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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![**Finding a Degree 3 Taylor Polynomial Approximation**
**Objective:**
Determine the degree 3 Taylor polynomial approximation of the function \( f(x) = (1 + x)^{1/2} \) centered at 0.
**Steps:**
1. **Identify the Function:**
The function given is \( f(x) = (1 + x)^{1/2} \).
2. **Taylor Series Formula:**
The Taylor series expansion of a function \( f(x) \) centered at \( a \) is:
\[ f(x) \approx \sum_{n=0}^{N} \frac{f^{(n)}(a)}{n!}(x-a)^n \]
For this problem, \( a = 0 \) and \( N = 3 \).
3. **Compute Derivatives:**
- \( f(x) = (1 + x)^{1/2} \)
- \( f'(x) = \frac{1}{2}(1 + x)^{-1/2} \)
- \( f''(x) = -\frac{1}{4}(1 + x)^{-3/2} \)
- \( f'''(x) = \frac{3}{8}(1 + x)^{-5/2} \)
4. **Evaluate at \( x = 0 \):**
- \( f(0) = 1 \)
- \( f'(0) = \frac{1}{2} \)
- \( f''(0) = -\frac{1}{4} \)
- \( f'''(0) = \frac{3}{8} \)
5. **Construct the Taylor Polynomial:**
\[
P_3(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3
\]
\[
P_3(x) = 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3
\]
**Conclusion:**
The degree 3 Taylor polynomial approximation of \( f(x) = (1 + x)^{1/2} \) centered at](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa3b47e7c-f6ee-4dde-9ef1-0f8b8a2d2829%2F082cbe0d-ef9f-4cd4-b9ec-9d3b039f781f%2Fogojfp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding a Degree 3 Taylor Polynomial Approximation**
**Objective:**
Determine the degree 3 Taylor polynomial approximation of the function \( f(x) = (1 + x)^{1/2} \) centered at 0.
**Steps:**
1. **Identify the Function:**
The function given is \( f(x) = (1 + x)^{1/2} \).
2. **Taylor Series Formula:**
The Taylor series expansion of a function \( f(x) \) centered at \( a \) is:
\[ f(x) \approx \sum_{n=0}^{N} \frac{f^{(n)}(a)}{n!}(x-a)^n \]
For this problem, \( a = 0 \) and \( N = 3 \).
3. **Compute Derivatives:**
- \( f(x) = (1 + x)^{1/2} \)
- \( f'(x) = \frac{1}{2}(1 + x)^{-1/2} \)
- \( f''(x) = -\frac{1}{4}(1 + x)^{-3/2} \)
- \( f'''(x) = \frac{3}{8}(1 + x)^{-5/2} \)
4. **Evaluate at \( x = 0 \):**
- \( f(0) = 1 \)
- \( f'(0) = \frac{1}{2} \)
- \( f''(0) = -\frac{1}{4} \)
- \( f'''(0) = \frac{3}{8} \)
5. **Construct the Taylor Polynomial:**
\[
P_3(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3
\]
\[
P_3(x) = 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3
\]
**Conclusion:**
The degree 3 Taylor polynomial approximation of \( f(x) = (1 + x)^{1/2} \) centered at
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