5. Sketch in the xy - plane the domain of the function f (x, y) = √25 - 25x² - y². Also, sketch the level curve for k = 0.
5. Sketch in the xy - plane the domain of the function f (x, y) = √25 - 25x² - y². Also, sketch the level curve for k = 0.
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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![**Transcription:**
5. Sketch in the *xy*-plane the domain of the function \( f(x, y) = \sqrt{25 - 25x^2 - y^2} \). Also, sketch the level curve for \( k = 0 \).
**Explanation:**
The function \( f(x, y) = \sqrt{25 - 25x^2 - y^2} \) is defined where the expression under the square root, \( 25 - 25x^2 - y^2 \), is non-negative (i.e., greater than or equal to zero). Thus, the domain consists of the set of points \((x, y)\) for which:
\[ 25 - 25x^2 - y^2 \geq 0 \]
This simplifies to:
\[ 25x^2 + y^2 \leq 25 \]
Dividing through by 25 gives:
\[ x^2 + \frac{y^2}{25} \leq 1 \]
This inequality represents an ellipse centered at the origin in the *xy*-plane, with a major axis of length 5 along the *y*-axis and a minor axis of length 1 along the *x*-axis.
For the level curve for \( k = 0 \), we set \( f(x, y) = 0 \), which implies:
\[ \sqrt{25 - 25x^2 - y^2} = 0 \]
This simplifies to:
\[ 25 - 25x^2 - y^2 = 0 \]
Or:
\[ 25x^2 + y^2 = 25 \]
Dividing through by 25 gives:
\[ x^2 + \frac{y^2}{25} = 1 \]
This equation represents the boundary of the domain: an ellipse centered at the origin in the *xy*-plane, identical to the ellipse defined by \( x^2 + \frac{y^2}{25} \leq 1 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ba91fa8-c4da-44a3-a037-923d1d08ef86%2Fab18f560-5f75-4b0c-853a-4626d3640fa6%2Fwq38xlh_processed.png&w=3840&q=75)
Transcribed Image Text:**Transcription:**
5. Sketch in the *xy*-plane the domain of the function \( f(x, y) = \sqrt{25 - 25x^2 - y^2} \). Also, sketch the level curve for \( k = 0 \).
**Explanation:**
The function \( f(x, y) = \sqrt{25 - 25x^2 - y^2} \) is defined where the expression under the square root, \( 25 - 25x^2 - y^2 \), is non-negative (i.e., greater than or equal to zero). Thus, the domain consists of the set of points \((x, y)\) for which:
\[ 25 - 25x^2 - y^2 \geq 0 \]
This simplifies to:
\[ 25x^2 + y^2 \leq 25 \]
Dividing through by 25 gives:
\[ x^2 + \frac{y^2}{25} \leq 1 \]
This inequality represents an ellipse centered at the origin in the *xy*-plane, with a major axis of length 5 along the *y*-axis and a minor axis of length 1 along the *x*-axis.
For the level curve for \( k = 0 \), we set \( f(x, y) = 0 \), which implies:
\[ \sqrt{25 - 25x^2 - y^2} = 0 \]
This simplifies to:
\[ 25 - 25x^2 - y^2 = 0 \]
Or:
\[ 25x^2 + y^2 = 25 \]
Dividing through by 25 gives:
\[ x^2 + \frac{y^2}{25} = 1 \]
This equation represents the boundary of the domain: an ellipse centered at the origin in the *xy*-plane, identical to the ellipse defined by \( x^2 + \frac{y^2}{25} \leq 1 \).
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