(10 points) 1. A particle passes through the point P = (5, 4, −2) at time t = 4, moving with constant velocity
!v = 2!i − 3!j + !k. Give a parametric equation (in vector form) for its motion.
(10 points) 2. Evaluate the following iterated integral:
!
0
1! 1
√
y
3 cos(x3 ) dx dy.
(10 points) 3. Set up, but do not!evaluate, an iterated integral to give the mass of the solid that lies
outside the cone z = x2 + y 2 and inside the sphere of radius 16 centered at the origin, where
2
2
2
the density at a point (x, y, z) in the solid is given by δ(x, y, z) = ex +y +z .
(Note: This solid has portions above and below the xy-plane.)
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(10 points) 4. The velocity of a flow at the point (x, y) is F! (x, y) = 2y!i + 4!j. If an object is at (1, 3) in the
flow at time t = 1, then where is the object at time t = 2?
(Hint: First find the object’s path of motion.)
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(15 points) 5. Use Lagrange multipliers to find the maximum value of the function f (x, y) = x2 + 2y 2 subject
to the constraint x2 + y 2 ≤ 4.
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(15 points) 6. Find the point on the surface y = x + 2z that is closest to the point (1, 0, 2). You do not need
to verify that the point you find is the closest.
(Hint: Minimize the square of the distance between a point (x, y, z) on the surface and
the point (1, 0, 2).)
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(15 points) 7. Evaluate the following triple integral:
!
1
−1
!
√
−
1−x2
√
1−x2
!
1
0
1
dz dy dx.
(x2 + y 2 )1/2
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(15 points) 8. Let R be the region in 2-space bounded by the curves x2 + y 2 = 1 and x2 + y 2 = 9. Find the
average distance from the origin of the points of R.
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