Tutorial 4 solution

Tutorial 4 solution - Name sin 4 5 ID(0(0...

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Hand in this sheet before leaving class. Name: ____________________ ID: _______________________ The equation: sin ݔ ൅ 4 ൌ 5ݔ Starting from the interval [a (0) , b (0) ] = [0.9, 1.0], use the bisection method to identify the next interval [a (1) , b (1) ] that brackets a solution of the equation. fሺݔሻ ൌ sin ݔ ൅ 4 െ 5ݔ ൌ 0 f(0.9) = 0.2833 > 0 f(1.0) = –0.1585 < 0 The signs are opposite, so this continuous function has a zero somewhere between 0.9 and 1.0. c (0) = (0.9 + 1.0)/2 = 0.95 f(0.95) = 0.0634 > 0 Choose the interval [a (1) , b (1) ] = [0.95, 1.0] because the corresponding f values have opposite signs; there is a zero somewhere between 0.95 and 1.0. Starting from the initial iterate x (0) = 1.0, use Newton’s method to identify the next iterate x (1) that approximates a solution of the equation. ݂ ሺݔሻൌcosݔെ5 x (1) = x (0) ௙ሺ௫ ሺబሻ ௙ᇱሺ௫ ሺబሻ = 1.0 – ି଴.ଵହ଼ହ ିସ.ସହଽ଻ = 0.9645
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