This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ntimes 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 ....
View
Full
Document
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.
 Summer '08
 Staff
 Economics

Click to edit the document details