Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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![### Parametric Equations Example
#### Consider the parametric equations:
\[ x = \sqrt{t} \]
\[ y = \sqrt{4 - t} \]
Complete each of the following (show your work):
**(a) Graph the equation of the parametric equation on the interval \(0 \leq t \leq 4\). Be sure to indicate the direction in your graph.**
- **Instructions**: Insert your graph in the designated area.
- **Graph Section**:
- *There is a placeholder for a graph below:*
- 
**(b) Eliminate the parameter.**
- **Instructions**: Show your work for eliminating the parameter in the designated area.
**(c) Write the domain of the parametric equations.**
- **Instructions**: In the designated space, write the domain of the parametric equations.
---
#### Detailed Steps and Explanation:
1. **Graphing the Parametric Equations**:
- To graph the equations \(x = \sqrt{t}\) and \(y = \sqrt{4 - t}\), use the interval \(0 \leq t \leq 4\).
- Plot the points by substituting values for \(t\) within the interval and calculating the corresponding \(x\) and \(y\) values.
- Indicate the direction of the graph as \(t\) increases from 0 to 4.
2. **Eliminating the Parameter**:
- Begin by expressing \(t\) in terms of \(x\):
\[ t = x^2 \]
- Substitute \(t = x^2\) into the equation for \(y\):
\[ y = \sqrt{4 - t} \Rightarrow y = \sqrt{4 - x^2} \]
- Now, the parametric equations are represented as the Cartesian equation \(y = \sqrt{4 - x^2}\).
3. **Domain of the Parametric Equations**:
- The range for \(t\) is from 0 to 4.
- Since \(x = \sqrt{t}\), \(x\) will be between \(0\) and \(2\) (\( \sqrt{4} = 2 \)).
- Hence, the domain for \(x\)](https://content.bartleby.com/qna-images/question/3945ad73-e6d6-4f80-98e3-ea2222211659/0d1a8289-49eb-467f-b92d-6b008efc476f/fi6kdgi_processed.jpeg)
Transcribed Image Text:### Parametric Equations Example
#### Consider the parametric equations:
\[ x = \sqrt{t} \]
\[ y = \sqrt{4 - t} \]
Complete each of the following (show your work):
**(a) Graph the equation of the parametric equation on the interval \(0 \leq t \leq 4\). Be sure to indicate the direction in your graph.**
- **Instructions**: Insert your graph in the designated area.
- **Graph Section**:
- *There is a placeholder for a graph below:*
- 
**(b) Eliminate the parameter.**
- **Instructions**: Show your work for eliminating the parameter in the designated area.
**(c) Write the domain of the parametric equations.**
- **Instructions**: In the designated space, write the domain of the parametric equations.
---
#### Detailed Steps and Explanation:
1. **Graphing the Parametric Equations**:
- To graph the equations \(x = \sqrt{t}\) and \(y = \sqrt{4 - t}\), use the interval \(0 \leq t \leq 4\).
- Plot the points by substituting values for \(t\) within the interval and calculating the corresponding \(x\) and \(y\) values.
- Indicate the direction of the graph as \(t\) increases from 0 to 4.
2. **Eliminating the Parameter**:
- Begin by expressing \(t\) in terms of \(x\):
\[ t = x^2 \]
- Substitute \(t = x^2\) into the equation for \(y\):
\[ y = \sqrt{4 - t} \Rightarrow y = \sqrt{4 - x^2} \]
- Now, the parametric equations are represented as the Cartesian equation \(y = \sqrt{4 - x^2}\).
3. **Domain of the Parametric Equations**:
- The range for \(t\) is from 0 to 4.
- Since \(x = \sqrt{t}\), \(x\) will be between \(0\) and \(2\) (\( \sqrt{4} = 2 \)).
- Hence, the domain for \(x\)
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