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

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. 17. 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 shows another application related to our Malthusian growth and radioactive decay problems in differential equations. No special techniques are required for this problem and all but the graphing part can be done by hand.

Question 2: This problem uses a special routine in Maple to better understand the behavior of differential equations. Many of you have expressed difficulty understanding how Euler's numerical method relates to the solution of the differential equation. I have chosen two important population models to try to give you a better geometric understanding of the solutions of these differential equations. (From earlier work in this course, you should already understand the Malthusian and logistic growth models from their discrete counterparts.)

The main idea behind this problem is called slope fields. Slope fields show at each point on a grid in the time versus population graph the direction of the solution to the differential equation. Thus, a slope pointing in the direction of the derivative (which represents the growth rate or right hand side of the differential equation) is drawn at each point on the grid, which shows the direction the solution proceeds through a particular grid point. This is very similar to Euler's method, which approximates the solution to a differential equation by drawing a short straight line segment from the approximate solution in the direction of the derivative for a time interval of h.

For specifics on this problem, you will begin by solving each of the differential equations either by hand in the case of Malthusian growth or using Maple's dsolve command for the logistic growth model. You will use the exact solutions to determine matters like the doubling of the population. For graphing the slope fields, you use the special Maple routine under DEtools with the DEplot routine. Below is an illustration of this command (and its output) for the Newton's Law of Cooling problem

The Maple commands are:

> with(DEtools):

> DEplot(diff(T(t),t)=-0.05*(T(t)-20),T(t),t=0..100,[[T(0)=50],[T(0)=30],[T(0)=5]],T=0..50,color=blue);

The DEplot command begins with the statement of the differential equation (diff(T(t),t)=-0.05*(T(t)-20)). This is followed by the variable we want to solve (T(t)) and the time interval for the graph (t=0..100). Next the different initial conditions are inserted ([[T(0)=50],[T(0)=30],[T(0)=5]]), which is followed by the range of the dependent variable (T=0..50). I simply added a color option at the end, but this is not necessary.

Question 3: This question examines how drugs can be absorbed into the body through different delivery systems. The advent of polymers that dissolve safely in the body allows the administration of drugs, such as Deprovera for extended birth control. Getting the appropriate doses with this new technology requires some careful analysis of both theoretical models and experimental tests. This problem examines some of the simplest theoretical models.

Part a should be very easy as it is like the radioactive decay problems. Part b has a differential equation that we will not be solving in class. This will once again require that you use Maple's dsolve command to solve this differential equation. Once you have the solution to this differential equation, the problem should be quite easy. You can find the maximum either by techniques learned in class or by lab techniques with Maple.