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204Lecture112006

# 204Lecture112006 - Economics 204 Lecture 11Monday Revised...

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Economics 204 Lecture 11–Monday, August 14, 2006 Revised 8/14/06, Revisions marked by ** Sections 4.1-4.3, Unified Treatment Definition 1 Let f : I R , where I R is an open interval. f is differentiable at x I if lim h 0 f ( x + h ) f ( x ) h = a for some a R . This is equivalent to** lim h 0 f ( x + h ) ( f ( x ) + ah ) h = 0 ⇔ ∀ ε> 0 δ> 0 0 < | h | < δ f ( x + h ) ( f ( x ) + ah ) h < δ ⇔ ∀ ε> 0 δ> 0 0 < | h | < δ | f ( x + h ) ( f ( x ) + ah ) | | h | < δ lim h 0 | 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 0 | 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

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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 0 means K,δ> 0 | h | < δ ⇒ | y | ≤ ∗ ∗ K | h | **read y is big-Oh of h y = o ( h ) as h 0 means lim h 0 | y | | h | = 0 **read y is little-oh 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 0 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 0 2
Let’s compute Df ( x ) = ( a ij ). Let { e 1 , . . . , e n } be the standard basis of R n . Look in direction e j ; | γe j | = | γ | . o ( γ ) = f ( x + γe j ) ( f ( x ) + T x ( γe j )) = f ( x + γe j ) f ( x ) + a 11 · · · a 1 j · · · a 1 n . . . . . . . . . . . . . . . a m 1 · · · a mj . . . a mn 0 . . . 0 γ 0 . . . 0 = f ( x + γe j ) f ( x ) + γa 1 j .

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