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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 4.8.21
Chapter 4, Problem 4.8.21

{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) = cos 2x - x² + 2x

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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) = cos(2x) - x^2 + 2x. Analyze the behavior of the function by considering its components: cos(2x), -x^2, and 2x. Consider the range and periodicity of cos(2x) and the parabolic nature of -x^2 + 2x.
Step 3: Graph the function f(x) = cos(2x) - x^2 + 2x to visually identify potential roots. Look for points where the graph crosses the x-axis, as these are the approximate locations of the 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 roots near x = a and x = b, use these as starting points.
Step 5: Apply Newton's method iteratively for each initial approximation. Calculate the derivative f'(x) = -2sin(2x) - 2x + 2. Use the formula x_{n+1} = x_n - \(\frac{f(x_n)}{f'(x_n)}\) to update the approximation until the change is sufficiently small, indicating convergence 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, making it effective for functions that are differentiable.
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Preliminary Analysis

Preliminary analysis involves examining the function's behavior to identify potential roots before applying numerical methods. This can include evaluating the function at various points, analyzing its continuity, and determining intervals where the function changes sign, which indicates the presence of roots according to the Intermediate Value Theorem.
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Graphing Functions

Graphing functions provides a visual representation of their behavior, helping to identify roots, intercepts, and critical points. By plotting the function, one can observe where it crosses the x-axis, indicating the roots, and assess the function's overall shape, which aids in selecting appropriate initial approximations for methods like Newton's.
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Related Practice
Textbook Question

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


lim_x→ 0 (eˣ - sin x - 1) / (x⁴ + 8x³ + 12x²)

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Textbook Question

Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.


f(x) = tan x

Textbook Question

{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) = e⁻ˣ - ((x + 4)/5)

Textbook Question

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.


f(x) = cos² x on [-π,π]

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Textbook Question

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.


∫ (3x ¹⸍³ + 4x ⁻¹⸍³ + 6) dx

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.


x⁴ - x² + y² = 0 (Figure-8 curve)