Mathematics - part one - Bozhinov 2011
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10.26 лв.
CONTENTS
Chapter 1. SETS AND FUNCTIONS
1.1. Sets and operations on sets
1.2. Functions in the set theory
1.3. Functions of one real variable .:
1. Even and odd functions
2. Periodic functions
3. Bounded functions
4. Monotone functions
5. Basic elementary functions
Chapter 2. ANALYTIC GEOMETRY
2.1. Coordinate systems
1. Coordinate systems in the straight line ..
2. Coordinate systems in the plane
3. Coordinate systems in three-dimensional space
4. Analytical representation of vector operations
2.2. Analytic representation of curves in the plane
1. Representation of curves in the plane by means of equations..
2. General equation of a straight line in the plane
3. Curves (A second degree. Conic sections
2.3. Representation of surfaces and curves in syniee
1. Representation of the surfaces in space by means of equations
2. General equation of a plane in space
3. Parametric representations of the curve's in space
4. Parametric and symmetric equations of a straight line
Chapter 3. DIFFERENTIAL CALCULUS
3.1. Infinite sequences
1. Basic concepts
2. -Limit of a sequence of real numbers
3. Arithmetic operations with convergent sequences
4. Limits in inequalities
5. Monotone sequences
6. The number e. Natural (Napierian) logarithms
7. Subsequences
8. Sequences tending to infinity
3.2. Limit of a function
1. Open and closed sets
2. Limit of a function.
3. Left-hand and right-hand limits •
4. Limit of a function, when the argument tends to infinity
5. Infinite limits of a function
6. Properties of the limits. Indeterminate forms 72
7. Some basic limits 75
3.3. Continuous functions 78
1. Continuity at a point 78
2. The continuity of the elementary functions 80
3. Global properties of the continuous functions • 82
3.4. Derivatives 84
1. Derivative of a function at a point 84
2. Geometric treatment of derivatives. Differential 85
3. Differentiation rules 87
4. Derivatives of basic elementary functions 88
5. Higher order derivatives and differentials 90
3.5. Basic theorems of differential calculus 93
1. Local extrema. Necessary condition for existence 93
2. The mean value theorems 94
3. L’Hospital rule 96
4. Taylor Formula 97
' 3.6. Investigation of functions behaviour 99
1. Monotone functions 99
2. Local extrema. Sufficient conditions for existence 100
3. Convex functions. Inflection 101
4. Asymptotes 105
5. Applications of curve tracing 107
6. Extremal problems for functions of one real variable Ill
Chapter 4. INTEGRAL CALCULUS , 116
4.1. The indefinite integral 116
1. The concept of the indefinite integral 116
2. Basic properties of indefinite integrals 118
3. Tal)le of the basic indefinite integrals 119
4. Direct integration 120
5. Integration by placing a function under the differential sign 120
6. Integration by parts 123
7. Integration by substitution (change of variable) 125
4.2. The definite integral 129
1. An intuitive approach to the concept of the definite integral 129
2. Calculation of definite integrals 134
3. Integration by parts. Change of the variable 136
4.3. Applications of the definite integral 140
1. Areas of plane figures 140
2. Calculation of volumes 142
3. Length of a curve 144
4. Area of rotational surface (surface of revolution) 145
4.4. Improper integrals 147
1. Improper integrals with infinite integration limits . 147
2. Improper integrals of unbounded functions 150
3. Principle of comparison of improper integrals 152
4.5. Infinite series 154
1. Basic concepts 154
2. Properties of infinite series 154
3. Absolute and conditional convergence 156
4. Power series 158
5. Expansions of the elementary functions in power series 159
