Use the binomial theorem to estimate the number, computing the smallest number of terms, n, necessary to obtain an estimate accurate to an error of at most 1 1,000 (15)1/4 using (16 - x)1/4 n 3=
Use the binomial theorem to estimate the number, computing the smallest number of terms, n, necessary to obtain an estimate accurate to an error of at most 1 1,000 (15)1/4 using (16 - x)1/4 n 3=
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
![**Problem Statement:**
Use the binomial theorem to estimate the number, computing the smallest number of terms, \( n \), necessary to obtain an estimate accurate to an error of at most \(\frac{1}{1000}\).
**Expression:**
\[
(15)^{1/4} \text{ using } (16 - x)^{1/4}
\]
**Find:**
\[
n = \_\_\_
\]
**Explanation for Educational Context:**
The task involves using the binomial theorem to approximate \((15)^{1/4}\) by expanding \((16 - x)^{1/4}\) and determining the smallest number of terms \( n \) required to achieve an accuracy where the error is no larger than \(\frac{1}{1000}\).
### Steps to Solve:
1. **Identify the Expansion**: Recognize that the expression \((16-x)^{1/4}\) can be expanded using the binomial theorem.
2. **Binomial Series Expansion**: Apply the binomial theorem to expand the expression up to the necessary number of terms.
3. **Determine \( x \)**: Set \( x \) such that \( 16 - x = 15 \), which simplifies the expression to \((15)^{1/4}\).
4. **Calculate Error Term**: Compute each term of the expansion and determine the point at which adding further terms results in an error less than \(\frac{1}{1000}\).
5. **Count Terms \( n \)**: Count the terms up to the point where the desired accuracy is achieved.
By following these steps, students can practice the application of the binomial theorem for approximation and understand error calculation in mathematical series.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fde5d5abe-d173-4c84-a521-ce5326c05407%2F9a4e47ff-3b6f-47da-a490-e3f6bda27ac8%2Fl8jwe76_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Use the binomial theorem to estimate the number, computing the smallest number of terms, \( n \), necessary to obtain an estimate accurate to an error of at most \(\frac{1}{1000}\).
**Expression:**
\[
(15)^{1/4} \text{ using } (16 - x)^{1/4}
\]
**Find:**
\[
n = \_\_\_
\]
**Explanation for Educational Context:**
The task involves using the binomial theorem to approximate \((15)^{1/4}\) by expanding \((16 - x)^{1/4}\) and determining the smallest number of terms \( n \) required to achieve an accuracy where the error is no larger than \(\frac{1}{1000}\).
### Steps to Solve:
1. **Identify the Expansion**: Recognize that the expression \((16-x)^{1/4}\) can be expanded using the binomial theorem.
2. **Binomial Series Expansion**: Apply the binomial theorem to expand the expression up to the necessary number of terms.
3. **Determine \( x \)**: Set \( x \) such that \( 16 - x = 15 \), which simplifies the expression to \((15)^{1/4}\).
4. **Calculate Error Term**: Compute each term of the expansion and determine the point at which adding further terms results in an error less than \(\frac{1}{1000}\).
5. **Count Terms \( n \)**: Count the terms up to the point where the desired accuracy is achieved.
By following these steps, students can practice the application of the binomial theorem for approximation and understand error calculation in mathematical series.
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 with 2 images
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