Math 1320 Lab 8
Name:
1. At what points does the helix
~r(t) = hsin(t), cos(t), ti
intersect the sphere x2 + y 2 + (z − 3)2 = 5 ?
2. (a) Identify the following surface by putting it in standard form. [Hint: complete the
square.]
9x2 + 4y 2 − 8y − 36z 2 + 216z = 356.
(b) Sketch the surface.
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3. Let (x, y, z) be a point in R3 . Determine the region of convergence of the following series:
∞
X
(x2 + y 2 + z 2 )n .
n=1
Describe the region of convergence in terms of surfaces.
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4. If two objects travel through space along two different curves, it is important to know
whether they will collide. The curves might intersect, but we need to know whether
the objects are in the same position at the same time. Suppose the trajectories of two
particles are given by the vector functions
r~1 (t) = ht3 , −2t, 6t2 + ti,
r~2 (t) = ht, 4t2 − 4t, 5t2 − 13t + 8i.
Do the two particles collide?
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