Practico De Alejandro
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1.- Determine las raices reales de f(x)=-0,4x²+2,2x+4,7 a) Graficamente b) Usando la formula cuadratica c) Usando el metodo de biseccion hasta 3 iteraciones para determinar Xl=5
Xu=10
calcule Ea y Ev
F(x)=-0,4x²+2,2 x
f(x) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
-57.3 -47.5 -38.5 -30.3 -22.9 -16.3 -10.5 -5.5 -1.3 2.1 4.7 6.5 7.5 7.7 7.1 5.7 3.5
GRAFICO 10 0 -10 -20 -30 -40 -50 -60 -10
-8
-6
-4
-2
0
2
4
-30 -40 7 8 9 10
0.5 -3.3 -7.9 -13.3
-50 -60 -10
-8
-6
-4
-2
0
2
F(x)=-0,4x²+2,2x+4,7 a= b= c=
-0.4 2.2 4.7
X1= X2=
-1.64 7.14
Xl=5 Xu=10 -------------->>>Calcule Ea y Ev. Valor verdadero= 7,14459896 Iteracion 1 2
Xl
Xu 5 5
Xr 10 7.5
F(Xl) 7.5 6.25
F(Xu) 5.7 5.7
-13.3 -1.3
4
3
6.25
7.5
6.88
2.83
-1.3
2.-Determine las raices reales de:
a) Graficamente b) Usando biseccion con un :
Es= 10% Xl= 0,5 Xu= 1,0
c)Realice el mismo calculo como en el inciso b), pero cuando el metodo d
x
f(x) -10 -210049 -9 -138422.15 -8 -87724.4 -7 -53019.85 -6 -30212.6 -5 -15968.75 -4 -7638.4 -3 -3177.65 -2 -1070.6 -1 -251.35 0 -26 1 5.35 2 26.6 3 83.65 4 162.4 5 266.75 6 496.6 7 1125.85 8 2680.4 9 6016.15 10 12397
GRAFICO 25000 0 -25000 -50000 -75000 -100000 -125000 -150000 -175000 -200000 -225000 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0
1
2
3
4
-150000 -175000 -200000 -225000 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0
1
Es= 10% Xl= 0,5 Xu= 1,0
Iteracion
Xl 1 2 3 4 5 6
Xu 0.5 0.5 0.5 0.56 0.56 0.58
Xr 1 0.75 0.63 0.63 0.59 0.59
Ea 0.75 0.63 0.56 0.59 0.58 0.59
F(Xl) 100 20 11.11 5.26 2.7 1.33
-1.72 -1.72 -1.72 -0.33 -0.33 -0.02
Es= 0,1%
Iteracion
Xl 1 2
Xu 0.5 -0.52
Xr 1 1
Ea -0.52 -0.09
F(Xl) 100 497.96
-1.72 -99.66
2
3
4
3 4 5 6 7 8 9
-0.09 -0.17 -0.15 -0.16 -0.15 -0.15 -0.15
1 1 1 1 1 1 1
-0.17 -0.15 -0.16 -0.15 -0.15 -0.15 -0.15
49.98 15.97 3.96 1.05 0.27 0.07 0.02
-33.85 -43.21 -40.47 -41.17 -40.98 -41.03 -41.02
para determinar la raiz mas grande.
F(x)=-0,4x²+2,2x+4,7
AFICO
0
2
4
6
8
10
0
2
4
6
8
lcule Ea y Ev.
F(Xr) -1.3 2.83
Ev -4.97 12.52
Ea 100 20
10
0.92
3.77
9.09
uando el metodo de falsa posicion con un Es=0,1%
RAFICO
-1
f(x)
0
1
2
3
4
5
6
7
8
9
10
-1
0
1
2
3
4
F(Xu)
5
6
F(Xr) 5.35 2.68 0.84 0.84 0.27 0.27
F(Xu)
2.68 0.84 -0.33 0.27 -0.02 0.13
F(Xr) 5.35 5.35
-99.66 -33.85
7
8
9
10
5.35 5.35 5.35 5.35 5.35 5.35 5.35
-43.21 -40.47 -41.17 -40.98 -41.03 -41.02 -41.02
3.- Determine la raiz real de:
a) Analiticamente b) Con el metodo de la falsa posicion con:
Es=0,1% use un valor inicial de 3,0 a 4,0.
X 3, 3 = 79 3, 3
X 3,3 = 3,3 79
X = 3,3 79 X = 3,758707344
Es=0,1
X=3,758707344 X-3,758707344=0
F(x)= x - 3,758707344 Xu=4,0 Xl=3,0
Iteracion
Xl 1 2 3 4 5 6 7
Xu
3 -4.24 -0.73 -1.09 -1.03 -1.04 -1.04
Xr 4 4 4 4 4 4 4
-4.24 -0.73 -1.09 -1.03 -1.04 -1.04 -1.04
Ea 100 483.67 33.18 5.84 0.91 0.14 0.02
F(Xl) -0.76 -8 -4.49 -4.85 -4.79 -4.8 -4.79
F(Xu) 0.24 0.24 0.24 0.24 0.24 0.24 0.24
4.- Determine la raiz real de :
a) Analiticamente b) Graficamente c)Usando tres iteraciones con el metodo de la falsa posicion con valores
0,9 − 0, 4 x F ( x) = x 0,9 − (0, 4 * 1 F (1) = 1 0,5 F ( x) = 1 F ( x) = 0,5
X
0,9 − (0, 4 * 1 F (1) = 1 0,5 F ( x) = 1 F ( x) = 0,5
F(X) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
-0.49 -0.5 -0.51 -0.53 -0.55 -0.58 -0.63 -0.7 -0.85 -1.3 #DIV/0! 0.5 0.05 -0.1 -0.18 -0.22 -0.25 -0.27 -0.29 -0.3 -0.31
GRAFICO 0.5 0.25 0 -0.25 -0.5 -0.75 -1 -1.25 -1.5 -10
-8
-6
-4
-2
0
2
Xl= 1 Xu= 3 Calcular Ea y Ev
Iteracion 1 2 3
Xl
Xu 1 1 1
3 4.67 5.53
Xr
Ea 4.67 5.53 6.04
100 15.54 8.52
F(xl) 0.5 0.5 0.5
F(xu) -0.1 -0.21 -0.24
4
ial de 3,0 a 4,0.
F(Xr) -8 -4.49 -4.85 -4.79 -4.8 -4.79 -4.79
on con valores iniciales 1 y 3 calcule Ea yEv.
9 − 0, 4 x x 9 − (0, 4 * 1) 1 5 1 5
9 − (0, 4 * 1) 1 5 1 5
ICO
0
2
4
6
8
10
VALOR VERDADERO= 0,5
F(xr) -0.21 -0.24 -0.25
Ev -833.33 -1005.05 -1108.03
5.- Use el metodo de iteracion simple de punto fijo para localizar la raiz de:
f ( x) = sen( x ) − x Donde Xo= 0,5 y Ea<= 0,01 %
f ( x) = sen( x ) − x sen
(
)
x −x =0
( x )* (−1) x =sen( x )
−x =−sen
xi +1 = sen
(
Donde ------>> Iteracion 1 2 3 4 5 6 7 8 9 1 11 12 13 14 15
Xi
)
xi = 0.5
Xi+1 0.5 0.01 0 0 0 0 0 0 0 0 0 0 0 0 0
xi
0.01 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Ea 100 536.5 152.29 58.84 26.03 12.26 5.95 2.93 1.46 0.73 0.36 0.18 0.09 0.04 0.02
16 17
6.- Determine la raiz de :
Usando Xo= 5
Ea< 0,01%
0 0
0 0
0.01 0.01
f ( x) = −0.9 x + 1.7 x + 2.5 2
Empleando
a) Iteracion punto fijo b) Newton c) Secante
f ( x) = −0.9 x + 1.7 x + 2.5 2
−9 x 2 + 1.7 x +2.5 =0 −0.9 x 2 =− 1.7 x −2.5 * ( − 1) 0.9 x 2 =1.7 +2.5 2 1.7 x +2.5 x = 0.9 1.7 x +2.5 x = 0.9
x i +1 = x i =5
1.7 x i −2.5 0.9
Iteracion
Xi
1 2 3 4 5 6 7 8 9 10
Xi+1 5 2.58 1.45 0.2 1.55 0 1.67 0.61 1.28 0.61
2.58 1.45 0.2 1.55 0 1.67 0.61 0.8 0.61 1.28
Ea 100 78.2 615.86 86.92 #DIV/0! 100 173.86 24 32.08 52.56
f ( x) = −0.9 x + 1.7 x + 2.5 2
xi +1 = xi −
f ( xi ) f ' ( xi )
f ' ( x) = −1.8 x +1.7 2
xi +1
( −0.9 xi +1.7 xi + 2.5) =5 − −1.8 xi +1.7
Iteracion
xi 1 2 3 4 5 6 7 8
xi+1 5 3.42 4.5 3.74 4.26 3.9 4.15 3.97
3.42 4.5 3.74 4.26 3.9 4.15 3.97 4.09
Ea 100 23.89 20.36 12.24 9.34 6.03 4.37 2.93
F(xi+1) -2.23 -8.07 -3.72 -6.59 -4.54 -5.92 -4.95 -5.61
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
4.09 4.01 4.07 4.03 4.05 4.04 4.05 4.04 4.05 4.04 4.04 4.04 4.04 4.04 4.04
4.01 4.07 4.03 4.05 4.04 4.05 4.04 4.05 4.04 4.04 4.04 4.04 4.04 4.04 4.04
2.07 1.41 0.98 0.67 0.47 0.32 0.22 0.15 0.11 0.07 0.05 0.04 0.02 0.02 0.01
-5.15 -5.47 -5.25 -5.4 -5.29 -5.37 -5.32 -5.35 -5.33 -5.34 -5.33 -2.23 -5.34 -5.34 -5.34
.7 x + 2.5
Xu=10
calcule Ea y Ev
F(x)=-0,4x²+2,2 x
f(x) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
-57.3 -47.5 -38.5 -30.3 -22.9 -16.3 -10.5 -5.5 -1.3 2.1 4.7 6.5 7.5 7.7 7.1 5.7 3.5
GRAFICO 10 0 -10 -20 -30 -40 -50 -60 -10
-8
-6
-4
-2
0
2
4
-30 -40 7 8 9 10
0.5 -3.3 -7.9 -13.3
-50 -60 -10
-8
-6
-4
-2
0
2
F(x)=-0,4x²+2,2x+4,7 a= b= c=
-0.4 2.2 4.7
X1= X2=
-1.64 7.14
Xl=5 Xu=10 -------------->>>Calcule Ea y Ev. Valor verdadero= 7,14459896 Iteracion 1 2
Xl
Xu 5 5
Xr 10 7.5
F(Xl) 7.5 6.25
F(Xu) 5.7 5.7
-13.3 -1.3
4
3
6.25
7.5
6.88
2.83
-1.3
2.-Determine las raices reales de:
a) Graficamente b) Usando biseccion con un :
Es= 10% Xl= 0,5 Xu= 1,0
c)Realice el mismo calculo como en el inciso b), pero cuando el metodo d
x
f(x) -10 -210049 -9 -138422.15 -8 -87724.4 -7 -53019.85 -6 -30212.6 -5 -15968.75 -4 -7638.4 -3 -3177.65 -2 -1070.6 -1 -251.35 0 -26 1 5.35 2 26.6 3 83.65 4 162.4 5 266.75 6 496.6 7 1125.85 8 2680.4 9 6016.15 10 12397
GRAFICO 25000 0 -25000 -50000 -75000 -100000 -125000 -150000 -175000 -200000 -225000 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0
1
2
3
4
-150000 -175000 -200000 -225000 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0
1
Es= 10% Xl= 0,5 Xu= 1,0
Iteracion
Xl 1 2 3 4 5 6
Xu 0.5 0.5 0.5 0.56 0.56 0.58
Xr 1 0.75 0.63 0.63 0.59 0.59
Ea 0.75 0.63 0.56 0.59 0.58 0.59
F(Xl) 100 20 11.11 5.26 2.7 1.33
-1.72 -1.72 -1.72 -0.33 -0.33 -0.02
Es= 0,1%
Iteracion
Xl 1 2
Xu 0.5 -0.52
Xr 1 1
Ea -0.52 -0.09
F(Xl) 100 497.96
-1.72 -99.66
2
3
4
3 4 5 6 7 8 9
-0.09 -0.17 -0.15 -0.16 -0.15 -0.15 -0.15
1 1 1 1 1 1 1
-0.17 -0.15 -0.16 -0.15 -0.15 -0.15 -0.15
49.98 15.97 3.96 1.05 0.27 0.07 0.02
-33.85 -43.21 -40.47 -41.17 -40.98 -41.03 -41.02
para determinar la raiz mas grande.
F(x)=-0,4x²+2,2x+4,7
AFICO
0
2
4
6
8
10
0
2
4
6
8
lcule Ea y Ev.
F(Xr) -1.3 2.83
Ev -4.97 12.52
Ea 100 20
10
0.92
3.77
9.09
uando el metodo de falsa posicion con un Es=0,1%
RAFICO
-1
f(x)
0
1
2
3
4
5
6
7
8
9
10
-1
0
1
2
3
4
F(Xu)
5
6
F(Xr) 5.35 2.68 0.84 0.84 0.27 0.27
F(Xu)
2.68 0.84 -0.33 0.27 -0.02 0.13
F(Xr) 5.35 5.35
-99.66 -33.85
7
8
9
10
5.35 5.35 5.35 5.35 5.35 5.35 5.35
-43.21 -40.47 -41.17 -40.98 -41.03 -41.02 -41.02
3.- Determine la raiz real de:
a) Analiticamente b) Con el metodo de la falsa posicion con:
Es=0,1% use un valor inicial de 3,0 a 4,0.
X 3, 3 = 79 3, 3
X 3,3 = 3,3 79
X = 3,3 79 X = 3,758707344
Es=0,1
X=3,758707344 X-3,758707344=0
F(x)= x - 3,758707344 Xu=4,0 Xl=3,0
Iteracion
Xl 1 2 3 4 5 6 7
Xu
3 -4.24 -0.73 -1.09 -1.03 -1.04 -1.04
Xr 4 4 4 4 4 4 4
-4.24 -0.73 -1.09 -1.03 -1.04 -1.04 -1.04
Ea 100 483.67 33.18 5.84 0.91 0.14 0.02
F(Xl) -0.76 -8 -4.49 -4.85 -4.79 -4.8 -4.79
F(Xu) 0.24 0.24 0.24 0.24 0.24 0.24 0.24
4.- Determine la raiz real de :
a) Analiticamente b) Graficamente c)Usando tres iteraciones con el metodo de la falsa posicion con valores
0,9 − 0, 4 x F ( x) = x 0,9 − (0, 4 * 1 F (1) = 1 0,5 F ( x) = 1 F ( x) = 0,5
X
0,9 − (0, 4 * 1 F (1) = 1 0,5 F ( x) = 1 F ( x) = 0,5
F(X) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
-0.49 -0.5 -0.51 -0.53 -0.55 -0.58 -0.63 -0.7 -0.85 -1.3 #DIV/0! 0.5 0.05 -0.1 -0.18 -0.22 -0.25 -0.27 -0.29 -0.3 -0.31
GRAFICO 0.5 0.25 0 -0.25 -0.5 -0.75 -1 -1.25 -1.5 -10
-8
-6
-4
-2
0
2
Xl= 1 Xu= 3 Calcular Ea y Ev
Iteracion 1 2 3
Xl
Xu 1 1 1
3 4.67 5.53
Xr
Ea 4.67 5.53 6.04
100 15.54 8.52
F(xl) 0.5 0.5 0.5
F(xu) -0.1 -0.21 -0.24
4
ial de 3,0 a 4,0.
F(Xr) -8 -4.49 -4.85 -4.79 -4.8 -4.79 -4.79
on con valores iniciales 1 y 3 calcule Ea yEv.
9 − 0, 4 x x 9 − (0, 4 * 1) 1 5 1 5
9 − (0, 4 * 1) 1 5 1 5
ICO
0
2
4
6
8
10
VALOR VERDADERO= 0,5
F(xr) -0.21 -0.24 -0.25
Ev -833.33 -1005.05 -1108.03
5.- Use el metodo de iteracion simple de punto fijo para localizar la raiz de:
f ( x) = sen( x ) − x Donde Xo= 0,5 y Ea<= 0,01 %
f ( x) = sen( x ) − x sen
(
)
x −x =0
( x )* (−1) x =sen( x )
−x =−sen
xi +1 = sen
(
Donde ------>> Iteracion 1 2 3 4 5 6 7 8 9 1 11 12 13 14 15
Xi
)
xi = 0.5
Xi+1 0.5 0.01 0 0 0 0 0 0 0 0 0 0 0 0 0
xi
0.01 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Ea 100 536.5 152.29 58.84 26.03 12.26 5.95 2.93 1.46 0.73 0.36 0.18 0.09 0.04 0.02
16 17
6.- Determine la raiz de :
Usando Xo= 5
Ea< 0,01%
0 0
0 0
0.01 0.01
f ( x) = −0.9 x + 1.7 x + 2.5 2
Empleando
a) Iteracion punto fijo b) Newton c) Secante
f ( x) = −0.9 x + 1.7 x + 2.5 2
−9 x 2 + 1.7 x +2.5 =0 −0.9 x 2 =− 1.7 x −2.5 * ( − 1) 0.9 x 2 =1.7 +2.5 2 1.7 x +2.5 x = 0.9 1.7 x +2.5 x = 0.9
x i +1 = x i =5
1.7 x i −2.5 0.9
Iteracion
Xi
1 2 3 4 5 6 7 8 9 10
Xi+1 5 2.58 1.45 0.2 1.55 0 1.67 0.61 1.28 0.61
2.58 1.45 0.2 1.55 0 1.67 0.61 0.8 0.61 1.28
Ea 100 78.2 615.86 86.92 #DIV/0! 100 173.86 24 32.08 52.56
f ( x) = −0.9 x + 1.7 x + 2.5 2
xi +1 = xi −
f ( xi ) f ' ( xi )
f ' ( x) = −1.8 x +1.7 2
xi +1
( −0.9 xi +1.7 xi + 2.5) =5 − −1.8 xi +1.7
Iteracion
xi 1 2 3 4 5 6 7 8
xi+1 5 3.42 4.5 3.74 4.26 3.9 4.15 3.97
3.42 4.5 3.74 4.26 3.9 4.15 3.97 4.09
Ea 100 23.89 20.36 12.24 9.34 6.03 4.37 2.93
F(xi+1) -2.23 -8.07 -3.72 -6.59 -4.54 -5.92 -4.95 -5.61
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
4.09 4.01 4.07 4.03 4.05 4.04 4.05 4.04 4.05 4.04 4.04 4.04 4.04 4.04 4.04
4.01 4.07 4.03 4.05 4.04 4.05 4.04 4.05 4.04 4.04 4.04 4.04 4.04 4.04 4.04
2.07 1.41 0.98 0.67 0.47 0.32 0.22 0.15 0.11 0.07 0.05 0.04 0.02 0.02 0.01
-5.15 -5.47 -5.25 -5.4 -5.29 -5.37 -5.32 -5.35 -5.33 -5.34 -5.33 -2.23 -5.34 -5.34 -5.34
.7 x + 2.5
