The abstract of lectures is developed in accordance with the work program of the discipline "Higher mathematics". It designated for undergraduate students of full-time education in specialty 185 - Oil and Gas Engineering. Конспект лекцій розроблено відповідно до робочої програми дисципліни "Вища математика". Призначено для студентів-бакалаврів очної форми навчання за спеціальністю 185 - Нафтогазова інженерія. CONTEXT Introduction……………………………………………………... 9 Part I Lecture 1 Sequence. Limit of Numeral Sequence. Limit's Theorems………………………………………………………... 10 1.1 Sequence………………………………………............. 10 1.2 Numerical Sequence Limit............................................. 10 1.3 Limit's Theorems……………………........................... 10 1.4 The Determinate and Indeterminate Expressions………. 11 Lecture 2 Concept of a Function. Function's Limit. Function Limit's Theorems……………………………………………….. 13 2.1 Concept of a Function…….……..……………………... 13 2.2 Limit of a Function……………………………………... 16 2.3 One-sided Limits……………………………………….. 17 2.4 Properties of Limits.……………………………………. 18 2.5 The First Honorable Limit. Number е. The Second Honorable Limit………………………………………………… 19 2.5.1 The First Honorable Limit……………………………. 19 2.5.2 The Second Honorable Limit………………………… 19 2.6 Comparison of Infinitesimals. Equivalent Infinitesimals. 20 2.7 Continuity of a Function at a Point. Classification of Points of Discontinuity………………………………………….. 20 2.7.1 Continuity of a Function at a Point…………………… 20 2.7.2 Classification of Points of Discontinuity…………….. 22 2.8 Some Problems Leading to Understanding of the Derivative Definition……………………………………………. 22 2. 9 Definition of Derivative……………………………….. 23 2.10 Geometrical, physical and mechanical concept of the derivative……………………………………………………….. 24 2.10.1 Geometrical Concept of Derivative………………… 24 2.10.2 Mechanical Concept of Derivative…………………. 27 2.11 The Main Rules of Differentiation…………………… 27 2.12 Table of Elementary Functions Derivatives………….. 28 2.13 The Derivative of Composite Function………………. 28 2. 14 Derivative of Inverse Function………………………. 29 2.15 Differentiation of implicit function…………………... 30 2.16 Differentiation of Functions Given Parametrically…... 30 2.17 Logarithmic Differentiation…………………………... 30 Lecture 3 Differential of a Function…………………………….. 32 3.1 Definition and Geometrical Interpretation of Differential 3.2 General Properties of Differential……………………… 33 3.3 Higher Order of Derivatives…………………………… 34 3.4 Leibniz Formula………………………………………... 34 3.5 Calculation of Higher Order Derivatives of Parametrical Functions………………………………………………………... 35 3.6 Higher Order Derivatives of Implicit Function………… 35 3.7 Higher Order Differentials……………………………... 36 Lecture 4 Fermat, Rolle, Lagrange, Cauchy Theorems………… 37 4.1 Fermat Theorem and Its Geometrical Interpretation…… 37 4.2 Rolle Theorem and Its Geometrical Interpretation……... 37 4.3 Lagrange Theorem and Its Geometrical Interpretation… 38 4.4 Cauchy Theorem……………………………………….. 39 Lecture 5 L'Hospital rule. Increasing and decreasing functions. Relative extremum of a function. Assimtots……………………. 40 5. 1 L'Hospital Rule………………………………………... 40 5.2 Increasing and Decreasing Functions. Extremum of a Function…………………………………………………………. 41 5.3 The Greatest and the Least Values of the Function…….. 44 5.4 The concavity of the Curve. The Inflection Points……... 44 5.5 Asymptotes of the Curve……………………………….. 46 5.6 Scheme of Investigation of the Graph………………….. 47 Part II Lecture 1 Definitions and Properties of Indefinite Integral…….. 48 1.1 Initial and Indefinite Integral…………………………… 48 1.2 Definitions and Properties of Indefinite Integral……….. 49 1.3 Properties of Indefinite Integral………………………… 50 1.4 Table of Integrals……………………………………….. 52 1.5 Replacement of Variable……………………….………. 55 1.6 Partial Integration………………………………………. 57 Lecture 2 Some Information about Complex Numbers and Polynomials……………………………………………………... 61 2.1 Complex Numbers……………………………………… 61 2.2 Polynomials…………………………………………….. 68 2.3 Polynomial with Real Coefficients……………………... 72 Lecture 3 Integration of Rational Functions…………………….. 73 3.1 Rational Fractions……………………………………….. 73 3.2 Integration of the Basic Rational Fractions……………... 81 Lecture 4 Integration of Irrational Functions…………………... 86 4.1 Preliminary remarks…………………………………….. 86 4.2 Integrals of Type ……... 87 4.3 Integrals Type . Euler Substitutions 89 4.4 Integrals of Differential Binomial……………………….. 92 Lecture 5 Integration of Some Trigonometric Functions. Trigonometric Substitution……………………………………… 96 5.1 Integrals of the Type ………………… 96 5.2 Integral Type ………………………….. 98 5.3 Integrals of the Type …………………. 100 5.4 Integration of Some Irrational Functions with Trigonometric Substitutions…………………………………… 101 Lecture 6 Definitions and Properties of Definite Integral…….. 104 6.1 Definition of Definite Integral……………………….. 104 6.2 Conditions of Existence of Definite Integral…………. 105 6.3 Basic Properties of Definite Integral…………………. 109 Lecture 7 Calculation of Definite Integral…………………….. 111 7.1 Definite Integral as the Limit of Integral Sum………... 113 7.2 Integral with Upper Boundary Variable……………… 114 7.3 Newton's Formula*-Gottfried Wilhelm Leibniz**….... 116 7.4 Replacing of Variable in a Specific integrals………… 117 7.5 Formula of Partial Integration………………………… 121 Part III Lecture 8 Improper integral……………………………………. 122 8.1 Definition of improper integrals………………………. 122 8.2 Formulas of integral calculus for non-proper integrals.. 125 8.3 The Cauchy criterion the convergence of improper integrals. Absolutely convergent integrals…………………….. 127 8.4 Improper integrals of non-essential functions……………. 129 Lecture 9 Some Geometric Applications of Definite Integral… 132 9.1. The Area of a Figure…………………………………. 132 9.2 Volume of a Rotation Body…………………………… 138 9.3 Length of a Curve…………………………………… .. 140 9.4 Area of a Rotation Surface……………………………. 143 Lecture 10 Some Physical Applications of Definite Integral…. 146 10.1. Work of Variable Power……………………………. 146 10.2 Coordinates of Center of Masses……………………. 147 10.3 Calculation of Moment of Inertia of a Line and Circle 153 Lecture 11 Approximate Calculus of Certain Integrals……….. 156 11.1 Formulas of Rectangles and Trapezes………………. 156 11.2 Simson Formula …………………………………….. 159 Part IV Lecture 1 Notion of Ordinary Differential Equations…………. 164 1.1 Common Notions and Determinations........................... 164 1.2 Solving Cauchy Task and Its Geometrical Interpretation…………………………166 1.3 Solving of Some Equations of the First Order, Untied in Relation to Derivative………………………..………………… 169 1.3.1 Equation with the Separated and Separated Variables and Panders to Them…………………………………………... 170 Lecture 2 Solving of Some Equations of the First Order, That Get Untied in Relation to Derivative……………………………….. 175 2.1 Equation in Complete Differentials Equation………… 175 2.2 Method of Integrating Cofactor………………..……… 178 2.3 Homogeneous Equations……………………………… 182 2.3.1 Equations That Are Taken to Homogeneous………... 186 2.4 Linear Differential Equations of the First Order……… 190 2.5 Bernoulli Equation…………………………………….. 197 Lecture 3 Differential Equations of Higher Orders…………… 204 3.1 Common Notions and Determinations………………... 204 3.2 Differential Equations of Higher Orders Which Assume the Decline of Order…………………………...………………. 207 3.2.1 Equations of a Kind y(n)=f(x)………………........... 207 3.2.2 Differential Equations of a Kind ………………………………….. 209 3.2.3 Differential Equation of the Kind ………………………………………… 214 Lecture 4 Linear Differential Equations of Higher Orders (elements of general theory)…………………………………… 219 4.1 Basic Concepts. Classification of Linear Differential Equations………………………………………………………. 219 4.2 Linear Dependence and Linear Independence of Systems of Functions……………………………………………………. 220 4.3 Linear Homogeneous Differential Equations……….. 222 4.3.1 Wronski-Determinante……………………………… 222 4.3.2 Structure of Common Decision of LHDE…………. 225 4.4 Linear Heterogeneous Differential Equations……….. 228 4.4.1 Structure of Common Decision of Linear Heterogeneous Differential Equation……………………................................... 228 Lecture 5 Solving of Homogeneous Linear Differential Equations (LHDE) with Constant Coefficients…………………………… 232 5.1 Linear homogeneous differential Equations with Constant Coefficients……………………………………......................... 232 5.1.1 Characteristic Equation……………………………... 232 5.1.2 The Roots of Characteristic Equation Real and Simple…………………………………………………………. 233 5.1.3 The Roots of Characteristic Equation are Real multiple………………………………………………………… 234 5.1.4 The Roots of Characteristic Equation Are Complex Conjugate………………………………………………………. 236 5.1.5 The Roots of Characteristic Equation are Arbitrary………………………………………………………. 238 Lecture 6 Solving of Heterogeneous Linear Differential Equations (LNDE) With Constant Coefficients………………………….. 241 6.1 Method of variation……………………………......... 241 6.2 Linear Heterogeneous Differential Equations of the n-th Order With Constant Coefficients and Special Type of the Right Part…………………………………………………………….. 245 6.3 Linear Heterogeneous Differential Equations of the Second Order With Constant Coefficients…………………….. 251 6.4 Principle of superposition of decisions………….......... 256 Lecture 7 Systems of Differential Equations………………… 260 7.1 Common Notions on Systems of Differential Equations………………………………………………............. 260 7.2 Solving of the Normal System of Differential Equations………………………………………………262 7.3 Solving of System of Linear Homogeneous Equations With Constant Coefficients…………………………………… 265 References................................................................................... 269