Find the Taylor polynomial Tn(x) for the function f at the number a. Graph f and T3 on the same paper. f (z) = In (3 z) 6x 1 .a =- n= 3 %3D T3(x) =
Find the Taylor polynomial Tn(x) for the function f at the number a. Graph f and T3 on the same paper. f (z) = In (3 z) 6x 1 .a =- n= 3 %3D T3(x) =
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 Statement:**
Find the Taylor polynomial \( T_n(x) \) for the function \( f \) at the number \( a \). Graph \( f \) and \( T_3 \) on the same paper.
Given:
\[ f(x) = \frac{\ln(3x)}{6x}, \ a = \frac{1}{3}, \ n = 3 \]
Find:
\[ T_3(x) = \]
**Instructions:**
1. **Identify the Function and Derivatives:**
- The function given is \( f(x) = \frac{\ln(3x)}{6x} \).
- The Taylor series expansion requires finding the first few derivatives of \( f(x) \).
2. **Compute the Derivatives:**
- Evaluate \( f(a) \), \( f'(a) \), \( f''(a) \), and \( f'''(a) \) at \( a = \frac{1}{3} \).
3. **Formulate the Taylor Polynomial:**
- Use the formula for Taylor polynomials:
\[
T_n(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3
\]
- Substitute \( a = \frac{1}{3} \) and \( n = 3 \) along with the derivative values into the formula.
4. **Graph the Function and Polynomial:**
- Plot \( f(x) \) and \( T_3(x) \) on the same graph to compare the function and its approximation.
**Note:** Ensure to verify the calculations for the derivatives and polynomial coefficients for accuracy.
**Additional Resources:**
- A step-by-step guide on how to find derivatives.
- Graph plotting tools or graphing calculators to visualize the function and polynomial.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5c647828-f044-4f1b-b879-19c790819471%2F3f88a314-44f8-49e2-bdad-29873431718d%2Fpvx66cp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the Taylor polynomial \( T_n(x) \) for the function \( f \) at the number \( a \). Graph \( f \) and \( T_3 \) on the same paper.
Given:
\[ f(x) = \frac{\ln(3x)}{6x}, \ a = \frac{1}{3}, \ n = 3 \]
Find:
\[ T_3(x) = \]
**Instructions:**
1. **Identify the Function and Derivatives:**
- The function given is \( f(x) = \frac{\ln(3x)}{6x} \).
- The Taylor series expansion requires finding the first few derivatives of \( f(x) \).
2. **Compute the Derivatives:**
- Evaluate \( f(a) \), \( f'(a) \), \( f''(a) \), and \( f'''(a) \) at \( a = \frac{1}{3} \).
3. **Formulate the Taylor Polynomial:**
- Use the formula for Taylor polynomials:
\[
T_n(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3
\]
- Substitute \( a = \frac{1}{3} \) and \( n = 3 \) along with the derivative values into the formula.
4. **Graph the Function and Polynomial:**
- Plot \( f(x) \) and \( T_3(x) \) on the same graph to compare the function and its approximation.
**Note:** Ensure to verify the calculations for the derivatives and polynomial coefficients for accuracy.
**Additional Resources:**
- A step-by-step guide on how to find derivatives.
- Graph plotting tools or graphing calculators to visualize the function and polynomial.
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