Show that the equation;x+3sinx=2 has a root between x=0.4 and x=0.6 using newton raphson's formular only

f(x) = x + 3sinx - 2

f'(x) = 1 + 3cosx

1: f(0.6000000) = 0.2939274
2: f(0.5154411) = -0.0058029
3: f(0.5170484) = -0.0000019
4: f(0.5170490) = -0.0000000

Farouk, It would be a good idea to check back the previous posts you have made before reposting. It may save you a long wait.

See answer to your previous post:
http://www.jiskha.com/display.cgi?id=1321390404

To show that the equation x + 3sin(x) = 2 has a root between x = 0.4 and x = 0.6 using Newton Raphson's formula, we need to follow these steps:

Step 1: Take the function f(x) = x + 3sin(x) - 2 and find its derivative, f'(x), which is 1 + 3cos(x). This derivative will be used to update the initial guess in the Newton Raphson iteration.

Step 2: Choose an initial guess, let's say x0 = 0.5, that lies between 0.4 and 0.6.

Step 3: Apply the Newton Raphson iteration formula to refine the initial guess and move closer to the root. The iteration formula is:
x(n+1) = x(n) - f(x(n))/f'(x(n))

Step 4: Iterate using the formula until the desired accuracy is achieved, or until a maximum number of iterations is reached.

Now, let's use these steps to find the root:

Step 1: The derivative of f(x) = x + 3sin(x) - 2 is f'(x) = 1 + 3cos(x).

Step 2: Choose an initial guess, let's take x0 = 0.5, which lies between the given interval.

Step 3: Apply the Newton Raphson iteration formula:
x(n+1) = x(n) - f(x(n))/f'(x(n))

Using x0 = 0.5, we plug it into f(x) and f'(x):
f(x0) = 0.5 + 3sin(0.5) - 2 = -0.139236
f'(x0) = 1 + 3cos(0.5) = 1.81567

Calculating x1 using the iteration formula:
x1 = 0.5 - (-0.139236) / 1.81567 ≈ 0.42871

Step 4: Repeat the iteration process until the desired accuracy is achieved or the maximum number of iterations is reached.

Continue iterating using the value obtained for x1 as the new guess in step 3. Repeat the process until the desired accuracy is achieved or a predetermined number of iterations is reached.

By following this iterative process, it can be determined whether the function converges to a root within the interval [0.4, 0.6].