# Newton-Raphson method

1 Each equation below is to be approximately solved using Newton-Raphson method with the given starting value xo. In each case, use technology to sketch the graph, draw the tangent at wo and find the next approximation to the root. a i 23 – 2×2 – 1=0, X0 = 2.5 ii 1 5.2 – 24 – 3 = 0, X0 = 1 2 x2 – 10 In x = 0, 20 = 4 bi ji x – 2 cos x = 0, X0 = 2 2 For each equation, use the Newton-Raphson method with the given starting value to find the root correct to three significant figures. Use the change of sign method to show that your root is correct to three significant figures. a i 24 – 3x + 1 = 0, X0 = 1.5 3.2 x3 +1 = 0, X0 = 0 ii 6) – bi sin – x + 1 = 0, xo = 0 ii e0.2x – 3V= 0, X0 = 11 The equation 1 – x2 + 2×3 = 0 has a root near -0.5. Using this value as the first approximation, use the Newton- Raphson method to find the next approximation to the root. 3 4 a Show that the equation 2×2 3 – 1= C can be written as 2×3 – X – 3 = 0. b Given that the equation has a root near 1.5, use the Newton-Raphson method to find the next two approximations. 2 The equation sin x = has a root near 6.5. Use the Newton-Raphson method to find the next two approximations to the root. 5 2 6 4 a Show that the equation 2 – x + = 0 has a root between 3 and 4. 2 b Using xo = 3 as the starting value, use the Newton-Raphson method to find the next approximation to the root. Give your answer correct to two decimal places. c Show that the approximation from part b is correct to one decimal place, but not to two decimal places. 1 20 For each equation below, carry out one iteration of the Newton-Raphson method starting with the given value of In each case, sketch the graph (using technology) to explain why zi is not a better approximation to the root than TO а і 2.23 – 5x + 2 = 0, X0 = 1 ii bi vã – 0.2×3 – 0.5 = 0, 20 = 1 tan c x – 1= 0, X0 = 0.5 In 2.0 + 2 = 0, 20 = 1 ii C 2 Let f(x) = 10×3 – 5.02 1. a Find the x-coordinates of the stationary points of f(x). b Show that the equation f(x) = 0 has a root between and 1. C The Newton-Raphson formula with xo = 0.35 is used to find x1. Explain why xı may not be an improved approximation for the root. The diagram shows the curve with equation f(x) = (x – 2) e-2 +1. 3 y x Find the x-coordinate of the stationary point of f(x) and, hence, explain, with an aid of a diagram, why a Newton- Raphson iteration with 20 = 3.5 will not converge to the root of f(x) = 0. 6 The function f(x) = 223 – 6x +1 has three zeros (a

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