A6_soln - a Øh7 A ×× igÒÑ eÒ Ø6Ó ÐÙ Ø iÓÒ × F iÒd Øh eÒdd eg ÖeeaÝ ÐÓ ÖÔÓ ÐÝÒÓÑ ia ÐfÓ ÖØh e fÓ ÐÐÓÛ iÒg

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Unformatted text preview: a Øh7 A ×× igÒÑ eÒ Ø6Ó ÐÙ Ø iÓÒ × F iÒd Øh eÒdd eg ÖeeaÝ ÐÓ ÖÔÓ ÐÝÒÓÑ ia ÐfÓ ÖØh e fÓ ÐÐÓÛ iÒg fÙÒ cØ iÓÒ ×a ØØh eg iÚ eÒÔÓ iÒ Ø a f ( x, y ) = x 2 y 3- xy a Ø (1 , 1) Ó ÐÙ Ø iÓÒ :ehaÚ e ∇ f = (2 xy 3- y, 3 x 2 y 2- x ) aÒd Hf = bracketleftbigg 2 y 3 6 xy 2- 1 6 xy 2- 1 6 x 2 y bracketrightbigg hÙ × P 2 , (1 , 1) ( x, y ) = f (1 , 1) + f x (1 , 1)( x- 1) + f y (1 , 1)( y- 1) + 1 2 f xx (1 , 1)( x- 1) 2 + f xy (1 , 1)( x- 1)( y- 1) + 1 2 f yy ( y- 1) 2 =0 + ( x- 1) + 2( y- 1) + ( x- 1) 2 + 5( x- 1)( y- 1) + 3( y- 1) 2 b f ( x, y ) = (1 + x ) y a Ø (0 , 0) Ó ÐÙ Ø iÓÒ :F Ó Ö ( x, y ) ×Ùc ieÒ Ø ÐÝc ÐÓ ×eØÓ (0 , 0) Û ehaÚ e ∇ f = ( y (1+ x ) y- 1 , (1+ x ) y ln(1+ x ) ) aÒd Hf ( x, y ) = parenleftbigg y ( y- 1)(1 + x ) y- 2 (1 + x ) y- 1 + y (1 + x ) y- 1 ln(1 + x ) (1 + x ) y- 1 + y (1 + x ) y- 1 ln(1 + x ) (1 + x ) y (ln(1 + x )) 2 parenrightbigg . hÙ × ∇ f (0 , 0) = (0 , 0) aÒd Hf (0 , 0) = parenleftbigg 1 1 parenrightbigg . h eÖe fÓ Öe P 2 , (0 , 0) ( x, y ) = f (0 , 0) + f x (0 , 0)( x- 0) + f y (0 , 0)( y- 0) + 1 2 f xx (0 , 0)( x- 0) 2 + f xy (0 , 0)( x- 0)( y- 0) + 1 2 f yy ( y- 0) 2 =1 + xy eØ f ( x, y ) = ln( x + 2 y ) a hÓÛ Øha ØfÓ Ö ( x, y ) ×Ùc ieÒ Ø ÐÝc ÐÓ ×eØÓ (3 ,- 1) Û ehaÚ e f ( x, y ) ≈ ( x- 3) + 2( y + 1) Ó ÐÙ Ø iÓÒ :ehaÚ e f x = 1 x +2 y aÒd f y = 2 x +2 y hÙ ×Øh e ÐiÒ ea ÖaÔÔ ÖÓÜ iÑ a Ø iÓÒg iÚ e× f ( x, y ) ≈ f (3 ,- 1) + f x (3 ,- 1)( x- 3) + f y (3 ,- 1)( y + 1) = ( x- 3) + 2( y + 1) . Ó ÖeÓÚ eÖÛ ekÒÓÛ Øha ØØh i× i×agÓÓdaÔÔ ÖÓÜ iÑ a Ø iÓÒ × iÒ ce f x aÒd f y a ÖebÓ Øh cÓÒ Ø iÒÙÓÙ × a Ø (3 ,- 1) aÒdh eÒ ce f i×d ieÖeÒ Ø iab Ðea Ø (3 ,- 1) b ÖÓÚ eØha Ø if x ≥ 3 aÒd y ≥ - 1 Øh eÒ Øh eeÖÖÓ Ö iÒ Øh eaÔÔ ÖÓÜ iÑ a Ø iÓÒ iÒa ×a Ø i×e× | f ( x, y )- [( x- 3) + 2( y + 1)] | ≤ 3[( x- 3) 2 + ( y + 1) 2 ] . Ó ÐÙ Ø iÓÒ :ehaÚ e f xx =- 1 ( x +2 y ) 2 f xy =- 2 ( x +2 y ) 2 f yx =- 2 ( x +2 y ) 2 aÒd f yy =- 4 ( x +2 y ) 2 eÒ ce × iÒ ce f ha ×cÓÒ Ø iÒÙÓÙ ××ecÓÒdÔa ÖØ ia Ðd eÖ iÚa Ø iÚ e× iÒaÒ e ighbÓ ÖhÓÓdÓ f (3 ,- 1) Û ecaÒaÔÔ ÐÝ aÝ ÐÓ Ö×Øh eÓ ÖeÑ ØÓg eØØha ØØh eÖeeÜ i×Ø×aÔÓ iÒ Ø c ÓÒ Øh e ÐiÒ e×egÑ eÒ ØjÓ iÒ iÒg ( x, y ) ØÓ (3 ,- 1) ×Ù ch Øha Ø | R 1 , (3 ,- 1) ( x, y ) | = 1 2 | f xx...
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This note was uploaded on 11/18/2010 for the course MATH 235 taught by Professor Celmin during the Spring '08 term at Waterloo.

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A6_soln - a Øh7 A ×× igÒÑ eÒ Ø6Ó ÐÙ Ø iÓÒ × F iÒd Øh eÒdd eg ÖeeaÝ ÐÓ ÖÔÓ ÐÝÒÓÑ ia ÐfÓ ÖØh e fÓ ÐÐÓÛ iÒg

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