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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblema 4.8.21
Capitolo 4, Problema 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

Guida verificata passo dopo passo
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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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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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Evaluating Composed Functions

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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Graph of Sine and Cosine Function
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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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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

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{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)

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


∫ (3x ¹⸍³ + 4x ⁻¹⸍³ + 6) dx

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


lim_x→ 0 (eˣ - 1) / (2x + 5)

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