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a2_soln

# a2_soln - a Øh7 A ×× igÒÑ eÒ ØÓ ÐÙ Ø iÓÒ × F...

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Unformatted text preview: a Øh7 A ×× igÒÑ eÒ ØÓ ÐÙ Ø iÓÒ × F iÒd Øh eÖ×ØaÒd ×ecÓÒdÔa ÖØ ia Ðd eÖ iÚa Ø iÚ e×Ó f a f ( x, y ) = x 3 + 3 x 2 y + 4 y 2 Ó ÐÙ Ø iÓÒ : f x = 3 x 2 + 6 xy f y = 3 x 2 + 8 y f xx = 6 x + 6 y, f xy = 6 x = f yx , f yy = 8 . b g ( x, y ) = radicalbig 2 x 2 + y 2 Ó ÐÙ Ø iÓÒ : g x = 2 x √ 2 x 2 + y 2 g y = y √ 2 x 2 + y 2 g xx = 2 √ 2 x 2 + y 2- 4 x 2 (2 x 2 + y 2 ) 3 / 2 , g xy =- 2 xy (2 x 2 + y 2 3 / 2 = g yx , g yy = 1 √ 2 x 2 + y 2- y 2 (2 x 2 + y 2 ) 3 / 2 A fÙÒ cØ iÓÒ f : R 2 → R i×d eÒ edbÝ f ( x, y ) = g ( x- 2 y ) Û h eÖe g : R → R f g prime (1) = 3 ca ÐcÙ Ða Øe f x (3 , 1) aÒd f y (3 , 1) hÓÛ Øha Ø f y ( x, y ) =- 2 f x ( x, y ) iÒg eÒ eÖa Ð Ó ÐÙ Ø iÓÒ :ehaÚ e f x ( x, y ) = g prime ( x- 2 y ) · (1) aÒd f y ( x, y ) = g prime ( x- 2 y ) · (- 2) h eÒ ceÛ ehaÚ e f y ( x, y ) =- 2 f x ( x, y ) , aÒd f x (3 , 1) = g prime (3- 2(1)) = g prime (1) = 3 f y (3 , 1) =- 2 g prime (3- 2(1)) =- 2(3) =- 6 . eØ f ( x, y ) = braceleftBigg xy √ x 2 + y 2 + 1 , if ( x, y ) negationslash = (0 , 0) 1 , if ( x, y ) = (0 , 0) ....
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