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 similar to the earlier problem you worked in lab for a yeast culture. You begin this problem by plotting the data in the second and third and fifth and sixth columns. Use Trendline as you have before to get the best quadratic through the data. Be sure to set the y-intercept equal to zero in the options and watch that you have at least 4 significant figures from the equation. Use that equation to compute equilibria (zero growth), find the vertex (maximum growth), and determine the sum of square errors between the model and the data.

The next part of the problem has you define variables a and b for the Gompertz model. The values you choose are not very sensitive, but taking a = 0.1 and b = 0.2 seems to work fine. You define the sum of square errors between the data and the Gompertz function, then use Solver as you have before to minimize this sum of square errors. You should graph the Gompertz function with about 50 evenly spaced points along the domain of your function (the p values). This is added to the graph from Part a.

The last two parts have you simulate the nonlinear discrete dynamical model starting with a variable that you define, p₀. Again you find the sum of square errors between the time series data given in the table and the simulation of your model using either the logistic or Gompertz growth function. With Solver you minimize the sum of square errors by only changing p₀. The rest of the questions that you need to answer are fairly routine.