Question

Use Newton's method to approximate a root of the equation cos(x² + 2) = x³ as follows.

Let x₁ = 1 be the initial approximation. The second approximation x₂ is_____

Answer 1

To approximate a root of the equation cos(x² + 2) = x³ using Newton's method, we start with an initial approximation x₁ = 1. The second approximation x₂ is approximately 1.1908.

Step-by-step explanation:

To approximate a root of the equation cos(x² + 2) = x³ using Newton's method, we start with an initial approximation x₁ = 1. We then use the formula x₂ = x₁ - f(x₁) / f'(x₁), where f(x) = cos(x² + 2) - x³ and f'(x) is the derivative of f(x). Evaluating f(x) and f'(x) at x = x₁ = 1, we find that f(1) = cos(3) - 1 = -0.9899 and f'(1) = -2sin(3) - 3. Plugging these values into the formula, we get x₂ = 1 - (-0.9899) / (-2sin(3) - 3). Solving this equation, x₂ is approximately 1.1908.