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...
Related questions
Question
![**Finding the Maclaurin Series for the Function**
We are tasked with finding the Maclaurin series for the function \( f(x) = x^8 \sin(x) \).
### Function Representation:
The function \( f(x) \) is given by:
\[ f(x) = x^8 \sin(x) \]
### Maclaurin Series Expression:
The Maclaurin series representation of a function is expressed using the summation notation:
\[ f(x) = \sum_{n = 0}^{\infty} \text{[expression]} \]
The summation index \( n \) ranges from 0 to infinity (\(\infty\)), indicating that the series continues indefinitely. A blank space is provided suggesting where the series expression is inserted after finding the series expansion.
### Explanation:
The Maclaurin series is essentially a Taylor series expansion of a function about the point \( x = 0 \), focusing on polynomial representations of the given function. It involves calculating derivatives of \( f(x) \) at 0 and using them in the series expansion formula:
\[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n \]
In our scenario, to solve for the series, we need to expand \( \sin(x) \) using its own Maclaurin series:
\[ \sin(x) = \sum_{k = 0}^{\infty} \frac{(-1)^k}{(2k+1)!} x^{2k+1} \]
By substituting this into \( f(x) = x^8 \sin(x) \), we obtain the Maclaurin series for the entire function. The resultant series can be represented in the summation notation provided in the diagram.
This approach is vital for approximations and is a foundational concept in calculus and mathematical analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15d21cd-3875-4270-92b4-6749e1bc7456%2F3b1fbf99-ed3b-4280-b90d-f09ba0c012b1%2F4ucxtrr_processed.png&w=3840&q=75)
Transcribed Image Text:**Finding the Maclaurin Series for the Function**
We are tasked with finding the Maclaurin series for the function \( f(x) = x^8 \sin(x) \).
### Function Representation:
The function \( f(x) \) is given by:
\[ f(x) = x^8 \sin(x) \]
### Maclaurin Series Expression:
The Maclaurin series representation of a function is expressed using the summation notation:
\[ f(x) = \sum_{n = 0}^{\infty} \text{[expression]} \]
The summation index \( n \) ranges from 0 to infinity (\(\infty\)), indicating that the series continues indefinitely. A blank space is provided suggesting where the series expression is inserted after finding the series expansion.
### Explanation:
The Maclaurin series is essentially a Taylor series expansion of a function about the point \( x = 0 \), focusing on polynomial representations of the given function. It involves calculating derivatives of \( f(x) \) at 0 and using them in the series expansion formula:
\[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n \]
In our scenario, to solve for the series, we need to expand \( \sin(x) \) using its own Maclaurin series:
\[ \sin(x) = \sum_{k = 0}^{\infty} \frac{(-1)^k}{(2k+1)!} x^{2k+1} \]
By substituting this into \( f(x) = x^8 \sin(x) \), we obtain the Maclaurin series for the entire function. The resultant series can be represented in the summation notation provided in the diagram.
This approach is vital for approximations and is a foundational concept in calculus and mathematical analysis.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning