Final answer:
Newton's method is used to find the root of a function through iteration. Given the function f(x) = x² - 2 and initial value x₀ = 2, we can calculate x₁, x₂, and x₃ using the formula xₙ = xₙ₋₁ - f(xₙ₋₁) / f'(xₙ₋₁). The values are x₁ = 1.5, x₂ ≈ 1.4167, and x₃ ≈ 1.4142.
Step-by-step explanation:
Newton's Method is an iterative method used to find the root of a function. To apply Newton's Method, we start with an initial value, x₀, and use the formula xₙ = xₙ₋₁ - f(xₙ₋₁) / f'(xₙ₋₁), where f(x) is the function and f'(x) is its derivative. In this case, we have the function f(x) = x² - 2 and initial value x₀ = 2.
Step 1: Calculate f(x₀) = f(2) = 2² - 2 = 2.
Step 2: Calculate f'(x₀) by finding the derivative of f(x), which is f'(x) = 2x. Therefore, f'(2) = 2 * 2 = 4.
Step 3: Calculate x₁ using the formula x₁ = x₀ - f(x₀) / f'(x₀) = 2 - 2 / 4 = 1.5.
Step 4: Repeat steps 1-3 to find x₂ and x₃:
x₂ = x₁ - f(x₁) / f'(x₁) = 1.5 - (1.5² - 2) / (2 * 1.5) = 1.4167.
x₃ = x₂ - f(x₂) / f'(x₂) = 1.4167 - (1.4167² - 2) / (2 * 1.4167) = 1.4142.