30122 - PRECORSO DI MATEMATICA / MATHEMATICS - PREPARATORY COURSE
CLEAM - CLEACC - BESS-CLES - BIEF - BIEM - BIG - BEMACS - CLEF - BIG
Per la lingua del corso verificare le informazioni sulle classi/
For the instruction language of the course see class group/s below
Classe 1: MARCO UGO BOELLA, Classe 2: FEDERICA ANDREANO, Classe 3: ANDREA GIUSSANI, Classe 4: GIANPAOLO MONTI
Classe/i impartita/e in lingua italiana
Il precorso ha l’obiettivo di consolidare alcuni argomenti di matematica a livello preuniversitario, per permettere allo studente di iniziare gli studi universitari con maggiore serenità e competenza. Durante i corsi di I semestre, tali argomenti sono dati per noti e non vengono più ripetuti.
- Ripasso di alcune nozioni preliminari. Insiemi. Insiemi numerici, potenze/radici. Equazioni razionali. Geometria analitica nel piano.Trigonometria.Disequazioni razionali.
- Funzioni reali di una variabile reale. Funzioni potenza e loro grafici. Funzioni esponenziali e loro grafici. Funzioni logaritmiche e loro grafici. Proprietà dei logaritmi. Funzioni trigonometriche e loro grafici. Trasformazioni di funzioni elementari. Funzioni definite a pezzi. Equazioni e disequazioni risolubili con metodi grafici. Disequazioni in due variabili risolubili con metodi grafici.
- La nozione di derivata. Derivate di 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 sui teoremi. Implicazione, equivalenza. Condizione sufficiente, condizione necessaria; condizione necessaria e sufficiente. Dimostrazioni dirette, dimostrazioni per assurdo.
- G. OSIMO, M. D’AMICO, M.B. ZAVELANI ROSSI, et al., Matematica. Precorsi, Milano, EGEA, 2011, seconda edizione, ISBN 978-88-238-2147-7.
Classe/i impartita/e in lingua italiana
- Insiemi, operazioni con gli insiemi. Numeri, operazioni con i numeri. Intervalli.
- Equazioni di primo/secondo grado, sistemi di equazioni.
- Disequazioni di primo/secondo grado, sistemi di disequazioni.
- Geometria analitica. Piano cartesiano. Retta, intersezione tra due rette. Parabola, intersezioni retta-parabola.
- Funzioni elementari e loro grafici: funzioni potenza, esponenziali, logaritmiche.
- Funzione inversa, funzione composta. Funzione modulo. Funzioni definite a tratti. Trasformazioni geometriche.
- Equazioni/disequazioni esponenziali e logaritmiche. Equazioni/disequazioni con termini in modulo.
- G. OSIMO, M. D’AMICO, M.B. ZAVELANI ROSSI, et al., Matematica. Precorsi, Milano, EGEA, 2011, II edizione, ISBN 978-88-238-2147-7.
Class group/s taught in English
- Sets, operations with sets. Numbers, operations with numbers. Intervals.
- First/second degree equations, systems of equations.
- First/second degree inequalities, systems of inequalities.
- Coordinate geometry. Cartesian plane. Straight line, intersection of two lines. Parabola, line-parabola intersections.
- Elementary functions and their graphs: power, exponential, logarithmic functions.
- Inverse function, composite function. Absolute value function. Piece-wise defined functions. Geometric transformations.
- Exponential and logarithmic equations/inequalities. Equations/inequalities with terms in absolute value.
- G. OSIMO, M. D’AMICO, M.B. ZAVELANI ROSSI, et al., Matematica. Precorsi, Milano, EGEA, 2011, II edizione, ISBN 978-88-238-2147-7.
Class group/s taught in English
The aim of the preparatory course is to consolidate some topics in pre-undergraduate mathematics, so as to help the student to begin University studies with more self-confidence and competence. During the I semester courses, such topics are considered as known and are not repeated.
- Review of some preliminary notions: sets; number sets, powers/roots; rational equations; Cartesian geometry in the plane; trigonometry; rational inequalities.
- Real functions of one real variable. Power functions and their graphs. Exponential functions and their graphs. Logarithmic functions and their graphs. Properties of logarithms. Trigonometric functions and their graphs. Transformations of elementary functions. Piece-wise defined functions. Solving equations and inequalities with graphical methods. Solving two-variable inequalities with graphical methods.
- The notion of 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. Terminology on theorems. Implication, equivalence. Sufficient condition, necessary condition, necessary and sufficient condition. Direct proofs, proofs by contradiction.
- Materials in electronic format is provided by the instructor.
Class 9: GUIDO OSIMO, Class 10: FEDERICO MARIO GIOVANNI VEGNI, Class 11: FEDERICA ANDREANO, Class 12: MARIA BEATRICE ZAVELANI ROSSI
Class group/s taught in English
The aim of the preparatory course is to consolidate some topics in pre-undergraduate mathematics, so as to help the student to begin University studies with more self-confidence and competence. During the I semester courses, such topics are considered as known and are not repeated.
- Review of some preliminary notions: sets; number sets, powers/roots; rational equations; Cartesian geometry in the plane; trigonometry; rational inequalities.
- Real functions of one real variable. Power functions and their graphs. Exponential functions and their graphs. Logarithmic functions and their graphs. Properties of logarithms. Trigonometric functions and their graphs. Transformations of elementary functions. Piece-wise defined functions. Solving equations and inequalities with graphical methods. Solving two-variable inequalities with graphical methods.
- The notion of 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. Terminology on theorems. Implication, equivalence. Sufficient condition, necessary condition, necessary and sufficient condition. Direct proofs, proofs by contradiction
- The course instructors gives indications on course materials.
- A suitable textbook in Italian is G. OSIMO, M. D’AMICO, M.B. ZAVELANI ROSSI, et al., Matematica. Precorsi, Milano, EGEA, 2011, II edizione, ISBN 978-88-238-2147-7.
Class group/s taught in English
The aim of the preparatory course is to consolidate some topics in pre-undergraduate mathematics, so as to help the student to begin University studies with more self-confidence and competence.Later on, in Mathematics, Statistics and Computer Science courses these topics are assumed as known.
- Review of some preliminary notions: sets; number sets, powers/roots; rational equations; Cartesian geometry in the plane; trigonometry; rational inequalities.
- Real functions of one real variable. Power functions and their graphs. Exponential functions and their graphs. Logarithmic functions and their graphs. Properties of logarithms. Trigonometric functions and their graphs. Transformations of elementary functions. Piece-wise defined functions. Solving equations and inequalities with graphical methods. Solving two-variable inequalities with graphical methods.
- The notion of 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. Terminology on theorems. Implication, equivalence. Sufficient condition, necessary condition, necessary and sufficient condition. Direct proofs, proofs by contradiction.
- An introduction to Matlab/Octave.
- Materials in electronic format are provided by the instructor.
Classe/i impartita/e in lingua italiana
Il precorso ha l’obiettivo di consolidare alcuni argomenti di matematica a livello preuniversitario, per permettere allo studente di iniziare gli studi universitari con maggiore serenità e competenza. Durante i corsi di I semestre, tali argomenti sono dati per noti e non vengono più ripetuti.
- Ripasso di alcune nozioni preliminari: insiemi; insiemi numerici, potenze/radici; equazioni razionali; geometria analitica nel piano; trigonometria; disequazioni razionali.
- Funzioni reali di una variabile reale. Funzioni potenza e loro grafici. Funzioni esponenziali e loro grafici. Funzioni logaritmiche e loro grafici. Proprietà dei logaritmi. Funzioni trigonometriche e loro grafici. Trasformazioni di funzioni elementari. Funzioni definite a pezzi. Equazioni e disequazioni risolubili con metodi grafici. Disequazioni in due variabili risolubili con metodi grafici.
- La nozione di derivata. Derivate di 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 sui teoremi. Implicazione, equivalenza. Condizione sufficiente, condizione necessaria; condizione necessaria e sufficiente. Dimostrazioni dirette, dimostrazioni per assurdo.
- G. OSIMO, M. D’AMICO, M.B. ZAVELANI ROSSI, et al., Matematica. Precorsi, Milano, EGEA, 2011, II edizione, ISBN 978-88-238-2147-7.
Class group/s taught in English
The aim of the preparatory course is to consolidate some topics in pre-undergraduate mathematics, so as to help the student to begin University studies with more self-confidence and competence. During the I semester courses, such topics are considered as known and are not repeated.
- Sets, numbers. Summation symbol and remarkable sums. Review on algebraic calculus.
- Elementary rational equations and inequalities. Exponential equations. Notion of logarithm. Properties of logarithms. Logarithmic equations.
- Coordinate geometry. Cartesian plane, distance formula. Straight line, parabola, circle and rectangular hyperbola.
- Powers with integer exponent. Roots. Powers with rational exponent. Powers with real exponent. Generalities on irrational equations and inequalities.
- Notion of real function of one real variable. Domain, codomain, range. Zeroes of a function. Graph of a function. Graphs of elementary functions: linear, affine linear, quadratic, power, exponential, logarithmic.
- Calculation of the derivatives of elementary functions. Algebra of derivatives: derivative of the (algebraic) sum, derivative of the product, derivative of the quotient. Derivatives of composite functions.
- Calculation of the second derivatives. Study of the graph of a function: domain, behaviour at the boundary points of the domain, monotonicity, points of maximum/minimum and concavity/convexity.
- Calculation of the antiderivatives of elementary functions. Integration methods: decomposition and substitution.
- The course instructors gives indications on course materials.