Math 122 Calculus for Biology II Fall Semester, 2000 Lab Help	26-Oct-00
	San Diego State University

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Laboratory Help Page

This page is designed to provide helpful information about the laboratory questions.

Because of the upcoming Exam, you will have two weeks to work this laboratory. Begin this lab and every lab by introducing yourself to your partner. Detemine the times when you can meet together during the week before the lab is due on Friday, Nov. 10. If your schedules are totally incompatible, then notify me immediately.

You will probably want to download your specific lab page (and may want to convert it to a Word document). On the cover page you begin by typing in the name of each team member and your group number.

Question 1: This problem is meant to give you practice with Euler's method and Improved Euler's method. You will be solving the differential equations using the Excel worksheet with numerical methods for differential equations that is provided via a hyperlink. There are instructions written on the Excel worksheet on how to use this worksheet. You will be using another new feature in Excel, the Macro Function through the Visual Basic Editor. You will compare your numerical solution to the actual solution, which in both cases is a separable differential equation. At the time of writing this lab, we have not solved these in class, but will be doing so very shortly. An alternate method of solving the differential equation (if you are in a hurry to get this done) is to let Maple solve the differential equation. In the last lab, you were introduced to Maple's dsolve routine. Below is a quick sketch of how to solve the initial value problem

Start by typing the differential equation, so you can check that Maple has the equation that you want:

de := diff(y(t),t) = f(t,y(t)); where you enter the function f(t, y(t))

Next you use Maple's dsolve routine. If you have entered the line above, you type

dsolve({de, y(0) = a}, y(t)); where you put in the appropriate initial condition

These are the only two lines that you need to obtain the solution of the differential equation that you want to solve. The rest of the problem is simply following the directions and putting the appropriate numbers into the appropriate tables.

This problem is very similar to the lake pollution problem that we will soon cover in class. In this case, you use actual data on the Great Lakes to determine how long it takes for a pollutant to build up in one of these lakes and how long it takes for the pollutants to be eliminated. This problem can largely be done by hand, except for Part c., where you will use the Improved Euler's method developed in Question 1.

Question 3: Parts a and b of this problem are like the murder problem we will soon work in class. Part c is a modification that makes the question more realistic, but causes certain complications. The solution to this differential equation can be easily found using Maple's dsolve command. The constant k₂ in the solution appears in a very nonlinear manner in this equation, so it is impossible to solve for k₂ exactly. Thus, you need to use a numerical routine to find the value of k₂. You can apply Newton's method as we did in the last Lab or more simply, you may use Maple's fsolve command. This is done by writing the equation with all the known information at time t = 2. The only unknown is now k₂. In Maple, you write your equation in k₂, say f(k₂) = 30, where you put in all the information for f(k₂). To solve this in Maple, you simply type

fsolve(f(k)=30,k);

and Maple should give you the value of k₂. If not, you could have to modify it to say

fsolve(f(k)=30,k=0..10);

At the end of this problem, your graph should show you the differences between the two solutions and show how well the easier equation approximates the time of death.