Use Newton\'s method to solve the equation 0 = 1 2 + 1 4x2 ?? x sin x ?? 1 2 cos

Question

Use Newton's method to solve the equation 0 = 1 2 + 1 4x2 ?? x sin x ?? 1 2 cos 2x; Iterate using Newton's method until an accuracy of 10??5 is obtained.

Explanation / Answer

18x (x2 + 9)(2x) - x2 (2x) = 2 . (x2 + 9)2 (x + 9)2 y = Since the denominator is always positive, the sign of y is the same as the sign of the numerator. Therefore, y < 0 when x < 0 and y > 0 when x > 0. Hence, y is decreasing for x < 0, y is increasing for x > 0 and, by the ?rst derivative test, y has a local minimum of 0 at x = 0. Taking the second derivative using the quotient rule, y = 1 Notice that y is positive for - 1 2 . Hence, 2 2 2 y is concave down on (-8, -1/2) and (1/2, 8), y is concave up on (-1/2, 1/2), and both -1/2 and 1/2 are in?ection points. Finally, notice that (x2 + 9)2 (18) - 18x(2(x2 + 9)(2x)) (x2 + 9)2 (1 - 4x2 ) 1 - 4x2 = 18 = 18 2 . (x2 + 9)4 (x2 + 9)4 (x + 9)2 and, likewise x2 1 = lim =1 x?8 x2 + 9 x?8 1 + 9/x2 lim 1 x2 = lim = 1, x?-8 1 + 9/x2 x?-8 x2 + 9 lim so y has a horizontal asymptote at y = 1 in both directions. Putting all the above information together yields a sketch of the curve:

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