Let's break down the problems:

Part (a): Newton-Raphson Scheme for Finding the Square Root of a Positive Number $N$

The Newton-Raphson method is an iterative method used to find successively better approximations to the roots (or zeroes) of a real-valued function. To find the square root of a positive number $N$, we can apply the Newton-Raphson method as follows:

1. Set up the equation: To find $\sqrt{N}$, we solve the equation $f(x) = x^2 - N = 0$.

2. Apply Newton-Raphson iteration: The iteration formula is given by: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$ For $f(x) = x^2 - N$, the derivative $f'(x)$ is $2x$. Therefore, the iteration formula becomes: $x_{n+1} = x_n - \frac{x_n^2 - N}{2x_n}$ Simplifying this, we get: $x_{n+1} = \frac{1}{2} \left(x_n + \frac{N}{x_n}\right)$ This is the Newton-Raphson iteration for finding the square root of $N$.

3. Evaluate $\sqrt{12}$ to four decimal places: Start with an initial guess $x_0$, for example, $x_0 = 3.5$ (since $\sqrt{12} \approx 3.464$).

   Iterating: $x_1 = \frac{1}{2} \left(3.5 + \frac{12}{3.5}\right) = 3.4643$ $x_2 = \frac{1}{2} \left(3.4643 + \frac{12}{3.4643}\right) \approx 3.4641$ $x_3 = \frac{1}{2} \left(3.4641 + \frac{12}{3.4641}\right) \approx 3.4641$ So, $\sqrt{12} \approx 3.4641$ (correct to four decimal places).

Part (b): Finding the Unique Polynomial of Degree 2 or Less

Given points are $f(0) = 1$, $f(1) = 3$, and $f(3) = 55$.

1. General form of the polynomial: A polynomial of degree 2 can be written as: $f(x) = ax^2 + bx + c$

2. Set up the system of equations: Using the given points:

   • For $x = 0$: $f(0) = c = 1$
   • For $x = 1$: $f(1) = a(1)^2 + b(1) + c = a + b + 1 = 3$
   • For $x = 3$: $f(3) = a(3)^2 + b(3) + c = 9a + 3b + 1 = 55$

   Simplifying these, we get: $a + b = 2 \quad \text{(Equation 1)}$ $9a + 3b = 54 \quad \text{(Equation 2)}$

3. Solve the system of equations:

   • From Equation 1: $b = 2 - a$.
   • Substitute $b = 2 - a$ into Equation 2: $9a + 3(2 - a) = 54$ Simplifying: $9a + 6 - 3a = 54 \implies 6a = 48 \implies a = 8$
   • Substituting $a = 8$ into Equation 1: $b = 2 - 8 = -6$.

   So, $a = 8$, $b = -6$, and $c = 1$.

   Therefore, the polynomial is: $f(x) = 8x^2 - 6x + 1$

4. Evaluate $f(2)$: $f(2) = 8(2)^2 - 6(2) + 1 = 32 - 12 + 1 = 21$

Thus, $f(2) = 21$.

---

Would you like further details or have any questions?

Here are some related questions:

1. How would you generalize Newton-Raphson's method for other functions?
2. How do you determine a good initial guess for the Newton-Raphson method?
3. Can the Newton-Raphson method fail? If so, under what circumstances?
4. How would you extend the polynomial interpolation method to a cubic polynomial?
5. What is the significance of using a second-degree polynomial for interpolation?

Tip: When using Newton-Raphson, always verify that the function and its derivative are well-behaved near the root to ensure convergence.