italian version

Aims :

We plan to give basic knowledge concewrning the theory of functions of a real variable with enphasis on the theory of limits, differentiable functions and Riemann integral and their application to the analysis of concrete problems.

Topics :

Sequences in R and their limits. Uniqueness of limits. Theorem del confronto. Algebraic operations. Basic limits. Limits of monotone sequences. Series and their convergence, divergence and indeterminateness. Series with positive terms. Convergence criteria. Absolute convergence. Leibnitz theorem. Functions from R to R. Limits. Right and left limit. Basic theorems about limits. Algebraic operations, indecision forms. Limits of monotone functions. o(f) and O(f) and their comparison. Baic limits. Continuous functions, types of discontinuities. Algebra of continuous functions. Theorems about continuous functions in intervals. Continuity of the inverse function. Differentiable function. Left and right derivative. Continuity and differentiability. Rules of differentiation. Chain rule, differential of the inverse function. Local and global max and min. Theorems of Fermat, Rolle, Lagrange. e consequences. De l'Hopital theorems. Taylor e di Mac Laurin expansions with remainder in the form of Peano and Lagrange. Using Taylor formula to compute limits. Convex functions. Graph of a function. Riemann integral. Linearity e monotonicity of the intagral. Mean value and fundamental theorem. Rules of integration. Improper integrals, convergence criteria. Integrability and absolute integrability.

Textbooks :

Marcellini, Sbordone; Elementi di Analisi Matematica 1, Liguori.

Exam :

Written and oral proof.

Tutorial Session :

To be defined.