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ɆɂɇɂɋɌȿɊɋɌȼɈ ɈȻɊȺɁɈȼȺɇɂə ɂ ɇȺɍɄɂ ɊɈɋɋɂɃɋɄɈɃ ɎȿȾȿɊȺɐɂɂ

ɆɈɋɄɈȼɋɄɂɃ ȽɈɋɍȾȺɊɋɌȼȿɇɇɕɃ ɌȿɏɇɂɑȿɋɄɂɃ ɍɇɂȼȿɊɋɂɌȿɌ ɆɂɊɗȺ

Ⱥ Ȼ ɋȺɆɈɏɂɇ ȼ ȼ ɑȿɊȾɕɇɐȿȼ Ⱥ Ⱥ ȼɈɊɈɇɐɈȼ

ɑɂɋɅȿɇɇɕȿ ɆȿɌɈȾɕ

ɍɑȿȻɇɈȿ ɉɈɋɈȻɂȿ

ɞɥɹ ɫɬɭɞɟɧɬɨɜ ɨɛɭɱɚɸɳɢɯɫɹ ɩɨ ɧɚɩɪɚɜɥɟɧɢɹɦ ɩɨɞɝɨɬɨɜɤɢ

«ɂɧɮɨɪɦɚɬɢɤɚ ɢ ɜɵɱɢɫɥɢɬɟɥɶɧɚɹ ɬɟɯɧɢɤɚ», «ɂɧɮɨɪɦɚɰɢɨɧɧɵɟ ɫɢɫɬɟɦɵ ɢ ɬɟɯɧɨɥɨɝɢɢ», «ɉɪɨɝɪɚɦɦɧɚɹ ɢɧɠɟɧɟɪɢɹ», «ɉɪɢɤɥɚɞɧɚɹ ɦɚɬɟɦɚɬɢɤɚ»

ɆɈɋɄȼȺ ɆȽɌɍ ɆɂɊɗȺ

2014

ɍȾɄ ȻȻɄ ɹ ɋ

ɍɬɜɟɪɠɞɟɧɨ ɪɟɞɚɤɰɢɨɧɧɨ-ɢɡɞɚɬɟɥɶɫɤɢɦ ɫɨɜɟɬɨɦ ɆȽɌɍ ɆɂɊɗȺ ɜ ɤɚɱɟɫɬɜɟ ɦɟɬɨɞɢɱɟɫɤɨɝɨ ɩɨɫɨɛɢɹ ɞɥɹ ɫɬɭɞɟɧɬɨɜ

ɉɨɞɝɨɬɨɜɥɟɧɨ ɧɚ ɤɚɮɟɞɪɟ ɩɪɢɤɥɚɞɧɨɣ ɦɚɬɟɦɚɬɢɤɢ

Ɋɟɰɟɧɡɟɧɬɵ ɞ ɮ -ɦ ɧ ɩɪɨɮɟɫɫɨɪ ȼɇɋ ɇɂȼɐ ɆȽɍ Ⱥ ȼ ɋɟɬɭɯɚ; ɞ ɮ -ɦ ɧ ɩɪɨɮɟɫɫɨɪ ɊɈɋɇɈɍ Ⱥ ɋ Ʉɪɸɤɨɜɫɤɢɣ

ɋɚɦɨɯɢɧ Ⱥ Ȼ

ɋ ɑɢɫɥɟɧɧɵɟ ɦɟɬɨɞɵ ɭɱɟɛɧɨɟ ɩɨɫɨɛɢɟ Ⱥ Ȼ ɋɚɦɨɯɢɧ ȼ ȼ ɑɟɪɞɵɧɰɟɜ Ⱥ Ⱥ ȼɨɪɨɧɰɨɜ. — Ɇ ɆȽɌɍ ɆɂɊɗȺ

2014. — 84 ɫ

ISBN 978-5-7339-1075-8

Ɋɚɫɫɦɨɬɪɟɧɵ ɱɢɫɥɟɧɧɵɟ ɦɟɬɨɞɵ ɪɟɲɟɧɢɹ ɩɪɢɤɥɚɞɧɵɯ ɦɚɬɟ ɦɚɬɢɱɟɫɤɢɯ ɡɚɞɚɱ ɍɱɟɛɧɨɟ ɩɨɫɨɛɢɟ ɧɚɩɢɫɚɧɨ ɞɥɹ ɫɬɭɞɟɧɬɨɜ ɮɚ ɤɭɥɶɬɟɬɚ ɢɧɮɨɪɦɚɰɢɨɧɧɵɯ ɬɟɯɧɨɥɨɝɢɣ Ɉɧɨ ɦɨɠɟɬ ɛɵɬɶ ɩɨɥɟɡɧɵɦ ɬɚɤɠɟ ɩɪɢ ɢɡɭɱɟɧɢɢ ɞɢɫɰɢɩɥɢɧ «ȼɵɱɢɫɥɢɬɟɥɶɧɚɹ ɦɚɬɟɦɚɬɢɤɚ», «Ɇɚɬɟɦɚɬɢɱɟɫɤɨɟ ɦɨɞɟɥɢɪɨɜɚɧɢɟ», «Ɇɟɬɨɞɵ ɨɩɬɢɦɢɡɚɰɢɢ» ɢ ɞɪ

ISBN 978-5-7339-1075-8	ɋɚɦɨɯɢɧ Ⱥ Ȼ ɑɟɪɞɵɧɰɟɜ ȼ ȼ
	ȼɨɪɨɧɰɨɜ Ⱥ Ⱥ , 2014

ȼȼȿȾȿɇɂȿ

ɉɹɬɢɞɟɫɹɬɢɥɟɬɧɹɹ ɷɜɨɥɸɰɢɹ ɗȼɆ ɨɬ ɩɟɪɜɵɯ ɥɚɦɩɨɜɵɯ ɞɨ ɫɨ

ɜɪɟɦɟɧɧɵɯ ɫɟɪɢɣɧɵɯ ɫ ɛɵɫɬɪɨɞɟɣɫɬɜɢɟɦ ɩɨɪɹɞɤɚ 1012 ɨɩɟɪɚɰɢɣ ɜ ɫɟɤɭɧɞɭ ɩɪɢɜɟɥɚ ɤ ɪɚɡɜɢɬɢɸ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɦɨɞɟɥɢɪɨɜɚɧɢɹ ɢ ɱɢɫɥɟɧɧɨɝɨ ɚɧɚɥɢɡɚ ɩɪɚɤɬɢɱɟɫɤɢ ɜɨ ɜɫɟɯ ɨɬɪɚɫɥɹɯ ɱɟɥɨɜɟɱɟɫɤɨɝɨ ɡɧɚɧɢɹ Ɋɚɡɜɢɬɢɟ ɬɟɯɧɢɱɟɫɤɢɯ ɜɨɡɦɨɠɧɨɫɬɟɣ, ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɢ ɩɪɨɝɪɚɦɦɧɨɝɨ ɨɛɟɫɩɟɱɟɧɢɹ ɗȼɆ ɩɨɤɚɡɚɥɢ ɧɟɫɨɜɟɪɲɟɧɫɬɜɨ ɧɟɤɨ ɬɨɪɵɯ ɤɥɚɫɫɢɱɟɫɤɢɯ ɦɟɬɨɞɨɜ ɪɟɲɟɧɢɹ ɢɧɠɟɧɟɪɧɵɯ ɢ ɧɚɭɱɧɨɬɟɯɧɢɱɟɫɤɢɯ ɡɚɞɚɱ ɱɬɨ ɨɛɭɫɥɨɜɢɥɨ ɪɚɡɜɢɬɢɟ ɧɨɜɵɯ ɦɟɬɨɞɨɜ ɢɯ ɱɢɫɥɟɧɧɨɝɨ ɪɟɲɟɧɢɹ ɉɪɨɛɥɟɦɚ ɜɵɛɨɪɚ ɨɩɬɢɦɚɥɶɧɨɝɨ ɱɢɫɥɟɧɧɨɝɨ ɦɟɬɨɞɚ ɪɟɲɟɧɢɹ ɫ ɬɨɱɤɢ ɡɪɟɧɢɹ ɤɚɤ ɷɤɨɧɨɦɢɢ ɪɟɫɭɪɫɨɜ ɗȼɆ ɬɚɤ ɢ ɫɧɢɠɟɧɢɹ ɪɟɡɭɥɶɬɢɪɭɸɳɟɣ ɩɨɝɪɟɲɧɨɫɬɢ ɬɪɟɛɭɟɬ ɨɩɪɟɞɟɥɟɧɧɨɝɨ ɨɩɵɬɚ ɢ ɜɵɱɢɫɥɢɬɟɥɶɧɨɣ ɩɪɚɤɬɢɤɢ

ɇɚɫɬɨɹɳɟɟ ɩɨɫɨɛɢɟ ɹɜɥɹɟɬɫɹ ɜɜɟɞɟɧɢɟɦ ɜ ɱɢɫɥɟɧɧɵɟ ɦɟɬɨɞɵ ȼ ɤɨɧɰɟ ɤɚɠɞɨɣ ɬɟɦɵ ɩɪɢɜɟɞɟɧɵ ɡɚɞɚɧɢɹ ɞɥɹ ɩɪɚɤɬɢɱɟɫɤɢɯ ɡɚɧɹ ɬɢɣ ɜɵɩɨɥɧɟɧɢɟ ɤɨɬɨɪɵɯ ɩɨɡɜɨɥɹɟɬ ɝɥɭɛɠɟ ɩɨɧɹɬɶ ɢ ɭɫɜɨɢɬɶ ɜɵ ɱɢɫɥɢɬɟɥɶɧɵɟ ɚɥɝɨɪɢɬɦɵ ɉɪɢ ɢɯ ɪɟɲɟɧɢɢ ɞɨɩɭɫɬɢɦɨ ɢɫɩɨɥɶɡɨ ɜɚɧɢɟ ɢɧɠɟɧɟɪɧɵɯ ɤɚɥɶɤɭɥɹɬɨɪɨɜ ɢ ɩɪɢɦɟɧɟɧɢɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɩɚɤɟɬɨɜ ɩɪɢɤɥɚɞɧɵɯ ɩɪɨɝɪɚɦɦ

ȺȻɋɈɅɘɌɇȺə ɂ ɈɌɇɈɋɂɌȿɅɖɇȺə ɉɈȽɊȿɒɇɈɋɌɂ

ɑɢɫɥɟɧɧɵɟ ɦɟɬɨɞɵ ɫɥɭɠɚɬ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ ɩɪɢɛɥɢɠɟɧɧɨɝɨ ɪɟɲɟɧɢɹ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɡɚɞɚɱ Ʌɸɛɨɟ ɩɪɢɛɥɢɠɟɧɧɨɟ ɪɟɲɟɧɢɟ ɫɜɹɡɚɧɨ ɫ ɨɲɢɛɤɨɣ ɩɨɝɪɟɲɧɨɫɬɶɸ ȼɢɞɵ ɨɲɢɛɨɤ

ɉɨɝɪɟɲɧɨɫɬɶ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɦɨɞɟɥɢ ɫɜɹɡɚɧɧɚɹ ɫ ɧɟɩɨɥɧɵɦɢ ɡɧɚɧɢɹɦɢ ɨ ɩɪɨɰɟɫɫɟ

1.ɉɨɝɪɟɲɧɨɫɬɶ ɭɩɪɨɳɟɧɢɹ ɦɨɞɟɥɢ

2.ɉɨɝɪɟɲɧɨɫɬɶ ɫɜɹɡɚɧɧɚɹ ɫ ɩɪɢɛɥɢɠɟɧɧɵɦ ɯɚɪɚɤɬɟɪɨɦ ɧɚ ɱɚɥɶɧɵɯ ɞɚɧɧɵɯ

3.ɉɨɝɪɟɲɧɨɫɬɶ ɨɤɪɭɝɥɟɧɢɹ ɩɪɢ ɪɚɫɱɟɬɚɯ

ɉɟɪɜɵɟ ɞɜɟ ɩɨɝɪɟɲɧɨɫɬɢ ɨɬɧɨɫɹɬɫɹ ɤ ɫɢɫɬɟɦɚɬɢɱɟɫɤɢɦ ɚ ɞɜɟ ɩɨɫɥɟɞɧɢɟ — ɤ ɫɬɚɬɢɫɬɢɱɟɫɤɢɦ ɨɲɢɛɤɚɦ Ⱦɥɹ ɢɯ ɨɰɟɧɤɢ ɜɜɨɞɢɬɫɹ ɚɛɫɨɥɸɬɧɚɹ ɢ ɨɬɧɨɫɢɬɟɥɶɧɚɹ ɩɨɝɪɟɲɧɨɫɬɢ

Ⱥɛɫɨɥɸɬɧɚɹ ɩɪɟɞɟɥɶɧɚɹ ɩɨɝɪɟɲɧɨɫɬɶ — ɨɩɪɟɞɟɥɹɟɬ ɢɧɬɟɪ ɜɚɥ ɜ ɤɨɬɨɪɨɦ ɥɟɠɢɬ ɬɨɱɧɨɟ ɡɧɚɱɟɧɢɟ ɜɟɥɢɱɢɧɵ ɉɭɫɬɶ Ⱥ — ɬɨɱ ɧɨɟ ɡɧɚɱɟɧɢɟ ɜɟɥɢɱɢɧɵ ɧɟɢɡɜɟɫɬɧɨ ɚ ɚ — ɩɪɢɛɥɢɠɟɧɧɨɟ ɡɧɚɱɟ ɧɢɟ ɜɟɥɢɱɢɧɵ ɢɡɜɟɫɬɧɨ Ɂɚ ɚɛɫɨɥɸɬɧɭɸ ɩɨɝɪɟɲɧɨɫɬɶ ' ɩɪɢɧɢ ɦɚɟɬɫɹ ɦɢɧɢɦɚɥɶɧɨɟ ɱɢɫɥɨ 'a , ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɭɫɥɨɜɢɸ

'a t		A a		.	(1.1)

ɉɪɢ ɫɬɚɬɢɫɬɢɱɟɫɤɢɯ ɢɡɦɟɪɟɧɢɹɯ ɩɨɝɪɟɲɧɨɫɬɶ 'a ɡɚɞɚɟɬɫɹ ɫ ɨɩ ɪɟɞɟɥɟɧɧɨɣ ɞɨɫɬɨɜɟɪɧɨɫɬɶɸ ɬ ɟ ɜɟɪɨɹɬɧɨɫɬɶ ɫɨɛɵɬɢɹ A a d 'a

ɛɨɥɶɲɟ ɨɩɪɟɞɟɥɟɧɧɨɣ ɜɟɥɢɱɢɧɵ J: P( A a d 'a) ! J d1 ɉɟɪɟ

ɩɢɲɟɦ ɨɩɪɟɞɟɥɟɧɢɟ a 'a d A d a 'a ɬɨ ɟɫɬɶ ɬɨɱɧɨɟ ɡɧɚɱɟɧɢɟ A ɥɟɠɢɬ ɜ ɡɚɞɚɧɧɨɦ ɢɧɬɟɪɜɚɥɟ Ⱦɥɹ ɨɰɟɧɤɢ ɤɚɱɟɫɬɜɚ ɢɡɦɟɪɟɧɢɣ

ɜɜɨɞɢɬɫɹ ɨɬɧɨɫɢɬɟɥɶɧɚɹ ɩɨɝɪɟɲɧɨɫɬɶ
Gɚ	'ɚ		\|	'ɚ .		(1.2)
		Ⱥ		ɚ
Ɂɚɞɚɧɧɵɟ ɜɟɥɢɱɢɧɵ 'ɚ	ɢɥɢ Gɚ				ɩɨɡɜɨɥɹɸɬ ɡɚɩɢɫɚɬɶ ɬɨɱɧɨɟ
ɡɧɚɱɟɧɢɟ Ⱥ ɜ ɫɢɦɜɨɥɢɱɟɫɤɨɦ ɜɢɞɟ A					a(1 r Ga) ɢɥɢ A	a r 'a .

ɑɢɫɥɨ ɜɟɪɧɵɯ ɡɧɚɤɨɜ ɩɪɢɛɥɢɠɟɧɧɨɝɨ ɱɢɫɥɚ

ɉɪɢɛɥɢɠɟɧɧɨɟ ɱɢɫɥɨ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ

a a 10m a	m 1	10m 1	... a	m n 1	10m n 1	,	(1.1.1)
m	m 1			m n 1

ɝɞɟ m — ɜɟɥɢɱɢɧɚ ɫɬɚɪɲɟɝɨ ɪɚɡɪɹɞɚ; n — ɬɟɤɭɳɢɣ ɧɨɦɟɪ ɡɧɚɤɚ ɨɬɫɱɢɬɵɜɚɟɦɵɣ ɫɥɟɜɚ ɧɚɩɪɚɜɨ Ƚɨɜɨɪɹɬ ɱɬɨ k ɩɟɪɜɵɯ ɡɧɚɤɨɜ ɩɪɢɛɥɢɠɟɧɧɨɝɨ ɱɢɫɥɚ ɜɟɪɧɵɟ ɟɫɥɢ ɚɛɫɨɥɸɬɧɚɹ ɩɨɝɪɟɲɧɨɫɬɶ 'a

ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɭɫɥɨɜɢɸ A a d 'a d 0,5 10m k 1 ɬɨ ɟɫɬɶ ɦɟɧɶɲɟ ɩɨɥɨɜɢɧɵ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɪɚɡɪɹɞɚ ɉɨɞɛɢɪɚɟɬɫɹ ɦɢɧɢɦɚɥɶɧɨɟ ɱɢɫɥɨ ɜɢɞɚ 0,5 10l ɛɨɥɶɲɟɟ ɱɟɦ 'a , ɢ ɫɪɚɜɧɢɜɚɸɬɫɹ ɪɚɡɪɹɞɵ

ɉɨɝɪɟɲɧɨɫɬɶ ɮɭɧɤɰɢɣ

ɉɭɫɬɶ ɞɚɧɚ ɮɭɧɤɰɢɹ f (a1, a2 ,..., an ) ɨɬ n ɩɪɢɛɥɢɠɟɧɧɵɯ ɡɧɚɱɟ ɧɢɣ ai ɩɨɝɪɟɲɧɨɫɬɢ ɤɨɬɨɪɵɯ ɢɡɜɟɫɬɧɵ Ɍɪɟɛɭɟɬɫɹ ɨɩɪɟɞɟɥɢɬɶ ɩɨ ɝɪɟɲɧɨɫɬɶ ɮɭɧɤɰɢɢ ('f ?) :

'f max f (a1 '1, a2 '2 ,..., an 'n ) f (a1,..., an ) ,

ɝɞɟ 'i — ɚɛɫɨɥɸɬɧɚɹ ɩɨɝɪɟɲɧɨɫɬɶ ɩɪɢɛɥɢɠɟɧɧɨɣ ɜɟɥɢɱɢɧɵ ai . ȿɫɥɢ 'i ai ɬɨ ɪɚɡɧɨɫɬɶ ɫɬɨɹɳɭɸ ɜ ɮɨɪɦɭɥɟ ɦɨɠɧɨ ɨɰɟɧɢɬɶ ɜ ɥɢɧɟɣɧɨɦ ɩɪɢɛɥɢɠɟɧɢɢ

f (a1 '1, a2 '2 ,..., an 'n ) f (a1, a2 ,..., an ) |

\|	w f	'	w f	'		...	w f	'	n	.
	w a		w a				w a
		1			2
	1		2				n

Ɉɬɫɸɞɚ ɫɥɟɞɭɟɬ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɩɨɝɪɟɲɧɨɫɬɢ ɮɭɧɤɰɢɢ:

w f

, Gf

w f

ln f

. (1.2.1)

' f ¦

i 1

w ai

i 1

w ai

i 1

w ai

ɉɨɝɪɟɲɧɨɫɬɶ ɩɪɨɫɬɟɣɲɢɯ ɮɭɧɤɰɢɣ ɞɜɭɯ ɩɟɪɟɦɟɧɧɵɯ

ɉɨɝɪɟɲɧɨɫɬɶ ɫɭɦɦɵ

f a a ,

1, 'f

'a ,Gf

a1 'a2

1 2 wa wa

a a

ɉɨɝɪɟɲɧɨɫɬɶ ɪɚɡɧɨɫɬɢ

f a a ,	wf	1, 'f 'a 'a ,G		f	'	a1 'a		2	.
f a a ,	wa	1, 'f 'a 'a ,G							.
1 2	wa	1	2			a a	2
	2					1	2

ɉɪɢ a1 | a2 ɤɚɱɟɫɬɜɨ ɢɡɦɟɪɟɧɢɣ ɪɚɡɧɨɫɬɢ ɭɯɭɞɲɚɟɬɫɹ

Ɂɚɦɟɱɚɧɢɟ Ⱥɛɫɨɥɸɬɧɚɹ ɩɨɝɪɟɲɧɨɫɬɶ ɫɭɦɦɵ ɢ ɪɚɡɧɨɫɬɢ n ɩɪɢ ɛɥɢɠɟɧɧɵɯ ɜɟɥɢɱɢɧ ɪɚɜɧɚ ɫɭɦɦɟ ɢɯ ɚɛɫɨɥɸɬɧɵɯ ɩɨɝɪɟɲɧɨɫɬɟɣ

ɉɨɝɪɟɲɧɨɫɬɶ ɩɪɨɢɡɜɟɞɟɧɢɹ

f a a

;

a ;

;

1 2

'a1

'a2

Ga Ga .

Ɍɨ ɟɫɬɶ ɩɪɟɞɩɨɱɬɢɬɟɥɶɧɟɣ ɫɧɚɱɚɥɚ ɧɚɣɬɢ ɨɬɧɨɫɢɬɟɥɶɧɭɸ ɩɨɝɪɟɲ ɧɨɫɬɶ ɚ ɡɚɬɟɦ ɢɫɤɚɬɶ ɚɛɫɨɥɸɬɧɭɸ 'f a1 a2 G f .

Ɂɚɦɟɱɚɧɢɹ:

x Ɉɬɧɨɫɢɬɟɥɶɧɚɹ ɩɨɝɪɟɲɧɨɫɬɶ ɫɬɟɩɟɧɢ ɟɫɬɶ ɩɪɨɢɡɜɟɞɟɧɢɟ ɦɨ ɞɭɥɹ ɩɨɤɚɡɚɬɟɥɹ ɧɚ ɨɬɧɨɫɢɬɟɥɶɧɭɸ ɩɨɝɪɟɲɧɨɫɬɶ ɨɫɧɨɜɚɧɢɹ ɫɬɟ ɩɟɧɢ f aX, G f X Ga .

x Ɉɬɧɨɫɢɬɟɥɶɧɚɹ ɩɨɝɪɟɲɧɨɫɬɶ ɩɪɨɢɡɜɟɞɟɧɢɹ n ɫɨɦɧɨɠɢɬɟɥɟɣ ɩɪɢɛɥɢɠɟɧɧɵɯ ɜɟɥɢɱɢɧ ɪɚɜɧɚ ɫɭɦɦɟ ɨɬɧɨɫɢɬɟɥɶɧɵɯ ɩɨɝɪɟɲɧɨ ɫɬɟɣ ɫɨɦɧɨɠɢɬɟɥɟɣ

f a1 a2 ...an , G f ¦ ai .

i 1

ɉɨɝɪɟɲɧɨɫɬɶ ɱɚɫɬɧɨɝɨ

;

G f

;

G f

;

Ga1

Ga2

a22

'a1

'a ;

'a1

'a2

G G

a22

ȼɫɟ ɡɚɦɟɱɚɧɢɹ, ɫɞɟɥɚɧɧɵɟ ɞɥɹ ɩɪɨɢɡɜɟɞɟɧɢɹ ɫɩɪɚɜɟɞɥɢɜɵ ɢ ɜ ɷɬɨɦ ɫɥɭɱɚɟ

ɉɪɢɦɟɪɵ ɢ ɡɚɞɚɧɢɹ

ɉɪɢɦɟɪ. Ⱦɚɧɨ ɩɪɢɛɥɢɠɟɧɧɨɟ ɱɢɫɥɨ ɩɨɝɪɟɲɧɨɫɬɶ — 0,6. ɇɚɣɬɢ ɱɢɫɥɨ ɜɟɪɧɵɯ ɡɧɚɤɨɜ ɐɢɮɪɚ ɜɯɨɞɢɬ ɜ ɱɢɫɥɨ ɫ ɜɟɫɨɦ 3,

(1.3), ɬɨ ɟɫɬɶ m=3; 0,6 0,5 10k ɦɢɧɢɦɚɥɶɧɨɟ k = 1, (3 n 1 1) ,

ɬɨ ɟɫɬɶ ɜɟɪɧɵ ɬɪɢ ɡɧɚɤɚ 345 7,0.

ɜɟɪɧɵ

ɉɪɢɦɟɪ. Ⱦɚɧ ɤɭɛ ɫɬɨɪɨɧɚ ɤɨɬɨɪɨɝɨ a 100 ɫɦ ɢɡɦɟɪɟɧɚ ɫ ɬɨɱɧɨ ɫɬɶɸ 'a 1 ɫɦ Ɉɩɪɟɞɟɥɢɬɶ ɩɨɝɪɟɲɧɨɫɬɢ ɢɡɦɟɪɟɧɢɹ ɩɨɜɟɪɯɧɨɫɬɢ ɢ ɨɛɴɟɦɚ ɤɭɛɚ

S 6a2 ;V a3.

da3

'a 3a2'a

30000 ɫɦ3

3a2'a

;

d 6a2

12a'a

1200ɫɦ2 GS

12a'a

ba2

ɉɪɢɦɟɪ. Ɋɚɫɱɟɬ ɩɨɝɪɟɲɧɨɫɬɢ ɮɭɧɤɰɢɢ ɬɪɟɯ ɩɟɪɟɦɟɧɧɵɯ

a1 a3D

;

G f

a2E a3

a2E a3D

a1 a3D

EaE 1 ,

(D 1)a3D a1

wa1

a3 (a1 a2E )2 wa2

a3 (a1 a2E )2

wa3

a32 (a1 a2E )

a2E a3D

a1 a3D

EaE 1

(D 1)a3D a1

a (a aE )2

a (a

aE )2

a2 (a

aE )

ɉɪɢɦɟɪ.

Ʉɚɬɟɬɵ

ɩɪɹɦɨɭɝɨɥɶɧɨɝɨ

ɬɪɟɭɝɨɥɶɧɢɤɚ

30 ɫɦ ɢ

b 40 ɫɦ

ɢɡɦɟɪɟɧɵ ɫ ɩɨɝɪɟɲɧɨɫɬɶɸ '

1 ɫɦ Ɉɩɪɟɞɟɥɢɬɶ ɩɨ

ɝɪɟɲɧɨɫɬɶ ɢɡɦɟɪɟɧɢɹ ɝɢɩɨɬɟɧɭɡɵ ɫ

wc a 3

wc b 4

a2 b2

50 ɫɦ,

1, 4 ɫɦ

ȼ ɤɚɠɞɨɦ ɜɚɪɢɚɧɬɟ ɡɚɞɚɧɢɹ ɬɪɢ ɡɚɞɚɱɢ ɧɢɠɟ ɩɪɢɜɟɞɟɧɵ ɩɨ ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɩɟɪɜɚɹ ɜɬɨɪɚɹ ɢ ɬɪɟɬɶɹ ɡɚɞɚɱɢ ɜɚɪɢɚɧɬɨɜ

Ⱥ ɇɚɣɬɢ ɚɛɫɨɥɸɬɧɭɸ ɢ ɨɬɧɨɫɢɬɟɥɶɧɭɸ ɨɲɢɛɤɢ ɜɵɪɚɠɟɧɢɹ ɝɞɟ a1, a2 ɢ a3 — ɩɪɢɛɥɢɠɟɧɧɵɟ ɜɟɥɢɱɢɧɵ ɞɚɧɧɵɟ ɫ

ɩɨɝɪɟɲɧɨɫɬɹɦɢ '1, '2 , '3 — ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ
		a		u a	D		1				(a D	a ) a		1
1)			1	2		, D		,E	3. 2)		1	2 3	, D		,E 4 .
			1	2
		E
												E
			a2 ) a3				3					E		5
	(a1		a2 ) a3						1		a1 a2
3) (a D a ) a a a E,								D	1	,E 3.
1				2	3	1	2		2		3
4) (a a				E ) / a a / a E , D					1	,E	3	.
1			2		3	1	2		5		2

	a D a a									1													a	u a D									2
5)		1			2 3		, D						2		,E 3. 6)								1		2				, D 3,E					.
5)					E		, D								,E 3. 6)								E						, D 3,E					.
					E																		E										7
				a1 a2																	(a1 a2 ) a3
7)	(a1D a2 ) a3							, D					2		5	,E 3. 8) (a D								a ) a a a								E	,
7)								, D					2			,E 3. 8) (a D								a ) a a a									,
					E																		1		3			2		1	2
				a1 a2								a a D
		2										a a D							, D 2,E 2
D					,E 3. 9)									1			2											.
D					,E 3. 9)																							.
		5											E		a2 ) a3												3
					a D ) a							(a1			a2 ) a3
				(a	a D ) a													4
10)				1	2			3			, D				3,E					.
10)						E					, D				3,E					.
						E												3
					a1 a2													D 1
11) (a D a						) / a				a				a			E ,	D 1				2	,E		3.
		1			2			3		1						2						2	1
12) (a D a ) a									a / a							E , D		4,E					1	3	.
		1			2			3		1					2									3
				a D	a	/ a							1												a		u a D						2
13)			1		2	3		, D								,E		3 .					14)		1			2			, D			,E 5.
13)						E		, D								,E		3 .					14)								, D			,E 5.
						E									2										E	a2 ) a3							7
				a1 a2																					(a1	a2 ) a3
15)				(a1D a3 ) a2						, D					2	5	,E	3.					16) (a D			a )				a	a		a	E ,
15)										, D					2		,E	3.					16) (a D			a )				a	a		a	E ,
						E																			1			2		3		1	3
					a1 a2
D				3,E	4	.
					3

Ȼ Ⱦɚɧɨ ɩɪɢɛɥɢɠɟɧɧɨɟ ɱɢɫɥɨ ɢ ɟɝɨ ɩɨɝɪɟɲɧɨɫɬɶ ɇɚɣɬɢ ɤɨɥɢ ɱɟɫɬɜɨ ɜɟɪɧɵɯ ɡɧɚɤɨɜ

1) 23,587; 0,08 . 2) 13,58; 0,07. 3) 103,58; 0,03. 4) 1655; 6.

5) 323,07; 0,06. 6) 43,837; 0,008. 7) 16,402; 0,009. 8) 13,540; 0,006. 9) 31,541; 0,003. 10) 13,42; 0,03. 11) 137,5; 0,08. 12) 134; 20.

13) 3457,0; 0,6. 14) 4657; 8. 15) 16,47; 0,07. 16) 130,6; 0,06.

ȼ Ⱦɚɧɚ ɝɟɨɦɟɬɪɢɱɟɫɤɚɹ ɮɢɝɭɪɚ Ɉɩɪɟɞɟɥɢɬɶ ɜ ɬɪɟɯɦɟɪɧɨɦ ɫɥɭɱɚɟ ɨɛɴɟɦ ɢ ɩɨɥɧɭɸ ɩɨɜɟɪɯɧɨɫɬɶ ɚ ɜ ɩɥɨɫɤɨɦ ɫɥɭɱɚɟ ɩɥɨɳɚɞɶ ɢ ɩɟɪɢɦɟɬɪ ɉɨɝɪɟɲɧɨɫɬɶ ɨɩɪɟɞɟɥɟɧɢɹ ɪɚɡɦɟɪɨɜ ɥɢɧɟɣɧɵɯ ɷɥɟ ɦɟɧɬɨɜ ɪɚɜɧɚ ɫɦ

1)Ɋɚɜɧɨɛɟɞɪɟɧɧɚɹ ɬɪɚɩɟɰɢɹ ɫɨ ɫɬɨɪɨɧɚɦɢ ɨɫɧɨɜɚɧɢɹ ɪɚɜɧɵ ɦɢ ɢ ɫɦ ɢ ɜɵɫɨɬɨɣ, ɪɚɜɧɨɣ ɫɦ

2)ɉɪɚɜɢɥɶɧɚɹ ɱɟɬɵɪɟɯɭɝɨɥɶɧɚɹ ɩɢɪɚɦɢɞɚ ɫɨ ɫɬɨɪɨɧɨɣ ɨɫɧɨ ɜɚɧɢɹ ɪɚɜɧɨɣ ɫɦ ɢ ɜɵɫɨɬɨɣ, ɪɚɜɧɨɣ ɫɦ

3)Ʉɨɧɭɫ ɫ ɜɵɫɨɬɨɣ, ɪɚɜɧɨɣ ɫɦ, ɢ ɪɚɞɢɭɫɨɦ, ɪɚɜɧɵɦ ɫɦ

4)ɉɪɹɦɨɭɝɨɥɶɧɵɣ ɩɚɪɚɥɥɟɥɟɩɢɩɟɞ ɫ ɜɵɫɨɬɨɣ ɫɦ, ɫɬɨɪɨ ɧɨɣ ɨɫɧɨɜɚɧɢɹ ɫɦ ɢ ɞɢɚɝɨɧɚɥɶɸ ɨɫɧɨɜɚɧɢɹ ɫɦ

5)ɐɢɥɢɧɞɪ ɫ ɝɥɚɜɧɨɣ ɞɢɚɝɨɧɚɥɶɸ, ɪɚɜɧɨɣ ɫɦ, ɢ ɪɚɞɢɭ ɫɨɦ, ɪɚɜɧɵɦ ɫɦ

6)Ɋɚɜɧɨɛɟɞɪɟɧɧɚɹ ɬɪɚɩɟɰɢɹ ɫɨ ɫɬɨɪɨɧɚɦɢ ɨɫɧɨɜɚɧɢɹ ɪɚɜɧɵ ɦɢ ɢ ɫɦ, ɢ ɜɵɫɨɬɨɣ, ɪɚɜɧɨɣ ɫɦ

7)ɉɪɚɜɢɥɶɧɚɹ ɱɟɬɵɪɟɯɭɝɨɥɶɧɚɹ ɩɢɪɚɦɢɞɚ ɫɨ ɫɬɨɪɨɧɨɣ ɨɫɧɨ ɜɚɧɢɹ, ɪɚɜɧɨɣ ɫɦ, ɢ ɜɵɫɨɬɨɣ, ɪɚɜɧɨɣ ɫɦ

8)ɉɪɹɦɨɭɝɨɥɶɧɵɣ ɩɚɪɚɥɥɟɥɟɩɢɩɟɞ ɫ ɜɵɫɨɬɨɣ ɫɦ ɫɬɨɪɨ ɧɨɣ ɨɫɧɨɜɚɧɢɹ ɢ ɞɢɚɝɨɧɚɥɶɸ ɨɫɧɨɜɚɧɢɹ ɫɦ

9)Ɋɚɜɧɨɛɟɞɪɟɧɧɚɹ ɬɪɚɩɟɰɢɹ ɫɨ ɫɬɨɪɨɧɚɦɢ ɨɫɧɨɜɚɧɢɹ ɪɚɜɧɵ ɦɢ ɢ ɫɦ, ɢ ɜɵɫɨɬɨɣ, ɪɚɜɧɨɣ ɫɦ

10)ɉɪɚɜɢɥɶɧɚɹ ɱɟɬɵɪɟɯɭɝɨɥɶɧɚɹ ɩɢɪɚɦɢɞɚ ɫɨ ɫɬɨɪɨɧɨɣ ɨɫ ɧɨɜɚɧɢɹ, ɪɚɜɧɨɣ ɫɦ, ɢ ɜɵɫɨɬɨɣ, ɪɚɜɧɨɣ ɫɦ

11)Ʉɨɧɭɫ ɫ ɜɵɫɨɬɨɣ, ɪɚɜɧɨɣ ɫɦ, ɢ ɪɚɞɢɭɫɨɦ ɨɫɧɨɜɚɧɢɹ ɪɚɜɧɵɦ ɫɦ

12)ɉɪɹɦɨɭɝɨɥɶɧɵɣ ɩɚɪɚɥɥɟɥɟɩɢɩɟɞ ɫ ɜɵɫɨɬɨɣ ɫɦ ɫɬɨɪɨ ɧɨɣ ɨɫɧɨɜɚɧɢɹ ɢ ɞɢɚɝɨɧɚɥɶɸ ɨɫɧɨɜɚɧɢɹ ɫɦ

13)ɐɢɥɢɧɞɪ ɫ ɨɛɪɚɡɭɸɳɟɣ, ɪɚɜɧɨɣ ɫɦ, ɢ ɝɥɚɜɧɨɣ ɞɢɚɝɨ ɧɚɥɶɸ, ɪɚɜɧɨɣ ɫɦ

14)Ɋɚɜɧɨɛɟɞɪɟɧɧɚɹ ɬɪɚɩɟɰɢɹ ɫɨ ɫɬɨɪɨɧɚɦɢ ɨɫɧɨɜɚɧɢɹ ɪɚɜ ɧɵɦɢ ɢ ɫɦ, ɢ ɜɵɫɨɬɨɣ, ɪɚɜɧɨɣ ɫɦ

15)ɉɪɚɜɢɥɶɧɚɹ ɱɟɬɵɪɟɯɭɝɨɥɶɧɚɹ ɩɢɪɚɦɢɞɚ ɫɨ ɫɬɨɪɨɧɨɣ ɨɫ ɧɨɜɚɧɢɹ, ɪɚɜɧɨɣ ɫɦ, ɢ ɜɵɫɨɬɨɣ, ɪɚɜɧɨɣ ɫɦ

16)ɉɪɹɦɨɭɝɨɥɶɧɵɣ ɩɚɪɚɥɥɟɥɟɩɢɩɟɞ ɫɨ ɫɬɨɪɨɧɨɣ ɨɫɧɨɜɚɧɢɹ 12 ɫɦ ɟɝɨ ɞɢɚɝɨɧɚɥɶɸ ɫɦ ɢ ɜɵɫɨɬɨɣ ɫɦ

ɉɊɂȻɅɂɀȿɇɂȿ ɎɍɇɄɐɂɃ

ȼɨ ɦɧɨɝɢɯ ɫɥɭɱɚɹɯ ɮɭɧɤɰɢɹ ɡɚɞɚɟɬɫɹ ɬɚɛɥɢɱɧɨ ɬ.ɟ. ɢɡɜɟɫɬɧɵ ɟɺ ɡɧɚɱɟɧɢɹ ɬɨɥɶɤɨ ɜ ɭɡɥɨɜɵɯ ɬɨɱɤɚɯ ɭɡɥɚɯ

Ɍɚɛɥɢɰɚ

N	0		1		...	n

X		X 0		X1	...	X n
Y		Y0		Y1	...	Yn

ɇɟɨɛɯɨɞɢɦɨ ɩɨɫɬɪɨɢɬɶ ɮɭɧɤɰɢɸ ɩɪɢɛɥɢɡɢɬɟɥɶɧɨ ɨɩɢɫɵɜɚɸ ɳɭɸ ɡɚɜɢɫɢɦɨɫɬɶ ɦɟɠɞɭ ɭɡɥɚɦɢ ɉɪɢɛɥɢɠɚɸɳɚɹ ɮɭɧɤɰɢɹ ɨɛɵɱɧɨ ɛɟɪɟɬɫɹ ɜ ɜɢɞɟ ɫɭɦɦɵ ɷɥɟɦɟɧɬɚɪɧɵɯ ɮɭɧɤɰɢɣ ɇɚ ɩɪɚɤɬɢɤɟ ɢɫ ɩɨɥɶɡɭɸɬɫɹ ɫɬɟɩɟɧɧɵɟ ɩɨɤɚɡɚɬɟɥɶɧɵɟ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɟ ɮɭɧɤɰɢɢ ȼ ɞɚɥɶɧɟɣɲɟɦ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɩɨɥɢɧɨɦɢɚɥɶɧɨɟ ɩɪɢ ɛɥɢɠɟɧɢɟ ɬ ɟ ɩɪɢɛɥɢɠɚɸɳɚɹ ɮɭɧɤɰɢɹ ɢɦɟɟɬ ɜɢɞ

P	x	a	a x a x2		... a xn .	(2.1)
n		0	1	2	n

ɋɭɳɟɫɬɜɭɸɬ ɞɜɚ ɨɫɧɨɜɧɵɯ ɤɪɢɬɟɪɢɹ ɭɫɥɨɜɢɹ ɩɨɫɬɪɨɟɧɢɹ ɩɪɢɛɥɢɠɚɸɳɢɯ ɮɭɧɤɰɢɣ Ʉɪɢɬɟɪɢɣ ɢɧɬɟɪɩɨɥɹɰɢɢ ɬɪɟɛɭɟɬ ɱɬɨɛɵ ɩɪɢɛɥɢɠɚɸɳɚɹ ɮɭɧɤɰɢɹ ɩɪɨɯɨɞɢɥɚ ɱɟɪɟɡ ɭɡɥɵ Ʉɪɢɬɟɪɢɣ ɚɩɩɪɨɤ ɫɢɦɚɰɢɢ ɬɪɟɛɭɟɬ ɦɢɧɢɦɢɡɚɰɢɢ ɧɟɤɨɬɨɪɨɝɨ ɮɭɧɤɰɢɨɧɚɥɚ

ɂɧɬɟɪɩɨɥɹɰɢɨɧɧɵɟ ɩɨɥɢɧɨɦɵ

ɉɨɥɢɧɨɦ ɫɬɟɩɟɧɢ n ɨɞɧɨɡɧɚɱɧɨ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɜɨɢɦɢ ɡɧɚɱɟ ɧɢɹɦɢ ɜ (n+1)-ɣ ɬɨɱɤɟ ɫ ɩɨɩɚɪɧɨ ɪɚɡɧɵɦɢ ɚɛɫɰɢɫɫɚɦɢ xi z x j ɟɫ

ɥɢ i z j Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ ɜɵɩɢɲɟɦ ɫɨɝɥɚɫɧɨ ɤɪɢɬɟɪɢɸ ɢɧɬɟɪɩɨ

ɥɹɰɢɢ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ Pn xi			yi ɢɥɢ ɜ ɪɚɡɜɟɪɧɭɬɨɦ ɜɢɞɟ
a	a x		... a	xn	y
° 0	1	0	n	0	0
°a0	a1	x1 ... an x1n			y1
®............................................ .
°
°a	a x		... a	xn	y	n
¯ 0	1	n	n	n		n

ɋɢɫɬɟɦɚ n+1)-ɝɨ ɭɪɚɜɧɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ai ,0 d i d n ɢɦɟɟɬ ɟɞɢɧɫɬɜɟɧɧɨɟ ɪɟɲɟɧɢɟ ɟɫɥɢ xi z x j ɬɚɤ ɤɚɤ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɨɩɪɟɞɟɥɢ

ɬɟɥɶ ɧɟ ɪɚɜɟɧ ɋɭɳɟɫɬɜɭɸɬ ɦɟɬɨɞɵ ɩɨɡɜɨɥɹɸɳɢɟ ɢɡɛɟɠɚɬɶ ɧɟɩɨ ɫɪɟɞɫɬɜɟɧɧɨɝɨ ɪɟɲɟɧɢɹ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ ai .

ɂɧɬɟɪɩɨɥɹɰɢɨɧɧɵɣ ɩɨɥɢɧɨɦ Ʌɚɝɪɚɧɠɚ

Ɋɚɫɫɦɨɬɪɢɦ ɜɧɚɱɚɥɟ n=1 (2.1):

Pn x	a0	a1x; ®a0 a1x0	y0 ;
		¯a0 a1x1	y1

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