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PS4_answers - Math 237 A2 = 2 A3 a Hf(2 3 = 9 6 6 4 Problem...

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Math 237 Problem Set 4 Answers A2. α = - β 2 . A3. a) Hf (2 , 3) = 9 - 6 - 6 4 . b) L a (2 , 3) = f (2 , 3) + f x (2 , 3)( x - 2) + f y (2 , 3)( y - 3) = 1 + 3( x - 2) - 2( y - 3). P 2 , (2 , 3) ( x, y ) = 1 + 3( x - 2) - 2( y - 3) + 9 2 ( x - 2) 2 - 6( x - 2)( y - 3) + 2( y - 3) 2 c) The level curves of f are straight lines. A4. i) P 2 , (1 , 0) ( x, y ) = ln 2 + 1 2 ( x - 1) + 1 2 y - 1 8 ( x - 1) 2 - 1 4 ( x - 1) y + 1 8 y 2 . ii) P 2 , (1 , 1) ( x, y ) = 1 + 2( x - 1) - ( y - 1) - 3 2 ( x - 1) 2 - 2( x - 1)( y - 1) + ( y - 1) 2 A5. i) P 2 , ( π,π ) ( x, y ) = ( x - π ) 2 - ( y - π ) 2 . A8. u xx + u yy = D 2 1 f + D 2 2 f , u xx - u yy = D 2 1 f · cos 2 θ - D 1 D 2 f · sin 2 θ - D 2 D 1 f · sin 2 θ - D 2 2 f · cos 2 θ . u xy
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