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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.