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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%
7/24/2019 Metodos Numericos Derivacion e Integracion
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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
4
256
16 +
5
4+
97
4
+Error
1
0
1 + 9x4dx= 1.178053613 +Error
1
0
1 + 9x4dx= 1.178053613 (ba) h
2 g()
= 1.178053613
(10)2 1
4g
1
3
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Error=(ba) h2
g()
Error=
(10)2
14
10
5
= 0.0790569415
Er% =
IanaticaInumerica|Ianatica 100%
Er% =
1.5478656551.1780536131.547865655 100%
Er% = 23.89
ba
g(x)dx=h
2
g(x0) + 2
m1i=0
g(xi) +g(xm)
+Error
m1i=0
hg(xi) = 2(g(x1) +g(x2)g(x3))
1
0
1 + 9x4dx=
1
2 1
4
1 + 2
256
16 +
5
4+
97
4
+
10
+Error
= 1.69833832 +Error
Error=(ba) h212
g()
Error=(10)
12
1
4
2(8.208)
Error= 3.29103
Er% =
IanaticaInumerica|Ianatica 100%
Er% = 1.5478656551.69833832
1.547865655
100%
Er% = 9.72
b
a
g(x)dx=h
3
g(x0) + 4
mi=1
g(x2i1) + 2m1i=1
g(x2i) +g(x2m)
+Error
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a= 0b= 1= 13h= ba2m =
1023 =
16
xk g(xk)
0 11/6 145/122/6
10/3
3/6 5/44/6 5/3
5/6
769/12
b
a
g(x)dx=h
3
g(x0) + 4
3i=1
g(x2i1) + 22
i=1
g(x2i) +g(x2m)
+Error
impar3
i=1g(x2i1) = g(x1) +g(x3) +g(x5) =
145
12 +
5
4+
769
12 = 4.564370319
par2
i=1
g(x2i) = g(x2) +g(x4) =
10
3 +
5
3=
5 +
10
3
1
0
1 + 9x4dx=
1
3 1
6
1 + 4(4.564370319) + 2
5 +
10
3
+
10
+Error
= 1.547848743 +Error
Error=(ba) h4
180 g(4)()
g
(x) = 36x3
2
1 + 9x4
g
(x) =54(3x6 +x2)
3
1 + 9x4
g
(x) =180x(19x4)
5
1 + 9x4
g(4)(x) =108(405x8 126x4 + 1)
7
1 + 9x4
g(4)() =108(4058 1264 + 1)
7
1 + 94g(4)(
1
3) =87.56099604
1
0
1 + 9x4dx= 1.547848743
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Er% =
IanaticaInumerica|Ianatica 100%
Er% =
1.5478656551.5478487431.547865655
100%
Er% = 1.09103
1.09103
longitud =
/4
0
1 + (sin(x))2dx= 0.8520033931
ba
g(x)dx=
/4
0
1 + (sin(x))2dx
ba
g(x)dx=m1i=0
hg(xi) +Error
g(x) =
1 + (sin(x))
2
g
(x) =sin(x)cos(x)
1 + (sin(x))2
a= 0b= 4= 16h= bam =
404 =
16
xk g(xk)
0 1/16 1.01885241
2/16 1.0707224713/16 1.143966033
4/1662
b
a
g(x)dx=m1i=0
hg(xi) +Error
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m1i=0
hg(xi) = 1h
(g(x1) +g(x2) +g(x3))
/4
01 + (sin(x))
2dx=
16(1.01885241 + 1.070722471 + 1.143966033)
/4
0
1 + (sin(x))
2dx= 0.6349042738
Er% =
IanaticaInumerica|Ianatica 100%
Er% =
0.85200339310.63490427380.8520033931
100%
Er% = 25.48
ba
g(x)dx=h
2
g(x0) + 2
m1i=0
g(xi) +g(xm)
+Error
m1i=0
hg(xi) = 2(g(x1) +g(x2) +g(x3))
/4
0
1 + (sin(x))2dx=
1
2
16
1 + 2 (1.01885241 + 1.070722471 + 1.143966033) +
6
2
/4
0
1 + (sin(x))
2dx= 0.8533180908
Er% =
IanaticaInumerica|Ianatica 100%
Er% = 0.85200339310.8533180908
0.8520033931
100%
Er% = 0.154
b
a
g(x)dx=h
3
g(x0) + 4
mi=1
g(x2i1) + 2m1i=1
g(x2i) +g(x2m)
+Error
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b
a
g(x)dx=h
3
g(x0) + 4
3i=1
g(x2i1) + 22
i=1
g(x2i) +g(x2m)
+Error
a= 0b= 4
m= 3
h= ba
2m =
4023 =
24
xk g(xk)0 1
/24 1.0084825662/24 1.0329507723/24 1.070722471
4/2452
5/24 1.170722203
6/2462
impar3
i=1
g(x2i1) = g(x1) +g(x3) = 1.008482566 + 1.070722471 = 2.079205037
par2
i=1
g(x2i) = g(x2) = 1.070722471
/4
0
1 + (sin(x))
2dx=1
3
16
1 + 4 (2.079205037) + 2 (1.070722471) +
6
2
/4
0
1 + (sin(x))
2dx= 0.8301010608
Er% =
IanaticaInumerica|Ianatica 100%
Er% =
0.85200339310.83010106080.8520033931 100%
Er% = 2.57
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ex
longitud =
1
0
1 +e2xdx= 1.192701402
ba
g(x)dx=
1
0
1 +e2xdx
b
a
g(x)dx=m1i=0
hg(xi) +Error
g(x) = 1 +e2xa= 0b= 1= 1/3h= bam =
16
xk g(xk)0 2
1/6 1.3101646122/6 1.2302101933/6 1.1695637824/6 1.1240983675/6 1.090355723
1 1.065521132b
a
g(x)dx=m1i=0
hg(xi) +Error
m1i=0
hg(xi) = 1h
(g(x1) +g(x2) +g(x3) +g(x4) +g(x5))
1
0
1 +e2xdx=1
6(1.310164612 + 1.230210193 + 1.169563782 + 1.124098367 + 1.090355723)
1
0
1 +e2xdx= 0.9873987795
Er% =
IanaticaInumerica|Ianatica 100%
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Er% =
1.1927014020.98739877951.192701402 100%
Er% = 17.213245
b
a
g(x)dx=h
2
g(x0) + 2
m1i=0
g(xi) +g(xm)
+Error
m1i=0
hg(xi) = 2(g(x1) +g(x2) +g(x3) +g(x4) +g(x5))
1
01 +e2xdx=
1
2
1
6
(2 + 2(1.310164612 + 1.230210193 + 1.169563782 + 1.124098367 + 1.090355723) + 1.0655
1
0
1 +e2xdx= 1.2428588738
Er% =
IanaticaInumerica|Ianatica 100%
Er% =
1.1927014021.24285887381.192701402 100
Er% = 4.2053670
b
a
g(x)dx=h
3
g(x0) + 4
mi=1
g(x2i1) + 2m1i=1
g(x2i) +g(x2m)
+Error
b
a
g(x)dx=h
3
g(x0) + 4
3i=1
g(x2i1) + 22
i=1
g(x2i) +g(x2m)
+Error
a= 0
b= 4
m= 3
h= ba
2m =
1023 =
1
6
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xk g(xk)0 2
1/6 1.3101646122/6 1.2302101933/6 1.169563782
4/6 1.1240983675/6 1.0903557231 1.065521132
impar3
i=1
g(x2i1) = g(x1) +g(x3) = 1.310164612 + 1.169563782 = 2.479728394
par2
i=1
g(x2i) = g(x2) = 1.230210193
1
0
1 +e2xdx=
1
3 1
6(2 + 4 (2.479728394) + 2 (1.230210193) + 1.065521132)
1
0
1 +e2xdx= 0.858047505
Er% =
IanaticaInumerica|Ianatica 100%
Er% =
0.85200339310.8580475050.8520033931 100%
Er% = 0.709399979.
20
g(t)dt= w0g(0) +w1g(1) +w2g(2)
t2
g(0) = 1, g(1) = 1, g(2) = 1 20
g(t)dt=
20
dt
=t|20 = 2w0+w1+w2 = 2
g(0) = 0, g(1) = 1, g(2) = 2
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20
tdt= t2
2 |20
20
tdt= 2
w1+ 2w2 = 2
t2g(0) = 0, g(1) = 1, g(2) = 4
20
t2dt= t3
3 |20
20
tdt=8
3
w1+ 4w2 =8
3
w0 + w1 + w2 = 20 + w1 + 2w2 = 2
0 + w1 + 4w2 = 83
w0 =1
3
w1 =4
3
w2 =1
3
f(x0 +ht) = g(t)
x= x0 +ht
dx= hdt
xi+1
xi
f(x)dx=
h
3
f(xi) + 4f(
xi+xi+12
) +f(xi+1)
x1+htx0+ht
g(t)hdt=
13
xi+1x0
t
f(xi) + 4f
2xi+ht
2
+f(xi+1)