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
![**Title: Taylor Series Expansion for Exponential Functions**
**Objective:**
Learn how to write the Taylor series for an exponential function and find the first five coefficients.
**Problem Statement:**
Write the Taylor series for \( f(x) = e^x \) about \( x = -3 \).
**Expression:**
The series is expressed as:
\[ \sum_{n=0}^{\infty} c_n (x+3)^n \]
**Task:**
Find the first five coefficients \( c_0, c_1, c_2, c_3, \) and \( c_4 \).
**Steps:**
1. **Identify the Function:**
- \( f(x) = e^x \)
2. **Center of Expansion:**
- At \( x = -3 \)
3. **Determine Coefficients:**
- Apply derivatives and evaluate at the center point to compute each coefficient.
4. **Table Layout:**
- The image contains placeholders for the coefficients:
- \( c_0 = \)
- \( c_1 = \)
- \( c_2 = \)
- \( c_3 = \)
- \( c_4 = \)
**Conclusion:**
Calculate \( c_0, c_1, c_2, c_3, \) and \( c_4 \) to complete the Taylor series representation for \( e^x \) about \( x = -3 \). This provides a polynomial approximation of the function in the vicinity of \( x = -3 \).
**Hint:**
Remember, the general formula for the nth coefficient of a Taylor series centered at \( a \) is:
\[ c_n = \frac{f^{(n)}(a)}{n!} \]
Use derivatives of \( e^x \) appropriately evaluated at \( x = -3 \).
**Note:**
- This exercise helps in understanding the process of approximating complex functions with polynomials, a fundamental concept in calculus and analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F09b3997c-2096-4573-b74b-055c69181c0d%2F28df6f57-5b65-42e0-be04-256b50dcd2b5%2Ffsbvz49_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Taylor Series Expansion for Exponential Functions**
**Objective:**
Learn how to write the Taylor series for an exponential function and find the first five coefficients.
**Problem Statement:**
Write the Taylor series for \( f(x) = e^x \) about \( x = -3 \).
**Expression:**
The series is expressed as:
\[ \sum_{n=0}^{\infty} c_n (x+3)^n \]
**Task:**
Find the first five coefficients \( c_0, c_1, c_2, c_3, \) and \( c_4 \).
**Steps:**
1. **Identify the Function:**
- \( f(x) = e^x \)
2. **Center of Expansion:**
- At \( x = -3 \)
3. **Determine Coefficients:**
- Apply derivatives and evaluate at the center point to compute each coefficient.
4. **Table Layout:**
- The image contains placeholders for the coefficients:
- \( c_0 = \)
- \( c_1 = \)
- \( c_2 = \)
- \( c_3 = \)
- \( c_4 = \)
**Conclusion:**
Calculate \( c_0, c_1, c_2, c_3, \) and \( c_4 \) to complete the Taylor series representation for \( e^x \) about \( x = -3 \). This provides a polynomial approximation of the function in the vicinity of \( x = -3 \).
**Hint:**
Remember, the general formula for the nth coefficient of a Taylor series centered at \( a \) is:
\[ c_n = \frac{f^{(n)}(a)}{n!} \]
Use derivatives of \( e^x \) appropriately evaluated at \( x = -3 \).
**Note:**
- This exercise helps in understanding the process of approximating complex functions with polynomials, a fundamental concept in calculus and analysis.
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