Consider the parametric equations: a = Vt y = V4 -t Complete each of the following (show your work): (a) Graph the equation of the parametric equation on the interval Os ts 4. Be sure to indicate th direction in your graph. (b) Eliminate the parameter. (c) In the designated space, write the domain of the parametric equations. a) Put your graph here. Draw or add images here b) Show your work here for removing the parameter. c) What are the valid x-values for these parametric equations?

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...
icon
Related questions
Question
### 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:*
    - ![Graph Placeholder](data:image/png;base64,iVBORw...)

**(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 \(x\)
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:* - ![Graph Placeholder](data:image/png;base64,iVBORw...) **(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 \(x\)
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 1 images

Blurred answer
Knowledge Booster
Area of a Circle
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning