Question

Use Newton's method to approximate a root of the equation 4 sin (x)=x as follows. Let x_1=2 be the initial approximation. The second approximation x_2 is:

Answer 1

To approximate a root of the equation 4sin(x) = x using Newton's method, the second approximation x2 is approximately 1.89549.

Step-by-step explanation:

The given equation is 4sin(x) = x. To approximate a root of the equation using Newton's method, we start with an initial approximation x1 = 2. Next, we use the formula:

xn+1 = xn - (f(xn) / f'(xn))

where f(x) = 4sin(x) - x and f'(x) is the derivative of f(x). Evaluating f(2) and f'(2) using calculus, we can calculate x2 as the second approximation.

In this case, x2 ≈ 2 - ((4sin(2) - 2)/(4cos(2) - 1)) ≈ 1.89549.