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Use differentials to approximate the number V6.022 + 2.972 + 5.99². (Round your answer to five decimal place 8.9967

Use differentials to approximate the number V6.022 + 2.972 + 5.99². (Round your answer to five decimal place 8.9967

Advanced Engineering Mathematics
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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**Problem Statement:** Use differentials to approximate the number \(\sqrt{6.02^2 + 2.97^2 + 5.99^2}\). (Round your answer to five decimal places.) **User Input:** 8.9967 (This answer is marked as incorrect with an "X") --- **Explanation:** In this exercise, you are asked to use differentials to approximate a numerical expression. Differentials are a way to estimate small changes in functions based on their derivatives. In this context, they can be applied to approximate the square root expression provided. When applying differentials to approximate \(\sqrt{x^2 + y^2 + z^2}\), you consider the small changes in \(x\), \(y\), and \(z\) as they slightly deviate from whole numbers (e.g., from 6.02 to 6, from 2.97 to 3, etc.). The expression implies calculating the distance in three-dimensional space from the origin using the modified coordinates, and the use of differentials involves approximating this result efficiently, particularly when the modifications to each coordinate are small. The emphasis on approximation suggests utililzing linear approximations or first-order Taylor expansions around a point (x = 6, y = 3, z = 6). After determining the appropriate calculations, results are rounded to five decimal places for precision.
Transcribed Image Text:**Problem Statement:** Use differentials to approximate the number \(\sqrt{6.02^2 + 2.97^2 + 5.99^2}\). (Round your answer to five decimal places.) **User Input:** 8.9967 (This answer is marked as incorrect with an "X") --- **Explanation:** In this exercise, you are asked to use differentials to approximate a numerical expression. Differentials are a way to estimate small changes in functions based on their derivatives. In this context, they can be applied to approximate the square root expression provided. When applying differentials to approximate \(\sqrt{x^2 + y^2 + z^2}\), you consider the small changes in \(x\), \(y\), and \(z\) as they slightly deviate from whole numbers (e.g., from 6.02 to 6, from 2.97 to 3, etc.). The expression implies calculating the distance in three-dimensional space from the origin using the modified coordinates, and the use of differentials involves approximating this result efficiently, particularly when the modifications to each coordinate are small. The emphasis on approximation suggests utililzing linear approximations or first-order Taylor expansions around a point (x = 6, y = 3, z = 6). After determining the appropriate calculations, results are rounded to five decimal places for precision.
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