### Teachings

Select Academic Year:     2017/2018 2018/2019 2019/2020 2020/2021 2021/2022 2022/2023
Professor
Period
First Semester
Teaching style
Convenzionale
Lingua Insegnamento
ITALIANO

Informazioni aggiuntive

Course Curriculum CFU Length(h)
[60/60]  PHYSICS [60/00 - Ord. 2012]  PERCORSO COMUNE 12 96

### Objectives

1. Knowledge and understanding

Knowledge and understanding of the terms and the notation in the theory of limits and differential and integral calculus for functions of one real variable. Knowledge of the basic notions of the theory of numerical sequences and infinite series, as well as the theory of functions of one real variable.

2. Applying knowledge and understanding

Ability in the usage of the algorithms for computing limits of numerical sequences as well as of functions of one real variable, and for the symbolical computation of derivatives and the integration by quadratures of such functions. Usage of integration theory to compute the area of simple plain regions. Usage of some criteria to establish the convergence of an infinite series.

3. Making judgements

Apply the differential calculus to discover the qualitative properties of the graph of a given function, and to determine its maxima and minima. Bring together and connect the information arising from different algorithms concerning a given function, and intelligently use such information to establish the nature of the function. Critically evaluate if an alleged proof, or, more generally, a given argument is correct, as well as if a given definition is well posed. Locate and correct one's own mistakes, without getting lost.

4. Communication skills

Ability in communicating information, ideas, problems and solutions by means of the current terminology, in order to effectively interact with other students and teachers, in the present, and prospectively with colleagues or in a possible teaching activity. Ability in supporting a claim by means of a logical argument (a proof), making a correct usage of terms as "if", "then", "that is to say".

5. Learning skills

Being able to consult and interpret textbooks to build and expand one's own knowledge. Being able to use the sources of information as a tool in problem-solving activities. Being able, if necessary, to use the sources of information in order to go slightly beyond a strict interpretation of the course boundaries.

### Prerequisites

Basic notions of set theory and numbers sets. Basic algebraic and symbolic calculus skills. 2nd degree polinomial equations and inequalities. Systems of equations and inequalities. Trigonometry. Basic notions of analytic geometry (lines and curves in the 2-dimensional Euclidean space)

### Contents

Contents (w.p. means “with proof”, w/p means “without proof”)
1.Preliminary concepts on sets and sets of real numbers. Real numbers properties: maximum, minimum, supremum, infimum.
2.Real line topology: different nature of the points of the real line, definitions and examples. Subsets of the real line: open sets, closed sets, bounded sets,, definitions and examples. Fundamental properties of open and closed sets (thm. 2.1, 2.2 e 2.3 (w.p.)). Bolzano-Weirstrass theorem (w.p.). Compact and connected sets, Heine-Borel theorem (w/p).
3.Functions between sets: injection, surjection, inverse. Composition of functions. Definitions and examples (pg. 24-29, 32-35). The principle of induction. Application (sum of the first N numbers).
4.Real functions: sign and symmetries, absolute value of f, even and odd functions. Bounded functions, local and global maxima and minima. Monotone functions. Examples: the elementary functions (sin(x), cos(x), log(x), ex, etc.).
5.Sequences and series: limit of a sequence, comparison theorem (due carabinieri), special limits (Nepero number e included), max and min lim. Sequences and topology, Cauchy principle, Weierstrass theorem.
Infinite sums: definitions and convergence. Method for establishing the convergence of a series (ratio, comparison, square, Cauchy), generalized harmonic series, absolute convergence, alternate series, Leibnitz rule.
6.Limits for functions: uniqueness, right and left limits. Definitions for limits at infinity. Invariance of the sign theorem (w.p.). Comparison theorem (w.p.). Special cases (e.g. sin(x)/x). Limits properties and operations (w.p.). Non existence of the limit. Indeterminate forms. Composed function limit (w/p). Monotone functions limit theorem (w/p). Limits of elementary functions. Hyperbolic functions, definitions and graphs.
7.Asymptotes. Comparison principles for infinite and infinitesimal functions. Landau symbols.
8.Continuous functions. Continuity for the composition of functions (w/p). Discontinuity, classification of three cases. Examples. Fundamental theorems for continuous functions (zero's, sign, Weirestrass, Darboux, all w.p.). Uniform continuity. Heine-Cantor theorem (w/p). Continuous functions on connected sets (all properties w/p).
9.Differential calculus: derivative of a function. Definition, geometric and physical meaning. Properties of the derivatives (w.p.). Composed and inverse functions. Elementary functions derivatives. Higher order derivatives. Differential df and dx.
10.Methods for finding maxima and minima using derivatives. Fermat, Rolle, Cauchy and Lagrange theorems (all w.p.). Monotone functions and derivatives (rate of growth). Monotonicity test. De L'Hopital theorems (w.p.). Concavity, convexity.
11.Scheme for approaching the analysis of the graph of a function. Taylor approximation formula (w.p.). Peano and Lagrange error formulas (w.p.) Fundamental properties, examples and applications.
12.Integral calculus. Riemann integral, definition and properties. Class of R-integrable functions. Integrability fundamental theorem, integral mean value theorem (all w.p.). Methods for calculating integrals. Generalized integrals and convergence.

13. Ordinary differential equations, by separation of variables, first order linear, second order linear with constant coefficients. The Cauchy problem, boundary value problems, initial conditions.

### Teaching Methods

Traditional lessons on the blackboard and, eventually, with online streaming. Didactic activities (for a total of 96 hours) consist of theoretic explaination of mathematical concepts (in terms of definitions, properties and theorems with proofs), closely linked to examples and applications.

### Verification of learning

First half: written tests (6 possibilities per year: January, 2 in February, June, July, September) with exercises. With a result of 18/30 one can apply for the second part of the exam (theory, definitions, theorems and proofs). The written test “lifespan” is, normally, 60 days. If a student fails to get a positive result during the second half of the exam he must apply again for the first part.

For previous exams tests see http://people.unica.it/luciocadeddu/

### Texts

C. D. Pagani, S. Salsa – “Analisi Matematica, Vol. 1” – zanichelli Editore.
Esercizi: P. Marcelllini e C. Sbordone, “Esercitazioni di Matematica, vol. 1”, parte
prima e parte seconda, Liguori Editore.
F. Buzzetti, E. Grassini Raffaglio, e A. Vasconi Ajroldi, “Esercitazioni di Analisi"