This page is designed to provide helpful information about the laboratory questions. You should start with your Cover Page and be sure that you have all the important contact information about your lab partner for coordination later in the week. On the cover page you begin by typing in the name of each team member and your computer numbers. For your WeBWork answers, it is important that you keep all calculations to 5 or 6 significant figures. Too few digits have been one of the leading causes of errors that we have found when approached by students.

This lab explores three applications of differential equations. The first problem considers two growth models for E. coli and compares them. The second looks at the toxic build up of lead in children. The last problem considers a time-varying growth rate in a Malthusian growth differential equation.

Problem 1: This question examines two models using relatively simple differential equations. The first one is a simple integration like the cat falling under gravity, while the second one uses a differential equation like the one for Malthusian growth. The only difference from what we have done in class is that you will need to use information on the doubling time to find the constants a and k. That is, you substitute twice the volume at the appropriate time and solve for the unknown. (The constant of integration from solving the differential equations can be obtained from the initial condition.)

In Part c. you want to take the difference between your two solutions. (These solutions should have a difference of zero at times 0 and the cell divide time, since they both should satisfy the same conditions at these times.) Your lab report gives you the form of the function, which can be easily graphed. Use your standard techniques to find the maximum of this function, giving you the greatest error on the interval.

I'll let you think about the Biology of Part d. However, you should think about the magnitude of the error found in Part c and what you know about errors in lab experiments. Also, you should remember how models are formulated or the processes involve in cell division.

Problem 2 : This problem is very similar to the ones we have recently done in class on Hg in fish or lead in children. This is a variation on the lead in children problem from the Worked Examples section and the homework problems. Mostly, this problem has you showing your expertise in graphing and solving differential equations that you have learned over the semester to date. There are no special instructions needed for this problem (though I might suggest using the Google search engine with key words lead and children to find more information for the last question).

Problem 3: This question uses our least squares techniques from before (Excel's solver). You begin this problem by solving the differential equation using either Maple or techniques from class. Next step up a spreadsheet in Excel with the data on the population from your country in Europe. If Column A contains the year, then you'll want to use Column B for the times after 1950 (i.e., t = 0, 10, 20,...) and Column C will contain the census data. In Column G, you should list your parameters, a, b, and P₀, then place the initial guesses for those parameters in Column H. Good initial guesses for your parameters are to have P₀ be the population in 1950, b = 0.01, and a = 0. You should name these parameters, then in Column D, you insert the formula for the model, which you found earlier, including the named parameters, a, b, and P₀. In column E, you'll put the square error between the data in column C and the model in Column D. Sum the square error, then use Excel's solver to minimize this error and find the best parameters (as you have done before. The remainder of the problem is simply graphing and writing up your results.