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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![Certainly! Below is the transcription and explanation for the content shown in the image suitable for an educational website:
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**Problem 10:**
Find the degree 2 Taylor polynomial of the function
\[ f(x) = \ln(x + 1) \]
at
\[ a = 1 \]
**Options:**
A. \(\ln 2 + \frac{1}{4}(x - 1) - \frac{1}{4}(x - 1)^2\)
B. \(\ln 2 + \frac{1}{2}(x - 1) - \frac{1}{8}(x - 1)^2\)
C. \(\ln 2 + \frac{1}{2}(x - 1) + \frac{1}{4}(x - 1)^2\)
D. \(\ln 2 + \frac{1}{2}(x - 1) - \frac{1}{4}(x - 1)^2\)
E. \(\ln 2 + \frac{1}{2}(x - 1) + \frac{1}{8}(x - 1)^2\)
**Explanation:**
In this problem, you are asked to determine the degree 2 Taylor polynomial for the natural logarithm function \( f(x) = \ln(x + 1) \) centered at \( a = 1 \). The Taylor polynomial provides an approximation of the function near the point \( a \).
To find the correct polynomial, consider the formula for the Taylor series expansion up to the second degree:
\[ T_2(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2}(x - a)^2 \]
Plug in \( a = 1 \) and calculate derivatives \( f'(x) \) and \( f''(x) \) to complete the polynomial. Compare your answer to the options given to find the correct one.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe8ae9bc3-07a4-434a-8bcc-febfe8cf279c%2F41677a4d-c159-446c-8eda-0bbc75f43dd2%2Fq4vr47m.jpeg&w=3840&q=75)
Transcribed Image Text:Certainly! Below is the transcription and explanation for the content shown in the image suitable for an educational website:
---
**Problem 10:**
Find the degree 2 Taylor polynomial of the function
\[ f(x) = \ln(x + 1) \]
at
\[ a = 1 \]
**Options:**
A. \(\ln 2 + \frac{1}{4}(x - 1) - \frac{1}{4}(x - 1)^2\)
B. \(\ln 2 + \frac{1}{2}(x - 1) - \frac{1}{8}(x - 1)^2\)
C. \(\ln 2 + \frac{1}{2}(x - 1) + \frac{1}{4}(x - 1)^2\)
D. \(\ln 2 + \frac{1}{2}(x - 1) - \frac{1}{4}(x - 1)^2\)
E. \(\ln 2 + \frac{1}{2}(x - 1) + \frac{1}{8}(x - 1)^2\)
**Explanation:**
In this problem, you are asked to determine the degree 2 Taylor polynomial for the natural logarithm function \( f(x) = \ln(x + 1) \) centered at \( a = 1 \). The Taylor polynomial provides an approximation of the function near the point \( a \).
To find the correct polynomial, consider the formula for the Taylor series expansion up to the second degree:
\[ T_2(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2}(x - a)^2 \]
Plug in \( a = 1 \) and calculate derivatives \( f'(x) \) and \( f''(x) \) to complete the polynomial. Compare your answer to the options given to find the correct one.
Expert Solution
Step 1
The Taylor polynomial about 1 of degree 2 is given by:
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