Lectures on Differential Calculus of Functions of One Variable (330,00 руб.)

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MINISTRY OF SCIENCE AND HIGHER EDUCATION OF THE RUSSIAN FEDERATION SOUTHERN FEDERAL UNIVERSITY Mikhail E. Abramyan LECTURES ON DIFFERENTIAL CALCULUS OF FUNCTIONS OF ONE VARIABLE Textbook for students of science and engineering Rostov-on-Don – Taganrog Southern Federal University Press 2020

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УДК 517.4(075.8) ББК 22.162я73 А164 Published by decision of the educational-methodical commission of the I. I. Vorovich Institute of Mathematics, Mechanics, and Computer Science of the Southern Federal University (minutes No. 2 dated February 14, 2020) Reviewers: Doctor of physical and mathematical sciences, professor of the Department of Applied Mathematics of the South Russian State Polytechnic University, Honorary official of higher professional education of the Russian Federation, Professor A. E. Pasenchuk; Doctor of physical and mathematical sciences, head of the Department of Computer Science and Computational Experiment of the I. I. Vorovich Institute of Mathematics, Mechanics, and Computer Science of the Southern Federal University, Professor V. S. Pilidi Abramyan, M. E. А164 Lectures on differential calculus of functions of one variable : Textbook / M. E. Abramyan ; Southern Federal University. — Rostov-onDon ; Taganrog : Southern Federal University Press, 2020. — 220 p. ISBN 978-5-9275-3494-4 The textbook contains lecture material for the first semester of the course on mathematical analysis and includes the following topics: the limit of a sequence, the limit of a function, continuous functions, differentiable functions (up to Taylor's formula, L'Hospital's rule, and the study of functions by differential calculus methods). A useful feature of the book is the possibility of studying the course material at the same time as viewing a set of 22 video lectures recorded by the author and available on youtube.com. Sections and subsections of the textbook are provided with information about the lecture number, the start time of the corresponding fragment and the duration of this fragment. In the electronic version of the textbook, this information is presented as hyperlinks, allowing reader to immediately view the required fragment of the lecture. The textbook is intended for students specializing in science and engineering. УДК 517.4(075.8) ББК 22.162я73 ISBN 978-5-9275-3494-4 © Southern Federal University, 2020 © Abramyan M. E., 2020

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Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Video lectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Using video lectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Using subtitles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Preliminary information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15 Mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Absolute value and the integer part of a real number . . . . . . . . . . . . . . . . .17 Principle of mathematical induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Mappings and functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1. Boundaries of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 The continuity axiom of real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20 Boundaries and exact boundaries of number sets . . . . . . . . . . . . . . . . . . . . . 20 Arithmetic operations on sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24 2. Limit of a sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Neighborhood and symmetric neighborhood of a point . . . . . . . . . . . . . . . 27 Definition of the limit of a sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 The simplest properties of the limit of a sequence . . . . . . . . . . . . . . . . . . . . 32 3. Properties of the limit of a sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Infinitesimal sequences: definition and properties . . . . . . . . . . . . . . . . . . . . 34 A criterion for convergence in terms of an infinitesimal . . . . . . . . . . . . . . .35 Arithmetic properties of the limit of a sequence . . . . . . . . . . . . . . . . . . . . . . 35 Passing to the limit in inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4. Infinite limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40 Neighborhoods of the points at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Infinitely large sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Arithmetic properties of infinitely large sequences . . . . . . . . . . . . . . . . . . . .41

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4 M. E.Abramyan. Lectures on differential calculus 5. Monotone sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44 Bounded and monotone sequences: definitions . . . . . . . . . . . . . . . . . . . . . . . 44 Convergence of monotone sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Examples of application of the convergence theorem for monotone sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 6. Nested segments theorem and Bolzano–Cauchy theorem on the limit point . . . . . . . . . . 53 Nested segments theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Limit points of a set. Bolzano–Cauchy theorem . . . . . . . . . . . . . . . . . . . . . . 54 7. Subsequences. Cauchy criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58 Subsequences. Bolzano–Weierstrass theorem . . . . . . . . . . . . . . . . . . . . . . . . . 58 Fundamental sequences. Cauchy criterion for sequence convergence . . 62 8. The limit of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Definition and uniqueness of the limit of a function . . . . . . . . . . . . . . . . . . 66 Criterion for the existence of the limit of a function in terms of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68 9. Properties of the limit of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Limit of a function and arithmetic operations . . . . . . . . . . . . . . . . . . . . . . . . 74 Passing to the limit of a function in inequalities . . . . . . . . . . . . . . . . . . . . . .75 The theorem on the limit of superposition of functions . . . . . . . . . . . . . . . 76 10. One-sided limits. Some important function limits . . . . . . . . . . . 78 Definition of one-sided limits of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Criterion for the existence of the limit of a function in terms of one-sided limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79 The first remarkable limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 The second remarkable limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 11. The limits of monotone bounded functions. Cauchy criterion for functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Monotone and bounded functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Cauchy criterion for the existence of the function limit . . . . . . . . . . . . . . . 89 12. Continuity of function at a point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Definition of a continuous function at a point . . . . . . . . . . . . . . . . . . . . . . . . 92 Examples of functions with and without limits . . . . . . . . . . . . . . . . . . . . . . . 70 Limits at the points at infinity and infinite limits . . . . . . . . . . . . . . . . . . . . 72

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Contents 5 Examples of continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Simplest properties of continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Arithmetic properties of continuous functions . . . . . . . . . . . . . . . . . . . . . . . . 95 Superposition of continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 13. Continuity of a function on a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Intermediate value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101 Weierstrass theorems on the properties of functions continuous on a segment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Uniform continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 14. Points of discontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Points of discontinuity of a function, their classification and examples 112 Discontinuity points for monotone functions . . . . . . . . . . . . . . . . . . . . . . . . 114 Criterion for the continuity of a monotone function . . . . . . . . . . . . . . . . . 117 Inverse function theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 15. O-notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Functions which are infinitesimal in comparison with other functions 121 Functions which are bounded in comparison with other functions . . . 123 Some properties related to O-notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Equivalent functions at a point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 16. Differentiable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Preliminary remarks and basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Continuity of a differentiable function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Differential of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .131 Derivatives of some elementary functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 17. Properties of differentiable functions . . . . . . . . . . . . . . . . . . . . . . . . .135 Arithmetic properties of derivatives and differentials . . . . . . . . . . . . . . . . 135 Differentiation of superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Differentiation of inverse function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .142 18. Hyperbolic and inverse hyperbolic functions . . . . . . . . . . . . . . . . 146 Hyperbolic functions and their properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Inverse hyperbolic functions and their properties . . . . . . . . . . . . . . . . . . . .147 19. Physical sense and geometric sense of the derivative . . . . . . . 151 Physical sense of the derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Geometric sense of the derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

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6 M. E.Abramyan. Lectures on differential calculus 20. Higher-order derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Higher-order derivatives: definition and examples . . . . . . . . . . . . . . . . . . . 155 Higher-order derivatives for sum and product of functions . . . . . . . . . . 157 Number of combinations: definition and properties . . . . . . . . . . . . . . . . . 158 The Leibniz rule for the differentiation of a product . . . . . . . . . . . . . . . . 159 21. The basic theorems of differential calculus . . . . . . . . . . . . . . . . . . 163 Local extrema of functions. Fermat’s theorem . . . . . . . . . . . . . . . . . . . . . . 163 Rolle’s theorem, Lagrange’s theorem, and Cauchy’s mean value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .165 22. Taylor’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 Taylor’s formula for polynomials and for arbitrary differentiable functions . . . . . . . . . . . . . . . . . . . . . . . . 172 Various representations of the remainder term in Taylor’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Expansions of elementary functions by Taylor’s formula in a neighborhood of zero . . . . . . . . . . . . . . . . . . . 180 23. L’Hospital’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 Formulation and proof of L’Hospital’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . 186 Examples of applying L’Hospital’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 Supplement. An example of a differentiable function whose derivative is not continuous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .191 24. Application of differential calculus to the study of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Local extrema of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Convex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Inflection points of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Location of the graph of a function relative to a tangent line . . . . . . . .205 Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 Example of a function study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .210 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

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