204Lecture122008Web

204Lecture122008Web - Economics 204 Lecture 12Tuesday,...

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Economics 204 Lecture 12–Tuesday, August 12, 2008 Section 4.4 (Cont.): Taylor’s Theorem in R n Defnition 1 X R n , X open, f : X R m . f is continuously diferentiable on X if f is diFerentiable on X and df x is a continuous function of x from X to L ( R n , R m ), with operator norm k x k f is C k if all partial derivatives of order k exist and are continuous in X . Theorem 2 (4.3) Suppose X R n , X open, f : X R m .T h en f is continuously diferentiable on X iF and only iF f is C 1 . Notational Problem in Taylor’s Theorem: If f : R n R m , the quadratic terms are OK for m =1 ;for m> 1, handle each component separately. ±or cubic and higher order terms, notation is a mess. Linear Terms: Theorem 3 Suppose X R n , X is open, x X .I F f : X R m is diferentiable, then f ( x + h )= f ( x )+ Df ( x ) h + o ( h ) as h 0 The previous theorem is essentially a restatement of the de²nition of diFerentiability. 1
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Theorem 4 (Corollary of 4.4) Suppose X R n , X is open, x X .I f f : X R m is C 2 ,then f ( x + h )= f ( x )+ Df ( x ) h + O ± | h | 2 ² as h 0 Quadratic Terms: Treat each component of the function separately, so consider f : X R , X R n an open set. Let D 2 f ( x 2 f ∂x 2 1 2 f 2 1 ··· 2 f n 1 2 f 1 2 2 f 2 2 2 f n 2 . . . . . . . . . . . . 2 f 1 n 2 f 2 n f C 2 2 f i j = 2 f j i D 2 f ( x ) is symmetric D 2 f ( x ) has an orthonormal basis of eigenvectors and thus can be diagonalized Theorem 5 (Stronger Version of 4.4) Let X R n be open, f : X R , f C 2 ( X ) , x X .Then f ( x + h f ( x Df ( x ) h + 1 2 h > ( D 2 f ( x )) h + o ± | h | 2 ² as h 0 If f C 3 , f ( x + h f ( x Df ( x ) h + 1 2 h > ( D 2 f ( x )) h + O ± | h | 3 ² as h 0 2
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Remark: De la Fuente assumes X is convex which he has not yet defned. X is said to be convex i±, ±or every x, y X and α [0 , 1], αx +(1 α ) y X . We don’t need this. Since X is open, x X ⇒∃ δ> 0 B δ ( x ) X and B δ ( x )isconvex . Defnition 6 We say f has a saddle at x Df ( x ) = 0 but x has neither a local maximum nor a local minimum at x .
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204Lecture122008Web - Economics 204 Lecture 12Tuesday,...

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