The image asks to evaluate $\sqrt{12}$ to four decimal places using the Newton-Raphson method.

Newton-Raphson Method Overview:

The Newton-Raphson method is an iterative technique used to find successively better approximations of the roots (or zeros) of a real-valued function.

For this problem, we want to solve $f(x) = x^2 - 12 = 0$ because the square root of 12 is the root of this equation.

Formula:

The general formula for Newton-Raphson is: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$ Where:

• $f(x) = x^2 - 12$
• $f'(x) = 2x$

Now, let's calculate $\sqrt{12}$ using this method.

1. Initial guess: We start with an initial guess, say $x_0 = 3.5$, because $\sqrt{12}$ is between 3 and 4.

I'll compute the first few iterations to approximate $\sqrt{12}$ to four decimal places.Using the Newton-Raphson method, the approximation for $\sqrt{12}$ to four decimal places is 3.4641. This result was obtained after 2 iterations.

Would you like more details on how the iterations were performed or any related information?

Here are five related questions you might find useful:

1. How does the Newton-Raphson method converge so quickly?
2. Can Newton-Raphson be used for complex numbers?
3. What are the limitations of the Newton-Raphson method?
4. How does this method compare to other root-finding algorithms, such as the bisection method?
5. How would you apply this method to functions other than square roots?

Tip: Always choose an initial guess close to the actual root to ensure faster convergence when using the Newton-Raphson method.