Final answer:
Newton's Method is used to calculate two iterations of approximating a zero of the function f(x) = x^2 - 2, starting with an initial guess of x₁ = 1.4. The first iteration yields x₁ = 1.414286 and the second one gives x₂ = 1.414287.
Step-by-step explanation:
The student is asking how to approximate a zero of a function using Newton's Method with a given initial guess. Newton's Method is an iterative numerical method for finding successively better approximations to the roots (or zeroes) of a real-valued function. The initial guess provided is x₁ = 1.4 for the function f(x) = x² - 2.
To apply Newton's Method, we need the function f(x) and its derivative f'(x). The derivative of f(x) is f'(x) = 2x. We then use the Newton's iteration formula:
x₁ = x₀ - f(x₀) / f'(x₀)
For the first iteration with x₀ = 1.4:
- f(1.4) = (1.4)² - 2 = 1.96 - 2 = -0.04
- f'(1.4) = 2(1.4) = 2.8
- x₁ = 1.4 - (-0.04/2.8) = 1.4 + 0.014286 = 1.414286
This gives us our first approximation x₁ = 1.414286. For the second iteration, we use x₁ as our new x₀:
- f(1.414286) = (1.414286)² - 2 ≈ -0.000004
- f'(1.414286) = 2(1.414286) ≈ 2.82857
- x₂ = 1.414286 - (-0.000004/2.82857) ≈ 1.414286 + 0.0000014 = 1.414287
Thus, the second approximation x₂ is approximately 1.414287. This is the Newton's Method process of iteratively getting closer to the actual zero of the function.