Question

Use the algebraic formula for Newton's Method to find the third approximation, X3,

for f(x) = x3 –5x2 + 6.25x, where the first approximation is Xo = 3.
Find the answer to three decimal places.

Answer 1

To find the third approximation X3 for the function using Newton's Method, start with the first approximation X0 = 3, and apply the iterative formula involving the function and its derivative. Repeat this process until the third approximation is obtained. Precision to at least three decimal places is important for accuracy.

Step-by-step explanation:

To find the third approximation (X3) for the function f(x) = x³ – 5x² + 6.25x using Newton's Method, we must first recognize that this method requires both the function f(x) and its derivative f'(x). The derivative of the function is f'(x) = 3x² - 10x + 6.25.

Newton's Method formula is Xn+1 = Xn - f(Xn)/f'(Xn), where Xn is the current approximation and Xn+1 is the next. Starting with X0 = 3, we can calculate the next approximation, X1, then X2, and finally X3. The calculations proceed as follows:

1. Compute f(X0) and f'(X0).
2. Calculate X1 = X0 - f(X0)/f'(X0).
3. Repeat the process using X1 to find X2, and then X2 to find X3.

It's important to perform each computation to at least three decimal places for precision.

The Newton's Method is an iterative process, and each step should refine the approximation of the root. After finding X3, we can check how much the approximation has improved from the initial guess by comparing it with the true root of the function, if known.