Domanda del libro di testo
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) = 1/x³
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Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 4, Problema 4.20.1
{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) = cos x - x/7
Guida verificata passo dopo passo1
Start by performing a preliminary analysis of the function \( f(x) = \cos x - \frac{x}{7} \). Consider the behavior of \( \cos x \) and \( -\frac{x}{7} \) to understand where the function might cross the x-axis.
Graph the function \( f(x) = \cos x - \frac{x}{7} \) to visually identify approximate locations of the roots. Look for points where the graph intersects the x-axis.
Choose initial approximations for the roots based on the graph. These are the x-values where the function appears to cross the x-axis.
Apply Newton's method, which uses the formula \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \), where \( f'(x) = -\sin x - \frac{1}{7} \). Calculate \( f'(x) \) for the function.
Iteratively apply Newton's method using the initial approximations. For each iteration, compute the next approximation until the values converge to a stable root. Repeat this process for each initial approximation to find all roots.
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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 and the function behaves well.
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4:26
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, checking for sign changes, and analyzing critical points. Understanding the function's continuity and differentiability is crucial for effective root-finding.
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06:29Derivatives Applied To Velocity
Graphing Functions
Graphing functions provides a visual representation of their behavior, helping to identify where roots may lie. By plotting the function, one can observe intersections with the x-axis, which indicate potential roots. This visual approach aids in selecting appropriate initial approximations for methods like Newton's Method.
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5:53Graph of Sine and Cosine Function
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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.
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