To find the real root of the equation 3x-cos(x)-1=0 using the Newton Raphson Method, you will need to follow these steps:
Step 1: Choose an initial guess for the root. Let's start with x0 = 0.
Step 2: Compute the function value at the initial guess. Plug x0 into the equation 3x-cos(x)-1=0 to get f(x0).
f(x0) = 3(0) - cos(0) - 1 = -2.
Step 3: Compute the derivative of the function. Differentiate the equation 3x-cos(x)-1=0 with respect to x to get f'(x).
f'(x) = 3 + sin(x).
Step 4: Check for convergence. Determine if the initial guess is close enough to the real root by checking if |f(x0)/f'(x0)| < ε, where ε is a small positive number (e.g., 0.001). If it is, then stop and report x0 as the approximate root. Otherwise, continue to the next step.
Let's calculate |f(x0)/f'(x0)|: |(-2)/(3+sin(0))| = 0.66667.
Since 0.66667 > 0.001, we need to continue to the next step.
Step 5: Compute the next guess for the root using the formula:
x1 = x0 - (f(x0)/f'(x0)).
Plug in the values to get:
x1 = 0 - (-2/(3+sin(0))) = 0.66667.
Step 6: Repeat steps 2-5 until convergence. Compute f(x1), f'(x1), check for convergence, and compute the next guess until |f(xn)/f'(xn)| < ε.
Let's calculate the values for x1:
f(x1) = 3(0.66667) - cos(0.66667) - 1 ≈ -0.0026,
f'(x1) = 3 + sin(0.66667) ≈ 3.63004014,
|f(x1)/f'(x1)| = |-0.0026/3.63004014| ≈ 0.00071696.
Since 0.00071696 < 0.001, we have reached the desired accuracy.
Therefore, the real root of the equation 3x-cos(x)-1=0, correct to four decimal places, is approximately x = 0.6667.