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Unformatted text preview: Economics 204 Lecture 11Monday, August 14, 2006 Revised 8/14/06, Revisions marked by ** Sections 4.14.3, Unified Treatment Definition 1 Let f : I R , where I R is an open interval. f is differentiable at x I if lim h f ( x + h ) f ( x ) h = a for some a R . This is equivalent to** lim h f ( x + h ) ( f ( x ) + ah ) h = 0 > > <  h  < f ( x + h ) ( f ( x ) + ah ) h < > > <  h  <  f ( x + h ) ( f ( x ) + ah )   h  < lim h  f ( x + h ) ( f ( x ) + ah )   h  = 0 Recall that the limit considers h near zero, but not h = 0 . Definition 2 If X R n is open, f : X R m is differentiable at x X if T x L ( R n , R m ) lim h  f ( x + h ) ( f ( x ) + T x ( h ))   h  = 0 (Recall   denotes the Euclidean distance.) ** f is differentiable if it is differentiable at all x X . 1 h is a small, nonzero element of R n ; h 0 from any direction, along a spiral, etc. One linear operator T x works no matter how h approaches zero. f ( x ) + T x ( h ) is the best linear approximation to f ( x + h ) for small h Notation: y = O ( h ) as h means K,>  h  <  y  K  h  **read y is bigOh of h y = o ( h ) as h means lim h  y   h  = 0 **read y is littleoh of h Therefore, f is differentiable at x T x L ( R n , R m ) f ( x + h ) = f ( x ) + T x ( h ) + o ( h ) as h Notation: df x is the linear transformation T x Df ( x ) is the matrix of df x with respect to the standard basis; called the Jacobian or Jacobian matrix of f at x E f ( h ) = f ( x + h ) ( f ( x ) + df x ( h )) (Error Term) f is differentiable at x E f ( h ) = o ( h ) as h 2 Lets compute Df ( x ) = ( a ij ). Let { e 1 , . . . , e n } be the standard basis of R n . Look in direction e j ;  e j  =   . o ( ) = f ( x + e j ) ( f ( x ) + T x ( e j )) = f ( x + e j ) f ( x ) + a 11 a 1 j a 1 n . . . . . . . . . . . . . . . a m 1 a mj . . . a mn ....
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 Summer '08
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 Economics

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