Math 265
Final Exam Fall 2015
Student name:
Instructor & Section:
This test is closed book and closed notes. A (graphing) calculator is allowed for this test but cannot
also be a communication device (i.e., your cellphones or tablets are not calculators). Answer each
question completely using exact values unless otherwise indicated. Show your work (legibly);
answers without work and/or justifications will not receive credit. Place your final
answer in the provided box. Each problem is worth 10 points for a total of 70 points.
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DO NOT BEGIN THIS TEST UNTIL INSTRUCTED TO START
1. Change the order of integration and then find the exact value of
Z 1 Z y1/3
4 sin(2x2 − x4 ) dx dy.
0
Z
Answer:
y
Z
4 sin(2x2 − x4 ) dy dx =
2. Find the unique point, (x, y, z), on the surface x2 −2xy+3y2 −z = 20 where the tangent
plane is parallel to the lines
x = t−2
y = t+3
z = −4t + 5
and
x = 3t − 4
y = 2t − 1 .
z = 2t + 7
(Hint: what relationship will the gradient vector have with the lines?)
Answer:
3. Find the center of mass for the region of points
with 1 6 x2 + y2 6 4 and y > 0 given
that δ(x, y) = y. (Recall: cos2 θ = 21 1 + cos(2θ) and sin2 θ = 21 1 − cos(2θ) .)
Answer:
4. For the curve r(t) = 13 t3 − 7t i + (5t2 )j + 73 t3 + t k, find aT , i.e., the amount of
acceleration pointing in the tangential direction of motion. Simplify your answer to a
polynomial function of t.
Answer:
p
x2 + y2 6 z 6 2 − (x2 + y2 ), and let
F = 5x sin2 z + arctan(y) i + 5y cos2 z − ln(1 + x2 ) j + 7z − ex−y k.
RR
Use the Divergence Theorem to find G F · n dσ, where G is the surface of T and n is
the outward pointing normal vector of the surface.
5. Let T be the solid given by
Answer:
6. Find and classify all three critical points of h(x, y) = 4x3 − 6x2 y − 3y2 + 12y − 7.
Answer:
7. After finishing your final you head to the theatre to catch the midnight screening
of the new Star Wars movie. As you wait to be seated you decide to pass the time by
determining the number of people in line. Your math adrenaline still pumping from
the calculus exam, you decide to do this by using a line integral. In particular,
the√wall
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against which people are lined up is given by the curve C with x(t), y(t) = (6t , 4 3t3 )
for 0 6 t 6 1 with distances measured in meters. Also the number of people in the line
13
5
at a given
R (x,5 y) is132 x + 2 people per meter. Determine the number of people in line,
i.e., find C ( 2 x + 2 ) ds. (Note: Your answer should be a whole number.)
Answer: