a3_soln

# a3_soln - a Øh7 A ×× igÒÑ eÒ ØÓ ÐÙ Ø iÓÒ ×...

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Unformatted text preview: a Øh7 A ×× igÒÑ eÒ ØÓ ÐÙ Ø iÓÒ × ×eØh e ÐiÒ ea ÖaÔÔ ÖÓÜ iÑ a Ø iÓÒ ØÓaÔÔ ÖÓÜ iÑ a Øe arctan(0 . 99(1 . 01) 3 ) Ó ÐÙ Ø iÓÒ :eØ f ( x, y ) = arctan( xy 3 ) aÒd ( a, b ) = (1 , 1) h eÒ f x = y 3 1 + ( xy 3 ) 2 f y = 3 xy 2 1 + ( xy 3 ) 2 . eÒ ce L (1 , 1) ( x, y ) = f (1 , 1) + f x (1 , 1)( x- 1) + f y (1 , 1)( y- 1) = π 4 + 1 2 ( x- 1) + 3 2 ( y- 1) eÒ ce f (0 . 99 , 1 . 01) ≈ L (1 , 1) (0 . 99 , 1 . 01) = π 4 + 1 2 (- . 01) + 3 2 ( . 01) = π 4 + 0 . 01 eØ f ( x, y, z ) = xyz x + y + z ×eØh e ÐiÒ ea ÖaÔÔ ÖÓÜ iÑ a Ø iÓÒ ØÓaÔÔ ÖÓÜ iÑ a Øe f (- 1 . 04 ,- 1 . 98 , 3 . 97) Ó ÐÙ Ø iÓÒ :eØ ( a, b, c ) = (- 1 ,- 2 , 4) ehaÚ e f x = yz ( x + y + z )- xyz ( x + y + z ) 2 = y 2 z + yz 2 ( x + y + z ) 2 , f y = x 2 z + xz 2 ( x + y + z ) 2 , f z = x 2 y + xy 2 ( x + y + z ) 2 eÒ ce L (- 1 ,- 2 , 4) ( x, y, z ) = f (- 1 ,- 2 , 4) + f x (- 1 ,- 2 , 4)( x + 1) + f y (- 1 ,- 2 , 4)( y + 2) + f z (- 1 ,- 2 , 4)( z- 4) = 8- 16( x + 1)- 12( y + 2)- 6( z- 4) hÙ × f (- 1 . 04 ,- 1 . 98 , 3 . 97) ≈ L (- 1 ,- 2 , 4) (- 1 . 04 ,- 1 . 98 , 3 . 97) = 8- 16(- . 04)- 12(0 . 02)- 6(- . 03) = 8 . 58 . F Ó Öea chÓ fØh e fÓ ÐÐÓÛ iÒg fÙÒ cØ iÓÒ × f : R 2 → R d eØeÖÑ iÒ e if f i×d ieÖeÒ Ø iab Ðea Ø (0 , 0) a f ( x, y ) = braceleftBigg x 4- y 4 x 2 + y 2 , if ( x, y ) negationslash...
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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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a3_soln - a Øh7 A ×× igÒÑ eÒ ØÓ ÐÙ Ø iÓÒ ×...

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