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Математика для менеджеров. Часть I. Учебное пособие

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				ɡɞɟɫɶ n=3, U													1 x4											2			2.
				ɡɞɟɫɶ n=3, U													1 x4											x2			2.
																	4									3		x2
														1											2								3 1						1													2
							y							1			x		4						2				2										1			x			4							2					2				.
				Ɍɨɝɞɚ
				Ɍɨɝɞɚ						3													3				2																							3			2
														4									3			x	2												4											3		x	2
				ɇɚɣɞɟɦ U/.										4												x													4													x
				ɇɚɣɞɟɦ U/.
			1		4		2													1					4												2																1			4									2
			1		4		2													1					4																												1			4
U				x							2													x								2 x					3								2										x							2x			3
U				x					2		2													x								2 x					3								2										x							2x			3
			4			3	x		2											4																																4
			4				x													4																																4
	1		4					2						1									4 1										2								2		1								3		4						5			3					4
	1		4											1									4 1										2										1								3		4									3					4
		x				2 x	3									4 x												2								x				3						x									x				3 x											.
		x				2 x	3									4 x												2								x				3						x									x				3 x							3			5	.
	4													4																			3																				3											3		3		x	5
	4													4																			3																				3											3				x
																		1										2						2																		4
																		1				4						2																3								4						.

ɋɥɟɞɨɜɚɬɟɥɶɧɨ										y				3							x					3				2			2								x											3
																		4								3		x		2																			3			3	x	5
																		4										x																					3				x
				2)		y 6tgx						x2 cos2x .
				Ɋɟɲɟɧɢɟ y 6tgx																																														x2			cos2x .
				Ɋɟɲɟɧɢɟ y 6tgx																		x2 cos2x																6tgx												x2			cos2x .
																	ɩɨ ɮɨɪɦɭɥɟ au)'=au·lna·U'.
				ɇɚɣɞɟɦ 6tgx													ɩɨ ɮɨɪɦɭɥɟ au)'=au·lna·U'.
																		tgx									tgx																					tgx													1
				Ȼɭɞɟɦ ɢɦɟɬɶ 6																				6						ln6(tg x)																	6					ln6											.
				Ȼɭɞɟɦ ɢɦɟɬɶ 6																				6						ln6(tg x)																	6					ln6						cos2 x					.
				ɉɪɨɢɡɜɨɞɧɭɸ ɯ2cos2x																															ɧɚɣɞɟɦ																	ɩɨ								ɮɨɪɦɭɥɟ
(Uāȣ)'=U ȣ Uȣ ɢ cosU)'=-sinU U Ɍɨɝɞɚ
x2										2 ) cos2x x2																	(cos2x) 2xcos2x x2 ( sin2x)(2x)
x2		cos2x) (x								2 ) cos2x x2																	(cos2x) 2xcos2x x2 ( sin2x)(2x)
2xcos2x 2x2 sin2x 2x(cos2x xsin2x).
ɋɥɟɞɨɜɚɬɟɥɶɧɨ										y				6	tgx						ln6									1			2x(cos2x																		xsin2x).
ɋɥɟɞɨɜɚɬɟɥɶɧɨ										y				6							ln6						cos2 x						2x(cos2x																		xsin2x).
				ɉɪɢɦɟɪ									ɇɚɣɬɢ														ɞɢɮɮɟɪɟɧɰɢɚɥ																								dy								ɮɭɧɤɰɢɢ
y arctg						1	arccos												x 2 .
y arctg					x	2	arccos												x 2 .
					x	2

Ɋɟɲɟɧɢɟ Ɍɚɤ ɤɚɤ ɞɢɮɮɟɪɟɧɰɢɚɥ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɵɪɚɠɟ ɧɢɟɦ dy=y'dx ɬɨɧɚɣɞɟɦɫɧɚɱɚɥɚɩɪɨɢɡɜɨɞɧɭɸy':

							1																								1
y			arctg								arccos										x		2				arctg								arccos						x 2 .
y			arctg				x		2		arccos										x		2				arctg			x					arccos						x 2 .
							x		2																					x				2
						Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɩɪɨɢɡɜɨɞɧɨɣ arctg																												1		ɩɪɢɦɟɧɢɦ ɮɨɪ
						Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɩɪɨɢɡɜɨɞɧɨɣ arctg																												x	2	ɩɪɢɦɟɧɢɦ ɮɨɪ
														U														1						x	2
														U														1
ɦɭɥɭ arctgU																			Ɂɞɟɫɶ U												Ɍɨɝɞɚ
ɦɭɥɭ arctgU													1 U2						Ɂɞɟɫɶ U									(x 2)			Ɍɨɝɞɚ
																1									x
					1																												( 1)(x 2)

													x				2											2 1												1 1
arctg
					x 2									1						2								1 2						1	1					2
											1													1
											1							2						1
														x				2									x	2							x 2
					1												1
				(x 2)2											(x		2)2											1										1				.
				(x 2)2											(x		2)2											1										1
				1				2					(x 2)2 1														(x 2)2				1				x2 4x 5
1				1				2					(x 2)2 1														(x 2)2				1				x2 4x 5

														(x				2)			2
																					2
				x			2
						Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɩɪɨɢɡɜɨɞɧɨɣ arccos																													x					ɩɪɢɦɟɧɢɦ
						Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɩɪɨɢɡɜɨɞɧɨɣ arccos																													x			2		ɩɪɢɦɟɧɢɦ
																					U'
ɮɨɪɦɭɥɭ arccos													U												.
ɮɨɪɦɭɥɭ arccos													U						1 U2						.
																			1 U2
																																					1
																									( x			2)'				x 2 2

						Ɍɨɝɞɚ arccos											x			2
																									1 ( x 2)2									1 x 2
	1					x 2 12 1										1		x 2							1						1
	1			2		x 2 12 1										1	2	x 2							2														.
				2													2
						x 1													x				1
						x 1													x				1					2 ( x 1)(x 2)
																											1									1
						ɋɥɟɞɨɜɚɬɟɥɶɧɨ y																x2					4x 5 2							( x 1)(x 2) .
						ɋɥɟɞɨɜɚɬɟɥɶɧɨ y																x2					4x 5 2							( x 1)(x 2) .
Ɉɬɫɸɞɚ												ɞɢɮɮɟɪɟɧɰɢɚɥ																	ɪɚɜɟɧ											dy=y'dx=
					1												1
																										dx .
			2																							dx .
		x	2	4x 5						2			( x 1)(x
		x		4x 5						2			( x 1)(x										2)

7.6. ɉɪɨɢɡɜɨɞɧɚɹ ɧɟɹɜɧɨɣ ɮɭɧɤɰɢɢ

ɉɪɟɞɩɨɥɨɠɢɦ ɱɬɨ ɡɧɚɱɟɧɢɹ ɞɜɭɯ ɩɟɪɟɦɟɧɧɵɯ x ɢyɫɜɹɡɚ ɧɵ ɦɟɠɞɭ ɫɨɛɨɣ ɭɪɚɜɧɟɧɢɟɦ F(x; y) = 0.

Ⱦɢɮɮɟɪɟɧɰɢɪɭɹ ɬɨɠɞɟɫɬɜɨ F(x; y ɩɨ ɩɟɪɟɦɟɧɧɨɣ x, ɧɚɣɞɟɦ ɩɪɨɢɡɜɨɞɧɭɸ ɧɟɹɜɧɨɣ ɮɭɧɤɰɢɢ

ɉɪɢɦɟɪ ɇɚɣɬɢ ɩɪɨɢɡɜɨɞɧɭɸ y (x ɧɟɹɜɧɨɣ ɮɭɧɤɰɢɢ xy 2 cos 3 x y 5x 0.

Ɋɟɲɟɧɢɟ ɉɪɨɞɢɮɮɟɪɟɧɰɢɪɭɟɦ ɞɚɧɧɨɟ ɪɚɜɟɧɫɬɜɨ ɩɨ ɩɟ ɪɟɦɟɧɧɨɣ x ɉɪɢ ɷɬɨɦ ɩɪɨɢɡɜɨɞɧɚɹ x ɚ ɩɪɨɢɡɜɨɞɧɚɹy ɩɨɤɚ ɧɚɦ ɧɟ ɢɡɜɟɫɬɧɚ ȿɟ ɦɵ ɤɚɤ ɪɚɡ ɢ ɧɚɣɞɟɦ ɢɡ ɩɨɥɭɱɟɧɧɨɝɨ ɪɚɜɟɧ ɫɬɜɚ

xy2

y 2 x 2y

cos 3	x y 5x 'x		0.
			1		2
y sin 3		x y	1	x	3	y	5x ln5	0 .
y sin 3		x y		x	3	y	5x ln5	0 .
			3
			3

Ɋɚɫɤɪɨɟɦ ɫɤɨɛɤɢ ɢ ɫɝɪɭɩɩɢɪɭɟɦ ɫɥɚɝɚɟɦɵɟ ɫɨɞɟɪɠɚɳɢɟ ɜ ɥɟɜɨɣ ɱɚɫɬɢ ɪɚɜɟɧɫɬɜɚ

y 2

2xy y sin 3

x y sin 3 x y y 5x ln5 0 .

3 3

x y y 2

y 2xy sin 3

x y sin 3

5x ln5.

3 3

ɂɡ ɩɨɫɥɟɞɧɟɝɨ ɪɚɜɟɧɫɬɜɚ ɧɚɯɨɞɢɦ ɩɪɨɢɡɜɨɞɧɭɸ ɧɟɹɜɧɨɣ ɮɭɧɤ

ɰɢɢ

x y

sin 3

ln3

sin 3 x y 33 x2

y2 33

x2 5x

ln3

2xy sin 3

x y

63 x5 y 33

x2 sin 3

x y

ɌȿɆȺ 8 ɉɊɂɅɈɀȿɇɂə ɉɊɈɂɁȼɈȾɇɈɃ 8.1.Ɉɫɧɨɜɧɵɟ ɬɟɨɪɟɦɵ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɢɫɱɢɫɥɟɧɢɹ

Ɍɟɨɪɟɦɚ Ɏɟɪɦɚ ɉɭɫɬɶ ɮɭɧɤɰɢɹ y = f(x ɨɩɪɟɞɟɥɟɧɚ ɧɚ ɧɟɤɨɬɨɪɨɦ ɢɧɬɟɪɜɚɥɟ a, b ɢ ɜ ɧɟɤɨɬɨɪɨɣ ɬɨɱɤɟ x0 (a; b ɢɦɟ ɟɬ ɷɤɫɬɪɟɦɭɦ Ɍɨɝɞɚ ɟɫɥɢ ɜ ɬɨɱɤɟ x0 ɫɭɳɟɫɬɜɭɟɬ ɩɪɨɢɡɜɨɞɧɚɹ ɬɨ ɨɧɚ ɪɚɜɧɚ ɧɭɥɸ f (x0) = 0).

Ɍɟɨɪɟɦɚ Ɋɨɥɥɹ ɉɭɫɬɶ ɧɚ ɨɬɪɟɡɤɟ [a; b] ɡɚɞɚɧɚ ɧɟɩɪɟ ɪɵɜɧɚɹ ɮɭɧɤɰɢɹ y = f(x ɩɪɢɱɟɦ

1)f(x ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚ ɧɚ ɢɧɬɟɪɜɚɥɟ a; b);

2)f(a) = f(b).

Ɍɨɝɞɚ ɫɭɳɟɫɬɜɭɟɬ ɬɨɱɤɚ x0 (a; b ɜ ɤɨɬɨɪɨɣ f (x0) = 0. Ɍɟɨɪɟɦɚ Ʌɚɝɪɚɧɠɚ ɉɭɫɬɶ ɧɚ ɨɬɪɟɡɤɟ >a; b@ ɡɚɞɚɧɚ ɧɟ

ɩɪɟɪɵɜɧɚɹ ɮɭɧɤɰɢɹ y = f(x ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚɹ ɧɚ ɢɧɬɟɪɜɚɥɟ (a; b Ɍɨɝɞɚ ɫɭɳɟɫɬɜɭɟɬ ɬɨɱɤɚ ɫ (a; b ɬɚɤɚɹ ɱɬɨ ɫɩɪɚɜɟɞɥɢɜɚ

ɮɨɪɦɭɥɚ

f b f a f c . b a

ɗɬɨ ɪɚɜɟɧɫɬɜɨ ɦɨɠɧɨ ɩɟɪɟɩɢɫɚɬɶ ɜ ɜɢɞɟ ɮɨɪɦɭɥɵ Ʌɚ

ɝɪɚɧɠɚ
f b f a		c (a;b).
f b f a	f c b a ,	c (a;b).

Ɏɨɪɦɭɥ Ʌɚɝɪɚɧɠɚ ɫɜɹɡɵɜɚɟɬ ɩɪɢɪɚɳɟɧɢɟ ɮɭɧɤɰɢɢ ɧɚ ɤɨɧɟɱɧɨɦ ɨɬɪɟɡɤɟ ɫ ɩɪɨɢɡɜɨɞɧɨɣ ɮɭɧɤɰɢɢ ɧɚ ɷɬɨɦ ɨɬɪɟɡɤɟ

ɋɥɟɞɫɬɜɢɟ ɢɡ ɬɟɨɪɟɦɵ Ʌɚɝɪɚɧɠɚ ȿɫɥɢ ɜɧɭɬɪɢ ɧɟɤɨɬɨɪɨ ɝɨ ɢɧɬɟɪɜɚɥɚ (a; b) ɩɪɨɢɡɜɨɞɧɚɹ ɮɭɧɤɰɢɢ ɬɨɠɞɟɫɬɜɟɧɧɨ ɪɚɜɧɚ ɧɭɥɸ ɬɨ ɨɧɚ ɹɜɥɹɟɬɫɹ ɤɨɧɫɬɚɧɬɨɣ ɧɚ ɨɬɪɟɡɤɟ [a; b].

8.2. ɂɫɫɥɟɞɨɜɚɧɢɟ ɮɭɧɤɰɢɢ ɢ ɩɨɫɬɪɨɟɧɢɟ ɟɟ ɝɪɚɮɢɤɚ

Ɉɞɧɢɦ ɢɡ ɜɚɠɧɟɣɲɢɯ ɩɪɢɥɨɠɟɧɢɣ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɧɨɝɨ ɢɫɱɢɫɥɟɧɢɹ ɹɜɥɹɟɬɫɹ ɢɫɫɥɟɞɨɜɚɧɢɟ ɮɭɧɤɰɢɢ ɫ ɰɟɥɶɸ ɩɨɫɬɪɨɟɧɢɹ ɟɟ ɝɪɚɮɢɤɚ

Ɉɩɪɟɞɟɥɟɧɢɟ Ɏɭɧɤɰɢɹ ɭ f(x) ɧɚɡɵɜɚɟɬɫɹ ɜɨɡɪɚɫɬɚɸ ɳɟɣ ɜ ɢɧɬɟɪɜɚɥɟ a,b ɟɫɥɢ ɩɪɢ ɯ2!ɯ1, f(x2)>f ɯ1) ɢ ɭɛɵɜɚɸ ɳɟɣ ɟɫɥɢ f(x2)<f ɯ1).

Ⱦɨɫɬɚɬɨɱɧɵɟ ɩɪɢɡɧɚɤɢ ɜɨɡɪɚɫɬɚɧɢɹ ɢ ɭɛɵɜɚɧɢɹ ɮɭɧɤ

ɰɢɢ

–ɟɫɥɢ ɮɭɧɤɰɢɹ f(x ɜ ɤɚɠɞɨɣ ɬɨɱɤɟ ɢɧɬɟɪɜɚɥɚ (a,b) ɢɦɟɟɬ ɩɨɥɨɠɢɬɟɥɶɧɭɸ ɩɪɨɢɡɜɨɞɧɭɸ ɬɨ ɫɚɦɚ ɮɭɧɤɰɢɹ ɜ ɷɬɨɦ ɢɧɬɟɪɜɚ ɥɟ ɜɨɡɪɚɫɬɚɟɬ

–ɟɫɥɢ ɮɭɧɤɰɢɹ f(x ɜ ɤɚɠɞɨɣ ɬɨɱɤɟ ɢɧɬɟɪɜɚɥɚ (a,b) ɢɦɟɟɬ ɨɬɪɢɰɚɬɟɥɶɧɭɸ ɩɪɨɢɡɜɨɞɧɭɸ ɬɨ ɮɭɧɤɰɢɹ ɜ ɷɬɨɦ ɢɧɬɟɪɜɚɥɟ ɭɛɵɜɚɟɬ

Ɏɭɧɤɰɢɹ ɭ f(x) ɢɦɟɟɬ ɷɤɫɬɪɟɦɭɦ ɦɚɤɫɢɦɭɦ ɢɥɢ ɦɢɧɢ ɦɭɦ ɜ ɬɨɱɤɟ ɯ ɯ0 ɟɫɥɢ f(x0) ɹɜɥɹɟɬɫɹ ɧɚɢɛɨɥɶɲɢɦ ɢɥɢ ɧɚɢɦɟɧɶɲɢɦ ɡɧɚɱɟɧɢɟɦ ɮɭɧɤɰɢɢ ɜ ɧɟɤɨɬɨɪɨɣ ɨɤɪɟɫɬɧɨɫɬɢ ɷɬɨɣ ɬɨɱɤɢ

ɇɟɨɛɯɨɞɢɦɨɟ ɭɫɥɨɜɢɟ ɷɤɫɬɪɟɦɭɦɚ ɮɭɧɤɰɢɢ

ȿɫɥɢ ɮɭɧɤɰɢɹ ɭ f(x) ɢɦɟɟɬ ɷɤɫɬɪɟɦɭɦ ɜ ɬɨɱɤɟ ɯ ɯ0 ɬɨ ɟɟ ɩɪɨɢɡɜɨɞɧɚɹ ɜ ɷɬɨɣ ɬɨɱɤɟ ɪɚɜɧɚ ɧɭɥɸ ɥɢɛɨ ɧɟ ɫɭɳɟɫɬɜɭɟɬ

Ɂɧɚɱɟɧɢɹ ɚɪɝɭɦɟɧɬɚ ɩɪɢ ɤɨɬɨɪɵɯ ɮɭɧɤɰɢɹ f(x ɫɨɯɪɚɧɹ ɟɬ ɧɟɩɪɟɪɵɜɧɨɫɬɶ ɚ ɟɟ ɩɪɨɢɡɜɨɞɧɚɹ f'(x ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɭɥɶ ɢɥɢ ɧɟ ɫɭɳɟɫɬɜɭɟɬ ɧɚɡɵɜɚɸɬɫɹ ɫɬɚɰɢɨɧɚɪɧɵɦɢ ɢɥɢ ɤɪɢɬɢɱɟɫɤɢ ɦɢ ɬɨɱɤɚɦɢ

ɉɟɪɜɵɣ ɞɨɫɬɚɬɨɱɧɵɣ ɩɪɢɡɧɚɤ ɷɤɫɬɪɟɦɭɦɚ ɮɭɧɤɰɢɢ

ȿɫɥɢ ɮɭɧɤɰɢɹ ɭ f(x) ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɫɬɚɰɢɨɧɚɪɧɨɣ ɬɨɱɤɢ ɯ0 ɢ ɟɟ ɩɪɨɢɡɜɨɞɧɚɹ ɫɥɟɜɚ ɨɬ ɷɬɨɣ ɬɨɱɤɢ ɩɨ ɥɨɠɢɬɟɥɶɧɚɹ ɚ ɫɩɪɚɜɚ ɨɬɪɢɰɚɬɟɥɶɧɚɹ ɬɨ ɜ ɬɨɱɤɟ ɯ0 ɮɭɧɤɰɢɹ ɞɨ ɫɬɢɝɚɟɬ ɦɚɤɫɢɦɭɦɚ ɟɫɥɢ ɩɪɨɢɡɜɨɞɧɚɹ ɫɥɟɜɚ ɨɬ ɫɬɚɰɢɨɧɚɪɧɨɣ ɬɨɱɤɢ ɯ0 ɨɬɪɢɰɚɬɟɥɶɧɚɹ ɚ ɫɩɪɚɜɚ – ɩɨɥɨɠɢɬɟɥɶɧɚɹ ɬɨ ɜ ɬɨɱɤɟ ɯ0 ɮɭɧɤɰɢɹ ɞɨɫɬɢɝɚɟɬ ɦɢɧɢɦɭɦɚ ɟɫɥɢ ɩɪɨɢɡɜɨɞɧɚɹ ɫɥɟɜɚ ɢ ɫɩɪɚɜɚ ɨɬ ɫɬɚɰɢɨɧɚɪɧɨɣ ɬɨɱɤɢ ɯ0 ɢɦɟɟɬ ɨɞɢɧɚɤɨɜɵɣ ɡɧɚɤ ɬɨ ɜ ɷɬɨɣ ɬɨɱ ɤɟ ɮɭɧɤɰɢɹ ɷɤɫɬɪɟɦɭɦɚ ɧɟ ɢɦɟɟɬ

ȼɬɨɪɨɣ ɞɨɫɬɚɬɨɱɧɵɣ ɩɪɢɡɧɚɤ ɷɤɫɬɪɟɦɭɦɚ

ȿɫɥɢ ɜ ɫɬɚɰɢɨɧɚɪɧɨɣ ɬɨɱɤɟ ɯ0 ɜɬɨɪɚɹ ɩɪɨɢɡɜɨɞɧɚɹ ɨɬ ɥɢɱɧɚ ɨɬ ɧɭɥɸ ɬɨ ɜ ɷɬɨɣ ɬɨɱɤɟ ɮɭɧɤɰɢɹ ɭ f(x) ɢɦɟɟɬ ɦɚɤɫɢ ɦɭɦ ɩɪɢ f''(x0)<0 ɢ ɦɢɧɢɦɭɦ ɩɪɢ f''(x0)>0.

Ʉɪɢɜɚɹ ɭ f(x) ɧɚɡɵɜɚɟɬɫɹ ɜɵɩɭɤɥɨɣ ɧɚ ɢɧɬɟɪɜɚɥɟ(a,b), ɟɫɥɢ ɩɪɢ a<x<b ɨɧɚ ɪɚɫɩɨɥɨɠɟɧɚ ɧɢɠɟ ɤɚɫɚɬɟɥɶɧɨɣ ɩɪɨɜɟɞɟɧ ɧɨɣ ɜ ɥɸɛɨɣ ɬɨɱɤɟ ɢɧɬɟɪɜɚɥɚ (a,b).

Ʉɪɢɜɚɹ ɭ f(x) ɧɚɡɵɜɚɟɬɫɹ ɜɨɝɧɭɬɨɣ ɧɚ ɢɧɬɟɪɜɚɥɟ(a,b), ɟɫɥɢ ɩɪɢ a<x<b ɨɧɚ ɪɚɫɩɨɥɨɠɟɧɚ ɜɵɲɟ ɤɚɫɚɬɟɥɶɧɨɣ ɩɪɨɜɟɞɟɧ ɧɨɣ ɜ ɥɸɛɨɣ ɬɨɱɤɟ ɢɧɬɟɪɜɚɥɚ (a,b).

Ɍɨɱɤɢ ɨɬɞɟɥɹɸɳɢɟ ɜɵɩɭɤɥɭɸ ɱɚɫɬɶ ɧɟɩɪɟɪɵɜɧɨɣ ɤɪɢɜɨɣ ɨɬ ɜɨɝɧɭɬɨɣ ɢɥɢ ɧɚɨɛɨɪɨɬ ɧɚɡɵɜɚɸɬɫɹ ɬɨɱɤɚɦɢ ɩɟɪɟɝɢɛɚ ɤɪɢ ɜɨɣ

ɉɪɢɡɧɚɤɢ ɜɵɩɭɤɥɨɫɬɢ ɢ ɜɨɝɧɭɬɨɫɬɢ ɤɪɢɜɨɣ

ȿɫɥɢ ɜɬɨɪɚɹ ɩɪɨɢɡɜɨɞɧɚɹ ɮɭɧɤɰɢɢ y''(x) ɩɨɥɨɠɢɬɟɥɶɧɚ ɜɨ ɜɫɟɯ ɬɨɱɤɚɯ ɢɧɬɟɪɜɚɥɚ (a,b) ɬɨ ɧɚ ɷɬɨɦ ɢɧɬɟɪɜɚɥɟ ɝɪɚɮɢɤ ɮɭɧɤɰɢɢ ɹɜɥɹɟɬɫɹ ɜɨɝɧɭɬɵɦ

ȿɫɥɢ ɜɬɨɪɚɹ ɩɪɨɢɡɜɨɞɧɚɹ ɮɭɧɤɰɢɢ y''(x) ɨɬɪɢɰɚɬɟɥɶɧɚɹ ɜɨ ɜɫɟɯ ɬɨɱɤɚɯ ɢɧɬɟɪɜɚɥɚ (a,b) ɬɨ ɧɚ ɷɬɨɦ ɢɧɬɟɪɜɚɥɟ ɝɪɚɮɢɤ ɮɭɧɤɰɢɢ ɹɜɥɹɟɬɫɹ ɜɵɩɭɤɥɵɦ

Ⱥɫɢɦɩɬɨɬɨɣ ɤɪɢɜɨɣ ɧɚɡɵɜɚɟɬɫɹ ɩɪɹɦɚɹ ɤ ɤɨɬɨɪɨɣ ɧɟ ɨɝɪɚɧɢɱɟɧɧɨ ɩɪɢɛɥɢɠɚɟɬɫɹ ɬɨɱɤɚ ɤɪɢɜɨɣ ɩɪɢ ɧɟɨɝɪɚɧɢɱɟɧɧɨɦ ɭɞɚɥɟɧɢɢ ɟɟ ɨɬ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ ɉɪɢ ɷɬɨɦ ɝɪɚɮɢɤ ɮɭɧɤɰɢɢ ɦɨɠɟɬ ɩɟɪɟɫɟɤɚɬɶ ɚɫɢɦɩɬɨɬɭ ɧɟ ɛɨɥɟɟ ɱɟɦ ɤɨɧɟɱɧɨɟ ɱɢɫɥɨ ɪɚɡ

	ȿɫɥɢ limf(x) ɢɥɢ limf(x) ɬɨ ɩɪɹɦɚɹ ɯ				ɚ ɹɜ
		x a	x a	f(x).
ɥɹɟɬɫɹ ɜɟɪɬɢɤɚɥɶɧɨɣ ɚɫɢɦɩɬɨɬɨɣ ɤɪɢɜɨɣ ɭ				f(x).
	ɉɪɹɦɚɹ ɭ b ɹɜɥɹɟɬɫɹ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɚɫɢɦɩɬɨɬɨɣ ɤɪɢ
ɜɨɣ	ɭ f(x) ɟɫɥɢ ɫɭɳɟɫɬɜɭɟɬ ɩɪɟɞɟɥ			lim f(x) b	ɢɥɢ
lim f(x) b .				x
lim f(x) b .
x	ȿɫɥɢ ɫɭɳɟɫɬɜɭɸɬ ɩɪɟɞɟɥɵ
	ȿɫɥɢ ɫɭɳɟɫɬɜɭɸɬ ɩɪɟɞɟɥɵ
lim	f (x)	k	ɢ lim f (x) kx b ɬɨ	ɩɪɹɦɚɹ y kx b
lim		k	ɢ lim f (x) kx b ɬɨ	ɩɪɹɦɚɹ y kx b
x	x		x

ɟɫɬɶ ɧɚɤɥɨɧɧɚɹ ɚɫɢɦɩɬɨɬɚ ɤɪɢɜɨɣ y f (x).

Ⱦɥɹ ɩɨɫɬɪɨɟɧɢɹ ɝɪɚɮɢɤɚ ɮɭɧɤɰɢɢ ɟɟ ɦɨɠɧɨ ɢɫɫɥɟɞɨɜɚɬɶ ɩɨ ɫɥɟɞɭɸɳɟɣ ɫɯɟɦɟ

ɇɚɣɬɢ ɨɛɥɚɫɬɶ ɨɩɪɟɞɟɥɟɧɢɹ ɮɭɧɤɰɢɢ ɢɧɬɟɪɜɚɥɵ ɧɟ ɩɪɟɪɵɜɧɨɫɬɢ ɢ ɬɨɱɤɢ ɪɚɡɪɵɜɚ ɮɭɧɤɰɢɢ ɇɚɣɬɢ ɜɟɪɬɢɤɚɥɶɧɵɟ

ɚɫɢɦɩɬɨɬɵ ɢɫɫɥɟɞɭɹ ɢɡɦɟɧɟɧɢɟ ɮɭɧɤɰɢɢ ɩɪɢ ɯ ɫɬɪɟɦɹɳɟɦɫɹ ɤ ɬɨɱɤɚɦ ɪɚɡɪɵɜɚ ɮɭɧɤɰɢɢ

ɂɫɫɥɟɞɨɜɚɬɶ ɮɭɧɤɰɢɸ ɧɚ ɱɟɬɧɨɫɬɶ ɧɟɱɟɬɧɨɫɬɶ ɩɟɪɢɨ ɞɢɱɧɨɫɬɶ

ɇɚɣɬɢ ɬɨɱɤɢ ɷɤɫɬɪɟɦɭɦɚ ɢ ɩɪɨɦɟɠɭɬɤɢ ɜɨɡɪɚɫɬɚɧɢɹ ɢ ɭɛɵɜɚɧɢɹ ɮɭɧɤɰɢɢ ȼɵɱɢɫɥɢɬɶ ɡɧɚɱɟɧɢɹ ɷɤɫɬɪɟɦɭɦɨɜ

Ɉɩɪɟɞɟɥɢɬɶ ɢɧɬɟɪɜɚɥɵ ɜɵɩɭɤɥɨɫɬɢ ɢ ɜɨɝɧɭɬɨɫɬɢ ɝɪɚ ɮɢɤɚ ɧɚɣɬɢ ɬɨɱɤɢ ɩɟɪɟɝɢɛɚ

ɇɚɣɬɢ ɝɨɪɢɡɨɧɬɚɥɶɧɵɟ ɢ ɧɚɤɥɨɧɧɵɟ ɚɫɢɦɩɬɨɬɵ ɤɪɢ ɜɨɣ ɟɫɥɢ ɨɧɢ ɫɭɳɟɫɬɜɭɸɬ ɇɚɣɬɢ ɬɨɱɤɢ ɩɟɪɟɫɟɱɟɧɢɹ ɤɪɢɜɨɣ ɫ ɨɫɹɦɢ ɤɨɨɪɞɢɧɚɬ ɟɫɥɢ ɨɧɢ ɫɭɳɟɫɬɜɭɸɬ

ɉɪɢɦɟɪ ɂɫɫɥɟɞɨɜɚɬɶ ɮɭɧɤɰɢɸ y	x3	x	2	3x	2	ɢ
	3

ɩɨɫɬɪɨɢɬɶ ɟɟ ɝɪɚɮɢɤ Ɋɟɲɟɧɢɟ Ɏɭɧɤɰɢɹ ɭ ɯ ɬɨɱɟɤ ɪɚɡɪɵɜɚ ɧɟ ɢɦɟɟɬ Ɉɛɥɚɫɬɶ

ɨɩɪɟɞɟɥɟɧɢɹ – ɜɫɹ ɱɢɫɥɨɜɚɹ ɨɫɶ ȼɟɪɬɢɤɚɥɶɧɵɯ ɚɫɢɦɩɬɨɬ ɧɟɬ Ƚɨɪɢɡɨɧɬɚɥɶɧɵɯ ɢ ɧɚɤɥɨɧɧɵɯ ɚɫɢɦɩɬɨɬ ɧɟɬ

ɂɫɫɥɟɞɭɟɦ ɞɚɧɧɭɸ ɮɭɧɤɰɢɸ ɧɚ ɷɤɫɬɪɟɦɭɦ Ɉɩɪɟɞɟɥɢɦ ɤɪɢɬɢɱɟɫɤɢɟ ɬɨɱɤɢ Ⱦɥɹ ɷɬɨɝɨ ɧɚɯɨɞɢɦ ɩɟɪɜɭɸ ɩɪɨɢɡɜɨɞɧɭɸ ɞɚɧɧɨɣ ɮɭɧɤɰɢɢ ɢ ɩɪɢɪɚɜɧɢɜɚɟɦ ɟɟ ɤ ɧɭɥɸ y'=x2–2x–3, x2–

2x–3=0.

Ɋɟɲɚɹ ɩɨɫɥɟɞɧɟɟ ɭɪɚɜɧɟɧɢɟ ɧɚɯɨɞɢɦ ɟɝɨ ɤɨɪɧɢ ɯ1=–1, ɯ2= Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɯ1=– ɢ ɯ2=3 – ɤɪɢɬɢɱɟɫɤɢɟ ɬɨɱɤɢ Ɍɚɤ ɤɚɤ ɩɪɨɢɡɜɨɞɧɚɹ f'(x)ɫɭɳɟɫɬɜɭɟɬ ɩɪɢ ɥɸɛɨɦ ɡɧɚɱɟɧɢɢ ɯ ɬɨ ɞɪɭɝɢɯ ɤɪɢɬɢɱɟɫɤɢɯɬɨɱɟɤɧɟɢɦɟɟɬɫɹ

ɂɫɫɥɟɞɭɟɦ ɤɪɢɬɢɱɟɫɤɭɸ ɬɨɱɤɭ ɯ = – ɉɪɨɢɡɜɨɞɧɭɸ y'(x) ɩɪɟɞɫɬɚɜɢɦ ɜ ɜɢɞɟ ɩɪɨɢɡɜɟɞɟɧɢɹ ɞɜɭɯ ɫɨɦɧɨɠɢɬɟɥɟɣ y'=(x+1)(x–3 ɂɡ ɷɬɨɝɨ ɪɚɜɟɧɫɬɜɚ ɜɢɞɧɨ ɱɬɨ ɩɪɢ x<– ɩɪɨɢɡ ɜɨɞɧɚɹ f'(x) ɩɨɥɨɠɢɬɟɥɶɧɚɹ ɚ ɩɪɢ –1<x<3 ɩɪɨɢɡɜɨɞɧɚɹ f'(x) ɨɬɪɢɰɚɬɟɥɶɧɚ ɋɥɟɞɨɜɚɬɟɥɶɧɨ ɜ ɢɧɬɟɪɜɚɥɟ (– ,–1) ɮɭɧɤɰɢɹ ɜɨɡ ɪɚɫɬɚɟɬ ɜ ɢɧɬɟɪɜɚɥɟ (–1,3) – ɭɛɵɜɚɟɬ ɚ ɜ ɢɧɬɟɪɜɚɥɟ - ɜɨɡ ɪɚɫɬɚɟɬ Ɍɚɤ ɤɚɤ ɩɪɨɢɡɜɨɞɧɚɹ y ɩɪɢ ɩɟɪɟɯɨɞɟ ɱɟɪɟɡ ɤɪɢɬɢɱɟ ɫɤɭɸ ɬɨɱɤɭ ɯ = – ɦɟɧɹɟɬ ɫɜɨɣ ɡɧɚɤ ɫ ɩɥɸɫɚ ɧɚ ɦɢɧɭɫ ɬɨ ɜ ɷɬɨɣ ɬɨɱɤɟ ɮɭɧɤɰɢɹ ɢɦɟɟɬ ɦɚɤɫɢɦɭɦ

Ⱥɧɚɥɨɝɢɱɧɨ ɢɫɫɥɟɞɭɟɦ ɬɨɱɤɭ ɯ2 ɢ ɭɛɟɠɞɚɟɦɫɹ ɱɬɨ ɜ ɷɬɨɣ ɬɨɱɤɟɮɭɧɤɰɢɹ ɢɦɟɟɬɦɢɧɢɦɭɦ

ɇɚɣɞɟɦ ɷɤɫɬɪɟɦɭɦ ymax 3	2	,	ymin 7 ɂɡ ɜɬɨɪɨɣ ɩɪɨ
	3

ɢɡɜɨɞɧɨɣ y''(x)=2x–2 ɧɚɣɞɟɦ ɬɨɱɤɭ ɯ ɩɨɞɨɡɪɢɬɟɥɶɧɭɸ ɧɚ ɬɨɱɤɭ ɩɟɪɟɝɢɛɚ Ɍɚɤ ɤɚɤ ɩɪɢ ɩɟɪɟɯɨɞɟ ɱɟɪɟɡ ɷɬɭ ɬɨɱɤɭ y''(x) ɦɟ ɧɹɟɬ ɡɧɚɤ ɬɨ ɬɨɱɤɚ ɯ ɹɜɥɹɟɬɫɹ ɬɨɱɤɨɣ ɩɟɪɟɝɢɛɚ ȼ ɢɧɬɟɪɜɚɥɟ (– ,1) ɝɪɚɮɢɤ ɮɭɧɤɰɢɢ ɹɜɥɹɟɬɫɹ ɜɵɩɭɤɥɵɦ ɬɚɤ ɤɚɤ y''(x)<0 ɜ ɢɧɬɟɪɜɚɥɟ (1,+ – ɝɪɚɮɢɤ ɜɨɝɧɭɬ ɬɚɤ ɤɚɤ y''(x)>0 ɋɬɪɨɢɦ ɝɪɚɮɢɤ

–4

–2

–2 –4 –6

ɉɪɢɦɟɪ ɂɫɫɥɟɞɨɜɚɬɶ ɮɭɧɤɰɢɸ y	(x3	4)	ɢ ɩɨɫɬɪɨɢɬɶ
	x2

ɟɟ ɝɪɚɮɢɤ

Ɋɟɲɟɧɢɟ ɇɚɣɞɟɦ ɩɪɨɢɡɜɨɞɧɵɟ y (x3 3 8);y 244 . x x

Ɉɛɥɚɫɬɶ ɨɩɪɟɞɟɥɟɧɢɹ D(y) ( ,0) (0, ) . ɋɜɨɣɫɬɜɚɦɢ ɱɟɬɧɨɫɬɢ ɧɟɱɟɬɧɨɫɬɢ ɮɭɧɤɰɢɹ ɧɟ ɨɛɥɚɞɚɟɬ

Ƚɪɚɮɢɤ ɩɟɪɟɫɟɤɚɟɬɫɹ ɫ ɨɫɶɸ Ɉɯ ɜ ɬɨɱɤɟ x 3 4 Ʉɪɢɬɢɱɟɫɤɢɟ ɬɨɱɤɢ ɯ ɯ ɉɨɫɥɟɞɧɹɹ ɧɟ ɜɯɨɞɢɬ ɜ ɨɛɥɚɫɬɶ ɨɩɪɟɞɟɥɟɧɢɹ ɩɨɷɬɨɦɭ ɟɟ ɧɟ ɪɚɫɫɦɚɬɪɢɜɚɟɦ ɇɚɣɞɟɦ ɭ ɢ ɧɚɧɟɫɟɦ ɬɨɱɤɭɧɚ ɩɥɨɫɤɨɫɬɶ Ɉɯɭ

ɂɫɫɥɟɞɭɟɦ ɩɨɜɟɞɟɧɢɟ ɭ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɬɨɱɤɢ ɯ ɉɪɢ 0<ɯ , y ɩɪɢ ɯ! , y ! ɋɥɟɞɨɜɚɬɟɥɶɧɨ ɜ ɬɨɱɤɟ ɯ ɮɭɧɤ ɰɢɹ ɢɦɟɟɬ ɦɢɧɢɦɭɦ ɇɚ ɩɪɨɦɟɠɭɬɤɟ(– ,0):y' >0 ɫɥɟɞɨɜɚɬɟɥɶɧɨ

ɮɭɧɤɰɢɹ ɜɨɡɪɚɫɬɚɟɬ ɂɫɫɥɟɞɭɟɦ ɧɚɩɪɚɜɥɟɧɢɟ ɜɵɩɭɤɥɨɫɬɢ ɝɪɚ ɮɢɤɚ ȼɫɸɞɭ y''>0 ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɬɨɱɟɤ ɩɟɪɟɝɢɛɚ ɧɟɬ ɢ ɤɪɢɜɚɹ ɜɫɸɞɭ ɜɨɝɧɭɬɚ ɂɫɫɥɟɞɭɟɦ ɮɭɧɤɰɢɸ ɜɛɥɢɡɢ ɬɨɱɤɢ ɪɚɡɪɵɜɚ ɧɟ ɩɪɟɪɵɜɧɨɫɬɢ ɯ ɢ ɩɪɢ ɯ ɯ –

lim

x3 4

lim

x3 4

x 0

ɉɪɹɦɚɹ ɯ

– ɜɟɪɬɢɤɚɥɶɧɚɹ ɚɫɢɦɩɬɨɬɚ

ɇɚɣɞɟɦ ɧɚɤɥɨɧɧɭɸ ɚɫɢɦɩɬɨɬɭ y=kx+b:

k lim

x x2 x

x3 4

x3 4 x3

b lim

x lim

lim

x x

lim	x3 4	.
	x2
x

ɉɨɥɭɱɢɦ ɧɚɤɥɨɧɧɭɸ ɚɫɢɦɩɬɨɬɭ ɭ ɯ ɋɬɪɨɢɦ ɝɪɚɮɢɤ ɮɭɧɤɰɢɢ

	8	y
	6
	6
	4
	2			x
	3 4			x
-4	-2	0 2	4	6

8.3. ɉɪɚɜɢɥɨ Ʌɨɩɢɬɚɥɹ

ɇɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ ɜɢɞɚ ɉɟɪɜɨɟ ɩɪɚɜɢɥɨ Ʌɨɩɢɬɚɥɹ

ȿɫɥɢ lim f(x) = lim g(x)		ɬɨ lim	f(x)	lim	f (x)	ɤɨ

x a	x a	x a g(x)		x a g (x)

ɝɞɚ ɩɨɫɥɟɞɧɢɣ ɫɭɳɟɫɬɜɭɟɬɇɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ ɜɢɞɚ / ȼɬɨɪɨɟ ɩɪɚɜɢɥɨ Ʌɨɩɢɬɚɥɹ


ȿɫɥɢ lim f(x) = lim g(x)= ɬɨ lim			f(x)	lim	f (x)	ɤɨ
ȿɫɥɢ lim f(x) = lim g(x)= ɬɨ lim				lim		ɤɨ
x a	x a	x a g(x)		x a g (x)

ɝɞɚ ɩɨɫɥɟɞɧɢɣ ɫɭɳɟɫɬɜɭɟɬɇɟɨɩɪɟɞɟɥɟɧɧɨɫɬɢ ɜɢɞɚ , - , 1 ɢ 0 ɫɜɨɞɹɬɫɹ ɤ ɧɟ

ɨɩɪɟɞɟɥɟɧɧɨɫɬɹɦ ɢ / ɩɭɬɟɦ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɩɪɟɨɛɪɚɡɨɜɚ ɧɢɣ

ɉɪɢɦɟɪ. ɇɚɣɬɢ ɩɪɟɞɟɥ ɮɭɧɤɰɢɢ \	1	1	ɩɪɢ [ 0.
	x	ex 1

Ɋɟɲɟɧɢɟ ɂɦɟɟɦ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ ɜɢɞɚ - ɋɧɚɱɚɥɚ ɩɪɟ ɨɛɪɚɡɭɟɦ ɟɟ ɤ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɢ ɜɢɞɚ ɞɥɹ ɱɟɝɨ ɞɨɫɬɚɬɨɱɧɨ ɩɪɢɜɟɫɬɢ ɞɪɨɛɢ ɤ ɨɛɳɟɦɭ ɡɧɚɦɟɧɚɬɟɥɸ Ʉ ɩɨɥɭɱɟɧɧɨɦɭ ɜɵɪɚ ɠɟɧɢɸ ɞɜɚ ɪɚɡɚ ɩɪɢɦɟɧɢɦ ɩɪɚɜɢɥɨ Ʌɨɩɢɬɚɥɹ Ɂɚɩɢɫɵɜɚɹ ɩɨɫɥɟ ɞɨɜɚɬɟɥɶɧɨ ɜɫɟ ɩɪɨɦɟɠɭɬɨɱɧɵɟ ɜɵɱɢɫɥɟɧɢɹ ɛɭɞɟɦ ɢɦɟɬɶ

lim(

)

= lim

ex 1 x

x(e

x 0

lim

(ex

1 x)

=lim

ex 1

(x(e

1))

1 e

x 0

=lim

(ex

=lim

(xe

1 e

)

x 0

ɉɪɢɦɟɪ. ɇɚɣɬɢ lim

lnx

Ɋɟɲɟɧɢɟ Ɋɚɫɤɪɵɜɚɹ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ ɜɢɞɚ / ɩɨ ɩɪɚɜɢɥɭ Ʌɨɩɢɬɚɥɹ ɩɨɥɭɱɚɟɦ

lim	lnx	=	lim	1/ x	= lim	2	=0.

x +	x		x + 1/ (2 x)		x +	x

100

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