(12 points) 1. For the function f (x, y, z) = sin(xy) + sin(yz), compute grad f (1, π, −1).
1.
(12 points) 2. Write but do not
! evaluate a triple integral that gives the volume of the region under the
hemisphere z = 4 − x2 − y 2 and above the region (in the xy-plane) x2 + y 2 ≤ 4, x ≥ 0.
(14 points) 3. Compute all second-order partial derivatives of f (x, y) = (x2 + y)ey . Verify that fxy = fyx .
!
(12 points) 4. Compute the double integral R (x2 + y 2 ) dA, where R is the triangular region in the xy-plane
with vertices (0, 0), (1, 0), and (1, 2).
4.
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(12 points) 5. For the function f (x, y) = x2 − 2xy + 3y 2 − 8y, find the critical points of f and classify them
as local maxima, local minima, or saddle points.
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6. Consider the double integral
!
0
1! 1
sin(x2 ) dx dy.
y
(6 points)
(a) Sketch the region of integration.
(8 points)
(b) Evaluate the integral by switching the order of integration.
(b)
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(12 points) 7. Use Lagrange multipliers to find the maximum and minimum values of f (x, y) = x + 3y + 2
subject to the constraint x2 + y 2 ≤ 10.
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(12 points) 8. Find the mass of the solid above the rectangle in the xy-plane 0 ≤ x ≤ 3, 0 ≤ y ≤ 2, and
below the plane (x/3) + (y/2) + (z/6) = 2, where density of the solid is given by δ(x, y, z) = x.
8.
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