Newton Raphson Method to Find Root of Equation 3x - cosx - 1 = 0

Step 1: Write the equation in the form of f(x) = 0
f(x) = 3x - cosx - 1

Step 2: Find the derivative of f(x)
f'(x) = 3 + sinx

Step 3: Choose an initial guess x0
Let x0 = 0.5

Step 4: Calculate the next approximation using the formula:
x1 = x0 - f(x0)/f'(x0)

Substituting the values:
x1 = 0.5 - (3(0.5) - cos(0.5) - 1)/(3 + sin(0.5))
x1 = 0.6974

Step 5: Calculate the error using the formula:
error = abs(x1 - x0)

Substituting the values:
error = abs(0.6974 - 0.5)
error = 0.1974

Step 6: Check if the error is less than the desired tolerance (e.g. 0.0001). If it is, stop and report the answer. If not, go back to Step 4 and repeat the process with x1 as the new x0.

Since the error (0.1974) is greater than the desired tolerance (0.0001), we repeat the process with x1 as the new x0.

Step 4 (repeated): Calculate the next approximation using the formula:
x2 = x1 - f(x1)/f'(x1)

Substituting the values:
x2 = 0.6974 - (3(0.6974) - cos(0.6974) - 1)/(3 + sin(0.6974))
x2 = 0.8241

Step 5 (repeated): Calculate the error using the formula:
error = abs(x2 - x1)

Substituting the values:
error = abs(0.8241 - 0.6974)
error = 0.1267

Step 6 (repeated): Check if the error is less than the desired tolerance (e.g. 0.0001). If it is, stop and report the answer. If not, go back to Step 4 and repeat the process with x2 as the new x0.

Since the error (0.1267) is greater than the desired tolerance (0.0001), we repeat the process with x2 as the new x0.

Step 4 (repeated again): Calculate the next approximation using the formula:
x3 = x2 - f(x2)/f'(x2)

Substituting the values:
x3 = 0.8241 - (3(0.8241) - cos(0.8241) - 1)/(3 + sin(0.8241))
x3 = 0.824