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Practice_MT_1_Math_33B_F08

Practice_MT_1_Math_33B_F08 - Mathematics 33B Practice...

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Mathematics 33B - Practice Midterm 1 Official exam to be administered Oct. 24th, 11:00am NAME (please print legibly): Your University ID Number: Your Discussion Section and TA: Signature: QUESTION VALUE SCORE 1 10 2 10 3 10 4 10 5 10 TOTAL 50 1
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1. (10 points) Determine which equilibrium solutions to the equation dy dt = y 3 (1+ y ) 2 (2 - y ) are asymptotically stable and which are unstable. Then make a rough sketch showing the behavior of the solutions to this equation. 2
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2. (10 points) Suppose that P ( t ) is the population (in millions) at time t (in years) of species that is being harvested” at a rate of h million per year. If the growth rate of the population without harvesting is r , then P ( t ) satisfies dP dt = rP - h. Suppose that r = 0 . 4 and h = 0 . 5. If the population at time t = 0 is 1 (million), at what time will the population reach zero? Leave your answer in logarithms. 3
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3. (10 points) (a) Find the solution to dy dx = 1 x (2 y - 1) that satisfies y (1) = 2, AND find the domain of existence for this solution. b) What happens at the endpoint(s) of the domain of existence? Why doesn’t the existence theorem say that you should be able to continue the solution beyond them?
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