Question

Use Newton's method to approximate the indicated root of the equation correct to six decimal places.

The root of x⁴ -2x³ +4x²-7 =0 in the interval [1,2]

Answer 1

To approximate the root of the equation x⁴ -2x³ +4x²-7 =0 using Newton's method, follow these steps: Choose an initial guess value, calculate the derivative of the function, and use the Newton's method formula for finding subsequent approximations until the desired level of accuracy is reached.

Step-by-step explanation:

To approximate the root of the equation x⁴ -2x³ +4x²-7 =0 using Newton's method, follow these steps:

1. Choose an initial guess value for the root within the given interval [1,2].
2. Calculate the derivative of the function, which is 4x³-6x²+8x.
3. Use the formula: xn+1 = xn - f(xn)/f'(xn) to find each subsequent approximation of the root, where xn+1 is the next approximation, xn is the current approximation, f(xn) is the value of the function at xn, and f'(xn) is the value of the derivative at xn.
4. Repeat step 3 until you reach the desired level of accuracy.