Answer to Question #142196 in Quantitative Methods for Mehwish

Question

2 Find the root of the equation xex = cosx in the interval (0, 1) using
Regula-Falsi method correct to four decimal places.

Answer 1

$\displaystyle xe^x = \cos{x}\\ f(x) = xe^x - \cos{x}\\ f(0.5) = 0.5e^{0.5} - \cos{0.5} = -0.0532 < 0\\ f(0.6) = 0.6e^{0.6} - \cos{0.6} = 0.2679 > 0\\ \textsf{By Regula-Falsi method}\\ x_1 = \frac{0.5 f(0.6) - 0.6f(0.5)}{f(0.6) - f(0.5)} = 0.5165719036 \\ f(0.5165719036) < 0\\ f(0.6) > 0\\ \textsf{Repeating the process}\\ x_2 = \frac{0.6f(0.5165719036 ) - 0.5165719036 f(0.6)}{f(0.5165719036) - f(0.6)} = 0.51767881731\\ \therefore x = 0.5177 \, \, \textsf{(to 4.d.p)}$