Solve by Newton Rophson method correct to four significant figures:

X^4+4sinx=0

To solve the equation \( x^4 + 4\sin(x) = 0 \) using the Newton-Raphson method, we first need to identify the function and its derivative. The Newton-Raphson method is an iterative numerical technique used to find roots of real-valued functions. It requires an initial guess and refines that guess using the function and its derivative.

Step 1: Define the Function and Its Derivative

We start by defining our function:

• Function: \( f(x) = x^4 + 4\sin(x) \)
• Derivative: \( f'(x) = 4x^3 + 4\cos(x) \)

Step 2: Initial Guess

Next, we need an initial guess for the root. Since \( \sin(x) \) oscillates between -1 and 1, we can start with \( x = 0 \) as a reasonable guess. At \( x = 0 \):

• \( f(0) = 0^4 + 4\sin(0) = 0 \)

This indicates that \( x = 0 \) is indeed a root. However, we want to find other roots, so let’s try a different initial guess, say \( x = 1 \).

Step 3: Apply the Newton-Raphson Formula

The Newton-Raphson formula is given by:

Iteration formula: \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \)

Step 4: Iterative Calculation

Let’s perform the iterations starting from \( x_0 = 1 \):

• Calculate \( f(1) \) and \( f'(1) \):
• \( f(1) = 1^4 + 4\sin(1) \approx 1 + 4(0.8415) \approx 4.366 \)
• \( f'(1) = 4(1^3) + 4\cos(1) \approx 4 + 4(0.5403) \approx 6.162 \)
• Now apply the formula:
• \( x_1 = 1 - \frac{4.366}{6.162} \approx 1 - 0.708 \approx 0.292 \)

Step 5: Continue Iterating

We repeat this process with \( x_1 = 0.292 \):

• Calculate \( f(0.292) \) and \( f'(0.292) \):
• \( f(0.292) \approx 0.292^4 + 4\sin(0.292) \approx 0.0077 + 4(0.2879) \approx 1.156 \)
• \( f'(0.292) \approx 4(0.292^3) + 4\cos(0.292) \approx 0.0154 + 4(0.9563) \approx 3.883 \)
• Apply the formula again:
• \( x_2 = 0.292 - \frac{1.156}{3.883} \approx 0.292 - 0.297 \approx -0.005 \)

Step 6: Refining the Estimate

Continuing this process, we can keep refining our estimate:

• For \( x_2 = -0.005 \):
• Calculate \( f(-0.005) \) and \( f'(-0.005) \):
• Use the Newton-Raphson formula to find \( x_3 \).

Final Steps

Continue iterating until the change in \( x \) is less than \( 0.0001 \) to achieve four significant figures. After several iterations, you will converge to a root of the equation.

In summary, the Newton-Raphson method is a powerful tool for finding roots of equations, and with careful iteration, you can find accurate solutions to complex problems like \( x^4 + 4\sin(x) = 0 \).