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UMTYMP Calculus III Math Fair at the Geometry Center
Posted:
Jun 21, 1995 2:16 PM
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UMTYMP Calculus III Math Fair at the Geometry Center
On May 23, third year UMTYMP calculus students studying at
the Geometry Center held a math fair. Two of the Center's
postdocs, Rick Wicklin and Davide Cervone are teaching
differential equations and multivariable calculus to the
class and the fair is part of the curriculum. (For more
information on the class and the UMTYMP program see the article
UMTYMP Multivariable Calculus and the Geometry Center.)
The Math Fair was structured as a science fair event with
students presenting their projects at stations. Students worked
in groups of three or four, either trying to model a physical
phenomena or delving further into a mathematical concept they had
studied. Their work ranged from models of heartbeats, plagues
and roller-coasters to methods of visualizing mathematical
concepts like involutes and evolutes. Some groups made their
projects into modules that may be used in following years' classes.
Modeling was a popular topic for the Math Fair. One group of
students, Martin Almlof, Tierre Christen, Sarah Fellows and Apurv
Kamath decided to model the beating of a heart using differential
equations. For the model to be credible, it needs to contain two
states, the diastole state (the state when the heart relaxes and
fills with blood) and the systole state (the state when the heart
contracts and pumps the blood out). They researched and found an
ordinary differential equation that models the change in
electrochemical activity and fiber length over time given by the
following system of differential equations:
e*x' = -(x^3-T*x+b)
b' = x - x0 + u*(x0-x1)
where
u = 1, if b0 <= b <= b1 and (x^3-T*x+b) > 0
or if b > b1
u = 0, otherwise.
The two variables b and x represent electrochemical activity and
muscle fiber length respectively. This model has two equilibrium
states, one at (b0,x0), the diastolic state, and the other at (b1,x1),
the systolic state.
Students plotted phase portraits of this system and showed how the
dynamics vary as parameters are changed. Students also plotted the
electrochemical activity versus time, which produces the famous
Electrocardiogram or EKG graph.
Students created a lab in hypertext form. In this lab one could alter
the above parameters to the differential equation and see by producing
phase portraits and EKG graphs, how that would change the model. The
lab contains questions about what happens in the model when various
parameters are changed, what happens to the EKG graph when various
values are changed, and what do some of the parameters represent.
Two groups did work on modeling epidemics. They too researched and
found differential equations to model outbreaks. They started with the
following model for the spread of a disease:
S' = -a*S*I
I' = a*S*I - b*I
R' = b*I
Where S represents the number of people susceptible to the disease, I
the number of people infected, and R the number of people recovering
or immune to the disease. The parameter a represents the
infectiveness of the disease, and b the rate of removal from the
infected population.
One group of students Mike Arvold, Bjorn Soderlund, and Jeremy
Tremblay, created a hypertext lab. In the lab they explained and or
showed mathematical phenomena such as equilibria states and
bifurcations. They plotted phase portraits of the above differential
equations and included plots of the rate of infection as a function
of time. They also made the model more intricate by altering the
differential equation to include seasonal changes, as well as allowing
births and non-disease deaths.
Another group of students, Anastacia Rohrman and Kelly Plummer,
compared their mathematical model to data from smallpox statistics in
Mexico from 1922-1951. They incorporated vaccination into an Ordinary
Differential Equation in order to model the effects of the actual
vaccination efforts occurring in Mexico during that time.
How does one make a safe roller coaster ride? This was a question
behind two groups' work. One approach to answer this question comes
from a roller coaster designer, Schwarzkopf. A result from his
work is that safe roller coaster loops are ones where the radius of
curvature of the loop decreases at a constant rate. The curvature K
of a curve at a point is the magnitude of the rate of change of the
direction of the curve with respect to arclength. Sandy Choi, Laura
Wenzel, and Kris Keller then created via Maple, a mathematical software
program, various parametrized curves. They had Maple color segments of
the curve according to the degree of its curvature. They also plotted
the curvature as a function to illustrate which curves would be safe
for roller coaster rides, and which would not.
What is the involute of a curve? What is an evolute? What happens
when you take an evolute of an evolute, or an evolute of an involute?
Chris Wyman, Erik Streed, and Tim McMurray tried to answer this in
their project. Roughly speaking the evolute of a curve x(t) at t0 is
another curve y(t) for which a circle of radius |y(t0) - x(t0)|
centered at y(t0) best fits the curve at x(t). The involute can be
thought of as a curve w(t) created by unwrapping a taught string
from another curve z(t). The students plotted various curves as well
as the involutes and evolutes of them, and made some interesting
discoveries such as the evolute of a cardiod is a cardiod, in fact one
that is similar to the original cardiod. They also found some curves
where the involute and evolutes are inverses of one another.
Overall the fair was an entertaining and enjoyable experience.
Students demonstrated not only their mathematical acumen, but also
were able to communicate their knowledge using several different
media in several different ways. My only disappointment is that I
wasn't able to see all of the exhibits. Fortunately, some of the
students' hypertext documents are accessible on the Web at the URL
address http://www.geom.umn.edu/docs/education/student/ of the
Geometry Center.
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