Università Roma Tre

Numerical sets (N, Z, Q and R), axiomatic construction of R, construction of N and principle of induction, complex numbers; elements of topology in R and Bolzano-Weierstrass theorem; real functions of real variable, limits of functions and their properties, limits of sequences, notable limits, the Napier number; continuous functions and their properties; derivative of functions and their properties, the fundamental theorems of differential calculus (Fermat, Rolle, Cauchy, Lagrange, de l'Hopital, Taylor's formula), convex / concave functions; function graph; Riemann integration and properties, integrability of continuous functions, fundamental theorem of integral calculus, integration by substitution and by parts, integration rules; numerical series, simple and absolute convergence, convergence criteria for series with positive terms and for series with any terms; Taylor series; improper integrals.

Analisi matematica I Marcellini-Sbordone
Analisi matematica I Pagani-Salsa
Esercitazioni di Matematica Marcellini-Sbordone

October
Definition of a function, domain, codomain, image, graph. Injective, surjective, bijective functions, goniometric functions, inverse goniometric functions. Complex numbers: definition, properties, equations, power. Sequences: definition, theorems, limit of a sequence. Limits of functions.

November
Derivative: definition, geometrical meaning, derivative of elementary functions and by composition, derivative of the inverse function. Taylor polynomial, approximation of a function by Taylor polynomial, approximation error, limits using Taylor polynomials, o – notation. Function study in order to draw the graph: domain, simmetries, sign, intersection with the axis, limits and asymptotes, first derivative, monotonicity, minimum and maximum points, second derivative, convexity, inflection points.


December
Primitive functions, definition e theorems. Integrals of elementary functions and by composition, linear property. Techniques of integration: rational function, irrational function (Euler substitution), trigonometric substitution, substitution, by parts, fundamental theorem of calculus, definite integral. Improper integral with finite and infinite interval: solution by limit and by convergence tests.

January
Series, convergence tests: limit comparison test, ratio test, root test, integral test, Leibniz test, absolute convergence test.

Exercises on the syllabus topics, midterm exam simulations.