23s08Exam2-OLD

# 23s08Exam2-OLD - x 2 y 2 z 2 = 25 and outside the cylinder...

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OLD MATHEMATICS 23 EXAMINATION 2 SPRING 2008 Reminder: The actual examination may be very diﬀerent. (1) let z = f ( x,y ) where f is diﬀerentiable, x = g ( t ), y = h ( t ), g (3) = 2, g 0 (3) = 5, h (3) = 7, h 0 (3) = - 4, f x (2 , 7) = 5, f y (2 , 7) = - 8. Find dz dt when t = 3. Justify your answer. (2) Find the absolute maximum and minimum values of f ( x,y ) = 2 x 3 + y 4 on the closed unit disk D = { ( x,y ) | x 2 + y 2 1 } . (3) Use Lagrange multipliers to ﬁnd the maximum value of f ( x,y,z ) = 2 x + 6 y + 10 z subject to x 2 + y 2 + z 2 = 35. (4) Set up a double integral giving the volume of the solid inside the sphere
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Unformatted text preview: x 2 + y 2 + z 2 = 25 and outside the cylinder x 2 + y 2 = 16. (5) Evaluate the above integral. (6) Find the area of the part of the surface z = xy lying within the cylinder x 2 + y 2 = 1. (7) Evaluate the integral Z a-a Z √ a 2-y 2 ( x 2 + y 2 ) 3 2 dxdy. (8) A lamina occupies the part of the disk x 2 + y 2 ≤ 4 which lies in the upper half plane. If the density is a positive constant, ﬁnd the center of mass of the lamina. 1...
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