Themen der Lehrveranstaltung
• Complex numbers. Definition, algebraic operations, trigonometric and exponential form, algebraic equations. • Real functions, limits and continuity in one variable. Review of the properties of real numbers and the basic concepts on real functions of one real variable. Sequences and limits for sequences. Limits and continuity of functions. Infinite and infinitesimal functions: Landau symbols and rate of convergence. • Differential calculus of real valued functions in one variable. Tangent to a graph and first derivative. Rules of differentiation. Differentiability and singular points. Theorems of Rolle and Lagrange. Absolute and relative extrema. Higher-order derivatives. Qualitative study of a function. •Local comparison of functions and Taylor expansions in one variable. Taylor formulas. Expansions of elementary functions and algebraic techniques to determine Taylor polynomials. Local analysis and limits calculation using Taylor expansions. • Real sequences and numerical series. Convergence criteria for numerical real series. Fundamentals of Taylor series • Integral calculus of real valued functions in one variable. Antiderivatives and rules of indefinite integration for functions in one real variable. Definite and improper integrals.

Unterrichtsform
The course is made up of a series of frontal lectures, both devoted to the presentation of theoretical concepts and to their application in exercises. Topics will be presented on the blackboard and explanations will be supported by the use of software, both in analysing calculations and for the graphical visualization. The reference textbook for theory is cited in the bibliography. During the course lists of exercises will be made available to the students. Each of the activities carried out during the course’s hours will be documented on the OLE web site.

Bildungsziele
The course is part of basic didactic activities and in particular in the disciplinary sector “Mathematics, Informatics and Statistics”. The course is compulsory. It is and introductory course, devoted to the basic notions of mathematical analysis, in particular of infinitesimal, differential and integral calculus for real functions of one real variable. The objective is not only to convey a set of notions and tools, but also to acquire a deeper understanding of the mathematical concepts from which they are derived. This is fundamental in order to be able to use learned techniques in a rational way and to apply them to other technical and scientific subjects. Intended Learning Outcomes (ILO) Knowledge and understanding 1. Knowledge of the fundamental concepts and basic tools of infinitesimal, differential and integral calculus for real functions of one real variable. Applying knowledge and understanding 2. Mastering of a methodology of non-mechanical, rational application of calculus techniques. Making judgements 3. Ability to evaluate the characteristics of the problem to be solved, set it in the correct theoretical framework and subsequently choose the most suitable method to get to its solution. Communication skills 4. Mastery of the use of the specific mathematical lexicon. Learning skills 5. Capability to a rational, analytical approach in handling problems

Art der Prüfung
The final exam consists of: - a written examination, with exercises about the topics covered during the course, that aims at verifying the understanding of the theory and the ability in using the relevant calculus techniques; - the subsequent discussion of the corrected test.

Bewertungskriterien
A single final mark will be given. The written examination is composed of a first part in which the achievement of the minimal requisites of understanding of concepts and ability in the application of calculus tools is proofed (ILOs 1-2) and a second part consisting in the solution of exercises where the mastery of different topics, the ability to combine calculus tools and to have a comprehensive vision of the problem are evaluated (ILOs 3-5). In the second part, the clarity of the solution scheme, the appropriateness of the methodology chosen in solving the problems (ILOs 3-4) and the correctness of the application of calculus techniques (ILOs 1-2) are assessed. Any errors and omissions underlined during the correction will be discussed with the student. Formative assessment Form: written exam Length /duration: Short answer test with pass threshold of 60% ILOs assessed: 1-2 Summative assessment Form: written exam + (optional) Discussion of the written exam Grade: 95% for the written exam + 5% for the discussion) Length/Duration: 4 exercises (120 minutes) for the written exam + 5-10 minutes for the discussion ILOs assessed: 1-5 for the written exam + 4-5 for the discussion