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) = 1, y 0 (0) = 3.
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EGR 265, Spring 2011, Final Exam 4 Problem 4 (12 points) A mass of 12 kg stretches a spring by 40 cm. 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 30 cm below the equilibrium position at a upward velocity of 2 m/s (choose the positive x -axis to be oriented downward). (c) Find the amplitude at which the mass oscillates.
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EGR 265, Spring 2011, Final Exam 5 Problem 5 (10 points) (a) Find the gradient of f ( x, y ) = x ln( x 2 + y ). (b) Evaluate the directional derivative of f ( x, y ) at the point P (2 , - 3) in the direction of the vector i - 2 j . (c) Find a unit vector in the direction of steepest decrease of f ( x, y ) at the point P (2 , - 3). Also find the rate of decrease in this direction.
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EGR 265, Spring 2011, Final Exam 6 Problem 6 (8 points) Find an equation of the tangent plane to the level surface sin( xyz ) - x - 2 y - 3 z = 0 through the point (2 , - 1 , 0). Problem 7 (8 points) Find the work done by the force field F ( x, y ) = e x i + xy j along the curve parameterized by x = t 2 , y = t 3 , 0 t 1.
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EGR 265, Spring 2011, Final Exam 7 Problem 8 (12 points)
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