The question from the image asks:

"Find a root of the equation $2x - 3\sin x - 5 = 0$ by Regula-Falsi Method."

Solution Outline:

1. Rewrite the equation in the form of $f(x) = 0$: $f(x) = 2x - 3\sin x - 5$

2. Regula-Falsi (False Position) Method is an iterative numerical method to find the root of a function. It uses a linear interpolation to approximate the root between two guesses $x_0$ and $x_1$ where: $f(x_0) \times f(x_1) < 0$ implying the root lies between $x_0$ and $x_1$.

3. Iteration Formula: The formula to update the estimate of the root is: $x_2 = x_1 - \frac{f(x_1) \times (x_1 - x_0)}{f(x_1) - f(x_0)}$ where $x_0$ and $x_1$ are the current estimates, and $f(x_0)$ and $f(x_1)$ are the values of the function at those points.

4. Procedure:

   • Choose two initial guesses, $x_0$ and $x_1$, such that $f(x_0)$ and $f(x_1)$ have opposite signs.
   • Use the formula to compute a new estimate $x_2$.
   • Check if $f(x_2)$ is close enough to zero; if not, repeat the process, replacing either $x_0$ or $x_1$ based on the sign of $f(x_2)$.
   • Continue iterating until the desired accuracy is achieved.

Would you like to go through a step-by-step numerical example of this process? Let me know!

Follow-up Questions:

1. What are the key differences between the Regula-Falsi method and the Bisection method?
2. How does the convergence of the Regula-Falsi method compare to other root-finding methods?
3. In which cases would Regula-Falsi be preferred over methods like Newton-Raphson?
4. Can you explain how the choice of initial guesses affects the outcome of the Regula-Falsi method?
5. How can we ensure that the root obtained is accurate to a certain degree of precision?

Tip:

In the Regula-Falsi method, choosing initial guesses that are closer to the root can significantly speed up convergence.