i NEED TO TURN (7X-3)^1/3 INTO AN ITERVATIVE FORMULA USING THE NEWTON RAPHSON FORMULA?
HOW DO I DO IT IN SIMPLE STEPS?
To convert the expression (7x-3)^(1/3) into an iterative formula using the Newton-Raphson method, follow these steps:
Step 1: Choose an initial approximation for the root. Let's call it x0.
Step 2: Write down the function that you want to find the root of. In this case, the function is f(x) = (7x-3)^(1/3)
Step 3: Calculate the derivative of the function. In this case, the derivative of f(x) is f'(x) = (1/3)(7x-3)^(-2/3)(7)
Step 4: Use the formula x₁ = x₀ - f(x₀)/f'(x₀) to find the approximation of the root. Repeat this step until you achieve the desired level of accuracy.
Step 5: Iterate the formula above, substituting x₁ as x₀ on the next iteration, until convergence is reached.
Here's a step-by-step example:
Let's start with an initial approximation, x₀ = 1.
Step 1: Choose x₀ = 1.
Step 2: The function is f(x) = (7x-3)^(1/3).
Step 3: The derivative is f'(x) = (1/3)(7x-3)^(-2/3)(7).
Step 4: Calculate x₁ using the formula x₁ = x₀ - f(x₀)/f'(x₀).
x₁ = 1 - [(7(1)-3)^(1/3)] / [(1/3)(7(1)-3)^(-2/3)(7)]
Step 5: Repeat step 4 until you achieve the desired level of accuracy.
Calculate x₂, x₃, x₄, and so on, by substituting the previous approximation into the formula.
Keep iterating until the values converge to a desired level of precision.
The Newton-Raphson method is an iterative process, so you'll continue to repeat step 4 until the desired level of accuracy is reached.