4. Applications of Derivatives
Differentials
Problem 4.8.5
Textbook Question
Let ƒ(x) = 2x³ - 6x² + 4x. Use Newton’s method to find x₁ given that x₀ = 1.4. Use the graph of f (see figure) and an appropriate tangent line to illustrate how x₁ is obtained from x₀ . <IMAGE>
Verified step by step guidance1
First, identify the function f(x) = 2x³ - 6x² + 4x and compute its derivative f'(x) to use in Newton's method.
Calculate f(1.4) by substituting x₀ = 1.4 into the function f(x).
Next, calculate f'(1.4) by substituting x₀ = 1.4 into the derivative f'(x).
Apply Newton's method formula: x₁ = x₀ - f(x₀) / f'(x₀) using the values obtained from the previous steps.
Finally, to illustrate the process, sketch the graph of f(x) and draw the tangent line at the point (1.4, f(1.4)), showing how x₁ is the x-intercept of this tangent line.
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