Cartesian product of two sets. Relations and functions between two sets. Fundamental concepts related to functions: domain, codomain, image

set, injective, surjective, invertible functions, inverse function, the composition of two functions, with examples. Binary order relations in a

set. Numerical sets: N, Z, Q, and induction principle with examples and applications. The complete ordered field R of real numbers (axiomatic

introduction) and its main properties.

Maximum and minimum of a set in R, infimum and supremum, and related properties. Absolute value function and floor function, graphs of a

function, maximum function and minimum function between two assigned functions. Sequences in R. Distance between points of R and

between points of the plane. Cartesian coordinate system, abscissa/ordinate. Intervals of the real line and neighbourhood of a point.

Properties of intervals and neighbourhoods. Isolated points and accumulation points of a set. Extension of the real number line with the

points at infinity.

Limits and continuity. Definitions and main theorems. Monotone functions (increasing/decreasing). Theorems on continuous functions: intermediate value theorem, Weierstrass theorem on maxima and minima. Continuous functions defined on an interval. Theorems about limits: sum, product, quotient, the composition of functions; squeeze theorem (comparison on limits). List of important limits. Elementary functions and their graphs: powers, polynomials, rational, exponential functions, logarithms, trigonometric functions and their inverse. Limits concerning these

functions. Local comparison of functions. Landau symbols. Infinite and infinitesimal functions. Order of an infinity and of an infinitesimal, principal part with respect to a test function. Asymptotes.

Derivatives, derivability and continuity, successive derivatives, theorems on derivatives. Critical points (stationary points), Rolle, Cauchy and

Lagrange theorems. Maxima and minima, monotonicity on intervals using the first derivative. Taylor’s and Taylor-MacLaurin’s formula. Property of

the remainder and estimation of the error in the use of Taylor’s formula. Convex functions: equivalent definitions and criterion of the second

derivative for the maximum/minimum points. Inflection points. The concept of primitive of a function and the problem of searching for primitives.

Integrals and area computations. Lower and upper sums, lower and upper integral of a limited function over an interval. Integrable (and nonintegrable) functions according to Riemann. The integral mean value theorem and the fundamental theorem of calculus. The Newton-Leibniz theorem. Examples of integration techniques: by parts, by decomposition, by substitution, with significant examples. Computation of areas and function study using the fundamental theorem of calculus and its corollaries. Volume of solids of revolution.

Numerical series and main properties, with particular regard to generalising harmonic series and power series. Series with positive terms and comparison theorems for series. Convergence criteria for series with positive terms. Absolutely converging series and alternating series.

Improper integrals and main properties. Relationships between improper integrals and series.

Introduction to ordinary differential equations, first-order equations with separable variables, and related integration techniques.