Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter6: Applications Of The Derivative
Section6.CR: Chapter 6 Review
Problem 48CR
Related questions
Question
37 please
![### Problem Statement
**Question 37:**
Find and sketch the domains of the following functions:
1. \( f(x, y) = \frac{\sqrt{x + y + 1}}{x - 1} \)
2. \( g(x, y) = x \ln(y^2 - x) \)
## Detailed Explanation
### Function 1: \( f(x, y) = \frac{\sqrt{x + y + 1}}{x - 1} \)
**Domain Conditions:**
1. **Square Root Condition:** The expression under the square root must be non-negative.
\[
x + y + 1 \geq 0 \quad \Rightarrow \quad x + y \geq -1
\]
2. **Denominator Condition:** The denominator cannot be zero.
\[
x - 1 \neq 0 \quad \Rightarrow \quad x \neq 1
\]
**Domain Description:**
The domain of \( f(x, y) \) is given by the set of all points \((x, y)\) in the Cartesian plane that satisfy \( x + y \geq -1 \) and \( x \neq 1 \).
### Function 2: \( g(x, y) = x \ln(y^2 - x) \)
**Domain Conditions:**
1. **Logarithm Condition:** The expression inside the logarithm must be positive.
\[
y^2 - x > 0 \quad \Rightarrow \quad y^2 > x \quad \Rightarrow \quad x < y^2
\]
**Domain Description:**
The domain of \( g(x, y) \) is the set of all points \((x, y)\) in the Cartesian plane that satisfy \( x < y^2 \).
---
### Sketching the Domains
#### Domain of \( f(x, y) \):
To sketch \( x + y \geq -1 \):
- Draw the line \( x + y = -1 \). This line will have a slope of -1 and intercepts at \((0, -1)\) on the y-axis and \((-1, 0)\) on the x-axis.
- The domain includes all points above and on this line, except the vertical line \( x = 1 \).
#### Domain of \( g(x,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb24b717-8ee5-4ca9-8061-74ddb7e91c1a%2F7dc2a969-411c-4d52-a108-323ad8af2cb8%2Fr5q5rf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**Question 37:**
Find and sketch the domains of the following functions:
1. \( f(x, y) = \frac{\sqrt{x + y + 1}}{x - 1} \)
2. \( g(x, y) = x \ln(y^2 - x) \)
## Detailed Explanation
### Function 1: \( f(x, y) = \frac{\sqrt{x + y + 1}}{x - 1} \)
**Domain Conditions:**
1. **Square Root Condition:** The expression under the square root must be non-negative.
\[
x + y + 1 \geq 0 \quad \Rightarrow \quad x + y \geq -1
\]
2. **Denominator Condition:** The denominator cannot be zero.
\[
x - 1 \neq 0 \quad \Rightarrow \quad x \neq 1
\]
**Domain Description:**
The domain of \( f(x, y) \) is given by the set of all points \((x, y)\) in the Cartesian plane that satisfy \( x + y \geq -1 \) and \( x \neq 1 \).
### Function 2: \( g(x, y) = x \ln(y^2 - x) \)
**Domain Conditions:**
1. **Logarithm Condition:** The expression inside the logarithm must be positive.
\[
y^2 - x > 0 \quad \Rightarrow \quad y^2 > x \quad \Rightarrow \quad x < y^2
\]
**Domain Description:**
The domain of \( g(x, y) \) is the set of all points \((x, y)\) in the Cartesian plane that satisfy \( x < y^2 \).
---
### Sketching the Domains
#### Domain of \( f(x, y) \):
To sketch \( x + y \geq -1 \):
- Draw the line \( x + y = -1 \). This line will have a slope of -1 and intercepts at \((0, -1)\) on the y-axis and \((-1, 0)\) on the x-axis.
- The domain includes all points above and on this line, except the vertical line \( x = 1 \).
#### Domain of \( g(x,
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