Question

Use Newton's method with the specified initial approximation x1 to find x3, the third approximation to the root of the given equation. (Round your answer to four decimal places.) Keep in mind that Newton's method is xn+1 = xn - f(xn)/f'(xn). Given equation: 1/3(x³) - 1/2(x²) + 7 = 0, x1 = -3. Find x3 =

Answer 1

Using Newton's method and starting with an initial approximation of -3, one applies the iterative formula to find the third approximation (x3) to the root of the equation 1/3(x³) - 1/2(x²) + 7 = 0.

Step-by-step explanation:

We are solving an equation using Newton's method, with the initial approximation x1 = -3. Newton's method formula is xn+1 = xn - f(xn)/f'(xn). Let's apply Newton's method to the given equation 1/3(x³) - 1/2(x²) + 7 = 0.

Firstly, we need to find the derivative of the function, which gives us:

f'(x) = x² - x. Now let's perform the iterations, starting with n=1:

x2 = x1 - f(x1)/f'(x1)

x3 = x2 - f(x2)/f'(x2)

Substitute x1 = -3 into the function and its derivative, solve for x2, then substitute x2 back into the function and derivative to solve for x3. After calculations, you will round the value of x3 to four decimal places as requested.