5. Polinomio De Taylor.pdf
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POLINOMIO DE TAYLOR. TEOREMA DE TAYLOR. Funciones polin´ omicas. Dado el polinomio P (x) =
k=n X
bk (x − a)k = b0 + b1 (x − a) + b2 (x − a)2 + . . . + bn (x − a)n
k=0
y derivando se ve que sus coeficientes se pueden obtener como bk = En particular, si
P (k) (a) k!
P (x) = c0 + c1 x + c2 x2 + . . . + cn xn =⇒ ck =
P (k) (0) k!
Polinomio de Taylor Definici´ on. Si la funci´on f es derivable hasta orden n en x = a, de Taylor de grado n de f en a como Pn,f,a (x) =
k=n (k) X f (a) k=0
k!
(x − a)k = f (a) +
a ∈ Domf , se define polinomio
f 0 (a) f 00 (a) (x − a) + (x − a)2 + 1! 2!
f (n) (a) +... + (x − a)n n! Ejemplo. Calcular el polinomio de Taylor de
f (x) = ln(x)
en a = 1 hasta el orden 5.
Teorema de Taylor. Sea f derivable hasta el orden n + 1 en un entorno de x = a, a ∈ Dom f , se verifica que para todo punto de dicho entorno f (x) = f (a) +
f 0 (a) f 00 (a) f (n) (a) (x − a) + (x − a)2 + . . . + (x − a)n + 1! 2! n! f (n+1) (α) (x − a)n+1 (n + 1)! a < α < x o´ x < α < a +
f (n+1) (α) A Rn,a (x) = (x − a)n+1 se le llama Resto de Lagrange de orden n+1 (n + 1)! Ejemplos 1. Funci´on Exponencial f (k) (x) = ex ,
f (x) = ex
f (k) (0) = 1,
en a = 0
∀ k ∈ IN
ex = 1 + x +
x2 x3 xn eα + + ... + + xn+1 2! 3! n! (n + 1)! 0 < α < x o´ x < α < 0 1
2. Funci´on Logaritmo
f (k) (x) =
f (x) = ln(x)
(−1)k+1 (k − 1)! , xk
en a = 1
f (k) (1) = (−1)k+1 (k − 1)! ∀ k ≥ 1
ln(x) = (x − 1) −
(x − 1)n (x − 1)2 (x − 1)3 + − . . . + (−1)n+1 + 2 3 n +(−1)n+2
α−(n+1) (x − 1)n+1 (n + 1)
1 < α < x o´ 0 < x < α < 1 3. Funci´on seno
f (x) = sen x
f (2k) (x) = (−1)k sen x,
f (2k) (0) = 0
f (2k+1) (x) = (−1)k cos x, sen x = x −
en a = 0
f (2k+1) (0) = (−1)k ,
∀ k ∈ IN
x3 x5 x2n+1 cos α + − . . . + (−1)n + (−1)n+1 x2n+3 3! 5! (2n + 1)! (2n + 3)! 0 < α < x o´ x < α < 0
4. Funci´on coseno
f (x) = cos x
f (2k) (x) = (−1)k cos x,
f (2k) (0) = (−1)k
f (2k+1) (x) = (−1)k+1 sen x, cos x = 1 −
en a = 0
f (2k+1) (0) = 0,
∀ k ∈ IN
x 2 x4 x2n cos α + − . . . + (−1)n + (−1)n+1 x2n+2 2! 4! (2n)! (2n + 2)! 0 < α < x o´ x < α < 0
2
k=n X
bk (x − a)k = b0 + b1 (x − a) + b2 (x − a)2 + . . . + bn (x − a)n
k=0
y derivando se ve que sus coeficientes se pueden obtener como bk = En particular, si
P (k) (a) k!
P (x) = c0 + c1 x + c2 x2 + . . . + cn xn =⇒ ck =
P (k) (0) k!
Polinomio de Taylor Definici´ on. Si la funci´on f es derivable hasta orden n en x = a, de Taylor de grado n de f en a como Pn,f,a (x) =
k=n (k) X f (a) k=0
k!
(x − a)k = f (a) +
a ∈ Domf , se define polinomio
f 0 (a) f 00 (a) (x − a) + (x − a)2 + 1! 2!
f (n) (a) +... + (x − a)n n! Ejemplo. Calcular el polinomio de Taylor de
f (x) = ln(x)
en a = 1 hasta el orden 5.
Teorema de Taylor. Sea f derivable hasta el orden n + 1 en un entorno de x = a, a ∈ Dom f , se verifica que para todo punto de dicho entorno f (x) = f (a) +
f 0 (a) f 00 (a) f (n) (a) (x − a) + (x − a)2 + . . . + (x − a)n + 1! 2! n! f (n+1) (α) (x − a)n+1 (n + 1)! a < α < x o´ x < α < a +
f (n+1) (α) A Rn,a (x) = (x − a)n+1 se le llama Resto de Lagrange de orden n+1 (n + 1)! Ejemplos 1. Funci´on Exponencial f (k) (x) = ex ,
f (x) = ex
f (k) (0) = 1,
en a = 0
∀ k ∈ IN
ex = 1 + x +
x2 x3 xn eα + + ... + + xn+1 2! 3! n! (n + 1)! 0 < α < x o´ x < α < 0 1
2. Funci´on Logaritmo
f (k) (x) =
f (x) = ln(x)
(−1)k+1 (k − 1)! , xk
en a = 1
f (k) (1) = (−1)k+1 (k − 1)! ∀ k ≥ 1
ln(x) = (x − 1) −
(x − 1)n (x − 1)2 (x − 1)3 + − . . . + (−1)n+1 + 2 3 n +(−1)n+2
α−(n+1) (x − 1)n+1 (n + 1)
1 < α < x o´ 0 < x < α < 1 3. Funci´on seno
f (x) = sen x
f (2k) (x) = (−1)k sen x,
f (2k) (0) = 0
f (2k+1) (x) = (−1)k cos x, sen x = x −
en a = 0
f (2k+1) (0) = (−1)k ,
∀ k ∈ IN
x3 x5 x2n+1 cos α + − . . . + (−1)n + (−1)n+1 x2n+3 3! 5! (2n + 1)! (2n + 3)! 0 < α < x o´ x < α < 0
4. Funci´on coseno
f (x) = cos x
f (2k) (x) = (−1)k cos x,
f (2k) (0) = (−1)k
f (2k+1) (x) = (−1)k+1 sen x, cos x = 1 −
en a = 0
f (2k+1) (0) = 0,
∀ k ∈ IN
x 2 x4 x2n cos α + − . . . + (−1)n + (−1)n+1 x2n+2 2! 4! (2n)! (2n + 2)! 0 < α < x o´ x < α < 0
2
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