Course program
Numeration systems and numerical sets. Integer numbers and numeration systems. Positional numeration systems. Operations with integers numbers. Properties. Relative integer numbers. Decimal numbers. Rational numbers. Real numbers. Sets power. First and second Cantor's theorem. Induction-proof method. Indirect proof (argument's sake). Sets of real numbers. Neighbourhood of a point. Sequences Definitions. Graphic representation. Limit of a sequence (all cases). Uniqueness of the limit. Sign-persistence Theorem (direct and inverse). Comparison theorem. Theorems on Monotone sequences. Cauchy's convergence criterion (demonstration of necessary condition only). Operations on limits of sequences. Series General definitions. Convergent, divergent, indeterminate series. Geometric series and harmonic series. Cauchy's condition. Series with constant-sign terms. Pertinent theorem. Series with alternating-sign terms. Convergence criteria: comparison, root, ratio. Generalized harmonic series. Limit of a function Limit of a function. Case of finite limit and finite limit-point. Generalization of the definition and other limit cases. Right and left limit. Theorems on limits of functions: uniqueness, Sign-persistence Theorem (direct and inverse), of comparison. Operations on limits. Operations with infinitives symbols. Continuous functions Continuity on right and on left. Continuity on an interval. Singular points. Theorems on continuous functions: sign-persistence, of the maximum and the minimum (Weierstrass's theorem), of existence of zeros, of fixed point. Composed function and inverse function. Infinitesimals and infinitives. Remarkable limits. Differential calculus Definition of derivative. Relation with the continuity. Geometric interpretation of derivative. Kinematics interpretation of derivative. Rules of derivation: pertinent theorems. Derivative of power, exponential and logarithmic functions. Increase-ness and decrease-ness on a point and pertinent theorems. Theorem of mean: Rolle's, Cauchy's, Lagrange's. Global increase-ness and decrease-ness and pertinent theorems (without proofs). Indeterminate forms. De l'Hospital's theorems. Differential calculus Differential of a function. Derivative of the compound function. Derivative of the inverse function. Second derivative and higher order of derivatives. Locally concave and convex function. Points of inflection. Theorems. Concave and convex function in an interval. Theorems. Taylor's formula. Lagrange form for the remainder in Taylor's formula. Derivatives method to study stationary and points of inflections. Theorems. Asymptotes. Graphical representation of a function. Integral calculus Integral sums, definition of integral and associated theorems. Integrals as an area. Proprieties of the integral. The mean value theorem for integrals.Definite integral. Theorems. Integral function. Fundamental theorem of integral calculus. Use of the primitive function for evaluation of the definite integral. Indefinite integral. Indefinite integration methods: by decomposition, by substitution and by parts. Rules for evaluation of definite integrals. Definite integration methods: integration by decomposition, by substitution and by parts. Linear algebra Vectors. Algebra with vectors. Linear combination of vectors. Convex linear combination of vectors. Linear spaces and subspaces. Linear dependence and independence. Theorems. Rank of a vector set. Generator set of a linear space. Basis. Theorem of unique representation. Fundamental theorem of linear spaces. Matrices. Algebra with matrices. Determinant. Evaluation of determinants. Sarrus's rule. First Laplace's theorem. Minors of a matrix. Rank of a matrix. Kronecker's theorem. Properties of determinants. Systems of linear equations. Solution of a system of linear equations. Rouche-Capelli's theorem. Cramer's theroem. Homogeneous systems of linear equations. Parametric systems of linear equations.
