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
Theres 2 of them that i need to figure out
![**Taylor Polynomials for Trigonometric Functions**
**Problem Statement:**
Find the Taylor polynomials \( p_4 \) and \( p_5 \) centered at \( a = \frac{\pi}{6} \) for the function \( f(x) = 4 \cos(x) \).
**Objective:**
To determine the specific forms of the 4th and 5th degree Taylor polynomials for the cosine function multiplied by a constant, and centered around a specified value of \( x \).
**Approach:**
The goal is to expand the function \( f(x) \) in terms of its derivatives evaluated at \( a = \frac{\pi}{6} \). This involves:
1. Calculating the necessary derivatives of \( f(x) \),
2. Evaluating these derivatives at \( a = \frac{\pi}{6} \),
3. Constructing the polynomial expressions for both \( p_4 \) and \( p_5 \) using the Taylor series formula:
\[
p_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n
\]
**Note:** No graphs or diagrams are included in this exemplary transcribing. However, visually representing the function and its approximations can enhance understanding through graphical plots.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5f1b7c10-7ca4-413c-bb43-fe1a15020101%2F5de0c1dd-c528-489f-8aa6-6c7a9fcc10d7%2Fulifukw_processed.png&w=3840&q=75)
Transcribed Image Text:**Taylor Polynomials for Trigonometric Functions**
**Problem Statement:**
Find the Taylor polynomials \( p_4 \) and \( p_5 \) centered at \( a = \frac{\pi}{6} \) for the function \( f(x) = 4 \cos(x) \).
**Objective:**
To determine the specific forms of the 4th and 5th degree Taylor polynomials for the cosine function multiplied by a constant, and centered around a specified value of \( x \).
**Approach:**
The goal is to expand the function \( f(x) \) in terms of its derivatives evaluated at \( a = \frac{\pi}{6} \). This involves:
1. Calculating the necessary derivatives of \( f(x) \),
2. Evaluating these derivatives at \( a = \frac{\pi}{6} \),
3. Constructing the polynomial expressions for both \( p_4 \) and \( p_5 \) using the Taylor series formula:
\[
p_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n
\]
**Note:** No graphs or diagrams are included in this exemplary transcribing. However, visually representing the function and its approximations can enhance understanding through graphical plots.
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