Question

Use Newton’s root finding method to determine the zeros for the following functions, from the starting points x0:

a) f (x) = sin(x) + x² cos(x) − x² − x, x0 = −1, and x0 = 0

Answer 1

To find the zeros of the given function using Newton's root finding method, we substitute the initial conditions into the formula and iterate until convergence.

Step-by-step explanation:

To determine the zeros of the function f(x) = sin(x) + x² cos(x) − x² − x using Newton's root finding method, we will use the initial conditions x0 = -1 and x0 = 0. We will iterate through the Newton's method formula until we converge to a root.

1. For x0 = -1, we substitute it into the equation f(x) and calculate f'(x) which is the derivative of f(x).
2. We then apply the Newton's method formula: x1 = x0 - f(x0)/f'(x0).
3. We repeat step 2 until we reach convergence, which means that the difference between subsequent approximations is below a certain threshold.
4. Following the same steps, we can find the zeros of f(x) for x0 = 0 as well.

Answer 2

To find the zeros of the function using Newton's root finding method, we need to apply the formula: x = x0 - f(x0)/f'(x0). For each starting point, calculate f(x0) and f'(x0), and use the formula to iteratively find the zeros of the function.

Step-by-step explanation:

To find the zeros of the function using Newton's root finding method, we need to apply the formula: x = x0 - f(x0)/f'(x0)

1. For the first starting point, x0 = -1, we have f(-1) = sin(-1) + (-1)²cos(-1) - (-1)² - (-1) = -0.1586
2. For the second starting point, x0 = 0, we have f(0) = sin(0) + (0)²cos(0) - (0)² - (0) = 0
3. Next, we need to calculate the derivative of the function f'(x) = cos(x) + 2xcos(x) - 2x - 1
4. Using the formula x = x0 - f(x0)/f'(x0), we can iteratively find the zeros of the function for both starting points.