Metodos Numericos Derivacion e Integracion

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  • 7/24/2019 Metodos Numericos Derivacion e Integracion

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    df(x)

    f(x)

    f(x) = sin(x) x = 2;13432 h h

    f(x) = f1f0h +O(h)f(x) = sin(x)x= 2, 13432

    f(x) =0, 53416f(x) = sin(2,13432+h)sin(2,13432)hh= 2 : f(x) =0, 84144 error= 57, 52%h= 1 : f(x) =0, 83810 error= 56, 90%h= 0, 5 : f(x) =0, 71916 error= 34, 63%h= 0, 2 : f(x) =0, 61487 error= 15, 10%h= 0, 1 : f(x) =0, 57551 error= 7, 74%

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    g(x) = 1/(1 +ex) x= 1/2

    f(x) = f1f0h +O(h) f(x) = f0f1h

    f(x) = sin(x) f(x) = sin(x) x= 2, 13432 x= 2, 13432 f(x) =0, 53416 f(x) =0, 53416f(x) = sin(2,13432+h)sin(2,13432)h f

    (x) = sin(2,13432)sin(2,13432h)hh= 2 : f(x) =0, 84144 h= 2 : f(x) = 0, 35573h= 1 : f(x) =0, 83810 h= 1 : f(x) =0, 06086h= 0, 5 : f(x) =0, 71916 h= 0, 5 : f(x) =0, 30520h= 0, 2 : f(x) =0, 61487 h= 0, 2 : f(x) =0, 44635h= 0, 1 : f(x) =0, 57551 h= 0, 1 : f(x) =0, 49104

    f(x) = f1f0h +O(h) f(x) = f0f1h

    g(x) = 1/(1 +ex) g(x) = 1/(1 +ex) x= 1/2 x= 1/2

    g(x) =0, 2350

    g(x) =0, 2350g(x) =

    1

    1+e0,5+h 1

    1+e0,5

    h g(x) =

    11+e0,5

    11+e0,5h

    h

    h= 2 : g(x) =0, 15084 h= 2 : g(x) =0, 22001h= 1 : g(x) =0, 19511 h= 1 : g(x) =0, 24491h= 0, 5 : g(x) =0, 21719 h= 0, 5 : g(x) =0, 24991h= 0, 2 : g(x) =0, 22864 h= 0, 2 : g(x) =0, 24008h= 0, 1 : g(x) =0, 23196 h= 0, 1 : g(x) =0, 23771

    h h

    f(x) = sin(x)

    x= 0, 6

    h = 0, 1

    h= 0, 01 h= 0, 0000000001

    f(x) = f1f0h +O(h)f(x) = sin(x)x= 0, 6f(x) = 0, 82533f(x) = sin(0,6+h)sin(0,6)hh= 0, 1 : f(x) = 0, 79575 error= 3, 58%h= 0, 01 : f(x) = 0, 82249 error= 0, 34%h= 0, 0000000001 : f(x) = 0, 82531 error= 0, 0024%

    h

    h h

    f(x) = tan(x) x = 3, 14 h = 0, 1 h = 0, 01

    f(x) = tan(x)x= 3, 14f(x) = 1, 00000f(x) = tan(3,14+h)tan(3,14)h

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    h= 0, 1 : f(x) = 1, 00318h= 0, 01 : f(x) = 1, 00001

    g(x)

    xi

    f(x) = (0,990025)+4(0,997502)3(1)0,2 =8, 5x105f(x) = 12(0,997502)+0,9900250,22 =0, 12447f(x) = 0,9975022(0,990025)+0.9603980,22 =0, 55375f(x) = 0,9900252(0,960398)+0,9406780,22 = 0, 2476

    f(x) = 3(0,940678)4(0,960398)+0,9900250,4 =0, 07383

    f(x) =arctan(x) x=

    2 h

    f(x) = f12f0+f1h2f(x) = arctan(x)x=

    2

    f(x) = 1/3

    f(x) = arctan(2+h)2arctan(

    2)+arctan(

    2h)

    h2

    h= 2 : f(x) =0, 28866 error= 13, 40%h= 1 : f(x) =0, 33983 error= 1, 94%h= 0, 5 : f(x) =0, 32254 error= 3, 23%h= 0, 2 : f(x) =0, 31565 error= 5, 305%h= 0, 1 : f(x) =0, 31461 error= 5, 617%

    x, x+h, x+ 2h, x+ 3h xh f(x) f(xh) = f(x)f(x)h+ 12f(x)h2 16f(3)(x)h3 + 124f(4)(x)(h)4 +O(h5)f(x+h) = f(x) +f(x)h+ 12f

    (x)h2 + 16f(3)(x)h3 + 124f

    (4)(x)h4 +O(h5)

    f(x+ 2h) = f(x) + 2f(x)h+ 12f(x)(2h)2 + 16f

    (3)(x)(2h)3 + 124f(4)(x)(2h)4 +O(h5)

    f(x+ 3h) = f(x) + 3f(x)h+ 12f(x)(3h)2 16f(3)(x)(3h)3 + 124f(4)(x)(3h)4 +O(h5)

    f(x+ 3h)f(x2h) = f(x)h+ 52f

    (x)h2 + 196f(3)(x)h3 +O(h4)

    f(x+h)f(xh) = 2f(x)h+ 26f(3)(x)h3 +O(h5)8(f(x+h)f(xh))(f(x+ 3h)f(x2h)) = 12f(x)h+O(h5)f

    (x) 112h [f(x+ 2h) + 8f(x+h)8f(xh) +f(x2h)] +O(h4)

    J0(x) J0(0, 0) = 1; 00000000, J0(0, 1) =0, 99750156, J0(0, 2) = 0, 99002497, J0(0, 3) = 0, 97762625, J0(0, 4) = 0, 96039823 J0(0, 5) = 0, 93846981.

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    x f0(x) f

    (x)

    0, 0 1, 00000000 0, 0000010, 1 0, 99750156 0, 0409360, 2 0, 99002497 0, 00495010, 3 0, 97762625 0, 1483180, 4 0, 96039823

    0, 146427

    0, 5 0, 93846981 0, 242786 f(x) =|x2| cos(x) x = 2

    h f0(x) f

    (x) f

    (x)

    1, 0 1, 00000000 0, 414847 0, 651840, 8 0, 99750156 0, 753147 0, 486090, 6 0, 99002497 0, 43594 0, 801190, 4 0, 97762625 0, 12915 0, 588500, 2 0, 96039823 0, 40778 0, 50484

    f(x) ={

    0,1

  • 7/24/2019 Metodos Numericos Derivacion e Integracion

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    f

    (x) =f2+16f130f0+16f1+f2

    12h2

    f(x) = 1/(1 +x2)

    x= 1

    f(x) = 0, 5

    f(x) = 1

    1+(1+(1+2h)2)+ 16

    1+(1+(1+h)2) 30

    1+(1+(1)2)+ 16

    1+(1+(1h)2)+ 1

    1+(1+(12h)2)

    12h2

    h= 0, 5 : f

    (x) = 0, 40305h= 0, 3 : f

    (x) = 0, 93093h= 0, 1 : f

    (x) = 13, 1865

    h= ban+ 2

    =120

    4 + 2 =

    12

    6 = 2

    x1 = a+hx2 = a+ 2hx3 = a+ 3hx4 = a+ 4h

    x5 = a+ 5h

    x1 = 2x2 = 4x3 = 6x4 = 8x5 = 10

    h= b

    a

    n =12

    0

    3 =12

    3 = 4

    x0 = ax1 = a+hx2 = a+ 2hx3 = a+ 3h= b

    x0 = 0

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    x1 = 4x2 = 8x3 = 12

    h= ba

    n =

    1202

    =12

    2 = 6

    x0 = ax1 = a+hx2 = a+ 2h= b

    x0 = 0x1 = 6x2 = 12

    0 sin xdx

    0 sin xdx= pi

    22 (sin(0) + 2sin(/2) +sin())0 sin xdx=

    pi

    22 (0 + 21 + 0)0 sin xdx=

    pi

    2 = 1.5708

    20

    3x2dx

    ba

    f(x)dx=3

    8h [f(x0) + 3f(x1) + 3f(x2) +f(x3)]

    a= 0b= 2h= ban =

    203 =

    23

    xk f(xk)0 0

    2/3 4/34/3 16/3

    2 12

    2

    0

    3x2dx=3

    8 2

    3

    0 + 3

    4

    3

    + 3

    16

    3

    + 12

    = 8

    ba sin xdx

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    Erect=(ba)2

    2 f ()

    ESimp =(ba)5

    90 =f(4) ()

    f (0) = cos (0) = 1

    fiv

    2

    = sin

    2

    = 1

    (ba)22

    =(ba)5

    90

    45 (ba)2 = (ba)5

    ba= 3

    45 = 3.556899

    10

    x2exdx

    1

    0

    x2exdx= 0.1606027941

    a= 0b= 1a+b2 =

    12

    ba

    f(x)dx= (ba) f

    a+b

    2

    1

    0

    x2exdx= (10)f

    1

    2

    1

    0

    x2exdx= 0.1516326649

    a= 0b= 1a+b2 =

    12

    b

    a

    f(x)dx= (f(a) +f(b)) ba

    2 1

    0

    x2exdx= (0 + 0.3678794412)

    102

    1

    0

    x2exdx= 0.1839397206

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    ba

    f(x)dx=

    ba

    6

    f(a) + 4f(

    a+b

    2 ) +f(b)

    1

    0

    x2exdx=

    106

    0 + 4f(

    1

    2) + 0.3678794412

    1

    0

    x2exdx=

    106

    [0 + 40.1516326649 + 0.3678794412]

    1

    0

    x2exdx= 0.1624016835

    f(x) = sin(x)f(x) = 1 +ex cos (4x)f(x) = sin (

    x)

    P1(x) = f(x0)(xx1)

    (x0x1)+ f(x1)

    (xx0)

    (x1x0)

    xk f(xk)x0 = a f(x0)x1 = b f(x1)

    ba

    f(x)dx=

    f(a)

    ba

    (xb)(ab)

    dx+f(b)

    ba

    (xa)(ba)

    dx

    b

    a

    f(x)dx=

    f(a)

    (ab)

    b

    a

    (xb)dx+ f(b)(ba)

    b

    a

    (xa)dx

    b

    a

    f(x)dx=

    f(a)

    (ab)

    x2

    2bx

    b

    |a

    + f(b)

    (ba)

    x2

    2ax

    b

    |a

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    ba

    f(x)dx= f(a)

    (ab)

    1

    2

    b2 a2+b(ab)+ f(b)

    (ba)

    1

    2

    b2 a2a(ba)

    ba

    f(x)dx=f(a)(ab)

    (ab) 1

    2(a+b) +b

    +

    f(b)(ba)(ba)

    1

    2(b+a)a

    ba

    f(x)dx= f(a)a

    2 b

    2+ b

    +f(b)

    a

    2+

    b

    2a

    ba

    f(x)dx=f(a)

    2 (ba) + f(b)

    2 (ba)

    ba

    f(x)dx= (f(a) +f(b))

    ba2

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  • 7/24/2019 Metodos Numericos Derivacion e Integracion

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    b

    a

    g(x)dx=

    1

    0

    1 + 9x4dx

    ba

    g(x)dx=m1i=0

    hg(xi) +Error

    g(x) =

    1 + 9x4

    g

    (x) = 36x3

    2

    1 + 9x4

    a= 0b= 1

    = 13h= bam =

    104 =

    14

    xk g(xk)0 1

    1/4

    256/161/2 5/4

    3/4

    97/4

    ba

    g(x)dx=m1i=0

    hg(xi) +Error

    m1i=0

    hg(xi) = 1h

    (g(x1) +g(x2)g(x3))

    1

    0

    1 + 9x4dx=

    1