5,758 research outputs found
On Kahan's Rules for Determining Branch Cuts
In computer algebra there are different ways of approaching the mathematical
concept of functions, one of which is by defining them as solutions of
differential equations. We compare different such approaches and discuss the
occurring problems. The main focus is on the question of determining possible
branch cuts. We explore the extent to which the treatment of branch cuts can be
rendered (more) algorithmic, by adapting Kahan's rules to the differential
equation setting.Comment: SYNASC 2011. 13th International Symposium on Symbolic and Numeric
Algorithms for Scientific Computing. (2011
High School 4th Mathematics: Precalculus for AP Calculus
ABSTRACT The purpose of this thesis is to provide the needed instructional materials to those who are teaching a Precalculus course following Algebra I, Geometry, and Algebra II. The recent adoption of the Common Core State Standards in Mathematics (CCSSM) has left many teachers scrambling to find instructional materials that meet the graduation requirement as well as insuring that our students are college and career ready when they leave high school. Furthermore, the College Board’s Advanced Placement (AP) Calculus curriculum is generally accepted as the model for a twenty-first century calculus course serving as prerequisite for STEM related fields of study at the college level. The path now needs to be set for a new precalculus course to align the AP goals and objectives with the CCSSM. For the 2014-2015 school year, high schools must offer AP courses in all four core content areas, math, ELA, science, and social studies (www.louisianabelieves.com). However, for students to be adequately prepared for AP Calculus there must be an effective precalculus course available to be taken first. This thesis, “High School 4th Mathematics: Precalculus for AP Calculus,” is written specifically with the goal of meeting this requirement. In Appendix C of this thesis, high school mathematics teachers are provided with comprehensive lecture notes that contain lesson plans and student activities that are aligned with AP Calculus ready, the CCSSM, and the Common Core State (+) Standards in Mathematics (CCS(+)SM). Each section of the lecture notes consists of a lesson plan that begins with a comprehensive overview of the major concepts, a list of the related CCSSM, a set of section learning objectives, lecture notes, and a variety of lesson activities that support the Common Core State Content Standards as well as Mathematical Practice Standards (MPS). Even though Appendix C can be used by any Precalculus teacher as a resource, it is designed specifically to go along with the textbook, Precalculus 8th Edition, written by Demana, Waits, Foley, and Kennedy, the textbook which will be used in 2014-2015 by Southeastern Louisiana University for its Dual Enrollment Precalculus course, Math 165
If Archimedes would have known functions
These are notes and slides from a Pecha-Kucha talk given on March 6, 2013.
The presentation tinkered with the question whether calculus on graphs could
have emerged by the time of Archimedes, if the concept of a function would have
been available 2300 years ago. The text first attempts to boil down discrete
single and multivariable calculus to one page each, then presents the slides
with additional remarks and finally includes 40 "calculus problems" in a
discrete or so-called 'quantum calculus' setting. We also added some sample
Mathematica code, gave a short overview over the emergence of the function
concept in calculus and included comments on the development of calculus
textbooks over time.Comment: 31 pages, 36 figure
Orthogonal polynomials of compact simple Lie groups
Recursive algebraic construction of two infinite families of polynomials in
variables is proposed as a uniform method applicable to every semisimple
Lie group of rank . Its result recognizes Chebyshev polynomials of the first
and second kind as the special case of the simple group of type . The
obtained not Laurent-type polynomials are proved to be equivalent to the
partial cases of the Macdonald symmetric polynomials. Basic relation between
the polynomials and their properties follow from the corresponding properties
of the orbit functions, namely the orthogonality and discretization. Recurrence
relations are shown for the Lie groups of types , , , ,
, , and together with lowest polynomials.Comment: 34 pages, some minor changes were done, to appear in IJMM
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