20e-lecture8

# 20e-lecture8 - Lecture 8 Taylor's Theorem Monday October...

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Lecture 8 Taylor s Theorem Monday October 13th

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Functions R R : f Single Variable Taylor s Theorem Let R R : f be a function which has n continuous derivatives on the closed interval ] , [ x a and suppose the th ) 1 ( + n derivative of f exists on the open interval ) , ( x a . Then there exists a real number ξ ) , ( x a such that = + + + + = n k n n k k n a x f k a x a f x f 0 1 ) 1 ( ) ( )! 1 ( ) ( ) ( ! ) ( ) ( ) ( ξ ± The last term is denoted by ) , ( a x R n and called the Lagrange form of the remainder . ± The sum is denoted ) , ( a x T n and is called the Taylor series of order n for f at a .
Functions R R : f ± Example 1 If ) 1 log( ) ( x x f + = then we compute n x x x x x T n n n 1 3 2 ) 1 ( 3 2 ) 0 , ( + + = L 1 1 ) 1 )( 1 ( ) 1 ( ) 0 , ( + + + + = n n n n n x x R ξ where x < < ξ 0 . Therefore we get a good approximation for ) 1 log( x + if 1 0 x for 1 1 ) 1 ( ) 0 , ( 1 + + + n n x x R n n and this tends to zero.

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Functions R R n f : ± To state Taylor s Theorem for several variables properly, we need some notation. ± If R R n f : is a function of variables ) ,..., , ( 2 1 n x x x and z is any unordered list of i x s, then ) ( x f z means the
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20e-lecture8 - Lecture 8 Taylor's Theorem Monday October...

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