Unit MATHEMATICAL ANALYSIS I
- Course
- Mathematics
- Study-unit Code
- GP006036
- Curriculum
- In all curricula
- Teacher
- Anna Salvadori
- Teachers
-
- Anna Salvadori
- Anna Salvadori
- Hours
- 30 ore - Anna Salvadori
- 43 ore - Anna Salvadori
- CFU
- 9
- Course Regulation
- Coorte 2022
- Offered
- 2022/23
- Learning activities
- Base
- Area
- Formazione matematica di base
- Academic discipline
- MAT/05
- Type of study-unit
- Obbligatorio (Required)
- Type of learning activities
- Attività formativa monodisciplinare
- Language of instruction
- Italian
- Contents
- Basic course of Mathematical Analysis for functions of one variable.
- Reference texts
- P.Brandi - A.Salvadori, Prima di iniziare, Aguaplano Officina del Libro (2015) - [reference text for basic knowledge]
P.Brandi - A. Salvadori, Percorsi di Matematica, 2 volums, Aguaplano-Officina del libro, Passignano s.T. (PG), (2015) (2015) (textbook)
William F. Trench, Andrew G. Cowles, Introduction to real analysis, Department of Mathematics
Trinity University, San Antonio, Texas, USA, http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF
Vladimir A. Zorich, Mathematical Analysis I, Moscow State University, Universitext, Springer
http://math.univ-lyon1.fr/~okra/2011-MathIV/Zorich1.pdf
G.C. Barozzi G.Dore E. Obrecht, Elementi di analisi matematica, Zanichelli Ed. (2011) - Educational objectives
- The course has the role of introducing students to the structures the demonstrative processes and the argumentative tools of the discipline.
The student will have acquired basic knowledge of Mathematical Analysis for functions of one variable, a good ability to conjecture, argue and prove. He will have also acquired basic skills on elementary modeling. - Prerequisites
- Mathematical Analysis 1 is a challenging and intensive course (CFU 9 - 13 weeks). In order to follow the lessons in a profitable way, it is essential to have a good preparation on the basic knowledge base (see below).
Basic knowledge. Order relation. Algebra of polynomials. Elements of analytic geometry. Elements of goniometry and trigonometry.
Elementary functions and their inverse or partial inverse.
Transformations of a function (translation and rescaling) and effect on the graph.
Elementary equations and inequalities.
Logic elements (conjunctions and, or, not, De Morgan's laws). - Teaching methods
- The course is organized as follows:
- Classroom lectures on all the topics of the course
- Exercises in the classroom with student involvement
- Two written exercises in the classroom as a simulation of the written exam - Learning verification modality
- The exam includes a written test and an oral interview.
Written test (2 hours)
- Type: resolution of some open questions; It is allowed the use of textbooks, manuals, graphic-symbolic calculators (strictly off-line);
- Aim: to assess the knowledge, skills and expertise in arguing, conjecturing and demonstrating.
It will be particularly appreciated not only the correctness of the procedures, but also the quality of the arguments used to support the answers.
Interview (30-45 minutes)
Aim: to assess the communication skills of the student, and the skill in organizing the exhibition content.
The exam will assess the knowledge on the course content and the skills acquired in the demonstrative and argumentative tools.
The final evaluation will be based on the results of the written test and the outcome of the interview.
Timing: the date of the written tests are fixed; that of the interview can be agreed with the teacher.
For information on support services for students with disabilities and / or SLD
visit the page http://www.unipg.it/disabilita-e-dsa. - Extended program
- Order structure in R. Maximum and minimum of a set. Dedekind’s extension (upper and lower bound). Sequences. Induction principle.
Topological structure of R and extended R. Concept of limit; basic properties. Indeterminate forms and fundamental limits. Infinite and infinitesimal.
Continuity and uniform continuity. Conservation of compactness. Weierstrass Theorem. Conservation of connection. Intermediate value theorem.
The derivative. Differentiable functions: local and global properties (Fermat, Rolle, Lagrange, Cauchy, Hospital). Higher order derivatives. Linearization methods. Polynomial approximation.
Qualitative study of the graph of a function. Optimization problems.
The Riemann integral. Antiderivatives. Torricelli-Barrow theorem. Darboux theorem. Integral function. Integration's techniques.
Generalized integrals and numerical series. Convergence criteria for numerical series. Taylor series. Asymptotic expansions.