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 2009, Final Exam 4 Problem 4 (12 points) A mass of 10 kg stretches a spring by 50 cm. Include the correct units in all your answers below. (a) Find the spring constant k , assuming that g = 10 m/s 2 . (b) What is the frequency at which the mass oscillates? (c) Find the equation of motion of the mass if it is released from rest at a position 20 cm below the equilibrium position (choose the positive x -axis to be oriented downward). (d) Find the first positive time at which the mass passes through the equilibrium position.
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EGR 265-6D, Fall 2009, Final Exam 5 Problem 5 (10 points) (a) Find the gradient of f ( x, y ) = p x 2 + y 3 . (b) Evaluate the directional derivative of f ( x, y ) at the point P (1 , 2) in the direction from P to the point Q (3 , 3). (c) Find a unit vector in the direction of steepest decrease of f ( x, y ) at the point (1 , 2). Also find the rate of increase in this direction.
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EGR 265-6D, Fall 2009, Final Exam 6 Problem 6 (8 points) Determine parametric equations of the normal line to the graph of z = x x + y at the point (1 , - 2 , - 1).
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EGR 265-6D, Fall 2009, Final Exam 7 Problem 7 (8 points) Find the line integral Z C x 2 y ds, where C is a quarter of a unit circle centered at the origin and contained in the first quadrant, starting at (1 , 0) and ending at (0 , 1).
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EGR 265-6D, Fall 2009, Final Exam 8 Problem 8 (12 points) (a) Show that the force field F ( x, y ) = (4 e y - 2 ye x ) i +(4 xe y - 2 e x ) j is conservative and find a potential function φ ( x, y ) for it. (b) Find the work done by the force field F from part (a) along the curve x ( t ) = t 2 , y = t 3 , 0 t 1.
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EGR 265-6D, Fall 2009, Final Exam 9 Problem 9 (10 points) A lamina of constant density ρ ( x, y ) = 1 is bounded by the curves y = x 2 and y = 1. (a) Find the lamina’s mass. (b) Find the lamina’s centroid. Use geometric considerations to simplify your work.
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EGR 265-6D, Fall 2009, Final Exam 10 Problem 10 (10 points) Find the double integral of the function f ( x, y ) = e x 2 + y 2 over the region in the first quadrant which is bounded by the circles r = 1 and r = 2.
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