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Assignment 3

Assignment 3 - f x y = x 2 y subject to the constraint x 2...

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Department of Mathematics MATHS 208 Assignment 3 Due:30 September 2008 Please hand in your assignment to the appropriate box outside the Student Resource Centre by 4 pm on the due date. SHOW ALL WORKING FOR CREDIT. 1. [ 6 marks ] (a) The equation x 3 - xy + 4 y 2 + xyz - z 3 = 11 defnes a ±unction z = f ( x, y ) implicitly. Use implicit di²erentiation to fnd the value ∂z ∂x when ( x, y, z ) = (1 , 2 , 2) . (b) I± f ( x, y, z ) = y 2 ze x + y - sin( xyz ) = 0, fnd ∂z ∂x and ∂y ∂z . 2. [ 5 marks ] Given w = x 2 y + y 2 z + z 2 x, veri±y that ∂w ∂x + ∂w ∂y + ∂w ∂z = ( x + y + z ) 2 . 3. [ 5 marks ] Given f ( x, y, z ) = 3 x 2 + xy - 2 y 2 - yz + z 2 , fnd the rate o± change o± f ( x, y, z ) at the point (1 , - 2 , - 1) in the direction o± the vector (2 , - 2 , - 1) . 4. [ 10 marks ] I± f ( x, y ) = x 4 + y 4 - 4 xy + 1 classi±y all relative extrema o± f . 5. [ 20 marks ] (a) Using the method o± Lagrange multipliers fnd the extreme values o±
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Unformatted text preview: f ( x, y ) = x 2 + y subject to the constraint x 2 + y 2 = 1. (b) Using the method o± Lagrange multipliers fnd the maximum value o± the ±unction f ( x, y, z ) = x + 2 y + 3 z on the curve o± intersection o± the plane x-y + z = 1 and the cylinder x 2 + y 2 = 1. 6. [ 6 marks ] Find the limits o± the ±ollowing sequences as n → ∞ i± (a) a n = 2 n + 5 5 n + 2 (b) a n = ln n e n (c) a n = n 1-e n 7. [ 8 marks ] Find the limits o± the ±ollowing sequences: (a) b (-1) n +1 √ n B ∞ n =1 (b) ± r n + 1 n ² ∞ n =1 1 (c) b 1-5 n 4 n 4 + 8 n 3 B ∞ n =1 (d) b sin n n B ∞ n =1 MATHS 208 Assignment 3 Page 2 of 2...
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Assignment 3 - f x y = x 2 y subject to the constraint x 2...

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