Sets and Numbers

Elements of set theory: concept of set, universal set, empty set, operations with sets and Venn diagrams, families of sets, partition of a set, Cartesian product.

The numbers and their properties: natural numbers, operations of sum and product and their properties. Integers, Rational and Real numbers. Cardinality of sets, order relation, axiom of completeness.

Extremes: R intervals, maxima, minima, suprema and infima of a set,

proposition on the uniqueness of the maximum (minimum). bounded sets.

Equations and Inequalities

Introductive notions: monomial, polynomials, operations with polynomials, equivalent equations, the fundamental theorem of algebra, inequalities, equivalent inequalities.

Linear Equations (and Inequalities): solution and examples.

Quadratic equations (and inequalities): solutions, discriminant, sign of a trinomial of second degree.

Higher order Equations (and Inequalities): factorization, Ruffini's theorem and its applications. Systems of inequalities. Rational inequalities.

Elementary Functions

Introductive notions: the concept of function, independent and dependent variables, graph, increasing (decreasing) function,

local and absolute maxima (minima), bounded functions, even functions, odd functions, convex (concave) function and inverse function.

Operations with functions: sum, product, multiplication by a

constant, translations, expansions and compressions, composition.

Elementary functions, graphs, geometrical properties, analytical properties: sign function, identical function, linear and affine function, absolute value function, power function, root function, power function with real exponent , exponential function, logarithm function, exponential equations and inequalities, logarithmic equations and inequalities.

Tools for applications:

Lines: slope of the linear function; equation of the line given two points; equation of the line given a point and the slope; conditions for parallelism and for perpendicularity of two lines.

Sequences and Limits: sequences, monotone sequences, convergent sequence, unbounded sequence, irregular sequence; uniqueness theorem for limits; theorem of sign permanence; limit point; limit of a function; characterization theorems of the limit.

Slope of a non-linear function: difference quotient; derivative of a function; differentiable function; continuous function; the relationship between differentiability and continuity; Intermediate value theorem; derivability of elementary functions; second order derivatives; Ck functions.

Algebra of derivatives: the derivative of a constant; the derivative of a power; sum rule; product rule; quotient rule; the chain rule. Derivative of the exponential function; derivative of the logarithmic function.

Applications

Tangent to a curve: secant, tangent, equation of the tangent line.

Differential: Differential of a function and its geometric interpretation.

Graphs of functions: Weierstrass theorem; Fermat theorem; Lagrange theorem; Rolle theorem; monotonicity criteria, convexity criteria.

Graph of polynomial functions: asymptotes; the procedure for the study of the graph.

Graph of Rational Functions: domain, asymptotes, procedure for the study of the graph.

Integration

Introduction: intuitive concept of integral.

Definite Integral: the area problem and the definite integral; splitting intervals; rectangles; upper and lower Riemann sum; lemma on the inequality between lower and upper sums related to different partitions and its corollary; Riemann integrable function; the Dirichlet function; Characterization of Riemann integrability (theorem); some proposition: homogeneity of the integral; the monotony of the absolute value; additivity of integration on intervals; linearity; non-negativity proposition; monotonicity proposition; mean value theorem for integral; mean value of a function.

Indefinite integral: Torricelli - Barrow theorem; primitive for a function; two primitives differ by a constant (proposition); indefinite integral; fundamental formula of integral calculus.

Methods of integration: some basic antiderivatives; integration by parts (proposition and application); integration by substitution (proposition and application); calculation of the definite integral .

Integration of rational functions: rational functions; quotient of polynomials; integration of rational function with quadratic denominator ( cases Δ > 0 , Δ = 0 , Δ < 0).

Elements of probability

Combinatorial analysis: the basic principle of counting; permutations (theorem and application); combinations (theorem and application); binomial coefficient, dispositions (theorem and application),

The algebra of Events: standard event, implication between events, equal events; complement of an event; the union event; the intersection event; mutually exclusive events; sample space; conditional events.

The different definitions of probability: classical, frequentist, axiomatic; some simple propositions on the probability of events. Sample spaces having equally likely outcomes; card games; the problem of birthdays.

Introduction to conditional probability: the multiplication rule; independent events; Bayes' Formula (proof and application).

Elements of statistics

Introduction: problems of descriptive and inferential statistics.

Elements of descriptive statistics: discrete variables and continuous variables, population; character; sample; absolute frequency; relative frequency; cumulative frequency; variable; statistics; dot plot; bar graph, pie chart.

Main statistics: mode, median, quartiles, quantiles, arithmetic mean, deviation; arithmetic mean as the center of gravity of the distribution (proposition); internality of the arithmetic mean (proposition); range of data, deviance, variance, computational formula for the variance (proposition); standard deviation.