Course 2024-2025 a.y.

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: definizione, esempi. Funzioni suriettive, iniettive, biunivoche. Funzioni reali di una variabile reale: definizione, grafico cartesiano, 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 trigonometriche e loro grafici. Funzione arcotangente e suo grafico. Funzioni definite a tratti. Funzione modulo. Trasformazioni di funzioni elementari. Funzioni limitate, funzioni monotone. Funzioni reali di n variabili reali. Funzioni reali di due variabili reali: definizione, grafico cartesiano, curve di livello, esempi.
• Funzioni reali di una variabile reale: rapporto incrementale, derivata; derivate delle funzioni elementari; algebra delle derivate; derivata della funzione composta; equazione della retta tangente. Funzioni reali di due variabili reali: derivate parziali.
• 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. Esempi di dimostrazioni per contronominale.

• Lezioni

Lezioni frontali: nulla da specificare.

Lezioni online: la prima parte del precorso si svolge online, sulla piattaforma didattica Bboard.

    La valutazione dell'apprendimento avviene all'interno del corso di matematica di primo anno, primo semestre, con le modalità definite da quel corso.

• Parte online: tutti i materiali didattici sono disponibili sulla piattaforma Bboard.
• Parte in presenza: sarà utilizzato il testo Corso preparatorio in Matematica, Guido Osimo, EGEA (2024).

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.

• Lezioni

Lezioni frontali: niente da specificare.

Lezioni online: la prima parte del precorso si svolge online, sulla piattaforma didattica Bboard.

  La valutazione dell'apprendimento avviene all'interno del corso di matematica di primo anno, primo semestre, con le modalità definite da quel corso.

• Parte online: tutti i materiali didattici sono disponibili sulla piattaforma Bboard.
• Parte in presenza: materiali didattici a cura del docente.

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.

• Lectures

Face-to-face lectures: nothing to specify.

Online lectures: the first part of the preparatory course takes place online, on the Bboard teaching platform.

   Assessment is included in the first year, first semester Mathematics course, with the methods used for that course.

• Online part: all teaching materials are available on the Bboard platform.
• Classroom part: teaching materials prepared by the instructor.

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:

• Introduction to propositional logic. Basic logical connectives, truth tables. Tautology and contradiction. De Morgan’s laws. Conditional and biconditional statements. Contrapositive and converse statements. Predicates. Universal quantifiers. Negating quantified statements.

• Sets. Operations on sets and logical connectives. Properties. Power set. Set of natural, integer, rational and irrational numbers. Factorial of a number. Binomial coefficient. Sum and product of numbers.
• Meaning of the following terms: definition, theorem, proposition, lemma, corollary, and proof. Direct and contrapositive proofs. Proofs by contradiction. Special forms of the premise or of the conclusion. Proof by induction.
• Order structure of R. Intervals. Lower and upper bounds. Bounded sets. Maxima and minima. Supremum and infimum. Least upper bound principle. The extended real line. Short review of powers and logarithms.
• Real functions of one real variable. Domain, codomain, and image of a function. Surjective, injective, and bijective functions. Bounded functions. Elementary functions. Geometric notion of derivative of a function at a point. Derivatives of elementary functions. Algebra of derivatives.
• Vectors of R^n. Real functions of n real variables. Domain, codomain, and image of a function. Separable functions. The calculation of partial derivatives.

• Lectures

Face-to-face lectures: nothing to specify.

Online lectures: the first part of the preparatory course takes place online, on the Bboard teaching platform.

 Assessment is included in the first year, first semester Mathematics course, with the methods used for that course.

• Online part: all teaching materials are available on the Bboard platform.
• Classroom part: teaching materials prepared by the instructor.

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:

• Functions: definition, examples. Surjective, injective, bijective functions. Real functions of one real variable: definition, Cartesian 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. Trigonometric functions and their graphs. The inverse tangent function and its graph. Piecewise defined functions. Absolute value function. Transformation of elementary functions. Bounded, monotone functions. Real functions of n real variables. Real functions of two real variables: definition, Cartesian graph, level curves, examples.
• Real functions of one real variable: difference quotient, derivative; derivatives of elementary functions; algebra of derivatives; chain rule; equation of the tangent line. Real functions of two real variables: partial derivatives.
• 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. Examples of proofs by contrapositive.

• Lectures

Face-to-face lectures: nothing to specify.

Online lectures: the first part of the preparatory course takes place online, on the Bboard teaching platform.

    Assessment is included in the first year, first semester Mathematics course, with the methods used for that course.

• Online part: all teaching materials are available on the Bboard platform.
• Classroom part: we use the textbook Preparatory Course in Mathematics, Guido Osimo, EGEA (2024).

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:

• Functions: definition, examples. Surjective, injective, bijective functions. Real functions of one real variable: definition, Cartesian 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. Trigonometric functions and their graphs. The inverse tangent function and its graph. Piecewise defined functions. Absolute value function. Transformation of elementary functions. Bounded, monotone functions. Real functions of n real variables. Real functions of two real variables: definition, Cartesian graph, level curves, examples. 
• Real functions of one real variable: difference quotient, derivative; derivatives of elementary functions; algebra of derivatives; chain rule; equation of the tangent line. Real functions of two real variables: partial derivatives.
• 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. Examples of proofs by contrapositive.

• Lectures

Face-to-face lectures: nothing to specify.

Online lectures: the first part of the preparatory course takes place online, on the Bboard teaching platform.

Assessment is included inside the first year, first semester Mathematics course, with the methods used for that course.

• Online part: all teaching materials are available on the Bboard platform.
• Classroom part: we use the textbook Preparatory Course in Mathematics, Guido Osimo, EGEA (2024).

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:

• Cartesian product. Relations. Equivalence relations, order relations. Functions. Surjective, injective, bijective functions. Internal operations, external operations. 
• Real functions of one real variable. 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. Trigonometric functions and their graphs. Inverse trigonometric functions and their graphs. Piecewise defined functions. Absolute value function. Transformations of elementary functions. Positive part and negative part of a function. Graphical solution of equations. Bounded functions. Increasing, decreasing, monotonic functions. Global maxima, global minima. 
• Mathematics as an axiomatic system: axioms and theorems, primitive notions and definitions. Basic terminology on theorems. Examples of proofs and proving techniques. Conjectures: proofs and counterexamples. Implication, equivalence. Sufficient condition, necessary condition, necessary and sufficient condition. The negation of a proposition. Proofs by contradiction. Proofs by contrapositive. Proofs by induction.
• Elements of combinatorics. Permutations, combinations. Binomial coefficients, Pascal's triangle, binomial theorem.

• Lectures

Face-to-face lectures: nothing to specify.

Online lectures: the first part of the preparatory course takes place online, on the Bboard teaching platform.

         Assessment is included in the first year Mathematics courses, with the methods used for those courses.

• Online part: all teaching materials are available on the Bboard platform.
• Classroom part: we use the textbook BAI Preparatory Course in Mathematics, Guido Osimo, EGEA (2022), ISBN 978-88-6407-473-3.

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:

• Functions: definition, examples. Surjective, injective, bijective functions. Real functions of one real variable: definition, Cartesian 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. Trigonometric functions and their graphs. The inverse tangent function and its graph. Piecewise defined functions. Absolute value function. Transformation of elementary functions. Bounded, monotone functions. Real functions of n real variables. Real functions of two real variables: definition, Cartesian graph, level curves, examples.
• Real functions of one real variable: difference quotient, derivative; derivatives of elementary functions; algebra of derivatives; chain rule; equation of the tangent line. Real functions of two real variables: partial derivatives.
• 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. Examples of proofs by contrapositive.

• Lectures

Face-to-face lectures: nothing to specify.

Online lectures: the first part of the preparatory course takes place online, on the Bboard teaching platform.

   Assessment is included inside the first year, first semester Mathematics course, with the methods used for that course.

• Online part: all teaching materials are available on the Bboard platform.
• Classroom part: we use the textbook Preparatory Course in Mathematics, Guido Osimo, EGEA (2024).

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:

A) BIG

• Real functions of one real variable: general concepts and examples. Composite and inverse functions. Graph of a function. Graphs of elementary functions: linear, power, exponential, and logarithmic functions. Piecewise-defined functions and their graphs. Transformations of elementary functions.

• Exponential equations and inequalities. Properties of logarithms. Logarithmic equations and inequalities.

• More on functions: increasing/decreasing, concave/convex, and bounded functions. Global and local extrema of a function.

• Behavior of elementary functions at the boundaries of the domain. Hierarchy of infinite functions.

• Graph of a function: domain, behavior at the boundaries, zeros (or x-intercepts), y-intercept, sign, monotonicity, points of maximum and minimum, concavity and convexity.

• Summation symbol and remarkable sums. Summation properties. 

• Limits: an intuitive idea.

• Basic language and examples of differential and integral calculus.

B) BIG-HEC

• Real functions of one real variable: general concepts and examples. Graph of a function. Graphs of elementary functions: linear, power, exponential, and logarithmic functions. Piecewise-defined functions and their graphs. Transformations of elementary functions. Composite and inverse functions.
• Exponential equations and inequalities. Properties of logarithms. Logarithmic equations and inequalities.
• More on functions: increasing/decreasing, bounded functions.
• Behavior of elementary functions at the boundaries of the domain, intuitive idea of limits.
• The rate of change of a function: the notion of difference quotient, the notion of derivative. Elementary derivatives. Rules on derivatives. The chain rule.
• Introduction to the mathematical language.
• Summation symbol. Summation properties.

• Lectures

Face-to-face lectures: nothing to specify.

Online lectures: the first part of the preparatory course takes place online, on the Bboard teaching platform.

Assessment is included inside the first year, first semester Mathematics course, with the methods used for that course.

Online part: all teaching materials are available on the Bboard platform.

Classroom part: teaching materials prepared by the instructor.