математика

ɋɈȾȿɊɀȺɇɂȿ

ȼȼȿȾȿɇɂȿ..................................................................................................................

6

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ɉɊɈȽɊȺɆɆȺ..............................................................................................................

7

1. ɇȿɈɉɊȿȾȿɅȿɇɇɕɃ ɂɇɌȿȽɊȺɅ.......................................................................

9

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1.1. ɉɨɧɹɬɢɟ ɧɟɨɩɪɟɞɟɥɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ.............................................................

9

1.2. Ɉɫɧɨɜɧɵɟ ɦɟɬɨɞɵ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ..............................................................

11

ɇ

1.2.1. ɇɟɩɨɫɪɟɞɫɬɜɟɧɧɨɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ ɮɭɧɤɰɢɣ ɢ ɦɟɬɨɞ ɩɨɞɧɟɫɟɧɢɹ ɩɨɞ

ɡɧɚɤ ɞɢɮɮɟɪɟɧɰɢɚɥɚ .......................................................................................

11

Ȼ

1.2.2. ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɡɚɦɟɧɨɣ ɩɟɪɟɦɟɧɧɨɣ (ɩɨɞɫɬɚɧɨɜɤɨɣ).....................

12

1.2.3. ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɩɪɢ ɩɨɦɨɳɢ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɯ ɩɨɞɫɬɚɧɨɜɨɤ...

13

1.2.4. ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɩɨ ɱɚɫɬɹɦ...................................................................

14

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1.2.5. ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɮɭɧɤɰɢɣ, ɫɨɞɟɪɠɚɳɢɯ ɤɜɚɞɪɚɬɧɵɣ ɬɪɟɯɱɥɟɧ ɜ

ɡɧɚɦɟɧɚɬɟɥɟ......................................................................................................

16

ɢ

1.2.6. ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɪɚɰɢɨɧɚɥɶɧɵɯ ɞɪɨɛɟɣ..............................................

16

1.2.7. ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɯ ɮɭɧɤɰɢɣ................................

20

1.2.8. ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɢɪɪɚɰɢɨɧɚɥɶɧɵɯ ɮɭɧɤɰɢɣ.......................................

23

ɨ

1.2.9. ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯɪ

ɛɢɧɨɦɨɜ..................................

23

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2. ɈɉɊȿȾȿɅȿɇɇɕɃ ɂɇɌȿȽɊȺɅ..........................................................................

26

2.1. Ɏɨɪɦɭɥɚɇɶɸɬɨɧɚ–Ʌɟɣɛɧɢɰɚ. Ɂɚɦɟɧɚɩɟɪɟɦɟɧɧɨɣɜɨɩɪɟɞɟɥɟɧɧɨɦ

ɢɧɬɟɝɪɚɥɟ. ɂɧɬɟɝɪɢɪɨɜɚɧɢɟɩɨɱɚɫɬɹɦ. ȼɵɱɢɫɥɟɧɢɟɩɥɨɳɚɞɟɣɩɥɨɫɤɢɯɮɢɝɭɪ..

26

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2.2. ȼɵɱɢɫɥɟɧɢɟ ɞɥɢɧɢɞɭɝ ɤɪɢɜɵɯ. ȼɵɱɢɫɥɟɧɢɟ ɨɛɴɟɦɨɜ................................

31

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2.3. ɇɟɫɨɛɫɬɜɟɧɧɵɟ ɢɧɬɟɝɪɚɥɵ ............................................................................

35

2.3.1.ɂɧɬɟɝɪɚɥɵ ɫ ɛɟɫɤɨɧɟɱɧɵɦɢ ɩɪɟɞɟɥɚɦɢ (ɧɟɫɨɛɫɬɜɟɧɧɵɟ ɢɧɬɟɝɪɚɥɵ

ɩ

ɩɟɪɜɨɝɨ ɪɨɞɚ) ...................................................................................................

35

2.3.2. ɂɧɬɟɝɪɚɥɵ ɨɬ ɧɟɨɝɪɚɧɢɱɟɧɧɵɯ ɮɭɧɤɰɢɣ (ɧɟɫɨɛɫɬɜɟɧɧɵɟ ɢɧɬɟɝɪɚɥɵ

ɜɬɨɪɨɝɨ ɪɨɞɚ) ...................................................................................................

36

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3.ɟɎɍɇɄɐɂɂ ɇȿɋɄɈɅɖɄɂɏ ɉȿɊȿɆȿɇɇɕɏ...................................................

37

3.1. ɉɨɧɹɬɢɟ ɮɭɧɤɰɢɢ ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɟɦɟɧɧɵɯ ................................................

37

3.2. ɉɪɟɞɟɥ ɢ ɧɟɩɪɟɪɵɜɧɨɫɬɶ ɮɭɧɤɰɢɢ ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɟɦɟɧɧɵɯ....................

37

3.3. Ⱦɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟ ɮɭɧɤɰɢɣ ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɟɦɟɧɧɵɯ...........................

38

3.3.1. ɑɚɫɬɧɨɟ ɢ ɩɨɥɧɨɟ ɩɪɢɪɚɳɟɧɢɹ ɮɭɧɤɰɢɢ............................................

38

3.3.2. ɑɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ...........................................................................

40

3.3.3. ɉɨɥɧɵɣ ɞɢɮɮɟɪɟɧɰɢɚɥ ɮɭɧɤɰɢɢ ........................................................

42

3.3.4. Ⱦɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟ ɫɥɨɠɧɵɯ ɢ ɧɟɹɜɧɵɯ ɮɭɧɤɰɢɣ

..........................

44

3.4. Ʉɚɫɚɬɟɥɶɧɚɹ ɩɥɨɫɤɨɫɬɶ ɢ ɧɨɪɦɚɥɶ ɤ ɩɨɜɟɪɯɧɨɫɬɢ......................................

46

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3.5. ɗɤɫɬɪɟɦɭɦ ɮɭɧɤɰɢɢ ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɟɦɟɧɧɵɯ............................................

47

3.6. ɇɚɢɛɨɥɶɲɟɟ ɢ ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɹ ɮɭɧɤɰɢɢ ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɟɦɟɧɧɵɯ ɜ

Ɍ

ɡɚɦɤɧɭɬɨɣ ɨɛɥɚɫɬɢ

.................................................................................................

48

4. ȾɂɎɎȿɊȿɇɐɂȺɅɖɇɕȿ ɍɊȺȼɇȿɇɂə ɉȿɊȼɈȽɈ ɉɈɊəȾɄȺ ...................

51

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4.1. Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɫ ɪɚɡɞɟɥɹɸɳɢɦɢɫɹ ɩɟɪɟɦɟɧɧɵɦɢ..........

52

4.2. Ɉɞɧɨɪɨɞɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ 1 ɩɨɪɹɞɤɚ.............................

53

Ȼ

4.3. Ʌɢɧɟɣɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ 1–ɝɨ ɩɨɪɹɞɤɚ...........................

55

4.4. ɍɪɚɜɧɟɧɢɹ Ȼɟɪɧɭɥɥɢ......................................................................................

57

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4.5. Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɜ ɩɨɥɧɵɯ ɞɢɮɮɟɪɟɧɰɢɚɥɚɯ......................

58

4.6. Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɜɵɫɲɢɯ ɩɨɪɹɞɤɨɜ. Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ

ɢ

ɭɪɚɜɧɟɧɢɹ, ɞɨɩɭɫɤɚɸɳɢɟ ɩɨɧɢɠɟɧɢɟ ɩɨɪɹɞɤɚ...................................................

60

5. ɅɂɇȿɃɇɕȿ ȾɂɎɎȿɊȿɇɐɂȺɅɖɇɕȿ ɍɊȺȼɇȿɇɂə ȼɕɋɒɂɏ ɉɈɊəȾɄɈȼ

ɪ

ɋ ɉɈɋɌɈəɇɇɕɆɂ ɄɈɗɎɎɂɐɂȿɇɌȺɆɂ ...................................................

62

5.1. Ʌɢɧɟɣɧɵɟ ɨɞɧɨɪɨɞɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ n–ɝɨ ɩɨɪɹɞɤɚ ɫ

ɩɨɫɬɨɹɧɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ...........................................................................

62

ɬ

5.2. Ʌɢɧɟɣɧɵɟ ɧɟɨɞɧɨɪɨɞɧɵɟɨɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɫ ɩɨɫɬɨɹɧɧɵɦɢ

ɢ

ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ....................................................................................................

64

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6. ɅɂɇȿɃɇɕȿ ɇȿɈȾɇɈɊɈȾɇɕȿ ȾɂɎɎȿɊȿɇɐɂȺɅɖɇɕȿ ɍɊȺȼɇȿɇɂə

ȼɕɋɒɂɏ ɉɈɊəȾɄɈȼ ɋ ɉɈɋɌɈəɇɇɕɆɂ ɄɈɗɎɎɂɐɂȿɇɌȺɆɂ ɂ

ɨ

ɋɉȿɐɂȺɅɖɇɈɃ ɉɊȺȼɈɃ ...............................................................ɑȺɋɌɖɘ

65

ɩ

7. ɋɂɋɌȿɆɕ ȾɂɎɎȿɊȿɇɐɂȺɅɖɇɕɏ ɍɊȺȼɇȿɇɂɃ. ɆȿɌɈȾ ɂɋɄɅɘɑȿɇɂə.

ɆȿɌɈȾ ɗɃɅȿɊȺ Ɋȿɒȿɇɂə ɅɂɇȿɃɇɕɏ ɋɂɋɌȿɆ ɋ ɉɈɋɌɈəɇɇɕɆɂ

ɟ

ɄɈɗɎɎɂɐɂȿɇɌȺɆɂ.........................................................................................

71

Ɋ

7.1 ɇɨɪɦɚɥɶɧɚɹ ɫɢɫɬɟɦɚ n–ɝɨ ɩɨɪɹɞɤɚ ɨɛɵɤɧɨɜɟɧɧɵɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ

ɭɪɚɜɧɟɧɢɣ...............................................................................................................

71

7.2. Ʌɢɧɟɣɧɚɹ ɨɞɧɨɪɨɞɧɚɹ ɫɢɫɬɟɦɚ n–ɝɨ ɩɨɪɹɞɤɚ ɫ ɩɨɫɬɨɹɧɧɵɦɢ

ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ....................................................................................................

74

7.3. Ɂɚɞɚɱɢɞɢɧɚɦɢɤɢ .., ɩɪɢɜɨɞɹɳɢɟɤɪɟɲɟɧɢɸɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯɭɪɚɜɧɟɧɢɣ

78

ȼɈɉɊɈɋɕ ȾɅə ɋȺɆɈɄɈɇɌɊɈɅə......................................................................

82

ɇɟɨɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ...................................................................................

82

Ɉɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ.......................................................................................

83

Ɏɭɧɤɰɢɢ ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɟɦɟɧɧɵɯ.......................................................................

84

Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɢ ɫɢɫɬɟɦɵ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ... 86

ɄɈɇɌɊɈɅɖɇȺə ɊȺȻɈɌȺ ʋ 2 ...............................................................................

88

Ɍ

ɅɂɌȿɊȺɌɍɊȺ...........................................................................................................

ɍ95

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Ȼ

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ɢ

ɪ

ɨ

ɬ

ɢ

ɡ

ɨ

ɩ

ɟ

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Ɍ

ɉɨɫɨɛɢɟ ɫɨɞɟɪɠɢɬ ɨɫɧɨɜɧɵɟ ɬɟɨɪɟɬɢɱɟɫɤɢɟ ɫɜɟɞɟɧɢɹ ɢɡ ɩɪɨɝɪɚɦɦɧɨɝɨ

ɦɚɬɟɪɢɚɥɚ, ɬɢɩɨɜɵɟ ɩɪɢɦɟɪɵ ɢ ɤɨɧɬɪɨɥɶɧɵɟ ɡɚɞɚɧɢɹ ɩɨ ɬɟɦɚɦ ɤɭɪɫɚ ɍɜɵɫɲɟɣ

ɦɚɬɟɦɚɬɢɤɢ (20 ɜɚɪɢɚɧɬɨɜ).

ɋɬɭɞɟɧɬ ɞɨɥɠɟɧ ɢɡɭɱɢɬɶ ɬɟɨɪɟɬɢɱɟɫɤɢɣ ɦɚɬɟɪɢɚɥ, ɪɚɡɨɛɪɚɬɶ ɩɪɢɜɟɞɟɧɧɵɟ

Ȼ

ɨɛɪɚɡɰɵ ɪɟɲɟɧɢɹ ɬɢɩɨɜɵɯ ɩɪɢɦɟɪɨɜ ɢ ɡɚɞɚɱ, ɜɵɩɨɥɧɢɬɶ

ɪɟɲɟɧɢɟɇɡɚɞɚɱ ɫɜɨɟɝɨ

ɜɚɪɢɚɧɬɚ, ɧɨɦɟɪ

ɤɨɬɨɪɨɝɨ ɫɨɜɩɚɞɚɟɬ ɫ ɞɜɭɦɹ ɩɨɫɥɟɞɧɢɦɢ ɰɢɮɪɚɦɢ ɡɚɱɟɬɧɨɣ

ɤɧɢɠɤɢ (ɲɢɮɪɚ).

ɣ

ȿɫɥɢ ɧɨɦɟɪ ɲɢɮɪɚ ɛɨɥɶɲɟ ɞɜɚɞɰɚɬɢ, ɬɨ ɫɥɟɞɭɟɬ ɨɬɧɹɬɶ ɨɬ

ɢ

ɧɨɦɟɪɚ ɲɢɮɪɚ ɱɢɫɥɨ, ɤɪɚɬɧɨɟ 20, ɢ ɩɨɥɭɱɟɧɧɚɹ ɪɚɡɧɨɫɬɶ (ɞɜɟ ɩɨɫɥɟɞɧɢɟ ɰɢɮɪɵ)

ɛɭɞɟɬ ɧɨɦɟɪɨɦ ɜɚɪɢɚɧɬɚ.

ɪ

ɇɚɩɪɢɦɟɪ:

ɨ

ɇɨɦɟɪ ɡɚɱɟɬɧɨɣ ɤɧɢɠɤɢ

ɇɨɦɟɪɚ ɡɚɞɚɱ

ɇɨɦɟɪ ɜɚɪɢɚɧɬɚ

301789/148

ɬ

8

8, 28, 48 ɢ ɬ. ɞ.

303700/194

ɢ

14

14, 34, 54 ɢ ɬ. ɞ.

300120/100

20

20, 40, 80 ɢ ɬ. ɞ.

ɡ

ɨ

ɩ

ɟ

Ɋ

ɉɊɈȽɊȺɆɆȺ

Ɍɟɦɚ 1. ɇɟɨɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ

ɍ

ɉɟɪɜɨɨɛɪɚɡɧɚɹ. ɇɟɨɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ ɢ ɟɝɨ ɫɜɨɣɫɬɜɚ. Ɍɚɛɥɢɰɚ

ɨɫɧɨɜɧɵɯ ɢɧɬɟɝɪɚɥɨɜ. Ɂɚɦɟɧɚ ɩɟɪɟɦɟɧɧɨɣ ɢ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ ɩɨ

ɱɚɫɬɹɦ ɜ

ɧɟɨɩɪɟɞɟɥɟɧɧɨɦ ɢɧɬɟɝɪɚɥɟ.

Ɍ

ɇ

ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɩɪɨɫɬɟɣɲɢɯ ɞɪɨɛɟɣ. ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɪɚɰɢɨɧɚɥɶɧɵɯ

ɮɭɧɤɰɢɣ. Ɇɟɬɨɞ ɪɚɰɢɨɧɚɥɢɡɚɰɢɢ. ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɯ ɮɭɧɤɰɢɣ.

ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɩɪɨɫɬɟɣɲɢɯ ɢɪɪɚɰɢɨɧɚɥɶɧɨɫɬɟɣ.

Ȼ

Ɍɟɦɚ 2. Ɉɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ

ɣ

Ɂɚɞɚɱɢ, ɩɪɢɜɨɞɹɳɢɟ ɤ ɩɨɧɹɬɢɸ ɨɩɪɟɞɟɥɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ. Ɉɩɪɟɞɟɥɟɧɧɵɣ

ɢ

ɢɧɬɟɝɪɚɥ ɢ ɟɝɨ ɫɜɨɣɫɬɜɚ. Ɏɨɪɦɭɥɚ ɇɶɸɬɨɧɚ–Ʌɟɣɛɧɢɰɚ.

Ɂɚɦɟɧɚ ɩɟɪɟɦɟɧɧɨɣ ɢ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ ɩɨ ɱɚɫɬɹɦ ɜ ɨɩɪɟɞɟɥɟɧɧɨɦ ɢɧɬɟɝɪɚɥɟ.

ɇɟɫɨɛɫɬɜɟɧɧɵɟ ɢɧɬɟɝɪɚɥɵ.

ɨ

ɉɪɢɥɨɠɟɧɢɹ ɨɩɪɟɞɟɥɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚɪɤ ɜɵɱɢɫɥɟɧɢɸ ɩɥɨɳɚɞɟɣ ɩɥɨɫɤɢɯ

ɬ

ɮɢɝɭɪ ɜ ɞɟɤɚɪɬɨɜɵɯ ɢ ɩɨɥɹɪɧɵɯ ɤɨɨɪɞɢɧɚɬɚɯ. ȼɵɱɢɫɥɟɧɢɟ ɨɛɴɟɦɨɜ ɢ ɞɥɢɧ ɞɭɝ.

ɉɪɢɛɥɢɠɟɧɧɵɟ ɦɟɬɨɞɵ ɜɵɱɢɫɥɟɧɢɹ ɨɩɪɟɞɟɥɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ.

ɡ

Ɍɟɦɚɢ3. Ɏɭɧɤɰɢɢ ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɟɦɟɧɧɵɯ

ɨ

Ɏɭɧɤɰɢɢ ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɟɦɟɧɧɵɯ. Ɉɛɥɚɫɬɶ ɨɩɪɟɞɟɥɟɧɢɹ. ɉɪɟɞɟɥ.

ɇɟɩɪɟɪɵɜɧɨɫɬɶ. ɑɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ.

ɩ

ɮɭɧɤɰɢɢ ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɟɦɟɧɧɵɯ,

ɩɨɥɧɵɣ

Ⱦɢɮɮɟɪɟɧɰɢɪɭɟɦɨɫɬɶ

ɟ

ɞɢɮɮɟɪɟɧɰɢɚɥ. ɉɪɨɢɡɜɨɞɧɵɟ ɨɬ ɫɥɨɠɧɨɣ ɮɭɧɤɰɢɢ. ɂɧɜɚɪɢɚɧɬɧɨɫɬɶ ɮɨɪɦɵ

Ɋ

ɩɟɪɜɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɚ. ɇɟɹɜɧɵɟ ɮɭɧɤɰɢɢ ɢ ɢɯ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟ.

Ɍ

Ɂɚɞɚɱɢ, ɩɪɢɜɨɞɹɳɢɟ ɤ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦ ɭɪɚɜɧɟɧɢɹɦ. Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ

ɭɪɚɜɧɟɧɢɹ 1–ɝɨ ɩɨɪɹɞɤɚ. Ɂɚɞɚɱɚ Ʉɨɲɢ. Ɍɟɨɪɟɦɚ ɫɭɳɟɫɬɜɨɜɚɧɢɹ ɢ ɟɞɢɧɫɬɜɟɧɧɨɫɬɢɍ

ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ Ʉɨɲɢ.

ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ

1–ɝɨ ɩɨɪɹɞɤɚ ɫ

Ȼ

ɪɚɡɞɟɥɹɸɳɢɦɢɫɹ ɩɟɪɟɦɟɧɧɵɦɢ, ɨɞɧɨɪɨɞɧɵɯ, ɥɢɧɟɣɧɵɯ, ɭɪɚɜɧɟɧɢɹɇȻɟɪɧɭɥɥɢ ɢ ɜ

ɩɨɥɧɵɯ ɞɢɮɮɟɪɟɧɰɢɚɥɚɯ.

ɣ

Ɍɟɨɪɟɦɚ

Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɜɵɫɲɢɯ ɩɨɪɹɞɤɨɜ. Ɂɚɞɚɱɚ Ʉɨɲɢ.

ɢ

ɫɭɳɟɫɬɜɨɜɚɧɢɹ ɢ ɟɞɢɧɫɬɜɟɧɧɨɫɬɢ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ Ʉɨɲɢ. ɍɪɚɜɧɟɧɢɹ, ɞɨɩɭɫɤɚɸɳɢɟ

ɩɨɧɢɠɟɧɢɟ ɩɨɪɹɞɤɚ.

ɪ

ɩɨɪɹɞɤɨɜ.

ɋɜɨɣɫɬɜɚ

Ʌɢɧɟɣɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɜɵɫɲɢɯ

ɨ

ɥɢɧɟɣɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɨɩɟɪɚɬɨɪɚ. Ʌɢɧɟɣɧɨ–ɡɚɜɢɫɢɦɵɟ ɢ ɥɢɧɟɣɧɨ–

ɬ

ɧɟɡɚɜɢɫɢɦɵɟ ɫɢɫɬɟɦɵ ɮɭɧɤɰɢɣ. Ɉɩɪɟɞɟɥɢɬɟɥɶ ȼɪɨɧɫɤɨɝɨ.

Ʌɢɧɟɣɧɵɟ ɨɞɧɨɪɨɞɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ, ɭɫɥɨɜɢɟ ɥɢɧɟɣɧɨɣ

ɢ

ɧɟɡɚɜɢɫɢɦɨɫɬɢ ɢɯ ɪɟɲɟɧɢɣ. Ɏɭɧɞɚɦɟɧɬɚɥɶɧɚɹ ɫɢɫɬɟɦɚ ɪɟɲɟɧɢɣ. ɋɬɪɭɤɬɭɪɚ

ɨɛɳɟɝɨ ɪɟɲɟɧɢɹ. Ʌɢɧɟɣɧɵɟ ɨɞɧɨɪɨɞɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɫ

ɨ

.

ɩɨɫɬɨɹɧɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢɡ

ɩ

Ʌɢɧɟɣɧɵɟ ɧɟɨɞɧɨɪɨɞɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ. ɋɬɪɭɤɬɭɪɚ ɨɛɳɟɝɨ

ɪɟɲɟɧɢɹ. Ɇɟɬɨɞ Ʌɚɝɪɚɧɠɚ ɜɚɪɢɚɰɢɢ ɩɪɨɢɡɜɨɥɶɧɵɯ ɩɨɫɬɨɹɧɧɵɯ. Ʌɢɧɟɣɧɵɟ

ɧɟɨɞɧɨɪɨɞɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɫ ɩɨɫɬɨɹɧɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ ɫɨ

Ɋ

ɩɪɚɜɨɣ ɱɚɫɬɶɸ.

ɫɩɟɰɢɚɥɶɧɨɣɟ

ɍ

Ɋɟɲɟɧɢɟ ɫɢɫɬɟɦ ɥɢɧɟɣɧɵɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɫ ɩɨɫɬɨɹɧɧɵɦɢ

ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ.

Ɍ

ɉɨɧɹɬɢɟ ɨ ɤɚɱɟɫɬɜɟɧɧɵɯ ɦɟɬɨɞɚɯ ɢɫɫɥɟɞɨɜɚɧɢɹ ɫɢɫɬɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ

ɭɪɚɜɧɟɧɢɣ.

ɇ

1. ɇȿɈɉɊȿȾȿɅȿɇɇɕɃ ɂɇɌȿȽɊȺɅ

Ȼ

1.1. ɉɨɧɹɬɢɟ ɧɟɨɩɪɟɞɟɥɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ

Ɉɩɪɟɞɟɥɟɧɢɟ 1. Ɏɭɧɤɰɢɹ F(x) ɧɚɡɵɜɚɟɬɫɹ ɩɟɪɜɨɨɛɪɚɡɧɨɣ ɞɥɹ ɮɭɧɤɰɢɢ f(x) ɧɚ

ɢɧɬɟɪɜɚɥɟ (a, b),

ɟɫɥɢ ɜɨ

ɜɫɟɯ

ɬɨɱɤɚɯ ɷɬɨɝɨ ɢɧɬɟɪɜɚɥɚ ɜɵɩɨɥɧɹɟɬɫɹ ɪɚɜɟɧɫɬɜɨ

Fc(x)= f(x).

ɣ

ɪ

Ɉɩɪɟɞɟɥɟɧɢɟ 2. ɋɨɜɨɤɭɩɧɨɫɬɶ ɜɫɟɯɢɩɟɪɜɨɨɛɪɚɡɧɵɯ {F(x)+ɋ}, ɝɞɟ ɋ –

ɩɪɨɢɡɜɨɥɶɧɚɹ ɩɨɫɬɨɹɧɧɚɹ,

ɞɥɹ ɮɭɧɤɰɢɢ f(x) ɧɚɡɵɜɚɟɬɫɹ ɧɟɨɩɪɟɞɟɥɟɧɧɵɦ

ɢɧɬɟɝɪɚɥɨɦ ɢ ɨɛɨɡɧɚɱɚɟɬɫɹ

ɨ

³ f (x)dx F(x) C

ɬ

Ɏɭɧɤɰɢɹ f(x) ɧɚɡɵɜɚɟɬɫɹ ɩɨɞɵɧɬɟɝɪɚɥɶɧɨɣ ɮɭɧɤɰɢɟɣ, ɜɵɪɚɠɟɧɢɟ f(x) dx –

ɩɨɞɵɧɬɟɝɪɚɥɶɧɵɦ ɜɵɪɚɠɟɧɢɟɦɢ.

ɨ

Ɉɬɵɫɤɚɧɢɟɡɞɥɹ ɮɭɧɤɰɢɢ f(x) ɜɫɟɯ ɟɟ ɩɟɪɜɨɨɛɪɚɡɧɵɯ F(x) + ɋ ɧɚɡɵɜɚɟɬɫɹ

ɩ

ɢɧɬɟɝɪɢɪɨɜɚɧɢɟɦ. ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɟɫɬɶ ɞɟɣɫɬɜɢɟ, ɨɛɪɚɬɧɨɟ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɸ.

Ɉɫɧɨɜɧɵɟ ɩɪɚɜɢɥɚ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ:

ɟ

³df (x )

f (x ) C ;

Ɋ

1)

³ f c(x )dx

³ f (x)dx d(F(x) C)

f (x)dx ;

2) ³( f (x ) r M(x ))dx ³ f (x )dx r ³M(x )dx ;

3)

³af (x )dx

a³ f (x )dx,

(a

const) ;

4) ɟɫɥɢ ³ f (x )dx

F(x ) C , ɬɨ ³ f (ax b)dx

1

F(ax b) C , ɩɪɢ ɭɫɥɨɜɢɢ,

a

ɱɬɨ a, b – ɩɨɫɬɨɹɧɧɵɟ ɱɢɫɥɚ, az0;

5) ɟɫɥɢ ³ f (x )dx

F(x ) C ɢ u=M (x) – ɥɸɛɚɹ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚɹ ɮɭɧɤɰɢɹ,

ɬɨ ³ f (u)du F(u) C ;

ɍ

Ɍɚɛɥɢɰɚ ɨɫɧɨɜɧɵɯ ɧɟɨɩɪɟɞɟɥɟɧɧɵɯ ɢɧɬɟɝɪɚɥɨɜ

Ɍ

ɇ

³du

u C ;

³

du

u

1.

10.

ln

tg

C

;

sin u

2

D

uD1

du

§ u

S·

2.

³

u du

C , ɝɞɟ D z 1;

11.

³

ln

Ȼ

¹

C ;

©

D

1

cos u

tg¨

2

4

¸

du

du

3.

³

u

ɢ

u

ln | u | C ;

12.

³

ɣ2 ctgu C ;

ɪ

sin

u

4.

³audu

a

C ;

du

13.

³

tgu C ;

ln a

ɨ

cos2 u

5.

³eu du

eu C ;

14.

du

1

u

³

arctg

C ;

a2 u2

a

a

ɢ

du

u

ɬ

6.

³sin udu

cosu C ;

15.

³

a2

u2

arcsin

a

C ;

ɨ

7.

³cosudu ɡsin u C ;

16.

³

du

1

ln

a u

C ;

a2 u2

2a

a u

³tgudu ln | cosu | C ;

³

du

u2 r D2

C .

8.

17.

ln

u

ɟ

ɩ

u2 r D2

9.

³ctg udu

ln|sin u| C ;

ȼ ɩɪɢɜɟɞɟɧɧɨɣ ɬɚɛɥɢɰɟ ɛɭɤɜɚ u ɦɨɠɟɬ ɨɛɨɡɧɚɱɚɬɶ ɤɚɤ ɧɟɡɚɜɢɫɢɦɭɸ

Ɋɩɟɪɟɦɟɧɧɭɸ, ɬɚɤ ɢ ɧɟɩɪɟɪɵɜɧɨ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɭɸ ɮɭɧɤɰɢɸ u=M(x) ɚɪɝɭɦɟɧɬɚ x.