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Math1 - Math 262 Fall 2009 Dec 18 ‘09 9:00 ~12200noon 1...

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Unformatted text preview: Math 262 Fall 2009 Dec 18, ‘09, 9:00 ~12200noon 1) Determine the center, radius and interval of convergence (including end points) of ” (2x+5)"' a 3 ' _ n (egg—F2 “)3, (b) 2» .(2x 3) .1 _ 2 2) Evaluate Ixze" dx (1ermrl<.001). 0 , 3) Find the curvature of the twisted cubic r(t) = ti + tzj + fit at a general point and at (0, 0, 0). Find also To), N0), 130:). 4) Let L(X, y) denote the local linear approximation to f(x, y) = «fxz + y2 at point (3, 4). Compute L(3'.01, 3.98). Give the second degree approximation PZ(x, y) 7' for f(x,y):\fx2+y2. . _ '5) Show that the equations ' xey +uz—cosv: 2 ucoservad—yz2 =1 can be solved for u and v as functions of x, y, 2 near point P0 where (x,y,z) = (2,0,1) and (u,v) = (1,0). Find also (6%2) at (x,y,z) é (2,0,1). My 6) Find and classify the critical points of f (x, y) = , 2 2 . 1 - x + y + x + y 7) Find the maximum and minimum values of for, y, z): xyz on the sphere x2+y27+zz=12 8) If z=f(x,y), where x=s+tand y=s—t-,_show-that (9212 _ a: 2 , @212 6x 8y as a: ' 9) Evaluate ”(336+ 4 y2 )giA, where; R is the region in the upper half—piane bounded R . ‘ . by the circles x2 +y2 =1 and 3:2 +3;2 =4_ ...
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