Introduction: Elements of logic and set theory. Numerical sets.
Linear Algebra: Vectors, operations with vectors and their properties, linear dependence. Matrices, operations with matrices and their properties, invertible matrices. Determinant of a square matrix and its properties. Algebraic complement of an element of a matrix. Rank of a matrix. Linear systems: Rouchè-Capelli and Cramer theorems. Parametric systems.
Numerical sequences: Limits and their properties. Unicity, permanence of sign and squeeze theorems. Limits and algebraic operations. Indeterminate forms.
Real functions of a real variable: Graphics. Limited, monotonic functions, composition of functions, invertible functions. Elementary functions: linear functions and exponential functions with application to financial capitalization lows. Limits. Asymptotes of a function. Continuity in a point. Classification of discontinuity points. Properties of the functions which are continuous on a limited and closed interval: Weirstrass, existence of zeros and intermediate values theorems.
Differential calculus: Derivability of a function on a point and its geometric meaning. Relationship between continuity and derivability. Elementary derivatives and derivative rules. Fermat, Rolle and Lagrange theorems. Test of monotonicity and (first) test for classification of stationary points . De l’Hopital theorem. Comparison between infinitesimals and infinities. Taylor theorem. (Second) test for classification of stationary points. Convex functions. Test for convexity. Flexes. Study of the graphic of a function.
Integral calculus: Integral of Riemann and its properties. Integral mean theorem and fundamental theorem of the integral calculus. Primitives of a function. Indefinite integral. Immediate integrals. Relationship between definite and indefinite integral. Integration methods: integration by parts and integration by substitution.