**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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Answered: Find the Taylor polynomial Tn(x) for…

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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Question

**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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