Hi, thanks for help in advance.

Using the Equation: f(w) = 10/w - 0.77ln(w) – 3.2

How do i use Newton-Raphson iterative to find the required value of w correct to 3 significant figures, starting from w0 = 3.

Thanks

enter your functions here and see the steps:

http://maccery.com/maths/#newton-raphson

To use the Newton-Raphson method, you need to follow these steps:

1. Start with an initial guess, w0, which in this case is given as 3.
2. Calculate the value of the function, f(w), using the given equation for the guess value w0.
3. Calculate the derivative of the function, f'(w), with respect to w.
4. Use the formula: w1 = w0 - f(w0) / f'(w0) to find the next approximation, w1.
5. Repeat steps 2-4 using the new approximation value, w1, until the desired level of accuracy is achieved.

Let's go through the steps to find the required value of w:

Step 1: The initial guess is given as w0 = 3.

Step 2: Calculate the value of the function, f(w0), using the given equation:
f(w0) = 10/w0 - 0.77ln(w0) – 3.2 = 10/3 - 0.77ln(3) – 3.2 ≈ -6.16615

Step 3: Calculate the derivative of the function, f'(w), with respect to w:
f'(w) = -10/w^2 - 0.77/w

For w0 = 3, f'(w0) = -10/3^2 - 0.77/3 ≈ -3.697

Step 4: Use the Newton-Raphson formula to find the next approximation, w1:
w1 = w0 - f(w0)/f'(w0)
= 3 - (-6.16615)/(-3.697)
≈ 4.666

Step 5: Now, with the new approximation value, w1, repeat steps 2-4 until the desired level of accuracy is achieved.

Repeat steps 2 and 3 using w1 = 4.666 to get the updated values of f(w) and f'(w):

f(4.666) ≈ -1.403
f'(4.666) ≈ -0.556

Using these values, we can find the next approximation, w2:
w2 = w1 - f(w1)/f'(w1)
= 4.666 - (-1.403)/(-0.556)
≈ 6.584

Repeat this process until the desired level of accuracy is achieved or until the number of significant figures required is reached.

It's important to note that the Newton-Raphson method may not always converge or may converge to a different root if there are multiple roots or oscillations. Therefore, it's essential to verify the accuracy and stability of the method in your specific case.