204Lecture112009web - Economics 204 Lecture 11Monday,...

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Economics 204 Lecture 11–Monday, August 10, 2009 Sections 4.1-4.3, Unifed Treatment Defnition 1 Let f : I R ,where I R is an open interval. f is diferentiable 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 | ± ± ± ± ± f ( x + h ) ( f ( x ah ) h ± ± ± ± ± ε> 0 0 0 < | h | | f ( x + h ) ( f ( x ah ) | | h | lim h 0 | f ( x + h ) ( f ( x ah ) | | h | Recall that the limit considers h near zero, but not h . Defnition 2 If X R n is open, f : X R m is diferentiable at x X if T x L ( R n , R m ) lim h 0 ,h R n | f ( x + h ) ( f ( x T x ( h )) | | h | (Recall |·| denotes the Euclidean distance.) f is diferentiable if it is diFerentiable 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 | n )as h 0 means K,δ> 0 | h | ⇒| y |≤ K | h | n read y is big-Oh of | h | n y = o ( | h | n h 0 means lim h 0 | y | | h | n =0 read y is little-oh of | h | n Note that the statement y = O ( | h | n +1 h 0 implies y = o ( | h | n h 0. Note that f is diFerentiable at x ⇔∃ T x L ( R n , R m ) f ( x + h )= f ( x T x ( h o ( h h 0 Notation: df x is the linear transformation T x Df ( x ) is the matrix of 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 x ( h )) (Error Term) f is diFerentiable at x E f ( h o ( h h 0 2
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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 + j ) ( f ( x )+ T x ( j )) = f ( x + 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 + j ) f ( x γa 1 j . . . mj For i =1 ,...,m ,let f i denote the i th component of the function f : f i ( x + j ) ± f i ( x ij ² = o ( γ ) so a ij = ∂f i ∂x j Theorem 3 (3.3) Suppose X R n is open and f : X R m is diferentiable at x X .Th en i j exists For 1 i m , 1 j n ,and ( Df )( x )= 1 1 1 n .
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204Lecture112009web - Economics 204 Lecture 11Monday,...

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