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Bivariate descriptive analysis

A:

B:

C:

Univariate descriptive analysis

A:

B:

C:

Random variable and multiple random variable

A:

B:

C:

Probability distributions

A:

B:

C:

Introduction to correlation and regression

A:

B:

C:

Financial mathematics

A:

B:

C:

Equations and inequalities solved by graphing functions

A:

B:

C:

Economy

A:

B:

C:

Introduction to estimation theory

A:

B:

C:

Binomial equations

A:

  • Solving binomial equations

B:

-

C:

-

Logic and sets

A:

  • Basic set operations (intersection, union, set difference, complement)
  • Sets specified by a characteristic property

B:

  • Math logic statements, truth values of statements, quantifiers
  • Set operations – complex problems

C:

  • Word problems – solvable with aid of Venn diagrams

Elementary arithmetics

A:

  • Calculations with fractions and decimals
  • Rounding
  • Writing numbers in exponential form

B:

  • Numbers divisibility

C:

-

Polynomials and fractions

A:

  • Basic operations with polynomials (addition, subtraction, multiplication, use of formulas for (a+b)^2 and (a-b)^2)
  • Simplifying algebraic expressions
  • Evaluating algebraic expressions

B:

  • Factoring polynomials into products
  • Simplifying algebraic expressions – complex problems
  • Problems solvable with use of formulas for (a+b)^3 and (a-b)^3
  • Finding of all values such that an expression is not defined
  • Finding of all values such that an expression equals zero
  • Word problems – isolating the variable out of a formula

C:

  • Division of two polynomials
  • Problems solved using Binomial theorem
  • Problems solved using formulas for a^3+b^3 and a^3-b^3, …

Expressions with exponents and radicals

A:

  • Powers with natural exponents
  • Second and third root
  • Simplifying fractions with roots in denominator

B:

  • Powers with integer or rational exponents
  • Higher roots
  • Comparing values of expressions

C:

  • Simplifying expressions with exponents and radicals – complex problems
  • Simplifying fractions with roots in denominator – complex problems
  • Comparing values of expressions - complex problems

Absolute value

A:

  • Evaluating absolute value of numerical expressions

B:

  • Geometric interpretation of the absolute value
  • Simplifying expressions with absolute value
  • Simple equations and inequalities with absolute value

C:

  • Properties of the absolute value 

Percent problems and financial mathematics

A:

  • Calculating percentages
  • Standard percent problems with financial themes

B:

  • Calculating a percentage of another percentage

C:

  • Calculating interest
  • More complex percent problems (price growth, inflation, interests)

Matrices and determinants

A:

  • Classification of matrices 
  • Addition and multiplication of matrices
  • Matrix equations
  • Applications

B:

  • Matrix calculations - complex problems
  • Rank of a matrix
  • Inverse matrices 

C:

  • Determinant of a matrix 
  • Determinant properties

Linear equations and inequalities

A:

  • Simple linear equations
  • Equivalent equations
  • Graphical solutions of linear equations
  • Linear equations specified by word description

B:

  • Simple linear inequalities
  • Graphical solutions of linear inequalities
  • Linear inequalities specified by word description

C:

  • Word problems leading to linear equations and inequalities

Quadratic equations and inequalities

A:

  • Quadratic equations

B:

  • Quadratic inequalities
  • Vieta’s formulas
  • Word problems leading to quadratic equations and inequalities

C:

  • Quadratic equations and inequalities with absolute value
  • Word problems – more complex

Higher degree equations and inequalities

A:

  • Equations solvable by factoring polynomials to products of linear and quadratic factors

B:

  • Equations solvable by substitution method
  • Cubic equations with one of the roots known
  • Inequalities solvable by factoring polynomials to products of linear and quadratic factors

C:

  • Equations of 4th degree with two of the roots known
  • Equations of higher degrees, guessing of some of the roots is necessary

Rational equations and inequalities

A:

  • Rational equations
  • Domains of rational equations

B:

  • Rational inequalities
  • Domains of rational inequalities

C:

  • Rational equations and inequalities with absolute values 

Absolute value equations and inequalities

A:

  • Linear equations and inequalities with single absolute value - solutions based on geometric interpretation of the absolute value

B:

  • Linear equations with one or more absolute values

C:

  • Linear inequalities with one or more absolute values
  • Linear equations and inequalities with absolute values inserted in absolute values

Radical equations and inequalities

A:

  • Equations with an unknown under a single radical
  • Domain of an equation
  • Domain of an expression with an unknown under a radical 

B:

  • Equations with unknown under several radicals
  • Inequalities with unknowns under radicals

C:

  • Word problems
  • More complex equations – combinations of radicals and absolute values

Equations and inequalities with parameters

A:

  • Linear equations with a parameter
  • Equations and inequalities with a parameter solved for a given value of the parameter

B:

  • Linear inequalities with a parameter
  • Quadratic equations and inequalities with a parameter

C:

  • Rational equations and inequalities with a parameter

Systems of linear equations and inequalities

A:

  • Systems of two equations with two unknowns
  • One equation with two unknowns

B:

  • Systems of three equations with three unknowns
  • Using matrices to solve systems of equations 
  • Solvability of systems of equations 
  • Cramer's rule 
  • Word problems 
  • Systems of equations with parameters

C:

  • Systems of inequalities
  • One inequality with two unknowns
  • Two inequalities with one unknown

Systems of nonlinear equations and inequalities

A:

  • Systems with linear and quadratic equations
  • One nonlinear equation with two unknowns
  • Systems with an unknown in the denominator

B:

  • Systems of polynomial equations
  • Graphical solutions of systems of polynomial equations

C:

  • Systems with the unknown in absolute value
  • Systems with the unknown under the square root
  • Systems with parameter
  • Systems of inequalities

Properties of functions

A:

  • Properties of functions given by a table or a graph (parity, monotonicity, minima, maxima)

B:

  • Properties of functions given by equations – practicing through various types of functions (linear, quadratic, with absolute values, rational)
  •  Domains of composite functions

C:

  • One-to-one function and inverse function

Linear functions

A:

  • Properties of linear functions and of their restrictions (domain, range, monotonicity, intercepts with axis, … )
  • Function values
  • Equation of a linear function
  • Verifying if the given point lies on a graph of a function

B:

  • Transformations of a graph of a linear function
  • Solving linear equations with aid of graphs of linear functions

C:

  • Finding equation of a linear function (complex problems)
  • Linear functions with parameter
  • Word problems

Quadratic functions

A:

  • Properties of quadratic functions (domain, range, intercepts with axis, monotonicity, …)
  • Determining of function values
  • Matching graphs to equations of corresponding functions

B:

  • Transformations of a graph of a quadratic function
  • Determining of the equation of a function given by three points
  • Determining of the vertex of a parabola
  • Solving quadratic equations and inequalities with an aid of graphs of quadratic functions

C:

  • Quadratic functions with parameter
  • Quadratic functions with absolute values
  • Solving quadratic equations and inequalities with absolute value with an aid of graphs of quadratic functions
  • Word problems

Linear functions with absolute values

A:

  • Properties of absolute value function
  • Function f(x)=a|x-b|+c, where a, b, c are real numbers

B:

  • Functions with absolute values and their graphs
  • Properties of functions with absolute value (domain, range, monotonicity, extremes, boundedness, parity)

C:

  • Function with an absolute value inside an absolute value

Power and radical functions

A:

Power functions with integer exponent
  • Determining function value
  • Graphs of functions and their transformations
  • Properties of functions (domain, range, monotonicity, extremes, boundedness, parity)
  • Inequalities assessments with an aid of graphs of functions

B:

  • Nth-root function

C:

  • Functions with absolute values
  • Word problems

Rational functions

A:

Inverse proportionality
  • Graph of the function
  • Function value
  • Word problems

B:

Linear rational function
  • Function’s graph and its transformations
  • Center of a hyperbola
  • Properties of functions (domain, range, monotonicity, extremes, boundedness, parity)

C:

  • Rational function
  • Functions with absolute values
  • Problems with parameters
  • Word problems

Calculations with logarithms

A:

  • Definition of logarithm
  • Domains of logarithmic expressions

B:

  • Logarithms counting rules

C:

  • Simplifying expressions with logarithms of various base

Exponential functions

A:

  • Definition of the exponential function
  • Graph and its transformations
  • Domain and range

B:

  • Exponential function properties – monotonicity, boundedness
  • Comparing of function values (with aid of graphs or monotonicity)

C:

  • Composite functions (with absolute values or radicals)
  • Practical word problems

Logarithmic functions

A:

  • Definition of the logarithmic function
  • Graph and its transformations
  • Domain and range

B:

  • Logarithmic function properties – monotonicity, boundedness
  • Comparing of function values (with aid of graphs or monotonicity)

C:

  • Composite functions (with absolute values or radicals)
  • Practical word problems

Exponential equations and inequalities

A:

  • Equations with the same base - solvable by comparing exponents

B:

  • Equations with the same base (more complex) - solvable by comparing exponents
  • Equations solvable by substitution

C:

  • Inequalities solvable by comparing exponents
  • Inequalities solvable by substitution
  • System of inequalities

Logarithmic equations and inequalities

A:

  • Equations with the logarithms of the same base - solvable by comparing arguments
  • Equations with the logarithms of the same base – solvable with use of logarithm counting rules

B:

  • Equations with the logarithms of the same base (more complex) – solvable with use of logarithm counting rules
  • Equations with logarithms of various bases
  • Equations solvable by substitution
  • Equations solvable by taking logarithm
  • System of equations

C:

  • Inequalities solvable by simplifying and comparing arguments
  • Inequalities solvable by substitution

Angles, arcs and sectors

A:

  • Conversions of degrees to radians and vice versa
  • Coterminal angles, coterminal angles between 0 and 360 degrees.
  • Correspondence between angles and quadrants
  • Adding and subtracting angles

B:

  • Angles specified by given conditions – arithmetic mean, enumeration, …
  • Computational problems involving clocks, calculation of marching angle (azimuth)
  • Coterminal angles – complex problems

C:

-

Sine, cosine, tangent and cotangent

A:

  • Trigonometric ratios of standard angles

B:

  • Properties of trigonometric functions – parity, periodicity, boundedness
  • Domains and ranges
  • Graphs of trigonometric functions
  • Sine and cosine relationships

C:

  • Simplifying expressions with trigonometric functions - use of trigonometric identities
  • Domains of trigonometric expressions
  • Trigonometric functions with absolute value

Trigonometric equations and inequalities

A:

  • Basic trigonometric equations
  • Using substitution for solving trigonometric equations
  • Using basic identities for solving trigonometric equations

B:

  • Basic trigonometric inequalities

C:

  • More complex trigonometric equations and inequalities (use of trigonometric identities, exponentiation, …)
  • Trigonometric equations and inequalities with absolute value

Positional problems

A:

  • Point, line, half-line, and line segment
  • Half-plane
  • Angle, pair of angles (corresponding, alternate, adjacent, vertical)
  • Relative position of two lines (parallel, intersecting, perpendicular)

B:

  • Mutual position of line and circle
  • Mutual position of two circles

C:

  • Sets of points with a given property 
  • Thales' circle

Triangles

A:

  • Computation of angle measures in a triangle where the angles satisfy the given condition
  • Relationships between sides and angles of a triangle
  • Properties of triangles, computational problems
  • Medians, altitudes, midlines
  • Circle circumscribed and inscribed in a triangle 

B:

  • Trigonometric functions in a right triangle
  • Application problems solved using trigonometric functions
  • Area of a triangle

C:

  • Law of sines and law of cosines
  • More complex application problems

Polygons

A:

Computational problems on angles, lengths and areas
  • Square
  • Rectangle
  • Rhombus

B:

Computational problems on angles, lengths and areas
  • Trapezoid
  • Parallelogram
  • Regular polygons

C:

Computational problems on angles, lengths and areas
  • Deltoid (Kite)
  • Complex computational problems on angles, lengths and areas

Circles

A:

  • Inscribed and central angle

B:

  • Angles between tangents
  • Polygons inscribed to a circle
  • Disc, annulus
  • Circular sector and circular segment

C:

  • Disc, circular sector and circular segment – complex problems
  • Segment angle

Geometric mappings

A:

  • Point symmetry
  • Line symmetry (reflection symmetry)

B:

  • Translation
  • Rotation

C:

  • Dilation

Positional properties

A:

  • Point, line, and plane in space
  • Mutual position of points, lines, and planes

B:

  • Imagination in space
  • Solid nets

C:

  • Cross-sections of cube and pyramid
  • Intersections of line with cube and pyramid surfaces

Metric properties

A:

  • Word description of angles in a cube
  • Cube – distances of points, lines, planes
  • Cube – angles between lines, planes
  • Cuboid - distances of points, lines, planes
  • Cuboid – angles between lines, planes

B:

  • Word description of angles in a pyramid
  • Square pyramid – distances of points, lines, planes
  • Square pyramid - angles between lines, planes
  • Cone - angles

C:

  • Regular right hexagonal prism - distances and angles
  • Hexagonal pyramid – distances and angles
  • Tetrahedron - distances and angles

Volume and surface of solids

A:

Computation of volumes and surfaces
  • Cube
  • Cuboid

B:

Computation of volumes and surfaces
  • Cone
  • Cylinder
  • Sphere
  • Three or four sided pyramid
  • Right triangular or rectangular prism

C:

Computation of volumes and surfaces
  • Truncated pyramid
  • Truncated cone
  • Regular right hexagonal prism
  • Regular hexagonal pyramid

Points and vectors

A:

  • Points and vectors in plane and in space
  • Length of a vector
  • Operations with vectors – sum, scalar multiple
  • Linear combination of vectors
  • Linear dependence of vectors
  • Line segment – center, length
  • Triangle – centroid, centers of sides, lengths of sides, perimeter

B:

  • Scalar product (dot product) of vectors in plane and in space
  • Perpendicular vectors
  • Angle of vectors
  • Applications – plane shapes, solids in coordinate system

C:

  • Vector product of vectors
  • Area of a plane region, area of a face of a solid
  • Volume of a solid (parallelepiped, pyramid, tetrahedron)
  • Complex problems covering whole topic 

Analytical plane geometry

A:

  • Line – parametric description, general equation, point-slope form equation
  • Direction vector and normal vector of a line
  • Line segment, half-line – parametric description
  • Relative position of two lines
  • Perpendicularity of lines
  • Parallelity of lines

B:

  • Distance of a point from a line
  • Distance of two parallel lines
  • Angle of two lines
  • Triangle – medians, heights (altitudes), side perpendicular bisectors
  • Line and point reflection, translation

C:

  • Angles and distances – more complex problems
  • Complex problems covering whole topic

Analytical space geometry

A:

  • Line – parametric description
  • Plane - parametric description, general equation
  • Intersection of two lines
  • Intersection of a line and a plane
  • Intersection of two planes
  • Relative position of points, lines and planes

B:

  • Intersection of two planes – more complex problems
  • Perpendicularity of lines and planes
  • Parallelity of lines and planes
  • Angles of lines and planes

C:

  • Complex problems on perpendicularity
  • Point, line and plane reflection
  • Distance of a point from a plane
  • Distance of a point from a line
  • Metric problems on solids

Conics

A:

  • Circle (center and radius)
  • Ellipse (center, semi-major and semi-minor axis, foci, vertex and co-vertex)

B:

  • Parabola (vertex, directrix, focus)
  • Hyperbola (center, foci, vertices, semi-major and semi-minor axis, eccentricity)

C:

  • Tangent line to a conic
  • Conic and a line
  • Conic passing through given points 

Complex numbers in algebraic and polar form

A:

  • Imaginary unit
  • Algebraic form of a complex number – addition, subtraction, multiplication, division
  • Complex conjugate of a complex number
  • Geometric representation of complex numbers in gaussian plain
  • Absolute value of a complex number

B:

  • Polar form of a complex number
  • Polar form of a complex number – multiplication, division
  • Polar and algebraic form conversion of complex numbers

C:

  • Simple equations of two variables with complex coefficients

Powers and roots of complex numbers

A:

  • Powers of complex numbers (de Moivre’s theorem)

B:

  • Roots of complex numbers - Binomial equations with real coefficients

C:

  • Roots of complex numbers - Binomial equations with complex coefficients

Quadratic equations with complex roots

A:

  • Quadratic equations with real coefficients
  • Factoring of quadratic trinomial

B:

  • Quadratic equations with real coefficients (complex problems)
  • Quadratic equations with real coefficients with parameter

C:

  • Quadratic equations with complex coefficients

Combinatorics

A:

  • Combinatorial product rule and sum rule
  • Arrangements without repetition / k-permutations without repetition
  • Arrangements with repetition / k-permutations with repetition
  • Permutations without repetition
  • Permutations with repetition
  • Selections without repetition / k-combinations without repetition

B:

  • Simplifying expressions with factorials and binomial coefficients
  • Combinatorial equations

C:

  • Selections with repetition/ k-combinations with repetition
  • Combinatorial inequalities
  • Binomial theorem

Probability

A:

  • Classical probability definition

B:

  • Geometrical probability
  • Probability of complementary event
  • Probability of union of events
  • Probability of intersection of independent events

C:

  • Binomial distribution (Bernoulli scheme)
  • Conditional probability

Statistics

A:

  • Measures of location (mean, median, mode)
  • Arithmetic, geometric and harmonic mean

B:

  • Measures of variability (variance, standard deviation, coefficient of variation)

C:

  • Summary statistics
  • Correlation coefficient

Introduction to sequences

A:

  • Ways to specify a sequence
  • Finding of one or more members of a sequence

B:

  • Defining a sequence (nth term formula or recurrent relation)
  • Properties of sequences (strictly increasing or strictly decreasing, nonincreasing, nondecreasing, upper or lower bounded, bounded)
  • Finding the nth term of a sequence

C:

-

Arithmetic sequences

A:

  • Defining a sequence (nth term formula or recurrent relation)
  • Finding the nth term of a sequence
  • Finding the common difference of a sequence

B:

  • Finding the nth term of a sequence – complex problems
  • Finding the common difference of a sequence – complex problems
  • Sum of the first n terms of a sequence
  • Systems of equations containing sequence terms

C:

  • Word problems
  • Equations and inequalities containing sums of sequences

Geometric sequences

A:

  • Defining a sequence (nth term formula or recurrent relation)
  • Finding the nth term of a sequence
  • Finding the common ratio of a sequence

B:

  • Finding the nth term of a sequence – complex problems
  • Finding the common ratio of a sequence – complex problems
  • Sum of the first n terms of a sequence
  • Systems of equations containing sequence terms

C:

  • Word problems
  • Combinations of arithmetic and geometric sequences

Limit of a sequence

A:

  • Evaluation of limits of sequences containing polynomials and rational expressions
  • Limit laws – sum of limits, difference of limits, product of limits and ratio of limits laws

B:

  • Evaluation of limits with trigonometric, exponential and logarithmic functions

C:

  • Use of the limit of the sequence (1+1/n)^n
  • Evaluation of limits with radicals
  • Evaluation of limits containing sums of sequences

Infinite series

A:

  • Summation notation
  • Finding of the first term and of the common ratio of a geometric sequence
  • The sum of an infinite geometric series

B:

  • Periodic numbers (repeating decimals)
  • Finding of all x for which a series diverges or converges
  • Solving equations with infinite series
  • Word problems

C:

-

Limits and continuity

A:

  • Calculating limits – polynomials and rational functions
  • One-sided limits
  • Finding limits of functions from graphs

B:

  • Calculating limits – trigonometric functions
  • Calculating limits – functions with radicals
  • Continuity, discontinuity points

C:

  • Theoretical aspects related to limits calculations

Derivative

A:

  • Geometric interpretation of the derivative
  • Derivatives of elementary functions

B:

  • Derivative of a product of functions
  • Derivative of a quotient function
  • Derivative of a composite function

C:

  • Derivative of a composite function – complex problems
  • Applications of derivatives in physics

Analyzing function behavior

A:

  • Function’s monotonicity
  • Local extrema

B:

  • Second derivative and its geometric interpretation
  • Concavity and convexity of a function
  • Inflection points

C:

  • Asymptotes of a graph of a function

Applications of derivatives

A:

  • Calculating limits using L'Hospital's rule  

B:

  • Tangent line to graph of a function
  • Normal line to graph of a function

C:

  • Global extrema
  • Optimization problems (global extrema)

Primitive function

A:

  • Geometric interpretation of the antiderivative (primitive function)
  • Solving simple indefinite integrals (Finding a primitive function)

B:

  • Solving integrals requiring simplification of expressions
  • Solving integrals by substitution
  • Solving integrals by Parts

C:

  • Integrals solved by substitution – complex problems
  • Integrals solved by Parts – complex problems
  • Solving integrals requiring partial fraction decomposition

Definite integral

A:

  • Evaluation of simple definite integrals

B:

  • Evaluating integrals requiring simplification of expressions
  • Evaluating integrals using substitution
  • Evaluating integrals by Parts

C:

  • Evaluating integrals using substitution – complex problems
  • Evaluating integrals by Parts – complex problems
  • Evaluating integrals requiring partial fraction decomposition

Applications of definite integral

A:

  • The area of a plane region

B:

  • The volume of a solid

C:

  • The area of a plane region – complex problems
  • The volume of a solid – complex problems
  • Applications to physics