Fall2009 - Find also ( ) , x y u z at (x,y,z) = (2,0,1). 6)...

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Math 262 Fall 2009 Dec 18, ‘09, 9:00 -12:00noon 1) Determine the center, radius and interval of convergence (including end points) of (a) 2 0 (2 5) ( 1)3 n n n x n = + + (b) 3 0 (2 3) n n n x = - 2) Evaluate 2 0.1 2 0 ( .001). x x e dx error - < 3) Find the curvature of the twisted cubic 2 3 ( ) t t t t = + + r i j k at a general point and at (0, 0, 0). Find also ( ), ( ), ( ) t t t T N B . 4) Let L(x, y) denote the local linear approximation to 2 2 ( , ) f x y x y = + at point (3, 4). Compute L(3.01, 3.98). Give the second degree approximation 2 ( , ) P x y for 2 2 ( , ) f x y x y = + . 5) Show that the equations 2 2 cos 2 cos 1 y xe uz v u y x v yz + - = + - = can be solved for u and v as functions of x, y, z near point P 0 where (x,y,z) = (2,0,1) and (u,v) = (1,0).
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Unformatted text preview: Find also ( ) , x y u z at (x,y,z) = (2,0,1). 6) Find and classify the critical points of 2 2 1 ( , ) 1 f x y x y x y =-+ + + . 7) Find the maximum and minimum values of ( , , ) f x y z xyz = on the sphere 2 2 2 12 x y z + + = 8) If ( , ), z f x y = where x s t = + and y s t =-, show that 2 2 z z z z x y s t -= . 9) Evaluate ( ) 2 3 4 , R x y dA + where R is the region in the upper half-plane bounded by the circles 2 2 1 x y + = and 2 2 4 x y + = ....
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