Give the first 5 non-zero terms in the power series expansion and evaluate the derivative of the following functions: f(x) = ³x² +.... f(6) (0) =
Give the first 5 non-zero terms in the power series expansion and evaluate the derivative of the following functions: f(x) = ³x² +.... f(6) (0) =
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
![**Power Series Expansion and Derivatives**
In this exercise, we explore the power series expansion and derivative evaluations for given functions. Specifically, you will:
1. Determine the first five non-zero terms in the power series expansion.
2. Evaluate the specified derivatives at \( x = 0 \).
### Functions:
1. **Function \( f(x) = e^{3x^2} \):**
- **Power Series Expansion:**
\[
f(x) = e^{3x^2} = \Box + \cdots
\]
- **6th Derivative at \( x = 0 \):**
\[
f^{(6)}(0) = \Box
\]
2. **Function \( g(x) = \tan^{-1}(3x) \):**
- **Power Series Expansion:**
\[
g(x) = \tan^{-1}(3x) = \Box + \cdots
\]
- **7th Derivative at \( x = 0 \):**
\[
g^{(7)}(0) = \Box
\]
### Explanation:
- **Power Series Expansion:** Break down the given functions into their corresponding power series to find the first five non-zero terms. Generally, the power series of a function \( f(x) \) about \( x = 0 \) is given by:
\[
f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n
\]
- **Derivative Evaluation:** Calculate the precise derivative value at \( x = 0 \). For instance, if you have the 6th or 7th derivative of a function, determine its value at zero. These derivatives provide valuable insights into the behavior of the function at specific points.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe8fabc15-e5db-480c-94d9-e42b67b656a8%2F5c011df2-76ae-4152-96a4-6ebecd28eb7f%2F0ulxd3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Power Series Expansion and Derivatives**
In this exercise, we explore the power series expansion and derivative evaluations for given functions. Specifically, you will:
1. Determine the first five non-zero terms in the power series expansion.
2. Evaluate the specified derivatives at \( x = 0 \).
### Functions:
1. **Function \( f(x) = e^{3x^2} \):**
- **Power Series Expansion:**
\[
f(x) = e^{3x^2} = \Box + \cdots
\]
- **6th Derivative at \( x = 0 \):**
\[
f^{(6)}(0) = \Box
\]
2. **Function \( g(x) = \tan^{-1}(3x) \):**
- **Power Series Expansion:**
\[
g(x) = \tan^{-1}(3x) = \Box + \cdots
\]
- **7th Derivative at \( x = 0 \):**
\[
g^{(7)}(0) = \Box
\]
### Explanation:
- **Power Series Expansion:** Break down the given functions into their corresponding power series to find the first five non-zero terms. Generally, the power series of a function \( f(x) \) about \( x = 0 \) is given by:
\[
f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n
\]
- **Derivative Evaluation:** Calculate the precise derivative value at \( x = 0 \). For instance, if you have the 6th or 7th derivative of a function, determine its value at zero. These derivatives provide valuable insights into the behavior of the function at specific points.
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