Download Final Projects

Final Projects
Here are some potential topics for your final project. In each case I give you a specific
reference from which you can get started.
1. Centers of Mass: One interesting application of the definite integral is to compute
centers of mass for physical objects. When Newton was first discovering
calculus, he conceived the idea that we could imagine gravitational effects by
replacing an object like the moon with a theoretical object, where all the mass is
concentrated at a single point (at the center of the moon). These ideas led him to
invent the integral! In this project you would describe the basic ideas of center of
mass, and do a few examples. Reference: Most “fat” calculus books have a
section on this. For example, Section 5.7 in Thomas & Finney’s calculus book
would do. There is also a brief Interlude in our text about this (pp. 454-458).
[Lee]
2. Population Growth and the AIDS epidemic: Epidemics can be modeled by
ordinary differential equations; an unrealistic model might lead to unfettered
exponential growth. A more realistic model might lead to the logistic curve (see
pp. 294-297 in our text). The AIDS epidemic has been modeled by a cubic curve
(see Applications in Calculus, P. Straffin, editor, pp. 215-221). In this project you
would report on these ideas.
3. Present Value: A dollar you might expect to get a year in the future is not really
worth as much to you as a dollar in your pocket today. How do we model this
economic phenomenon? The economists have a notion of present value to
describe this (see section 7.5 in our text). In this project you would describe the
idea from economics, and explain how an integral helps us compute this. [Janet]
4. Euler’s Remarkable Formula: In class I reported that Euler had shown that the
sum of the reciprocals of squares of positive integers is actually equal to 2/6. In
this project you will describe how Euler reached this conclusion, and also look at
a more modern argument. (See Calculus by George Simmons, sections A12 and
A13). [Jette]
5. The Brachistochrone: Suppose we connect two points with a smooth curve, and
we imagine a bead sliding from the first point to the second. Under the effects of
gravity, which curve will lead to the fastest bead? This is the famous
brachistrchrone problem. In this project you do a little physics and calculus to
describe which curve is best. (See Calculus by George Simmons, Section 17.2).
6. Splines – fitting a curve to data: If you have some data points, and you’d like to
give a curve passing through them, you could certainly draw a line segment
between each successive point – but this leads to a curve with many sharp corners
(places where there is no derivative). To model real-world phenomena we’d like
a smooth curve. This leads to the idea of quadratic and cubic splines. (See pp.
517-523 in our text).
7. Safe and Effective Dosage: A single dose of drug becomes less effective as the
drug is eliminated from the body. Consequently, we’d like to administer repeated
doses, which keep the amount of the drug in the blood stream both effective and
safe. What is a mathematical model for this? (See exercises 37 and 38, in
Thomas & Finney’s Calculus.) [Stephanie]
8. Euler’s Method: Many ordinary differential equations modeling real-world
phenomena cannot be solved exactly. But we can nevertheless plot a solution,
using a numerical method discovered by Euler. In this project you will describe
how this algorithm works (See section 6.3 in our text).
9. Where should you sit at the movies? If you sit too close to the movie screen in a
typical theater, the angle subtended by the screen at your eyes is too small for
optimal viewing. But the same thing occurs if you sit too far back! Finding the
optimal solution is a max-min problem that requires some computation. See p.
463 in Stewart’s Calculus.
10. The Catenary: If a flexible chain with uniform density is hung on two points, the
curve it describes is called a catenary. We can describe this curve
mathematically, with a little physics and a little bit of integration (a trig
substitution!) See Section A9 in Simmons’ Calculus.