C solve the initial value problem given by 1 and

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(c) Solve the initial value problem given by (1) and initial conditions y (0) = 0, y 0 (0) = 0.
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EGR 265-6D, Fall 2010, Final Exam 4 Problem 4 (12 points) A 100 cm spring measures 140 cm long after a mass of 10 kg is attached to it. The medium through which the mass moves offers a damping force with damping coefficient β = 100 kg/s. Include the correct units in all your answers below. (a) Find the spring constant k , assuming that g = 10 m/s 2 . (b) Find the equation of motion of the mass if it is released from a position 10 cm below the equilibrium position with an upward velocity of 50 cm/s (choose the positive x -axis to be oriented downward). (c) Will the mass return to the equilibrium position? If yes, when is the first time? If no, why not?
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EGR 265-6D, Fall 2010, Final Exam 5 Problem 5 (10 points) (a) Find the gradient of f ( x, y ) = p 2 x 2 + 2 xy + y 2 . (b) Evaluate the directional derivative of f ( x, y ) at the point P (1 , 2) in the direction from P (1 , 2) to Q (2 , 3). (c) Find a unit vector in the direction of steepest decrease of f ( x, y ) at the point P (1 , 2). Also find the rate of decrease in this direction.
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EGR 265-6D, Fall 2010, Final Exam 6 Problem 6 (8 points) Determine the equation of the tangent plane to the level surface xyz = 6 through the point (1 , 2 , 3).
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EGR 265-6D, Fall 2010, Final Exam 7
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