This is your Lab Final worth double a normal lab and should be worked individually. However, over the semester, you have developed partnerships that have worked with varying degrees of success. You may want to team up with others with whom you worked well, then work in parallel as the problems are essentially the same with only minor changes in the numbers. This lab begins with a graphing and derivative problem for which Maple should help. The second question is an allometric model very similar to some of your earlier lab questions. The third problem examines drug treatment regimes and allows you to use the derivative to find peak drug concentration. The last problem is another discrete dynamical model for studying insects.

Question 1: This problem extends the work that we are doing in class on finding minima and maxima of a function to a more complicated function. You will also find points of inflection. You will want to use Maple to differentiate this function and find values of the extrema. Maple will help to find the points of inflection by taking the second derivative. The Maple commands that you need are listed below (and you may want to review previous help sheets). First, define the function.

> f : = x -> (16-x^2)*exp(-x^2);

Next you should plot the function on the intervals listed to have an idea of what you are examining.

> plot(f(x), x = -10..10);

To find the derivative you type:

> df := diff(f(x), x);

You can simplify the expression by typing either of the following commands.

> simplify(%); factor(%);

To graph the derivative you type

> plot(df, x = -10..10);

To find when the derivative is zero, you first look on the graph to get an approximate idea. Then you type

> xc := fsolve(df = 0, x = 1..10); f(xc);

This second command gives the function value of the maximum or minimum.To find a point of inflection we take the second derivative, simplify it, then set it equal to zero. Finally, we do a function evaluation at the point of inflection.

> sdf := diff(df, x); factor(%);

> xpoi := fsolve(sdf = 0, x = 1..10); f(xpoi);

Question 2: This problem is very similar to other allometric problems that we worked early in the semester. You fit the data using Excel's Trendline and the Power Law option to get the best fit. The second part has an applet where you need to minimze the sum of squares error. You will probably want to start the applet near the values that you got from Excel in Part a.. To add the graph of the function that you found with the applet, you will need to create a series of points with which to track the function.You need a large number of points for small values of w, but fewer as w increases to 500. (I would suggest starting with 0.015, 0.02, 0.04, 0.06, 0.08, 0.1, 0.15, 0.2, 0.3, 0.5, 0.7, 1, 2, etc. until you reach 500.) The remainder of the problem is just doing an error analysis and writing up your results. 

Question 3: In this problem you will be graphing the given functions, then finding when they are equal to certain values.The exponentially decaying amount of drug should be a problem you can do by hand. The second part requires differentiating exponentials to find the maximum. You can have Maple help you with this or practice the techniques by hand. The most difficult part is trying to find when the second drug level is at 3 mg. This is most easily done using Maple's fsolve command, where you define the function B(t), then type fsolve(B(t) = 3, t = 0..20); You adjust the second part of the command to include the t value that you expect for when B(t) = 3 from examining the graph.

Question 4: This is another problem using real data. The first part of this problem has you finding a discrete logistic growth model for these beetles based on the data. Follow the directions carefully to obtain the right alignment of the data to use Excel's trendline polynomial fit and to get the right quadratic function here. Note that the function you are producing here is the updating function for finding the next population. Its Part b. where you actually simulate the logistic growth model to see how well it fits the population data as a function of time as appears in the table. In Part b. you find equilibria much as we have done before. I'm also asking about stability of the two equilibria you find by having you put the equilibrium values into the derivative of the growth function. If the derivative at this point is less than one in absolute value, then the equilibrium is said to be stable. Give your value of the derivative to verify if the equilibrium is stable or not.

Part c. corresponds to the calculations in Part a. with a different model (Ricker's model), but in this case you are only plotting the function to see how it looks. You should notice that it is shaped similar to the quadratic in Part a over the range of the data, but this function never goes negative, which is a complaint biologists have with the logistic growth equation. Part d. is similar to Part b., but this function is a little harder to solve for its equilibria and for taking its derivative. Maple might help you here!