1) Sets, numbers, absolute value, powers and logarithms
Numerical sets. Fractions. Arithmetic and geometric means. Ratio and percentage. Permutations and combinations. Absolute value. Intervals. Powers and roots. Exponential and logarithm in a given basis. Powers of ten.
2) Elements of Geometry, Trigonometry and analytic Geometry
Geometric objects and loci. The real line. Polygons and polyhedra. Perimeter length and area of some regular polygons. Surface area and volume of some solids (cube, parallelepiped, pyramid, sphere, cone, ellipsoid).
Angles. Sine, cosine, tangent and their properties. Sum and difference, duplication and bisection formulas. Werner's and prosthaphaeresis identities. Solving triangles.
Cartesian plane and coordinate system. Rotations and translations. Straight line. Circumference. Ellipse. Parabola. Hyperbola.
3) Functions
Monotonicity and invertibility. Maximum, minimum and inflection points. Linear functions (affinities). Parametric linear functions: the state equation of a gas, the uniform motion and other examples.
Power functions and n-root functions. Algebraic functions. Exponential and logarithmic functions with applications to logarithmic scales. Periodic functions. Trigonometric and hyperbolic functions. Examples from natural sciences.
4) Equations, inequalities, systems
Polynomials. Division and factorization of polynomials. Integer and rational equations.
Higher order equations. Integer and rational inequalities. Absolute-value equations and inequalities. Irrational equations and inequalities. Exponential equations and inequalities. Logarithmic equations and inequalities. Trigonometric equations and inequalities.
How to solve linear systems. Second order systems. Symmetrical systems. Fourth order systems.
5) Limits and derivatives
Limit and convergence of a sequence, sum and convergence of a series. Special sequences and series. Functions: continuity, points of discontinuity. Newton's difference quotients and growth rates. First and second derivatives of a regular function and their geometrical meaning. Detection of critical (maximum, minimum and inflection) points for regular single variable functions. Examples and applications.
6) Integrals and differential equations
Integration of a function: definition and geometrical meaning. Examples and applications. Derivative and antiderivative: fundamental theorem of calculus. Mean value of a function on a fixed interval.
First and second-order linear differential equations. Solutions to the Cauchy problem. Special first-order nonlinear differential equations. Examples from natural sciences.