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Национальный исследовательский университет «МИЭТ»

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Численные методы

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Алгоритм решения БДЗ / 06c

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f (x) = x2 sin x на отрезке [2 ; 3 ] с точностью " = 10 3

n = 9, "практич. = 9:0027 10 5, f (9) = x2 cos(x) 72 cos(x)+18 x sin(x) , x1 = 7:854+1:5708 cos 1718 = 6:3070,

= 7:854+1:5708 cos

= 6:4936, x3 = 7:854+1:5708 cos

= 6:8443, x4

= 7:854+1:5708 cos

= 7:3167,

= 7:854 + 1:5708 cos

= 7:8540,

x6 = 7:854 + 1:5708 cos

= 8:3912, x7

= 7:854 + 1:5708 cos

= 8:8637,

= 7:854 + 1:5708 cos

= 9:2143,

x9 = 7:854 + 1:5708 cos

= 9:4009

f (x) =

+ cos x

на отрезке [ 2 ; 32 ] с точностью " = 10 2

1 + x

(5)

= sin(x)

945

n = 5

"практич. = 0:0058

x1 = 3:1416 + 1:5708 cos 10 = 1:6477

x2 = 3:1416 +

32 (x+1)

1:5708 cos

= 2:2183, x3 = 3:1416 + 1:5708 cos

= 3:1416, x4 = 3:1416 + 1:5708 cos

= 4:0649, x5 =

3:1416 + 1:5708 cos

= 4:6355

f (x) = e x cos x на отрезке [ 2 ; ] с точностью " = 10 3

n = 4,

"практич. = 5:3128 10 4, f (4) = 4 e x cos(x) ,

x1 = 2:3562 + 0:7854 cos

= 1:6306,

x2 = 2:3562 +

0:7854 cos

= 2:0556, x3

= 2:3562 + 0:7854 cos

= 2:6568, x4 = 2:3562 + 0:7854 cos

= 3:0818

f (x) = ch( x) cos x

на отрезке [0; ] с точностью " = 10 3

n = 9, "практич. = 1:1581 10 4,

f (9) = 16 cos(x) sinh(x) 16 cosh(x) sin(x) , x1 = 1:5708 + 1:5708 cos

0:0239,

= 1:5708 + 1:5708 cos

= 0:2104,

= 1:5708 + 1:5708 cos

= 0:5611,

= 1:5708 +

1:5708 cos

= 1:0336, x5

= 1:5708 + 1:5708 cos

= 1:5708,

= 1:5708 + 1:5708 cos

= 2:1080, x7

1:5708 + 1:5708 cos

= 2:5805,

x8 = 1:5708 + 1:5708 cos

= 2:9311,

= 1:5708 + 1:5708 cos

= 3:1177

f (x) =

sin x

на отрезке [ ; 2 ] с точностью " = 10 2

n = 4, "практич. = 0:0056, f (4) =

4 cos(x)

24 cos(x)

sin(x)

12 sin(x)

24 sin(x)

, x1 = 4:7124 + 1:5708 cos 78 =

3:2612, x2 = 4:7124+1:5708 cos

= 4:1113, x3 = 4:7124+1:5708 cos

= 5:3135, x4 = 4:7124+1:5708 cos

6:1636

sin x

на отрезке [ ; 2 ] с точностью " = 10 2

f (x) =

n = 5, "практич. = 0:0018, f (5)

cos(x)

15 cos(x)

525 cos(x)

5 sin(x)

75 sin(x)

945 sin(x)

, x1 = 4:7124 +

2 x

16 x 2

2 x 2

4 x 2

32 x 2

1:5708 cos

= 3:2185, x2

= 4:7124 + 1:5708 cos

= 3:7891,

= 4:7124 + 1:5708 cos

= 4:7124, x4

4:7124 + 1:5708 cos

= 5:6357,

x5 = 4:7124 + 1:5708 cos

= 6:2063

f (x) =

cos(2x)

на отрезке [ ; ] с точностью " = 10 2

n = 9, "практич. = 0:0075, f (9)

32256 cos(2 x)

2304 cos(2 x)

241920 cos(2 x)

725760 cos(2 x)

362880 cos(2 x)

x10

512 sin(2 x)

9216 sin(2 x)

96768 sin(2 x)

483840 sin(2 x)

725760 sin(2 x)

, x1 = 1:0472 + 0:5236 cos 1718 = 0:5316, x2 =

1:0472 + 0:5236 cos

= 0:5937, x3 = 1:0472 + 0:5236 cos

= 0:7106, x4

= 1:0472 + 0:5236 cos

= 0:8681,

x5 = 1:0472 + 0:5236 cos

= 1:0472, x6 = 1:0472 + 0:5236 cos

= 1:2263, x7

= 1:0472 + 0:5236 cos

1:3838,

x8 = 1:0472 + 0:5236 cos

= 1:5006, x9 = 1:0472 + 0:5236 cos

= 1:5628

f (x) = ln x sin x + 1

на отрезке [ ; 2 ] с точностью " = 10 3

n = 7,

"практич. = 1:7231 10 4

, f (7)

210 cos(x)

21 cos(x)

840 cos(x)

cos(x) log(x)

7 sin(x)

70 sin(x)

504 sin(x)

720 sin(x)

= 4:7124 + 1:5708 cos

13 = 3:1810,

= 4:7124 + 1:5708 cos

= 3:4843,

4:7124 + 1:5708 cos

= 4:0308,

x4 = 4:7124 + 1:5708 cos

= 4:7124,

x5 = 4:7124 + 1:5708 cos

= 5:3939,

x6 = 4:7124 + 1:5708 cos

= 5:9405,

x7 = 4:7124 + 1:5708 cos

= 6:2438

) + p

на отрезке [0; =2] с точностью " = 10 3

f (x) = sin(x +

n = 10

"практич. = Inf

(10)

sin

+ x

34459425 ,

x2 = 0:7854+

1024 x 2

= 0:7854+0:7854 cos 20 = 0:0097

+ 0:7854 cos

= 0:2300, x

= 0:7854 + 0:7854 cos

= 0:4288,

0:7854 cos

= 0:0856

= 0:7854

(

)

0:7854 + 0:7854 cos

= 0:6625, x6 = 0:7854 + 0:7854 cos

= 0:9083,

x7 = 0:7854 + 0:7854 cos

= 1:1420,

x8 = 0:7854 + 0:7854 cos

= 1:3408,

x9 = 0:7854 + 0:7854 cos

= 1:4852, x10 = 0:7854 + 0:7854 cos

1:5611

f (x) = cos(2x) p

на отрезке [

;

] с точностью " = 10 3

n = 6, "практич. = 6:5171

10 4, f (6)

945

64 cos(2 x) , x1

= 3:1416 + 0:7854 cos

= 2:3830,

64 x 2

3:1416 + 0:7854 cos

= 2:5862,

x3 = 3:1416 + 0:7854 cos

= 2:9383,

x4 = 3:1416 + 0:7854 cos

= 3:3449,

x5 = 3:1416 + 0:7854 cos

= 3:6970,

x6 = 3:1416 + 0:7854 cos

= 3:9002

f (x) = x2 sin(x=2)

на отрезке [ 4 ; ] с точностью " = 10 3

(

2 )

n = 4, "практич. = 6:2705

10 4, f (4) =

sin

= 1:9635 + 1:1781 cos 8 = 0:8751,

= 1:9635 +

1:1781 cos

= 1:5127,

x3 = 1:9635 + 1:1781 cos

= 2:4143, x4 = 1:9635 + 1:1781 cos

= 3:0519

f (x) = x2 sin(3x) на отрезке [1; 5; 2] с точностью " = 10 3

n = 5, "практич. = 6:5335 10 4, f (5) = 810 x sin(3 x) 540 cos(3 x)+243 x2 cos(3 x) , x1 = 1:75+0:25 cos 910 =

1:5122, x2

= 1:75 + 0:25 cos

= 1:6031,

= 1:75 + 0:25 cos

= 1:7500, x4 = 1:75 + 0:25 cos

= 1:8969,

x5 = 1:75 + 0:25 cos

= 1:9878

f (x) = sh x cos(3x)

на отрезке [2; 2; 5] с точностью " = 10 3

n = 5, "практич. = 7:0658 10 4, f (5) = 316 cos(3 x) cosh(x) + 12 sin(3 x) sinh(x) ,

x1 = 2:25 + 0:25 cos

2:0122, x2

= 2:25 + 0:25 cos

= 2:1031,

= 2:25 + 0:25 cos

= 2:2500, x4 = 2:25 + 0:25 cos

= 2:3969,

x5 = 2:25 + 0:25 cos

= 2:4878

f (x) =

sh x

на отрезке [0; 5; 2; 5] с точностью " = 10 3

n = 5, "практич. = 1:2683 10 4, f (5) =

cosh(x)

20 cosh(x)

40 cosh(x)

5 sinh(x)

20 sinh(x)

40 sinh(x)

, x1 =

3 x

3 x3

3 x2

1:5 + 1 cos

= 0:5489, x2 = 1:5 + 1 cos

= 0:9122, x3 = 1:5 + 1 cos

= 1:5000, x4

= 1:5 + 1 cos

= 2:0878,

x5 = 1:5 + 1 cos 110 = 2:4511

15
			f (x) = arccos x							на отрезке [ 21 ; 21 ] с точностью " = 10 2
	,		,			(4)	=		15 x3				9 x	,					7	= 0:4619	,			5
n = 4			"практич. = 0:0047 f												x1 = 0 + 0:5 cos 8							x2 = 0 + 0:5 cos		8	=
n = 4			"практич. = 0:0047 f					(1 x2)		2		(1 x2) 2			x1 = 0 + 0:5 cos 8							x2 = 0 + 0:5 cos		8	=
										7	5
0:1913, x3 = 0 + 0:5 cos				3	= 0:1913, x4 = 0 + 0:5 cos										1	= 0:4619
0:1913, x3 = 0 + 0:5 cos				8	= 0:1913, x4 = 0 + 0:5 cos										8	= 0:4619
16
			f (x) = arcsin x							на отрезке [ 52 ; 53 ] с точностью " = 10 2
						(5)			90 x2				105 x4		9							9
		,	,															,				10 = 0:3755	,
n = 5		,	,			f =				7	+ (1 x2)			9			5	,	x1 = 0:1 + 0:5 cos				,	x2	=
n = 5			"практич. = 0:0046			f =			(1 x2) 2		+ (1 x2)			2	+ (1 x2) 2				x1 = 0:1 + 0:5 cos					x2	=

0:1+0:5 cos 710 = 0:1939, x3 = 0:1+0:5 cos 510 = 0:1000, x4 = 0:1+0:5 cos 310 = 0:3939, x5 = 0:1+0:5 cos 110 = 0:5755

f (x) =

[0; 1] с точностью " = 10 3

на отрезке

1 + x

"практич. = 4:8828 10 4,

17280 x2

720

57600 x4

46080 x6

= 6,

f (6) = (x2+1)5

(x2+1)6 +

(x2+1)7 , x1

= 0:5 + 0:5 cos 12 =

(x2+1)4

0:0170,

x2 = 0:5 + 0:5 cos

= 0:1464,

= 0:5 + 0:5 cos

= 0:3706,

x4 = 0:5 + 0:5 cos

= 0:6294,

= 0:5 + 0:5 cos

= 0:8536,

x6 = 0:5 + 0:5 cos

= 0:9830

f (x) =

[ 21 ; 1] с точностью " = 10 3

на отрезке

1 + x

n = 5, "практич. = 2:3312 10 4, f (5)

6720 x3

74880 x7

184320 x11

122880 x15

= (x4+1)3

(x4+1)4

+ (x4+1)5

(x4+1)6

, x1 = 0:75+0:25 cos 10

= 0:5122,

= 0:75 + 0:25 cos

= 0:6031,

x3 = 0:75 + 0:25 cos

= 0:7500,

= 0:75 + 0:25 cos

= 0:8969,

= 0:75 + 0:25 cos

= 0:9878

f (x) =

на отрезке [0; 0; 6] с точностью " = 10 3

1 + x4

n = 5, "практич. = 5:5794 10 4, f (5)

28800 x5

720 x

151680 x9

245760 x13

122880 x17

= (x4+1)3

(x4+1)4

(x4+1)5 (x4+1)6

, x1 = 0:3+0:3 cos 10 =

(x4+1)2

0:0147,

x2 = 0:3 + 0:3 cos

= 0:1237,

= 0:3 + 0:3 cos

= 0:3000,

x4 = 0:3 + 0:3 cos

= 0:4763,

= 0:3 + 0:3 cos

= 0:5853

f (x) =

[0; 1] с точностью " = 10 3

на отрезке

1 + x

"практич. = 4:1807 10 4,

40320 x3

5040 x

80640 x5

46080 x7

= 6,

f (6) = (x2+1)5

(x2+1)6 +

(x2+1)7 , x1

= 0:5 + 0:5 cos 12 =

(x2+1)4

0:0170,

x2 = 0:5 + 0:5 cos

= 0:1464,

= 0:5 + 0:5 cos

= 0:3706,

x4 = 0:5 + 0:5 cos

= 0:6294,

= 0:5 + 0:5 cos

= 0:8536,

x6 = 0:5 + 0:5 cos

= 0:9830

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