Taylor 5 Gaia
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P ( x) = n k=0 a k x k = a 0 + a 1 x + a 2 x 2 + . .. + a n x n P (x) = P (x) = ··· P (n) (x) = P (n+1) (x) = ··· x = 0 P (k) (0) = k !a k k = 0,...,n a k = P (k) (0) k! k = 0,...,n. P ( x) = n k=0 a k (x − a) k = a 0 + a 1 (x − a) + a 2 (x − a) 2 + . .. + a n (x − a) n x = a a k = P (k) (a) k! k = 0,...,n x = a n f (x) n. a P n,a (x) = n k=0 f (k) (a) k! (x − a) k = = f (a) + f (a) (x − a) + f (a) 2! (x − a) 2 + . .. + f (n) (a) n! (x − a) n .
Transcript of Taylor 5 Gaia
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P (x) =
n
k=0
akxk = a0 + a1x + a2x2 + . . . + anx
n
P (x) =
P (x) =
· · ·
P (n) (x) =
P (n+1) (x) =
· · ·
x = 0 P (k) (0) = k!ak k = 0, . . . , n
ak = P (k) (0)
k! k = 0, . . . , n .
P (x) =n
k=0
ak (x− a)k
= a0 + a1 (x− a) + a2 (x− a)2
+ . . . + an (x− a)n
x = a
ak = P (k)
(a)k!
k = 0, . . . , n
x = a n f (x)
n. a
P n,a (x) =n
k=0
f (k) (a)
k! (x− a)
k=
= f (a) + f (a) (x− a) + f (a)
2! (x− a)
2+ . . . +
f (n) (a)
n! (x− a)
n.
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a = 0
f (x) = ex
f (x) = ln (1 + x)
f (x) = sin x
f (x) = cosx
f (x) = ex n = 1, 2, 3,...
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P (k) (a) = f (k) (a) k = 0, . . . , n .
f (x) a
a
f (x) = ex
lımx→0
f (x)−P 1,0(x)x
= 0 x ex
1 +x
x
x2
lımx→0
f (x)−P 1,0(x)x2
= 0
lımx→0
f (x)−P 2,0(x)x2
= 0
lımx→0
f (x)−P 2,0(x)x3
= 0 limx→0
f (x)−P 3,0(x)x3
= 0
limx→a
f (x) − P n,a (x)
(x− a)n = 0 .
e
n = 3 n = 4 n = 5 n = 6
