Requirements for the Synthesis Paper

You may earn up to 0.4 grade points for writing a synthesis paper as described below. The amount earned will depend on the depth of your analysis and the quality of your writing. Your paper should be about 4 to 6 typed pages long (single-spaced, extra line between paragraphs, in a reasonable font size). The paper is due on Monday, June 6, 2011, and should be submitted to the M422 Sp11 Journals and Papers dropbox.)

The overall goal of the paper is to communicate how your understanding of the mathematical ideas we've studied has developed during this course. The development of ideas in the text splits naturally into two sections. The first section, Chapters 1 through 3, deals with sequences. The objects considered there - step-sizes, areas, partial sums, etc. - are indexd by the natural numbers 1, 2, 3, ... . In Chapters 4 and 5, these discrete sequences are replaced by functions of a continuous variable t or x.

Most of your paper should focus Chapters 1-3 by discussing the table on p. 164 at the end of Chapter 3. The goal of your analysis should be not to explain the specific concepts or examples from these chapters, but rather to explain the overall idea of the table. In doing so, you should address several (but probably not all) of the following issues.

• What is the basic idea of a pairing of discrete and "infinite" objects? Illustrate what you are saying by some examples.
• What is gained by such a table? Be as specific as you can here.
• The notion of a limiting case comes up in connection with many of these pairs. Explain and illustrate this idea in terms of statements such as "a circle is an infinite-sided polygon." What is gained by this idea?
• The notion of a limit comes up in the discussion of many of these pairs. Explain and illustrate how and why this is.
• Whenever the notion of limit comes up, its cousin "approximation" also comes up. How and why is this? How are approximation and limit connected in most people's minds? How accurate, valid, or helpful is this?
• What are some of the interesting distinctions and contrasts that occur to you as you look down through this list? These pairs are supposed to be parallel, in some cases even the same. But how are they different?
• What are some of the confusions and misconceptions that tend to develop around the items on this table among students in M422? How does one avoid falling into them?

The last part of your paper, which should be at least one page long, should discuss the relationship between the mathematical ideas expressed in the table on p. 164 and the material in Chapters 4-5 of the book. What connections, if any, exist between the mathematical ideas of Chapters 1-3 and those of Chapters 4-5? Should there be a table for Chapters 4-5? If so, what should it include and how should it be organized? If not, what is the core idea of these chapters and how is this related to the ideas in the table? How would you summarize the main message of these last two chapters, and of the course?

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Most recently updated on May 19, 2011.