Computer Lab Help 8

This page is designed to provide helpful information about the laboratory questions. You will find more details in the Lab Manual that accompanies this course. Begin this lab and every lab by introducing yourself to your partner. Determine the times when you can meet together during the week before the lab is due at your next Lab session. You should start this lab and each lab by typing the name of each team member and your computer number on the Lab Cover Page (or a copy of it).

The first WeBWorK problem asks questions about this help page and appropriate lecture material. This should help you work through the Lab more smoothly. In this Lab, the first problem studies the growth of fish using von Bertalanffy's equation to connect the length to age. An allometric model relates length and weight. These models are combined by a composite function, then differentiation is applied to find when the fish are gaining weight most rapidly as function of age. The second problem examines the logistic and Gompertz growth models for tumor cells. These growth models are fit to data, then simulated to fit the time data. The last problem  looks at creating an open box from a sheet of paper by cutting the corners and folding the edges. You determine the optimal volume possible by making the appropriate cuts. 

Problem 1: This problem is very similar to the material in the lecture notes. We fit data to find the length of a fish as a function of its age using von Bertalanffy's equation. Then we create an allometric model for the weight of a fish as a function of its length. The questions that you answer are very similar to the ones in the Chain Rule homework problems. The fit to the von Bertalanffy equation is done using Excel's Solver very much the same way as you have done before, while the fitting to the allometric model follows the techniques learned in earlier Labs using Excel's Trendline. This is readily accessed by right clicking on the data. You may need to increase the number of significant digits given to you by Trendline. To obtain 5 significant figures, we right click on the formula, selecting Format Trendline Label, choose Number and Scientific, and finally select 5 Decimal Places. The use of Maple might be valuable for helping with the differentiations (though you can practice your skills by doing them by hand). For this problem, the first and second derivatives are zero at t = 0, because the curve is very flat there. To get the point of inflection and the value of the function and the derivative at the point of inflection you need the following commands (assuming you see the point of inflection is between t = 1 and t = 20:

> p := t -> model goes here;

> dp := diff(p(t),t); # This gives the derivative

> sdp := diff(dp,t); # This gives the second derivative

> tp := fsolve(sdp=0, t = 1..20); # This gives the point of inflection

> p(tp); evalf(%); # This gives the population at the point of inflection

> subs(t = tp, dp); # This gives the derivative at the point of inflection

Problem 2: This problem begins by fitting a quadratic function to tumor growth data. This is the classic logistic growth model, and we are attempting to find the best fitting quadratic function passing through the origin. 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.

Problem 3: This problem is one of the classic problems in using Calculus to find an optimum. You may want to take a sheet of paper and cut squares out of the 4 corners to give you some intuition into this problem. Recall that the volume of a rectangular box is the product of the length times the height times the width. The surface area will clearly be the area of the sheet of cardboard minus the area of the pieces cut out of the cardboard. By trying to cut out square corners from a sheet of paper, you should be able to figure the limits of the domain. (Obviously, one is x = 0 by not doing any cutting from the paper.)