X36 taylor's remainder theorem ii
Transcript of X36 taylor's remainder theorem ii
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Taylor's Remainder Theorem II
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Taylor's Remainder Theorem IIWe state the general form of the Taylor's remainder formula.
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Taylor's Remainder Theorem IIWe state the general form of the Taylor's remainder formula. Taylor's Remainder Theorem (General Form): Let f(x) be an infinitely differentiable function over some open interval that contains [a, b]
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Taylor's Remainder Theorem IIWe state the general form of the Taylor's remainder formula. Taylor's Remainder Theorem (General Form): Let f(x) be an infinitely differentiable function over some open interval that contains [a, b]
a( )[ ]
b
f(x) is infinitely differentiable in here
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Taylor's Remainder Theorem IIWe state the general form of the Taylor's remainder formula. Taylor's Remainder Theorem (General Form): Let f(x) be an infinitely differentiable function over some open interval that contains [a, b] and
pn(x) =
be the n'th Taylor-poly expanded around at a.
a( )[ ]
b
f(x) is infinitely differentiable in here
Σk=0
n (x – a)k
k! f(k)(a)
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Taylor's Remainder Theorem IIWe state the general form of the Taylor's remainder formula. Taylor's Remainder Theorem (General Form): Let f(x) be an infinitely differentiable function over some open interval that contains [a, b] and
pn(x) =
be the n'th Taylor-poly expanded around at a. Then there exists a "c" between a and b such that
f(b) = pn(b) + (b – a)n+1(n + 1)! f(n+1)(c)
a( )[ ]
bc
f(x) is infinitely differentiable in here
Σk=0
n (x – a)k
k! f(k)(a)
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or in full detail,
f '(a)(b – a) f(2)(a)+ 2!= f(a) + (b – a)2f(b) ..
f(n)(a)n! (b – a)n+
Taylor's Remainder Theorem II
+ Rn(b)
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or in full detail,
where Rn(b) = (b – a)n+1
(n + 1)! f(n+1)(c)
is the Lagrange form of the Taylor-remainder.
f '(a)(b – a) f(2)(a)+ 2!= f(a) + (b – a)2f(b) ..
f(n)(a)n! (b – a)n+
Taylor's Remainder Theorem II
+ Rn(b)
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or in full detail,
where Rn(b) = (b – a)n+1
(n + 1)! f(n+1)(c)
is the Lagrange form of the Taylor-remainder.
f '(a)(b – a) f(2)(a)+ 2!= f(a) + (b – a)2f(b) ..
f(n)(a)n! (b – a)n+
Again, we note the following
Taylor's Remainder Theorem II
+ Rn(b)
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or in full detail,
where Rn(b) = (b – a)n+1
(n + 1)! f(n+1)(c)
is the Lagrange form of the Taylor-remainder.
f '(a)(b – a) f(2)(a)+ 2!= f(a) + (b – a)2f(b) ..
f(n)(a)n! (b – a)n+
Again, we note the following
* the theorem also works for the interval [b, a]
Taylor's Remainder Theorem II
+ Rn(b)
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or in full detail,
where Rn(b) = (b – a)n+1
(n + 1)! f(n+1)(c)
is the Lagrange form of the Taylor-remainder.
f '(a)(b – a) f(2)(a)+ 2!= f(a) + (b – a)2f(b) ..
f(n)(a)n! (b – a)n+
Again, we note the following
* the theorem also works for the interval [b, a]
* the value c changes if the value of b or n changes
Taylor's Remainder Theorem II
+ Rn(b)
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or in full detail,
where Rn(b) = (b – a)n+1
(n + 1)! f(n+1)(c)
is the Lagrange form of the Taylor-remainder.
f '(a)(b – a) f(2)(a)+ 2!= f(a) + (b – a)2f(b) ..
f(n)(a)n! (b – a)n+
Again, we note the following
* the theorem also works for the interval [b, a]
* the value c can't be easily determined, we just know there is at least one c that fits the description
* the value c changes if the value of b or n changes
Taylor's Remainder Theorem II
+ Rn(b)
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Example: A. Find the Taylor-series of f(x) = sin(x) at x = .π
2
Taylor's Remainder Theorem II
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Example: A. Find the Taylor-series of f(x) = sin(x) at x = .π
2sin(x)
cos(x)
-sin(x)
-cos(x)
Taylor's Remainder Theorem II
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Example: A. Find the Taylor-series of f(x) = sin(x) at x = .π
2sin(x)
cos(x)
-sin(x)
-cos(x)
At x = π2 , we get the sequence of coefficients
1, 0, -1, 0, 1, 0, -1, …
Taylor's Remainder Theorem II
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Example: A. Find the Taylor-series of f(x) = sin(x) at x = .π
2sin(x)
cos(x)
-sin(x)
-cos(x)
At x = π2 , we get the sequence of coefficients
1, 0, -1, 0, 1, 0, -1, … So the Taylor expansions is
= 1P(x)
Taylor's Remainder Theorem II
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Example: A. Find the Taylor-series of f(x) = sin(x) at x = .π
2sin(x)
cos(x)
-sin(x)
-cos(x)
At x = π2 , we get the sequence of coefficients
1, 0, -1, 0, 1, 0, -1, … So the Taylor expansions is
– = 1P(x)(x – π/2)2
2!
Taylor's Remainder Theorem II
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Example: A. Find the Taylor-series of f(x) = sin(x) at x = .π
2sin(x)
cos(x)
-sin(x)
-cos(x)
At x = π2 , we get the sequence of coefficients
1, 0, -1, 0, 1, 0, -1, … So the Taylor expansions is
– = 1P(x)(x – π/2)2
2!1(x – π/2)4
+ 4!
Taylor's Remainder Theorem II
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Example: A. Find the Taylor-series of f(x) = sin(x) at x = .π
2sin(x)
cos(x)
-sin(x)
-cos(x)
At x = π2 , we get the sequence of coefficients
1, 0, -1, 0, 1, 0, -1, … So the Taylor expansions is
– = 1P(x)(x – π/2)2
2!1(x – π/2)4
+ 4!– 1(x – π/2)6
6!..1(x – π/2)8
+ 8!
Taylor's Remainder Theorem II
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Example: A. Find the Taylor-series of f(x) = sin(x) at x = .π
2sin(x)
cos(x)
-sin(x)
-cos(x)
At x = π2 , we get the sequence of coefficients
1, 0, -1, 0, 1, 0, -1, … So the Taylor expansions is
– = 1P(x)(x – π/2)2
2!1(x – π/2)4
+ 4!– 1(x – π/2)6
6!..
or P(x) = Σ(-1)n(x – π/2)2n
(2n)!n=0
n =∞
1(x – π/2)8
+ 8!
Taylor's Remainder Theorem II
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Example:
B. Describe the Taylor remainder Rn(b) and show that f(b) = P(b) for all values b. (a = ) π/2
Taylor's Remainder Theorem II
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Example:
B. Describe the Taylor remainder Rn(b) and show that f(b) = P(b) for all values b. (a = )
The remainder Rn(b) = (b – a)n+1
(n + 1)! f(n+1)(c)
π/2
Taylor's Remainder Theorem II
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Example:
B. Describe the Taylor remainder Rn(b) and show that f(b) = P(b) for all values b. (a = )
The remainder Rn(b) = (b – a)n+1
(n + 1)! f(n+1)(c)
=
π/2
(b – )n+1
(n + 1)! f(n+1)(c) π
2
Taylor's Remainder Theorem II
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Example:
B. Describe the Taylor remainder Rn(b) and show that f(b) = P(b) for all values b. (a = )
The remainder Rn(b) = (b – a)n+1
(n + 1)! f(n+1)(c)
=
π/2
(b – )n+1
(n + 1)! f(n+1)(c) π
2
where f(n+1) may be ±sin(x) or ±cos(x) and for some c that is between a and b.
Taylor's Remainder Theorem II
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Example:
B. Describe the Taylor remainder Rn(b) and show that f(b) = P(b) for all values b. (a = )
The remainder Rn(b) = (b – a)n+1
(n + 1)! f(n+1)(c)
=
π/2
(b – )n+1
(n + 1)! f(n+1)(c) π
2
where f(n+1) may be ±sin(x) or ±cos(x) and for some c that is between a and b.
Since |f(n+1)(c)| < 1,
Taylor's Remainder Theorem II
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Example:
B. Describe the Taylor remainder Rn(b) and show that f(b) = P(b) for all values b. (a = )
The remainder Rn(b) = (b – a)n+1
(n + 1)! f(n+1)(c)
=
π/2
(b – )n+1
(n + 1)! f(n+1)(c) π
2
where f(n+1) may be ±sin(x) or ±cos(x) and for some c that is between a and b.
Since |f(n+1)(c)| < 1, we've (b – )n+1
(n + 1)! f(n+1)(c) π
2 <(n + 1)!
(b – )n+1π2
Taylor's Remainder Theorem II
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Example:
B. Describe the Taylor remainder Rn(b) and show that f(b) = P(b) for all values b. (a = )
The remainder Rn(b) = (b – a)n+1
(n + 1)! f(n+1)(c)
=
π/2
(b – )n+1
(n + 1)! f(n+1)(c) π
2
where f(n+1) may be ±sin(x) or ±cos(x) and for some c that is between a and b.
Since |f(n+1)(c)| < 1, we've (b – )n+1
(n + 1)! f(n+1)(c) π
2 <(n + 1)!
(b – )n+1π2
Again, as n we've (n + 1)!
(b – )n+1π2 0∞,
Taylor's Remainder Theorem II
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Example:
B. Describe the Taylor remainder Rn(b) and show that f(b) = P(b) for all values b. (a = )
The remainder Rn(b) = (b – a)n+1
(n + 1)! f(n+1)(c)
=
π/2
(b – )n+1
(n + 1)! f(n+1)(c) π
2
where f(n+1) may be ±sin(x) or ±cos(x) and for some c that is between a and b.
Since |f(n+1)(c)| < 1, we've (b – )n+1
(n + 1)! f(n+1)(c) π
2 <(n + 1)!
(b – )n+1π2
Again, as n we've (n + 1)!
(b – )n+1π2 0∞,
Hence Rn(b) 0 and that f(b) = P(b) for all b.
Taylor's Remainder Theorem II
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Differentiation and Integration of Power Series
The Taylor series of f(x) is the only power series that could be the same as f(x).
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Σk=0
∞
Differentiation and Integration of Power Series
Let f(x) = ck(x – a)k for all x in an open interval
(a – R, a + R) for some R, then the series ck(x – a)k
is the Taylor series P(x) of f(x).
Σk=0
∞
Theorem (Uniqueness theorem for Taylor series) :
The Taylor series of f(x) is the only power series that could be the same as f(x).