying exponentials. One decay rate comes from the analysis of the first part of the problem. The other two parameters are fit to the data with Excel's Solver, obtaining the other decay rate and the constant multiplying the exponentials. This maximum and point of inflection can be done by hand or you can use Maple to find the values more quickly. Again the techniques for this problem are similar to the ones using Excel's Trendline and Solver and Maple, as you have done in the past. No new computer techniques are introduced.

Problem 2: You start this problem by taking the two columns of data labeled P and G(P), and plotting the P data on the horizontal axis and the G(P) data on the vertical axis. Excel's trendline is applied using the polynomial fit with order 2 . It is very important that in the options section besides having Excel Display the Equation on the chart you check the option Set intercept = 0 . This gives you the best quadratic fit that you will be using for the G(P) that is used in the rest of the problem. You should be able to easily find the P-intercepts and vertex for this quadratic function.

The second part of this question uses the function G(P) found above to insert into the discrete logistic growth model. You simulate this model much as you have other discrete dynamical models. In the first column you enter the initial time, t = 0. Suppose this entry is in A2 on your spreadsheet, then in A3 you enter =A2+1 and fill down for 17 iterations. In Column B, insert the population data labeled P in the table. Create a named variable p0, by going to the Formula menu tab and choosing Create from Selection. Estimate p0 to be the initial population given in your table. In the column next to the population data, you enter this initial population, p0 , which would be in C2 on the spreadsheet. In C3, much as you have done before, you enter the equation for the model below the entry for p0, then fill down to have Excel simulate this discrete logistic growth model. In Column D find the square error between the data, P, and the logistic growth model. Compute the sum of square errors, and let Solver minimize this to find the best value of p0 for the simulation. Finally, you graph the simulation and add the actual data. The remainder of this problem is simply answering the questions asked about the model.

Problem 3:  This problem is similar to the earlier problem you worked in lab for a yeast culture. You begin this problem by plotting the data in the second and third and fifth and sixth columns. Use Trendline as you have before to get the best quadratic through the data. Be sure to set the y-intercept equal to zero in the options and watch that you have at least 4 significant figures from the equation. Use that equation to compute equilibria (zero growth), find the vertex (maximum growth), and determine the sum of square errors between the model and the data.

The next part of the problem has you define variables a and b for the Gompertz model. The values you choose are not very sensitive, but taking a = 0.1 and b = 0.2 seems to work fine. You define the sum of square errors between the data and the Gompertz function, then use Solver as you have before to minimize this sum of square errors. You should graph the Gompertz function with about 50 evenly spaced points along the domain of your function (the p values). This is added to the graph from Part a.

The last two parts have you simulate the nonlinear discrete dynamical model starting with a variable that you define, p0. Again you find the sum of square errors between the time series data given in the table and the simulation of your model using either the logistic or Gompertz growth function. With Solver you minimize the sum of square errors by only changing p0. The rest of the questions that you need to answer are fairly routine.