n exact form. Use symbolic notation and fractions where needed.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Problem Statement:**
Let \( S \) be the surface in \( \mathbb{R}^3 \) defined by the equation \( x^2 \cdot y^7 - 4z = 9 \). Find a real-valued function \( g(x, y) \) of two variables such that \( S \) is the graph of \( g \).
(Express numbers in exact form. Use symbolic notation and fractions where needed.)
**Solution Space:**
\[ g(x, y) = \underline{\hspace{5cm}} \]
**Explanation:**
To find the function \( g(x, y) \), we need to express \( z \) in terms of \( x \) and \( y \) from the given equation of the surface:
\[
x^2 \cdot y^7 - 4z = 9
\]
Rearranging the equation to solve for \( z \), we get:
\[
- 4z = 9 - x^2 \cdot y^7
\]
\[
z = \frac{x^2 \cdot y^7 - 9}{4}
\]
Thus, the real-valued function \( g(x, y) \) is:
\[
g(x, y) = \frac{x^2 \cdot y^7 - 9}{4}
\]
This function represents the surface \( S \) in \( \mathbb{R}^3 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F51ff2e68-0bd1-44c4-8696-122fa89f1551%2F4098f340-6458-4984-98c3-bb17494375de%2F1av751n_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Let \( S \) be the surface in \( \mathbb{R}^3 \) defined by the equation \( x^2 \cdot y^7 - 4z = 9 \). Find a real-valued function \( g(x, y) \) of two variables such that \( S \) is the graph of \( g \).
(Express numbers in exact form. Use symbolic notation and fractions where needed.)
**Solution Space:**
\[ g(x, y) = \underline{\hspace{5cm}} \]
**Explanation:**
To find the function \( g(x, y) \), we need to express \( z \) in terms of \( x \) and \( y \) from the given equation of the surface:
\[
x^2 \cdot y^7 - 4z = 9
\]
Rearranging the equation to solve for \( z \), we get:
\[
- 4z = 9 - x^2 \cdot y^7
\]
\[
z = \frac{x^2 \cdot y^7 - 9}{4}
\]
Thus, the real-valued function \( g(x, y) \) is:
\[
g(x, y) = \frac{x^2 \cdot y^7 - 9}{4}
\]
This function represents the surface \( S \) in \( \mathbb{R}^3 \).
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