1. **State the problem:** We need to find roots of the equations using the Regula-Falsi Method correct to four decimal places. 2. **Equations:** i) $3x + \sin x - e = 0$ ii) $\sin x - \cosh x + 1 = 0$ 3. **Regula-Falsi Method formula:** Given a continuous function $f(x)$ and two initial points $a$ and $b$ such that $f(a)f(b) < 0$, the root is approximated by: $$x_r = b - \frac{f(b)(a - b)}{f(a) - f(b)}$$ We update either $a$ or $b$ depending on the sign of $f(x_r)$ and repeat until desired accuracy. 4. **Important rules:** - Choose initial $a$ and $b$ so that $f(a)$ and $f(b)$ have opposite signs. - Stop when $|x_{r,new} - x_{r,old}| < 0.0001$ for four decimal places. --- ### i) Solve $3x + \sin x - e = 0$ 5. Choose initial interval: Try $a=0$, $b=1$ - $f(0) = 3(0) + \sin 0 - e = -e \approx -2.718 < 0$ - $f(1) = 3(1) + \sin 1 - e = 3 + 0.8415 - 2.718 = 1.1235 > 0$ 6. Compute $x_r$: $$x_r = 1 - \frac{1.1235(0 - 1)}{-2.718 - 1.1235} = 1 - \frac{-1.1235}{-3.8415} = 1 - 0.2927 = 0.7073$$ 7. Evaluate $f(0.7073)$: $$3(0.7073) + \sin 0.7073 - e = 2.1219 + 0.649 - 2.718 = 0.0529 > 0$$ 8. Since $f(0.7073) > 0$ and $f(0) < 0$, update $b = 0.7073$ 9. Repeat steps 6-8 until $|x_{r,new} - x_{r,old}| < 0.0001$: - Next $x_r = 0.7073 - \frac{0.0529(0 - 0.7073)}{-2.718 - 0.0529} = 0.7073 - \frac{-0.0374}{-2.7709} = 0.7073 - 0.0135 = 0.6938$ - $f(0.6938) = 3(0.6938) + \sin 0.6938 - e = 2.0814 + 0.640 - 2.718 = 0.0034 > 0$ - Update $b = 0.6938$ - Next $x_r = 0.6938 - \frac{0.0034(0 - 0.6938)}{-2.718 - 0.0034} = 0.6938 - \frac{-0.0024}{-2.7214} = 0.6938 - 0.0009 = 0.6929$ - $f(0.6929) = 3(0.6929) + \sin 0.6929 - e = 2.0787 + 0.639 - 2.718 = 0.0001 > 0$ - Update $b = 0.6929$ - Next $x_r = 0.6929 - \frac{0.0001(0 - 0.6929)}{-2.718 - 0.0001} = 0.6929 - \frac{-0.00007}{-2.7181} = 0.6929 - 0.00003 = 0.69287$ - Difference $|0.6929 - 0.69287| = 0.00003 < 0.0001$ stop. **Root for i):** $x \approx 0.6929$ --- ### ii) Solve $\sin x - \cosh x + 1 = 0$ 10. Choose initial interval: Try $a=0$, $b=1$ - $f(0) = \sin 0 - \cosh 0 + 1 = 0 - 1 + 1 = 0$ - $f(1) = \sin 1 - \cosh 1 + 1 = 0.8415 - 1.5431 + 1 = 0.2984 > 0$ 11. Since $f(0) = 0$, root is exactly at $x=0$. 12. To confirm, check values near zero: - $f(-0.1) = \sin(-0.1) - \cosh(-0.1) + 1 = -0.0998 - 1.005 + 1 = -0.1048 < 0$ 13. So root lies between $-0.1$ and $0$. 14. Apply Regula-Falsi: $$x_r = 0 - \frac{0(-0.1 - 0)}{-0.1048 - 0} = 0 - 0 = 0$$ 15. Since $x_r = 0$ and $f(0) = 0$, root is $x=0$. **Root for ii):** $x = 0.0000$ --- **Final answers:** - i) $x = 0.6929$ - ii) $x = 0.0000$