The number of spam attacks seems to approximately follow a logistic model (see insert). Here is a list of the number of attacks P (in millions), starting on January 2001 (t = 0) through october 2002 (t = 21 ): 0.74, 0.81, 0.89, 1.01, 1.14, 1.301, 1.50, 1.74, 2.01, 2.31, 2.64, 2.98, 3.33, 3.67, 4.00, 4.30, 4.57, 4.80, 5.00, 5.16, 5.30, 5.41 Your job will be to find a suitable model for this data, and to use it to determine some trends. |

**Write a report that has the following parts.**

- Title, Author, Date, Course and Section, Instructor.
- Calculate numerical estimates of for t =1 through t = 20. For this, use the "central difference " approximation
- Plot of (vertical axis) versus (horizontal axis) for the points you have information on (see previous question)
- Use Maple's
**fit**command to fit a polynomial curve

to the points in the previous question by choosing suitable values of the parameters , , and . Plot together the points and the curve that you have chosen, so that it can be verified (visually) that you have a ``reasonable fit''.

- The differential equation

is satisfied approximately by the number of spam attacks (in millions) as a function of time (in months). Explain why.

- Use algebra and Maple to determine if there exist equilibrium values of . Verify your answer by producing a slope field plot with 1 < t < 30 and 0 < P < 10.. Comment on whether the plot confirms or not your conclusion on equilibrium values.
- Generate a slope field plot with 0< t < 100 and 0 < P < 10 . What does the model predict the number of attacks will be in January of 2003? What will be tthe number of attacks in January of 2008?

**Tips, comments, and additional information**

- In this example, points and a curve are plotted together:

with(plots):

p1 :=PLOT(POINTS([1,2],[2,4],[1.5,3]));

p2 := plot(x^2,x=0..2):

display([a,b]);

To use differential equations related commands in Maple you must first load the package DEtools. Here is an example of how one plots in Maple a direction field and a solution of the logistic differential equation . Note the x(t) term:

with(DEtools):

de1 := diff(x(t),t) = 3*x(t)*(1-x(t)/300);

DEplot(de1,x(t),t=0..3,x=0..400);

Here is how to plot the slope field and a particular solution:

DEplot(de1,x(t),t=0..3,x=0..400,[[x(0)=50]]);

Maple can solve some differential equations, for example:

dsolve({de1,x(0)=50},x(t));

Curve fitting in Maple may be done with the fit command. Here is one example:

with(stats):

xvals = [-2.,-1.,0.,1.,2.];

yvals = [-3,0,1,0,-3];

fit[leastsquare[[x,y], y=a*x^2+b*x+c]]( [xvals,yvals]); - The final project should have only one author. You may discuss the project with your classmates, but what you turn in should contain your own original designs.
*Before each calculation, describe in English what you are about to do.*- Use the electronic submission system to turn in your work.
- Maple should be used in all calculations and plots.
- For basic information on Plotting, solving equations and
calculating integrals in Maple: see the Maple worksheet
*`` Introduction to Maple in Calculus II''(intro142.mws)*, located in*www.math.uri.edu/Center/* - MAPLE HELP is be available. The schedule and location is
announced in
*www.math.uri.edu*.