Algebra 11th Grade
(4 hours per week, total 136 hours)
I. Polynomials
Expressions and types of expressions. Identical transformations of entire rational expressions.
Formulas for the shortened multiplication. The square of the sum of several terms and formulas
for
and,
, n being an odd number.
Complete and incomplete induction. The method of mathematical induction. Proof of identities
and inequalities using mathematical induction. Polynomials with one variable. Canonical form
of entire rational expressions. Division of a polynomial with a remainder. Bezout's theorem.
Polynomial roots. The generalized Vieta's theorem. Identical equality of rational expressions.
Canonical form of rational expressions.
II. Graphs of Functions
Complex functions. Plotting graphs of functions. Transformation of graphs (parallel translation
(shifting graphs), stretching and compressing the graph along the y-axis, stretching and
compressing the graph along the x-axis, symmetry with respect to the origin and coordinate
axes). Graph of a fractional-linear function, vertical and horizontal asymptotes. Graphs of
functions containing the modulus sign. Inversely related functions and their graphs. The
condition for the existence of an inverse function.
III. Exponential and Logarithmic Functions
Exponential function, its properties, and graphs. Concept of exponent with a real exponent.
Properties of exponent with a real exponent. Logarithm of a number and its properties.
Logarithmic function, its properties, and graph. Methods for solving exponential and
logarithmic equations and inequalities. Approximate solutions to equations. The method of
successive approximations.
IV. Limit and Continuity
Limit of a sequence. Existence of the limit of a monotonic and bounded sequence. The number
𝑒�. Natural logarithm. Some limits related to the number 𝑒�.�Limit of a function at a point and
its properties. Continuous functions. Continuity of sine and cosine. Points of discontinuity.
Vertical asymptotes. Calculating limits related to inverse trigonometric functions. Infinitely
small functions. Operations with infinitely small functions. Limit of a function and its properties
when approaching infinity.
. Infinitely large functions. Horizontal and oblique
asymptotes. Arithmetic operations on continuous functions. The theorem on intermediate
values for functions continuous on a segment.
V. Derivative and Its Applications
Function increment. Differentiable functions. Derivative. The physical meaning of the
derivative. Differential. Approximate calculations. Geometric meaning of the derivative.
Tangent to the graph of a function and its equation. Continuity of a differentiable function.
Differentiation rules. Differentiation of a complex function. Differentiation of fractions.
Differentiation of trigonometric functions. Differentiation of inverse trigonometric functions.
Differentiation of exponential and logarithmic functions. Differentiation of a power function.
The second derivative. Studying the function for increase and decrease. Sufficient condition for
an extremum. Studying the graph for convexity. Points of inflection. Application of the
derivative in studying and plotting graphs, finding the maximum and minimum values of a
function. Review and problem solving.
VI. Integral. Differential Equations
Antiderivative and indefinite integral. Properties of the indefinite integral. Integration rules.
Area of a curvilinear trapezoid. Definite integral. Properties of the definite integral. NewtonLeibniz formula. Approximate calculation of definite integrals. Application of integrals.
Calculating areas of geometric figures and volumes of bodies. Calculating the length of an arc.
Application of integrals to solving physical problems. Problems leading to differential
equations. Initial conditions. Equations with separable variables. Differential equation of
harmonic oscillations. Differential equation of radioactive decay. Application of differential
equations.
VII. Elements of Combinatorics
Basic concepts of combinatorics. Sum rule. Product rule. Permutations with repetitions.
Permutations without repetitions. Combinations without repetitions. Combinations with
repetitions. Newton's binomial. Some properties of binomial coefficients. Solving
combinatorial problems.
VIII. Elements of Probability Theory and Mathematical Statistics
Random events. Probability. Addition theorem. Independent random events. Conditional
probability. Multiplication formula. Bernoulli formula. Random variables. Mathematical
expectation and variance. The concept of the law of large numbers. Statistical characteristics.
Arithmetic mean, range, and mode. Median of an ordered series. Statistical research. Collection
and grouping of statistical data. Visual representation of statistical information.
IX. Complex Numbers
Complex numbers and operations with them. Geometric representation of complex numbers.
Polar coordinate system and trigonometric form of complex numbers. Multiplication,
exponentiation, and division of complex numbers in trigonometric form. De Moivre's formula.
Extraction of the root of a complex number. Complex roots of algebraic equations. Concept of
the fundamental theorem of algebra. Application of complex numbers.
X. Systems of Equations and Inequalities
General methods for solving equations. Theorems on the equivalence of equations. Extraneous
roots and loss of roots. Trial and error method. Equations with several variables. Examples of
solving equations in integers. Inequalities. Replacing an inequality with an equivalent
inequality or system of inequalities. Inequality with a modulus. Systems of irrational equations.
Irrational inequalities. Basic methods for solving systems of equations. Gauss's method.
Solving inequalities with two variables. Representation on the coordinate plane of the set of
solutions to inequalities with two variables. Solving word problems and mathematical
modeling. Equations and inequalities with parameters.
Review and problem solving.