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. =. Compute the percent error in the approximation, 100. approximation - exact exact where the exact value is given by a calculator. f(x)=9-x² at a = 1; f(0.9) a. L(x) =
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. =. Compute the percent error in the approximation, 100. approximation - exact exact where the exact value is given by a calculator. f(x)=9-x² at a = 1; f(0.9) a. L(x) =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Linear Approximation and Percent Error Calculation
#### Problem Statement
Consider the function \( f(x) = 9 - x^2 \) and the point \( a = 1 \). You are given \( f(0.9) \).
a. **Equation of the Line for Linear Approximation:**
Write the equation of the line that represents the linear approximation to the function \( f(x) \) at the given point \( a \).
b. **Estimating the Quantity:**
Use the linear approximation to estimate \( f(0.9) \).
c. **Percent Error Calculation:**
Compute the percent error in the approximation using the formula:
\[
\text{Percent Error} = 100 \cdot \frac{|\text{approximation} - \text{exact}|}{\text{exact}}
\]
where the exact value is given by a calculator.
#### Given Function
\[ f(x) = 9 - x^2 \]
#### Given Point
\[ a = 1 \]
#### Exact Value
\[ f(0.9) \]
#### Required Calculations
a. **Linear Approximation Equation**:
\[ L(x) = \]
This involves finding the first derivative \( f'(x) \) and using the formula for linear approximation.
b. **Estimate at \( x = 0.9 \)**:
Use the linear approximation equation \( L(x) \) to calculate the value at \( x = 0.9 \).
c. **Percent Error Calculation**:
Substitute the approximation and the exact values into the percent error formula.
This is a step-by-step guide to solving problems involving linear approximation and calculating percent error in approximations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fec7a888d-f205-4cdb-8106-1ac64a94a20a%2Fdfd3590e-16f8-464f-ba00-8c503b226332%2F6g819aq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Approximation and Percent Error Calculation
#### Problem Statement
Consider the function \( f(x) = 9 - x^2 \) and the point \( a = 1 \). You are given \( f(0.9) \).
a. **Equation of the Line for Linear Approximation:**
Write the equation of the line that represents the linear approximation to the function \( f(x) \) at the given point \( a \).
b. **Estimating the Quantity:**
Use the linear approximation to estimate \( f(0.9) \).
c. **Percent Error Calculation:**
Compute the percent error in the approximation using the formula:
\[
\text{Percent Error} = 100 \cdot \frac{|\text{approximation} - \text{exact}|}{\text{exact}}
\]
where the exact value is given by a calculator.
#### Given Function
\[ f(x) = 9 - x^2 \]
#### Given Point
\[ a = 1 \]
#### Exact Value
\[ f(0.9) \]
#### Required Calculations
a. **Linear Approximation Equation**:
\[ L(x) = \]
This involves finding the first derivative \( f'(x) \) and using the formula for linear approximation.
b. **Estimate at \( x = 0.9 \)**:
Use the linear approximation equation \( L(x) \) to calculate the value at \( x = 0.9 \).
c. **Percent Error Calculation**:
Substitute the approximation and the exact values into the percent error formula.
This is a step-by-step guide to solving problems involving linear approximation and calculating percent error in approximations.
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