Write the Linear Approximation to f(x, y) = x(1 + y)−¹ at (a, b) = (18, 2) in the form f(a+h, b + k) ≈ f(a, b) + fx(a, b)h + fy(a, b)k (Give an exact answer. Use decimal notation and fractions where needed.) f(18+ h, 2+k) ~ 17.97 3.04 Use it to estimate 17.97 3.04 (Use decimal notation. Give your answer to three decimal places.) 3.91 Incorrect 6+ 1/ h the percentage error: h - 2k Compare this approximation with the value obtained using the calculator. Calculate the percentage error obtained by using the approximation. Percentage error is the absolute value of the ratio of the error to the actual value of the expression, multiplied by 100%. (Use decimal notation. Give your answer to three decimal places.) 3.261 Incorrect %
Write the Linear Approximation to f(x, y) = x(1 + y)−¹ at (a, b) = (18, 2) in the form f(a+h, b + k) ≈ f(a, b) + fx(a, b)h + fy(a, b)k (Give an exact answer. Use decimal notation and fractions where needed.) f(18+ h, 2+k) ~ 17.97 3.04 Use it to estimate 17.97 3.04 (Use decimal notation. Give your answer to three decimal places.) 3.91 Incorrect 6+ 1/ h the percentage error: h - 2k Compare this approximation with the value obtained using the calculator. Calculate the percentage error obtained by using the approximation. Percentage error is the absolute value of the ratio of the error to the actual value of the expression, multiplied by 100%. (Use decimal notation. Give your answer to three decimal places.) 3.261 Incorrect %
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 Error Analysis
#### Problem Statement:
**Objective:** Write the Linear Approximation to \( f(x, y) = x(1 + y)^{-1} \) at \( (a, b) = (18, 2) \) in the form:
\[ f(a + h, b + k) \approx f(a, b) + f_x(a, b)h + f_y(a, b)k \]
**Provide an exact answer. Use decimal notation and fractions where needed.**
#### Solution:
The linear approximation at \( (18, 2) \) is:
\[ f(18 + h, 2 + k) \approx 6 + \frac{1}{3}h - 2k \]
#### Estimation:
Use this approximation to estimate \( \frac{17.97}{3.04} \).
**Instructions:** Use decimal notation. Provide your answer to three decimal places.
\[ \frac{17.97}{3.04} \approx 3.91 \]
**Feedback:** The given answer is marked as incorrect.
#### Error Analysis:
**Task:** Compare this approximation with the value obtained using a calculator. Calculate the percentage error obtained by using the approximation. The percentage error is the absolute value of the ratio of the error to the actual value of the expression, multiplied by 100%.
\[ \text{Percentage Error} = \left| \frac{\text{Error}}{\text{Actual Value}} \right| \times 100\% \]
**Instructions:** Use decimal notation. Provide your answer to three decimal places.
\[ \text{The percentage error: } 3.261 \% \]
**Feedback:** The given percentage error is marked as incorrect.
#### Explanation of Textboxes and Feedback:
1. The estimated value of \( \frac{17.97}{3.04} \approx 3.91 \) is entered in a textbox highlighted in red, indicating an incorrect answer.
2. The calculated percentage error \( 3.261\% \) is also entered in a textbox highlighted in red, indicating an incorrect answer.
The provided mathematical expressions and methods illustrate the process of linear approximation and error analysis, important concepts in calculus and numerical methods. These examples highlight the significance of accuracy and precision in mathematical computations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa40558bc-afda-4e2b-be1a-c4211c27f73c%2F3c13fffa-c485-4f5b-a3f3-8fcdab2e3b22%2Fkjcn9c_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Approximation and Error Analysis
#### Problem Statement:
**Objective:** Write the Linear Approximation to \( f(x, y) = x(1 + y)^{-1} \) at \( (a, b) = (18, 2) \) in the form:
\[ f(a + h, b + k) \approx f(a, b) + f_x(a, b)h + f_y(a, b)k \]
**Provide an exact answer. Use decimal notation and fractions where needed.**
#### Solution:
The linear approximation at \( (18, 2) \) is:
\[ f(18 + h, 2 + k) \approx 6 + \frac{1}{3}h - 2k \]
#### Estimation:
Use this approximation to estimate \( \frac{17.97}{3.04} \).
**Instructions:** Use decimal notation. Provide your answer to three decimal places.
\[ \frac{17.97}{3.04} \approx 3.91 \]
**Feedback:** The given answer is marked as incorrect.
#### Error Analysis:
**Task:** Compare this approximation with the value obtained using a calculator. Calculate the percentage error obtained by using the approximation. The percentage error is the absolute value of the ratio of the error to the actual value of the expression, multiplied by 100%.
\[ \text{Percentage Error} = \left| \frac{\text{Error}}{\text{Actual Value}} \right| \times 100\% \]
**Instructions:** Use decimal notation. Provide your answer to three decimal places.
\[ \text{The percentage error: } 3.261 \% \]
**Feedback:** The given percentage error is marked as incorrect.
#### Explanation of Textboxes and Feedback:
1. The estimated value of \( \frac{17.97}{3.04} \approx 3.91 \) is entered in a textbox highlighted in red, indicating an incorrect answer.
2. The calculated percentage error \( 3.261\% \) is also entered in a textbox highlighted in red, indicating an incorrect answer.
The provided mathematical expressions and methods illustrate the process of linear approximation and error analysis, important concepts in calculus and numerical methods. These examples highlight the significance of accuracy and precision in mathematical computations.
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