Basic set theory
Universe, proposition, set, the complement of a set, Operations on sets: intersection, union, subsets, the Cartesian product of sets. Common number sets N, Z, Q, R. Intervals of R.
Elements of analytic geometry
The Cartesian plane and space: Cartesian product, representation of R2 on the Cartesian plane, subsets of R2 and subsets of the Cartesian plane, R3 and its representation on the Cartesian space. Lines on the plane: the 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. Quadratic functions: equation of a quadratic function with the vertical axis of symmetry, vertex, convexity and concavity of a quadratic function. Equation of a quadratic function with the horizontal axis of symmetry.
Elementary Functions
Introductive notions: the concept of function, independent and dependent variables, composite function, inverse function, restriction of a function, prolongation of a function.
Real Numbers
Order relation and algebraic structure of R, bounded sets and extremes of a set, metric properties of the real numbers, elements of topology in R.
Functions from R to R
Introductive notions: the concept of function, independent and dependent variables, graph, the image of a function, the inverse image of a function, increasing (decreasing) function, local and absolute maxima (minima), bounded functions, extremes of a function, even functions, odd functions, inverse function. 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.
Limits
Possible cases: Finite limit at a finite point, limit from the right, limit from the left, bilateral limit, finite limit at infinity, infinite limit at a finite point, infinite limit at infinity. Some theorems: existence theorem for monotone functions, squeeze theorem, limits of elementary functions, theorem of the operations with limits, indeterminate forms, limit and comparison of elementary functions. Comparisons among limits, order of infinity, principles of elimination and substitution.
Continuous functions
Generalities: definition of continuity of a function, discontinuity points of a function and their classification, continuity of elementary functions, continuity with respect to algebraic operations. Continuous function on intervals: Intermediate value theorem, Bolzano theorem for continuous functions, Weierstrass theorem. Composite functions: limits and continuity. Limits for common functions: logarithm, exponential, indices.
Derivatives
The slope of a non-linear function: difference quotient; derivative of a function; differentiable function; the relationship between differentiability and continuity; derivability of elementary functions; second-order derivatives; Ck functions, points of non-derivability of a function. Algebra of derivatives: the derivative of a constant; the derivative of indices, 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: Fermat theorem; Lagrange theorem; Rolle theorem; monotonicity criteria, convex (concave) function, convexity criteria. Graph of polynomial functions: asymptotes; the procedure for the study of the graph. Graph of Rational Functions: domain, asymptotes, the procedure for the study of the graph. Graph of non-elementary Functions: procedure for the study of the graph, De L'Hopital's theorem. Taylor’s formula.
Integration
Indefinite integral: Torricelli - Barrow theorem; primitive for a function; two primitives differ by a constant (proposition); indefinite integral; linearity of integration. Methods of integration: some basic antiderivatives; integration by parts (proposition and application); integration by substitution (proposition and application). Definite integral: geometrical aspects, introduction to the theory of Riemann integration, the fundamental formula of integral calculus, calculation of the definite integral.
Vector space Rn
Set R^n, vector spaces, linear dependence, linear combinations, bases, dimension, subspaces, dimension of subspaces, inner product, norms.
Linear transformations and matrices
Linear Transformations, matrices, the image of a linear transformation, inverses.
Determinant and Rank
Definition of the determinant of a matrix, computation of the determinant, inverse of a matrix, geometric interpretation of the determinant, computation of the rank.
Sistems of linear equations
Systems of m equations in n unknowns, the matrix form of a linear system, solution of a linear system, homogeneous system, The Rouché-Capelli theorem, Cramer’s theorem and his rule.
