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Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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![### Mathematical Functions
#### Problem Set
Consider the following functions of two variables, \(x\) and \(y\):
**(a)**
The function \(f(x, y)\) is defined as:
\[
f(x, y) = \frac{x + y}{\sqrt{x + y}}
\]
**(b)**
The function \(f(x, y)\) is given by:
\[
f(x, y) = \frac{\sin(x + y)}{\sqrt{y^2 - 1}}
\]
### Analysis
- **Function (a):** The expression \(\frac{x + y}{\sqrt{x + y}}\) involves a rational function where the numerator is a linear expression \((x + y)\) and the denominator is the square root of this linear expression.
- **Function (b):** The expression \(\frac{\sin(x + y)}{\sqrt{y^2 - 1}}\) combines trigonometric and algebraic aspects. The numerator is the sine of the sum of \(x\) and \(y\), while the denominator involves the square root of \(y^2 - 1\).
These functions can be explored for their continuity, differentiability, and to study their behavior based on the values of \(x\) and \(y\).](https://content.bartleby.com/qna-images/question/1079eff7-cd9f-4a9b-a6b7-8587fcd056cc/c783ac3f-e152-42b8-9f09-d17604c06b51/n00hj3_thumbnail.png)
Transcribed Image Text:### Mathematical Functions
#### Problem Set
Consider the following functions of two variables, \(x\) and \(y\):
**(a)**
The function \(f(x, y)\) is defined as:
\[
f(x, y) = \frac{x + y}{\sqrt{x + y}}
\]
**(b)**
The function \(f(x, y)\) is given by:
\[
f(x, y) = \frac{\sin(x + y)}{\sqrt{y^2 - 1}}
\]
### Analysis
- **Function (a):** The expression \(\frac{x + y}{\sqrt{x + y}}\) involves a rational function where the numerator is a linear expression \((x + y)\) and the denominator is the square root of this linear expression.
- **Function (b):** The expression \(\frac{\sin(x + y)}{\sqrt{y^2 - 1}}\) combines trigonometric and algebraic aspects. The numerator is the sine of the sum of \(x\) and \(y\), while the denominator involves the square root of \(y^2 - 1\).
These functions can be explored for their continuity, differentiability, and to study their behavior based on the values of \(x\) and \(y\).
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