MATHEMATICS

1. In about one page, discuss the theme “Analysis is all about approximation.” Be sure to address the following five important calculus topics and any connection to approximation: Differentiability Integrability Convergence of Infinite Series Taylor Polynomials Uniform Continuity Also address at least one of the following important calculus topics and any connection to approximation: Density of Rationals in R. Least Upper Bound Sequence Convergence and Divergence Limits of functions Continuity 2. Other than Approximation, what is the most important definition, axiom, theorem, or concept in all of Real Analysis? Justify your response in an essay that is approximately one page long.

Scenario: Interpret the charts (link below) showing the history of # IPOs from 1975 to 2008: What additional perspective is gained by presenting these data as behavior charts? What action could be taken in terms of Deming’s insistence on using data for “prediction” and why? Week4_DQ 1_charts.doc [“Inherited” from past instructor — doesn’t mean it’s right…] Helpful clarifying information: · An initial public offering, or IPO, is the first sale of stock by a company to the public. A company can raise money by issuing either debt or equity. If the company has never issued equity to the public, it’s known as an IPO. · As such the number of IPOs can be used as one indicator of the degree of business expansion in a given time period. For your dialogue, ponder: Do you see “patterns” in the time periods corresponding to economic events during these years? Are they the types of things that could happen again? – if so, can you “predict” consequences? Bottom line: Don’t concentrate on individual points or individual special cause tests, but, rather, keep the focus when possible on the process “needle.” Just because a special cause test is triggered doesn’t necessarily mean that the special cause happened there!

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