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Fall10Pre1 - Math 1920 Prelim I 7:30 PM to 9:00 PM You are...

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Unformatted text preview: Math 1920, Prelim I September 30, 2010. 7:30 PM to 9:00 PM You are NOT allowed calculators, the text or any other boolc or'notes. SHOW ALL WORK! Writing clearly and legibly will improve your chances of receiving the maximum credit that your sblution deserves. Please label the questions in you answer booklet clearly. Write your name and section number on each booklet you use. You may leave when you have finished, but if you have not handed in your exam booklet and~left the room by 8:45pm, please remain in your seat so as not to disturb others who are still working. ' 1. Consider the vectors V = i+ Zj + ak and W = i+j + k. (a) (10 pts) Find all values of the number a (if any) such that V is perpendicular to w. (b) (10 pts) Find all the values of the number a (if any) such that the area of the parallelogram determined by v and W is equal to V0. 2. (10 pts) Find the plane through the origin perpendicular to the plane 22: + 2y + z = 1 and perpendicular to the vector V = (1,1, —4). 3; Consider the function 97(55, y) = My? - 932. (a) (9 pts) Sketch the level curves g(ac,y) = c, for c =071, 2. (Ware-a rs itsdomam D of g? (c) (4 pts) What is the bonndary of D? (d) (4 pts) ls D a closed set, an open set, both or neither? '1} 4. (9 pts) The wave equation, where a2 is constant, is given by It describes the motion of a waveform; examples of such waves could include fluid, sound, «light. Suppose u(a:, t) represents the displacement of a vibrating guitar string at time t at a distance a: from one end of the string. If u(:z:, t) = sin(a° — at), show that it satisfies the wave equation. in: r 5. Calculate each of the following limits or show it doesth exist. , — l (a) (10 pts) lim —‘/————— “4+. (am—44.3) a: — y — 1 x7+y+1 \,. y ‘ b\ 10 pts hm ‘ ——-————. < ’ < ) (mm—40.0) x2 + y2 6. Consider the force vector field given by F = sci +ryj + zk. (a) (10 pts) Calculate the work done on a particle by the force F when the particle . moves along the conical helix r(9) =, (6.00s o9)i + {6’ sin 0)j + 6k from 9 =7 0 to ’ 0 = 27r. ,, , (b) (10 pts) Calculate the work done on a particle by F when the particle moves along / " “‘ "my” 'a‘str’aight‘iinerfrom the*'origin"tamer”point“with‘co-‘orriinatesf‘Zir:Oflfl.’13088133?” A‘M’H" work done along these two paths depend on the path taken? ...
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