PRACTICE MIDTERM 2
Math 1320 sec 004
November XXX, 2013
Name: |
{z
by writing my name I swear by the honor code
}
Read all of the following information before starting the exam:
• Show all work, clearly and in order, if you want to get full credit. I may take off points if
I cannot see how you arrived at your answer (even if your final answer is correct).
• No calculators/books/notes are allowed for this test.
• Please keep your written answers brief; be clear and to the point. You may lose points for
incorrect or irrelevant statements.
• Please turn off cellphones, turn hats backwards, take out headphones and sit every other
seat, if possible.
• You have 50 minutes to work on this exam.
• Good luck!
Page (value)
1 (20 points)
2 (15 points)
3 (20 points)
4 (20 points)
Total (75 points)
Score
1.
(20 points) Short Answer. For True or False you must write ‘True’ or ‘False’ and justify
your answer to receive credit.
a. (4 pts)
Set up but do not evaluate an integral that represents the arc length of the
curve segment parameterized by ~r(t) = 2t sin tx̂ + 8t3/2 ŷ + ln tẑ
b. (4 pts) The point P is described in Cartesian coordinates as (0,
spherical coordinates?
c. (4 pts)
d. (4 pts)
e. (4 pts)
√
3, 1). What is P in
True or False. The vectors u~1 = x̂ + ŷ and u~2 = x̂ − ŷ are orthogonal.
True or False. The planes 2x + 6y + 3z = 9 and 4x − 12y + 6z = 10 are parallel.
Describe the shape of the curve given by ~r(t) = cos tx̂ + sin tŷ + 3tẑ.
2.
(7 points) Calculate the Taylor polynomial of degree 3 about x = 0 for the function
f (x) = (1 − x)2/3 .
3.
(8 points)

a. (4 pts)



1
2
Let ~u =  3  and ~v =  7 . Calculate ~u · ~v .
4
-5
b. (4 pts)
A force F~ = 6x̂ − 2ẑ is applied to the end of a lever arm located at ~r = 8ŷ.
Calculate the applied torque.
4.
(20 points)
a. (6 pts)
Write down the parametric equations for the line that passes through the
points (6, 1, −3) and (2, 4, 5).
b. (7 pts)
v~2 = 2ŷ + 6ẑ.
c. (7 pts)
Find an equation for the plane that contains the vectors v~1 = x̂ + 2ŷ + 3ẑ and
Find the angle between the planes x + y + z = 1 and x − 2y + 3z = 1.
5.
(20 points)
a. (10 pts)
b. (6 pts)
point (0, 2, 1).
Calculate the curvature for the curve ~r(t) = 3tx̂ + 4 sin tŷ + 4 cos tẑ.
√
Find the unit tangent vector to the curve x = ln t, y = 2 t, z = t2 at the
c. (4 pts)
Consider a curve parametrized by x = f (t) and y = g(t). Write down a
formula for the curvature in terms of f and g.