Write the Linear Approximation to f(x, y) = x(1 + y)-¹ at (a, b) = (8, 1) 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(8 + h, 1+k) ≈ 7.97 Use it to estimate 2.05 (Use decimal notation. Give your answer to three decimal places.) 7.97 2.05 22
Write the Linear Approximation to f(x, y) = x(1 + y)-¹ at (a, b) = (8, 1) 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(8 + h, 1+k) ≈ 7.97 Use it to estimate 2.05 (Use decimal notation. Give your answer to three decimal places.) 7.97 2.05 22
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
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#### Linear Approximation
Write the Linear Approximation to \( f(x, y) = x(1 + y)^{-1} \) at \( (a, b) = (8, 1) \) in the form:
\[ f(a + h, b + k) \approx f(a, b) + f_x(a, b)h + f_y(a, b)k \]
(Give an exact answer. Use decimal notation and fractions where needed.)
\[ f(8 + h, 1 + k) \approx \]
---
#### Estimation
Use it to estimate \( \frac{7.97}{2.05} \).
(Use decimal notation. Give your answer to three decimal places.)
\[ \frac{7.97}{2.05} \approx \]
---
#### Error Analysis
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.)
\[ \text{The percentage error:} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \% \]
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faccc7427-2c55-44a5-b68e-19581b7568be%2F85594fe5-d855-446b-bfd7-980c8bb92540%2Fxo9gzd_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Approximation and Error Analysis
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#### Linear Approximation
Write the Linear Approximation to \( f(x, y) = x(1 + y)^{-1} \) at \( (a, b) = (8, 1) \) in the form:
\[ f(a + h, b + k) \approx f(a, b) + f_x(a, b)h + f_y(a, b)k \]
(Give an exact answer. Use decimal notation and fractions where needed.)
\[ f(8 + h, 1 + k) \approx \]
---
#### Estimation
Use it to estimate \( \frac{7.97}{2.05} \).
(Use decimal notation. Give your answer to three decimal places.)
\[ \frac{7.97}{2.05} \approx \]
---
#### Error Analysis
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.)
\[ \text{The percentage error:} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \% \]
---
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