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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.8.9
Chapter 4, Problem 4.8.9

{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

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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-f(x_n)f'(x_n).
First, we need to find the derivative of the function f(x)=x2-6. The derivative is f'(x)=2x.
Using the initial approximation x_0=3, calculate f(x_0) and f'(x_0).
Substitute these values into the Newton's method formula to find x_1: x_1=x_0-f(x_0)f'(x_0).
Repeat the process using x_1 to find x_2: x_2=x_1-f(x_1)f'(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 find approximate solutions to equations. It starts with an initial guess and refines it using the formula x₁ = x₀ - f(x₀)/f'(x₀), where f'(x) is the derivative of the function. This method is particularly effective for functions that are continuous and differentiable near the root.
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Derivative

The derivative of a function measures how the function's output changes as its input changes. It is a fundamental concept in calculus that provides the slope of the tangent line to the function at any given point. In the context of Newton's Method, the derivative is crucial for determining the direction and magnitude of the adjustments made to the initial approximation.
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Initial Approximation

An initial approximation is the starting value used in iterative methods like Newton's Method. The choice of this value can significantly affect the convergence of the method to the actual root. A good initial approximation should be close to the actual root to ensure that the iterations lead to a successful and rapid convergence.
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Related Practice
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