204Lecture112009

204Lecture112009 - Economics 204 Lecture 11–Monday...

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Unformatted text preview: Economics 204 Lecture 11–Monday, August 10, 2009 Revised 8/10/09, Revisions indicated by ** and Sticky Notes 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 → 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 ⇔ ∀ ε> ∃ δ> < | h | < δ ⇒ f ( x + h ) − ( f ( x ) + ah ) h < ε ⇔ ∀ ε> ∃ δ> < | h | < δ ⇒ | f ( x + h ) − ( f ( x ) + ah ) | | h | < ε ⇔ 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 → ,h ∈ R n | f ( x + h ) − ( f ( x ) + T x ( h )) | | h | = 0 (1) (Recall | · | denotes the Euclidean distance.) f is differentiable if it is differentiable at all x ∈ X . 1 ** T x is uniquely determined by Equation (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 | n ) as h → means ∃ K,δ> | h | < δ ⇒ | y | ≤ K | h | n read y is big-Oh of | h | n y = o ( | h | n ) as h → means lim h → | y | | h | n = 0 read y is little-oh of | h | n Note that the statement y = O ( | h | n +1 ) as h → 0 implies y = o ( | h | n ) as h → 0. Note that 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 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 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ . . . γ . . . ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ = f ( x + γe j ) − ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ f ( x ) + ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ γa 1 j ....
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This note was uploaded on 10/04/2010 for the course ECON 204 taught by Professor Anderson during the Summer '08 term at Berkeley.

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204Lecture112009 - Economics 204 Lecture 11–Monday...

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