Textbook Question
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→0 x csc x
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Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
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]
Verified step by step guidance1
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
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