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The spring 2000 exam 1 for math 550, a university-level mathematics course. The exam covers topics from vector calculus, including problem-solving and calculations based on vector notation, equations of planes, and calculus. Students are required to work logically, show their work, and indicate their reasoning. The exam includes instructions, problem inspirations, and examples from class and homework.
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Prof. Girardi Math 550 12 February Spring 2000 Exam 1
MARK BOX problem points 1 10 2 10 3 10 4 10 5 10 6 10 total 60 % 100
Problem Inspiration:
1
P 1 : 3 x + y + z = 5 P 2 : x − 2 y + 3z = 1
and passing through the point (4, 2 , 1) is:
R^ ~(t) = 〈 , , 〉 + t 〈 , , 〉
where t varies as:.
f (x, y) = x^2 + y^4 + exy
at the point (1, 0 , f (1, 0) ) is:
Remark: your solution should be of the form ax + by + cz = d.
constant speed of e^6 meters per second. The temperature of the ship’s hull when he is at location (x, y, z) will be given by
T (x, y, z) = exp (−x^2 − y^2 − z^2 )
where x, y, and z are measured in meters. He is currently at (1, 2 , 1). De- scribe the set of possible directions in which he may proceed to bring the ship’s hull temperature down at exactly a rate of 3
2 degrees per second. Box your answer.
with position vector ~r(t) =
t^2 , 3 t^2 , 4 t
If the power thrusters are turned off at time t, the Atlantis will coast off, with constant speed along a straight path tangent to the vector ~r(t). The Atlantis is almost out of fuel when astronaut John Grunsfeld notices the Mir space station off ahead of them at the position (220, 660 , 64). John realizes that their only hope is to turn the thrusters off, just at the proper time, so that the Atlantis will safely coast to dock with the Mir; but, John is not sure if his plan will work. So John quickly calls Tom and Ray for advice. Tom claims that John’s plan will work; Ray claims that John’s plan will not work. Who is right: Tom or Ray? Why? be sure to mathematically support your answer, explaining your thought process. If so needed, continue on the next (blank) page.