-

5

:

: ______________: _____

: 14

: ..

__________________

" ___ " ______________ 2006 .

2006

5.1. )(xfy = ],[ ba .

b

a

dxxf )( 10,...,2,1=n (

)n

abh -= , :

) ; ) ; ) ; D) .

A)-D) 10,...,2,1=n ( h ) 10,...,2,1=n ( h ). .

:

. . :

: S= -

=

1

0)(

n

iixfh ;

n

abMR2

)( 21 -=

: S= =

n

iixfh

1)( ;

n

abMR2

)( 21 -=

: S= -

=

++1

0

1 )2

(n

i

ii xxfh ;

23

2

24)(

nabMR -=

: S=

+

+ -

=

)(2

)()( 1

1

0n

ii

n xfxfxfh ;

23

2

12)(

nabMR -=

:

f x( ) 1

9 x2-:= [2;2] a 2-:= b 2:=

4 2 0 2 4

2

4

f x( )

x

:

:

32arcsin

32arcsin

3arcsin

9

1 2

2

2

2 2

--==

- --

xdxx

true asin23

asin2-

3

-:= 1.45945531245393( )=

A) :

n 0 9..:=IntLeftRect f a, b, n,( ) hb a-

n

xi a h i+

i 0 n..for

S 0

S S f xi( )+i 0 n 1-..for

S h S

S

:=

integraln IntLeftRect f a, b, n 1+,( ):=

integral

001

2

3

4

5

6

7

8

9

1.788854381999831.56109385766658

1.50796940567705

1.48765371001984

1.47782348555966

1.47234586341758

1.468990009447

1.46678873636108

1.46526836423323

1.46417502415523

=

E_Leftn true integraln-:=

E_Left

001

2

3

4

5

6

7

8

9

0.3293990695458990.10163854521265

0.048514093223115

0.028198397565906

0.01836817310573

0.012890550963647

9.5346969930636410 -3

7.3334239071438710 -3

5.8130517792942510 -3

4.7197117013022510 -3

=

df x( )xf x( )d

d:= absdf x( ) df x( ):=

2 0 20

0.1absdf z( )

z

M1 absdf 2( ):= n 0 9..:=

M1 0.178885=

R_Left f a, b, n,( )b a-( )2

2 nM1:=

0 2 4 6 8 100

1

2

R_Left f a, b, n,( )

E_Leftn

n

, .

B) :

n 0 9..:=Intentr f a, b, n,( ) hb a-

n

xi a h i+

i 0 n..for

S 0

S S f xi 1-h2

+

+

i 1 n..for

S h S

S

:=

integraln Intentr f a, b, n 1+,( ):=

integral

001

2

3

4

5

6

7

8

9

1.333333333333331.41421356237309

1.43672232115811

1.44592376270231

1.45052656275081

1.45314060011584

1.45476123518253

1.45583272997773

1.45657688052445

1.45711415927405

=

E_Centrn true integraln-:=

E_Centr

001

2

3

4

5

6

7

8

9

0.1261219791205990.045241750080838

0.022732991295821

0.013531549751618

8.9287497031258510 -3

6.3147123380942210 -3

4.6940772713983710 -3

3.6225824762019510 -3

2.8784319294852310 -3

2.3411531798835910 -3

=

df2 x( )2xf x( )d

d

2:= absdf2 x( ) df2 x( ):=

2 0 20

0.2

0.4

absdf2 z( )

z

M2 absdf2 2( ):= n 0 9..:=

M2 0.304105=

R_Centr f a, b, n,( )b a-( )3

24 n2M2:=

0 2 4 6 8 100

0.5

1

R_Centr f a, b, n,( )

E_Centrn

n

, .

C) :

n 0 9..:=IntRightRect f a, b, n,( ) hb a-

n

xi a h i+

i 0 n..for

S 0

S S f xi( )+i 1 n..for

S h S

S

:=

integraln IntRightRect f a, b, n 1+,( ):=

integral

001

2

3

4

5

6

7

8

9

1.788854381999831.56109385766658

1.50796940567705

1.48765371001984

1.47782348555966

1.47234586341758

1.468990009447

1.46678873636108

1.46526836423323

1.46417502415523

=

E_Rightn true integraln-:=

E_Right

001

2

3

4

5

6

7

8

9

0.3293990695458990.10163854521265

0.048514093223115

0.028198397565906

0.01836817310573

0.012890550963647

9.5346969930636410 -3

7.3334239071436410 -3

5.8130517792942510 -3

4.7197117013020310 -3

=

df x( )xf x( )d

d:= absdf x( ) df x( ):=

2 0 20

0.1absdf z( )

z

M1 absdf 2( ):= n 0 9..:=

M1 0.178885=

R_Right f a, b, n,( )b a-( )2

2 nM1:=

0 2 4 6 8 100

1

2

R_Right f a, b, n,( )

E_Rightn

n

, .

D) :

n 0 9..:=Trapeia f a, b, n,( ) hb a-

n

xi a h i+

i 0 n..for

Sf x0( ) f xn( )+

2

S S f xi( )+i 1 n 1-..for

S h S

S

:=

integraln Trapeia f a, b, n 1+,( ):=

integral

001

2

3

4

5

6

7

8

9

5.36656314599951.56109385766658

1.50796940567705

1.48765371001984

1.47782348555966

1.47234586341758

1.468990009447

1.46678873636108

1.46526836423323

1.46417502415523

=

E_Trapecian true integraln-:=

E_Trapecia

001

2

3

4

5

6

7

8

9

3.907107833545560.10163854521265

0.048514093223115

0.028198397565906

0.01836817310573

0.012890550963647

9.5346969930636410 -3

7.3334239071438710 -3

5.8130517792942510 -3

4.7197117013022510 -3

=

2 0 20

0.2

0.4

absdf2 z( )

z

df2 x( )2xf x( )d

d

2:= absdf2 x( ) df2 x( ):=

M2 absdf2 2( ):= n 0 9..:=

M2 0.304105=

R_Trapecia f a, b, n,( )b a-( )3

12 n2M2:=

0 2 4 6 8 100

2

4

R_Trapecia f a, b, n,( )

E_Trapecian

n

, n.

:

n 9:=

norma_1 V n,( )

0

n

i

Vi=

:= norma_2 V n,( )

0

n

i

Vi( )2=

:= norma_c V n,( )

si Vi

i 0 n..for

max s( )

:=

:

max_P V n,( ) s1 norma_1 V n,( )

s2 norma_2 V n,( )

s3 norma_c V n,( )

max s( )

:=

E_Simpsonn1 true integraln1-:=

integral

001

2

3

4

5

6

7

8

9

2.081458476888781.46317366080426

1.46047134933109

1.45983374514149

1.45962553702043

1.45954235454975

1.45950415993736

1.45948473210551

1.45947404176071

1.45946778090111

=

integral n2

1-

Simpson f a, b, n,( ):=

n1 0 9..:=n 2 4, 20..:=

Simpson f a, b, n,( ) hb a-

n

xi a h i+

i 0 n..for

S f x0( ) f xn( )+

S S 4 f x2 i 1-( )+

i 1n2

..for

S S 2 f x2 i( )+

i 1n 1-

2..for

Sh3

S

S

:= integral 0:=

:

5.2. 5.1 b

a

dxxf )(

20,...,4,2=n (

)n

abh -= .

10,...,4,2=n ( h ) 10,...,4,2=n ( h ). , , 5.1.

c n .

:

min E_Trapecia( ) 4.71971170130225 10 3-=max_P E_Trapecia n,( ) 4.14411847799742=

min E_Right( ) 4.71971170130203 10 3-=max_P E_Right n,( ) 0.56640971399775=

min E_Centr( ) 2.34115317988359 10 3-=max_P E_Centr n,( ) 0.236407977147066=

min E_Left( ) 4.71971170130225 10 3-=max_P E_Left n,( ) 0.56640971399775=

, :

E_Simpson

001

2

3

4

5

6

7

8

9

0.6220031644348443.718348350324710 -3

1.016036877157910 -3

3.784326875564410 -4

1.702245664929610 -4

8.7042095819445110 -5

4.8847483422598310 -5

2.9419651579765110 -5

1.8729306774600810 -5

1.2468447178504110 -5

=

min E_Simpson( ) 1.24684471785041 10 5-=max_P E_Simpson 9,( ) 0.627482713901151=

min E_Trapecia( ) 4.71971170130225 10 3-=max_P E_Trapecia 9,( ) 4.14411847799742=

min E_Right( ) 4.71971170130203 10 3-=max_P E_Right 9,( ) 0.56640971399775=

min E_Centr( ) 2.34115317988359 10 3-=max_P E_Centr 9,( ) 0.236407977147066=

min E_Left( ) 4.71971170130225 10 3-=max_P E_Left 9,( ) 0.56640971399775=

:

, .

0 2 4 6 8 100

10

20

R_Simpson f a, b, n,( )

E_Simpsonn

n

R_Simpson f a, b, n,( )b a-( )5

180 n4M4:=

M4 2.651082=

n 0 9..:=M4 absdf4 2( ):=

2 0 20

1

2

3

absdf4 z( )

z

absdf4 x( ) df4 x( ):=df4 x( )4xf x( )d

d

4:=

:

n 2 4, 20..:= E_Centr 0:=

integraln Intentr f a, b, n,( ):=

E_Centr n2

1-

true integraln-:=

max_P E_Centr 9,( ) 0.076141573677683= min E_Centr( ) 5.9341029956772 10 4-=

max_P E_Simpson 9,( ) 0.627482713901151= min E_Simpson( ) 1.24684471785041 10 5-=

n2 1 9..:=

0 5 100

0.005

0.01

0.015

E_Centrn2

E_Simpsonn2

n2

: c .

5.3. , N (N=1,2,3,4) 12 - Nm .

: 1) ( , , ),,,(IntGauss Nbaf ),

],[ ba N (N=1,2,3,4).

2) mmm xcxcxccxP ++++= ...)(2

210 7..0=m .

3) ( -)

bam

m

b

amm m

xcxcxcxcdxxPI |)1

...32

()(13

2

2

10 +++++==

+

. 4)

),,,(IntGauss, NbaPIG mNm = , 7..0=m , N=1,2,3,4. 5) N (N=1,2,3,4) mI

NmIG , . , c Nmm IGI ,= 12 - Nm .

6) .

:

t

0

0

0

0

0.5773502692-

0.5773502692

0

0

0.7745966692-

0

0.7745966692

0

0.8611363116-

0.3399810436-

0.3399810436

0.8611363116

:=A

2

0

0

0

1

1

0

0

59

89

59

0

0.3478548451

0.6521451549

0.6521451549

0.3478548451

:=

IntGauss f a, b, N,( ) hb a-

2

S 0

S S Ai N 1-, fa b+

2b a-

2ti N 1-,+

+

i 0 N 1-..for

S h S

S

:=

IP7 10 b 8b2

2+ 4

b3

3+ 5

b4

4+ 7

b7

7+ 8

b8

8+

10 a 8a2

2+ 4

a3

3+ 5

a4

4+ 7

a7

7+ 8

a8

8+

-:=

IP6 32b4

4 6

b6

6+ 3

b7

7-

32a4

4 6

a6

6+ 3

a7

7-

-:=

IP5 31 b 2b2

2- 7

b3

3+

b4

4- 12

b5

5+ 5

b6

6+

31 a 2a2

2- 7

a3

3+

a4

4- 12

a5

5+ 5

a6

6+

-:=

IP4b2

24

b3

3+ 5

b4

4+

b5

5+

a2

24

a3

3+ 5

a4

4+

a5

5+

-:=

IP3 8 b 2b2

2+ 9

b3

3+ 7

b4

4-

8 a 2a2

2+ 9

a3

3+ 7

a4

4-

-:=

IP2 5 b 9b2

2+

b3

3-

5 a 9a2

2+

a3

3-

-:=

IP1 11 bb2

2+

11 aa2

2+

-:=

IP0 5 b 5 a-:=

:

P7 x( ) 10 8 x+ 4 x2+ 5 x3+ 7 x6+ 8 x7+:=

P6 x( ) 32 x3 6 x5+ 3 x6-:=

P5 x( ) 31 2 x- 7 x2+ x3- 12 x4+ 5 x5+:=

P4 x( ) x 4 x2+ 5 x3+ x4+:=

P3 x( ) 8 2 x+ 9 x2+ 7 x3-:=

P2 x( ) 5 9 x+ x2-:=

P1 x( ) 11 x+:=

P0 x( ) 5 x0:=

x a a 0.01+, b..:=

b 1:=a 0:= 8 :

0 5 100

20

40

IPm

IGm N 1-,

m

N 1:=

m 0 7..:=

, .

IGm N 1-, IntGauss P7 a, b, N,( ):=

m 7:=

IG

5

11.5

9.25

10.375

2.1875

32.53125

4.140625

5

11.5

9.16666666666367

10.2499999999955

3.27777777781721

35.1805555556569

8.55555555571503

5

11.5

9.16666666667559

10.2500000000134

3.28333333321462

35.3166666663355

8.57249999952149

5

11.5

9.16666666666996

10.2500000000049

3.28333333328971

35.3166666665484

8.57142857125246

=IGm N 1-, IntGauss P6 a, b, N,( ):=

m 6:=

IGm N 1-, IntGauss P5 a, b, N,( ):=

m 5:=

IGm N 1-, IntGauss P4 a, b, N,( ):=

m 4:= , .

IGm N 1-, IntGauss P3 a, b, N,( ):=

m 3:=

IGm N 1-, IntGauss P2 a, b, N,( ):=

m 2:=

IGm N 1-, IntGauss P1 a, b, N,( ):=

m 1:=

IGm N 1-, IntGauss P0 a, b, N,( ):=

m 0:=

N 1 4..:=

N m:

N 2:=

0 5 100

20

40

IPm

IGm N 1-,

mN 3:=

0 5 100

20

40

IPm

IGm N 1-,

m

N 4:=

0 5 100

20

40

IPm

IGm N 1-,

m

:

m 0 7..:= N 1 4..:=

Em N 1-, IPm IGm N 1-,-:=

E0

0

0.083333333333334

0.125

1.09583333333333

2.78541666666667

4.43080357142857

2.78645833333334

0

0

2.99493763122882 10 12-

4.49240644684323 10 12-

5.55555551612397 10 3-

0.136111111009789

0.015873015713542

0.398148148045578

0

0

8.92619311798626 10 12-

1.33884014985597 10 11-

1.18712151220279 10 10-

3.31148442000995 10 10-

1.07142809292071 10 3-

0.012500000393182

0

0

3.29158922340866 10 12-

4.9382720135327 10 12-

4.36197744591027 10 11-

1.18234311230481 10 10-

1.7610979341498 10 10-

1.3245227137304 10 10-

=

, m, , m

E_SimpsonN 1-

001

2

3

0.6220031644348443.718348350324710 -3

1.016036877157910 -3

3.784326875564410 -4

=E_LeftN 1-

001

2

3

0.3293990695458990.10163854521265

0.048514093223115

0.028198397565906

=

1 2 3 40

0.1

0.2

E_GaussN 1-

N

integral

1.33333333333333

1.44463023703374

1.45759141942188

1.45921307015357

=E_Gauss

0.126121979120599

0.014825075420188

1.86389303204892 10 3-

2.4224230036074 10 4-

=

E_GaussN 1- true integralN 1--:=

integralN 1- IntGauss f a, b, N,( ):=

integral 0:=

N 1 4..:=

true asin23

asin2-

3

-:= 1.45945531245393( )=

b 2:=a 2-:= [2;2]f x( ) 1

9 x2-:=

:

5.4. b

a

dxxf )( 5.1,

, , , . . . , , 5.1 5.2.

E_RightN 1-

00123

0.3293990695458990.10163854521265

0.0485140932231150.028198397565906

= E_CentrN 1-

00123

0.0452417500808380.013531549751618

6.3147123380942210 -3

3.6225824762019510 -3

=

E_TrapeciaN 1-

00123

3.907107833545560.10163854521265

0.0485140932231150.028198397565906

= E_Gauss

0.126121979120599

0.014825075420188

1.86389303204892 10 3-

2.4224230036074 10 4-

=

: c .

5.5. )(xfy = ],[ ba .

b

a

dxxf )( 610-=e :

) ; ) .

n . , MathCad.

:

n .

A) : e

e12

2)((12

)( 32

32 Mabceiln

nabM -

-

B) : e-45

4

180)(

nabM

)180

4)((45

eMabceiln -

:

e 10 6-:= a 3:= b 6:=

f x( )atan x( )

x 2-:=

A)

df2 x( )2xf x( )d

d

2:= absdf2 x( ) df2 x( ):=

3 4 5 60

0.5

1

df2 z( )

z

M2 absdf2 3( ):= n 0 9..:=

M2 0.77678432929884=

n ceilb a-( )3

12 eM2

:= n 1323=

Trapeia f a, b, n,( ) hb a-

n

xi a h i+

i 0 n..for

Sf x0( ) f xn( )+

2

S S f xi( )+i 1 n 1-..for

S h S

S

:=

integral_T Trapeia f a, b, n,( ):= integral_T 2.67094=

integral_M 2.67094=integral_Ma

bxf x( )

d:=

MathCAD:

integral_S 2.67094=integral_S Simpson f a, b, n,( ):=

Simpson f a, b, n,( ) hb a-

n

xi a h i+

i 0 n..for

S f x0( ) f xn( )+

S S 4 f x2 i 1-( )+

i 1n2

..for

S S 2 f x2 i( )+

i 1n 1-

2..for

Sh3

S

S

:=

n 34:= n 33=n ceil

4b a-( )5

180 eM4

:=

M4 0.77678432929884=

n 0 9..:=M4 absdf2 3( ):=

3 4 5 60

0.5

1

absdf4 z( )

z

absdf4 x( ) df2 x( ):=df4 x( )2xf x( )d

d

2:=

B)

: , .. 39 . , .. . , MathCAD, , .

5.6. )(xfy = ( 5.1) ],[ ba

bxxxa n =

)(xfy = ],[ ba

bxxxa n =

DF_DX f a, b, n,( ) hb a-

n

n n 1-

xi a h i+

i 0 n..for

D03- f x0( ) f x1( )+ f x2( )-

2 h

Dif xi 1+( ) f xi 1-( )-

2 h

i 1 n 1-..for

Dn3 f xn( ) 4 f xn 1-( )- f xn 2-( )+

2 h

D

:=

DF2_DX f a, b, n,( ) hb a-

n

n n 1-

xi a h i+

i 0 n..for

D02 f x0( ) 5 f x1( )- 4 f x2( )+ f x3( )-

h2

Dif xi 1+( ) 2 f xi( )- f xi 1-( )+

h2

i 1 n 1-..for

Dn2 f xn( ) 5 f xn 1-( )- 4 f xn 2-( )+ f xn 3-( )-

h2

D

:=

A) n 10:=

i 0 n 1-..:=

hb a-

n:= h 0.4=

xi a h i+:=

Fi f xi( ):=

DF_DX_ARR_1 DF_DX f a, b, n,( ):=

DF2_DX_ARR_1 DF2_DX f a, b, n,( ):=

2 1 0 1 2

0.1

0

0.1

0.2

DF_DX_A z( )

DF2_DX_A z( )

DF_DX_ARR_1i

DF2_DX_ARR_1i

z z, xi,

DF2_DX_ARR_3 DF2_DX f a, b, n,( ):=

DF_DX_ARR_3 DF_DX f a, b, n,( ):=

Fi f xi( ):=xi a h i+:=

h 0.2=hb a-

n:=

i 0 n 1-..:=

n 20:=C)

2 1 0 1 2

0.1

0

0.1

0.2

DF_DX_A z( )

DF2_DX_A z( )

DF_DX_ARR_2i

DF2_DX_ARR_2i

z z, xi,

DF2_DX_ARR_2 DF2_DX f a, b, n,( ):=

DF_DX_ARR_2 DF_DX f a, b, n,( ):=

Fi f xi( ):=

xi a h i+:=

h 0.266666666666667=hb a-

n:=

i 0 n 1-..:=

n 15:=B)

2 1 0 1 2

0.1

0

0.1

0.2

DF_DX_A z( )

DF2_DX_A z( )

DF_DX_ARR_3i

DF2_DX_ARR_3i

z z, xi,

: , , .. . , , . , , , . , .