4. Applications of Derivatives
Differentials
Problem 4.8.9
Textbook Question
{Use of Tech} Write the formula for Newton’s method and use the given initial approximation to compute the approximations x₁ and x₂.
f(x) = x² - 6; x₀ = 3
Verified step by step guidance1
Write down the formula for Newton's method, which is given by the equation: x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}.
Identify the function f(x) = x² - 6 and compute its derivative f'(x) = 2x.
Substitute the initial approximation x₀ = 3 into the function to find f(x₀): f(3) = 3² - 6.
Calculate f'(x₀) by substituting x₀ into the derivative: f'(3) = 2(3).
Use the values of f(x₀) and f'(x₀) in the Newton's method formula to find the first approximation x₁, and then repeat the process to find x₂.
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