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.
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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