1. First approaches to real numbers and functions:
Axioms of real numbers, and related properties. Set theory: generalities and representation; intersection, union, set of parts, difference; Cartesian product. Natural, integer, rational, real numbers. Upper and lower extremes, maximum and minimum of a number set. Natural, integer and rational numbers. Functions: domain, invertibility, increasing and decreasing. Elementary functions.
2. Equations and inequalities:
Equations of first and second degree. Disequations of first and second degree. Disequations of degree greater than two, fraternal inequalities and systems of inequalities. Irrational, exponential and logarithmic inequalities.
3. Complex numbers:
Complex numbers in Cartesian form, trigonometric form, exponential form and polar form. Rationalization of complex numbers in Cartesian form. Properties of product and ratio between complex numbers in trigonometric, exponential and polar form. Powers and roots of complex numbers. Solving equations in the complex field.
4. Matrices:
Operations between matrices: addition, subtraction, multiplication between matrices and properties of noncommutativity; multiplication between a scalar and a matrix. Determinants of square matrices: generalities; calculation rules; properties of determinants. Rank of matrices: generalities; computation in the case of square and nonsquare matrices; relationship between rank of matrices and linear independence/dependence of row/column vectors.
5. Linear systems:
Matrix formulation of a linear system. Characterization of the solutions of a linear system: uniqueness of solution, infinite solutions, incompatibility; geometric meaning of a linear system and relation to its solutions. Solving rules: inverse matrix method; Cramer's method. Homogeneous linear systems. Formalization of solutions of a linear system by procedures involving elementary operations between rows and columns of complete and incomplete matrices.

6. Domains and limits of real functions of a real variable:
Real functions of real variable: techniques for calculating the domain. Intuitive approach to limit definition: geometric meaning of a finite limit when the independent variable tends to a finite/infinite value; geometric meaning of an infinite limit when the independent variable tends to a finite/infinite value. Analytical formalization of finite/infinite limit when the independent variable tends to a finite/infinite value. Limits of compound functions. Uniqueness theorem of the limit. Comparison theorem. Indeterminate forms. Infinities and infinitesimals. Asymptotes of a function.
7. Calculation of limits and continuous functions:
Calculation procedures for limits occurring in indeterminate form. Notable limits and their applications. Continuous functions at a point and in an interval. Points of discontinuity of a function. Weierstrass theorem, permanence of sign theorem, zero theorem, intermediate value theorem.


8. Derivatives of real functions of a real variable:
Incremental ratio limit and definition of derivative. Geometric meaning of the derivative. Derivability and continuity. Derivatives of elementary functions. Derivatives of compound functions. Relative maxima and minima for real functions of real variable. Fermat's theorem. Rolle's theorem. Lagrange's theorem. Concave, convex functions and points of inflection. De L'Hospital's theorem.
9. Function study:
Procedures for studying a function and plotting its qualitative graph. Insights into rational fraternal functions.
10. Indefinite and definite integrals:
Primitives and indefinite integral. Immediate integrals of elementary and compound functions. Rules and methods of integration. Integrals by parts and by substitution. Definite integral and its geometric meaning. Averaging theorem. Integral function and fundamental theorem of integral calculus. Fundamental formula of integral calculus.
11. Numerical series:
Link between numerical successions and numerical series. Converging, diverging and indeterminate series. Geometric, harmonic and telescopic series. Convergence criteria for series with positive terms: comparison, ratio and root criteria. Leibniz criterion for convergence of series with alternate sign.

12. Functions of two variables:
Domains and topology for functions of two variables. Limits, continuity and derivability. Differentiability. Relative maxima and minima for functions of two variables: generalities; necessary condition of first order. Sufficient condition of second order.

- Paolo Marcellini, Carlo Sbordone, Elements of Mathematical Analysis 1, Simplified version for new undergraduate courses, Liguori Editore, 2016.
- Ciro D'Apice, Rosanna Manzo, Toward the Mathematics 1 exam, Maggioli Editore, 2015.
- Ciro D'Apice, Tiziana Durante, Rosanna Manzo, Toward the Mathematics 2 exam, Maggioli Editore, 2015.
Supplementary teaching materials will be available in the teaching section within the university platform