Lectures cover both theoretical arguments and solution techniques for the main problems that students are required to be able to handle. Practice hours are devoted to additional exercises to sharpen students calculus skills.
Active participation to lectures and practice hours is highly recommended.

Lectures will mostly cover the following topics:
Number sets.
The set of real numbers: algebraic and metric structure. Distance, ordering, supremum and infimum, interior, exterior and boundary points, cluster points, maximum and minimum.
Real functions of one real variable.
Definition of function. Elementary functions, graphic, geometric transformations, graphical methods to solve inequalities. Domain, bounded functions, composition of function, monotonicity, invertibility, concavity and convexity.
Limit.
Theorems (Uniqueness of the limit, sign permanence - proofs) and calculus techniques, notable limits, infinite and infinitesimal.
Continuous functions.
Theorems (Weierstrass, Intermediate Zero Theorem, Intermediate Value Theorem - proofs).
Sequences.
Recursively defined sequences, limits of sequences.
Linear Algebra.
Vectors and matrices algebra, determinant (Sarrus’s Rule, Laplace expansion), inverse matrix (proof of the uniqueness of the inverse matrix), transposed, rank, linear systems (study of solutions and resolution).
One variable differential calculus.
Difference Quotient, derivative and its geometrical interpretation, points of non differentiability, calculus of derivatives, differentiability and continuity, higher order derivatives, Taylor's expansion (order n), De Hospital’s Theorem. Rolle's Theorem (proof), Mean Value Theorem (proof), Fermat's Theorem for stationary points (proof). Testing monotonicity for differentiable functions (proof), Testing convexity for twice
differentiable functions, Second order sufficient condition for stationary points (proof), deducing the graph of a function.
Integral Calculus.
Indefinite integral, primitives of elementary functions, primitives of rational functions, integration by parts and by substitution. Definite integral, integral function. Mean Value Theorem of integral calculus (proof), Fundamental Theorem of integral calculus (proof). Generalized integrals.
Series.
Convergence of a series, convergence of the geometric series (proof). Necessary condition for convergence (proof). Series with positive terms: harmonic series, convergence criteria: comparison, asymptotic comparison, ratio, root, integral. A dash on absolute convergence of a general series.