А.Н.Шыныбеков. Алгебра және анализ бастамалары 10-сынып, 2014

Preface

Exercises for reviewing Grade 9 material

Chapter I. Function and its properties

§ 1. Numerical functions and their graphs

1.1. Emergence of the concept of function

1.2. Correspondences

1.3. Functions

§ 1 есептер

§ 2. Ways of defining a function and operations on functions

2.1. Ways of defining a function

2.2. Operations on functions

§ 2 есептер

§ 3. Composite and inverse functions

3.1. Composite function

3.2. Inverse function

§ 3 есептер

§ 4. Simple plan for studying a function

§ 4 есептер

Chapter II. Trigonometric functions

§ 1. Transforming trigonometric expressions (review)

1.1. Definition of trigonometric functions of a numerical argument

1.2. Basic formulas of trigonometry

§ 1 есептер

§ 2. Main properties of trigonometric functions

2.1. Function y = sin x

2.2. Function y = cos x

2.3. Function y = tan x

2.4. Function y = cot x

§ 2 есептер

§ 3. Examples of plotting graphs of some trigonometric functions

§ 3 есептер

§ 4. Inverse trigonometric functions

4.1. Function y = arcsin x

4.2. Function y = arccos x

4.3. Function y = arctan x

4.4. Function y = arccot x

§ 4 есептер

§ 5. Trigonometric equations

5.1. Solving simple trigonometric equations

5.2. Methods for solving trigonometric equations

§ 5 есептер

§ 6. Solving systems of trigonometric equations

6.1. Systems of the form sin x · sin y = a, cos x · cos y = b and sin x · cos y = a, cos x · sin y = b

6.2. Systems of the form sin x + sin y = a, sin²x + sin²y = b

6.3. Systems of the form sin x + sin y = a, x + y = α

6.4. Systems of the form a sin x + b sin y = c, p cos x + q cos y = r

§ 6 есептер

§ 7. Inverse trigonometric equations

7.1. Solving simple inverse trigonometric equations

7.2. Methods for solving inverse trigonometric equations

§ 7 есептер

§ 8. Trigonometric inequalities

8.1. Solving simple trigonometric inequalities

8.2. Proving trigonometric inequalities

§ 8 есептер

Chapter III. Limit and continuity

§ 1. Limit of a numerical sequence

1.1. Numerical sequence

1.2. Limit of a numerical sequence

1.3. Limit of a monotone sequence. Infinitely large and infinitesimal quantities

1.4. Main theorems on limits of sequences

§ 1 есептер

§ 2. Limit of a function at a point

2.1. Limit of a function at a point

2.2. Main theorems on limits of functions

2.3. Limit of a function at infinity

2.4. Asymptotes of a function

2.5. First remarkable limit

§ 2 есептер

§ 3. Continuity of a function

3.1. Continuity of a function at a point and its properties

3.2. Properties of functions continuous on a segment

§ 3 есептер

Chapter IV. Derivative and its applications

§ 1. Derivative and differential of a function

1.1. Problems leading to the derivative of a function

1.2. Derivative of a function

1.3. Differential of a function and its geometric meaning

§ 1 есептер

§ 2. Rules for differentiating functions

2.1. Rules for finding derivatives

2.2. Derivatives of elementary functions

§ 2 есептер

§ 3. Derivative of composite and inverse functions

3.1. Derivative of a composite function

3.2. Derivative of an inverse function

3.3. Concept of higher-order derivative. Mechanical meaning of the second derivative

§ 3 есептер

§ 4. Mean value theorems

4.1. Fermat's and Rolle's theorems

4.2. Lagrange's theorem

§ 4 есептер

§ 5. Applying the derivative to studying a function

5.1. Intervals of increase and decrease of a function

5.2. Necessary and sufficient conditions for extrema of a function

§ 5 есептер

§ 6. Simple plan for studying a function and plotting its graph

6.1. Greatest and least values of a function on a segment

6.2. Simple plan for studying a function and plotting its graph

§ 6 есептер

§ 7. Full plan for studying a function and plotting its graph

7.1. Convexity, concavity and inflection points of a function graph

7.2. Full plan for studying a function and plotting its graph

§ 7 есептер

§ 8*. L'Hôpital's rule. Applying the derivative to find limits of functions

§ 8 есептер

§ 9. Applying the derivative in approximate calculations

9.1. Concept of Taylor's formula

9.2. Approximate solution of equations using the derivative

§ 9 есептер

Chapter V. Elements of combinatorics and their use in calculating probabilities

§ 1. Elements of combinatorics. Newton's binomial theorem

1.1. Elements of combinatorics

1.2. Newton's binomial theorem

1.3*. Arrangements of a given composition

1.4*. Combinations with repetitions

§ 1 есептер

§ 2. Algebra of events and classical definition of probability

2.1. Algebra of events

2.2. Classical definition of probability. Properties of probability

§ 2 есептер

§ 3*. Total probability of an event. Bayes' formula

3.1. Total probability of an event

3.2. Bayes' formula

§ 3 есептер

§ 4. Bernoulli formula. Concept of the law of large numbers

4.1. Bernoulli formula

4.2. Concept of large numbers

§ 4 есептер

Chapter VI*. Field of real numbers

§ 1*. Theory of natural numbers. Peano axioms. Principle of mathematical induction

1.1. Emergence of the concept of natural numbers. Historical information

1.2. Axiomatic method in mathematics

1.3. Peano axioms

1.4. Principle of mathematical induction

§ 1 есептер

§ 2*. Ring of integers. Field of rational and real numbers

2.1. Ring of integers

2.2. Field of rational numbers

2.3. Field of real numbers

§ 2 есептер

Chapter VII*. Polynomials

§ 1*. Polynomials in one variable

1.1. Polynomials and operations on them

1.2. Roots of polynomials. Bézout's theorem. Horner's scheme

1.3. Vieta's formulas

§ 1 есептер

§ 2*. Rational expressions and their simplest form (review)

2.1. Factoring polynomials into irreducible factors. Concept of the fundamental theorem of polynomial algebra

2.2. Rational expression and its simplest form

§ 2 есептер

§ 3*. Identities and equations

3.1. General concepts of equations and identities

3.2. Equivalent transformations

3.3. Sophisms

3.4. Methods for solving equations

Problems for 3.4

3.5. Some special methods for solving equations

Problems for 3.5

§ 4*. Inequalities

4.1. Main properties of inequalities

4.2. Proving inequalities

Problems for 4.2

4.3. Solving inequalities

Problems for 4.3

Answers to problems