204Lecture112006

204Lecture112006 - Economics 204 Lecture 11–Monday...

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Unformatted text preview: 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 → 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 < δ ⇔ ∀ ε> ∃ δ> < | h | < δ ⇒ | f ( x + h ) − ( f ( x ) + ah ) | | h | < δ ⇔ 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,δ> | h | < δ ⇒ | y | ≤ ∗ ∗ K | h | **read y is big-Oh of h y = o ( h ) as h → means lim h → | 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 → 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 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ....
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204Lecture112006 - Economics 204 Lecture 11–Monday...

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