This reviews the lecture material from the Linear Discrete Dynamical Models section and starts with the intuition behind the derivative, reviewing our allometric models. The first question has you simulate the breathing model discussed in lecture. The second question extends our population growth studies to a model with immigration and the logistic grrowth model. This question reviews the use of Solver to fit these models to population data. The third lab problem examines the growth of children, which begins our discussion to introducing the derivative. This lab has you reproducing similar results to the ones in the lecture notes, so you need to understand the material in the lecture notes to work this lab.

Question 1: This question introduces you to the material for linear discrete dynamical systems. The lecture notes have most of the material that you will need to help you through this problem. In Part a., you begin by using the formula to calculate q, then you simply iterate the discrete model much as you have been doing for the last few lab problems. You will want to perform 40-50 iterations (which is very easy in Excel) to get far enough out to answer the question about how many breaths it takes to reduce the level of Ar to 0.01. Part c requires that you use some of your skills in algebra (or skills in Maple to let it do the algebra). You are given c₀ and c₁, which you use in the formula for the model as c_n and c_n+1, respectively. You know g and V_i also, so you are only lacking V_r, which you must solve for. After you have V_r, the problem is very much like Part a.

Question 2: This problem is another examination of census data for some country. Part a is very similar to problems you have done before with simple Malthusian growth. Rather than actually using the solution of the discrete Malthusian growth model, you simulate the model with your named parameters for growth rate, r, and initial population, P₀. You use Excel's Solver to find the best fitting (least sum of square errors) parameters growth rate, r, and initial population, P₀. In Part b, the discrete Malthusian growth model adds a term for immigration, m. Once again, you name the 3 parameters in this model (use unique names with initial guesses suggested in the question), then simulate the discrete linear model of population growth with immigration. You should find Excel very good at simulating the model by entering the equation in the line below the initial population and filling down. Again, you use Excel's Solver to find the best fitting parameters for this model by letting it compute the least sum of square errors. The third model in this question uses a logistic growth model to simulate the census data. This is performed just like the immigration model, but uses the logistic growth equation given in your Lab question with its 3 parameters (with reasonable initial guesses for the parameters). Again Excel's Solver will find the parameters for this model that give you the least sum of square errors. If you want to be safe with your answer from Solver, then run Solver twice. The second run should leave an answer that is very close to your first execution of Solver.

The last part of the question with the logistic growth model asks for equilibria. These are found by substituting P_e for P_n and P_n+1 in the logistic growth equation, then solving for P_e. Since this is a quadratic equation, it should be easy for you to find the equilibria. One of the equilibria should be obvious from what you know should be true of population models.

Question 3: This problem is meant to help motivate the idea of a derivative, which we will be studying over most of the rest of this course. The first part of this problem gives you more experience with allometric modeling using heights and weights of children. The remainder of the problem is designed to build intuition on the rate of gain of height or weight, which is a derivative. The lecture notes have details on computing the average rate of change in height, and this lab problem repeats a very similar calculation and adds the calculations for the average rate of gain of weight.