The course is a studio video recording of the first half of the first semester of lectures on mathematical analysis in the form in which they are given at the Academic University. Over 4 modules, students will become familiar with the basic concepts of mathematical analysis: sequences, limits and continuity. We will limit ourselves only to real numbers and functions of one variable. The presentation will be conducted at a fairly elementary level without possible generalizations that do not change the main ideas of the evidence, but significantly complicate perception. All statements (except for some boring formal justifications at the very beginning of the course and in the definition of elementary functions) will be strictly proven. The video recordings are accompanied by a large number of tasks for students to work independently.
Who is this course for
Junior students of technical specialties
Students must have a good command of the school mathematics curriculum. Namely, you need to know what the graphs of basic elementary functions look like, know the basic formulas for trigonometric, exponential and logarithmic functions, for arithmetic and geometric progressions, and also confidently be able to do algebraic transformations with equalities and inequalities. For several problems you also need to know the simplest properties of rational and irrational numbers.
The course is aimed at bachelors and masters specializing in mathematical, economic or natural science disciplines, as well as secondary school mathematics teachers and university professors. It will also be useful for schoolchildren who study mathematics in depth.
The course structure is traditional. The course covers classical material on mathematical analysis, studied in the first year of university in the first semester. Sections “Elements of set theory and real numbers”, “Theory of number sequences”, “Limit and continuity of a function”, “Differentiability of a function”, “Applications of differentiability” will be presented. We will get acquainted with the concept of a set, give a strict definition of a real number and study the properties of real numbers. Then we'll talk about number sequences and their properties. This will allow us to consider the concept of a numerical function, well known to schoolchildren, at a new, more rigorous level. We will introduce the concept of limit and continuity of a function, discuss the properties of continuous functions and their application to solve problems.
In the second part of the course, we will define the derivative and differentiability of a function of one variable and study the properties of differentiable functions. This will allow you to learn how to solve such important applied problems as approximate calculation of function values and solving equations, calculating limits, studying the properties of a function and constructing its graph.
Format
The form of study is correspondence (distance).
Weekly classes will include watching thematic video lectures and completing test tasks with automated verification of results.
An important element of studying the discipline is the independent solution of computational problems and proof problems. The solution will have to contain rigorous and logically correct reasoning that leads to the correct answer (in the case of a computational problem) or completely proves the required statement (for theoretical problems).
Requirements
The course is designed for 1st year bachelors. Knowledge of elementary mathematics at the level of high school (grade 11) is required.
Course program
Lecture 1. Elements of set theory.
Lecture 2. The concept of a real number. Exact faces of numerical sets.
Lecture 3. Arithmetic operations on real numbers. Properties of real numbers.
Lecture 4. Number sequences and their properties.
Lecture 5. Monotonous sequences. Cauchy criterion for sequence convergence.
Lecture 6. The concept of a function of one variable. Function limit. Infinitely small and infinitely large functions.
Lecture 7. Continuity of function. Classification of break points. Local and global properties of continuous functions.
Lecture 8. Monotonous functions. Inverse function.
Lecture 9. The simplest elementary functions and their properties: exponential, logarithmic and power functions.
Lecture 10. Trigonometric and inverse trigonometric functions. Remarkable limits. Uniform continuity of function.
Lecture 11. The concept of derivative and differential. Geometric meaning of derivative. Rules of differentiation.
Lecture 12. Derivatives of basic elementary functions. Function differential.
Lecture 13. Derivatives and differentials of higher orders. Leibniz's formula. Derivatives of parametrically defined functions.
Lecture 14. Basic properties of differentiable functions. Rolle's and Lagrange's theorems.
Lecture 15. Cauchy's theorem. L'Hopital's first rule of disclosing uncertainty.
Lecture 16. L'Hopital's second rule for disclosing uncertainties. Taylor's formula with a remainder term in Peano form.
Lecture 17. Taylor's formula with a remainder term in general form, in Lagrange and Cauchy form. Expansion according to the Maclaurin formula of the main elementary functions. Applications of Taylor's formula.
Lecture 18. Sufficient conditions for an extremum. Asymptotes of the graph of a function. Convex.
Lecture 19. Inflection points. General scheme of function research. Examples of plotting graphs.
Learning outcomes
As a result of mastering the course, the student will gain an understanding of the basic concepts of mathematical analysis: set, number, sequence and function, become familiar with their properties and learn to apply these properties when solving problems.
Questions for the exam in “Mathematical Analysis”, 1st year, 1st semester.
1. Multitudes. Basic operations on sets. Metric and arithmetic spaces.
2. Numerical sets. Sets on the number line: segments, intervals, semi-axes, neighborhoods.
3. Definition of a bounded set. Upper and lower bounds of number sets. Postulates about the upper and lower bounds of numerical sets.
4. Method of mathematical induction. Bernoulli and Cauchy inequalities.
5. Definition of a function. Function graph. Even and odd functions. Periodic functions. Methods for specifying a function.
6. Consistency limit. Properties of convergent sequences.
7. Limited sequences. Theorem on a sufficient condition for the divergence of a sequence.
8. Definition of a monotonic sequence. Weierstrass's theorem on a monotone sequence.
9. Number e.
10. Limit of a function at a point. Limit of a function at infinity. One-sided limits.
11. Infinitesimal functions. Limit of sum, product and quotient of functions.
12. Theorems on the stability of inequalities. Passage to the limit in inequalities. Theorem about three functions.
13. The first and second are wonderful limits.
14. Infinitely large functions and their connection with infinitesimal functions.
15. Comparison of infinitesimal functions. Properties of equivalent infinitesimals. Theorem on replacing infinitesimals with equivalent ones. Basic equivalences.
16. Continuity of a function at a point. Actions with continuous functions. Continuity of basic elementary functions.
17. Classification of function discontinuity points. Definition by continuity
18. Definition of a complex function. Limit of a complex function. Continuity of a complex function. Hyperbolic functions
19. Continuity of a function on a segment. Cauchy's theorems on the vanishing of a continuous function on an interval and on the intermediate value of the function.
20. Properties of functions continuous on an interval. Weierstrass's theorem on the boundedness of a continuous function. Weierstrass's theorem on the largest and smallest values of a function.
21. Definition of a monotonic function. Weierstrass's theorem on the limit of a monotone function. Theorem on the set of values of a function that is monotonic and continuous on an interval.
22. Inverse function. Graph of the inverse function. Theorem on the existence and continuity of the inverse function.
23. Inverse trigonometric and hyperbolic functions.
24. Determination of the derivative of a function. Derivatives of basic elementary functions.
25. Definition of a differentiable function. Necessary and sufficient condition for differentiability of a function. Continuity of a differentiable function.
26. Geometric meaning of derivative. Equation of tangent and normal to the graph of a function.
27. Derivative of the sum, product and quotient of two functions
28. Derivative of a complex function and its inverse function.
29. Logarithmic differentiation. Derivative of a function given parametrically.
30. The main part of the function increment. Function linearization formula. Geometric meaning of differential.
31. Differential of a complex function. Invariance of the shape of the differential.
32. Theorems of Rolle, Lagrange and Cauchy on the properties of differentiable functions. Finite increment formula.
33. Application of derivative to the disclosure of uncertainties within limits. L'Hopital's rule.
34. Definition of derivative nth order. Rules for finding the nth order derivative. Leibniz's formula. Differentials of higher orders.
35. Taylor's formula with a remainder term in Peano form. Residue terms in Lagrange and Cauchy forms.
36. Increasing and decreasing functions. Extremum points.
37. Convexity and concavity of function. Inflection points.
38. Endless function breaks. Asymptotes.
39. Scheme for constructing a graph of a function.
40. Definition of antiderivative. Basic properties of the antiderivative. The simplest rules of integration. Table of simple integrals.
41. Integration by change of variable and formula for integration by parts in the indefinite integral.
42. Integrating expressions of the form e ax cos bx and e ax sin bx using recurrence relations.
43. Fraction Integration |
using recurrence relations. |
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a 2 n |
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44. Indefinite integral of a rational function. Integration of simple fractions.
45. Indefinite integral of a rational function. Decomposition of proper fractions into simple ones.
46. Indefinite integral of an irrational function. Integrating Expressions
R x, m |
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47. Indefinite integral of an irrational function. Integration of expressions of the form R x , ax 2 bx c . Euler's substitutions.
48. Integrating expressions of the form |
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ax2 bx c |
ax2 bx c |
2 bx c |
49. Indefinite integral of an irrational function. Integration of binomial differentials.
50. Integrating trigonometric expressions. Universal trigonometric substitution.
51. Integration of rational trigonometric expressions in the case when the integrand is odd with respect to sin x (or cos x) or even with respect to sin x and cos x.
52. Integrating Expressions sin n x cos m x and sin nx cos mx .
53. Integrating Expressions tg m x and ctg m x .
54. Integrating Expressions R x , x 2 a 2 , R x , a 2 x 2 and R x , x 2 a 2 using trigonometric substitutions.
55. Definite integral. The problem of calculating the area of a curved trapezoid.
56. Integral sums. Darboux sums. Theorem on the condition for the existence of a definite integral. Classes of integrable functions.
57. Properties of a definite integral. Mean value theorems.
58. Definite integral as a function of the upper limit. Formula Newton-Leibniz.
59. Formula for changing a variable and formula for integrating by parts in a definite integral.
60. Application of integral calculus to geometry. Volume of the figure. Volume of rotation figures.
61. Application of integral calculus to geometry. Area of a flat figure. Area of a curved sector. Curve length.
62. Definition of an improper integral of the first kind. Formula Newton-Leibniz for improper integrals of the first kind. The simplest properties.
63. Convergence of improper integrals of the first kind for a positive function. 1st and 2nd comparison theorems.
64. Absolute and conditional convergence of improper integrals of the first kind from an alternating function. Tests for Abel and Dirichlet convergence.
65. Definition of an improper integral of the second kind. Formula Newton-Leibniz for improper integrals of the second kind.
66. Connection of improper integrals 1st and 2nd kind. Improper integrals in the sense of principal value.