204Lecture122006 - Economics 204 Lecture 12Tuesday Revised...

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Economics 204 Lecture 12–Tuesday, August 15, 2006 Revised 8/16/06, Revisions indicated by ** Section 4.4, Taylor’s Theorem Theorem 1 (1.9, Taylor’s 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 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 + λh ) h n n ! for some λ (0 , 1) Motivation: Let T n ( h ) = f ( x ) + n 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
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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 + λh is not known in advance. Theorem 2 (Alternate Taylor’s 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 k =1 f ( k ) ( x ) h k k ! + o ( h n ) as h 0 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 k =1 f ( k ) ( x ) h k k ! + O h n +1 as h 0 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 . Section 4.4: Taylor’s Theorem in R n Definition 3 X R n , X open, f : X R m . f is continu- ously differentiable on X if f is differentiable on X and 2
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** df x is a continuous function of x from X to L ( R n , R m ), with operator norm df x f is C k if all partial derivatives of order k
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