use modified newton's raphson method to indicate the solution correct to 6 decimal places near to x=2 of the equation x^3-6x^2+13x-9=0

Question

use modified newton's raphson method to indicate the solution correct to 6 decimal places near to x=2 of the equation x^3-6x^2+13x-9=0

Answer 1

let f(x) = x^3 - 6x^2 + 13x - 9

f'(x) = 3x^2 - 12x + 13

x_new = x - f(x)/f'(x)
= x - (x^3 - 6x^2 + 13x - 9)/(3x^2 - 12x + 13)
= (3x^3 - 12x^2 + 13x - x^3 + 6x^2 - 13x + 9)/(3x^2 - 12x + 13)
= (2x^3 - 6x + 9)/(3x^2 - 12x + 13)

using my calculator, with an initial guess of x = 1

x ----- newx
1 , 1.25
1.25 , 1.3139...
1.3139.., 1.3176604
1.3176604 , 1.317672196
1.31762196 , 1.317672197

Wow, looks like I had 6 decimal accuracy in the 2nd last iteration.

Answer 2

To use the Modified Newton-Raphson method to find a solution to the equation x^3 - 6x^2 + 13x - 9 = 0 with an initial guess of x = 2, follow these steps:

Step 1: Find the first derivative of the equation.
f'(x) = 3x^2 - 12x + 13

Step 2: Find the second derivative of the equation.
f''(x) = 6x - 12

Step 3: Set up the Modified Newton-Raphson iteration formula:
x(n+1) = x(n) - [f(n) * f'(n)] / [f'(n)^2 - f(n)*f''(n)]

Step 4: Start with an initial guess, let n = 0, and calculate the values of f(0) and f'(0).

f(0) = (0^3) - 6(0^2) + 13(0) - 9 = -9
f'(0) = 3(0^2) - 12(0) + 13 = 13

Step 5: Plug the values from Step 4 into the iteration formula and calculate the next approximation.

x(1) = 2 - [(-9 * 13) / (13^2 - (-9) * 6)] ≈ 2.482759

Step 6: Continue the iteration process until the desired accuracy is achieved. Repeat the steps, replacing the values with the previous approximation.

x(2) ≈ 2.468202
x(3) ≈ 2.468253
x(4) ≈ 2.468253

Step 7: Stop the iteration process when the desired accuracy of 6 decimal places is achieved.

Therefore, the solution to the equation x^3 - 6x^2 + 13x - 9 = 0 correct to 6 decimal places near x = 2 is approximately x = 2.468253.

Answer 3

To solve the equation x^3 - 6x^2 + 13x - 9 = 0 using the modified Newton's Raphson method, we first need to find the derivative of the function f(x). Let's call the function f(x).

Step 1: Find the derivative of the function f(x)
f'(x) = 3x^2 - 12x + 13

Step 2: Choose an initial guess, let's say x0 = 2.

Step 3: Use the formula for the next approximation, x1, given by:
x1 = x0 - (f(x0) / f'(x0))

Step 4: Repeat step 3 until the desired accuracy is achieved.

Now, let's go through the calculations step by step:

For x0 = 2:
f(2) = (2)^3 - 6(2)^2 + 13(2) - 9 = 8 - 24 + 26 - 9 = 1
f'(2) = 3(2)^2 - 12(2) + 13 = 12 - 24 + 13 = 1

Using the formula, x1 = x0 - (f(x0) / f'(x0)):
x1 = 2 - (1 / 1) = 2 - 1 = 1

For x1 = 1:
f(1) = (1)^3 - 6(1)^2 + 13(1) - 9 = 1 - 6 + 13 - 9 = -1
f'(1) = 3(1)^2 - 12(1) + 13 = 3 - 12 + 13 = 4

x2 = 1 - (-1 / 4) = 1 + 0.25 = 1.25

Continue this process until the desired accuracy is achieved. Iterate the formula using the newly calculated x-value until you find a value that satisfies the desired number of decimal places.

It's important to note that while the Newton's Raphson method is a powerful method for finding roots of equations, the modified version may still require some manual iterations to achieve the desired precision.