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Математика для менеджеров. Часть I. Учебное пособие

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101

ɌȿɆȺ ɎɍɇɄɐɂɂ ɇȿɋɄɈɅɖɄɂɏ ɉȿɊȿɆȿɇɇɕɏ

Ɇɧɨɝɢɦ ɷɤɨɧɨɦɢɱɟɫɤɢɦ ɹɜɥɟɧɢɹɦ ɩɪɢɫɭɳɚ ɦɧɨɝɨɮɚɤɬɨɪ ɧɚɹ ɡɚɜɢɫɢɦɨɫɬɶ ɂɫɫɥɟɞɨɜɚɧɢɟ ɬɚɤɢɯ ɡɚɜɢɫɢɦɨɫɬɟɣ ɩɨɬɪɟɛɨɜɚɥɨ ɫɨɜɟɪɲɟɧɫɬɜɨɜɚɧɢɹ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɚɩɩɚɪɚɬɚ ɜ ɱɚɫɬɧɨɫɬɢ ɜɜɟɞɟɧɢɹ ɩɨɧɹɬɢɹ ɮɭɧɤɰɢɢ ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɟɦɟɧɧɵɯ

9.1. Ɉɫɧɨɜɧɵɟ ɩɨɧɹɬɢɹ

ɉɟɪɟɦɟɧɧɚɹ z ɧɚɡɵɜɚɟɬɫɹ ɮɭɧɤɰɢɟɣ ɩɟɪɟɦɟɧɧɵɯ ɯ ɢ ɭ, ɟɫɥɢ ɤɚɠɞɨɣ ɩɚɪɟ ɡɧɚɱɟɧɢɣ ɯ ɢ ɭ ɜ ɧɟɤɨɬɨɪɨɣ ɨɛɥɚɫɬɢ ɢɯ ɢɡɦɟ ɧɟɧɢɹ ɩɨɫɬɚɜɥɟɧɨ ɜ ɫɨɨɬɜɟɬɫɬɜɢɟ ɨɞɧɨ ɡɧɚɱɟɧɢɟ z Ɏɭɧɤɰɢɨ ɧɚɥɶɧɭɸ ɡɚɜɢɫɢɦɨɫɬɶ z ɨɬ ɯ ɢ ɭ ɡɚɩɢɫɵɜɚɸɬ ɜ ɜɢɞɟ z=f(x,ɭ).

ɗɬɨ ɭɪɚɜɧɟɧɢɟ ɨɩɪɟɞɟɥɹɟɬ ɧɟɤɨɬɨɪɭɸ ɩɨɜɟɪɯɧɨɫɬɶ ɜ ɩɪɨɫɬɪɚɧ ɫɬɜɟ R3.

ȼ ɷɤɨɧɨɦɢɱɟɫɤɢɯ ɢɫɫɥɟɞɨɜɚɧɢɹɯ ɱɚɫɬɨ ɢɫɩɨɥɶɡɭɟɬɫɹ ɩɪɨɢɡ ɜɨɞɫɬɜɟɧɧɚɹ ɮɭɧɤɰɢɹ Ʉɨɛɛɚ-Ⱦɭɝɥɚɫɚ z = Ax y ɝɞɟ z - ɜɟɥɢ

ɱɢɧɚ ɨɛɳɟɫɬɜɟɧɧɨɝɨ ɩɪɨɞɭɤɬɚ x - ɡɚɬɪɚɬɵ ɬɪɭɞɚ y - ɨɛɴɟɦ ɩɪɨ ɢɡɜɨɞɫɬɜɟɧɧɵɯ ɮɨɧɞɨɜ ɨɛɵɱɧɨ z ɢ y ɢɡɦɟɪɹɸɬɫɹ ɜ ɫɬɨɢɦɨɫɬɧɵɯ ɟɞɢɧɢɰɚɯ x - ɜ ɱɟɥɨɜɟɤɨ-ɱɚɫɚɯ $ , - ɩɨɫɬɨɹɧɧɵɟ Ɏɭɧɤɰɢɹ Ʉɨɛɛɚ-Ⱦɭɝɥɚɫɚ ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɟɣ ɞɜɭɯ ɩɟɪɟɦɟɧɧɵɯ ] I [ \

Ƚɪɚɮɢɤɨɦ ɮɭɧɤɰɢɢ ɞɜɭɯ ɩɟɪɟɦɟɧɧɵɯ ݖ = (ݔ,ݕ) ɧɚɡɵ ɜɚɟɬɫɹ ɦɧɨɠɟɫɬɜɨ ɬɨɱɟɤ ɩɪɨɫɬɪɚɧɫɬɜɚ (ݔ,ݕ,ݖ) ɚɩɩɥɢɤɚɬɚ ݖ ɤɨɬɨ ɪɵɯ ɫɜɹɡɚɧɚ ɫ ɚɛɫɰɢɫɫɨɣ ݔ ɢ ɨɪɞɢɧɚɬɨɣ ݕ ɮɭɧɤɰɢɨɧɚɥɶɧɵɦ ɫɨ ɨɬɧɨɲɟɧɢɟɦ ݖ = (ݔ,ݕ).

Ʌɢɧɢɟɣ ɭɪɨɜɧɹ ɮɭɧɤɰɢɢ ɞɜɭɯ ɩɟɪɟɦɟɧɧɵɯ

ɧɚɡɵɜɚɟɬɫɹ ɦɧɨɠɟɫɬɜɨ ɬɨɱɟɤ ɧɚ ɩɥɨɫɤɨɫɬɢ ɬɚɤɢɯ ɱɬɨ ɜɨ ɜɫɟɯ

ݖ = (ݔ,ݕ)

ɷɬɢɯ ɬɨɱɤɚɯ ɡɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ ɨɞɧɨ ɢ ɬɨ ɠɟ ɢ ɪɚɜɧɨ

ɬɨ

ɟɫɬɶ

ɨɛ

Ƚɟɨɦɟɬɪɢɱɟɫɤɢɦ

(ݔ,ݕ) =

ɪɚɡɨɦ ɮɭɧɤɰɢɢ z=x

ɹɜɥɹɟɬɫɹ ɩɚɪɚɛɨɥɨɢɞ ɉɭɫɬɶ

z x2

z=a ɬɨɝɞɚ

x2+y2=a,

ɬ ɟ

ɥɢɧɢɹ ɩɟɪɟɫɟɱɟɧɢɹ ɩɥɨɫɤɨ

ɫɬɢ z=a

ɩɨɜɟɪɯɧɨɫɬɶɸ

z=x2+y2

ɟɫɬɶ ɨɤɪɭɠɧɨɫɬɶ

x2+y2=a ɪɚɞɢɭɫɚ R a ɉɭɫɬɶ ɭ ɬɨɝɞɚ z=x2 ɢ ɫɥɟɞɨɜɚɬɟɥɶ

ɧɨ ɩɪɢ ɩɟɪɟɫɟɱɟɧɢɢ ɩɥɨɫɤɨɫɬɢ Oɯz ɫ ɩɨɜɟɪɯɧɨɫɬɶɸ ɩɨɥɭɱɚɟɬɫɹ ɩɚɪɚɛɨɥɚ Ɇɟɬɨɞ ɫɟɱɟɧɢɣ ɞɚɟɬ ɜɨɡɦɨɠɧɨɫɬɶ ɥɭɱɲɟ ɩɪɟɞɫɬɚɜɢɬɶ ɫɟɛɟ ɝɟɨɦɟɬɪɢɱɟɫɤɢɣ ɨɛɪɚɡ ɞɚɧɧɨɣ ɮɭɧɤɰɢɢ

ɑɢɫɥɨ Ⱥ ɧɚɡɵɜɚɟɬɫɹ ɩɪɟɞɟɥɨɦ ɮɭɧɤɰɢɢz=f(x,ɭ) ɜ ɬɨɱɤɟ Ɇ0(ɯ0, ɭ0 ɟɫɥɢ ɞɥɹ ɤɚɠɞɨɝɨ ɱɢɫɥɚ İ! ɧɚɣɞɟɬɫɹ ɬɚɤɨɟ ɱɢɫɥɨ ȕ! ɱɬɨ ɞɥɹ ɜɫɟɯ ɬɨɱɟɤ Ɇ(ɯ,ɭ ɞɥɹ ɤɨɬɨɪɵɯ ɜɵɩɨɥɧɹɟɬɫɹ ɧɟɪɚ ɜɟɧɫɬɜɨ _ɆɆ0|<ȕ ɛɭɞɟɬ ɜɵɩɨɥɧɹɬɶɫɹ ɧɟɪɚɜɟɧɫɬɜɨ |f(x,ɭ)–A|< .

Ɉɛɨɡɧɚɱɢɦ lim f(x, y) A.

x x0 y y0

Ɏɭɧɤɰɢɹ z=f(x,ɭ) ɧɚɡɵɜɚɟɬɫɹ ɧɟɩɪɟɪɵɜɧɨɣ ɜ ɬɨɱɤɟ

Ɇ0(ɯ0,ɭ0 ɟɫɥɢ ɢɦɟɟɬ ɦɟɫɬɨ ɪɚɜɟɧɫɬɜɨ lim f(x, y) f(x0, y0) .

x x0 y y0

9.2. ɑɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɢ ɩɨɥɧɵɣ ɞɢɮɮɟɪɟɧɰɢɚɥ -ɝɨ ɩɨɪɹɞɤɚ

ɉɪɨɢɡɜɨɞɧɚɹ ɨɬ ɮɭɧɤɰɢɢ z=f(x,ɭ) ɩɨ ɯ ɧɚɣɞɟɧɧɚɹ ɜ ɩɪɟɞ ɥɨɠɟɧɢɢ ɱɬɨ ɭ ɨɫɬɚɟɬɫɹ ɩɨɫɬɨɹɧɧɵɦ ɧɚɡɵɜɚɟɬɫɹ ɱɚɫɬɧɨɣ ɩɪɨ

ɢɡɜɨɞɧɨɣ ɨɬ z ɩɨ ɯ ɢ ɨɛɨɡɧɚɱɚɟɬɫɹ xz ɢɥɢ f'x (x,ɭ) Ⱥɧɚɥɨɝɢɱɧɨ

ɨɩɪɟɞɟɥɹɟɬɫɹ ɢ ɨɛɨɡɧɚɱɚɟɬɫɹ ɱɚɫɬɧɚɹ ɩɪɨɢɡɜɨɞɧɚɹ z ɩɨ ɭ. ȼɵɪɚɠɟɧɢɟ dxz x dyz y ɹɜɥɹɟɬɫɹ ɝɥɚɜɧɨɣ ɱɚɫɬɶɸ ɩɨɥɧɨɝɨ

ɩɪɢɪɚɳɟɧɢɹ z ɢ ɧɚɡɵɜɚɟɬɫɹ ɩɨɥɧɵɦ ɞɢɮɮɟɪɟɧɰɢɚɥɨɦ ɮɭɧɤ ɰɢɢz=f(x,ɭ) ɢ ɨɛɨɡɧɚɱɚɟɬɫɹ dz:

dz xz x yz y .

ɉɨɥɚɝɚɹ z ɪɚɜɧɵɦ ɯ ɧɚɣɞɟɦ dx x ɚ ɩɪɢ z=y dy y .

ɉɨɷɬɨɦɭ dz	z	dx	z	dy .

	dx		dy
Ɂɚɞɚɱɚ		ɇɚɣɬɢ ɩɨɥɧɵɣ ɞɢɮɮɟɪɟɧɰɢɚɥ ɮɭɧɤɰɢɢ
z 5x2 6y 3x2 y3 .

Ɋɟɲɟɧɢɟ ɋɧɚɱɚɥɚ ɧɚɣɞɟɦ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ

102

z 5x2 6y 3x2 y3 x 10x 0 3 2xy3 10x 6xy3,

z 5x2 6y 3x2 y3 y 0 6 3x2 3y2 6 9x2 y2.y

	ɉɪɨɢɡɜɨɞɧɚɹ					z	ɧɚɣɞɟɧɚ ɜ ɩɪɟɞɩɨɥɨɠɟɧɢɢ ɱɬɨ ɭ ɩɨɫɬɨ
	ɉɪɨɢɡɜɨɞɧɚɹ						ɧɚɣɞɟɧɚ ɜ ɩɪɟɞɩɨɥɨɠɟɧɢɢ ɱɬɨ ɭ ɩɨɫɬɨ
			z			x
ɹɧɧɚ ɚ			z	ɧɚɣɞɟɧɚ ɜ ɩɪɟɞɩɨɥɨɠɟɧɢɢ ɱɬɨ ɯ ɩɨɫɬɨɹɧɧɚ ɉɨɥɭɱɚɟɦ
ɹɧɧɚ ɚ			y	ɧɚɣɞɟɧɚ ɜ ɩɪɟɞɩɨɥɨɠɟɧɢɢ ɱɬɨ ɯ ɩɨɫɬɨɹɧɧɚ ɉɨɥɭɱɚɟɦ
			y
dz	z	dx		z	dy (10x 6xy3 )dx (9x2 y2 6)dy .
	x			y

9.3. Ƚɪɚɞɢɟɧɬ ɮɭɧɤɰɢɢ ɉɪɨɢɡɜɨɞɧɚɹ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ

ɉɭɫɬɶ ɮɭɧɤɰɢɹz=f(x,ɭ) ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚɹ ɮɭɧɤɰɢɹ ɞɜɭɯ

ɩɟɪɟɦɟɧɧɵɯ Ɍɨɝɞɚ ɜɟɤɬɨɪ gradz

ɧɚɡɵɜɚɟɬɫɹ ɝɪɚɞɢ

ɟɧɬɨɦ ɮɭɧɤɰɢɢ z=f(x,ɭ).

Ɉɧ ɨɛɥɚɞɚɟɬ ɫɥɟɞɭɸɳɢɦɢ ɫɜɨɣɫɬɜɚɦɢ:

gradC 0,

c const,

grad(U ) gradU grad ,

grad(CU ) CgradU, C const,

grad(U ) gradU Ugrad ,

grad

gradU Ugrad .

ɉɭɫɬɶ cos ,

cos – ɧɚɩɪɚɜɥɹɸɳɢɟ ɤɨɫɢɧɭɫɵ ɧɟɤɨɬɨɪɨɝɨ

ɜɟɤɬɨɪɚ ɬ ɟ cos i cos j Ɍɨɝɞɚ

cos

cos –

ɩɪɨɢɡɜɨɞɧɚɹ ɮɭɧɤɰɢɢ z=f(x,ɭ) ɜ ɞɚɧɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ .

9.4. ɗɤɫɬɪɟɦɭɦ ɮɭɧɤɰɢɢ ɞɜɭɯ ɩɟɪɟɦɟɧɧɵɯ

Ɏɭɧɤɰɢɹ z=f(x,ɭ) ɢɦɟɟɬ ɜ ɬɨɱɤɟ Ɇ0(ɯ0,ɭ0) ɦɚɤɫɢɦɭɦ ɟɫɥɢ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɷɬɨɣ ɬɨɱɤɢ ɜɵɩɨɥɧɹɟɬɫɹ ɪɚɜɟɧɫɬɜɨ f(x,ɭ)<f(x0,ɭ0).

103

Ⱥɧɚɥɨɝɢɱɧɨ ɨɩɪɟɞɟɥɹɟɬɫɹ ɦɢɧɢɦɭɦ ɮɭɧɤɰɢɢ z=f(x,ɭ) ɜ ɬɨɱɤɟ Ɇ0(ɯ0, ɭ0).

ɇɟɨɛɯɨɞɢɦɵɣ ɩɪɢɡɧɚɤ ɷɤɫɬɪɟɦɭɦɚ

ȿɫɥɢ Ɇ(ɯ0,ɭ0) – ɬɨɱɤɚ ɷɤɫɬɪɟɦɭɦɚ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɨɣ ɮɭɧɤɰɢɢ z=f(x,ɭ) ɬɨ

	f(x0, y0) 0	ɢ	f(x0, y0) 0, ɬɨ ɟɫɬɶ df(x		, y ) 0.
	x		y	0	0

Ⱦɨɫɬɚɬɨɱɧɵɣ ɩɪɢɡɧɚɤ ɷɤɫɬɪɟɦɭɦɚ
ɉɭɫɬɶz=f(x,ɭ)		– ɮɭɧɤɰɢɹ ɞɥɹ ɤɨɬɨɪɨɣ ɫɭɳɟɫɬɜɭɸɬ ɩɪɨ
ɢɡɜɨɞɧɵɟ ɩɟɪɜɨɝɨ		ɢ	ɜɬɨɪɨɝɨ	ɩɨɪɹɞɤɚ ɜ ɬɨɱɤɟ Ɇ(ɯ0,ɭ0):
				(x0, y0 ) ɋɨɫɬɚɜɢɦ ɜɵɪɚɠɟɧɢɟ
A fxx	(x0, y0), B fxy(x0, y0 ), C fyy

ǻ Ⱥɋ–ȼ2.

ȿɫɥɢ ǻ! ɬɨ Ɇ(ɯ0, ɭ0) – ɬɨɱɤɚ ɷɤɫɬɪɟɦɭɦɚ ɚ ɢɦɟɧɧɨ ɬɨɱɤɚ ɦɚɤɫɢɦɭɦɚ ɩɪɢ A ɟɫɥɢ C ɬɨɱɤɚ ɦɢɧɢɦɭɦɚ ɩɪɢ A! ɢɥɢ ɋ! ȿɫɥɢ ǻ ɬɨ ɜ ɬɨɱɤɟ Ɇ ɧɟɬ ɷɤɫɬɪɟɦɭɦɚ

Ɂɚɞɚɱɚ ɂɫɫɥɟɞɨɜɚɬɶ ɧɚ ɷɤɫɬɪɟɦɭɦ ɡɚɞɚɧɧɭɸ ɮɭɧɤɰɢɸ

ݖ = 2ݔ ݔݕ + 3ݕ 2ݔ 11ݕ +1.

Ɋɟɲɟɧɢɟ ɇɚɯɨɞɢɦ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ

’ = (2ݔ ݔݕ +3ݕ 2ݔ 11ݕ + 1)’ = 4ݔ ݕ 2;

’ = (2ݔ ݔݕ +3ݕ 2ݔ 11ݕ + 1)’ = ݔ + 6ݕ 11;

ȼɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɧɟɨɛɯɨɞɢɦɵɦ ɭɫɥɨɜɢɟɦ ɷɤɫɬɪɟɦɭɦɚ ɧɚɯɨɞɢɦ ɫɬɚɰɢɨɧɚɪɧɵɟ ɬɨɱɤɢ Ⱦɥɹ ɷɬɨɝɨ ɪɟɲɚɟɦ ɫɢɫɬɟɦɭ ɭɪɚɜ ɧɟɧɢɣ

					’		= 0,		4ݔ ݕ 2 = 0,
								ݔ +6ݕ 11 = 0,
						’	= 0,
							= 0,
ɨɬɤɭɞɚ							ɨɛɪɚɡɨɦ ɫɬɚɰɢɨɧɚɪɧɨɣ ɹɜɥɹɟɬɫɹ ɬɨɱ
ɨɬɤɭɞɚ					Ɍɚɤɢɦ
ɤɚ		ݔ =	1;ݕ = 2.
ɤɚ	(1;2)		ɇɚɯɨɞɢɦ ɡɧɚɱɟɧɢɹ ɱɚɫɬɵɯ ɩɪɨɢɡɜɨɞɧɵɯ ɜɬɨɪɨɝɨ ɩɨ
	(1;2)			:					= 4; = ’’ ( ) = 4;
ɪɹɞɤɚ ɜ ɬɨɱɤɟ
’’			’’ = (4ݔ ݕ 2)’
’’		= (4ݔ ݕ 2)’			= 1; = ’’ ( ) = 1.
’’		= (ݔ +6ݕ 11)’					= 6; = ’’ ( ) = 6.

ɋɨɫɬɚɜɥɹɟɦ ɜɵɪɚɠɟɧɢɟ = · = 4 ·6 ( 1) = 23. Ɍɚɤ ɤɚɤ > 0ˋ > 0, ɞɟɥɚɟɦ ɜɵɜɨɞ ɨ ɧɚɥɢɱɢɢ ɦɢɧɢɦɭɦɚ ɜ ɬɨɱɤɟ(1;2) ɉɪɢ ɷɬɨɦ ɦɢɧɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ = 11.

104

ɌȿɆȺ 10. ɈɉɊȿȾȿɅȿɇɇɕɃ ɂ ɇȿɈɉɊȿȾȿɅȿɇɇɕɃ ɂɇɌȿȽɊȺɅ

10 ɇɟɨɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ

Ɏɭɧɤɰɢɹ F(x) ɧɚɡɵɜɚɟɬɫɹ ɩɟɪɜɨɨɛɪɚɡɧɨɣ ɮɭɧɤɰɢɟɣ ɞɥɹ ɮɭɧɤɰɢɢ f(x) ɟɫɥɢ ɩɪɨɢɡɜɨɞɧɚɹ ɟɟ F'(x)= f(x).

ɋɨɜɨɤɭɩɧɨɫɬɶ ɜɫɟɯ ɩɟɪɜɨɨɛɪɚɡɧɵɯ ɮɭɧɤɰɢɣ F(x ɋɞɥɹ ɮɭɧɤɰɢɢ f(x) ɧɚɡɵɜɚɟɬɫɹ ɧɟɨɩɪɟɞɟɥɟɧɧɵɦ ɢɧɬɟɝɪɚɥɨɦ ɮɭɧɤ

ɰɢɢ f(x) ɢ ɨɛɨɡɧɚɱɚɟɬɫɹ f(x)dx F(x) C.

Ɍɚɛɥɢɰɚ ɨɫɧɨɜɧɵɯ ɢɧɬɟɝɪɚɥɨɜ

1. 1 dx x c . 3. 1x dx ln x c .

2. x dx		x 1		c	( 1) .
	1

4. axdx			ax	c	(a 0,a 1).
			lna

ɋɥɟɞɫɬɜɢɟ ex dx ex c .

5. sinxdx cosx c .						6. cosxdx sinx c .
7.	1			dx tgx c.		8.					1			dx ctgx c .
	2	x									2		x
	cos				arcsinx c;					sin				arctgx c;
9.	1					10.			1
					dx							dx
	1 x		2				1	x			2				c.
					arccosx c.									arcctgx

Ɉɫɧɨɜɧɵɟ ɫɜɨɣɫɬɜɚ ɢɧɬɟɝɪɚɥɨɜ

1.(U )dx Udx dx;

2.cUdx c Udx.

ɉɪɢɦɟɪ. ȼɵɱɢɫɥɢɬɶ ɢɧɬɟɝɪɚɥɵ

1. x3dx	x3 1	c	x4	c.
	3 1
		4

105

ɉɪɨɜɟɪɤɚ

(C)

x .

3sinx

xdx

dx 2

sin

3 sinxdx

3cosx ctgx

ln2

sin

10 Ɂɚɦɟɧɚ ɩɟɪɟɦɟɧɧɨɣ ɜ ɧɟɨɩɪɟɞɟɥɟɧɧɨɦ ɢɧɬɟɝɪɚɥɟ

Ɂɚɦɟɧɚ ɩɟɪɟɦɟɧɧɨɣ ɜ ɧɟɨɩɪɟɞɟɥɟɧɧɨɦ ɢɧɬɟɝɪɚɥɟ ɩɪɨɢɡ ɜɨɞɢɬɫɹ ɫ ɩɨɦɨɳɶɸ ɩɨɞɫɬɚɧɨɜɨɤ ɞɜɭɯ ɜɢɞɨɜ

ɚ x (t) ɝɞɟ (t) – ɦɨɧɨɬɨɧɧɚɹ ɧɟɩɪɟɪɵɜɧɨ ɞɢɮɮɟ ɪɟɧɰɢɪɭɟɦɚɹ ɮɭɧɤɰɢɹ ɧɨɜɨɣ ɩɟɪɟɦɟɧɧɨɣ t Ɏɨɪɦɭɥɚ ɡɚɦɟɧɵ ɩɟ


ɪɟɦɟɧɧɨɣ ɜ ɷɬɨɦ ɫɥɭɱɚɟ f(x)dx f (t) (t)dt ;
ɛ U (x) ɝɞɟ U – ɧɨɜɚɹ ɩɟɪɟɦɟɧɧɚɹ Ɏɨɪɦɭɥɚ ɡɚɦɟɧɵ
ɩɟɪɟɦɟɧɧɨɣ	ɩɪɢ	ɬɚɤɨɣ	ɩɨɞɫɬɚɧɨɜɤɟ
		f(U)dU .
f(x)dx f[ (x)] (x)dx		f(U)dU .

ɉɪɢɦɟɪ

ɇɚɣɬɢ ɢɧɬɟɝɪɚɥ 2lnxx 3 3 dx .

Ɋɟɲɟɧɢɟ ɉɟɪɟɩɢɲɟɦ ɞɚɧɧɵɣ ɢɧɬɟɝɪɚɥ ɜ ɜɢɞɟ

2lnx 3 3 1x dx Ɍɚɤ ɤɚɤ ɩɪɨɢɡɜɨɞɧɚɹ ɜɵɪɚɠɟɧɢɹ 2lnx 3

ɪɚɜɧɚ ɯ ɚ ɜɬɨɪɨɣ ɦɧɨɠɢɬɟɥɶ ɯ ɨɬɥɢɱɚɟɬɫɹ ɨɬ ɷɬɨɣ ɩɪɨɢɡɜɨɞ ɧɨɣ ɬɨɥɶɤɨ ɩɨɫɬɨɹɧɧɵɦ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɬɨ ɩɪɢɦɟɧɹɟɦ ɩɨɞ

ɫɬɚɧɨɜɤɭ 2lnx 3 t Ɍɨɝɞɚ 2 dx

dt,

1 dt . ɉɨɥɭɱɚɟɦ

2ln x 3 3

dx t3

t3dt

2ln x 3 4

ɇɚɣɬɢ ɢɧɬɟɝɪɚɥ

e2x

dx .

106

Ɋɟɲɟɧɢɟ.e2x t ɬɨɝɞɚ e2xdx 12dt ɢ


	e2x		dx	1			1	dt	1	arctgt C	1	arctge	2x	C .
e	4x	1	dx	2	t	2	1	dt	2	arctgt C	2	arctge		C .
e		1		2	t		1		2		2

10 ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɩɨ ɱɚɫɬɹɦ

ɇɚɯɨɠɞɟɧɢɟ ɢɧɬɟɝɪɚɥɚ Ud ɩɨ ɮɨɪɦɭɥɟ

Ud U dU ɧɚɡɵɜɚɟɬɫɹ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟɦ ɩɨ ɱɚɫɬɹɦ

Ɂɞɟɫɶ U=U(ɯ) ȣ ȣ x ɧɟɩɪɟɪɵɜɧɨ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɵɟ ɮɭɧɤɰɢɢ ɨɬ ɯ ɋ ɩɨɦɨɳɶɸ ɷɬɨɣ ɮɨɪɦɭɥɵ ɧɚɯɨɠɞɟɧɢɟ ɢɧɬɟɝɪɚɥɚ ɫɜɨɞɢɬɫɹ

ɤ ɨɬɵɫɤɚɧɢɸ ɞɪɭɝɨɝɨ ɢɧɬɟɝɪɚɥɚ dU ɟɟ ɩɪɢɦɟɧɟɧɢɟ ɰɟɥɟɫɨ

ɨɛɪɚɡɧɨ ɜ ɬɟɯ ɫɥɭɱɚɹɯ ɤɨɝɞɚ ɩɨɫɥɟɞɧɢɣ ɢɧɬɟɝɪɚɥ ɥɢɛɨ ɩɪɨɳɟ ɢɫɯɨɞɧɨɝɨ ɥɢɛɨ ɟɦɭ ɩɨɞɨɛɟɧ

ɉɪɢ ɷɬɨɦ ɡɚ U ɛɟɪɟɬɫɹ ɬɚɤɚɹ ɮɭɧɤɰɢɹ ɤɨɬɨɪɚɹ ɩɪɢ ɞɢɮ ɮɟɪɟɧɰɢɪɨɜɚɧɢɢ ɭɩɪɨɳɚɟɬɫɹ ɚ ɡɚ dȣ – ɬɚ ɱɚɫɬɶ ɩɨɞɵɧɬɟɝɪɚɥɶ ɧɨɝɨ ɜɵɪɚɠɟɧɢɹ ɢɧɬɟɝɪɚɥ ɨɬ ɤɨɬɨɪɨɣ ɢɡɜɟɫɬɟɧ ɢɥɢ ɦɨɠɟɬ ɛɵɬɶ ɧɚɣɞɟɧ


Ɍɚɤ		ɧɚɩɪɢɦɟɪ		ɞɥɹ ɢɧɬɟɝɪɚɥɨɜ ɜɢɞɚ			P(x)e xdx ,
P(x)sin xdx ,			P(x)cos xdx ɝɞɟ P(x) – ɦɧɨɝɨɱɥɟɧ ɡɚ Uɫɥɟ
ɞɭɟɬ ɩɪɢɧɹɬɶ P(x ɚ ɡɚ dȣ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɜɵɪɚɠɟɧɢɟ								e xdx ,
sin xdx,	cos xdx			Ⱦɥɹ	ɢɧɬɟɝɪɚɥɨɜ			ɜɢɞɚ
P(x)lnxdx,		P(x)arcsin xdx,			P(x)arccos xdx		ɡɚ	U ɩɪɢ
ɧɢɦɚɸɬɫɹ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɮɭɧɤɰɢɢ lnx,						arcsin x,	arccos x ,
ɚ ɡɚ dȣ – ɜɵɪɚɠɟɧɢɟ P(x)dx.
ɉɪɢɦɟɪ ɇɚɣɬɢ ɢɧɬɟɝɪɚɥ x sinxdx .
Ɋɟɲɟɧɢɟ			ɉɨɥɨɠɢɦ		U x,	d sindx		ɬɨɝɞɚ
dU dx,	cos x .

Ɉɬɫɸɞɚ

xsinxdx xcosx cosxdx xcosx sinx C.

107

10 ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɪɚɰɢɨɧɚɥɶɧɵɯ ɞɪɨɛɟɣ

Ɋɚɰɢɨɧɚɥɶɧɨɣ ɞɪɨɛɶɸ ɧɚɡɵɜɚɟɬɫɹ ɞɪɨɛɶ ɜɢɞɚ QP((xx)) ɝɞɟ

P(x ɢ Q(x) – ɦɧɨɝɨɱɥɟɧɵ Ɋɚɰɢɨɧɚɥɶɧɚɹ ɞɪɨɛɶ ɧɚɡɵɜɚɟɬɫɹ ɩɪɚ ɜɢɥɶɧɨɣ ɟɫɥɢ ɫɬɟɩɟɧɶ ɦɧɨɝɨɱɥɟɧɚ P(x ɧɢɠɟ ɫɬɟɩɟɧɢ ɦɧɨɝɨ ɱɥɟɧɚ Q(x ɜ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ ɞɪɨɛɶ ɧɚɡɵɜɚɟɬɫɹ ɧɟɩɪɚɜɢɥɶ ɧɨɣ.

ɉɭɫɬɶ ɧɟɨɛɯɨɞɢɦɨ ɧɚɣɬɢ ɢɧɬɟɝɪɚɥ ɨɬ ɧɟɩɪɚɜɢɥɶɧɨɣ ɪɚ ɰɢɨɧɚɥɶɧɨɣ ɞɪɨɛɢ ɉɪɢ ɩɨɦɨɳɢ ɞɟɥɟɧɢɹ ɩɨ ɩɪɚɜɢɥɭ ɞɟɥɟɧɢɹ ɦɧɨɝɨɱɥɟɧɨɜ ɧɟɩɪɚɜɢɥɶɧɭɸ ɪɚɰɢɨɧɚɥɶɧɭɸ ɞɪɨɛɶ ɦɨɠɧɨ ɩɪɟɞ ɫɬɚɜɢɬɶ ɜ ɜɢɞɟ ɫɭɦɦɵ ɰɟɥɨɣ ɪɚɰɢɨɧɚɥɶɧɨɣ ɮɭɧɤɰɢɢ ɢ ɩɪɚɜɢɥɶ

ɧɨɣ			ɪɚɰɢɨɧɚɥɶɧɨɣ					ɞɪɨɛɢ	ɇɚɩɪɢɦɟɪ
	x3	6	x 2		3x 4			.
	x2	2x 1		x2	2x
						1
		Ɂɚɬɟɦ ɡɧɚɦɟɧɚɬɟɥɶ ɩɪɚɜɢɥɶɧɨɣ ɞɪɨɛɢ ɪɚɡɥɚɝɚɟɬɫɹ ɧɚ
ɦɧɨɠɢɬɟɥɢ ɜɢɞɚ (x a) ɢ							(x2 x q) ɚ ɩɪɚɜɢɥɶɧɚɹ ɞɪɨɛɶ

ɪɚɡɥɚɝɚɟɬɫɹ ɧɚ ɫɭɦɦɭ ɷɥɟɦɟɧɬɚɪɧɵɯ ɞɪɨɛɟɣ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚ ɡɨɦ

P(x)

...

(x a) (x2 x q)

x a

(x a)2

(x a)

x N

...

M x N

x q

(x2 x q)2

(x2

x q)

ɉɪɢɦɟɪ ɇɚɣɬɢ ɢɧɬɟɝɪɚɥ

x3 x2

dx .

6x 5

Ɋɟɲɟɧɢɟ ȼɵɞɟɥɢɦ ɰɟɥɭɸ ɱɚɫɬɶ ɞɚɧɧɨɣ ɧɟɩɪɚɜɢɥɶɧɨɣ

ɞɪɨɛɢ

x3 x2

37x 35

x 7

x2 6x 5

x2 6x

Ɋɚɡɥɨɠɢɦ ɡɧɚɦɟɧɚɬɟɥɶ ɧɚ ɥɢɧɟɣɧɵɟ ɦɧɨɠɢɬɟɥɢ ɩɨ ɮɨɪ

ɦɭɥɟ ax2 bx c a(x x1)(x x2 ) ɝɞɟ ɯ1

ɢ ɯ2 – ɤɨɪɧɢ ɤɜɚɞ

ɪɚɬɧɨɝɨ

ɭɪɚɜɧɟɧɢɹ

ax2 bx c 0,

ɬɨ

ɟɫɬɶ

x2 6x 5 (x 1)(x 5).

108

37x 35

Ax 5A Bx B

x2 6x

(x 1)(x 5)

x 1

(x 1)(x 5)

(A B)x 5A B

(x 1)(x 5)

ɨɬɤɭɞɚ ɩɨɥɭɱɚɟɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ ɩɪɢɪɚɜɧɢɜɚɹ ɤɨɷɮɮɢɰɢɟɧ

A B 37;

ɬɵ ɩɪɢ ɨɞɢɧɚɤɨɜɵɯ ɫɬɟɩɟɧɹɯ ɫɥɟɜɚ ɢ ɫɩɪɚɜɚ

5A B 35.

Ɋɟɲɚɹɟɟ ɢɦɟɟɦ

4A 2,		A			1	,		B 37 A 37								1	37		1	,
					2											2			2
ɡɧɚɱɢɬ 4A 2,												1					37 1
		x	3	x	2							1		2			37 1		2
		x		x										2					2
x2 6x 5dx x 7											x 1							x 5 dx

		x2	7x				1	ln	x 1	37	1	ln	x		5			C.
		x2					1				1
	2						2				2
	2						2				2

10 Ɉɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ

ɉɭɫɬɶ ɧɚ ɨɬɪɟɡɤɟ >D E@ ɨɩɪɟɞɟɥɟɧɚ ɮɭɧɤɰɢɹ I [ Ɋɚɡɨɛɶɟɦ ɨɬɪɟɡɨɤ >D E@ ɧɚ n ɱɚɫɬɟɣ ɬɨɱɤɚɦɢ D [0 < x1<...<xn E ɂɡ ɤɚɠ ɞɨɝɨ ɢɧɬɟɪɜɚɥɚ [i 1, xi ɜɨɡɶɦɟɦ ɩɪɨɢɡɜɨɥɶɧɭɸ ɬɨɱɤɭ i ɢ ɫɨɫɬɚ

ɜɢɦ ɫɭɦɦɭ f( i) xi, ɝɞɟ xi = xi - xi 1.

i=1

ɋɭɦɦɚ ɜɢɞɚ f( i) xi ɧɚɡɵɜɚɟɬɫɹ ɢɧɬɟɝɪɚɥɶɧɨɣ ɫɭɦɦɨɣ,

i=1

ɚ ɟɟ ɩɪɟɞɟɥ ɩɪɢ = max xi ɟɫɥɢ ɨɧ ɫɭɳɟɫɬɜɭɟɬ ɢ ɤɨɧɟɱɟɧ ɧɚɡɵɜɚɟɬɫɹ ɨɩɪɟɞɟɥɟɧɧɵɦ ɢɧɬɟɝɪɚɥɨɦ ɮɭɧɤɰɢɢ I [ ɨɬ a ɞɨ b ɢ ɨɛɨɡɧɚɱɚɟɬɫɹ

b	n
f(x)dx lim f( i) xi.
a	0 i=1

109

Ɏɭɧɤɰɢɹ I [ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɧɚɡɵɜɚɟɬɫɹ ɢɧɬɟɝɪɢɪɭɟɦɨɣ ɧɚ ɨɬɪɟɡɤɟ >D E@ ɱɢɫɥɚ a ɢ b ɧɨɫɹɬ ɧɚɡɜɚɧɢɟ ɧɢɠɧɟɝɨ ɢ ɜɟɪɯɧɟɝɨ ɩɪɟɞɟɥɚ ɢɧɬɟɝɪɚɥɚ

Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ ɫɩɪɚɜɟɞɥɢɜɵ ɫɥɟɞɭɸɳɢɟ ɫɜɨɣɫɬɜɚ

b	b	b
1) f(x)dx f(z)dz f(t)dt ;
a	a	a

2) f(x)dx 0;

b a

3)f(x)dx - f(x)dx ;

b b

4)kf(x)dx k f(x)dx , (k = const, k R);

	b		b	b
5)	(f(x) g(x))dx		f(x)dx+ g(x)dx;
	a		a	a
	b	c	b
6)	f(x)dx	f(x)dx f(x)dx;
	a	a	c
	b
7)	f(x)dx	f( )(b-a) ( [a,b]).
	a
ɉɨɫɥɟɞɧɟɟ ɫɜɨɣɫɬɜɨ ɧɚɡɵɜɚɟɬɫɹ ɬɟɨɪɟɦɨɣ ɨ ɫɪɟɞɧɟɦ ɡɧɚɱɟ
ɧɢɢ.
	10.6. Ɏɨɪɦɭɥɚ ɇɶɸɬɨɧɚ – Ʌɟɣɛɧɢɰɚ
	b		b
	f(x)dx F(x) \| F(b)			F(a),

a	a

ɝɞɟ F(x) – ɩɟɪɜɨɨɛɪɚɡɧɚɹ ɞɥɹ f(x) ɬ ɟ F'(x)= f(x).

ɉɪɢɦɟɪ


ȼɵɱɢɫɥɢɬɶ 4	dx		ɩɨ ɮɨɪɦɭɥɟ ɇɶɸɬɨɧɚ – Ʌɟɣɛɧɢɰɚ
ȼɵɱɢɫɥɢɬɶ 4	2	x	ɩɨ ɮɨɪɦɭɥɟ ɇɶɸɬɨɧɚ – Ʌɟɣɛɧɢɰɚ
	cos	x
6

110

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