Aproximacion polinomial

ANÁLISIS NUMÉRICO
APROXIMACIÓN POLINOMIAL




Andrés Felipe Solano Romero
2081633


Profesor:
Alfonso Mendoza Castellanos

UNIVERSIDAD INDUSTRIAL DE SANTANDER
Bucaramanga, 15 de octubre de 2010

1. Establecer la serie de Taylor correspondiente.

Si fx= e x entonces f(n)x = e x para toda x, por tanto, f(n)0 = 1 para toda n, así, de la ecuación demaclaurin se tiene la serie de maclaurin:
ex=1+x1!+x22!+x33!+…+xnn!, -∞<x<∞

2. Aplique la fórmula de Taylor deducida alrededor de x0=0 en series de potencia (x-0)k = x k hasta el polinomio de grado 15.

ex=n=0∞xnn!

ex=n=015xnn!

n | fkx | fk0 | Expresión | Pnx |
0 | ex | 1 | 1 | 1 |
1 | ex | 1 | x | 1+x |
2 | ex | 1 | x22! | 1+ x + x22! |
3 | ex | 1 | x33! | 1+x + x33! |
4 | ex | 1 | x44! | 1+ x + x33!+ x44! |
5 | ex | 1 | x55! | 1+ x + x33!+ x44!+ x55! |
6 | ex | 1 | x66! | 1+ x + x33!+ x44!+ x55!+ x66! |
7 | ex | 1 | x77! | 1+ x + x33!+ x44!+ x55!+ x66!+ x77! |
8 | ex | 1 | x88! | 1+ x + x33!+ x44!+ x55!+ x66!+ x77!+ x88! |
9 | ex | 1 | x99! | 1+ x + x33!+ x44!+ x55!+ x66!+ x77!+ x88!+ x99! |
10 | ex | 1 | x1010! | 1+ x + x33!+ x44!+x55!+ x66!+ x77!+ x88!+ x99!+ x1010! |
11 | ex | 1 | x1111! | 1+ x + x33!+ x44!+ x55!+ x66!+ x77!+ x88!+ x99!+ x1010!+ x1111! |
12 | ex | 1 | x1212! | 1+ x + x33!+ x44!+ x55!+ x66!+ x77!+ x88!+ x99!+ x1010!+ x1111!+ x1212! |
13 | ex | 1 | x1313! | 1+ x + x33!+ x44!+ x55!+ x66!+ x77!+ x88!+ x99!+ x1010!+ x1111!+ x1212!+ x1313! |
14 | ex | 1 | x1414! | 1+ x + x33!+ x44!+ x55!+ x66!+ x77!+ x88!+x99!+ x1010!+ x1111!+ x1212!+ x1313!+ x1414! |
15 | ex | 1 | x1515! | 1+ x + x33!+ x44!+ x55!+ x66!+ x77!+ x88!+ x99!+ x1010!+ x1111!+ x1212!+ x1313!+ x1414!+ x1515! |

3. Haga un listado de los resultados obtenidos de las sumas de cada polinomio, el valor de exp se aproxima a : 2.718281828459.

N | Pnx | Valor Pn(1) |
0 | 1 | 1 |
1 | 1+x | 2 |
2 | 1+ x + x22! | 2,500000000000|
3 | 1+ x + x22!+ x33! | 2,666666666667 |
4 | 1+ x + x22!+ x33!+ x44! | 2,708333333333 |
5 | 1+ x + x22!+ x33!+ x44!+ x55! | 2,716666666667 |
6 | 1+ x + x22!+ x33!+ x44!+ x55!+ x66! | 2,718055555556 |
7 | 1+ x + x22!+ x33!+ x44!+ x55!+ x66!+ x77! | 2,718253968254 |
8 | 1+ x+ x22! + x33!+ x44!+ x55!+ x66!+ x77!+ x88! | 2,718278769841 |
9 | 1+ x + x22!+ x33!+ x44!+ x55!+ x66!+ x77!+x88!+ x99! | 2,718281525573 |
10 | 1+ x + x22!+ x33!+ x44!+ x55!+ x66!+ x77!+ x88!+ x99!+ x1010! | 2,718281801146 |
11 | 1+ x+ x22! + x33!+ x44!+ x55!+ x66!+ x77!+ x88!+ x99!+ x1010!+ x1111! | 2,718281826198 |
12 | 1+ x + x22!+ x33!+ x44!+ x55!+ x66!+ x77!+ x88!+ x99!+ x1010!+ x1111!+ x1212! | 2,718281828286 |
13 | 1+ x + x22!+ x33!+ x44!+ x55!+ x66!+ x77!+ x88!+ x99!+ x1010!+ x1111!+x1212!+ x1313! | 2,718281828447 |
14 | 1+ x + x22!+ x33!+ x44!+ x55!+ x66!+ x77!+ x88!+ x99!+ x1010!+ x1111!+ x1212!+ x1313!+ x1414! | 2,718281828458 |
15 | 1+ x + x22!+ x33!+ x44!+ x55!+ x66!+ x77!+ x88!+ x99!+ x1010!+ x1111!+ x1212!+ x1313!+ x1414!+ x1515! | 2,718281828459 |
N | Valor Pn(1) | Pn(1) – exp(1) |
0 | 1 | 1,718281828 |
1 | 2 | 0,718281828 |
2 | 2,500000000000 | 0,218281828 |3 | 2,666666666667 | 0,051615162 |
4 | 2,708333333333 | 0,009948495 |
5 | 2,716666666667 | 0,001615162 |
6 | 2,718055555556 | 0,000226273 |
7 | 2,718253968254 | 0,000027860 |
8 | 2,718278769841 | 0,000003059 |
9 | 2,718281525573 | 0,000000303 |
10 | 2,718281801146 | 0,000000027 |
11 | 2,718281826198 | 0,000000002 |
12 | 2,718281828286 | 0,000000000 |
13 | 2,718281828447 |0,000000000 |
14 | 2,718281828458 | 0,000000000 |
15 | 2,718281828459 | 0,000000000 |

4. Grafique al menos 4 polinomios para establecer las diferencias en sus aproximaciones. De conclusiones.
Función Euler

P0x :

P1x :

P2x :

P3x :

P4x :

P5x :

P6x :

P7x :

Fórmula para encontrar el máximo

En=fn+1RRn+1n+1!(R)n+1

n | fn+1x |...