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# pexam2 - t y(a Sketch the slope ﬁeld using a scale of-3...

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Practice Exam for the second Midterm (Time: 90 minutes) 1. Determine all critical points of the function f ( x ) = x 2 e 3 x . For each critical point decide whether it is a minimum or maximum. 2. (a) Compute the linear approximation for the function f ( x ) = 1 x around x = 1, i.e., ﬁnd a simple (linear) expression for the value of the function at 1 + Δ x . (b) If you are you increase x = 1 by 12% what is the relative change in f ( x ) = 1 x ? 3. Solve the diﬀerential equation y 0 ( t ) = 3 t 2 + sin(2 t ) for the initial condition y (0) = 0. 4. A culture of bacteria consists of N ( t ) bacteria. N ( t ) satisﬁes the diﬀerential equation N 0 ( t ) = 0 . 2 N ( t ) where t is measured in hours. If I now have 5 times as many bacteria as I started with, how long ago did I start the experiment? 5. Consider the diﬀerential equation y 0 =
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Unformatted text preview: t + y . (a) Sketch the slope ﬁeld using a scale of -3 to 3 on both axes, and showing the ﬁve isocline for c =-2 ,-1 , , 1 , 2. (b) Sketch the two solution curves with initial conditions y (0) = 1 and y (0) =-1. 6. Consider the autonomous diﬀerential equation y = y 3-y . (a) Compute the critical points, and the corresponding stationary solutions. (b) Sketch the phase line. (c) Sketch the graphs of some solutions. Make sure you include at least one solutions with values in each interval above, below, and between the critical points. (d) What is the long term behavior of the solution with the initial condition y (0) = 1 2 . ( Hint: You answer should be a number.)...
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