### Math. 450-01, Fall, 2004 Assignments

This list is updated as the assignments are made.
R. Baker Kearfott

For Friday, August 27:
§1.1 of Giordano, Weir, and Fox, pp. 7--9: 3, 4, 5, 8.
§1.2 of Giordano, Weir, and Fox, pp. 15--17: 1, 3, 6, 7
For Monday, August 30:
§1.3 of Giordano, Weir, and Fox, pp. 31--34: 4, 11.
For Wednesday, September 1:
§1.3 of Giordano, Weir, and Fox, pp. 34--35: Do either Project no. 1 or Project no. 3.
For Friday, September 24, 2004:
1. Create a Mathematica notebook for exploration of the competing species model in §1.4 of the Giordano, Weir, and Fox text.
2. Solve the discrete problem, and create a graph like the graph on page 43 of the text.  You may use the Mathematica notebook illustrating a discrete model and a continuous model of decay with replenishment presented in class as a template.
3. Write down a corresponding continuous model, solve it with initial conditions corresponding to the initial conditions in the discrete model.  You may use the Mathematica notebook illustrating the predator-prey model on page 504 of Boyce and DiPrima (seventh edition)  from the Fall, 2002 course notes as a template.
4. (added later during class discussion) Experiment with your models, with different initial conditions, etc.  Compare the two models.  Draw a direction field for the continuous model, and discuss equilibria and stability of the equilibria.
For Monday, October 11, 2004:
Do problems 1, 2, and 4 from page 113 (§3.2) of the Giordano, Weir, and Fox text.  In problems 2 and 4, fit not only according to the largest deviation (minimax) criterion, but also according to the least squares and the sum of absolute deviations criteria.
For Monday, November 1, 2004:
Considering the population data as in Problem 3, page 172 of Giordano, Weir, and Fox (third edition),
do the following.
• Form a difference table from the data.  Do the following, based on the difference table.
1. Determine possible "outliers," based on the pattern of signs in the difference table.
2. After removing the outliers, speculate on a reasonable highest degree polynomial model for your data.
• Scale the data in a reasonable way, to avoid possible numerical problems.  (You may wish to refer to the Mathematica notebook from our October 11 class.  You may also need to actually experiment with some fits, as in the next problem.)
• Fit the data with polynomials of various orders, and answer the following.
1. What is the best order from the point of view of a plausible, smooth graph?
2. Does the "best order" correspond to your analysis of highest reasonable order from the difference table?
3. In addition to comparing the graphs within the interval [1790,2000] (or corresponding scaled interval), compare the predictions each model gives for the populations in 2010 and 2020.  (I suggest both  graphing the extrapolation and printing values.)
4. Repeat this problem both with and without any outliers you may have identified from your difference table analysis.  Comment on any differences you may have observed.
• Fit the problem with a cubic spline.  (There is a Mathematica function "Spline" that is accessible by loading: "<<Graphics`Spline`".)  Fit with and without outliers, and answer the same questions (including prediction of population at 2010 and 2020) as in your analysis of the polynomial fits.
• Based on all of this analysis, what is your best guess at the population in 2020?
For Wednesday, November 17, 2004:
The torque of an automobile racing engine was measured at 24 degrees centigrade, 10 meters of altitude, resulting in the following table:
 RPM (rotation rate in revolutions per minute) Torque (Newton-meters) 7000 600 7250 607 7500 611 7750 611 8000 609 8250 601 8500 590 8750 578 9000 561 9250 535 9500 501
Using the analysis we did for Problem 2, page 303  of the Giordano / Weir / Fox text, third edition (and repeating our analysis in your report):
1. develop a mathematical model for the torque of that particular engine as a function of the rotation rate, temperature, and altitude.
2. Fit any constants in your model using the tabular data above.
3. Analyze how well your model fits the data.  Try alternate models if your model does not fit the data well.
4. Use your model to predict the torque of the engine at 4,000 meters and 0 degrees centigrade, when it is revolving at 8000 RPM.
The analysis we did in class did not include temperature and altitude as variables.  However, the ideal gas law (Boyle's law)  implies that, for gas of a uniform composition, the density is inversely proportional to the temperature in degrees Kelvin (degrees Kelvin = degrees centigrade + 273.15) and is directly proportional to the pressure.  In turn, an analysis of a column of air gives that the pressure is proportional to e-kh, where h is the altitude;  you can find k by knowing that the pressure at 5,500 meters is about half the pressure at sea level (0 meters).  (We are not taking account of humidity and weather phenomena, but we can ignore those factors in this initial analysis.)  You can combine these considerations to replace density by a term that contains temperature and altitude.
You may also wish to study section 8.3 of the text to look at how submodels are included and how one might choose between competing models.
Final Project, Due Monday, December 6 (counts as one project):
On Wednesday, November 10, we studied the resonance phenomenon in damped and undamped oscillating pendula, both using a nonlinear model (with a sin(y) term) and linear model (with sin(y) replaced by y).  (See the corresponding Mathematica notebook that has been posted.)  In that study, when either the exact model or the approximate model (which corresponds to a more nearly exact model of a vibrating spring) had a non-zero damping, we used a forcing term with frequency that corresponded to the undamped problem.  However, the damping, besides introducing an exponential decay, changes the frequency of the oscillation.
For this project, you should experiment, both with the approximate and the "exact" model, by trying various sizes of the damping term, and by using a frequency for the forcing term that corresponds to the intrinsic oscillation frequency of the damped model.  (Note: You can compute the intrinsic frequency of the damped oscillation directly in the Mathematica notebook.)  Use a sufficiently wide range of damping parameters to get a good idea of how the damped oscillation behaves when driven at the resonance frequency.  Present your notebook (and explanations) in a complete and orderly manner.