30122 - PRECORSO DI MATEMATICA / MATHEMATICS - PREPARATORY COURSE
Department of Decision Sciences
For the instruction language of the course see class group/s below
GUIDO OSIMO
Classe 1: GUIDO OSIMO, Classe 2: FEDERICA ANDREANO, Classe 3: FEDERICO MARIO GIOVANNI VEGNI, Classe 4: MAURO D'AMICO, Classe 5: FABRIZIO IOZZI, Classe 6: MARCO UGO CLAUDIO BOELLA, Classe 7: FABIO TONOLI, Classe 8: GIANPAOLO MONTI
Classe/i impartita/e in lingua italiana
Mission e Programma sintetico
MISSION
PROGRAMMA SINTETICO
- Parte online:
- Insiemi. Operazioni tra insiemi. Insiemi numerici. Rappresentazione degli insiemi numerici sulla retta.
- Potenze a esponente intero. Radici. Potenze a esponente razionale e a esponente reale.
- Calcolo letterale.
- Equazioni di primo e secondo grado. Equazioni di grado superiore al secondo. Equazioni frazionarie. Sistemi di equazioni.
- Coordinate cartesiane nel piano. Rette. Parabole. Altre curve.
- Elementi di trigonometria.
- Disequazioni di primo e secondo grado. Disequazioni di grado superiore al secondo. Disequazioni frazionarie. Sistemi di disequazioni. Equazioni e disequazioni con termini in valore assoluto.
- Parte in presenza:
- Funzioni reali di una variabile reale: definizione, grafico, esempi. Funzione composta. Funzione inversa. Funzioni potenza e loro grafici. Funzioni esponenziali e loro grafici. Introduzione ai logaritmi. Proprietà dei logaritmi. Funzioni logaritmiche e loro grafici. Equazioni e disequazioni esponenziali/logaritmiche. Funzioni trigonometriche e loro grafici. Funzioni definite a tratti. Funzione modulo. Trasformazioni di funzioni elementari.
- Rapporto incrementale, derivata. Derivate delle funzioni elementari. Algebra delle derivate. Derivata della funzione composta. Equazione della retta tangente.
- La matematica come sistema assiomatico: nozioni primitive e definizioni, assiomi e teoremi. Terminologia di base sui teoremi. Esempi di dimostrazioni e di tecniche dimostrative. Congetture: dimostrazioni e controesempi. Implicazione, equivalenza. Condizione sufficiente, condizione necessaria, condizione necessaria e sufficiente. La negazione di una proposizione. Esempi di dimostrazioni per assurdo.
Modalità didattiche
- Lezioni frontali
- Lezioni online
DETTAGLI
Lezioni online: la prima parte del precorso si svolge online, sulla piattaforma didattica Bboard.
Materiali didattici
STUDENTI FREQUENTANTI E NON FREQUENTANTI
- Parte online: tutti i materiali didattici sono disponibili sulla piattaforma Bboard.
- Parte in presenza: viene utilizzata una dispensa, che è disponibile all’inizio di questa parte.
GUIDO OSIMO
Classe/i impartita/e in lingua italiana
Mission e Programma sintetico
MISSION
PROGRAMMA SINTETICO
- Parte online:
- Insiemi. Operazioni tra insiemi. Insiemi numerici. Rappresentazione degli insiemi numerici sulla retta.
- Potenze a esponente intero. Radici. Potenze a esponente razionale e a esponente reale.
- Calcolo letterale.
- Equazioni di primo e secondo grado. Equazioni di grado superiore al secondo. Equazioni frazionarie. Sistemi di equazioni.
- Coordinate cartesiane nel piano. Rette. Parabole. Altre curve.
- Elementi di trigonometria.
- Disequazioni di primo e secondo grado. Disequazioni di grado superiore al secondo. Disequazioni frazionarie. Sistemi di disequazioni. Equazioni e disequazioni con termini in valore assoluto.
- Parte in presenza:
- Funzioni reali di una variabile reale: definizione, grafico, esempi. Funzione composta. Funzione inversa. Funzioni potenza e loro grafici. Funzioni esponenziali e loro grafici. Introduzione ai logaritmi. Proprietà dei logaritmi. Funzioni logaritmiche e loro grafici. Equazioni e disequazioni esponenziali/logaritmiche. Funzioni trigonometriche e loro grafici. Funzioni definite a tratti. Funzione modulo. Trasformazioni di funzioni elementari.
- Rapporto incrementale, derivata. Derivate delle funzioni elementari. Algebra delle derivate. Derivata della funzione composta. Equazione della retta tangente.
- La matematica come sistema assiomatico: nozioni primitive e definizioni, assiomi e teoremi. Terminologia di base sui teoremi. Esempi di dimostrazioni e di tecniche dimostrative. Congetture: dimostrazioni e controesempi. Implicazione, equivalenza. Condizione sufficiente, condizione necessaria, condizione necessaria e sufficiente. La negazione di una proposizione. Esempi di dimostrazioni per assurdo.
Modalità didattiche
- Lezioni frontali
- Lezioni online
DETTAGLI
Lezioni online: la prima parte del precorso si svolge online, sulla piattaforma didattica Bboard.
Materiali didattici
STUDENTI FREQUENTANTI E NON FREQUENTANTI
- Parte online: tutti i materiali didattici sono disponibili sulla piattaforma Bboard.
- Parte in presenza: viene utilizzata una dispensa, che è disponibile all’inizio di questa parte.
GUIDO OSIMO
Class group/s taught in English
Mission & Content Summary
MISSION
CONTENT SUMMARY
- Online part:
- Sets. Operations with sets. Number sets. Representation of number sets on the line.
- Powers with integer exponents. Roots. Powers with rational exponents and with real exponents.
- Polynomial algebra.
- First and second degree equations. Higher degree equations. Fractional equations. Systems of equations.
- Cartesian coordinates in the plane. Straight lines. Parabolas. Other curves.
- First and second degree inequalities. Higher degree inequalities. Fractional inequalities. Systems of inequalities. Equations and inequalities with terms in absolute value.
- Classroom part:
- Real functions of one real variable: definition, graph, examples. Composite function. Inverse function. Power functions and their graphs. Exponential functions and their graphs. Introduction to logarithms. Properties of logarithms. Logarithmic functions and their graphs. Piecewise defined functions. Absolute value function. Transformation of elementary functions.
- Exponential/logarithmic equations and inequalities. Simple irrational equations and inequalities.
Teaching methods
- Face-to-face lectures
- Online lectures
DETAILS
Online lectures: the first part of the preparatory course takes place online, on the Bboard teaching platform.
Teaching materials
ATTENDING AND NOT ATTENDING STUDENTS
- Online part: all teaching materials are available on the Bboard platform.
- Classroom part: teaching materials prepared by the instructor.
GUIDO OSIMO
Class group/s taught in English
Mission & Content Summary
MISSION
CONTENT SUMMARY
- Online part:
- Sets. Operations with sets. Number sets. Representation of number sets on the line.
- Powers with integer exponents. Roots. Powers with rational exponents and with real exponents.
- Number systems.
- Polynomial algebra.
- First and second degree equations. Higher degree equations. Fractional equations. Systems of equations.
- Cartesian coordinates in the plane. Straight lines. Parabolas. Other curves.
- Elements of trigonometry.
- First and second degree inequalities. Higher degree inequalities. Fractional inequalities. Systems of inequalities. Equations and inequalities with terms in absolute value.
- Classroom part:
- Introduction to proofs. Connectives, quantifiers. Demonstration techniques: direct, contrapositive and indirect (or by contradiction). Principle of induction.
- Real functions of one real variable: definition, graph, examples. Bounded and unbounded functions. Increasing and decreasing functions. Global extrema of a function. Concave and convex functions. Operations with functions (sum, difference, product and quotient). Linear combination of functions. Odd and even functions. Periodic functions. Composite function. Elementary functions and their graphs: constant, linear, affine linear, power, exponential and logarithm. Piecewise defined functions. Elementary notion of continuity.
- Derivative. Derivatives of elementary functions. Algebra of derivatives. Chain rule. Equation of the tangent line. Stationary points. Second derivative.
Teaching methods
- Face-to-face lectures
- Online lectures
DETAILS
Online lectures: the first part of the preparatory course takes place online, on the Bboard teaching platform.
Teaching materials
ATTENDING AND NOT ATTENDING STUDENTS
- Online part: all teaching materials are available on the Bboard platform.
- Classroom part: teaching materials prepared by the instructor.
GUIDO OSIMO
Classe/i impartita/e in lingua italiana
Mission e Programma sintetico
MISSION
PROGRAMMA SINTETICO
- Parte online:
- Insiemi. Operazioni tra insiemi. Insiemi numerici. Rappresentazione degli insiemi numerici sulla retta.
- Potenze a esponente intero. Radici. Potenze a esponente razionale e a esponente reale.
- Calcolo letterale.
- Equazioni di primo e secondo grado. Equazioni di grado superiore al secondo. Equazioni frazionarie. Sistemi di equazioni.
- Coordinate cartesiane nel piano. Rette. Parabole. Altre curve.
- Disequazioni di primo e secondo grado. Disequazioni di grado superiore al secondo. Disequazioni frazionarie. Sistemi di disequazioni. Equazioni e disequazioni con termini in valore assoluto.
- Parte in presenza:
- Funzioni reali di una variabile reale: definizione, grafico, esempi. Funzione composta. Funzione inversa. Funzioni potenza e loro grafici. Funzioni esponenziali e loro grafici. Introduzione ai logaritmi. Proprietà dei logaritmi. Funzioni logaritmiche e loro grafici. Funzioni definite a tratti. Funzione modulo. Trasformazioni di funzioni elementari.
- Equazioni e disequazioni esponenziali/logaritmiche. Semplici equazioni e disequazioni irrazionali.
Modalità didattiche
- Lezioni frontali
- Lezioni online
DETTAGLI
Lezioni online: la prima parte del precorso si svolge online, sulla piattaforma didattica Bboard.
Materiali didattici
STUDENTI FREQUENTANTI E NON FREQUENTANTI
- Parte online: tutti i materiali didattici sono disponibili sulla piattaforma Bboard.
- Parte in presenza: materiali didattici a cura del docente.
GUIDO OSIMO
Class group/s taught in English
Mission & Content Summary
MISSION
CONTENT SUMMARY
- Online part:
- Sets. Operations with sets. Number sets. Representation of number sets on the line.
- Powers with integer exponents. Roots. Powers with rational exponents and with real exponents.
- Polynomial algebra.
- First and second degree equations. Higher degree equations. Fractional equations. Systems of equations.
- Cartesian coordinates in the plane. Straight lines. Parabolas. Other curves.
- Elements of trigonometry.
- First and second degree inequalities. Higher degree inequalities. Fractional inequalities. Systems of inequalities. Equations and inequalities with terms in absolute value.
- Classroom part:
- Mathematics as an axiomatic system: primitive notions and definitions, axioms and theorems. Set axioms, number sets.
- Elements of logics: propositions, quantifiers. Implication, equivalence. Sufficient condition, necessary condition, necessary and sufficient condition. The negation of a proposition. The contrapositive proposition.
- Basic terminology on theorems. Examples of proofs and demonstration techniques. Proofs by contradiction. The principle of mathematical induction; proofs by induction. Conjectures: proofs and counterexamples.
- Countable and uncountable sets. Combinatorics.
Teaching methods
- Face-to-face lectures
- Online lectures
DETAILS
Online lectures: the first part of the preparatory course takes place online, on the Bboard teaching platform.
Teaching materials
ATTENDING AND NOT ATTENDING STUDENTS
- Online part: all teaching materials are available on the Bboard platform.
- Classroom part: teaching materials prepared by the instructor.
GUIDO OSIMO
Class 15: GUIDO OSIMO, Class 16: FEDERICA ANDREANO, Class 17: JACOPO GIUSEPPE DE TULLIO, Class 18: FEDERICO MARIO GIOVANNI VEGNI
Class group/s taught in English
Mission & Content Summary
MISSION
CONTENT SUMMARY
- Online part:
- Sets. Operations with sets. Number sets. Representation of number sets on the line.
- Powers with integer exponents. Roots. Powers with rational exponents and with real exponents.
- Polynomial algebra.
- First and second degree equations. Higher degree equations. Fractional equations. Systems of equations.
- Cartesian coordinates in the plane. Straight lines. Parabolas. Other curves.
- Elements of trigonometry.
- First and second degree inequalities. Higher degree inequalities. Fractional inequalities. Systems of inequalities. Equations and inequalities with terms in absolute value.
- Classroom part:
- Real functions of one real variable: definition, graph, examples. Composite function. Inverse function. Power functions and their graphs. Exponential functions and their graphs. Introduction to logarithms. Properties of logarithms. Logarithmic functions and their graphs. Exponential/logarithmic equations and inequalities. Trigonometric functions and their graphs. Piecewise defined functions. Absolute value function. Transformation of elementary functions.
- Difference quotient, derivative. Derivatives of elementary functions. Algebra of derivatives. Chain rule. Equation of the tangent line.
- Mathematics as an axiomatic system: primitive notions and definitions, axioms and theorems. Basic terminology on theorems. Examples of proofs and demonstration techniques. Conjectures: proofs and counterexamples. Implication, equivalence. Sufficient condition, necessary condition, necessary and sufficient condition. The negation of a proposition. Examples of proofs by contradiction.
Teaching methods
- Face-to-face lectures
- Online lectures
DETAILS
Online lectures: the first part of the preparatory course takes place online, on the Bboard teaching platform.
Teaching materials
ATTENDING AND NOT ATTENDING STUDENTS
- Online part: all teaching materials are available on the Bboard platform.
- Classroom part: we use a booklet, which are available at the beginning of this part.
GUIDO OSIMO
Class group/s taught in English
Mission & Content Summary
MISSION
CONTENT SUMMARY
- Online part:
- Sets. Operations with sets. Number sets. Representation of number sets on the line.
- Powers with integer exponents. Roots. Powers with rational exponents and with real exponents.
- Polynomial algebra.
- First and second degree equations. Higher degree equations. Fractional equations. Systems of equations.
- Cartesian coordinates in the plane. Straight lines. Parabolas. Other curves.
- Elements of trigonometry.
- First and second degree inequalities. Higher degree inequalities. Fractional inequalities. Systems of inequalities. Equations and inequalities with terms in absolute value.
- Classroom part:
- Real functions of one real variable: definition, graph, examples. Composite function. Inverse function. Power functions and their graphs. Exponential functions and their graphs. Introduction to logarithms. Properties of logarithms. Logarithmic functions and their graphs. Exponential/logarithmic equations and inequalities. Trigonometric functions and their graphs. Piecewise defined functions. Absolute value function. Transformation of elementary functions.
- Difference quotient, derivative. Derivatives of elementary functions. Algebra of derivatives. Chain rule. Equation of the tangent line.
- Mathematics as an axiomatic system: primitive notions and definitions, axioms and theorems. Basic terminology on theorems. Examples of proofs and demonstration techniques. Conjectures: proofs and counterexamples. Implication, equivalence. Sufficient condition, necessary condition, necessary and sufficient condition. The negation of a proposition. Examples of proofs by contradiction.
Teaching methods
- Face-to-face lectures
- Online lectures
DETAILS
Online lectures: the first part of the preparatory course takes place online, on the Bboard teaching platform.
Teaching materials
ATTENDING AND NOT ATTENDING STUDENTS
- Online part: all teaching materials are available on the Bboard platform.
- Classroom part: we use a booklet, which are available at the beginning of this part.
GUIDO OSIMO
Class group/s taught in English
Mission & Content Summary
MISSION
CONTENT SUMMARY
- Online part:
- Sets. Operations with sets. Number sets. Representation of number sets on the line.
- Powers with integer exponents. Roots. Powers with rational exponents and with real exponents.
- Polynomial algebra.
- First and second degree equations. Higher degree equations. Fractional equations. Systems of equations.
- Cartesian coordinates in the plane. Straight lines. Parabolas. Other curves.
- First and second degree inequalities. Higher degree inequalities. Fractional inequalities. Systems of inequalities. Equations and inequalities with terms in absolute value.
- Classroom part:
- Summation symbol and remarkable sums.
- Real functions of one real variable: definition, graph, examples. Composite function. Inverse function. Elementary functions and their graphs: linear, affine linear, quadratic, power, exponential, logarithmic functions. Piecewise defined functions. Transformation of elementary functions.
- Behaviour of elementary functions at the boundary of their domain. Hierarchy of infinite functions.
- Exponential equations and inequalities. Properties of logarithms. Logarithmic equations and inequalities. Solution to equations and inequalities by graphical methods.
- Difference quotient, derivative. Derivatives of elementary functions. Algebra of derivatives. Chain rule. Equation of the tangent line. Stationary points. Derivative and monotonicity. Second derivative. Convex and concave functions.
- Graph of a function: domain, behaviour at the boundary, zeros (or x-intercepts), y-intercept, sign, monotonicity, points of maximum and minimum, concavity and convexity.
- Antiderivatives of elementary functions. Integration methods: decomposition, substitution, by parts.
Teaching methods
- Face-to-face lectures
- Online lectures
DETAILS
Online lectures: the first part of the preparatory course takes place online, on the Bboard teaching platform.
Teaching materials
ATTENDING AND NOT ATTENDING STUDENTS
- Online part: all teaching materials are available on the Bboard platform.
- Classroom part: teaching materials prepared by the instructor.