Newton\'s Method Application; (from University Calculus, by Hass, Weir and Thoma

Question

Newton's Method Application; (from University Calculus, by Hass, Weir and Thomas) The Sonobuov Problem: In submarine location problems, it is often necessary to find a submarine's closest point of approach (CPA) to a songbuoy (sound detector) in the water. Suppose that the submarine travels on the parabolic path y -x'and that the buoy is located at the point (2,-1/2 y' Submarine track in two dimensions CPA Sonobuoy2,-2 a. Show that the value of x that minimizes the distance between the submarine and the buoy is a solution to the equation x- . Find a solution to this equation to four decimal places using Newton's Method. List all of your iterations.

Explanation / Answer

given;

x = 1/(2(x^2+1))

d^2 = (x-2)^2 + (x^2 + 1/2)^2
take the derivative
2(x-2) + 4x(x^2+1/2)

then equal to 0 and simplified

2x^3 + 2x - 1 = 0

that reduces to x = 1/(2(x^2+1))

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