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: Understanding Taylor Series at a Point**
**Objective:**
Find the Taylor series generated by the function \( f \) at \( x = a \).
**Given Function:**
\[ f(x) = 4x^4 + x^2 + 4 \]
**Point of Expansion:**
\[ a = -3 \]
**Task:**
Determine the Taylor series generated by \( f \) at \( a = -3 \).
**Solution:**
The Taylor series expansion of a function \( f(x) \) around a point \( a \) is given by:
\[
f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots
\]
**Steps to Solve:**
1. Calculate the derivatives \( f'(x) \), \( f''(x) \), etc., as needed.
2. Evaluate each derivative at \( x = -3 \).
3. Plug values into the Taylor series formula.
---
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Transcribed Image Text:**Problem: Understanding Taylor Series at a Point**
**Objective:**
Find the Taylor series generated by the function \( f \) at \( x = a \).
**Given Function:**
\[ f(x) = 4x^4 + x^2 + 4 \]
**Point of Expansion:**
\[ a = -3 \]
**Task:**
Determine the Taylor series generated by \( f \) at \( a = -3 \).
**Solution:**
The Taylor series expansion of a function \( f(x) \) around a point \( a \) is given by:
\[
f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots
\]
**Steps to Solve:**
1. Calculate the derivatives \( f'(x) \), \( f''(x) \), etc., as needed.
2. Evaluate each derivative at \( x = -3 \).
3. Plug values into the Taylor series formula.
---
For illustration and understanding, additional resources such as examples and help can be accessed through:
- **View an example**
- **Get more help**
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