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Macroeconomics Exam Review 214

Macroeconomics Exam Review 214 - c 2001 Michael Carter All...

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4.52 The partial derivatives of the quadratic function are 𝐷 1 𝑓 = 2 𝑎𝑥 1 + 2 𝑏𝑥 2 𝐷 2 𝑓 = 2 𝑏𝑥 1 + 2 𝑐𝑥 2 The second-order partial derivatives are 𝐷 11 𝑓 = 2 𝑎 𝐷 21 𝑓 = 2 𝑏 𝐷 12 𝑓 = 2 𝑏 𝐷 22 𝑓 = 2 𝑐 4.53 Apply Exercise 4.37 to each partial derivative 𝐷 𝑖 𝑓 [ x ]. 4.54 𝐻 ( x 0 ) = ( 𝐷 11 𝑓𝑓 [ x 0 ] 𝐷 12 𝑓𝑓 [ x 0 ] 𝐷 21 𝑓𝑓 [ x 0 ] 𝐷 22 𝑓𝑓 [ x 0 ] ) = 2 ( 𝑎 𝑏 𝑐 𝑑 ) 4.55 4.56 For any 𝑥 1 𝑆 , define 𝑔 : 𝑆 → ℜ by 𝑔 ( 𝑡 ) = 𝑓 ( 𝑡 ) + 𝑓 [ 𝑡 ]( 𝑥 1 𝑡 ) + 𝑎 2 ( 𝑥 1 𝑡 ) 2 𝑔 is differentiable on 𝑆 with 𝑝 ( 𝑡 ) = 𝑓 [ 𝑡 ] 𝑓 [ 𝑡 ] + 𝑓 ′′ [ 𝑡 ]( 𝑥 1 𝑡 ) 2 𝑎 2 ( 𝑥 1 𝑡 ) = 𝑓 ′′ [ 𝑡 ]( 𝑥 1 𝑡 ) 2 𝑎 2 ( 𝑥 1 𝑡 ) Note that 𝑔 ( 𝑥 1 ) = 𝑓 ( 𝑥 1 ) and 𝑔 ( 𝑥 0 ) = 𝑓 ( 𝑥 0 ) + 𝑓 ( 𝑥 0 )( 𝑥 1 𝑥 0 ) + 𝑎 2 ( 𝑥 1 𝑥 0 ) 2 (4.50) is a quadratic approximation for 𝑓 near 𝑥 0 . If we require that this be exact at 𝑥 1 = 𝑥 0 , then 𝑔 ( 𝑥 0 ) = 𝑓 ( 𝑥 1 ) = 𝑔 (
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