General Mathematics

1910 Submissions

[2] viXra:1910.0165 [pdf] replaced on 2019-10-12 16:29:46

Tutorial: Continuous-Function on Closed Interval Basics, with Mean-Value and Taylor Theorem Upshots

Authors: Steven Kenneth Kauffmann
Comments: 4 Pages.

This tutorial explores the relation of the local concept of a function's continuity to its global consequences on closed intervals, such as a continuous function's unavoidable boundedness on a closed interval, its attainment of its least upper and greatest lower bounds on that interval, and its unavoidable assumption on that closed interval of all of the values which lie between that minimum and maximum. In a nutshell, continuous functions map closed intervals into closed intervals. It is understandable that verifying this local-to-global fact involves subtle and very intricate manipulation of the least-upper-bound/greatest-lower-bound postulate for the real numbers. In conjunction with the basic inequality properties of integrals, this continuous-function fact immediately implies the integral form of the mean-value theorem, which is parlayed into its differential form by the fundamental theorem of the calculus. Taylor expansion and its error estimation are further developments which are intertwined with these fascinating concepts.
Category: General Mathematics

[1] viXra:1910.0022 [pdf] replaced on 2019-10-08 04:49:08

Using the Rational Root Test to Factor with the TI-83

Authors: Timothy W. Jones
Comments: 5 Pages. Further corrections and amplifications.

The rational root test gives a way to solve polynomial equations. We apply the idea to factoring quadratics (and other polynomials). A calculator speeds up the filtering through possible rational roots.
Category: General Mathematics