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midterm1 - Math 108 First Midterm Examination NAME(Please...

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Math 108 First Midterm Examination September 21, 2007 NAME (Please print) Page Possible Score 2 20 3 10 4 20 5 10 6 10 7 20 8 10 Total 100 Instructions: 1. Do all computations on the examination paper. You may use the backs of the pages if necessary. 2. Put answers inside the boxes (when applicable). 3. Provide arguments for functions, and bounds on integrals. 4. Write equations and inequalities, not disconnected mathematical expressions. 5. Please signify your adherence to the honor code: I, , have neither given nor received unauthorized help on this exam and I have conducted myself within the guidelines of the Duke Community Standard. 1
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(20 Points) Score 1. (6 points) Determine the equilibrium points of y ( x ) = y ( x )[ y ( x ) + 1][ y ( x ) + 2] . Equilibrium points occur where 0 = y [ y + 1][ y + 2]. Solving for y gives y = 0 , - 1 , - 2. equilibrium points are y = 0 , - 1 , - 2 2. (6 points) Sketch the phase line for y ( x ) = y ( x )[ y ( x ) + 1][ y ( x ) + 2] . 3. (4 points) Which of these equilibrium points are stable? From the phase line, we see that y = - 1 is the only stable equilibrium point. We could 2
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also compute df ( y ) dy at each of the equilibrium points, and notice that this is negative only for y = - 1. stable equilibrium points are y = - 1. 4. (4 points) If y (0) = 1 , perform one step of Euler’s method to approximate y (1 / 4) . We approximate y (1 / 4) = y (0) + 1 / 4 f ( y (0)) = 1 + 1 / 4 × 1 × (1 + 1) × (1 + 2) = 1 + 3 / 2 = 5 / 2 .
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