Math 317, Fall 2012, Section 101
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Midterm Exam I
October 3, 2008
No books. No notes. No calculators. No electronic devices of any kind.
Name
Student Number
Problem 1. (5 points)
The cone with equation y 2 + z 2 = x2 and the plane with equation x + z = 4 intersect
in a curve C. Find the curvature of C at the point h2, 0, 2i.
Math 317, Midterm Exam I
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1
2
3
4
5
6
total/22
Problem 2. (6 points)
The spiral C in the plane is parametrized by the vector funtion
~r(t) = et hcos t, sin ti
(a) Find the arclength of the part of C which is parametrized by the interval
(−∞, 0 ].
(b) Reparametrize C using arc-length measured from t = −∞.
Math 317, Midterm Exam I
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Problem 3. (3 points)
True or false? (Assumme that a curve C is parametrized by a twice continuously
differentiable vector function ~r(t).)
(a) at a time t where |~v (t)| reaches a maximum, we necessarely have ~a(t) ⊥ ~v (t).
(b) at a time t where |~v (t)| is not zero and ~a(t) k ~v (t), the curvature κ(t) vanishes.
(c) at a time t where |~v (t)| vanishes, we must have that ~a(t) is tangent to the
curve.
Math 317, Midterm Exam I
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Problem 4. (4 points)
The curve C is parametrized by the vector funtion
~r(t) = ht, e−t , cos ti
Find an equation for the normal plane to C at the point corresponding to the
parameter value t = 0.
Math 317, Midterm Exam I
Problem 5. (4 points)
d
The derivative |~r 0 (t)| is given by
dt
00
(a) |~r (t)|,
(b) 2~r 0 (t) · ~r 00 (t),
(c)
~r 0 (t) · ~r 00 (t)
,
|~r 0 (t)|
(d) 0
(e) non of the above.
d
0
The derivative
~r(t) × ~r (t) is equal to
dt
(a) ~r 0 (t) × ~r 0 (t),
(b) ~r(t) × ~r 00 (t),
(c) ~r 0 (t) × ~r 00 (t) + ~r(t) × ~r 00 (t)
(d) ~r(t) × ~r 0 (t) + ~r(t) × ~r 00 (t)
(e) none of the above.
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Math 317, Midterm Exam I
Overflow space I.
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Math 317, Midterm Exam I
Overflow space II.
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