### Computation of Taylor Series#### Problem StatementCompute the Taylor series of the function around $ x = 1 $.#### Function\[ f(x) = \frac{x}{4x - 2x^2 - 1} \]#### Taylor Series Representation\[ f(x) = \sum_{n=0}^{\infty} \left( \text{[Expression to be determined]} \right) \]### ExplanationThe task is to determine the Taylor series expansion for the given function $ f(x) $ centered at $ x = 1 $. The Taylor series is represented as an infinite sum of terms calculated from the values of the function's derivatives at a single point.### Steps for Solution1. **Determine the point of expansion ($ x = 1 $).**2. **Calculate the derivatives of $ f(x) $ at $ x = 1 $.**3. **Substitute these derivatives into the Taylor series formula:** \[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(1)}{n!} (x - 1)^n \]4. **Fill in the expression in the series representation with the calculated coefficients.**This provides the structure to follow for computing the Taylor series. The placeholder in the series expression will be populated with specific coefficients derived from the function's derivatives evaluated at $ x = 1 $.### NoteAdditional materials may be provided to help understand Taylor series expansion and its applications, including video tutorials and sample problems.

the given function is f(x)=x4x-2x2-1. find the first,second and third order derivative of the…

Answered: Compute the Taylor series of the…

Math

Calculus

Compute the Taylor series of the function around x 1. (x) 4x - 2x² - 1

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

### Computation of Taylor Series

#### Problem Statement
Compute the Taylor series of the function around $ x = 1 $.

#### Function
\[ f(x) = \frac{x}{4x - 2x^2 - 1} \]

#### Taylor Series Representation
\[ f(x) = \sum_{n=0}^{\infty} \left( \text{[Expression to be determined]} \right) \]

### Explanation
The task is to determine the Taylor series expansion for the given function $ f(x) $ centered at $ x = 1 $. The Taylor series is represented as an infinite sum of terms calculated from the values of the function's derivatives at a single point.

### Steps for Solution
1. **Determine the point of expansion ($ x = 1 $).**
2. **Calculate the derivatives of $ f(x) $ at $ x = 1 $.**
3. **Substitute these derivatives into the Taylor series formula:**
\[
f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(1)}{n!} (x - 1)^n
\]
4. **Fill in the expression in the series representation with the calculated coefficients.**

This provides the structure to follow for computing the Taylor series. The placeholder in the series expression will be populated with specific coefficients derived from the function's derivatives evaluated at $ x = 1 $.

### Note
Additional materials may be provided to help understand Taylor series expansion and its applications, including video tutorials and sample problems.

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