Sets of numbers: natural numbers and induction principle, Bernoulli inequality, integers and rational numbers, square root of two is not rational, axiomatic introduction of real numbers (4 h.). Introduction to real functions of one variable: domain, range, injectivity and surjectivity, invertibility, monotone and bounded functions (2 h.). Numerical sequences, convergent, divergent, and irregular sequences, theorem on monotone sequence, Nepier's number, operations with limits, permanence of sign (4 h.). Subsequences and uniqueness of the limit, sandwich theorem, Landau symbols, order of infinite and infinitesimal quantities, asymptotic comparison theorem. Ratio test for sequences (2 h.). Numerical series and necessary condition for the convergence, geometric series, Mengoli series and telescopic series. Comparison criteria, harmonic series. Root and ratio test for series. Changing sign series and the Leibniz theorem (4 h.). Completeness of real numbers and the real powers. Elementary functions, graphs, symmetry, compositions, inverse (4 h.). Transformations in the plane and deduction of the graph following elementary transformations (2 h.). Limits for functions: extending the results proved for sequences, Landau symbols and infinite, infinitesimal functions. Continuity and global properties of continuous functions: zeros theorem, Weierstrass' theorem, Darboux theorem, invertibility of continuous functions, continuous extensions (4 h.). Introduction to derivatives, relationship between continuity and derivability, operations with derivatives. The chain rule and derivative of the inverse function (4 h.). Local optimization, Fermat's theorem, Lagrange's theorem, monotonicity criterium and characterizing functions with zero derivative. The Hospital theorem (4 h.). Higher order derivatives, Taylor expansion (with Lagrange and Peano remainders) (4 h.).
Introduction to the Riemann integral, Riemann summations and geometric interpretation, integrability of continuous functions (2 ore). Properties of the Riemann integral, primitives (antiderivatives). The mean value theorem and the fundamental theorem of Calculus (4 h.). Generalized integral and the study of integral functions (4 h.). First order Ordinary Differential Equations, population dynamics, separated variables equations, Cauchy problems, linear equations and the variation of constants formula (4 h.). Second order equations, Cauchy problems, the theorem of structure of solutions, homogeneous equations with constant coefficients, variation of constants formula and similarity methods, Euler's equation (4 h.).