Macroeconomics Exam Review 204

Macroeconomics Exam - c 2001 Michael Carter All rights reserved Solutions for Foundations of Mathematical Economics 4.16 The directional derivative

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4.16 The directional derivative ? x ± ( x 0 ) measures the rate of increase of ± in the di- rection x . Using Exercises 4.10, 4.14 and 3.61, assuming x has unit norm, ? x ± ( x 0 )= [ x 0 ]( x < ± ( x 0 ) , x > ± ( x 0 ) This bound is attained when x = ± ( x 0 ) / ± ( x 0 ) since ? x ± ( x 0 < ± ( x 0 ) , ± ( x 0 ) ∥∇ ± ( x 0 ) > = ± ( x 0 ) 2 ∥∇ ± ( x 0 ) = ± ( x 0 ) The directional derivative is maximized when ± ( x 0 )and x are aligned. 4.17 Using Exercise 4.14 ² = { x ³ : < ± [ x 0 ] , x > =0 } 4.18 Assume each ± ? is diFerentiable at x 0 and let [ x 0 ]=( 1 [ x 0 ] ,?± 2 [ x 0 ] ,...,?± ± [ x 0 ]) Then f ( x 0 + x ) f [ x 0 ] ? f [ x 0 ] x = ± 1 ( x 0 + x ) ± 1 [ x 0 ] 1 [ x 0 ] x ± 2 ( x 0 + x ) ± 2 [ x 0 ] 2 [ x 0 ] x . . . ± ± ( x 0 + x ) ± ± ( x 0 ) ± [ x 0 ] x and ± ? ( x 0 + x ) ± ? ( x 0 ) ? [ x 0 ] x x 0as x ∥→ 0 for every ´ implies f ( x 0 + x ) f ( x 0 ) ? f [ x 0 ]( x ) x x 0 (4.43) Therefore f is diFerentiable with derivative ? f [ x 0 ]= µ =( 1 ( x 0 ) 2 [ x 0 ] ± [ x 0 ]) Each ? [ x 0 ] is represented by the gradient ± ? [ x 0 ] (Exercise 4.13) and therefore [ x 0 ] is represented by the matrix = ± 1 [ x 0 ] ± 2 [ x 0 ] . . . ± ± [ x 0 ] = ? ² 1 ± 1 [ x 0 ] ? ² 2 ± 1 [ x 0 ] ... ? ² ? ± 1 [ x 0 ] ? ² 1 ± 2 [ x 0 ] ? ² 2 ± 2 [ x 0 ] ? ² ? ± 2 [ x 0 ] . . . . . . . . . . . . ? ² 1 ± ± [ x 0 ] ? ²
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This note was uploaded on 01/16/2012 for the course ECO 2024 taught by Professor Dr.dumond during the Fall '10 term at FSU.

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