Propositional calculus and calculus with predicates. Natural numbers, integers and rationals with algebraic properties. Partial and linear order on rationals.
Greatest an smallest elemnt in subset of linear order. Supremum of the set notion. Axioms of set of real numbers - together with Dedekind continuity axiom.
Axioms of extensionality, pairing and union of set theory. Notion of function, domain and range of function. Injection of function notion. Inverse function existence theorem.
Proof of the theorem that between any two diffreant reals one can find rational and irrational number see link.
Definition of the sequence. The concept of boundedness and monotonicity of a sequence. An example of monotonicity and proof that this sequence
$$
e_n = \left(1 + \frac{1}{n} \right)^n
$$
is bounded. The other examples are given also.
Combinatorial proof of the binomial formula:
$$
(\forall n\in\mathbb{N})(\forall a,b \in \mathbb{R})\;\; (a+b)^n = \sum_{k=0}^n \binom{n}{k} \cdot a^k \cdot b^{n-k}
$$
Notion of function, domain and rank function. Notion of injection, surrjection of function.
The concept of an inverse function and the theorem on the existence of an inverse function to a given function. Monotonicity of the function
Sequence notion, bounded, monotonic sequence. Limit of the sequence, theorems and examples. Improper limits, theorems and examples.
Notion neighborhood of a point on the real line. Limit of function in the sense of Heine at the point on real line and in infinity.
Theorems of limits of functions (arithmetic, about 3 functions for example). One-sided limit function definitions and relationship with two-sided limit of function.
The concept of the derivative of a function at a point. Theorem on the existence of derivatives and one-sided derivatives and its application.
Determination of derivatives for basic functions. Theorem on the derivative of a composite function and
theorem about the derivative of the sum of the product and the quotient of differentiable functions at a given point.
The algorithm determines the global extremum for a continuous function on a closed and bounded interval. Theorem of Rolle, Cauchy and Lagrange.
Theorem on a sufficient condition for the monotonic behavior of a function on an interval. A sufficient condition for the existence of a local extremum of a function.
D'Hopital's rule as a corollary of Cauchy's theorem and applications of this rule.
Indefinite integrals - definition and basic properties. Integration by parts, by substitution theorems.
Integration of rational and trigonometric functions.
Integration of the integrals of the form $\int R(x,\sqrt{a^2\pm x^2})\, dx$, $R$ is rational function.
The Riemann integrals - definition and basic properties. Proof of the Leibniz-Newton theorem.
Integration by parts and by substitution theorems.
Applications in geometry of the Riemann integrals.
Improper integral - definition and basic properties. Comparative and quotient criteria about convergence of the improper integrals.