204Lecture122006 - Economics 204 Lecture 12Tuesday, August...

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Unformatted text preview: Economics 204 Lecture 12Tuesday, August 15, 2006 Revised 8/16/06, Revisions indicated by ** Section 4.4, Taylors Theorem Theorem 1 (1.9, Taylors Theorem in R 1 ) Let f : I R be n-times differentiable, where I R is an open interval. If x, x + h I , then f ( x + h ) = f ( x ) + n 1 X k =1 f ( k ) ( x ) h k k ! + E n where f ( k ) is the k th derivative of f E n = f ( n ) ( x + h ) h n n ! for some (0 , 1) Motivation: Let T n ( h ) = f ( x ) + n X k =1 f ( k ) ( x ) h k k ! = f ( x ) + f ( x ) h + f ( x ) h 2 2 + + f ( n ) ( x ) h n n ! T n (0) = f ( x ) T n ( h ) = f ( x ) + f ( x ) h + + f ( n ) ( x ) h n 1 ( n 1)! T n (0) = f ( x ) T n ( h ) = f ( x ) + + f ( n ) ( x ) h n 2 ( n 2)! T n (0) = f ( x ) . . . T ( n ) n (0) = f ( n ) ( x ) 1 so T n ( h ) is the unique n th degree polynomial such that T n (0) = f ( x ) T n (0) = f ( x ) . . . T ( n ) n (0) = f ( n ) ( x ) The proof of the formula for the remainder E n is essentially the Mean Value Theorem; the problem in applying it is that the point x + h is not known in advance. Theorem 2 (Alternate Taylors Theorem in R 1 ) Let f : I R be n times differentiable, where I R is an open in- terval and x I . Then f ( x + h ) = f ( x ) + n X k =1 f ( k ) ( x ) h k k ! + o ( h n ) as h If f is ( n +1) times continuously differentiable (i.e. all deriva- tives up to order n + 1 exist and are continuous), then f ( x + h ) = f ( x ) + n X k =1 f ( k ) ( x ) h k k ! + O h n +1 as h Remark: The first equation in the statement of the theorem is essentially a restatement of the definition of the n th derivative. The second statement is proven from Theorem 1.9, and the continuity of the derivative, hence the boundedness of the derivative on a neighborhood of x ....
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This note was uploaded on 03/06/2009 for the course ECON 0204 taught by Professor Staff during the Summer '08 term at University of California, Berkeley.

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204Lecture122006 - Economics 204 Lecture 12Tuesday, August...

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