Agronomia

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  • Publicado : 31 de mayo de 2011
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1. Probar que la ecuación ex-2e-x=0 tiene una única solución real. Obtenerla mediante el método de Newton-Raphson (3 iteraciones). Utiliza 5 cifras decimales en los cálculos.
ex-2e-x=0 e2x-2=0 e2x=2 2x=ln2
x=ln22 x=0,34657359
Luego la ecuación tiene una única solución real
Para Newton-Raphson
x0=1
f(x)=ex-2e-x
f´(x)=ex+2e-x
n | pn| f(pn) | f´(pn) | E(%) |
0 | 1 | 1,98252 | 3,454041 | |
1 | 0,42603 | 0,22497 | 2,83736 | 134,726402 |
2 | 0,34674 | 0,00047 | 2,828427 | 22,8665469 |
3 | 0,34657 | 4,1E-12 | 2,828427 | 0,04812165 |

2. Aproximar mediante el método de la regula falsi la raíz de la ecuación x3-2x2-5=0 en el intervalo [1; 4], realizando 5 iteraciones y utilizando cinco cifras decimales.

n | Xa |Xb | f(Xa) | f(Xb) | Xr | m | f(Xr) | E |
0 | 1,00000 | 4,00000 | -6,00000 | 27,00000 | 1,54545 | 11,00000 | -6,08564 |   |
1 | 1,54545 | 4,00000 | -6,08564 | 27,00000 | 1,99693 | 13,47933 | -5,01222 | 22,60864 |
2 | 1,99693 | 4,00000 | -5,01222 | 27,00000 | 2,31055 | 15,98161 | 26,50152 | 13,57353 |
3 | 2,31055 | 4,00000 | -3,34202 | 27,00000 | 2,49664 | 17,95979 | 36,85307 | 7,45335 |
4| 2,49664 | 4,00000 | -1,90431 | 27,00000 | 2,595687 | 19,226506 | 43,39526 | 3,81580 |

3. Encontrar un valor aproximado de 32 mediante el método de bisección y el método de la secante

a. 32=1,259921 x=32 x3=2 x3-2=0
n | an | bn | pn | f(pn) | E (%) |
1 | 0 | 2 | 1 | -1 | |
2 | 1 | 2 | 1,5 | 1,375 | 33,3333333 |
3 | 1 | 1,5 | 1,25 | -0,046875 | 20 |
4 | 1,25 | 1,5| 1,375 | 0,59960938 | 9,09090909 |
5 | 1,25 | 1,375 | 1,3125 | 0,26098633 | 4,76190476 |
6 | 1,25 | 1,3125 | 1,28125 | 0,103302 | 2,43902439 |
7 | 1,25 | 1,28125 | 1,265625 | 0,02728653 | 1,2345679 |
8 | 1,25 | 1,265625 | 1,2578125 | -0,01002455 | 0,62111801 |
9 | 1,2578125 | 1,265625 | 1,26171875 | 0,00857323 | 0,30959752 |
10 | 1,2578125 | 1,26171875 | 1,25976563 | -0,00074007 |0,15503876 |
11 | 1,25976563 | 1,26171875 | 1,26074219 | 0,00391297 | 0,07745933 |
12 | 1,25976563 | 1,26074219 | 1,26025391 | 0,00158555 | 0,03874467 |
13 | 1,25976563 | 1,26025391 | 1,26000977 | 0,00042251 | 0,01937609 |
14 | 1,25976563 | 1,26000977 | 1,2598877 | -0,00015884 | 0,00968898 |
15 | 1,2598877 | 1,26000977 | 1,25994873 | 0,00013182 | 0,00484426 |
16 | 1,2598877 | 1,25994873| 1,25991821 | -1,351E-05 | 0,00242219 |
17 | 1,25991821 | 1,25994873 | 1,25993347 | 5,9156E-05 | 0,00121108 |
18 | 1,25991821 | 1,25993347 | 1,25992584 | 2,2822E-05 | 0,00060554 |
19 | 1,25991821 | 1,25992584 | 1,25992203 | 4,656E-06 | 0,00030277 |
20 | 1,25991821 | 1,25992203 | 1,25992012 | -4,4272E-06 | 0,00015139 |

b.
n | ani | bni-1 | F(ani) | F(bni-1) | pn | Ea |
0 | 0 | 2 |-2 | 6 | 0,5 |   |
1 | 0,5 | 2 | -1,875 | 6 | 0,85714286 | 17,8571429 |
2 | 0,85714286 | 2 | -1,37026239 | 6 | 1,06962025 | 10,6238698 |
3 | 1,06962025 | 2 | -0,77626085 | 6 | 1,17620077 | 5,32902579 |
4 | 1,06962025 | 1,17620077 | -0,77626085 | -0,37278711 | 1,27467519 | 4,92372085 |
5 | 1,06962025 | 1,17620077 | -0,77626085 | -0,37278711 | 1,27467519 | 0 |
6 | 1,06962025 | 1,17620077| -0,77626085 | -0,37278711 | 1,27467519 | 0 |
7 | 1,06962025 | 1,17620077 | -0,77626085 | -0,37278711 | 1,27467519 | 0 |
8 | 1,06962025 | 1,17620077 | -0,77626085 | -0,37278711 | 1,27467519 | 0 |
9 | 1,06962025 | 1,17620077 | -0,77626085 | -0,37278711 | 1,27467519 | 0 |
10 | 1,06962025 | 1,17620077 | -0,77626085 | -0,37278711 | 1,27467519 | 0 |
11 | 1,06962025 | 1,17620077 |-0,77626085 | -0,37278711 | 1,27467519 | 0 |
12 | 1,06962025 | 1,17620077 | -0,77626085 | -0,37278711 | 1,27467519 | 0 |
13 | 1,06962025 | 1,17620077 | -0,77626085 | -0,37278711 | 1,27467519 | 0 |
14 | 1,06962025 | 1,17620077 | -0,77626085 | -0,37278711 | 1,27467519 | 0 |
15 | 1,06962025 | 1,17620077 | -0,77626085 | -0,37278711 | 1,27467519 | 0 |
16 | 1,06962025 | 1,17620077 | -0,77626085 |...
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