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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![### Problem Statement:
Differentiate the function defined by the following infinite series:
\[ y = \sum_{k=0}^{\infty} (k + 4)x^{k+8}. \]
### Solution:
Compute the derivative \( y' \) of the given function. The general form of the series should be maintained in the derivative.
\[ y' = \sum_{k=0}^{\infty} \frac{d}{dx} \left( (k+4)x^{k+8} \right). \]
### Steps to Differentiate:
1. **Identify the term to differentiate**:
The term in the series to differentiate is \( (k+4)x^{k+8} \).
2. **Apply the power rule for differentiation**:
\[\frac{d}{dx}(x^n) = n \cdot x^{n-1}.\]
3. **Differentiate the term in the context of the series**:
\[ \frac{d}{dx}\left((k+4)x^{k+8}\right) = (k+4) \cdot (k+8)x^{k+8-1} = (k+4) \cdot (k+8)x^{k+7}. \]
4. **Express the differentiated term back into the series form**:
\[ y' = \sum_{k=0}^{\infty} (k + 4)(k + 8)x^{k+7}. \]
Therefore, the derivative of the given function is:
\[ y' = \sum_{k=0}^{\infty} (k + 4)(k + 8)x^{k+7}. \]
### Explanation of Symbols:
- \(\sum\) (Sigma) represents the summation over an infinite series starting from \( k = 0 \).
- \( k \) is the index of summation.
- \( x \) is the variable with respect to which differentiation is performed.
- \( k + 4 \) and \( k + 8 \) are constants with respect to \( x \) within each term of the series.
This completes the differentiation of the infinite series \( y \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc5134d69-32d8-4659-bfc6-df6862919637%2F14f8c75f-3894-4188-ac18-aeebdf11455f%2F12fwib_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
Differentiate the function defined by the following infinite series:
\[ y = \sum_{k=0}^{\infty} (k + 4)x^{k+8}. \]
### Solution:
Compute the derivative \( y' \) of the given function. The general form of the series should be maintained in the derivative.
\[ y' = \sum_{k=0}^{\infty} \frac{d}{dx} \left( (k+4)x^{k+8} \right). \]
### Steps to Differentiate:
1. **Identify the term to differentiate**:
The term in the series to differentiate is \( (k+4)x^{k+8} \).
2. **Apply the power rule for differentiation**:
\[\frac{d}{dx}(x^n) = n \cdot x^{n-1}.\]
3. **Differentiate the term in the context of the series**:
\[ \frac{d}{dx}\left((k+4)x^{k+8}\right) = (k+4) \cdot (k+8)x^{k+8-1} = (k+4) \cdot (k+8)x^{k+7}. \]
4. **Express the differentiated term back into the series form**:
\[ y' = \sum_{k=0}^{\infty} (k + 4)(k + 8)x^{k+7}. \]
Therefore, the derivative of the given function is:
\[ y' = \sum_{k=0}^{\infty} (k + 4)(k + 8)x^{k+7}. \]
### Explanation of Symbols:
- \(\sum\) (Sigma) represents the summation over an infinite series starting from \( k = 0 \).
- \( k \) is the index of summation.
- \( x \) is the variable with respect to which differentiation is performed.
- \( k + 4 \) and \( k + 8 \) are constants with respect to \( x \) within each term of the series.
This completes the differentiation of the infinite series \( y \).
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