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L.V. Kurpa, . V. Shmatko
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DIFFERENTIAL CALCULUS FOR ONE VARIABLE FUNCTIONS

The Educational Textbook for Students of Technical Universities

 

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ϲ 2008

 

22.161.1

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ISBN 978-966-593-659-6

 

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CONTENT

 

INTRODUCTION IN THE MATHEMATICAL ANALYSIS

CHAPTER I. FUNCTIONS OF ONE VARIABLE

1.1.Basic Logic Symbols

1.2.The Simplest Concepts 0and Designations of the Sets Theory

1.3.The Absolute Value (Modulus) of a Real Number and its Properties

1.4.Numerical Sets. Interval. A Neighborhood of a Point

1.5.Constants and Variables Quantities. Classification of Variables

1.6.Function

1.7.Ways of Function Representation

1.8.Implicit and Explicit Functions

1.9.Even and Odd Functions. Periodic Functions

1.10.Inverse Function

1.11.Basic Elementary Functions

1.12.Composite Function

1.13.Elementary Functions

 

CHAPTER II. LIMIT THEORY

2.1. Limit of Numerical Sequence

2.2. The Simplest Properties of the Limits

2.3. Limit of Monotonic Sequence

2.4. Infinitesimals and their Main Properties

2.5. Infinitely Large Values and their Main Properties

2.6. The Connection Between Infinitely Large and Infinitesimals

2.7. Properties of Limits Connected with Arithmetic Operations

2.8. The Limit of a Function at a Point and on Infinity

2.9. One-Sided Limits of a Function at a Point

2.10.Properties of the Function Limits

2.11.The Second Definition of a Function Limit at a Point and on Infinity

2.12.First Remarkable Limit

2.13.The Number e as Limit of the Numerical Sequence

2.14.The Second Remarkable Limit

2.15.Comparison of the Infinitesimals and Infinitely Large Valurs

2.16.Equivalent Infinitesimal Values

 

CHAPTER III. CONTINUITY OF A FUNCTION

3.1. Definition of Continuous Function at a Point

3.2. Another Definitions of the Function Continuity

3.3. Arithmetic Operations on Continuous Functions

3.4. Continuity of the Composite Function

3.5. Theorem about Preservation Sign of a Continuous Function

3.6. Continuity of the Inverse Function

3.7. Continuity of the Basic Elementary Functions

3.8. Classification of Discontinuity Points

3.9. Consequences of the Second Remarkable Limit

3.10.Limit of Power-Exponential Function

3.11.Lemma about Contracting Segments

3.12.Lemma By Boltsano- Weierstrass

3.13.Properties of the Functions Which are Continuous on Closed Interval

3.14.Uniform Continuity of a Function

 

CHAPTER IV. FUNDAMENTALS OF THE DIFFERENTIAL CALCULUS FOR ONE VARIABLE FUNCTIONS

4.1. Derivative of a Function and its Geometric Sense

4.2. The Connection Between Continuity and Differentiability of a Function

4.3. The Basic Rules for Finding Derivatives

4.4. Derivative of the Inverse Function

4.5. Derivatives of the Inverse Trigonometric Functions

4.6. Derivatives of the Hyperbolic and Inverse Hyperbolic Functions

4.7. The Table of the Basic Formulas and Rules of Differentiation

4.8. Derivative of the Composite Function

4.9. Differentiation of the Implicit Functions

4.10.Logarithmic Differentiation

4.11.Geometric and Physical Applications of the Derivatives

4.12.Parametric Representation of a Function

4.13.The Equations of Some Curves in Parametric Form

4.14.The Derivative of a Function Represented Parametrically

4.15.The Differential

4.16.The Geometric Meaning of the Differential

4.17.Derivatives of Different Orders

4.18.Differentials of Different Orders

4.19.Derivatives of the Higher Order of Functions Represented Parametrically

 

CHAPTER V. INVESTIGATION OF THE BEHAVIOUR OF FUNCTIONS

5.1. The Basic Theorems of Differential Calculus

5.2. L'Hospital's Rule for Evaluating Indeterminate Forms of the Type

5.3. Evaluating Indeterminate form by L'Hospital's Rule

5.4. Evaluating Power-Exponential Indeterminate Forms

5.5. Theorems about Increase and Decrease of a Function on an Interval

5.6. Extremum of a function

5.7. Sufficient Condition for Existence of an Extremum (the First Rule)

5.8. Testing a Differentiable Function for Maximum and Minimum with help the First Derivative

5.9. Testing a Function for Maximum and Minimum by a Second Derivative

5.10.The Greatest and the Smallest Values of a Function on an Interval

5.11.Applying the Theory of Maxima and Minima of Functions to the Solving Problems

5.12.Convexity and Concavity of a Curve. Points of Inflection

5.13.Asymptotes

5.14.General Plan for Investigating Functions and Constructing Graphs

5.15.Taylor's Formula

 

REFERENCES

APPENDIX 1. ENGLISH-UKRAINIAN-RUSSIAN VOCABULARY OF MATHEMATICAL TERMS

APPENDIX 2. RUSSIAN-ENGLISH VOCABULARY OF MATHEMATICAL TERMS

APPENDIX 3. -