MATH 208 Section 003
Practice Test 2
Name:
You must show me enough work that I can reproduce your results.
1. Let f (x, y) = y 2 − 18x2 + x4 .
(a) (8 points) Find all of the critical points of f . Determine, if possible, whether each one is a local
maximum, a local minimum, or a saddle point.
(b) (6 points) Find the quadratic approximation to f at one of the critical points that you found in
part (a). Indicate clearly which point you choose. [Hint: Re-use your work from part (a).]
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2. (10 points) Let x and y denote the measures of the acute angles of a right triangle. Find the maximum
value of the product sin(x) sin(y).
3. (8 points) Let f (x, y, z) and g(x, y, z) be functions whose gradients exist. Let P be a local maximum of
f subject to the constraint g(x, y, z) = c, where c is a constant. Suppose that grad g(P ) 6= ~0. Explain
why grad f (P ) and grad g(P ) are parallel. [Hint: Start by explaining why both grad f (P ) and grad g(P )
are perpendicular to the constraint g = c.]
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4. (10 points) A (closed) rectangular box has volume 27 cm3 . What is the minimum possible surface area
of the box?
5. (10 points) Evaluate the following double integral by reversing the order of integration:
Z
0
8
Z
2
√
3
4
ex dxdy.
y
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6. (10 points) An object is located between the sphere of radius 3 cm (centered at the origin) and the
sphere of radius 5 cm (centered at the origin) and below the xy-plane. The density of the object is 2
g/cm3 . Set up, but DO NOT EVALUATE, a triple integral expressing the mass of the object.
7. (10 points) Set up, but DO NOT EVALUATE, a triple integral
expressing thepvolume of the region
p
2
2
2
inside the cylinder x + y = 12 and between the cones z = x + y 2 and z = − x2 + y 2 .
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8. A particle rotates clockwise (from the perspective of the positive x-axis) around the circle of radius 4 m
that is centered at the point (0, 3, 0) and parallel to the yz-plane. The particle starts at the point (0, 7, 0)
at time t = 0, completes one revolution every 10 seconds, and moves at a constant speed.
(a) (6 points) Give a parameterization for the motion of the particle.
(b) (6 points) Find the velocity vector of the particle and find the speed of the particle in meters per
second.
Question:
1
2
3
4
5
6
7
8
Total
Points:
12
10
8
10
10
10
10
12
82
Score:
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