Mathematics module: Analysis for high school mathematics teachers
Lecturers and Teaching Assistants
Prof. Sergei Yakovenko
Course Schedule and Location
Tuesday, 09:15 - 12:00, Mausher, Conference Room
Field of Study, Course Type and Credit Points
Science Teaching (non thesis MSc Track): Lecture; Obligatory; Regular; 4.00 points
Science Teaching: Lecture; 2.00 points
1st year + 2nd year
For students in the Rothschild-Weizmann program only
Attendance and participation
Estimated Weekly Independent Workload (in hours)
The course will have an interactive component in the form of a "blog", where students are encouraged to ask questions and discuss matters between each other (visit the site http://yakovenko.wordpress.com for a previous albeit not very popular attempt). The class language is Hebrew, yet most of the written notes, books, and correspondence will be in English. Basic writing proficiency in any dialect of TEX/LATEX is a very desirable requisite.
Tentative syllabus of the course
- Introductory lecture. Infinity is a distinctive feature of analysis. Examples and counterexamples.
- Infinite sets and their comparison. "Paradoxes" of infinite sets. Cantor-Bernstein theorem. Diagonal process. Continuum hypothesis.
- Number systems. Natural, integer, rational, algebraic numbers. Completion: real numbers as the result of gap sealing. "Existence" and abundance of irrational and non-algebraic numbers. Complex numbers. Completeness and closure of the complex numbers.
- The notion of limit. Limits of sequences and functions. Other types of limits (exercises). Continuous functions. Tabulation.
- Series. Number series, series of functions.
- Differentiability and linearization. Differential: the meaning of dx and df. Chain rule.
- Polynomial approximation of functions. Taylor polynomials. Analytic functions.
- Integration: anti-derivation and limit of various integral sums (Riemann and Lebesgue approaches).
Upon successful completion of the course students will be able to:
- Demonstrate understanding of the main concept of mathematical infinity in various disguises (real numbers, limits, constructions involving countably many steps).
- Develop proficiency in working with logical definitions formalizing intuitive constructions.
- Great a link with the operational techniques of analysis in its applications to the study of functions and equations.
- W. Rudin, Principles of Mathematical Analysis
- R. Courant, H. Robbins, What is Mathematics
- Lecture notes available at the site http://yakovenko.wordpress.com/category/lecture/rothschild-course-analysis-for-high-school-teachers/