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p1spring10 pages 1 to 6

p1spring10 pages 1 to 6 - Prelim 1 Math 2930 show work 6...

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Prelim 1 Math 2930 Spring 2010 show work, 6 problems, no calculators 1a) (4 points) Write your name and section number at the top of this page. 1b) (4 points) The chef wants to put a 100C dish into a 200C oven at time 0, and he thinks the temperature T ( t ) of the dish will evolve according to T = 1 20 ( T - 200) But he made a mistake: Explain why there ought to be a minus sign on the 1 20 coefficient. 1c) (12 points) Solve the corrected equation T = - 1 20 ( T - 200) using the chef’s initial condition. 1
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2) A population p ( t ) [thousands] is described by the differential equation p ( t ) = 1 10 p ( t ) - 1 1000 ( p ( t )) 3 a) (4 points) Find three equilibrium (constant) solutions. b) (6 points) Determine the stability of the positive equilibrium. c) (6 points) Use two steps of Euler’s numerical method to estimate the population at time t = 0 . 2, if we assume p (0) = 3 . 0. You should find the approximations p 0 and p 1 exactly, but you do not have to simplify the ex- pression for p 2 . 2
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3a) (6 points) Consider the second order differential equation y +5 y +6 y = 0.
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