This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Economics 204 Lecture 11–Monday, August 10, 2009 Revised 8/10/09, Revisions indicated by ** and Sticky Notes 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 ⇔ ∀ 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 bigOh of  h  n y = o (  h  n ) as h → means lim h →  y   h  n = 0 read y is littleoh 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 ....
View
Full
Document
This note was uploaded on 10/04/2010 for the course ECON 204 taught by Professor Anderson during the Summer '08 term at Berkeley.
 Summer '08
 ANDERSON
 Economics

Click to edit the document details