Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Title: Finding and Sketching the Domain of a Function**
**Objective:**
Determine and illustrate the domain of the given function \( f(x, y) \).
**Function:**
\[ f(x, y) = \sqrt{4 - x^2} - \sqrt{25 - y^2} \]
**Instructions:**
To find the domain of the function \( f(x, y) \), we need to ensure that the expressions under both square roots are non-negative, i.e.,
\[ 4 - x^2 \geq 0 \]
and
\[ 25 - y^2 \geq 0 \].
**Steps:**
### Step 1: Determine the Domain Restrictions for Each Square Root
1. \( \sqrt{4 - x^2} \) is defined when \( 4 - x^2 \geq 0 \).
\[ \rightarrow x^2 \leq 4 \]
\[ \rightarrow -2 \leq x \leq 2 \]
2. \( \sqrt{25 - y^2} \) is defined when \( 25 - y^2 \geq 0 \).
\[ \rightarrow y^2 \leq 25 \]
\[ \rightarrow -5 \leq y \leq 5 \]
Thus, the domain of \( f(x, y) \) is the set of all points \( (x, y) \) such that \( -2 \leq x \leq 2 \) and \( -5 \leq y \leq 5 \).
### Step 2: Visual Representation of the Domain
Below are diagrams to illustrate various regions of interest:
1. **First Diagram**: Shows a circle with radius 2 centered at the origin, which represents the region where \( \sqrt{4 - x^2} \) is defined (\( -2 \leq x \leq 2 \)).
2. **Second Diagram**: Displays a rectangle extending horizontally from \(-6\) to \(6\) and vertically from \(-2\) to \(2\), which illustrates an incorrect region.
3. **Third Diagram**: Depicts a similar incorrect region but with dashed boundaries that also extend improperly.
4. **Fourth Diagram**: Correctly illustrates a circle with radius 2, centered at the origin, representing the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fab6b79a9-c663-4f87-993a-883a678be91b%2Fa99b9940-03cf-4687-b3ff-96ac2a0563bf%2Fprybg4p_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Finding and Sketching the Domain of a Function**
**Objective:**
Determine and illustrate the domain of the given function \( f(x, y) \).
**Function:**
\[ f(x, y) = \sqrt{4 - x^2} - \sqrt{25 - y^2} \]
**Instructions:**
To find the domain of the function \( f(x, y) \), we need to ensure that the expressions under both square roots are non-negative, i.e.,
\[ 4 - x^2 \geq 0 \]
and
\[ 25 - y^2 \geq 0 \].
**Steps:**
### Step 1: Determine the Domain Restrictions for Each Square Root
1. \( \sqrt{4 - x^2} \) is defined when \( 4 - x^2 \geq 0 \).
\[ \rightarrow x^2 \leq 4 \]
\[ \rightarrow -2 \leq x \leq 2 \]
2. \( \sqrt{25 - y^2} \) is defined when \( 25 - y^2 \geq 0 \).
\[ \rightarrow y^2 \leq 25 \]
\[ \rightarrow -5 \leq y \leq 5 \]
Thus, the domain of \( f(x, y) \) is the set of all points \( (x, y) \) such that \( -2 \leq x \leq 2 \) and \( -5 \leq y \leq 5 \).
### Step 2: Visual Representation of the Domain
Below are diagrams to illustrate various regions of interest:
1. **First Diagram**: Shows a circle with radius 2 centered at the origin, which represents the region where \( \sqrt{4 - x^2} \) is defined (\( -2 \leq x \leq 2 \)).
2. **Second Diagram**: Displays a rectangle extending horizontally from \(-6\) to \(6\) and vertically from \(-2\) to \(2\), which illustrates an incorrect region.
3. **Third Diagram**: Depicts a similar incorrect region but with dashed boundaries that also extend improperly.
4. **Fourth Diagram**: Correctly illustrates a circle with radius 2, centered at the origin, representing the
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