Programme

Logic

Syntax and semantics. Logical connectives. Sufficient condition, necessary condition, necessary and sufficient condition. Direct and constructive proofs. Proof by contradiction.

Set Theory

Sets and set membership. Primitive and derived concepts. Quantifiers. Euler-Venn diagrams. Subsets. Empty set and universal set. Union and intersection of sets. Disjoint sets. Difference between sets.

Relations and Functions – introductory concepts

Relations and functions. Tabular representation of relations and functions.

Number Systems

Natural numbers, integers, rational numbers, irrational numbers and real numbers. The irrationality of the square root of two, with proof. Order and order relations. Basic order axioms. Completeness axiom and the real line. Intervals of real numbers: closed and open intervals. Upper and lower bounds of a set. Infimum and supremum. Maximum and minimum of a set. Dedekind's axiom, its properties and implications for real intervals. Sets unbounded above and below. The mathematical notion of infinity.

Summations

Definition of summation and its properties. Sum of the reciprocals of the natural numbers and Nicola d'Oresme's result. Sum of the first n natural numbers, Gauss's sum with proof. Geometric sum with proof.

Real-Valued Functions

Definition of a real-valued function. Domain and range. Image and preimage. Cartesian product of two sets and the real plane. Graph of a function. Inverse function. Injectivity, surjectivity and invertibility. Monotonicity and strict monotonicity and their relationship with invertibility. Even and odd functions.

Elementary functions and their properties, including domain, range, monotonicity and invertibility: straight-line functions; power functions of even and odd degree; nth-root functions; exponential functions; logarithmic functions. Composition of functions. Piecewise-defined functions. Absolute value function. Transformations of graphs. Determination of the domain of a function. Sequences.

Topology of the Real Line

Complete neighbourhoods, circular neighbourhoods, right and left neighbourhoods. Neighbourhoods of infinity. Accumulation points of a set. Isolated points. Complement of a set. Interior points and boundary points. Closed and open sets.

Limits and Continuity

Finite limit at a finite point, infinite limit at a finite point, finite limit at infinity and infinite limit at infinity. Right-hand and left-hand limits. Verification of a limit. Vertical and horizontal asymptotes.

Theorems on limits: uniqueness of the limit, with proof; equality between right-hand and left-hand limits, with proof; sign-preservation theorem, with proof.

Computation of limits of functions at the endpoints of their domains. Algebraic operations on limits. Oblique asymptotes: formula and proof. Indeterminate forms of the type infinity over infinity, zero times infinity, zero over zero and infinity minus infinity. Indeterminate forms for polynomials and general functions. Hierarchy of infinities. The zero over zero indeterminate form for polynomials.

Continuity at a point and on a set. Continuity of elementary functions, with particular reference to the logarithmic function and its proof. Algebraic operations on continuous functions, with proof. Continuity of composite functions, with proof. Discontinuities of the first, second and third kind. Continuity of piecewise-defined functions. Continuity and invertibility.

Standard limits, with proof. Infinitesimals: definition, reference infinitesimal, order of an infinitesimal and comparison between infinitesimals. Cancellation theorem for higher-order infinitesimals, with proof, and its application to zero over zero indeterminate forms. Infinite quantities: definition, reference infinity, order of infinity, comparison between infinities and hierarchy of infinities.

Intermediate value theorem for zeros of a continuous function, with proof. Absolute maxima and minima. Compact sets. Weierstrass theorem and the necessity of its assumptions. Study of functions: domain, intercepts with the coordinate axes, sign and limits.

Derivatives

Secant line and tangent line to a function or curve. Difference quotient. Derivative of a function at a point. Equation of the tangent line to the graph of a function at a point. Differentiability at a point and on an interval. Derivative function. Right-hand and left-hand derivatives.

Derivatives of elementary functions: constant function, power function, exponential function and logarithmic function, with proofs. The theorem stating that differentiability implies continuity, with proof. Points of non-differentiability: corner points, vertical tangents and cusps.

Differentiation rules: multiplication by a scalar, sum of functions, product and quotient of two functions, with proofs. Derivative of a composite function, with proof.

Local maxima and minima. Fermat's theorem, with proof. Joint application of Weierstrass's and Fermat's theorems to compact sets. Rolle's theorem, with proof. Lagrange's mean value theorem, with proof, and its geometric interpretation. Cauchy's mean value theorem, with proof. Monotonicity test theorem, with proof. De l'Hôpital's theorem, with proof.

Higher-order derivatives. Functions of class C k. Convex and concave functions. Global and local convexity. Taylor and Maclaurin polynomials. Convexity test, with proof.

Integrals

Area of the region of the plane bounded by the horizontal axis and the graph of a function. Sequences and partitions. Upper and lower integral sums. Riemann integrable functions. Definite integral of a function over a closed and bounded interval. Relationship between continuity and integrability. Properties of the definite integral.

Antiderivative of a function. Set of antiderivatives of a function and indefinite integral. Basic integrals. Signed area. Integration by parts. Integration by substitution.

Vectors and Matrices

Geometric vectors, free vectors and applied vectors. Length, orientation and direction of a vector. Vectors in n-dimensional real space. Operations on vectors: vector addition and scalar multiplication, together with their geometric interpretation. Real vector spaces. Linear combinations of vectors. Linearly dependent and linearly independent vectors.

Matrices. Row and column vectors. Zero matrix and square matrices. Matrix addition and multiplication of a matrix by a real number. Transpose matrix. Symmetric matrices. Diagonal matrix and identity matrix. Scalar product. Matrix multiplication and its properties.

Area of a parallelogram in the real plane. Determinant of a square matrix of order two. Determinants of square matrices of order n and their properties. Relationship between determinants and linear dependence or independence of vectors. Determinant of a matrix of order three using Sarrus' rule.

Inverse matrix. Theorem on the inverse matrix, with proof. Computation of the inverse matrix.

Systems of Linear Equations

Cramer's rule. Homogeneous systems. Rank of a matrix. Rouché-Capelli theorem. Parametric systems.

Core Documentation

Suggested materials:
- Notes can be downloaded online from the course Matematica Generale on Moodle at the website: https://economia.el.uniroma3.it/
Optional materials:
- Loretta Mastroeni, Alessandro Mazzoccoli, Pierluigi Vellucci. Esercizi di matematica generale. Esculapio, 2023
- Loretta Mastroeni, Alessandro Mazzoccoli. Matematica generale. Teoria. Esculapio, 2025

Attendance

Attendance at the course is optional. Students may choose to attend classes and participate in classroom activities, but it is not mandatory. However, participation is strongly recommended for a better understanding of the concepts covered.

Type of evaluation

The exam will consist of a written test and an oral test, both mandatory.

Programme

Propositions. Logical operations with propositions. Logical implication. Sets. Operations with sets. Cartesian product. Applications. Injective and surjective applications. One-to-one correspondence. Inverse application.

Numeric numbers and sets: Natural numbers. Integer or relative numbers. Rational numbers. Real numbers and representation on the line. Bounded sets. Upper and lower extreme of sets of rational and real numbers. Intervals and neighborhoods. Accumulation, internal, isolated points. Open sets and closed sets.

Summations and products: Definition of summation. Properties. Special sums. Sum of the first n natural numbers. Arithmetic and geometric progressions and sum of their first n terms. Factorial.

Real functions of a real variable:

Definition of a real function of a real variable. The Euclidean plane and the graph of a function. Injective and surjective functions and graph. Even and odd functions. Increasing and decreasing functions. Concave and convex functions. Bounded functions. Composition function. Inverse function, monotonicity and invertibility, inverse function graph. Elementary functions. Functions with two laws. Transforming graphs. Domain of a function. Definition of a sequence.

Limits: Definition of limit. Convergence and divergence. Right limit and left limit. Vertical and horizontal asymptotes. Limit uniqueness theorem (w.p.). Sign permanence theorem in direct and inverse form (w.p.). Comparison theorem. Limit checks. Operations with limits. Indeterminate forms.

Infinitesimals and infinities: Definition of infinitesimal and infinite. Comparing infinitesimals and infinities. Order of infinitesimals and infinities. Propagation of the order. Computing limits with infinitesimals and infinities (w.p.).

Continuity and discontinuity: Definition of continuity. Limits and continuity. Classification of discontinuity points. Continuity of rational functions. Continuity of the inverse. Continuity of composition functions. Theorem of zeros (w.p.). Weierstrass theorem. Darboux's theorem (w.p.).

Differential calculus: Derivative of a function. Geometric interpretation. Derivability and continuity (w.p.) Points of non-derivability. Higher order derivatives. Derivatives of elementary functions. Rules of derivation. Chain rule. Derivative of the inverse function. Differential. First order approximation (w.p.). Taylor and McLaurin polynomial. Approximations of higher order. Stationary points. Local maxima and minima. First order necessary conditions for the existence of local maxima and minima. Fermat's theorem (w.p.). Rolle's theorem (w.p.). Lagrange theorem (w.p.). Lagrange theorem’s corollaries: zero-derivative functions (w.p.). Relations between monotonicity and derivative sign (w.p.). Local concave and convex functions. Relationship between the second order derivative and the concavity (w.p.). Points of inflection. Sufficient second order conditions for the existence of relative maxima and minima (w.p.). Sufficient conditions of order n for the existence of relative maxima and minima or inflections points (w.p.). De L'Hôpital theorem and application to limit calculus.

Graph of a function: Representation of the graph of a function on the Euclidean plane. Oblique asymptotes.

Linear algebra:

Vectors and vector spaces. Geometric representation of vectors. Linear combination of vectors. Linearly dependent and independent vectors. Rank of a set of vectors. Matrices. Operations with matrices. Product rows by columns. Particular matrices. Transposed matrix. Determinant of a matrix of order n. Properties of the determinant. Rank of a matrix. Rank and linear independence of vectors. Systems of linear equations. Cramer's theorem. Rouché-Capelli theorem. Homogeneous systems. Parametric systems.

Integral calculus:

Primitive functions. Indefinite integral. Characterization of the set of primitives (w.p.). Properties of the indefinite integral. Integral of elementary functions. Integration by parts (w.p.). Integration by substitution (w.p.). Definite integral. Properties of the definite integral. Integral function. Integral mean theorem(w.p.). Fundamental theorem of integral calculus (w.p.). Corollary to Torricelli-Barrow's theorem: relationship between the definite integral and the indefinite integral (w.p.). Applications.

(w.p.) = “with proof”

Core Documentation

Suggested materials:

- Mastroeni, Mazzoccoli, Vellucci: "Esercizi di matematica generale". Società editrice Esculapio. ISBN 9788893853910

Optional materials:

-Peccati, Salsa, Squellati. "Matematica per l'economia e l'azienda". Egea.

-Bramanti, Pagani, Salsa. Calcolo infinitesimale e algebra lineare Seconda edizione

- Alberto Bersani, Francesco Manzini, Loretta Mastroeni. Esercizi di Matematica Generale: Per i corsi del nuovo ordinamento delle Facoltà di Economia. Società Editrice Esculapio, 2009.

Reference Bibliography

- Mastroeni, Mazzoccoli, Vellucci: "Esercizi di matematica generale". Società editrice Esculapio. ISBN 9788893853910 Suggested materials: Optional materials: -Peccati, Salsa, Squellati. Matematica per l'economia e l'azienda. Egea. - Alberto Bersani, Francesco Manzini, Loretta Mastroeni. Esercizi di Matematica Generale: Per i corsi del nuovo ordinamento delle Facoltà di Economia. Società Editrice Esculapio, 2009.

Type of delivery of the course

Frontal lesson

Attendance

Attendance is not compulsory

Type of evaluation

The exam consists of a written and an oral test.

Programme

Logic

Syntax and semantics. Logical connectives. Sufficient condition, necessary condition, necessary and sufficient condition. Direct and constructive proofs. Proof by contradiction.

Set Theory

Sets and set membership. Primitive and derived concepts. Quantifiers. Euler-Venn diagrams. Subsets. Empty set and universal set. Union and intersection of sets. Disjoint sets. Difference between sets.

Relations and Functions – introductory concepts

Relations and functions. Tabular representation of relations and functions.

Number Systems

Natural numbers, integers, rational numbers, irrational numbers and real numbers. The irrationality of the square root of two, with proof. Order and order relations. Basic order axioms. Completeness axiom and the real line. Intervals of real numbers: closed and open intervals. Upper and lower bounds of a set. Infimum and supremum. Maximum and minimum of a set. Dedekind's axiom, its properties and implications for real intervals. Sets unbounded above and below. The mathematical notion of infinity.

Summations

Definition of summation and its properties. Sum of the reciprocals of the natural numbers and Nicola d'Oresme's result. Sum of the first n natural numbers, Gauss's sum with proof. Geometric sum with proof.

Real-Valued Functions

Definition of a real-valued function. Domain and range. Image and preimage. Cartesian product of two sets and the real plane. Graph of a function. Inverse function. Injectivity, surjectivity and invertibility. Monotonicity and strict monotonicity and their relationship with invertibility. Even and odd functions.

Elementary functions and their properties, including domain, range, monotonicity and invertibility: straight-line functions; power functions of even and odd degree; nth-root functions; exponential functions; logarithmic functions. Composition of functions. Piecewise-defined functions. Absolute value function. Transformations of graphs. Determination of the domain of a function. Sequences.

Topology of the Real Line

Complete neighbourhoods, circular neighbourhoods, right and left neighbourhoods. Neighbourhoods of infinity. Accumulation points of a set. Isolated points. Complement of a set. Interior points and boundary points. Closed and open sets.

Limits and Continuity

Finite limit at a finite point, infinite limit at a finite point, finite limit at infinity and infinite limit at infinity. Right-hand and left-hand limits. Verification of a limit. Vertical and horizontal asymptotes.

Theorems on limits: uniqueness of the limit, with proof; equality between right-hand and left-hand limits, with proof; sign-preservation theorem, with proof.

Computation of limits of functions at the endpoints of their domains. Algebraic operations on limits. Oblique asymptotes: formula and proof. Indeterminate forms of the type infinity over infinity, zero times infinity, zero over zero and infinity minus infinity. Indeterminate forms for polynomials and general functions. Hierarchy of infinities. The zero over zero indeterminate form for polynomials.

Continuity at a point and on a set. Continuity of elementary functions, with particular reference to the logarithmic function and its proof. Algebraic operations on continuous functions, with proof. Continuity of composite functions, with proof. Discontinuities of the first, second and third kind. Continuity of piecewise-defined functions. Continuity and invertibility.

Standard limits, with proof. Infinitesimals: definition, reference infinitesimal, order of an infinitesimal and comparison between infinitesimals. Cancellation theorem for higher-order infinitesimals, with proof, and its application to zero over zero indeterminate forms. Infinite quantities: definition, reference infinity, order of infinity, comparison between infinities and hierarchy of infinities.

Intermediate value theorem for zeros of a continuous function, with proof. Absolute maxima and minima. Compact sets. Weierstrass theorem and the necessity of its assumptions. Study of functions: domain, intercepts with the coordinate axes, sign and limits.

Derivatives

Secant line and tangent line to a function or curve. Difference quotient. Derivative of a function at a point. Equation of the tangent line to the graph of a function at a point. Differentiability at a point and on an interval. Derivative function. Right-hand and left-hand derivatives.

Derivatives of elementary functions: constant function, power function, exponential function and logarithmic function, with proofs. The theorem stating that differentiability implies continuity, with proof. Points of non-differentiability: corner points, vertical tangents and cusps.

Differentiation rules: multiplication by a scalar, sum of functions, product and quotient of two functions, with proofs. Derivative of a composite function, with proof.

Local maxima and minima. Fermat's theorem, with proof. Joint application of Weierstrass's and Fermat's theorems to compact sets. Rolle's theorem, with proof. Lagrange's mean value theorem, with proof, and its geometric interpretation. Cauchy's mean value theorem, with proof. Monotonicity test theorem, with proof. De l'Hôpital's theorem, with proof.

Higher-order derivatives. Functions of class C k. Convex and concave functions. Global and local convexity. Taylor and Maclaurin polynomials. Convexity test, with proof.

Integrals

Area of the region of the plane bounded by the horizontal axis and the graph of a function. Sequences and partitions. Upper and lower integral sums. Riemann integrable functions. Definite integral of a function over a closed and bounded interval. Relationship between continuity and integrability. Properties of the definite integral.

Antiderivative of a function. Set of antiderivatives of a function and indefinite integral. Basic integrals. Signed area. Integration by parts. Integration by substitution.

Vectors and Matrices

Geometric vectors, free vectors and applied vectors. Length, orientation and direction of a vector. Vectors in n-dimensional real space. Operations on vectors: vector addition and scalar multiplication, together with their geometric interpretation. Real vector spaces. Linear combinations of vectors. Linearly dependent and linearly independent vectors.

Matrices. Row and column vectors. Zero matrix and square matrices. Matrix addition and multiplication of a matrix by a real number. Transpose matrix. Symmetric matrices. Diagonal matrix and identity matrix. Scalar product. Matrix multiplication and its properties.

Area of a parallelogram in the real plane. Determinant of a square matrix of order two. Determinants of square matrices of order n and their properties. Relationship between determinants and linear dependence or independence of vectors. Determinant of a matrix of order three using Sarrus' rule.

Inverse matrix. Theorem on the inverse matrix, with proof. Computation of the inverse matrix.

Systems of Linear Equations

Cramer's rule. Homogeneous systems. Rank of a matrix. Rouché-Capelli theorem. Parametric systems.

Core Documentation

Suggested materials:
- Notes can be downloaded online from the course Matematica Generale on Moodle at the website: https://economia.el.uniroma3.it/
Optional materials:
- Loretta Mastroeni, Alessandro Mazzoccoli, Pierluigi Vellucci. Esercizi di matematica generale. Esculapio, 2023
- Loretta Mastroeni, Alessandro Mazzoccoli. Matematica generale. Teoria. Esculapio, 2025

Attendance

Attendance at the course is optional. Students may choose to attend classes and participate in classroom activities, but it is not mandatory. However, participation is strongly recommended for a better understanding of the concepts covered.

Type of evaluation

The exam will consist of a written test and an oral test, both mandatory.

Programme

Propositions. Logical operations with propositions. Logical implication. Sets. Operations with sets. Cartesian product. Applications. Injective and surjective applications. One-to-one correspondence. Inverse application.

Numeric numbers and sets: Natural numbers. Integer or relative numbers. Rational numbers. Real numbers and representation on the line. Bounded sets. Upper and lower extreme of sets of rational and real numbers. Intervals and neighborhoods. Accumulation, internal, isolated points. Open sets and closed sets.

Summations and products: Definition of summation. Properties. Special sums. Sum of the first n natural numbers. Arithmetic and geometric progressions and sum of their first n terms. Factorial.

Real functions of a real variable:

Definition of a real function of a real variable. The Euclidean plane and the graph of a function. Injective and surjective functions and graph. Even and odd functions. Increasing and decreasing functions. Concave and convex functions. Bounded functions. Composition function. Inverse function, monotonicity and invertibility, inverse function graph. Elementary functions. Functions with two laws. Transforming graphs. Domain of a function. Definition of a sequence.

Limits: Definition of limit. Convergence and divergence. Right limit and left limit. Vertical and horizontal asymptotes. Limit uniqueness theorem (w.p.). Sign permanence theorem in direct and inverse form (w.p.). Comparison theorem. Limit checks. Operations with limits. Indeterminate forms.

Infinitesimals and infinities: Definition of infinitesimal and infinite. Comparing infinitesimals and infinities. Order of infinitesimals and infinities. Propagation of the order. Computing limits with infinitesimals and infinities (w.p.).

Continuity and discontinuity: Definition of continuity. Limits and continuity. Classification of discontinuity points. Continuity of rational functions. Continuity of the inverse. Continuity of composition functions. Theorem of zeros (w.p.). Weierstrass theorem. Darboux's theorem (w.p.).

Differential calculus: Derivative of a function. Geometric interpretation. Derivability and continuity (w.p.) Points of non-derivability. Higher order derivatives. Derivatives of elementary functions. Rules of derivation. Chain rule. Derivative of the inverse function. Differential. First order approximation (w.p.). Taylor and McLaurin polynomial. Approximations of higher order. Stationary points. Local maxima and minima. First order necessary conditions for the existence of local maxima and minima. Fermat's theorem (w.p.). Rolle's theorem (w.p.). Lagrange theorem (w.p.). Lagrange theorem’s corollaries: zero-derivative functions (w.p.). Relations between monotonicity and derivative sign (w.p.). Local concave and convex functions. Relationship between the second order derivative and the concavity (w.p.). Points of inflection. Sufficient second order conditions for the existence of relative maxima and minima (w.p.). Sufficient conditions of order n for the existence of relative maxima and minima or inflections points (w.p.). De L'Hôpital theorem and application to limit calculus.

Graph of a function: Representation of the graph of a function on the Euclidean plane. Oblique asymptotes.

Linear algebra:

Vectors and vector spaces. Geometric representation of vectors. Linear combination of vectors. Linearly dependent and independent vectors. Rank of a set of vectors. Matrices. Operations with matrices. Product rows by columns. Particular matrices. Transposed matrix. Determinant of a matrix of order n. Properties of the determinant. Rank of a matrix. Rank and linear independence of vectors. Systems of linear equations. Cramer's theorem. Rouché-Capelli theorem. Homogeneous systems. Parametric systems.

Integral calculus:

Primitive functions. Indefinite integral. Characterization of the set of primitives (w.p.). Properties of the indefinite integral. Integral of elementary functions. Integration by parts (w.p.). Integration by substitution (w.p.). Definite integral. Properties of the definite integral. Integral function. Integral mean theorem(w.p.). Fundamental theorem of integral calculus (w.p.). Corollary to Torricelli-Barrow's theorem: relationship between the definite integral and the indefinite integral (w.p.). Applications.

(w.p.) = “with proof”

Core Documentation

Suggested materials:

- Mastroeni, Mazzoccoli, Vellucci: "Esercizi di matematica generale". Società editrice Esculapio. ISBN 9788893853910

Optional materials:

-Peccati, Salsa, Squellati. "Matematica per l'economia e l'azienda". Egea.

-Bramanti, Pagani, Salsa. Calcolo infinitesimale e algebra lineare Seconda edizione

- Alberto Bersani, Francesco Manzini, Loretta Mastroeni. Esercizi di Matematica Generale: Per i corsi del nuovo ordinamento delle Facoltà di Economia. Società Editrice Esculapio, 2009.

Reference Bibliography

- Mastroeni, Mazzoccoli, Vellucci: "Esercizi di matematica generale". Società editrice Esculapio. ISBN 9788893853910 Suggested materials: Optional materials: -Peccati, Salsa, Squellati. Matematica per l'economia e l'azienda. Egea. - Alberto Bersani, Francesco Manzini, Loretta Mastroeni. Esercizi di Matematica Generale: Per i corsi del nuovo ordinamento delle Facoltà di Economia. Società Editrice Esculapio, 2009.

Type of delivery of the course

Frontal lesson

Attendance

Attendance is not compulsory

Type of evaluation

The exam consists of a written and an oral test.

Programme

Logic

Syntax and semantics. Logical connectives. Sufficient condition, necessary condition, necessary and sufficient condition. Direct and constructive proofs. Proof by contradiction.

Set Theory

Sets and set membership. Primitive and derived concepts. Quantifiers. Euler-Venn diagrams. Subsets. Empty set and universal set. Union and intersection of sets. Disjoint sets. Difference between sets.

Relations and Functions – introductory concepts

Relations and functions. Tabular representation of relations and functions.

Number Systems

Natural numbers, integers, rational numbers, irrational numbers and real numbers. The irrationality of the square root of two, with proof. Order and order relations. Basic order axioms. Completeness axiom and the real line. Intervals of real numbers: closed and open intervals. Upper and lower bounds of a set. Infimum and supremum. Maximum and minimum of a set. Dedekind's axiom, its properties and implications for real intervals. Sets unbounded above and below. The mathematical notion of infinity.

Summations

Definition of summation and its properties. Sum of the reciprocals of the natural numbers and Nicola d'Oresme's result. Sum of the first n natural numbers, Gauss's sum with proof. Geometric sum with proof.

Real-Valued Functions

Definition of a real-valued function. Domain and range. Image and preimage. Cartesian product of two sets and the real plane. Graph of a function. Inverse function. Injectivity, surjectivity and invertibility. Monotonicity and strict monotonicity and their relationship with invertibility. Even and odd functions.

Elementary functions and their properties, including domain, range, monotonicity and invertibility: straight-line functions; power functions of even and odd degree; nth-root functions; exponential functions; logarithmic functions. Composition of functions. Piecewise-defined functions. Absolute value function. Transformations of graphs. Determination of the domain of a function. Sequences.

Topology of the Real Line

Complete neighbourhoods, circular neighbourhoods, right and left neighbourhoods. Neighbourhoods of infinity. Accumulation points of a set. Isolated points. Complement of a set. Interior points and boundary points. Closed and open sets.

Limits and Continuity

Finite limit at a finite point, infinite limit at a finite point, finite limit at infinity and infinite limit at infinity. Right-hand and left-hand limits. Verification of a limit. Vertical and horizontal asymptotes.

Theorems on limits: uniqueness of the limit, with proof; equality between right-hand and left-hand limits, with proof; sign-preservation theorem, with proof.

Computation of limits of functions at the endpoints of their domains. Algebraic operations on limits. Oblique asymptotes: formula and proof. Indeterminate forms of the type infinity over infinity, zero times infinity, zero over zero and infinity minus infinity. Indeterminate forms for polynomials and general functions. Hierarchy of infinities. The zero over zero indeterminate form for polynomials.

Continuity at a point and on a set. Continuity of elementary functions, with particular reference to the logarithmic function and its proof. Algebraic operations on continuous functions, with proof. Continuity of composite functions, with proof. Discontinuities of the first, second and third kind. Continuity of piecewise-defined functions. Continuity and invertibility.

Standard limits, with proof. Infinitesimals: definition, reference infinitesimal, order of an infinitesimal and comparison between infinitesimals. Cancellation theorem for higher-order infinitesimals, with proof, and its application to zero over zero indeterminate forms. Infinite quantities: definition, reference infinity, order of infinity, comparison between infinities and hierarchy of infinities.

Intermediate value theorem for zeros of a continuous function, with proof. Absolute maxima and minima. Compact sets. Weierstrass theorem and the necessity of its assumptions. Study of functions: domain, intercepts with the coordinate axes, sign and limits.

Derivatives

Secant line and tangent line to a function or curve. Difference quotient. Derivative of a function at a point. Equation of the tangent line to the graph of a function at a point. Differentiability at a point and on an interval. Derivative function. Right-hand and left-hand derivatives.

Derivatives of elementary functions: constant function, power function, exponential function and logarithmic function, with proofs. The theorem stating that differentiability implies continuity, with proof. Points of non-differentiability: corner points, vertical tangents and cusps.

Differentiation rules: multiplication by a scalar, sum of functions, product and quotient of two functions, with proofs. Derivative of a composite function, with proof.

Local maxima and minima. Fermat's theorem, with proof. Joint application of Weierstrass's and Fermat's theorems to compact sets. Rolle's theorem, with proof. Lagrange's mean value theorem, with proof, and its geometric interpretation. Cauchy's mean value theorem, with proof. Monotonicity test theorem, with proof. De l'Hôpital's theorem, with proof.

Higher-order derivatives. Functions of class C k. Convex and concave functions. Global and local convexity. Taylor and Maclaurin polynomials. Convexity test, with proof.

Integrals

Area of the region of the plane bounded by the horizontal axis and the graph of a function. Sequences and partitions. Upper and lower integral sums. Riemann integrable functions. Definite integral of a function over a closed and bounded interval. Relationship between continuity and integrability. Properties of the definite integral.

Antiderivative of a function. Set of antiderivatives of a function and indefinite integral. Basic integrals. Signed area. Integration by parts. Integration by substitution.

Vectors and Matrices

Geometric vectors, free vectors and applied vectors. Length, orientation and direction of a vector. Vectors in n-dimensional real space. Operations on vectors: vector addition and scalar multiplication, together with their geometric interpretation. Real vector spaces. Linear combinations of vectors. Linearly dependent and linearly independent vectors.

Matrices. Row and column vectors. Zero matrix and square matrices. Matrix addition and multiplication of a matrix by a real number. Transpose matrix. Symmetric matrices. Diagonal matrix and identity matrix. Scalar product. Matrix multiplication and its properties.

Area of a parallelogram in the real plane. Determinant of a square matrix of order two. Determinants of square matrices of order n and their properties. Relationship between determinants and linear dependence or independence of vectors. Determinant of a matrix of order three using Sarrus' rule.

Inverse matrix. Theorem on the inverse matrix, with proof. Computation of the inverse matrix.

Systems of Linear Equations

Cramer's rule. Homogeneous systems. Rank of a matrix. Rouché-Capelli theorem. Parametric systems.

Core Documentation

Suggested materials:
- Notes can be downloaded online from the course Matematica Generale on Moodle at the website: https://economia.el.uniroma3.it/
Optional materials:
- Loretta Mastroeni, Alessandro Mazzoccoli, Pierluigi Vellucci. Esercizi di matematica generale. Esculapio, 2023
- Loretta Mastroeni, Alessandro Mazzoccoli. Matematica generale. Teoria. Esculapio, 2025

Attendance

Attendance at the course is optional. Students may choose to attend classes and participate in classroom activities, but it is not mandatory. However, participation is strongly recommended for a better understanding of the concepts covered.

Type of evaluation

The exam will consist of a written test and an oral test, both mandatory.

Programme

Propositions. Logical operations with propositions. Logical implication. Sets. Operations with sets. Cartesian product. Applications. Injective and surjective applications. One-to-one correspondence. Inverse application.

Numeric numbers and sets: Natural numbers. Integer or relative numbers. Rational numbers. Real numbers and representation on the line. Bounded sets. Upper and lower extreme of sets of rational and real numbers. Intervals and neighborhoods. Accumulation, internal, isolated points. Open sets and closed sets.

Summations and products: Definition of summation. Properties. Special sums. Sum of the first n natural numbers. Arithmetic and geometric progressions and sum of their first n terms. Factorial.

Real functions of a real variable:

Definition of a real function of a real variable. The Euclidean plane and the graph of a function. Injective and surjective functions and graph. Even and odd functions. Increasing and decreasing functions. Concave and convex functions. Bounded functions. Composition function. Inverse function, monotonicity and invertibility, inverse function graph. Elementary functions. Functions with two laws. Transforming graphs. Domain of a function. Definition of a sequence.

Limits: Definition of limit. Convergence and divergence. Right limit and left limit. Vertical and horizontal asymptotes. Limit uniqueness theorem (w.p.). Sign permanence theorem in direct and inverse form (w.p.). Comparison theorem. Limit checks. Operations with limits. Indeterminate forms.

Infinitesimals and infinities: Definition of infinitesimal and infinite. Comparing infinitesimals and infinities. Order of infinitesimals and infinities. Propagation of the order. Computing limits with infinitesimals and infinities (w.p.).

Continuity and discontinuity: Definition of continuity. Limits and continuity. Classification of discontinuity points. Continuity of rational functions. Continuity of the inverse. Continuity of composition functions. Theorem of zeros (w.p.). Weierstrass theorem. Darboux's theorem (w.p.).

Differential calculus: Derivative of a function. Geometric interpretation. Derivability and continuity (w.p.) Points of non-derivability. Higher order derivatives. Derivatives of elementary functions. Rules of derivation. Chain rule. Derivative of the inverse function. Differential. First order approximation (w.p.). Taylor and McLaurin polynomial. Approximations of higher order. Stationary points. Local maxima and minima. First order necessary conditions for the existence of local maxima and minima. Fermat's theorem (w.p.). Rolle's theorem (w.p.). Lagrange theorem (w.p.). Lagrange theorem’s corollaries: zero-derivative functions (w.p.). Relations between monotonicity and derivative sign (w.p.). Local concave and convex functions. Relationship between the second order derivative and the concavity (w.p.). Points of inflection. Sufficient second order conditions for the existence of relative maxima and minima (w.p.). Sufficient conditions of order n for the existence of relative maxima and minima or inflections points (w.p.). De L'Hôpital theorem and application to limit calculus.

Graph of a function: Representation of the graph of a function on the Euclidean plane. Oblique asymptotes.

Linear algebra:

Vectors and vector spaces. Geometric representation of vectors. Linear combination of vectors. Linearly dependent and independent vectors. Rank of a set of vectors. Matrices. Operations with matrices. Product rows by columns. Particular matrices. Transposed matrix. Determinant of a matrix of order n. Properties of the determinant. Rank of a matrix. Rank and linear independence of vectors. Systems of linear equations. Cramer's theorem. Rouché-Capelli theorem. Homogeneous systems. Parametric systems.

Integral calculus:

Primitive functions. Indefinite integral. Characterization of the set of primitives (w.p.). Properties of the indefinite integral. Integral of elementary functions. Integration by parts (w.p.). Integration by substitution (w.p.). Definite integral. Properties of the definite integral. Integral function. Integral mean theorem(w.p.). Fundamental theorem of integral calculus (w.p.). Corollary to Torricelli-Barrow's theorem: relationship between the definite integral and the indefinite integral (w.p.). Applications.

(w.p.) = “with proof”

Core Documentation

Suggested materials:

- Mastroeni, Mazzoccoli, Vellucci: "Esercizi di matematica generale". Società editrice Esculapio. ISBN 9788893853910

Optional materials:

-Peccati, Salsa, Squellati. "Matematica per l'economia e l'azienda". Egea.

-Bramanti, Pagani, Salsa. Calcolo infinitesimale e algebra lineare Seconda edizione

- Alberto Bersani, Francesco Manzini, Loretta Mastroeni. Esercizi di Matematica Generale: Per i corsi del nuovo ordinamento delle Facoltà di Economia. Società Editrice Esculapio, 2009.

Reference Bibliography

- Mastroeni, Mazzoccoli, Vellucci: "Esercizi di matematica generale". Società editrice Esculapio. ISBN 9788893853910 Suggested materials: Optional materials: -Peccati, Salsa, Squellati. Matematica per l'economia e l'azienda. Egea. - Alberto Bersani, Francesco Manzini, Loretta Mastroeni. Esercizi di Matematica Generale: Per i corsi del nuovo ordinamento delle Facoltà di Economia. Società Editrice Esculapio, 2009.

Type of delivery of the course

Frontal lesson

Attendance

Attendance is not compulsory

Type of evaluation

The exam consists of a written and an oral test.

Programme

Logic

Syntax and semantics. Logical connectives. Sufficient condition, necessary condition, necessary and sufficient condition. Direct and constructive proofs. Proof by contradiction.

Set Theory

Sets and set membership. Primitive and derived concepts. Quantifiers. Euler-Venn diagrams. Subsets. Empty set and universal set. Union and intersection of sets. Disjoint sets. Difference between sets.

Relations and Functions – introductory concepts

Relations and functions. Tabular representation of relations and functions.

Number Systems

Natural numbers, integers, rational numbers, irrational numbers and real numbers. The irrationality of the square root of two, with proof. Order and order relations. Basic order axioms. Completeness axiom and the real line. Intervals of real numbers: closed and open intervals. Upper and lower bounds of a set. Infimum and supremum. Maximum and minimum of a set. Dedekind's axiom, its properties and implications for real intervals. Sets unbounded above and below. The mathematical notion of infinity.

Summations

Definition of summation and its properties. Sum of the reciprocals of the natural numbers and Nicola d'Oresme's result. Sum of the first n natural numbers, Gauss's sum with proof. Geometric sum with proof.

Real-Valued Functions

Definition of a real-valued function. Domain and range. Image and preimage. Cartesian product of two sets and the real plane. Graph of a function. Inverse function. Injectivity, surjectivity and invertibility. Monotonicity and strict monotonicity and their relationship with invertibility. Even and odd functions.

Elementary functions and their properties, including domain, range, monotonicity and invertibility: straight-line functions; power functions of even and odd degree; nth-root functions; exponential functions; logarithmic functions. Composition of functions. Piecewise-defined functions. Absolute value function. Transformations of graphs. Determination of the domain of a function. Sequences.

Topology of the Real Line

Complete neighbourhoods, circular neighbourhoods, right and left neighbourhoods. Neighbourhoods of infinity. Accumulation points of a set. Isolated points. Complement of a set. Interior points and boundary points. Closed and open sets.

Limits and Continuity

Finite limit at a finite point, infinite limit at a finite point, finite limit at infinity and infinite limit at infinity. Right-hand and left-hand limits. Verification of a limit. Vertical and horizontal asymptotes.

Theorems on limits: uniqueness of the limit, with proof; equality between right-hand and left-hand limits, with proof; sign-preservation theorem, with proof.

Computation of limits of functions at the endpoints of their domains. Algebraic operations on limits. Oblique asymptotes: formula and proof. Indeterminate forms of the type infinity over infinity, zero times infinity, zero over zero and infinity minus infinity. Indeterminate forms for polynomials and general functions. Hierarchy of infinities. The zero over zero indeterminate form for polynomials.

Continuity at a point and on a set. Continuity of elementary functions, with particular reference to the logarithmic function and its proof. Algebraic operations on continuous functions, with proof. Continuity of composite functions, with proof. Discontinuities of the first, second and third kind. Continuity of piecewise-defined functions. Continuity and invertibility.

Standard limits, with proof. Infinitesimals: definition, reference infinitesimal, order of an infinitesimal and comparison between infinitesimals. Cancellation theorem for higher-order infinitesimals, with proof, and its application to zero over zero indeterminate forms. Infinite quantities: definition, reference infinity, order of infinity, comparison between infinities and hierarchy of infinities.

Intermediate value theorem for zeros of a continuous function, with proof. Absolute maxima and minima. Compact sets. Weierstrass theorem and the necessity of its assumptions. Study of functions: domain, intercepts with the coordinate axes, sign and limits.

Derivatives

Secant line and tangent line to a function or curve. Difference quotient. Derivative of a function at a point. Equation of the tangent line to the graph of a function at a point. Differentiability at a point and on an interval. Derivative function. Right-hand and left-hand derivatives.

Derivatives of elementary functions: constant function, power function, exponential function and logarithmic function, with proofs. The theorem stating that differentiability implies continuity, with proof. Points of non-differentiability: corner points, vertical tangents and cusps.

Differentiation rules: multiplication by a scalar, sum of functions, product and quotient of two functions, with proofs. Derivative of a composite function, with proof.

Local maxima and minima. Fermat's theorem, with proof. Joint application of Weierstrass's and Fermat's theorems to compact sets. Rolle's theorem, with proof. Lagrange's mean value theorem, with proof, and its geometric interpretation. Cauchy's mean value theorem, with proof. Monotonicity test theorem, with proof. De l'Hôpital's theorem, with proof.

Higher-order derivatives. Functions of class C k. Convex and concave functions. Global and local convexity. Taylor and Maclaurin polynomials. Convexity test, with proof.

Integrals

Area of the region of the plane bounded by the horizontal axis and the graph of a function. Sequences and partitions. Upper and lower integral sums. Riemann integrable functions. Definite integral of a function over a closed and bounded interval. Relationship between continuity and integrability. Properties of the definite integral.

Antiderivative of a function. Set of antiderivatives of a function and indefinite integral. Basic integrals. Signed area. Integration by parts. Integration by substitution.

Vectors and Matrices

Geometric vectors, free vectors and applied vectors. Length, orientation and direction of a vector. Vectors in n-dimensional real space. Operations on vectors: vector addition and scalar multiplication, together with their geometric interpretation. Real vector spaces. Linear combinations of vectors. Linearly dependent and linearly independent vectors.

Matrices. Row and column vectors. Zero matrix and square matrices. Matrix addition and multiplication of a matrix by a real number. Transpose matrix. Symmetric matrices. Diagonal matrix and identity matrix. Scalar product. Matrix multiplication and its properties.

Area of a parallelogram in the real plane. Determinant of a square matrix of order two. Determinants of square matrices of order n and their properties. Relationship between determinants and linear dependence or independence of vectors. Determinant of a matrix of order three using Sarrus' rule.

Inverse matrix. Theorem on the inverse matrix, with proof. Computation of the inverse matrix.

Systems of Linear Equations

Cramer's rule. Homogeneous systems. Rank of a matrix. Rouché-Capelli theorem. Parametric systems.

Core Documentation

Suggested materials:
- Notes can be downloaded online from the course Matematica Generale on Moodle at the website: https://economia.el.uniroma3.it/
Optional materials:
- Loretta Mastroeni, Alessandro Mazzoccoli, Pierluigi Vellucci. Esercizi di matematica generale. Esculapio, 2023
- Loretta Mastroeni, Alessandro Mazzoccoli. Matematica generale. Teoria. Esculapio, 2025

Attendance

Attendance at the course is optional. Students may choose to attend classes and participate in classroom activities, but it is not mandatory. However, participation is strongly recommended for a better understanding of the concepts covered.

Type of evaluation

The exam will consist of a written test and an oral test, both mandatory.

Programme

Propositions. Logical operations with propositions. Logical implication. Sets. Operations with sets. Cartesian product. Applications. Injective and surjective applications. One-to-one correspondence. Inverse application.

Numeric numbers and sets: Natural numbers. Integer or relative numbers. Rational numbers. Real numbers and representation on the line. Bounded sets. Upper and lower extreme of sets of rational and real numbers. Intervals and neighborhoods. Accumulation, internal, isolated points. Open sets and closed sets.

Summations and products: Definition of summation. Properties. Special sums. Sum of the first n natural numbers. Arithmetic and geometric progressions and sum of their first n terms. Factorial.

Real functions of a real variable:

Definition of a real function of a real variable. The Euclidean plane and the graph of a function. Injective and surjective functions and graph. Even and odd functions. Increasing and decreasing functions. Concave and convex functions. Bounded functions. Composition function. Inverse function, monotonicity and invertibility, inverse function graph. Elementary functions. Functions with two laws. Transforming graphs. Domain of a function. Definition of a sequence.

Limits: Definition of limit. Convergence and divergence. Right limit and left limit. Vertical and horizontal asymptotes. Limit uniqueness theorem (w.p.). Sign permanence theorem in direct and inverse form (w.p.). Comparison theorem. Limit checks. Operations with limits. Indeterminate forms.

Infinitesimals and infinities: Definition of infinitesimal and infinite. Comparing infinitesimals and infinities. Order of infinitesimals and infinities. Propagation of the order. Computing limits with infinitesimals and infinities (w.p.).

Continuity and discontinuity: Definition of continuity. Limits and continuity. Classification of discontinuity points. Continuity of rational functions. Continuity of the inverse. Continuity of composition functions. Theorem of zeros (w.p.). Weierstrass theorem. Darboux's theorem (w.p.).

Differential calculus: Derivative of a function. Geometric interpretation. Derivability and continuity (w.p.) Points of non-derivability. Higher order derivatives. Derivatives of elementary functions. Rules of derivation. Chain rule. Derivative of the inverse function. Differential. First order approximation (w.p.). Taylor and McLaurin polynomial. Approximations of higher order. Stationary points. Local maxima and minima. First order necessary conditions for the existence of local maxima and minima. Fermat's theorem (w.p.). Rolle's theorem (w.p.). Lagrange theorem (w.p.). Lagrange theorem’s corollaries: zero-derivative functions (w.p.). Relations between monotonicity and derivative sign (w.p.). Local concave and convex functions. Relationship between the second order derivative and the concavity (w.p.). Points of inflection. Sufficient second order conditions for the existence of relative maxima and minima (w.p.). Sufficient conditions of order n for the existence of relative maxima and minima or inflections points (w.p.). De L'Hôpital theorem and application to limit calculus.

Graph of a function: Representation of the graph of a function on the Euclidean plane. Oblique asymptotes.

Linear algebra:

Vectors and vector spaces. Geometric representation of vectors. Linear combination of vectors. Linearly dependent and independent vectors. Rank of a set of vectors. Matrices. Operations with matrices. Product rows by columns. Particular matrices. Transposed matrix. Determinant of a matrix of order n. Properties of the determinant. Rank of a matrix. Rank and linear independence of vectors. Systems of linear equations. Cramer's theorem. Rouché-Capelli theorem. Homogeneous systems. Parametric systems.

Integral calculus:

Primitive functions. Indefinite integral. Characterization of the set of primitives (w.p.). Properties of the indefinite integral. Integral of elementary functions. Integration by parts (w.p.). Integration by substitution (w.p.). Definite integral. Properties of the definite integral. Integral function. Integral mean theorem(w.p.). Fundamental theorem of integral calculus (w.p.). Corollary to Torricelli-Barrow's theorem: relationship between the definite integral and the indefinite integral (w.p.). Applications.

(w.p.) = “with proof”

Core Documentation

Suggested materials:

- Mastroeni, Mazzoccoli, Vellucci: "Esercizi di matematica generale". Società editrice Esculapio. ISBN 9788893853910

Optional materials:

-Peccati, Salsa, Squellati. "Matematica per l'economia e l'azienda". Egea.

-Bramanti, Pagani, Salsa. Calcolo infinitesimale e algebra lineare Seconda edizione

- Alberto Bersani, Francesco Manzini, Loretta Mastroeni. Esercizi di Matematica Generale: Per i corsi del nuovo ordinamento delle Facoltà di Economia. Società Editrice Esculapio, 2009.

Reference Bibliography

- Mastroeni, Mazzoccoli, Vellucci: "Esercizi di matematica generale". Società editrice Esculapio. ISBN 9788893853910 Suggested materials: Optional materials: -Peccati, Salsa, Squellati. Matematica per l'economia e l'azienda. Egea. - Alberto Bersani, Francesco Manzini, Loretta Mastroeni. Esercizi di Matematica Generale: Per i corsi del nuovo ordinamento delle Facoltà di Economia. Società Editrice Esculapio, 2009.

Type of delivery of the course

Frontal lesson

Attendance

Attendance is not compulsory

Type of evaluation

The exam consists of a written and an oral test.

Programme

Logic

Syntax and semantics. Logical connectives. Sufficient condition, necessary condition, necessary and sufficient condition. Direct and constructive proofs. Proof by contradiction.

Set Theory

Sets and set membership. Primitive and derived concepts. Quantifiers. Euler-Venn diagrams. Subsets. Empty set and universal set. Union and intersection of sets. Disjoint sets. Difference between sets.

Relations and Functions – introductory concepts

Relations and functions. Tabular representation of relations and functions.

Number Systems

Natural numbers, integers, rational numbers, irrational numbers and real numbers. The irrationality of the square root of two, with proof. Order and order relations. Basic order axioms. Completeness axiom and the real line. Intervals of real numbers: closed and open intervals. Upper and lower bounds of a set. Infimum and supremum. Maximum and minimum of a set. Dedekind's axiom, its properties and implications for real intervals. Sets unbounded above and below. The mathematical notion of infinity.

Summations

Definition of summation and its properties. Sum of the reciprocals of the natural numbers and Nicola d'Oresme's result. Sum of the first n natural numbers, Gauss's sum with proof. Geometric sum with proof.

Real-Valued Functions

Definition of a real-valued function. Domain and range. Image and preimage. Cartesian product of two sets and the real plane. Graph of a function. Inverse function. Injectivity, surjectivity and invertibility. Monotonicity and strict monotonicity and their relationship with invertibility. Even and odd functions.

Elementary functions and their properties, including domain, range, monotonicity and invertibility: straight-line functions; power functions of even and odd degree; nth-root functions; exponential functions; logarithmic functions. Composition of functions. Piecewise-defined functions. Absolute value function. Transformations of graphs. Determination of the domain of a function. Sequences.

Topology of the Real Line

Complete neighbourhoods, circular neighbourhoods, right and left neighbourhoods. Neighbourhoods of infinity. Accumulation points of a set. Isolated points. Complement of a set. Interior points and boundary points. Closed and open sets.

Limits and Continuity

Finite limit at a finite point, infinite limit at a finite point, finite limit at infinity and infinite limit at infinity. Right-hand and left-hand limits. Verification of a limit. Vertical and horizontal asymptotes.

Theorems on limits: uniqueness of the limit, with proof; equality between right-hand and left-hand limits, with proof; sign-preservation theorem, with proof.

Computation of limits of functions at the endpoints of their domains. Algebraic operations on limits. Oblique asymptotes: formula and proof. Indeterminate forms of the type infinity over infinity, zero times infinity, zero over zero and infinity minus infinity. Indeterminate forms for polynomials and general functions. Hierarchy of infinities. The zero over zero indeterminate form for polynomials.

Continuity at a point and on a set. Continuity of elementary functions, with particular reference to the logarithmic function and its proof. Algebraic operations on continuous functions, with proof. Continuity of composite functions, with proof. Discontinuities of the first, second and third kind. Continuity of piecewise-defined functions. Continuity and invertibility.

Standard limits, with proof. Infinitesimals: definition, reference infinitesimal, order of an infinitesimal and comparison between infinitesimals. Cancellation theorem for higher-order infinitesimals, with proof, and its application to zero over zero indeterminate forms. Infinite quantities: definition, reference infinity, order of infinity, comparison between infinities and hierarchy of infinities.

Intermediate value theorem for zeros of a continuous function, with proof. Absolute maxima and minima. Compact sets. Weierstrass theorem and the necessity of its assumptions. Study of functions: domain, intercepts with the coordinate axes, sign and limits.

Derivatives

Secant line and tangent line to a function or curve. Difference quotient. Derivative of a function at a point. Equation of the tangent line to the graph of a function at a point. Differentiability at a point and on an interval. Derivative function. Right-hand and left-hand derivatives.

Derivatives of elementary functions: constant function, power function, exponential function and logarithmic function, with proofs. The theorem stating that differentiability implies continuity, with proof. Points of non-differentiability: corner points, vertical tangents and cusps.

Differentiation rules: multiplication by a scalar, sum of functions, product and quotient of two functions, with proofs. Derivative of a composite function, with proof.

Local maxima and minima. Fermat's theorem, with proof. Joint application of Weierstrass's and Fermat's theorems to compact sets. Rolle's theorem, with proof. Lagrange's mean value theorem, with proof, and its geometric interpretation. Cauchy's mean value theorem, with proof. Monotonicity test theorem, with proof. De l'Hôpital's theorem, with proof.

Higher-order derivatives. Functions of class C k. Convex and concave functions. Global and local convexity. Taylor and Maclaurin polynomials. Convexity test, with proof.

Integrals

Area of the region of the plane bounded by the horizontal axis and the graph of a function. Sequences and partitions. Upper and lower integral sums. Riemann integrable functions. Definite integral of a function over a closed and bounded interval. Relationship between continuity and integrability. Properties of the definite integral.

Antiderivative of a function. Set of antiderivatives of a function and indefinite integral. Basic integrals. Signed area. Integration by parts. Integration by substitution.

Vectors and Matrices

Geometric vectors, free vectors and applied vectors. Length, orientation and direction of a vector. Vectors in n-dimensional real space. Operations on vectors: vector addition and scalar multiplication, together with their geometric interpretation. Real vector spaces. Linear combinations of vectors. Linearly dependent and linearly independent vectors.

Matrices. Row and column vectors. Zero matrix and square matrices. Matrix addition and multiplication of a matrix by a real number. Transpose matrix. Symmetric matrices. Diagonal matrix and identity matrix. Scalar product. Matrix multiplication and its properties.

Area of a parallelogram in the real plane. Determinant of a square matrix of order two. Determinants of square matrices of order n and their properties. Relationship between determinants and linear dependence or independence of vectors. Determinant of a matrix of order three using Sarrus' rule.

Inverse matrix. Theorem on the inverse matrix, with proof. Computation of the inverse matrix.

Systems of Linear Equations

Cramer's rule. Homogeneous systems. Rank of a matrix. Rouché-Capelli theorem. Parametric systems.

Core Documentation

Suggested materials:
- Notes can be downloaded online from the course Matematica Generale on Moodle at the website: https://economia.el.uniroma3.it/
Optional materials:
- Loretta Mastroeni, Alessandro Mazzoccoli, Pierluigi Vellucci. Esercizi di matematica generale. Esculapio, 2023
- Loretta Mastroeni, Alessandro Mazzoccoli. Matematica generale. Teoria. Esculapio, 2025

Attendance

Attendance at the course is optional. Students may choose to attend classes and participate in classroom activities, but it is not mandatory. However, participation is strongly recommended for a better understanding of the concepts covered.

Type of evaluation

The exam will consist of a written test and an oral test, both mandatory.

Programme

Propositions. Logical operations with propositions. Logical implication. Sets. Operations with sets. Cartesian product. Applications. Injective and surjective applications. One-to-one correspondence. Inverse application.

Numeric numbers and sets: Natural numbers. Integer or relative numbers. Rational numbers. Real numbers and representation on the line. Bounded sets. Upper and lower extreme of sets of rational and real numbers. Intervals and neighborhoods. Accumulation, internal, isolated points. Open sets and closed sets.

Summations and products: Definition of summation. Properties. Special sums. Sum of the first n natural numbers. Arithmetic and geometric progressions and sum of their first n terms. Factorial.

Real functions of a real variable:

Definition of a real function of a real variable. The Euclidean plane and the graph of a function. Injective and surjective functions and graph. Even and odd functions. Increasing and decreasing functions. Concave and convex functions. Bounded functions. Composition function. Inverse function, monotonicity and invertibility, inverse function graph. Elementary functions. Functions with two laws. Transforming graphs. Domain of a function. Definition of a sequence.

Limits: Definition of limit. Convergence and divergence. Right limit and left limit. Vertical and horizontal asymptotes. Limit uniqueness theorem (w.p.). Sign permanence theorem in direct and inverse form (w.p.). Comparison theorem. Limit checks. Operations with limits. Indeterminate forms.

Infinitesimals and infinities: Definition of infinitesimal and infinite. Comparing infinitesimals and infinities. Order of infinitesimals and infinities. Propagation of the order. Computing limits with infinitesimals and infinities (w.p.).

Continuity and discontinuity: Definition of continuity. Limits and continuity. Classification of discontinuity points. Continuity of rational functions. Continuity of the inverse. Continuity of composition functions. Theorem of zeros (w.p.). Weierstrass theorem. Darboux's theorem (w.p.).

Differential calculus: Derivative of a function. Geometric interpretation. Derivability and continuity (w.p.) Points of non-derivability. Higher order derivatives. Derivatives of elementary functions. Rules of derivation. Chain rule. Derivative of the inverse function. Differential. First order approximation (w.p.). Taylor and McLaurin polynomial. Approximations of higher order. Stationary points. Local maxima and minima. First order necessary conditions for the existence of local maxima and minima. Fermat's theorem (w.p.). Rolle's theorem (w.p.). Lagrange theorem (w.p.). Lagrange theorem’s corollaries: zero-derivative functions (w.p.). Relations between monotonicity and derivative sign (w.p.). Local concave and convex functions. Relationship between the second order derivative and the concavity (w.p.). Points of inflection. Sufficient second order conditions for the existence of relative maxima and minima (w.p.). Sufficient conditions of order n for the existence of relative maxima and minima or inflections points (w.p.). De L'Hôpital theorem and application to limit calculus.

Graph of a function: Representation of the graph of a function on the Euclidean plane. Oblique asymptotes.

Linear algebra:

Vectors and vector spaces. Geometric representation of vectors. Linear combination of vectors. Linearly dependent and independent vectors. Rank of a set of vectors. Matrices. Operations with matrices. Product rows by columns. Particular matrices. Transposed matrix. Determinant of a matrix of order n. Properties of the determinant. Rank of a matrix. Rank and linear independence of vectors. Systems of linear equations. Cramer's theorem. Rouché-Capelli theorem. Homogeneous systems. Parametric systems.

Integral calculus:

Primitive functions. Indefinite integral. Characterization of the set of primitives (w.p.). Properties of the indefinite integral. Integral of elementary functions. Integration by parts (w.p.). Integration by substitution (w.p.). Definite integral. Properties of the definite integral. Integral function. Integral mean theorem(w.p.). Fundamental theorem of integral calculus (w.p.). Corollary to Torricelli-Barrow's theorem: relationship between the definite integral and the indefinite integral (w.p.). Applications.

(w.p.) = “with proof”

Core Documentation

Suggested materials:

- Mastroeni, Mazzoccoli, Vellucci: "Esercizi di matematica generale". Società editrice Esculapio. ISBN 9788893853910

Optional materials:

-Peccati, Salsa, Squellati. "Matematica per l'economia e l'azienda". Egea.

-Bramanti, Pagani, Salsa. Calcolo infinitesimale e algebra lineare Seconda edizione

- Alberto Bersani, Francesco Manzini, Loretta Mastroeni. Esercizi di Matematica Generale: Per i corsi del nuovo ordinamento delle Facoltà di Economia. Società Editrice Esculapio, 2009.

Reference Bibliography

- Mastroeni, Mazzoccoli, Vellucci: "Esercizi di matematica generale". Società editrice Esculapio. ISBN 9788893853910 Suggested materials: Optional materials: -Peccati, Salsa, Squellati. Matematica per l'economia e l'azienda. Egea. - Alberto Bersani, Francesco Manzini, Loretta Mastroeni. Esercizi di Matematica Generale: Per i corsi del nuovo ordinamento delle Facoltà di Economia. Società Editrice Esculapio, 2009.

Type of delivery of the course

Frontal lesson

Attendance

Attendance is not compulsory

Type of evaluation

The exam consists of a written and an oral test.

Programme

Logic

Syntax and semantics. Logical connectives. Sufficient condition, necessary condition, necessary and sufficient condition. Direct and constructive proofs. Proof by contradiction.

Set Theory

Sets and set membership. Primitive and derived concepts. Quantifiers. Euler-Venn diagrams. Subsets. Empty set and universal set. Union and intersection of sets. Disjoint sets. Difference between sets.

Relations and Functions – introductory concepts

Relations and functions. Tabular representation of relations and functions.

Number Systems

Natural numbers, integers, rational numbers, irrational numbers and real numbers. The irrationality of the square root of two, with proof. Order and order relations. Basic order axioms. Completeness axiom and the real line. Intervals of real numbers: closed and open intervals. Upper and lower bounds of a set. Infimum and supremum. Maximum and minimum of a set. Dedekind's axiom, its properties and implications for real intervals. Sets unbounded above and below. The mathematical notion of infinity.

Summations

Definition of summation and its properties. Sum of the reciprocals of the natural numbers and Nicola d'Oresme's result. Sum of the first n natural numbers, Gauss's sum with proof. Geometric sum with proof.

Real-Valued Functions

Definition of a real-valued function. Domain and range. Image and preimage. Cartesian product of two sets and the real plane. Graph of a function. Inverse function. Injectivity, surjectivity and invertibility. Monotonicity and strict monotonicity and their relationship with invertibility. Even and odd functions.

Elementary functions and their properties, including domain, range, monotonicity and invertibility: straight-line functions; power functions of even and odd degree; nth-root functions; exponential functions; logarithmic functions. Composition of functions. Piecewise-defined functions. Absolute value function. Transformations of graphs. Determination of the domain of a function. Sequences.

Topology of the Real Line

Complete neighbourhoods, circular neighbourhoods, right and left neighbourhoods. Neighbourhoods of infinity. Accumulation points of a set. Isolated points. Complement of a set. Interior points and boundary points. Closed and open sets.

Limits and Continuity

Finite limit at a finite point, infinite limit at a finite point, finite limit at infinity and infinite limit at infinity. Right-hand and left-hand limits. Verification of a limit. Vertical and horizontal asymptotes.

Theorems on limits: uniqueness of the limit, with proof; equality between right-hand and left-hand limits, with proof; sign-preservation theorem, with proof.

Computation of limits of functions at the endpoints of their domains. Algebraic operations on limits. Oblique asymptotes: formula and proof. Indeterminate forms of the type infinity over infinity, zero times infinity, zero over zero and infinity minus infinity. Indeterminate forms for polynomials and general functions. Hierarchy of infinities. The zero over zero indeterminate form for polynomials.

Continuity at a point and on a set. Continuity of elementary functions, with particular reference to the logarithmic function and its proof. Algebraic operations on continuous functions, with proof. Continuity of composite functions, with proof. Discontinuities of the first, second and third kind. Continuity of piecewise-defined functions. Continuity and invertibility.

Standard limits, with proof. Infinitesimals: definition, reference infinitesimal, order of an infinitesimal and comparison between infinitesimals. Cancellation theorem for higher-order infinitesimals, with proof, and its application to zero over zero indeterminate forms. Infinite quantities: definition, reference infinity, order of infinity, comparison between infinities and hierarchy of infinities.

Intermediate value theorem for zeros of a continuous function, with proof. Absolute maxima and minima. Compact sets. Weierstrass theorem and the necessity of its assumptions. Study of functions: domain, intercepts with the coordinate axes, sign and limits.

Derivatives

Secant line and tangent line to a function or curve. Difference quotient. Derivative of a function at a point. Equation of the tangent line to the graph of a function at a point. Differentiability at a point and on an interval. Derivative function. Right-hand and left-hand derivatives.

Derivatives of elementary functions: constant function, power function, exponential function and logarithmic function, with proofs. The theorem stating that differentiability implies continuity, with proof. Points of non-differentiability: corner points, vertical tangents and cusps.

Differentiation rules: multiplication by a scalar, sum of functions, product and quotient of two functions, with proofs. Derivative of a composite function, with proof.

Local maxima and minima. Fermat's theorem, with proof. Joint application of Weierstrass's and Fermat's theorems to compact sets. Rolle's theorem, with proof. Lagrange's mean value theorem, with proof, and its geometric interpretation. Cauchy's mean value theorem, with proof. Monotonicity test theorem, with proof. De l'Hôpital's theorem, with proof.

Higher-order derivatives. Functions of class C k. Convex and concave functions. Global and local convexity. Taylor and Maclaurin polynomials. Convexity test, with proof.

Integrals

Area of the region of the plane bounded by the horizontal axis and the graph of a function. Sequences and partitions. Upper and lower integral sums. Riemann integrable functions. Definite integral of a function over a closed and bounded interval. Relationship between continuity and integrability. Properties of the definite integral.

Antiderivative of a function. Set of antiderivatives of a function and indefinite integral. Basic integrals. Signed area. Integration by parts. Integration by substitution.

Vectors and Matrices

Geometric vectors, free vectors and applied vectors. Length, orientation and direction of a vector. Vectors in n-dimensional real space. Operations on vectors: vector addition and scalar multiplication, together with their geometric interpretation. Real vector spaces. Linear combinations of vectors. Linearly dependent and linearly independent vectors.

Matrices. Row and column vectors. Zero matrix and square matrices. Matrix addition and multiplication of a matrix by a real number. Transpose matrix. Symmetric matrices. Diagonal matrix and identity matrix. Scalar product. Matrix multiplication and its properties.

Area of a parallelogram in the real plane. Determinant of a square matrix of order two. Determinants of square matrices of order n and their properties. Relationship between determinants and linear dependence or independence of vectors. Determinant of a matrix of order three using Sarrus' rule.

Inverse matrix. Theorem on the inverse matrix, with proof. Computation of the inverse matrix.

Systems of Linear Equations

Cramer's rule. Homogeneous systems. Rank of a matrix. Rouché-Capelli theorem. Parametric systems.

Core Documentation

Suggested materials:
- Notes can be downloaded online from the course Matematica Generale on Moodle at the website: https://economia.el.uniroma3.it/
Optional materials:
- Loretta Mastroeni, Alessandro Mazzoccoli, Pierluigi Vellucci. Esercizi di matematica generale. Esculapio, 2023
- Loretta Mastroeni, Alessandro Mazzoccoli. Matematica generale. Teoria. Esculapio, 2025

Attendance

Attendance at the course is optional. Students may choose to attend classes and participate in classroom activities, but it is not mandatory. However, participation is strongly recommended for a better understanding of the concepts covered.

Type of evaluation

The exam will consist of a written test and an oral test, both mandatory.

Programme

Propositions. Logical operations with propositions. Logical implication. Sets. Operations with sets. Cartesian product. Applications. Injective and surjective applications. One-to-one correspondence. Inverse application.

Numeric numbers and sets: Natural numbers. Integer or relative numbers. Rational numbers. Real numbers and representation on the line. Bounded sets. Upper and lower extreme of sets of rational and real numbers. Intervals and neighborhoods. Accumulation, internal, isolated points. Open sets and closed sets.

Summations and products: Definition of summation. Properties. Special sums. Sum of the first n natural numbers. Arithmetic and geometric progressions and sum of their first n terms. Factorial.

Real functions of a real variable:

Definition of a real function of a real variable. The Euclidean plane and the graph of a function. Injective and surjective functions and graph. Even and odd functions. Increasing and decreasing functions. Concave and convex functions. Bounded functions. Composition function. Inverse function, monotonicity and invertibility, inverse function graph. Elementary functions. Functions with two laws. Transforming graphs. Domain of a function. Definition of a sequence.

Limits: Definition of limit. Convergence and divergence. Right limit and left limit. Vertical and horizontal asymptotes. Limit uniqueness theorem (w.p.). Sign permanence theorem in direct and inverse form (w.p.). Comparison theorem. Limit checks. Operations with limits. Indeterminate forms.

Infinitesimals and infinities: Definition of infinitesimal and infinite. Comparing infinitesimals and infinities. Order of infinitesimals and infinities. Propagation of the order. Computing limits with infinitesimals and infinities (w.p.).

Continuity and discontinuity: Definition of continuity. Limits and continuity. Classification of discontinuity points. Continuity of rational functions. Continuity of the inverse. Continuity of composition functions. Theorem of zeros (w.p.). Weierstrass theorem. Darboux's theorem (w.p.).

Differential calculus: Derivative of a function. Geometric interpretation. Derivability and continuity (w.p.) Points of non-derivability. Higher order derivatives. Derivatives of elementary functions. Rules of derivation. Chain rule. Derivative of the inverse function. Differential. First order approximation (w.p.). Taylor and McLaurin polynomial. Approximations of higher order. Stationary points. Local maxima and minima. First order necessary conditions for the existence of local maxima and minima. Fermat's theorem (w.p.). Rolle's theorem (w.p.). Lagrange theorem (w.p.). Lagrange theorem’s corollaries: zero-derivative functions (w.p.). Relations between monotonicity and derivative sign (w.p.). Local concave and convex functions. Relationship between the second order derivative and the concavity (w.p.). Points of inflection. Sufficient second order conditions for the existence of relative maxima and minima (w.p.). Sufficient conditions of order n for the existence of relative maxima and minima or inflections points (w.p.). De L'Hôpital theorem and application to limit calculus.

Graph of a function: Representation of the graph of a function on the Euclidean plane. Oblique asymptotes.

Linear algebra:

Vectors and vector spaces. Geometric representation of vectors. Linear combination of vectors. Linearly dependent and independent vectors. Rank of a set of vectors. Matrices. Operations with matrices. Product rows by columns. Particular matrices. Transposed matrix. Determinant of a matrix of order n. Properties of the determinant. Rank of a matrix. Rank and linear independence of vectors. Systems of linear equations. Cramer's theorem. Rouché-Capelli theorem. Homogeneous systems. Parametric systems.

Integral calculus:

Primitive functions. Indefinite integral. Characterization of the set of primitives (w.p.). Properties of the indefinite integral. Integral of elementary functions. Integration by parts (w.p.). Integration by substitution (w.p.). Definite integral. Properties of the definite integral. Integral function. Integral mean theorem(w.p.). Fundamental theorem of integral calculus (w.p.). Corollary to Torricelli-Barrow's theorem: relationship between the definite integral and the indefinite integral (w.p.). Applications.

(w.p.) = “with proof”

Core Documentation

Suggested materials:

- Mastroeni, Mazzoccoli, Vellucci: "Esercizi di matematica generale". Società editrice Esculapio. ISBN 9788893853910

Optional materials:

-Peccati, Salsa, Squellati. "Matematica per l'economia e l'azienda". Egea.

-Bramanti, Pagani, Salsa. Calcolo infinitesimale e algebra lineare Seconda edizione

- Alberto Bersani, Francesco Manzini, Loretta Mastroeni. Esercizi di Matematica Generale: Per i corsi del nuovo ordinamento delle Facoltà di Economia. Società Editrice Esculapio, 2009.

Reference Bibliography

- Mastroeni, Mazzoccoli, Vellucci: "Esercizi di matematica generale". Società editrice Esculapio. ISBN 9788893853910 Suggested materials: Optional materials: -Peccati, Salsa, Squellati. Matematica per l'economia e l'azienda. Egea. - Alberto Bersani, Francesco Manzini, Loretta Mastroeni. Esercizi di Matematica Generale: Per i corsi del nuovo ordinamento delle Facoltà di Economia. Società Editrice Esculapio, 2009.

Type of delivery of the course

Frontal lesson

Attendance

Attendance is not compulsory

Type of evaluation

The exam consists of a written and an oral test.