MATHEMATICAL ANALYSIS I
Academic year and teacher
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- Versione italiana
- Academic year
- 2016/2017
- Teacher
- ELISABETTA LORENZETTI
- Credits
- 9
- Didactic period
- Primo semestre (primi anni)
- SSD
- MAT/05
Training objectives
- Expected achievements (Dublin Descriptors)
• Knowledge and understanding
Aim of the course is to provide the basic tools of differential and integral calculus for functions of one real variable: limits, derivatives and integrals. Moreover, it emphasizes the use of logical-deductive methods in the setting and solving of problems based on mathematical models.
• Applying knowledge and understanding - Making judgements
At the end of the unit, students will be able to properly apply the concepts learnt to the resolution of different kinds of (applicative and not) problems and to identify the most appropriate approach to do so. Students will also be able to justify the choices made.
• Communication skills
At the end of the unit, students will be able to communicate in an effective and proper way and will demonstrate logical argumentation and synthesis skills.
• Lifelong learning skills
At the end of the unit, students not only will know the theoretical contents and the methods proper of mathematical analysis calculus but will also again a deep understanding of the problems encountered, which will support them to overcome issues in the working and/or research environment. Prerequisites
- Good knowledge of:
• elements of set theory;- elementary properties of Natural, Integer and Rational numbers;
• algebraic equation and inequalities of first and second degree;
• algebra of polynomials: factorization of a polynomial given its roots, division of polynomials;
• Pythagorean theorem and Euclid's theorems for right-angled triangles;
• trigonometry: definition of cosine, sine, tangent and addition formulae for cosine and sine;
• elementary properties of exponential and logarithmic functions;
• plane analytical geometry. Course programme
- Numerical sets: natural, integers, reals, complex numbers and their properties.
Elements of combinatorial calculus.
Elementary functions of one real variable and their graphics.
Limits and continuity of functions and asymptotic relations.
Numerical sequences and series.
Differential and integral calculus (computation of derivatives and primitives).
Properties of differentiable functions and local approximation of differential functions; Taylor polynomial and its applications.
The Riemann integral and the fundamental theorem of calculus.
Generalized integrals.
Study of the graph of a function in one real variable and optimization problems.
Functions of complex variable: exponential and logarithmic functions in complex field.
Differential equations od first and second order. Didactic methods
- * 72 hours of frontal lectures (48 theoretical and 24 of exercises) during the fall semester
* 2 weekly office hours of group and individual tutoring
* use of an electronic mailing list for discussions about course topics Learning assessment procedures
- Learning assessment procedures
Aim of the examination is to test the students with regard to the above mentioned expected achievements. The examination is comprised of a written test and, once this test is passed, of an oral test.
With reference to the Dublin Descriptors:
¿ The written test is 2 hours and 30 minutes long and is comprised of 8 exercises regarding the contents of the unit. The exercises proposed are similar to those solved during the lectures and tutorials or that have been proposed on the internet. The written test aims at verifying the student’s capacity to: understand the problems encountered during the unit, properly apply the theoretical knowledge, formulate the most appropriate strategy towards the resolution of the proposed exercises, communicate in written form in an effective and appropriate fashion.
During the examination it is neither possible to consult books, nor to use personal computers or smart phones; it is however possible to use a personal calculator. The student is allowed to bring with himself a paper (A4) containing ONLY the formulas deemed as useful; the paper is to be signed by the student and to be delivered to the professor at the end of the examination itself. It is severely forbidden to consult any other kind of notes.
The maximum score for each exercise is of 5 points. Students who gain an overall score of 20 (at least) are admitted to the oral test.
¿ The oral test is about 40 minutes long and aims at verifying:
• the degree of knowledge of theoretical contents and, in case mistakes are present in the written test or fundamental parts of it are not approached, of applicative skills;
• the degree of competency in the oral exposition: appropriateness in the terminology used, logical-deductive and synthesis skills, ability in argumentation and reasoning, autonomy in the evaluation of the most opportune approach in answering to the questions and problems proposed.
The final score will generally result from the average of the scores obtained in the two tests (written and oral). Reference texts
- M.Bramanti, C.D.Pagani, S.Salsa-"Analisi matematica 1" con elementi di geometria e algebra lineare - Zanichelli (ISBN 978-88-08-25421-4)
M.Bertsch, R.Dal Passo, L.Giacomelli - "Analisi Matematica" -McGraw-Hill (ISBN978-88-386-6234-8).
Internet notes available at: materiale didattico/a.a.2016-2017/Dispense
Esercizi: Marco Bramanti Esercitazioni di Analisi Matematica I SocietàEditrice ESCULAPIO ISBN 978-88-7488-444-5
Exercise:Marco Bramanti Esercitazioni di Analisi Matematica I SocietàEditrice ESCULAPIO ISBN 978-88-7488-444-5
