Question

Write the formula for Newton's method and use the given initial approximation to compute the approximations x₁ and x₂

f(x)=x² −4x−5,x₀=3

Answer 1

The formula for Newton's method is x_n+1 = x_n - f(x_n)/f'(x_n). To compute the approximations x₁ and x₂ for the given function f(x) = x² - 4x - 5 and initial approximation x₀ = 3, substitute the values into the formula and calculate x₁ and x₂.

Step-by-step explanation:

The formula for Newton's method is:

xn+1 = xn - f(xn)⁄f'(xn)

Now, let's use the given initial approximation x₀ = 3 to compute the approximations x₁ and x₂.

Given f(x) = x² - 4x - 5 and x₀ = 3:

1. Calculate f(x₀) = x₀² - 4x₀ - 5
2. Calculate f'(x₀) = 2x₀ - 4
3. Substitute the values into the formula: x₁ = x₀ - f(x₀)⁄f'(x₀) and calculate x₁
4. Repeat the process to compute x₂ using x₁ as the initial approximation.