Question

(a) Use bracketing and bisection to approximate a solution of sin² (x)=1−x² in the interval [0,1] to an absolute error of less than 0.1 .

(b) Now that you are within 0.1 of the correct answer, use two step of Newton's method to polish the estimate.

Answer 1

To approximate the solution of sin² (x)=1−x² in the interval [0,1] to an absolute error of less than 0.1, we can use bracketing and bisection method followed by two steps of Newton's method to polish the estimate.

Step-by-step explanation:

To approximate a solution of sin² (x)=1−x² in the interval [0,1] to an absolute error of less than 0.1 using bracketing and bisection method, we can start by dividing the interval [0,1] into smaller sub-intervals. Since the function is monotonically increasing in this interval, we can use bracketing to find two points, say a and b, in the interval such that sin²(a) > 1−a² and sin²(b) < 1−b². Once we have these points, we can use the bisection method to find the root of the equation within the sub-interval that satisfies the given absolute error condition.

Now that we are within 0.1 of the correct answer, we can use two steps of Newton's method to polish the estimate. Newton's method uses the derivative of the function to iteratively refine the estimate of the root. We start with the estimate obtained from bracketing and bisection, and use the formula xn+1 = xn - f(xn)/f'(xn) to update the estimate. We repeat this process two times to further refine the estimate.