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Macroeconomics Exam Review 204

Macroeconomics Exam Review 204 - c 2001 Michael Carter All...

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4.16 The directional derivative 𝐷 x 𝑓 ( x 0 ) measures the rate of increase of 𝑓 in the di- rection x . Using Exercises 4.10, 4.14 and 3.61, assuming x has unit norm, 𝐷 x 𝑓 ( x 0 ) = 𝐷𝑓 [ x 0 ]( x ) = < 𝑓 ( x 0 ) , x > 𝑓 ( x 0 ) This bound is attained when x = 𝑓 ( x 0 ) / 𝑓 ( x 0 ) since 𝐷 x 𝑓 ( x 0 ) = < 𝑓 ( x 0 ) , 𝑓 ( x 0 ) ∥∇ 𝑓 ( x 0 ) > = 𝑓 ( x 0 ) 2 ∥∇ 𝑓 ( x 0 ) = 𝑓 ( x 0 ) The directional derivative is maximized when 𝑓 ( x 0 ) and x are aligned. 4.17 Using Exercise 4.14 𝐻 = { x 𝑋 : < 𝑓 [ x 0 ] , x > = 0 } 4.18 Assume each 𝑓 𝑗 is differentiable at x 0 and let 𝐷𝑓 [ x 0 ] = ( 𝐷𝑓 1 [ x 0 ] , 𝐷𝑓 2 [ x 0 ] , . . . , 𝐷𝑓 𝑚 [ x 0 ]) Then f ( x 0 + x ) f [ x 0 ] 𝐷 f [ x 0 ] x = 𝑓 1 ( x 0 + x ) 𝑓 1 [ x 0 ] 𝐷𝑓 1 [ x 0 ] x 𝑓 2 ( x 0 + x ) 𝑓 2 [ x 0 ] 𝐷𝑓 2 [ x 0 ] x . . . 𝑓 𝑚 ( x 0 + x ) 𝑓 𝑚 ( x 0 ) 𝐷𝑓 𝑚 [ x 0 ] x and 𝑓 𝑗 ( x 0 + x ) 𝑓 𝑗 ( x 0 ) 𝐷𝑓 𝑗 [ x 0 ] x x 0 as x ∥ → 0 for every 𝑗 implies f ( x 0 + x ) f ( x 0 ) 𝐷 f [ x 0 ]( x ) x 0 as x ∥ → 0 (4.43)
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