Matemática

I. The content of science

     Fannie from the school target - students school , high school​ special education in institutions mathematics chair scientific the basics to teach​ and him/her effective to teach​ and current time mathematics to learn​ for enough mathematician knowledge , experience​​ and qualifications from forming consists of .

      Fannie of the lesson task – general middle​​ and middle​​ special education institutions in mathematics included mathematician to analysis relevant concepts scientific justification ; mathematics to analysis introduction , sequence and function limit , continuous functions and their properties to teach ; a​ о ' variable function differential and integral calculus and his/her applications to teach ; practical​ and practical importance had​​​​ rows theory with introduction , Taylor and Fourier rows functions in the study important that mathematical apparatus to recite ; two and three o ' dimensional integrals , curves linear from integrals next who does​ sciences for necessary in volume knowledge to give and their geometric and physicist sizes in measurement​ implementation to teach . logical​ reflection and scientific and literary the speech from development consists of .

2. Home part ( lecture training )​

II.I. Science content following topics includes :

Topic 1. Mathematics analysis about start​ information

Mathematician analysis science Subject : Historical data . Development of science Trends in Uzbekistan​​ mathematician analysis fan development . Mathematics analysis academic high school and general education in schools readable​​ mathematics course with relevance .

Topic 2. Real set of numbers​​

Rational set of numbers​​ and his/her properties , rational of the set of numbers t о ' section , concept of irrational number , real of the set of numbers t о ' main properties . Real the modulus of the number and his/her properties . From above and from below bounded sets , their​​​ Boundaries . Intervals .

Topic 3. Approaching sequence and his/her properties

Numerical sequence about concept . Sequences surrender methods . Bounded sequences , monotone sequences . Sequence limit Definition : Approaching of sequences properties . Unlimited small sequences and their properties . Approaching sequence limitedness , limit uniqueness . Infinite big sequences . Interval o ' variable limit about theorem . Sequences sum , summation​​​ and the division​​ limit . Uncertainties and them open .

Topic 4. Approach principle

Monotonous sequence limit , e number . Inside - inside is located segments principle . Partial sequence . Bolsano-Weierstrass Theorem . Sequence of approaching Koshi criterion .

Topic 5. Function and his/her important classes

Function definition of a function surrender methods . Function graph . Functions on arithmetic operations . Even , odd , bounded , monotone , periodic functions . Inverse function , functions composition .

Topic 6. Function limit

Function on point limit Heine and Koshi Definitions . To the limit had​​​​ of functions simple properties . One- sided limits . One- sided limits based on function limited to the limit to have​​​ condition . Two function sum , summation​​​ and the division​​ limit . Complex function limit . Monotone function limit . Cauchy criterion . Some one wonderful limits . Unlimited small functions and them comparison .

Topic 7. Continuous function and his/her properties

Function on point and in the set​ continuity . Gather , gather .​​ and the division​​ continuity . Functions composition continuity . One- sided continuity and interruption points . Monotone function continuity and interruption points . In the intersection continuous​​​​ of functions limited , most small and the most big values . Continuous of functions intermediate values about theorems . Monotone function continuity . Reverse function existence and continuity . Flat continuity concept . In the cross section continuous​​​​ function flat continuity . Continuous function properties equation and inequalities to solve applications .

Topic 8. Home elementary functions and their continuity

With real indicators​​ degree . K о ' exponential , logarithmic , power functions and their properties . Trigonometric functions . Inverse trigonometric functions and their properties .

Topic 9. Harvest concept , derivative calculation rules

Harvest to the concept take coming issues . Get it. definition , its geometric and mechanic Meanings . Curve line attempt and normal equations . Differentiable function continuity . Gather , gather .​​ and the division​​ derivative . Compound function derivative . Inverse function derivative . Logarithmic derivative . Degree k о ' exponent function derivative . Main elementary of functions derivatives .

Topic 10. Function differential

Differential sensitivity and derivative existence between connection . Differential , its​ geometric meaning . Differential uniform invariance of the differential approximate to calculate applications .

Topic 11. High orderly derivatives and differentials

High orderly derivatives . Second orderly derivative mechanic meaning . High orderly differentials . In parametric form​​ given functions differentiation .

Topic 12. Differential of the account main theorems

Roll, Lagrange , Cauchy theorems . Hospital Taylor 's rule formula . Some-some elementary functions for Taylor formulas . Taylor formula limits to calculate , approximate into account applications .

Topic 13. Derivatives applications

Function permanence condition . The function on point and in the set​ monotony condition . Maximum and minima . Extremum necessary condition . Extremum enough conditions . The largest and the most small values search . Function convexity , curvature Point . Asymptotes . Derivative function graph to make Application . Derived equation and inequalities to solve , inequality and the facts to prove applications .

Topic 14. Indefinite integral.

Start drinking function and indefinite integral. Main integrals table .

Topic 15. Uncertain integral calculation methods .

 Uncertain in the integral o ' variable replacement method . B о ' lacquer integration .

Topic 16. Rational functions integration

Simple rational fractions and them integration . True​​ rational fractions integration .

Topic 17. Fraction rational functions integration .

Topic 18. Simple irrational and transcendent functions integration Simple irrational expressions integration . Binomial differentials integration . Euler replacements .

Topic 19. Trigonometric expressions integration .

Topic 20. Precise of the integral definition , its existence conditions

To the concept of definite integral take coming problems . Integral sum , definition of definite integral . The existence of definite integral​ necessary condition . Strike meetings​ and their properties of the definite integral .​​​ necessary and enough condition , sufficient condition . Integrable functions class ( Continuous function , monotone function , finite on the thigh to break had​​​​ functions ).

Topic 21. Precise of the integral properties and him/her calculation

Precise of the integral equality and inequality with expressible properties . Medium​​ value about theorems . Top border It was a variable .​​ definite integral. Newton-Leibnitz formula . The variable О ' replacement and then​​​ integration methods . Precise integrals approximate calculation​

Topic 22. Integration field unlimited improper integral

Integration field unlimited The concept of the proper integral . The proper integral of the integral properties . Absolute approaching integrals .

Topic 23. Khosmas integrals Counting . Unlimited function unusual integral . Unbounded function unusual integral . Unbounded function unusual integral properties . Unlimited function unusual integral calculation​

Topic 24. Precise of the integral geometric sizes to calculate and to physics implementation Surface calculation formulas . Polar coordinates in the system figure his face Calculate the volume of a spatial object . calculation . Curve line or year length calculation . Arc length differential . Rotation surface his face calculation​

Topic 25 . Precise of the integral to physics implementation .

Oh , the variable. of power completed work and him/her using definite integral calculation . Flat bow and figure weight​ centers coordinates , inertia moment calculation formulas .

Topic 26. Numerical rows .

Numerical row concept , approaching row and his/her sum of the series remainder . Geometric​ row . Row of approaching necessary condition . Harmonic row​

Topic 27. Approaching rows and their properties Approaching of rows simple properties .​ criterion .

Topic 28. Positive rows

Positive of rows approach condition . Positive row of approaching necessary and enough condition . Comparison theorems . Cauchy and D'Alembert signs . Cauchy's integral sign . Hint on duty rows .

Topic 29. Hints on duty and variable rows .

Leibniz theorem . Absolute and conditional approaching rows , their properties .

Topic 30. Functional sequences

Functional sequence concept , approaching sequence , its limit . Flat approaching functional sequence . Flat approach sign . Flat approaching functional sequence properties (of the limit function) continuity , it differentiation and integration ).

Topic 31. Functional rows .

 Functional rows and his/her sum , flat​ approaching rows , straight approach condition . Flat approaching of the row properties ( array of the assembly continuity , line hadma -had differentiation and integration ).

Topic 32. Level rows

Level row concept . Abel's theorem . Degree of rows approach radius , approach interval and field . Level of the row flat Approach . Topic 33. Taylor row

Functions level in line spread Taylor​​ The series . sinx , cosx , e ^x , ln(1+x) and (1+x) ^a  functions level in line spread . Level of rows approximate into account implementation .

Topic 34. Fourier row

Fourier coefficients and Fourier The function​ Fourier in a row spread Dirichlet 's problem theorem ( without proof ). Periodic , even and odd functions for Fourier row .

Topic 35. Multivariable​​​​ functions

The number of variables​​​ function about concept . R ⁿ of space parts sets m o ' variable​​​ function determination field as . Two о ' variable function Graph . Level lines and surfaces concepts .

Topic 36. Multivariable​​​​ function limit

R ^m in space of the point area . R ⁿ in space points sequence and his/her limit , m o ' variable function limit . Repeat limits .

Topic 37. Multivariable​​​​ continuous functions

Continuity Definitions of multivariable​​​​​ continuous function properties . Complex function continuity . Multivariable​​​​​ function intermediate values about theorems . Weierstrass Theorems . Plain continuity and Cantor's theorem .

Topic 38. Multivariable​​​​ functions differentiation

Private derivatives . Multivariable​​​​​ differentiable functionY . Differentiable division​​​ necessary , sufficient conditions . Multivariable​​​​​ the total differential of a function . plain . Two о ' variable function differential geometric meaning . Complex function differentiation . Differential uniform invariance of the differential approximate to calculations applications .

Topic 39. Multivariable​​​​ functions high orderly private derivatives and differential

High​ orderly private derivatives . High orderly differentials . Two о ' variable function for Taylor formula .

Topic 40. Unrevealed functions differentiation

Single and multivariable​​​​​​  indiscreet functions . Unrevealed function existence and differentiability .

Topic 41. Multivariable​​​​ of functions extrema

Multivariable​​​​​​  function extremums . Extremum necessary condition . Two о ' variable function for extremum enough condition . The largest and the most small values search . Conditional extrema . By direction​​​​​ derivative . Gradient .

Topic 42. Two o ' dimensional integrals

Two The concept of a two - dimensional integral . o ' dimensional of the integral properties . Continuous of functions integrability . Repetition integrals . Two o ' dimensional integral count . Two o ' dimensional in the integral о ' variable Substitution . In polar coordinates two ω ' dimensional integral. Two o ' dimensional of the integral applications .

Topic 43. Three o ' dimensional integrals

Three The concept of a dimensional integral . Three o ' dimensional of the integral properties . Three o ' dimensional integral count . Three o ' dimensional in the integral o ' variables replacement .

Topic 44. Cylindrical and spherical in coordinates three ω - dimensional integral. Three measurable integral cylindrical and spherical coordinates to the system passing by calculation​

Topic 45. Sagittarius length by​​​ taken curve linear integrals

Bow length by​​​ taken curve linear to the integral take coming issues . Bow length by​​​ taken curve linear integral, its properties . Bow length by​​​ taken curve linear integral calculation . Arc length by​​​ taken curve Applications of linear integrals .

3. Practical Exercise instructions​​​​​​​​ and recommendations

Practical training​ for following topics recommendation is being done :

1. Real set of numbers​​ and his/her Properties of finite sets​​​ borders

2. Sequences surrender methods , limited , monotone sequences .

3. Approaching sequence , properties . Sequence limit calculation

4. Approach principle

5. Function , definition domain , set of values , graph​​

6. Function limit , it calculation , amazing limits , uncertainties open

7. Continuous function and his/her properties of the function interruption points , their types .

8. Home elementary functions and their continuity

9. Harvest concept , one sided derivatives . Derivative calculation rules . Home derivatives schedule

10. Function differential , its approximate into account implementation

11. High orderly derivatives and differentials

12. Indefinite integral and him/her to find simple methods

13. Rational functions integration

14. Simple irrational and transcendent functions integration

15. Precise of the integral properties and him/her calculation

16. Limit of integration indefinite integral, it Counting . Unlimited function unusual integral , it calculation

17. Precise of the integral geometric sizes to calculate and to physics implementation

18. Approaching rows and their properties

19. Positive rows , approach symptoms . Voluntary​​ situation rows , conditional and absolute approaching rows

20. Functional sequence , determination , convergence domains , limit function , properties . Functional of the row approach field

21. Level row , approach radius , area

22. Taylor as a string , functions Taylor in a row spread . Function Fourier in a row spread

23. Multivariable​​​​​  function determination area , two о ' variable function graph

24. Multivariable​​​​​  function limit

25. Multivariable​​​​​  functions private derivatives . Multivariable​​​​​ the final differential of a function

26. Unrevealed functions differentiation

27. Multivariable​​​​​  of functions extrema

28. Two o ' dimensional integrals , calculus , variables​​ replacement , pole coordinates in the system calculation , applications

29. Three o ' dimensional integrals , calculus , variables​​ replacement , cylindrical , spherical coordinates in the system calculation , applications

30. Sagittarius length by​​​ taken curve linear integrals , calculus , applications

4. Independent education and independent affairs

Independent of education content students by

· lecture and practical to practice preparation ;

· house tasks to do ;

· theoretical knowledge to master ;​

· differentiated alone in order assignments to do ;

· independent education intended for​​​ topics from mastering​ consists of .

Independent education for recommendation attainable topics :

1. Real the modulus of the number and his/her properties .

2. Sequence limit by definition​​​ calculation​

3. Unlimited big sequences , their endless small sequences with connection .

4. Sequences circle length and circle his face to calculate applications .

5. Uncertainties and them to open circle examples solution​

6. Various functions classes between relationships to learn .​

7. Unlimited small functions and them comparison .

8. Equivalent endless from the small ones function limit in finding and function graph in drawing use .

9. Unlimited big functions .

10. Functions sum , summation​​​ and the division​​ continuity evidence .

11. Functions composition continuity proof .

12. Monotonous function continuity proof .

13. Continuous function properties equation and inequalities to solve applications .

14. Two line between corner , it calculation​

15. One- sided derivatives , one sided attempts .

16. Home elementary n- order functions derivatives formula brought release​

17. Two n- order function of the k - order interval derivative​

18. Parametric equation with given line attempt and normal equations .

19. Taylor formula approximate to calculate applications .

20. High orderly derivative with the help of functions to the extreme  check .

21. In parametric view​​ given full functionality check .​​

22. Harvest equation and inequalities to solve , inequality and the facts to prove applications .

23. Euler replacements .

24. Binomial differential integration .

25. Limited on the thigh first round of interruption has of functions integrable that is .

26. Precise in the integral b о ' laklab integration .

27. Hosmas integrals to approach check .

28. To the parameter related​ integrals about concept​

29.Definite integral natural in the sciences , economics , etc. applications , to them circle examples solution​

30. Generalized harmonic row​

31. Multiply rows .​​​

32.Smoothly approaching functional of rows properties evidence

33. Level of the row flat approach proof .

34. Flat approaching level row of the assembly continuity proof .

35. Level the line hadma -had differentiation and integration .

36. [- l; l ] and in the intervals [0; l] given functions Fourier in a row spread​

37. Multivariable​​​​​  continuous function local properties .

38.K o ' p o ' variable function intermediate values about theorems evidence .

39. Multivariable​​​​​  indiscreet function existence and differentiability .

40. Three о ' variable function to the extreme check .

41. Two o ' dimensional of the integral to physics applications .

42. Three multiple of the integral properties evidence .

43. Sagittarius length by​​​ curve linear of the integral properties .

44. By coordinate b​​ curve linear of the integral properties .

V. Science education results ( formed) competencies )

Fannie to master​ as a result student :

– real of the set of numbers t о ' main properties ; sequence and function limit of a function continuity , derivative and differential and them calculation methods ; unclear and definite integral, them calculation methods ; numerical rows , functional sequence and series , degree , Taylor , Fourier rows , k о 'p о ' variables function limit and continuity , private derivatives , differentials , extrema ; two​​ and three o ' dimensional integrals , curves linear integrals about imagination and to knowledge to have ;​​​

– real of the set of numbers t о ' clear lower and clear high borders to find ; any event or the process descriptive function analytical expression and him/her check ; sequence limit calculation ; function limit calculation ; function continuity or to break showing that he / she has​​​ get ; function product and differential calculate them​ related science issues to solve implementation to make ; to make with the help of to fully verify the function and graph  drawing ; unclear integrals find the definite integral and him/her geometric and physicist sizes to calculate implementation do to take ; finite and functional rows to approach check ; function Taylor and Fourier to the rows spread ; from rows approximate in calculation use ; k о 'p о ' variable function limit calculation , continuity justification ; multivariable​​​​ function product and differential count , two , three multiple and curve linear integrals calculation ; application to do , mathematics analysis main concepts definitions , theorems and their proof analysis to do skills​​ to have ;​​​

– function properties related example and issues solution ; middle​​ special mathematics in education occurring of functions continuity basically to take ; their product and differential calculation ; unclear and clear integrals calculation ; rows , k о 'p о ' variables function differential and integral calculus related main examples solution , simple mathematician models to compose and him/her analysis to do to the qualification to have​​​ need .

VI. Education technologies and methods :

• Lectures ;
• Practical training ;
• Interactive case studies ;
• Presentations ;
• In groups work ;
• Team with work and protection to do ;

VII. Credit to take for requirements :

To science related concept definitions , theorems complete mastery of the content , problem solutions , theorems​​ correct proofs​​​ reflection bring to get , to study theory about independent under observation ongoing , current , interim control in the forms given task and assignments completion , final under control​​​ writing work or verbal submission .

Home literature

1. Adams, Robert A. (Robert Alexander), Calculus: a complete course. Textbooks. Christopher Essex. - 7th ed. Copyright @ 2010, 2006, 2003 Pearson Education Canada, a division of Pearson Canada Inc., Toronto, Ontario.- 1077 p. 2. Claudia Canuto , Anita Tabacco Mathematical analysis. I. Springer-Verlag. Italy, Milan. 2008.-435p.

3. Claudia Canuto , Anita Tabacco Mathematical analysis. II. Springer-Verlag. Italy, Milan. 2010.-534 p.

4. G. Khudaiberganov , AK Vorisov , HT Mansurov, BA Shoimkulov . Mathematics​ from analysis lectures . I. T.,« Voris-nashriyot ». 2010, -37 6 p.

5. G. Khudayberganov , AK Vorisov , HT Mansurov, BA Shoimkulov . Mathematics​ from analysis lectures . II. T.,« Voris-nashriyot ». 2010, -35 2 p.

Additional  literature

1. Turgunbaev R.M. Mathematical analysis part I. T.: ―Innovation- zi yo‖ . 2019- 340 p.

2. Turgunbaev R., Kadirov K., Bakirov T. Mathematical analysis ( Rows theory ).T. : ―Innovation ‖ . 2019-156 p.

3. Turgunbaev R., Kadirov K., Bakirov T. Mathematical analysis ( Much variable differential and integral calculus of a function ). 2020. Ferghana . ―Polygraph Super Service .

4 . Azlarov . T., Mansurov. Kh., Mathematical analysis. T.: " Uzbekistan ". 1 t: 1994 y.-416 p.

5. Azlarov . T., Mansurov. Kh., Mathematical analysis. T.: " Uzbekistan ". 2 t. 1995 - 436 p.

6. Gaziev A., Israilov I., Yakhabaev M. ―Mathematical analysis example and issues ‖ T.: ―New century " generation " 2006

7. Tashmetov O', Turgunbaev R. From mathematical analysis example and issues collection . 1st q. TDPU. 2006 - 140 p.

8. Tashmetov O', Turgunbaev R. From mathematical analysis example and issues collection , 2nd q. TDPU. 2010 - 48 p.

9. Toshmetov O'., Turgunbaev R., Saidamatov E., Madirimov M. Mathematical analysis part I.​ T.: ― Extremum -Press‖, 2015. -408 p.

Information sources

1. www.tdpu.uz

2. www.pedagog.uz

3. www.edu.uz

4. www.nadlib.uz ( A. Navoi named after  UzMK )