4. Applications of Derivatives
Differentials
Problem 4.8.3
Textbook Question
A graph of ƒ and the lines tangent to ƒ at x = 1, 2 and 3 are given. If x₀ = 3, find the values of x₁, x₂, and x₃, that are obtained by applying Newton’s method. <IMAGE>
Verified step by step guidance1
Identify the function ƒ(x) and its derivative ƒ'(x) from the graph provided, particularly at the points x = 1, 2, and 3.
Apply Newton's method formula: x_{n+1} = x_n - rac{f(x_n)}{f'(x_n)} starting with x₀ = 3.
Calculate f(3) and f'(3) using the values obtained from the graph.
Substitute f(3) and f'(3) into the Newton's method formula to find x₁.
Repeat the process using x₁ to find x₂ and then x₂ to find x₃, ensuring to calculate f(x_n) and f'(x_n) at each step.
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