Course materials
Here are some "non-standard" materials I have developed in various courses over the years. I'm including these materials for the free use of anyone who is interested.Calculus II projects:
Developed for Calculus II courses at Haverford College (some for "Accelerated Calculus", which is calculus II with a bit more emphasis on the theoretical aspects of calculus, others for our normal Calculus II course). At the end of the course, each student had to do a multi-week project, either individually or in pairs, with one mandatory consultation with me or preliminary written report after 2-3 weeks to check on progress: For each project, I've tried to note the sources I used in developing the material.The Ultimate Water Slide
Concepts covered: setting up integrals, improper integrals, optimization via Mathematica or calculator(This is essentially original material, although the fundamental problem is the classic brachistochrone)
"Accelerated Calculus" version (Mathematica based): PDF or TeX
Standard version (calculator based): PDF or TeX
The Floating Space Capsule
Concepts covered: setting up integrals, center of mass(adapted from "Student Research Projects in Calculus", M. Cohen, E. Gaughan, A. Knoebel, D. Kurtz, and D. Pengelley, Mathematical Association of America, 1991) The integration and algebra here is a bit messy. Please contact me if you would be interested in solutions.
"Accelerated Calculus" version: PDF or TeX
Standard version: PDF or TeX
The Happy Salmon Fishery
Concepts covered: recursive sequences, differential equations, max/min(adapted from various books on biomathematics) This is the Calculus II version of "Bioeconomics of Fisheries" in Calculus III below; the difference is more emphasis on recursive sequences (including some proofs of convergence), the addition of a differential equation version of the recursive sequence, but only a one-variable optimization for profit.
"Accelerated Calculus" version: PDF or TeX
The Happy Pumpkin Candy Factory
Concepts covered: setting up integrals, optimization via calculator, economic modeling(adapted from scheduling problems in the calculus of variations)
Standard version: PDF or TeX
Calculus III projects:
In a Calculus III course I taught at Williams College, I replaced one of the written midterms by a project. Several of the projects which worked out best can be found below. A detailed evaluation of the positive aspects (generally high student enthusiasm, frequent interaction between students and me, experience with technical and creative writing) and negative aspects (student procrastination, extra work for the students and me) is beyond the scope of this Web page, but I'd be happy to email-discuss my experience with anyone who's considering implementing projects such as these (rmanning@haverford.edu). For each project below, I've tried to note the sources I used in developing the material.Sports projectiles
Concepts covered: integration of vector functions(adapted from project in Edwards and Penney, "Multivariable Calculus", 4th edition, Prentice-Hall, 1994, p. 721 and "A Progression of Projectiles: Example from Sports", by Roland Minton, College Mathematics Journal, Nov. 1994, p. 436).
PDF or TeX
Designing an optimal stock portfolio
Concepts covered: multivariate max/min(adapted from a project in "Applications of Calculus", MAA Notes Number 29, Vol. 3, p. 251).
PDF or TeX
Bioeconomics of Fisheries
Concepts covered: recursive sequences, multivariate max/min(adapted from various books on biomathematics)
PDF or TeX
Computing digits of Pi
Concepts covered: infinite series -- yes, I know it's not a standard topic in Calc III, but it was at Williams(adapted, basically copied, from "Student Research Projects in Calculus", M. Cohen, E. Gaughan, A. Knoebel, D. Kurtz, and D. Pengelley, Mathematical Association of America, 1991)
PDF or TeX
Applied analysis:
This is a course I developed at Williams College, primarily geared toward sophomores and juniors who may be considering majoring in math, physics, chemistry, economics, etc. There are two possible analysis courses in the Williams sequence, this one and a standard analysis course. The specs for Applied Analysis at Williams are that it (1) should give students exposure to some basic ideas of analysis (e.g. convergence) and (2) should cover mathematical tools useful to applied mathematicians, physicists, chemists, economists, etc. I decided on the themes of optimization and approximation as two topics which satisfied both of these requirements.Course Syllabus: PDF or TeX
Linear Programming in-class computer lab (for Matlab's Optimization
Toolbox, running in Williams' Macintosh lab):
PDF
or TeX
Nonlinear Programming in-class computer lab (for Matlab's Optimization
Toolbox, running in Williams' Macintosh lab):
PDF
or TeX
Real analysis:
I developed an in-class project (4 half-days of student work in class, followed by 3-4 lectures) on the application of real analysis to fixed point theorems, especially in game theory. This occurred at the end of the first semester of the course (due in part to the presence of a significant population of mathematical economics minors who were unlikely to continue on to Analysis II). The underlying material is Milnor's proof of the Brouwer Fixed Point theorem, as outlined (and connected to game theory) in Methods of mathematical economics: linear and nonlinear programming by Joel Franklin (Springer-Verlag, 1980).Quite a bit of adaptation was required to yield manageable questions for students with just one semester of analysis under their belts. As it was, there were parts of the story that I felt I had to leave out of the project and just sketch in lecture. Overall, the level of some of the material may have been too intense.
The outline of the material:
The contraction mapping theorem, including application to Newton's method
The hairy ball theorem
Brouwer's fixed point theorem, based on the hairy ball theorem (in lecture only, based on Milnor's paper,
requires stereographic projection and some other interesting but nontrivial steps)
Topological equivalence, to show that given
Brouwer's Theorem on the unit ball, it holds on the convex sets appearing in game theory
Game theory application
Project in TeX or PDF
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