(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.
Determine possible "outliers," based on the pattern of signs in the difference
table.
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.
What is the best order from the point of view of a plausible, smooth graph?
Does the "best order" correspond to your analysis of highest reasonable
order from the difference table?
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.)
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):
develop a mathematical model for the torque of that particular engine as
a function of the rotation rate, temperature, and altitude.
Fit any constants in your model using the tabular data above.
Analyze how well your model fits the data. Try alternate models if
your model does not fit the data well.
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.