Н.Темірғалиев, Алгебра және анализ бастамалары 10-11-сынып, 2002

I. Prerequisite knowledge: review and clarification

Chapter I. Numbers

§1. Real numbers and their representation

1. Sets and sets of numbers

2. Mathematical notation and symbolic abbreviations

3. Operations on sets

4. Digits as number symbols and the positional numeral system

5. Representation of real numbers: common, decimal, and algebraic fractions

§2. Common fractions as a representation of rational numbers

1. Definition of common fractions

2. Meaning of the fraction bar

3. Addition of common fractions

§3. Decimal fractions

1. Definition of decimal fractions

2. Representation of numbers as infinite decimals

3. Repeating decimals as representations of rational numbers

4. Converting repeating decimals to common fractions

5. Nonrepeating decimals as representations of irrational numbers

§4. Number line

1. The measurement problem. Measuring length

2. Insufficiency of rational numbers for measuring segment length

3. Constructing a segment incommensurable with a given segment

4. Number line

5. Meaning of each place-value digit in a real number

§5. Algebraic fractions

1. Concept of an algebraic fraction

2. Applications of algebraic fractions

§6. Other numeral systems

§7. Complex numbers. Development of the number concept

Chapter II. Powers and logarithms

§8. Powers of numbers and their properties

1. Repeated addition and multiplication of a number and their notation

2. Positive integer powers of real numbers and their properties

3. Extending the concept of powers

§9. Integer powers of a number

1. Zero power of a number

2. Negative integer power of a nonzero real number

§10. Rational powers of a number

1. The 1/n power of a positive number: arithmetic root

2. Rational power of a positive number

3. Rational power of a negative number

§11. Irrational power of a positive number

§12. Logarithm of a number

1. Definition of the logarithm of a number

2. Logarithm of a product, quotient, and power

3. Change-of-base formula for logarithms

Chapter III. Functions and sequences

§13. Function: definition and analysis

1. General definition of a function

2. Argument of a function

3. The rule in the definition of a function

4. Mapping a given set into its subset

5. On the use of letter indices

6. Set operations and the domain of a function

§14. Numerical functions

1. Definition of numerical functions of a numerical argument

2. Examples of numerical functions

3. Numerical functions of several variables

§15. Numerical intervals

1. Numerical intervals

2. Table of numerical intervals

3. Neighborhoods

§16. Graphs of functions

1. Coordinate plane

2. Graphs of functions

3. Transforming function graphs: the quadratic case

4. Transforming function graphs: the general case

§17. Sequences

1. Numerical sequences a(n)=aₙ

2. Arithmetic and geometric progressions

3. Mathematical induction

4. Binomial theorem

5. Simple and compound interest

6. The number e

II. Basic elementary and elementary functions

Chapter IV. Properties of functions

§18. Geometric properties of a function

1. Introductory remarks

2. Evenness and oddness of a function

3. Monotonicity of a function

4. Boundedness of a function

§19. Inverse functions

1. One-to-one correspondence

2. Definition of an inverse function

§20. Continuity and limit of a function

1. Continuity on an interval and discontinuity at a point

2. Definition of continuity at a point and on an interval

3. Continuity for a monotonic function

4. Examples of proving continuity

5. Discontinuity point of a function

6. General definition of a function limit

7. The idea underlying limits and continuity

Chapter V. Basic elementary functions: power, exponential, and logarithmic functions

§21. Power function

1. Definition

2. Properties

§22. Power function with a rational exponent

§23. Exponential function

1. Definition and graph

2. Main properties

§24. Logarithmic function

1. Main properties of the logarithmic function

Chapter VI. Basic elementary functions: trigonometric and inverse trigonometric

§25. Trigonometric functions as functions of distances along a circle

1. Moving along a circle in a given direction by a given distance

2. Definition of cosine and sine as functions of a numerical argument

3. Definition of tangent and cotangent as functions of a numerical argument

§26. Periodic functions

1. Periodic functions and period. Periodicity of cos x and sin x with period 2π

2. Ways to represent all points where a property of a periodic function holds

§27. Numbers and angles in degree and radian measure as arguments of trigonometric functions

1. Degree and radian measures of angles

2. Converting between degree and radian measures

3. Cosine and sine as functions of a real number and an angle

4. Rotation in a given direction by a given radian or degree measure

5. Arguments of trigonometric functions

§28. Main properties and graph sketches of trigonometric functions

1. Properties and graph sketch of cos x

2. Properties and graph sketch of sin x

3. Properties and graph sketch of tan x

4. Properties and graph sketch of cot x

§29. Basic trigonometric identities

1. Fundamental trigonometric identities

2. Addition formulas

3. Addition formula for cos x

4. Basic trigonometric identities

§30. Main trigonometric formulas

1. Cosine of the difference of two angles

2. Formulas for complementary angles

3. Sine of a sum and difference

4. Double-angle and half-angle formulas

5. Tangent and cotangent formulas

6. Sums and differences of cosines and sines

§31. Relations between trigonometric functions of the same argument

§32. Reduction formulas

§33. Inverse trigonometric functions

1. Inverse function

2. Inverse trigonometric functions

3. The inverse cosine function

4. The inverse sine function

5. The inverse tangent function

6. The inverse cotangent function

Chapter VII. Elementary functions

§34. Definition of elementary functions

1. Replacing the argument of one function by another: composite functions

2. Using arithmetic operations to define new functions

3. Definition of elementary functions

4. Formulas and rules in the notation of elementary functions

5. Domain of an elementary function from its expression

6. Remarks on the definition of an elementary function

§35. Operations on functions producing new functions

1. Defining new functions by arithmetic operations on numerical functions

2. Defining a new function by successive application of rules

3. Defining a new function from functions on intersecting sets

III. Equations and inequalities

Chapter VIII. Equations and inequalities

§36. General information on solving equations

1. On the use of the identity sign

2. Equation as a conditional equality

3. Forming equations as a method of mathematical modeling

4. Formulation of the equation-solving problem

5. Equation as finding an argument from a given function value

6. On the content and aims of the topic “Equations”

7. Domain of the equation f(x)=g(x)

8. Systems and collections of equations

9. Equivalent equations

10. Transformations and transformed equations

11. Equivalent transformations

12. Equivalent and nonequivalent transformations involving power functions

13. Equivalent equations on a set

14. Classification of equations

§37. Polynomial equations

1. Analysis of a linear equation

2. Analysis of a quadratic equation

3. Some special types of higher-degree equations

4. Degree-reduction method for polynomial equations

V. Differential and integral calculus

Chapter XI. Derivative

§55. Problems leading to the concept of derivative

1. Velocity at an instant: model case

2. Velocity at an instant: general case

3. Tangent to a function graph at a point: the case y=x²

4. Tangent to a function graph at a point: general case

§56. Derivative

1. Definition of the derivative

2. Derivatives of linear and quadratic functions

§57. Derivatives of differentiable elementary functions

1. On calculating derivatives

2. Techniques for differentiating elementary functions

3. Table of derivatives of basic elementary functions

4. Differentiation rules

5. Table for differentiating composite functions with a basic elementary outer function

6. Differentiating elementary functions built by repeated composition

§38. Rational algebraic equations

§39. Irrational equations

§40. Exponential and logarithmic equations

§41. Equations involving absolute value

§42. Trigonometric equations

1. Families of equation solutions and their notation

2. Basic trigonometric equations

§43. Inequalities

§44. Solving inequalities by the interval method

1. Alternation of polynomial signs on intervals

2. Changes of polynomial signs on intervals

3. Interval method for solving inequalities

4. Cases reducible to the interval method

5. Irrational inequalities

IV. Combinatorics, probability theory, and mathematical statistics

Chapter IX. Combinatorics

§58. Indeterminate forms and their resolution

1. Indeterminate form 0/0

2. Remarkable limits

3. Factorization method

4. Rationalization method

5. Method based on the definition of e

§46. Combinatorics

1. Fundamental counting principle: special case

2. Fundamental counting principle: general case

3. Formulation of a combinatorial problem

4. Sampling with and without replacement

5. Unordered and ordered selections

6. Ordered selections with replacement

7. Ordered selections without replacement: arrangements

8. Permutations

9. Unordered selections of k elements from n without replacement: combinations

Chapter X. Probability theory and mathematical statistics

§47. Probability space: auxiliary and probabilistic models

1. Auxiliary model

2. Experiment and its outcome

3. Probability model

4. Probability model of tossing a coin

§48. General rule of probability theory

1. General scheme for constructing a finite probability model

2. An experiment was performed and the outcome is known. Did the event occur?

3. Practical meaning of event probability. Law of large numbers

§49. Introductory probability problems and methods of solving them

1. Formulation of a probability problem

2. Probability space of equally likely elementary outcomes. Classical definition of probability

3. Examples of experiments with equally likely outcomes

§50. Event and its probability

1. Algebra of events

2. Conditional probability of an event

3. Independent events

4. Additive and multiplicative properties of probability

§51. Fundamental formulas of probability theory

1. Multiplication rule for probabilities

2. Law of total probability

3. Bayes’ formulas

§52. Random variables

1. Definition of a random variable

2. Distribution of a random variable

§53. Expected value and variance of a random variable

1. Mean value

2. Expected value of a random variable

3. Variance

4. Approaches to defining the spread of random-variable values

§54. Mathematical statistics

1. Purpose of mathematical statistics

2. Estimating probability

3. Population and a sample of random-variable values

4. Empirical distribution law of a random variable

5. Estimating the expected value and variance of a random variable