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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 4.22.1
Chapter 4, Problem 4.22.1

{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.


f(x) = x/6 - sec x on [0,8]

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Step 1: Understand Newton's Method. Newton's method is an iterative numerical technique used to find approximate roots of a real-valued function. The formula for Newton's method is: x_{n+1} = x_n - \(\frac{f(x_n)}{f'(x_n)}\).
Step 2: Perform a preliminary analysis of the function f(x) = \(\frac{x}{6}\) - \(\sec\)(x). Analyze the behavior of the function on the interval [0, 8] by considering the properties of the secant function and the linear term \(\frac{x}{6}\).
Step 3: Graph the function f(x) = \(\frac{x}{6}\) - \(\sec\)(x) on the interval [0, 8] to visually identify approximate locations of the roots. Look for points where the graph crosses the x-axis, as these indicate potential roots.
Step 4: Choose initial approximations for the roots based on the graph. These initial guesses should be close to where the graph crosses the x-axis. For example, if the graph suggests a root near x = a, use x_0 = a as the initial guess.
Step 5: Apply Newton's method iteratively using the initial approximations. For each initial guess x_0, compute the derivative f'(x) = \(\frac{1}{6}\) + \(\sec\)(x)\(\tan\)(x), and use the formula x_{n+1} = x_n - \(\frac{f(x_n)}{f'(x_n)}\) to find successive approximations until the values converge to a root.

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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 roots of a real-valued function. It starts with an initial guess and refines it using the formula x_{n+1} = x_n - f(x_n)/f'(x_n), where f' is the derivative of f. This method converges quickly if the initial guess is close to the actual root and the function behaves well.
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Preliminary Analysis

Preliminary analysis involves examining the function's behavior, such as identifying intervals where the function changes sign, which indicates the presence of roots. This can include evaluating the function at specific points, checking for continuity, and determining critical points to understand the function's overall shape and potential root locations.
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Graphing Functions

Graphing functions provides a visual representation of the function's behavior, helping to identify roots, intercepts, and critical points. By plotting the function over a specified interval, one can observe where the function crosses the x-axis, which indicates the roots. This visual approach aids in selecting effective initial approximations for iterative methods like Newton's.
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Related Practice
Textbook Question

{Use of Tech} Special curves The following classical curves have been studied by generations of mathematicians. Use analytical methods (including implicit differentiation) and a graphing utility to graph the curves. Include as much detail as possible.


y = 8/(x² + 4) (Witch of Agnesi)

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Textbook Question

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Textbook Question

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Textbook Question

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Textbook Question

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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>