Theory of functions of one variable. Mathematical analysis. Theory of functions of one variable Mathematical analysis lectures 1 course 1 semester online

Questions for the exam in "Mathematical Analysis", 1st year, 1st semester.

1. Sets. Basic operations on sets. Metric and arithmetic spaces.

2. Numeric sets. Sets on the number line: segments, intervals, semiaxes, neighborhoods.

3. Definition of a bounded set. Upper and lower bounds of numerical sets. Postulates about upper and lower bounds of numerical sets.

4. Method of mathematical induction. Bernoulli and Cauchy inequalities.

5. Function definition. Function graph. Even and odd functions. Periodic functions. Ways to set a function.

6. Sequence limit. Properties of convergent sequences.

7. limited sequences. A theorem on a sufficient condition for the divergence of a sequence.

8. Definition of a monotonic sequence. Weierstrass' monotone sequence theorem.

9. Number e.

10. Limit of a function at a point. The limit of a function at infinity. Unilateral limits.

11. Infinitely small functions. Limit of sum, product and quotient functions.

12. Theorems on the stability of inequalities. Passage to the limit in inequalities. Theorem about three functions.

13. The first and second wonderful limits.

14. Endlessly great features and their relation to infinitesimal functions.

15. Comparison of infinitesimal functions. Properties of equivalent infinitesimals. The theorem on the replacement of infinitesimals by equivalent ones. Basic equivalences.

16. Continuity of a function at a point. Actions with continuous functions. Continuity of basic elementary functions.

17. Classification of breakpoints of a function. Extension 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 function continuous on an interval and on the intermediate value of a function.

20. Properties of functions continuous on a segment. The Weierstrass theorem on the boundedness of a continuous function. Weierstrass' theorem on the largest and smallest value of a function.

21. Definition of a monotonic function. Weierstrass' theorem on the limit of a monotone function. Theorem on the set of values ​​of a function that is monotone and continuous on an interval.

22. Inverse function. Schedule inverse function. Theorem on the existence and continuity of the inverse function.

23. Inverse trigonometric and hyperbolic functions.

24. Definition of the derivative of a function. Derivatives of basic elementary functions.

25. Definition of a differentiable function. A necessary and sufficient condition for the differentiability of a function. Continuity of a differentiable function.

26. The geometric meaning of the derivative. The equation of the tangent and normal to the graph of the function.

27. Derivative of the sum, product and quotient of two functions

28. Derivative of a compound function and an inverse function.

29. Logarithmic differentiation. Derivative of a function given parametrically.

30. The main part of the function increment. Function linearization formula. The geometric meaning of the differential.

31. Differential of a compound function. Invariance of the differential form.

32. Rolle's, Lagrange's and Cauchy's theorems on the properties of differentiable functions. Formula of finite increments.

33. Application of the derivative to the disclosure of uncertainties within. L'Hopital's rule.

34. Derivative definition nth order. Rules for finding the derivative of the nth order. Leibniz formula. Higher order differentials.

35. Taylor formula with remainder term in Peano form. Residual terms in the form of Lagrange and Cauchy.

36. Increasing and decreasing functions. extremum points.

37. Convexity and concavity of a function. Inflection points.

38. Endless function breaks. Asymptotes.

39. Scheme for plotting a function graph.

40. Definition of antiderivative. The main properties of the antiderivative. The simplest integration rules. Table of simple integrals.

41. Integration by change of variable and the formula for integration by parts in the indefinite integral.

42. Integration of expressions of the form e ax cos bx and e ax sin bx using recursive relations.

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. Expression Integration

47. Indefinite integral of an irrational function. Integration of expressions of the form R x , ax 2 bx c . Euler substitutions.

49. Indefinite integral of an irrational function. Integration of binomial differentials.

50. Integration of 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. Expression Integration sin n x cos m x and sin n x cos mx .

53. Expression Integration tg m x and ctg m x .

54. Expression Integration 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 curvilinear 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. Theorems on the mean value.

58. Definite integral as a function of the upper limit. Formula Newton-Leibniz.

59. Change of variable formula and formula for integration by parts in a definite integral.

60. Application of integral calculus to geometry. The volume of the figure. The volume of figures of rotation.

61. Application of integral calculus to geometry. The area of ​​a plane figure. The area of ​​the curvilinear 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 of an alternating function. Convergence criteria for Abel and Dirichlet.

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.

Let the variable x n takes an infinite sequence of values

x 1 , x 2 , ..., x n , ..., (1)

and the law of change of the variable is known x n, i.e. for every natural number n you can specify the corresponding value x n. Thus it is assumed that the variable x n is a function of n:

x n = f(n)

Let us define one of the most important concepts of mathematical analysis - the limit of a sequence, or, what is the same, the limit of a variable x n running sequence x 1 , x 2 , ..., x n , ... . .

Definition. constant number a called sequence limit x 1 , x 2 , ..., x n , ... . or the limit of a variable x n, if for an arbitrarily small positive number e there exists such a natural number N(i.e. number N) that all values ​​of the variable x n, beginning with x N, differ from a less in absolute value than e. This definition briefly written like this:

| x n -a |< (2)

for all n  N, or, which is the same,

Definition of the Cauchy limit. A number A is called the limit of a function f (x) at a point a if this function is defined in some neighborhood of the point a, except perhaps for the point a itself, and for each ε > 0 there exists δ > 0 such that for all x satisfying condition |x – a|< δ, x ≠ a, выполняется неравенство |f (x) – A| < ε.

Definition of the Heine limit. A number A is called the limit of a function f (x) at a point a if this function is defined in some neighborhood of the point a, except perhaps for the point a itself, and for any sequence such that converging to the number a, the corresponding sequence of values ​​of the function converges to the number A.

If the function f(x) has a limit at the point a, then this limit is unique.

The number A 1 is called the left limit of the function f (x) at the point a if for each ε > 0 there exists δ >

The number A 2 is called the right limit of the function f (x) at the point a if for each ε > 0 there exists δ > 0 such that the inequality

The limit on the left is denoted as the limit on the right - These limits characterize the behavior of the function to the left and right of the point a. They are often referred to as one-way limits. In the notation of one-sided limits as x → 0, the first zero is usually omitted: and . So, for the function

If for each ε > 0 there exists a δ-neighborhood of a point a such that for all x satisfying the condition |x – a|< δ, x ≠ a, выполняется неравенство |f (x)| >ε, then we say that the function f (x) has an infinite limit at the point a:

Thus, the function has an infinite limit at the point x = 0. Limits equal to +∞ and –∞ are often distinguished. So,

If for each ε > 0 there exists δ > 0 such that for any x > δ the inequality |f (x) – A|< ε, то говорят, что предел функции f (x) при x, стремящемся к плюс бесконечности, равен A:

Existence theorem for the least upper bound

Definition: AR mR, m - upper (lower) face of A, if аА аm (аm).

Definition: The set A is bounded from above (from below), if there exists m such that аА, then аm (аm) is satisfied.

Definition: SupA=m, if 1) m - upper bound of A

2) m’: m’ m' is not an upper face of A

InfA = n if 1) n is the infimum of A

2) n’: n’>n => n’ is not an infimum of A

Definition: SupA=m is a number such that: 1)  aA am

2) >0 a  A, such that a  a-

InfA = n is called a number such that:

2) >0 a  A, such that a E a+

Theorem: Any non-empty set АR bounded from above has a best upper bound, and a unique one at that.

Proof:

We construct a number m on the real line and prove that this is the least upper bound of A.

[m]=max([a]:aA) [[m],[m]+1]A=>[m]+1 - upper face of A

Segment [[m],[m]+1] - split into 10 parts

m 1 =max:aA)]

m 2 =max,m 1:aA)]

m to =max,m 1 ...m K-1:aA)]

[[m],m 1 ...m K , [m],m 1 ...m K + 1 /10 K ]A=>[m],m 1 ...m K + 1/ 10 K - top face A

Let us prove that m=[m],m 1 ...m K is the least upper bound and that it is unique:

to: .

Rice. 11. Graph of the function y arcsin x.

Let us now introduce the concept of a complex function ( display compositions). Let three sets D, E, M be given and let f: D→E, g: E→M. Obviously, it is possible to construct a new mapping h: D→M, called a composition of mappings f and g or a complex function (Fig. 12).

A complex function is denoted as follows: z =h(x)=g(f(x)) or h = f o g.

Rice. 12. Illustration for the concept of a complex function.

The function f (x) is called internal function, and the function g ( y ) - external function.

1. Internal function f (x) = x², external g (y) sin y. Complex function z= g(f(x))=sin(x²)

2. Now vice versa. Inner function f (x)= sinx, outer g (y) y 2 . u=f(g(x))=sin²(x)