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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![### 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F42913c3e-eecc-4769-9036-a58bbea00ee0%2F5db5d4ad-8cae-4d9f-ab46-efcbc23d32d5%2Fr71uini_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### 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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