Università Roma Tre

Sets and elementary mathematics: logical symbols and basics of logics, notion of set and difficulties with simple set theory, axiomatic theory of sets, natural numbers, relative numbers, rational and real numbers, extrema and infinites, Functions: function as an association, injections, surjections, bijections, invertibility, composition, graphs, monotonicity and strict monotonicity, elementary functions, linear and parabolic f., power f., trigonometry.
Limits: neighbourhood, definition of limit, theorem of unicity, persistence of the sign, comparison, operations on limits, limits of the kind L/0, limits of composite functions, Neper's constant e as a limit, exponential and logarithm function, infinite and infinitesimal order.
Derivatives: intuitive notion as a variation rate and rigorous definition, geometric interpretation, derivation rules, notable cases, theorem about the continuity of derivable f., monotonicity and sign of the derivative, relative minima and maxima, convexity and concavity, de l'Hôpital rule
Study of function: tracing the graph given the expression of the function.
Integrals: antiderivative of a function, indefinite integral, integration rules, integrals by substitution and by parts, contiguous sets, rectangoloid of a non-negative function, Riemann's integral, definite integral and its properties, theorem of the average, Torricelli-Barrow's theorem, fundamental theorem of integral calculus, integrability criteria, absolute integrability.
Linear systems: definitions, study of the coefficient matrix, Cramer and Gauss-Jordan techniques for the solutions.

C. Sbordone e M. Sbordone: Matematica per le Scienze della Vita, EDISES