### Linear Approximation and Percent Error Calculation#### Problem Statement1. **Linear Approximation Setup:** - Write the equation of the line that represents the linear approximation to the following function at the given point $ a $. - Use the linear approximation to estimate the given quantity.2. **Percent Error Calculation:** - Compute the percent error in the approximation using the formula: \[ 100 \cdot \frac{\text{approximation} - \text{exact}}{\text{exact}} \] where the exact value is given by a calculator.#### Given Data- Function: $ f(x) = \frac{1}{x+2} $- Point of approximation: $ a = 0 $- Exact value: $ \frac{1}{2.2} $### Step-by-Step Solution#### a. Write the Linear Approximation EquationDetermine the linear approximation $ L(x) $:\[ L(x) = \]#### b. Estimation Using Linear ApproximationEstimate $ \frac{1}{2.2} $ using the linear approximation:\[ \frac{1}{2.2} \approx \](Round to three decimal places as needed.)#### c. Compute Percent ErrorCalculate the percent error in the approximation:\[ \text{The percent error in the approximation is} \, \, \%\](Round to three decimal places as needed.)

Answered: Image /qna-images/answer/8284506b-f24c-4f27-8af1-ee0a1260dce8.jpg

a. Write the equation of the line that represents the linear approximation to the following function at the given point a. b. Use the linear approximation to estimate the given quantity. approximation-exact |exact c. Compute the percent error in the approximation, 100 • where the exact value is given by a calculator. f(x) = x+2 a=0; ,2 a. L(x) = || b. Using the linear approximation, 22 2.2 (Round to three decimal places as needed.) C. The percent error in the approximation is %. (Round to three decimal places as needed.)

a. Write the equation of the line that represents the linear approximation to the following function at the given point a. b. Use the linear approximation to estimate the given quantity. approximation-exact |exact c. Compute the percent error in the approximation, 100 • where the exact value is given by a calculator. f(x) = x+2 a=0; ,2 a. L(x) = || b. Using the linear approximation, 22 2.2 (Round to three decimal places as needed.) C. The percent error in the approximation is %. (Round to three decimal places as needed.)

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.

Question

### Linear Approximation and Percent Error Calculation

#### Problem Statement

1. **Linear Approximation Setup:**
- Write the equation of the line that represents the linear approximation to the following function at the given point $ a $.
- Use the linear approximation to estimate the given quantity.

2. **Percent Error Calculation:**
- Compute the percent error in the approximation using the formula:
\[
100 \cdot \frac{\text{approximation} - \text{exact}}{\text{exact}}
\]
where the exact value is given by a calculator.

#### Given Data

- Function: $ f(x) = \frac{1}{x+2} $
- Point of approximation: $ a = 0 $
- Exact value: $ \frac{1}{2.2} $

### Step-by-Step Solution

#### a. Write the Linear Approximation Equation
Determine the linear approximation $ L(x) $:

\[ L(x) = \]

#### b. Estimation Using Linear Approximation
Estimate $ \frac{1}{2.2} $ using the linear approximation:

\[ \frac{1}{2.2} \approx \]

(Round to three decimal places as needed.)

#### c. Compute Percent Error
Calculate the percent error in the approximation:

\[ \text{The percent error in the approximation is} \, \, \%\]

(Round to three decimal places as needed.)

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