Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problem 4.8.5

Chapter 4, Problem 4.8.5

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>

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Newton's method is an iterative process used to approximate the roots of a real-valued function. The formula for Newton's method is: x_{n+1} = x_n - \(\frac{f(x_n)}{f'(x_n)}\).

First, we need to find the derivative of the function f(x) = 2x^3 - 6x^2 + 4x. The derivative, f'(x), is calculated as follows: f'(x) = \(\frac{d}{dx}\)(2x^3 - 6x^2 + 4x) = 6x^2 - 12x + 4.

Next, substitute x_0 = 1.4 into the function f(x) and its derivative f'(x) to find f(1.4) and f'(1.4).

Calculate f(1.4) = 2(1.4)^3 - 6(1.4)^2 + 4(1.4) and f'(1.4) = 6(1.4)^2 - 12(1.4) + 4.

Finally, use the Newton's method formula to find x_1: x_1 = 1.4 - \(\frac{f(1.4)}{f'(1.4)}\). This will give you the next approximation of the root, x_1.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Newton's Method

Newton's Method is an iterative numerical technique used to approximate the roots of a real-valued function. It starts with an initial guess and uses the function's derivative to find successively better approximations. The formula for updating the guess is x₁ = x₀ - f(x₀)/f'(x₀), where f' is the derivative of f. This method is particularly effective when the initial guess is close to the actual root.

Derivative

The derivative of a function measures how the function's output changes as its input changes. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In the context of Newton's Method, the derivative is crucial for determining the slope of the tangent line at the current guess, which guides the next approximation towards the root.

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point and has the same slope as the curve at that point. In Newton's Method, the tangent line at the point (x₀, f(x₀)) is used to find the next approximation x₁. The intersection of this tangent line with the x-axis provides a new estimate for the root, illustrating how the method converges to the actual solution.

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