Математика для менеджеров. Часть I. Учебное пособие

ɑɬɨɛɵ ɜɜɟɫɬɢ ɮɨɪɦɭɥɵ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɤɨɨɪɞɢɧɚɬ ɩɪɢ ɩɚ ɪɚɥɥɟɥɶɧɨɦ ɩɟɪɟɧɨɫɟ ɨɫɟɣ ɜɵɞɟɥɢɦ ɜɵɪɚɠɟɧɢɟ x ɫɩɪɚɜɚ

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4.B = C = 0, Ax + D = 0 - ɩɥɨɫɤɨɫɬɶ ɩɚɪɚɥɥɟɥɶɧɚ ɩɥɨɫɤɨɫɬɢ 2\] ɍɪɚɜɧɟɧɢɹ ɤɨɨɪɞɢɧɚɬɧɵɯ ɩɥɨɫɤɨɫɬɟɣ [ \ ] ɉɪɹɦɚɹ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɦɨɠɟɬ ɛɵɬɶ ɡɚɞɚɧɚɤɚɤ ɥɢɧɢɹ ɩɟɪɟɫɟɱɟɧɢɹ ɞɜɭɯ ɩɥɨɫɤɨɫɬɟɣ ɬ ɟ ɫɢɫɬɟɦɨɣ

ɭɪɚɜɧɟɧɢɣ

A1 x + B1 y + C1 z + D1 = 0, A2 x + B2 y + C2 z + D2 = 0.

2) ɞɜɭɦɹ ɫɜɨɢɦɢ ɬɨɱɤɚɦɢ M1(x1, y1, z1 ɢ 02(x2, y2, z2 ɬɨ ɝɞɚ ɩɪɹɦɚɹ ɱɟɪɟɡ ɧɢɯ ɩɪɨɯɨɞɹɳɚɹ ɡɚɞɚɟɬɫɹ ɭɪɚɜɧɟɧɢɹɦɢ

ɬɨɱɤɨɣ 01(x1, y1, z1 ɟɣ ɩɪɢɧɚɞɥɟɠɚɳɟɣ ɢ ɜɟɤɬɨɪɨɦ a (m, Q ɪ ɟɣ ɤɨɥɥɢɧɟɚɪɧɵɦ Ɍɨɝɞɚ ɩɪɹɦɚɹ ɨɩɪɟɞɟɥɹɟɬɫɹ ɭɪɚɜɧɟɧɢɹ

ɗɬɨ ɤɚɧɨɧɢɱɟɫɤɢɟ ɭɪɚɜɧɟɧɢɹ ɩɪɹɦɨɣ.

ȼɟɤɬɨɪ a ɧɚɡɵɜɚɟɬɫɹ ɧɚɩɪɚɜɥɹɸɳɢɦ ɜɟɤɬɨɪɨɦ ɩɪɹɦɨɣ ɉɚɪɚɦɟɬɪɢɱɟɫɤɢɟ ɭɪɚɜɧɟɧɢɹ ɩɪɹɦɨɣ ɩɨɥɭɱɢɦ ɩɪɢɪɚɜɧɹɜ

ɤɚɠɞɨɟ ɢɡ ɨɬɧɨɲɟɧɢɣ ɤɚɧɨɧɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ ɩɪɹɦɨɣ ɤ ɩɚɪɚ ɦɟɬɪɭ t:

x = x1 +mt, y = y1 + nt, z = z1 ɪW

ɉɪɢɦɟɪ ɋɨɫɬɚɜɢɬɶ ɭɪɚɜɧɟɧɢɟ ɩɥɨɫɤɨɫɬɢ ɡɧɚɹ ɱɬɨ ɬɨɱɤɚ Ⱥ - ɫɥɭɠɢɬ ɨɫɧɨɜɚɧɢɟɦ ɩɟɪɩɟɧɞɢɤɭɥɹɪɚ ɩɪɨɜɟɞɟɧɧɨɝɨ ɢɡ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ ɤ ɷɬɨɣ ɩɥɨɫɤɨɫɬɢ

Ɋɟɲɟɧɢɟ ɉɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ɜɟɤɬɨɪ ɈȺ(1,- ɹɜɥɹɟɬɫɹ ɧɨɪɦɚɥɶɧɵɦ ɜɟɤɬɨɪɨɦ ɩɥɨɫɤɨɫɬɢ ɬɨɝɞɚ ɟɟ ɭɪɚɜɧɟɧɢɟ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ [-\ ] ' ɉɨɞɫɬɚɜɢɜ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ Ⱥ -ɩɪɢɧɚɞɥɟɠɚɳɟɣ ɩɥɨɫɤɨɫɬɢ ɧɚɣɞɟɦ ' -(-1)+3 3+D = 0 D = - ɂɬɚɤ [-y+3z-11=0.

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ɌȿɆȺ5. ɎɍɇɄɐɂə ɈȾɇɈɃ ɉȿɊȿɆȿɇɇɈɃ

Ɇɚɬɟɦɚɬɢɱɟɫɤɢɣ ɚɧɚɥɢɡ ɞɚɟɬ ɪɹɞ ɮɭɧɞɚɦɟɧɬɚɥɶɧɵɯ ɩɨ ɧɹɬɢɣ ɤɨɬɨɪɵɦɢ ɨɩɟɪɢɪɭɟɬ ɷɤɨɧɨɦɢɫɬ - ɷɬɨ ɮɭɧɤɰɢɹ ɩɪɟɞɟɥ ɩɪɨɢɡɜɨɞɧɚɹ ɢɧɬɟɝɪɚɥ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɇɚɩɪɢ ɦɟɪ ɜɬɨɪɨɣ ɡɚɦɟɱɚɬɟɥɶɧɵɣ ɩɪɟɞɟɥ ɩɪɢɦɟɧɹɟɬɫɹ ɩɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱ ɨ ɪɨɫɬɟ ɛɚɧɤɨɜɫɤɨɝɨ ɜɤɥɚɞɚ ɩɨ ɡɚɤɨɧɭ ɫɥɨɠɧɵɯ ɩɪɨɰɟɧɬɨɜ ɢɫɩɨɥɶɡɨɜɚɧɢɟ ɩɨɧɹɬɢɹ ɩɪɨɢɡɜɨɞɧɨɣ ɩɪɢɜɨɞɢɬ ɤ ɬɚɤɨɣ ɫɩɟɰɢ ɚɥɶɧɨɣ ɞɢɫɰɢɩɥɢɧɟ ɤɚɤ ɩɪɟɞɟɥɶɧɵɣ ɚɧɚɥɢɡ ɜ ɷɤɨɧɨɦɢɤɟ ɢ ɬ ɞ

5 ɉɨɧɹɬɢɟ ɦɧɨɠɟɫɬɜɚ

ɉɨɧɹɬɢɟ ɦɧɨɠɟɫɬɜɚ ɹɜɥɹɟɬɫɹ ɨɞɧɢɦ ɢɡ ɨɫɧɨɜɧɵɯ ɩɟɪ ɜɢɱɧɵɯ ɧɟɨɩɪɟɞɟɥɹɟɦɵɯ ɩɨɧɹɬɢɣ ɦɚɬɟɦɚɬɢɤɢ

Ɇɧɨɠɟɫɬɜɨ – ɫɨɜɨɤɭɩɧɨɫɬɶ ɨɛɴɟɤɬɨɜ ɨɛɴɟɞɢɧɟɧɧɵɯ ɩɨ ɤɚɤɨɦɭ-ɥɢɛɨ ɩɪɢɡɧɚɤɭ Ȼɭɞɟɦ ɫɱɢɬɚɬɶ ɦɧɨɠɟɫɬɜɨ ɡɚɞɚɧɧɵɦ ɟɫ ɥɢ ɢɡɜɟɫɬɟɧ ɡɚɤɨɧ ɩɨ ɤɨɬɨɪɨɦɭ ɦɨɠɧɨ ɫɤɚɡɚɬɶ ɩɪɢɧɚɞɥɟɠɢɬ ɧɟ ɤɨɬɨɪɵɣ ɨɛɴɟɤɬ ɞɚɧɧɨɦɭ ɦɧɨɠɟɫɬɜɭ ɢɥɢ ɧɟɬ Ɉɛɴɟɤɬɵ ɢɡ ɤɨɬɨ ɪɵɯ ɫɨɫɬɨɢɬ ɦɧɨɠɟɫɬɜɨ ɧɚɡɵɜɚɸɬɫɹ ɟɝɨ ɷɥɟɦɟɧɬɚɦɢ Ɇɧɨɠɟ ɫɬɜɚ ɩɪɢɧɹɬɨ ɨɛɨɡɧɚɱɚɬɶ ɡɚɝɥɚɜɧɵɦɢ ɛɭɤɜɚɦɢ ɚɥɮɚɜɢɬɚ A , B,ɚ ɢɯ ɷɥɟɦɟɧɬɵ – ɦɚɥɵɦɢ a , b, ȿɫɥɢ ɷɥɟɦɟɧɬ x ɩɪɢɧɚɞ ɥɟɠɢɬ ɦɧɨɠɟɫɬɜɭ X ɬɨ ɩɢɲɭɬ x X ɡɚɩɢɫɶ x X - ɨɡɧɚɱɚɟɬ ɱɬɨ ɷɥɟɦɟɧɬ x ɧɟ ɩɪɢɧɚɞɥɟɠɢɬ ɦɧɨɠɟɫɬɜɭ X .

Ɇɧɨɠɟɫɬɜɨ ɧɟ ɫɨɞɟɪɠɚɳɟɟ ɧɢ ɨɞɧɨɝɨ ɷɥɟɦɟɧɬɚ ɧɚɡɵɜɚ ɟɬɫɹ ɩɭɫɬɵɦ ɢ ɨɛɨɡɧɚɱɚɟɬɫɹ ɫɢɦɜɨɥɨɦ ɗɥɟɦɟɧɬɵ ɦɧɨɠɟɫɬɜɚ ɡɚɩɢɫɵɜɚɸɬɫɹ ɜ ɮɢɝɭɪɧɵɯ ɫɤɨɛɤɚɯ ɜɧɭɬɪɢ ɤɨɬɨɪɵɯ ɨɧɢ ɩɟɪɟ ɱɢɫɥɟɧɵ ɥɢɛɨ ɭɤɚɡɚɧɨ ɨɛɳɟɟ ɫɜɨɣɫɬɜɨ ɤɨɬɨɪɵɦ ɨɛɥɚɞɚɸɬ ɜɫɟ ɷɥɟɦɟɧɬɵ ɞɚɧɧɨɝɨ ɦɧɨɠɟɫɬɜɚ ɇɚɩɪɢɦɟɪ A 1, 2, 3 ,

A x: 0 x 2 .

Ɇɧɨɠɟɫɬɜɨ A ɧɚɡɵɜɚɟɬɫɹ ɩɨɞɦɧɨɠɟɫɬɜɨɦ ɦɧɨɠɟɫɬɜɚ

Bɟɫɥɢ ɤɚɠɞɵɣ ɷɥɟɦɟɧɬ ɦɧɨɠɟɫɬɜɚ A ɩɪɢɧɚɞɥɟɠɢɬ ɦɧɨɠɟɫɬɜɭ

Bɨɛɨɡɧɚɱɚɟɬɫɹ ɬɚɤ A B.

Ɇɧɨɠɟɫɬɜɚ A ɢ B ɧɚɡɵɜɚɸɬɫɹ ɪɚɜɧɵɦɢ ɟɫɥɢ A B ɢ

B A .

ȿɫɥɢ ɜ ɪɚɦɤɚɯ ɧɟɤɨɬɨɪɨɝɨ ɪɚɫɫɭɠɞɟɧɢɹ ɪɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɩɨɞɦɧɨɠɟɫɬɜɚ ɧɟɤɨɬɨɪɨɝɨ ɦɧɨɠɟɫɬɜɚ ɬɨ ɨɧɨ ɧɚɡɵɜɚɟɬɫɹ ɭɧɢ ɜɟɪɫɚɥɶɧɵɦ ɢɥɢ ɭɧɢɜɟɪɫɭɦɨɦ ɢ ɨɛɨɡɧɚɱɚɟɬɫɹ (

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Ɇɧɨɠɟɫɬɜɨ ɦɨɠɟɬ ɛɵɬɶ ɡɚɞɚɧɨ ɪɚɡɥɢɱɧɵɦɢ ɫɩɨɫɨɛɚɦɢ ɩɟɪɟɱɢɫɥɟɧɢɟɦ ɷɥɟɦɟɧɬɨɜ ɜ ɫɤɨɛɤɚɯ ɞɥɹ ɤɨɧɟɱɧɵɯ ɦɧɨ

ɠɟɫɬɜ ɢɥɢ ɭɤɚɡɚɧɢɟɦ ɢɯ ɫɜɨɣɫɬɜ ɨɞɧɨɡɧɚɱɧɨ ɨɩɪɟɞɟɥɹɸɳɢɯ ɩɪɢɧɚɞɥɟɠɧɨɫɬɶ ɷɥɟɦɟɧɬɨɜ ɞɚɧɧɨɦɭ ɦɧɨɠɟɫɬɜɭ ɩɪɢ ɷɬɨɦ ɢɫ

3 [ ` ɜɵɪɚɠɟɧɢɟ ɜ ɫɤɨɛɤɚɯ ɱɢɬɚɟɬɫɹ ɦɧɨɠɟɫɬɜɨ ɜɫɟɯ ɷɥɟɦɟɧɬɨɜ [ ɤɨɬɨɪɵɟ ɨɛɥɚɞɚɸɬ ɫɜɨɣɫɬɜɨɦ 3 [ Ɍɚɤ ɦɧɨɠɟɫɬɜɨ ɧɚɬɭɪɚɥɶ ɧɵɯ ɱɢɫɟɥ 1 ^ ` ɦɨɠɟɬ ɛɵɬɶ ɨɩɢɫɚɧɨ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ

N={n ɟɫɥɢ ɰɟɥɨɟ n N, ɬɨ n 1 N,n 1}.

5 Ɉɩɟɪɚɰɢɢ ɧɚɞ ɦɧɨɠɟɫɬɜɚɦɢ

Ⱦɥɹ ɩɨɥɭɱɟɧɢɹ ɧɨɜɵɯ ɦɧɨɠɟɫɬɜ ɢɡ ɭɠɟ ɫɭɳɟɫɬɜɭɸɳɢɯ ɢɫɩɨɥɶɡɭɸɬ ɨɩɟɪɚɰɢɢ ɧɚɞ ɦɧɨɠɟɫɬɜɚɦɢ Ɋɚɫɫɦɨɬɪɢɦ ɨɫɧɨɜɧɵɟ ɢɡ ɧɢɯ

ɋɭɦɦɨɣ ɢɥɢ ɨɛɴɟɞɢɧɟɧɢɟɦ ɦɧɨɠɟɫɬɜ ; ɢ < ɧɚɡɵɜɚɟɬɫɹ ɦɧɨɠɟɫɬɜɨ ɢɥɢ X Y ɜɫɟ ɷɥɟɦɟɧɬɵ ɤɨɬɨɪɨɝɨ ɹɜɥɹɸɬɫɹ ɷɥɟɦɟɧɬɚɦɢ ɦɧɨɠɟɫɬɜɚ ; ɢɥɢ <

ɉɪɨɢɡɜɟɞɟɧɢɟɦ ɢɥɢ ɩɟɪɟɫɟɱɟɧɢɟɦ ɦɧɨɠɟɫɬɜ ; ɢ < ɧɚɡɵɜɚɟɬɫɹ ɦɧɨɠɟɫɬɜɨ ɢɥɢ X Y ɷɥɟɦɟɧɬɵ ɤɨɬɨɪɨɝɨ ɹɜɥɹɸɬɫɹ ɷɥɟɦɟɧɬɚɦɢ ɨɛɨɢɯ ɦɧɨɠɟɫɬɜ ; ɢ <

={x | x ; ɢ \ Y}.

Ⱦɨɩɨɥɧɟɧɢɟɦ ɦɧɨɠɟɫɬɜɚ ; ɧɚɡɵɜɚɟɬɫɹ ɦɧɨɠɟɫɬɜɨ ɜɫɟɯ ɬɟɯ ɷɥɟɦɟɧɬɨɜ [ ɤɨɬɨɪɵɟ ɧɟ ɩɪɢɧɚɞɥɟɠɚɬ ɦɧɨɠɟɫɬɜɭ ;

={x | x E ɢ x X}.

Ɋɚɡɧɨɫɬɶɸ ɦɧɨɠɟɫɬɜ ; ɢ < ɧɚɡɵɜɚɟɬɫɹ ɦɧɨɠɟɫɬɜɨ ;\Y ɜɫɟɯ ɬɟɯ ɷɥɟɦɟɧɬɨɜ ; ɤɨɬɨɪɵɟ ɧɟ ɩɪɢɧɚɞɥɟɠɚɬ <

X\Y={x x ɢ x }=X Y .

Ⱦɨɩɨɥɧɟɧɢɟ ɦɧɨɠɟɫɬɜɚ ɏ ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɫ ɩɨɦɨɳɶɸ ɨɩɟɪɚɰɢɢ ɪɚɡɧɨɫɬɢ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ =E\X.

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ɋɢɦɦɟɬɪɢɱɧɨɣ ɪɚɡɧɨɫɬɶɸ ɦɧɨɠɟɫɬɜɚ ; ɢ < ɧɚɡɵɜɚɟɬɫɹ ɦɧɨɠɟɫɬɜɨ XǻY ɜɫɟɯ ɷɥɟɦɟɧɬɨɜ ; ɢ < ɤɨɬɨɪɵɟ ɧɟ ɜɯɨɞɹɬ ɨɞ

ɧɨɜɪɟɦɟɧɧɨ ɜ ɷɬɢ ɨɛɚ ɦɧɨɠɟɫɬɜɚ

( \ ) ( \ ) (X Y) \ (X Y) .

ɉɪɢɦɟɪ Ⱦɚɧɵ ɦɧɨɠɟɫɬɜɚ

A {2, 4, 5, 8, 9},B {0, 3, 4, 6, 7, 9}.

ɇɚɯɨɞɢɦ A B {0, 2, 3, 4, 5, 6, 7, 8, 9}, A B {4, 9}, A \ B {2, 5, 8},B \ A {0, 3, 6, 7, 8},A B {0, 2, 3, 5, 6, 7, 8}.

ɇɚɝɥɹɞɧɨ ɝɪɚɮɢɱɟɫɤɢ ɩɪɟɞɫɬɚɜɢɬɶ ɨɩɟɪɚɰɢɢ ɧɚɞ ɦɧɨɠɟ ɫɬɜɚɦɢ ɩɨɡɜɨɥɹɸɬ ɤɪɭɝɢ ɗɣɥɟɪɚ ɤɨɬɨɪɵɟ ɜ ɥɢɬɟɪɚɬɭɪɟ ɟɳɟ ɧɚɡɵɜɚɸɬɫɹ ɞɢɚɝɪɚɦɦɚɦɢ ȼɟɧɧɚ ȼɧɭɬɪɢ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɨɛɥɚ ɫɬɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɭɧɢɜɟɪɫɚɥɶɧɨɦɭ ɦɧɨɠɟɫɬɜɭ ɢɡɨɛɪɚɠɚɸɬ ɫɹ ɩɟɪɟɫɟɤɚɸɳɢɟɫɹ ɨɤɪɭɠɧɨɫɬɢ ɤɚɠɞɚɹ ɢɡ ɤɨɬɨɪɵɯ ɫɨɨɬɜɟɬ ɫɬɜɭɟɬ ɬɨɦɭ ɢɥɢ ɢɧɨɦɭ ɦɧɨɠɟɫɬɜɭ ɉɪɢɦɟɪɵ ɢɡɨɛɪɚɠɟɧɢɹ ɨɩɟ ɪɚɰɢɣ ɧɚɞ ɦɧɨɠɟɫɬɜɚɦɢ ɫ ɩɨɦɨɳɶɸ ɤɪɭɝɨɜ ɗɣɥɟɪɚ

ɉɪɢɦɟɪ. ɂɡ ɬɭɪɢɫɬɨɜ ɨɬɩɪɚɜɥɹɸɳɢɯɫɹ ɜ ɡɚɝɪɚɧɢɱ ɧɨɟ ɩɭɬɟɲɟɫɬɜɢɟ ɧɟɦɟɰɤɢɦ ɹɡɵɤɨɦ ɜɥɚɞɟɸɬ ɱɟɥɨɜɟɤ ɚɧ ɝɥɢɣɫɤɢɦ - ɮɪɚɧɰɭɡɫɤɢɦ - Ⱥɧɝɥɢɣɫɤɢɦ ɢ ɧɟɦɟɰɤɢɦ ɨɞɧɨ ɜɪɟɦɟɧɧɨ ɜɥɚɞɟɸɬ ɱɟɥɨɜɟɤ ɚɧɝɥɢɣɫɤɢɦ ɢ ɮɪɚɧɰɭɡɫɤɢɦ - 10, ɧɟɦɟɰɤɢɦ ɢ ɮɪɚɧɰɭɡɫɤɢɦ - ɜɫɟɦɢ ɬɪɟɦɹ ɹɡɵɤɚɦɢ - ɋɤɨɥɶɤɨ ɢɡ ɬɭɪɢɫɬɨɜ ɧɟ ɜɥɚɞɟɸɬ ɧɢ ɨɞɧɢɦ ɹɡɵɤɨɦ"

66

Ɋɟɲɟɧɢɟ ȼɵɪɚɡɢɦ ɭɫɥɨɜɢɟ ɷɬɨɣ ɡɚɞɚɱɢ ɝɪɚɮɢɱɟɫɤɢ Ɉɛɨɡɧɚɱɢɦ ɤɪɭɝɨɦ ɬɟɯ ɤɬɨ ɡɧɚɟɬ ɚɧɝɥɢɣɫɤɢɣ ɞɪɭɝɢɦ ɤɪɭɝɨɦ - ɬɟɯ ɤɬɨ ɡɧɚɟɬ ɮɪɚɧɰɭɡɫɤɢɣ ɢ ɬɪɟɬɶɢɦ ɤɪɭɝɨɦ - ɬɟɯ ɤɬɨ ɡɧɚɸɬ ɧɟɦɟɰɤɢɣ

ȼɫɟɦɢ ɬɪɟɦɹ ɹɡɵɤɚɦɢ ɜɥɚɞɟɸɬ ɬɪɢ ɬɭɪɢɫɬɚ ɡɧɚɱɢɬ ɜ ɨɛɳɟɣ ɱɚɫɬɢ ɤɪɭɝɨɜ ɜɩɢɫɵɜɚɟɦ ɱɢɫɥɨ Ⱥɧɝɥɢɣɫɤɢɦ ɢ ɮɪɚɧɰɭɡ ɫɤɢɦ ɹɡɵɤɨɦ ɜɥɚɞɟɸɬ ɱɟɥɨɜɟɤ ɚ ɢɡ ɧɢɯ ɜɥɚɞɟɸɬ ɟɳɟ ɢ ɧɟɦɟɰɤɢɦ ɋɥɟɞɨɜɚɬɟɥɶɧɨ ɬɨɥɶɤɨ ɚɧɝɥɢɣɫɤɢɦ ɢ ɮɪɚɧɰɭɡɫɤɢɦ ɜɥɚɞɟɸɬ - ɱɟɥɨɜɟɤ Ⱥɧɚɥɨɝɢɱɧɨ ɩɨɥɭɱɚɟɦ ɱɬɨ ɬɨɥɶɤɨ ɚɧɝɥɢɣɫɤɢɦ ɢ ɧɟɦɟɰɤɢɦ ɜɥɚ ɞɟɸɬ - ɱɟɥɨɜɟɤ ɚ ɧɟɦɟɰɤɢɦ ɢ ɮɪɚɧɰɭɡɫɤɢɦ - ɬɭɪɢɫɬɚ ȼɧɨɫɢɦ ɷɬɢ ɞɚɧɧɵɟ ɜ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɱɚɫɬɢ

Ɉɩɪɟɞɟɥɢɦ ɬɟɩɟɪɶ ɫɤɨɥɶɤɨ ɱɟɥɨɜɟɤ ɜɥɚɞɟɸɬ ɬɨɥɶɤɨ ɨɞ ɧɢɦ ɢɡ ɩɟɪɟɱɢɫɥɟɧɧɵɯ ɹɡɵɤɨɜ ɇɟɦɟɰɤɢɣ ɡɧɚɸɬ ɱɟɥɨɜɟɤ ɧɨɢɡ ɧɢɯ ɜɥɚɞɟɸɬ ɢ ɞɪɭɝɢɦɢ ɹɡɵɤɚɦɢ ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɬɨɥɶɤɨ ɧɟɦɟɰɤɢɣ ɡɧɚɸɬ ɱɟɥɨɜɟɤ Ⱥɧɚɥɨɝɢɱɧɨ ɩɨɥɭɱɚɟɦ ɱɬɨ ɨɞɧɢɦ ɚɧɝɥɢɣɫɤɢɦ ɜɥɚɞɟɸɬ ɱɟɥɨɜɟɤ ɚ ɨɞɧɢɦ ɮɪɚɧɰɭɡɫɤɢɦ -ɱɟɥɨɜɟɤ

ɉɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ɜɫɟɝɨ ɬɭɪɢɫɬɨɜɬɭɪɢɫɬɨɜ ɡɧɚɸɬ ɯɨɬɹ ɛɵ ɨɞɢɧ ɹɡɵɤ ɫɥɟ ɞɨɜɚɬɟɥɶɧɨ ɱɟɥɨɜɟɤ ɧɟ ɜɥɚɞɟɸɬ ɧɢ ɨɞɧɢɦ ɢɡ ɞɚɧɧɵɯ ɹɡɵɤɨɜ

5 ɑɢɫɥɨɜɵɟ ɦɧɨɠɟɫɬɜɚ

ȼ ɦɚɬɟɦɚɬɢɱɟɫɤɨɦ ɚɧɚɥɢɡɟ ɜ ɨɫɧɨɜɧɨɦ ɢɫɩɨɥɶɡɭɸɬ ɦɧɨ ɠɟɫɬɜɚ ɫɨɫɬɨɹɳɢɟ ɢɡ ɱɢɫɟɥ Ɉɧɢ ɧɚɡɵɜɚɸɬɫɹ ɱɢɫɥɨɜɵɦɢ ɦɧɨ ɠɟɫɬɜɚɦɢ

67

ɇɚɢɛɨɥɟɟ ɱɚɫɬɨ ɢɫɩɨɥɶɡɭɸɬɫɹ ɫɥɟɞɭɸɳɢɟ ɱɢɫɥɨɜɵɟ ɦɧɨɠɟɫɬɜɚ

–N 1, 2, 3, , n, - ɦɧɨɠɟɫɬɜɨ ɧɚɬɭɪɚɥɶɧɵɯ ɱɢɫɟɥ

–Z 0, 1, 2, , n, - ɦɧɨɠɟɫɬɜɨ ɰɟɥɵɯ ɱɢɫɟɥ

–Q m , m Z, n N - ɦɧɨɠɟɫɬɜɨ ɪɚɰɢɨɧɚɥɶɧɵɯ ɱɢɫɟɥ

n

–R – ɦɧɨɠɟɫɬɜɨ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɱɢɫɟɥ

–I x: x R, x Q - ɦɧɨɠɟɫɬɜɨ ɢɪɪɚɰɢɨɧɚɥɶɧɵɯ ɱɢɫɟɥ

Ɇɟɠɞɭ ɷɬɢɦɢ ɦɧɨɠɟɫɬɜɚɦɢ ɫɭɳɟɫɬɜɭɟɬ ɫɨɨɬɧɨɲɟɧɢɟ

N Z Q R .

5.4. Ɇɨɞɭɥɶ ɞɟɣɫɬɜɢɬɟɥɶɧɨɝɨ ɱɢɫɥɚ

Ɇɨɞɭɥɟɦ ɚɛɫɨɥɸɬɧɨɣ ɜɟɥɢɱɢɧɨɣ ɞɟɣɫɬɜɢɬɟɥɶɧɨɝɨ ɱɢɫɥɚ a ɧɚɡɵɜɚɟɬɫɹ ɱɢɫɥɨ a , ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɫɨɨɬɧɨɲɟɧɢɸ

5.5. Ɉɤɪɟɫɬɧɨɫɬɶ ɬɨɱɤɢ

ȼɫɹɤɢɣ ɢɧɬɟɪɜɚɥ ɫɨɞɟɪɠɚɳɢɣ ɬɨɱɤɭ x0 ɧɚɡɵɜɚɟɬɫɹ

ɨɤɪɟɫɬɧɨɫɬɶɸ ɬɨɱɤɢ x0 .

- ɨɤɪɟɫɬɧɨɫɬɶɸ U (x0 )ɬɨɱɤɢ x0 ɢɦɟɸɳɟɣ ɪɚɞɢɭɫ , ɧɚɡɵɜɚɟɬɫɹ ɢɧɬɟɪɜɚɥ ( x0 ; x0 ).

ȿɫɥɢ ɨɤɪɟɫɬɧɨɫɬɢ ɧɟ ɩɪɢɧɚɞɥɟɠɢɬ ɫɚɦɚ ɬɨɱɤɚ x0 ɬɨ ɬɚ

ɤɚɹ ɨɤɪɟɫɬɧɨɫɬɶ ɧɚɡɵɜɚɟɬɫɹ ɩɪɨɤɨɥɨɬɨɣ ɢ ɨɛɨɡɧɚɱɚɟɬɫɹ

U (x0 ) x0 ;x0 x0 ;x0 = x0 ;x0 \ x0 ; ( –

ɷɩɫɢɥɨɧ ɛɭɤɜɚ ɝɪɟɱɟɫɤɨɝɨ ɚɥɮɚɜɢɬɚ

68

ɉɪɢɦɟɪ. Ɂɚɩɢɫɚɬɶ ɨɤɪɟɫɬɧɨɫɬɢ U1 2 ; U5 13 ɩɪɨɦɟɠɭɬ

5.6. Ɉɩɪɟɞɟɥɟɧɢɟ ɮɭɧɤɰɢɢ

ɉɭɫɬɶ X ɢY– ɩɪɨɢɡɜɨɥɶɧɵɟ ɦɧɨɠɟɫɬɜɚ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɱɢɫɟɥ ȿɫɥɢ ɧɚ ɦɧɨɠɟɫɬɜɟX ɤɚɠɞɨɦɭ ɱɢɫɥɭ x X ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɞɟɣɫɬɜɢɬɟɥɶɧɨɟ ɱɢɫɥɨ y Y ɬɨ ɝɨɜɨɪɹɬ ɱɬɨ ɧɚ ɦɧɨɠɟɫɬɜɟX

ɨɩɪɟɞɟɥɟɧɚ ɞɟɣɫɬɜɢɬɟɥɶɧɚɹ ɮɭɧɤɰɢɹy f x ɞɟɣɫɬɜɢɬɟɥɶɧɨɣ

ɩɟɪɟɦɟɧɧɨɣx. ɆɧɨɠɟɫɬɜɨX ɧɚɡɵɜɚɟɬɫɹ ɨɛɥɚɫɬɶɸ ɨɩɪɟɞɟɥɟɧɢɹ ɚ ɦɧɨɠɟɫɬɜɨY– ɦɧɨɠɟɫɬɜɨɦ ɡɧɚɱɟɧɢɣ ɱɢɫɥɨɜɨɣ ɮɭɧɤɰɢɢf x .

ɉɪɢɦɟɪ. ɇɚɣɬɢ ɨɛɥɚɫɬɶ ɨɩɪɟɞɟɥɟɧɢɹ ɮɭɧɤɰɢɢ

Ɋɟɲɟɧɢɟ ɗɬɨ ɜɵɪɚɠɟɧɢɟ ɢɦɟɟɬ ɱɢɫɥɨɜɨɟ ɡɧɚɱɟɧɢɟ ɟɫɥɢ 3x 6 0 , 7x 56 0 ɢ x2 3x 2 0 ɂɧɵɦɢ ɫɥɨɜɚɦɢ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ ɨɛɥɚɫɬɢ ɨɩɪɟɞɟɥɟɧɢɹ ɧɚɞɨ ɢɫɤɥɸɱɢɬɶ ɢɡRɤɨɪɧɢ ɭɪɚɜɧɟɧɢɣ 3x 6 0, 7x 56 0 ɢ x2 3x 2 0 Ɋɟɲɚɹ ɷɬɢ ɭɪɚɜɧɟɧɢɹ ɩɨɥɭɱɚɟɦ ɤɨɪɧɢ - ɢ ɡɚɩɢɫɵɜɚɟɦ ɨɛɥɚɫɬɶ ɨɩɪɟɞɟɥɟɧɢɹ ɞɚɧɧɨɣ ɮɭɧɤɰɢɢ

x , 2 2,1 1,2 2,8 8, .

ȼ ɧɟɤɨɬɨɪɵɯ ɫɥɭɱɚɹɯ ɮɭɧɤɰɢɹ ɡɚɞɚɟɬɫɹ ɧɚ ɪɚɡɥɢɱɧɵɯ ɱɢɫɥɨɜɵɯ ɦɧɨɠɟɫɬɜɚɯ ɪɚɡɧɵɦɢ ɜɵɪɚɠɟɧɢɹɦɢ ɧɚɩɪɢɦɟɪ

ɇɚ ɩɪɚɤɬɢɤɟ ɱɚɫɬɨ ɭɞɨɛɧɵɦ ɨɤɚɡɵɜɚɟɬɫɹ ɬɚɛɥɢɱɧɵɣ ɫɩɨ ɫɨɛ ɡɚɞɚɧɢɹ ɮɭɧɤɰɢɣ ɧɚɩɪɢɦɟɪ ɩɪɢ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɢɡɦɟ ɪɟɧɢɹɯ ɫɨɰɢɨɥɨɝɢɱɟɫɤɢɯ ɨɩɪɨɫɚɯ ɬ ɞ ɇɚ ɬɚɛɥɢɱɧɨɦ ɫɩɨɫɨɛɟ

69

ɡɚɞɚɧɢɹ ɯɪɚɧɟɧɢɹ ɢ ɨɛɪɚɛɨɬɤɢ ɢɧɮɨɪɦɚɰɢɢ ɨɫɧɨɜɚɧɵ ɛɚɡɵ ɞɚɧ ɧɵɯ ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɬɚɛɥɢɰɚ ɢɦɟɟɬ ɜɢɞ

Ɉɧɚ ɩɨɡɜɨɥɹɟɬ ɧɚɯɨɞɢɬɶ ɡɧɚɱɟɧɢɹ ɮɭɧɤɰɢɢ ɞɥɹ ɜɵɛɪɚɧ ɧɵɯ ɡɧɚɱɟɧɢɣ ɚɪɝɭɦɟɧɬɚ Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɬɚɛɥɢɰɚ ɧɟ ɡɚɞɚɟɬ ɮɭɧɤɰɢɢ ɩɨɫɤɨɥɶɤɭ ɞɥɹ ɡɚɞɚɧɢɹ ɮɭɧɤɰɢɢ ɧɚɞɨ ɡɧɚɬɶ ɟɟ ɡɧɚɱɟ ɧɢɹ ɞɥɹ ɜɫɟɯ x X ɚ ɧɟ ɬɨɥɶɤɨ ɞɥɹ ɧɟɤɨɬɨɪɵɯ ɋɭɳɟɫɬɜɭɸɬ ɦɟɬɨɞɵ ɩɨɡɜɨɥɹɸɳɢɟ ɩɨ ɬɚɤɨɣ ɬɚɛɥɢɰɟ ɩɨɞɛɢɪɚɬɶ ɜɵɪɚɠɟɧɢɟ f x ɪɚɡɭɦɟɟɬɫɹ ɫ ɨɩɪɟɞɟɥɟɧɧɨɣ ɬɨɱɧɨɫɬɶɸ

M x, y ɧɚ ɤɨɨɪɞɢɧɚɬɧɨɣ ɩɥɨɫɤɨɫɬɢ ɋɥɟɞɨɜɚɬɟɥɶɧɨ ɝɪɚɮɢɤ

ɱɢɫɥɨɜɨɣ ɮɭɧɤɰɢɢ ɦɨɠɟɬ ɛɵɬɶ ɧɚɝɥɹɞɧɨ ɢɡɨɛɪɚɠɟɧ ɦɧɨɠɟɫɬɜɨɦ ɬɨɱɟɤ ɤɨɨɪɞɢɧɚɬɧɨɣ ɩɥɨɫɤɨɫɬɢ

5.7 ɋɜɨɣɫɬɜɚ ɮɭɧɤɰɢɣ

1.Ɉɝɪɚɧɢɱɟɧɧɨɫɬɶ Ɏɭɧɤɰɢɹ f ɡɚɞɚɧɧɚɹ ɧɚ ɦɧɨɠɟ ɫɬɜɟX ɧɚɡɵɜɚɟɬɫɹ ɨɝɪɚɧɢɱɟɧɧɨɣ ɫɜɟɪɯɭ ɫɧɢɡɭ ɧɚ ɷɬɨɦ ɦɧɨɠɟ ɫɬɜɟ ɟɫɥɢ ɫɭɳɟɫɬɜɭɟɬ ɬɚɤɨɟ ɱɢɫɥɨM ɱɬɨ ɞɥɹ ɜɫɟɯ x X ɜɵɩɨɥ ɧɹɟɬɫɹ ɧɟɪɚɜɟɧɫɬɜɨf x M f x M .

2.Ɇɨɧɨɬɨɧɧɨɫɬɶ Ɏɭɧɤɰɢɹ f ɧɚɡɵɜɚɟɬɫɹ

–ɜɨɡɪɚɫɬɚɸɳɟɣ ɧɚ X ɟɫɥɢ ɛɨɥɶɲɢɦ ɡɧɚɱɟɧɢɹɦ ɚɪɝɭɦɟɧ ɬɚ x ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɛɨɥɶɲɢɟ ɡɧɚɱɟɧɢɹ ɮɭɧɤɰɢɢ y.

–ɭɛɵɜɚɸɳɟɣ ɧɚ X ɟɫɥɢ ɛɨɥɶɲɢɦ ɡɧɚɱɟɧɢɹɦ ɚɪɝɭɦɟɧɬɚ [ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɦɟɧɶɲɢɟ ɡɧɚɱɟɧɢɹ ɮɭɧɤɰɢɢ \

3. ɑɟɬɧɨɫɬɶ ɢ ɧɟɱɟɬɧɨɫɬɶ Ɏɭɧɤɰɢɹ f ɡɚɞɚɧɧɚɹ ɧɚ ɦɧɨɠɟɫɬɜɟ X ɧɚɡɵɜɚɟɬɫɹ ɱɟɬɧɨɣ ɟɫɥɢ ɞɥɹ ɥɸɛɵɯ ɚɪɝɭɦɟɧɬɨɜ x

ɫɩɪɚɜɟɞɥɢɜɨ I –[ I [ Ɏɭɧɤɰɢɹ f ɡɚɞɚɧɧɚɹ ɧɚ ɦɧɨɠɟɫɬɜɟ X, ɧɚɡɵɜɚɟɬɫɹ ɧɟɱɟɬɧɨɣ ɟɫɥɢ ɞɥɹ ɥɸɛɵɯ ɚɪɝɭɦɟɧɬɨɜ ɯ ɫɩɪɚɜɟɞɥɢɜɨ f(–x) = –f(x).

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