Math 223
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#„
1. (8) Consider two vectors #„
v =4i
#„ perpendicular?
a are #„
v and w
#„
#„
#„
#„ = a #„
j + a k and w
i +5j
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#„
k . For what values of
2. (8) Find an equation for the plane passing through the points
(0, 2, 1), (1, 1, 5), and (2, 0, 11).
3. (8) Find a vector of magnitude 10 normal to the plane 5x + 3y = 6z + 1.
Math 223
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4. (8) At which point (or points) on the ellipsoid x2 + 4y 2 + z 2 = 9 is the tangent plane
parallel to the plane z = 0?
5. (8) Find a parametric equation for a line through the point (1, 3, 5) and parallel to the
#„
#„ #„
vector 5 i + 3 j
k
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6. (8) Find the directional derivative of f (x, y) = x2 y + y 2 x at the point (1, 1) in the
#„
#„
direction of 3 i 4 j .
7. (7) Compute the flux integral
z = x2
Z
S
#„ # „
#„
#„ #„
F · dA where F = i + j
y 2 , 0  x  3, 0  y  3, oriented upwards.
#„
k and S is the surface
Math 223
8. (20) Consider the integral
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Z
1
0
Z
2
(8y)1/3
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1
dxdy.
1 + x4
(a) Interchange the order of integration. Show your work by including a sketch of the
region of integration.
(b) Evaluate the integral.
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#„
#„
#„
#„
9. (20) Let F = x i + y j + z k . Evaluate the following:
Z
#„ #„
(a)
F · dr where C is the line from (0, 0, 0) to (1, 1, 1).
C
(b)
Z
S
#„ # „
F · dA where S is the triangle in the plane y = 10 with vertices (0, 10, 0), (4, 10, 0),
S
#„ # „
F · dA where S is the sphere of radius 2 centered at (5, 5, 0), oriented outward.
and (0, 10, 1), oriented in the direction of increasing y.
(c)
Z
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10. (15) Let H(x, y, z) = sin (2x + y) + z. Find the equation of the tangent plane to the
level surface H(x, y, z) = 5 at the point (⇡, ⇡, 5).
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#„
#„
#„
#„
11. (20) Let F = (y + z)x i + y j + xyz k .
#„
(a) Find curl( F ).
(b) Let S be the surface x2 +Z y 2 + z = 25, with 0  z  25, oriented upward. Find the
#„ # „
value of the flux integral
curl( F ) · dA.
S
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1
12. (15) Let a be a constant, a 6= 2, and consider function f (x, y) = x2 + 2y + 2xy + ay 2 .
2
(a) Find the critical point of f .
(b) Find all values of a so that the critical point is a global minimum.
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13. (20) Consider the contour diagram for the function f (x, y) sketched below.
y
0.125
2
5.35938
1.95313
0.015625
-2
1
-1
0.421875
1
8
x
3.375
-1
0
-2
1
(a) Sketch a graph of f (x, 0).
(b) Determine whether the following quantities are positive, negative, or equal to zero.
fxx (0, 0) is
fxy (0, 0) is
(c) If all contour lines are parallel to the line 2x + y = 0, then determine the direction in
which the gradient of f points, as a unit vector.
Math 223
14. (15) Rewrite the integral
drical coordinates.
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Z
3
3
Z
0
p
9 y2
Z p18
p
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x2 y 2
xy dz dx dy in spherical and cylin-
x2 +y 2
(a) In spherical coordinates, use the order of integration d⇢ d✓ d .
(b) In cylindrical coordinates, use the order of integration dz dr d✓.
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Page 11 of 11
#„
#„
#„
2
2
15. (20) Consider the 2-dimensional force field F = 2xex 5y i 5ex 5y j .
#„
#„
(a) Is F conservative? If so, find a potential function f (x, y) whose gradient is F .
#„
(b) Find the work done by the force field F in moving an object from P (0, 2) to Q( 2, 0)
along the path formed by C1 followed by C2 as shown in the figure below. C1 and C2
may be parametrized as follows:
C1 : x = t, y = 2 t, 0  t  2,
C2 : x = 2 cos t, y = 2 sin t, 0  t  ⇡ .