Question

Calculate two iterations of Newton's Method to approximate a zero of the function using the given initial guess: f(x) = cos(x), x₁ = 1.9.

Answer 1

To approximate a zero of the function f(x) = cos(x) using Newton's Method, start with an initial guess x₁ = 1.9. Two iterations of Newton's Method will give an approximation of x = 1.717 as the zero of the function.

Step-by-step explanation:

To approximate a zero of the function f(x) = cos(x) using Newton's Method, we start with an initial guess x₁ = 1.9. Newton's Method uses the formula x(i+1) = x(i) - f(x(i))/f'(x(i)), where f'(x) is the derivative of f(x). In this case, f'(x) = -sin(x).

1. Substitute x₀ = 1.9 into f(x) and f'(x) to get f(1.9) = cos(1.9) = -0.151 and f'(1.9) = -sin(1.9) = -0.982.
2. Apply the formula: x₁ = x₀ - f(x₀)/f'(x₀) = 1.9 - (-0.151)/(-0.982) = 1.750.
3. Repeat the process using x₁ as the new guess:
4. Substitute x₁ = 1.750 into f(x) and f'(x) to get f(1.750) = cos(1.750) = -0.033 and f'(1.750) = -sin(1.750) = -0.999.
5. Apply the formula: x₂ = x₁ - f(x₁)/f'(x₁) = 1.750 - (-0.033)/(-0.999) = 1.717.

Therefore, two iterations of Newton's Method give us an approximation of x = 1.717 as a zero of the function f(x) = cos(x) using the initial guess x₁ = 1.9.