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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 4.8.19
Chapter 4, Problem 4.8.19

{Use of Tech} Finding roots with Newton’s method For the given function f and initial approximation x₀, use Newton’s method to approximate a root of f. Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in Example 1.


f(x) = cos⁻¹ x - x; x₀ = 0.75

Verified step by step guidance
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Step 1: Understand Newton's Method. Newton's method is an iterative process used to find successively better approximations to the roots (or zeroes) of a real-valued function. The formula for Newton's method is: x=x-f(x)f'(x), where f'(x) is the derivative of f(x).
Step 2: Calculate the derivative of the function. Given f(x)=cos-1(x)-x, find f'(x). The derivative of cos-1(x) is -11-x2, and the derivative of x is 1. Therefore, f'(x)=-11-x2-1.
Step 3: Apply Newton's method using the initial approximation x=0.75. Substitute x=0.75 into the Newton's formula: x=0.75-f(0.75)f'(0.75). Calculate f(0.75) and f'(0.75).
Step 4: Compute the next approximation. Use the values from Step 3 to find the next approximation x=x-f(x)f'(x). Repeat this process iteratively, updating x each time.
Step 5: Continue iterations until two successive approximations agree to five decimal places. Create a table to track each iteration, showing the values of x, f(x), and f'(x) for each step. Stop when the difference between successive approximations is less than 0.00001.

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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 x₁ = x₀ - f(x₀)/f'(x₀), where f' is the derivative of f. This process continues until the difference between successive approximations is sufficiently small, indicating convergence to a root.
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Convergence Criteria

In numerical methods, convergence criteria determine when to stop the iterative process. For Newton's Method, this often involves checking if the absolute difference between two successive approximations is less than a specified tolerance level, such as 0.00001 for five decimal places. This ensures that the approximations are accurate enough for practical purposes.
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Derivatives

Function and Derivative Evaluation

To apply Newton's Method, one must evaluate both the function f and its derivative f' at each iteration. For the given function f(x) = cos⁻¹ x - x, the derivative f'(x) = -1/√(1-x²) - 1 must be computed. Accurate evaluation of these functions is crucial for the method's success, as errors in these calculations can lead to incorrect approximations of the root.
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Related Practice
Textbook Question

Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.


ƒ(x) = 3x² - 4x + 2

Textbook Question

Particular antiderivatives For the following functions f, find the antiderivative F that satisfies the given condition.


f(x) = 8x³ + sin x; F(0) = 2

Textbook Question

{Use of Tech} Absolute maxima and minima


a. Find the critical points of f on the given interval.

b. Determine the absolute extreme values of f on the given interval.

c. Use a graphing utility to confirm your conclusions.


f(x) = 2ᶻ sin x on [-2,6]

Textbook Question

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→ ∞ (e³ˣ ) / (3e³ˣ + 5)

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Textbook Question

Use the following graphs to identify the points (if any) on the interval [a, b] at which the function has an absolute maximum or an absolute minimum value <IMAGE>

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Textbook Question

Sketching curves Sketch a graph of a function f that is continuous on (-∞,∞) and has the following properties.


f'(x) < 0 and f"(x) > 0 on (-∞,0); f'(x) > 0 and f"(x) < 0 on (0,∞)