Subject: CALCULUS 1 (A.A. 2018/2019)
Unit analisi matematica I
Mathematics, Information Technology and Statistics (lesson)
OFA - Obblighi formativi aggiuntivi
Learning of the basic notions of the calculus for one variable function. Use of asuitable mathematical language. Mastery of the deductive reasoning. Capability of tackling problems with a strict logical planning and to fit the learned techniques to them.
We require students to be familiar with the following topics: elementary algebra, basic notions in Analytic Geometry, equations and inequalities, trigonometry, main properties of the exponential and logarithmic functions.
Real numbers, notions about order: maximum, minimum, least upper bound, greatest lower bound. Induction principle. Basics of combinatorial calculus, Newton's binomial expansion. Complex numbers: operations, powers, roots, De Moivre's formula. Basic notions about real functions of one real variable: monotonicity, one-to-one functions, inverse function, local and absolute extrema, limits, asymptotes, continuity, behaviour of limits with respect to order, elementary operations and composition. Numerical sequences, Nepero's number. Properties of continuous functions: the Weierstrass and Darboux theorems. Numerical series: series with positive terms, the geometric and harmonic series, absolute convergence, series with alternating signs. Differential calculus for one variable functions: the notions of derivative and tangent line, fundamental derivatives, differentiation rules. The theorems by Fermat, Rolle, Cauchy, Lagrange. Integral calculus for one variable functions:
Riemann integral, integration methods, generalized integrals, integral test for numerical series. De l'Hopital rule, comparison between infinitesimal functions, Taylor's and McLaurin's formulas and series, the complex exponential function.
Classes, exercise sessions
Written and oral examination
1) Knowledge and understanding of the basic notions of Mathematical Analysis for functions of one variable, and the main techniques for tackling problems such as: computation of limits, derivatives and integrals, qualitative study of numerical series, graphic representation of functions.
2) Capability of tackling problems with logical accuracy, and finding the most suitable techniques for solving them.
3) To focus (even new) problems in the most suitable framework, evaluate the several involved features and realize the necessary relations with the learned theoretical results.
4) To know how to explain a given subject, proof or problem with logical precision and a suitable mathematical language.
5) To be able to develop topics which are related to the course, to choose and consult new texts. To be ready in finding relations among the different theoretical results which were learned.
M. Bertsch, R. Dal Passo, L. Giacomelli, "Analisi Matematica", Seconda edizione, McGraw-Hill.
P. Marcellini, C. Sbordone, "Elementi di Analisi Matematica uno", Liguori Editore.
M. Bramanti, "Esercitazioni di Analisi Matematica I",
Società Editrice Esculapio.