To solve $f(x) = x - 2\sin(x) = 0$ using the Regula Falsi (False Position) Method, we will compute the real root correct to 6 decimal places. The formula for updating the value of $x$ is:

$x_2 = x_0 - \frac{(x_1 - x_0)}{f(x_1) - f(x_0)} \cdot f(x_0)$

Step-by-Step Solution

Step 1: Initial interval

From the given information:

• $x_0 = 1$
• $x_1 = 2$

Evaluate $f(x)$ at these points: $f(1) = 1 - 2\sin(1) \approx 1 - 1.6829 \approx -0.6829$ $f(2) = 2 - 2\sin(2) \approx 2 - 1.8186 \approx 0.1814$

Since $f(x_0) \cdot f(x_1) < 0$, the root lies in the interval $[1, 2]$.

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Step 2: Compute $x_2$ using the formula

Using the Regula Falsi formula: $x_2 = x_0 - \frac{(x_1 - x_0)}{f(x_1) - f(x_0)} \cdot f(x_0)$

Substitute the values: $x_2 = 1 - \frac{(2 - 1)}{0.1814 - (-0.6829)} \cdot (-0.6829)$

Simplify: $x_2 = 1 - \frac{1}{0.1814 + 0.6829} \cdot (-0.6829)$ $x_2 = 1 - \frac{1}{0.8643} \cdot (-0.6829)$ $x_2 = 1 + 0.7901$ $x_2 \approx 1.7901$

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Step 3: Recalculate $f(x_2)$

$f(x_2) = x_2 - 2\sin(x_2)$ Substitute $x_2 \approx 1.7901$: $f(1.7901) = 1.7901 - 2\sin(1.7901)$ Using a calculator: $\sin(1.7901) \approx 0.9773$ $f(1.7901) = 1.7901 - 2(0.9773) = 1.7901 - 1.9546 \approx -0.1645$

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Step 4: Update the interval

Since $f(x_2) \cdot f(x_1) < 0$, the root lies in $[1.7901, 2]$. Update $x_0 = 1.7901$ and $f(x_0) = -0.1645$.

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Step 5: Compute the next approximation $x_3$

Use the formula again: $x_3 = x_0 - \frac{(x_1 - x_0)}{f(x_1) - f(x_0)} \cdot f(x_0)$

Substitute: $x_3 = 1.7901 - \frac{(2 - 1.7901)}{0.1814 - (-0.1645)} \cdot (-0.1645)$ $x_3 = 1.7901 - \frac{0.2099}{0.1814 + 0.1645} \cdot (-0.1645)$ $x_3 = 1.7901 - \frac{0.2099}{0.3459} \cdot (-0.1645)$ $x_3 = 1.7901 + 0.0998$ $x_3 \approx 1.8899$

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Step 6: Recalculate $f(x_3)$

$f(x_3) = x_3 - 2\sin(x_3)$ Substitute $x_3 \approx 1.8899$: $\sin(1.8899) \approx 0.9497$ $f(1.8899) = 1.8899 - 2(0.9497) = 1.8899 - 1.8994 \approx -0.0095$

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Step 7: Update the interval and iterate

Since $f(x_3) \cdot f(x_1) < 0$, the root lies in $[1.8899, 2]$. Update $x_0 = 1.8899$ and $f(x_0) = -0.0095$.

Repeat the process until the difference between successive approximations is less than $10^{-6}$. After a few more iterations:

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Final Answer

The root converges to: $x \approx 1.895494 \, \text{(correct to 6 decimal places).}$

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Would you like additional details or a breakdown of any steps?

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Related Questions:

1. What are the advantages of the Regula Falsi Method compared to the Bisection Method?
2. How do you determine convergence in the Regula Falsi Method?
3. Can the Regula Falsi Method fail? If so, under what conditions?
4. What is the geometric interpretation of the Regula Falsi Method?
5. How would you modify this method for a higher degree of accuracy?

Tip: Always check the sign of $f(x) \cdot f(y)$ to ensure the root lies in the chosen interval.