A pair of parametric equations is given. x = 4 cos(t), y = 5 sin(t), Osts 2n (a) Sketch the curve represented by the parametric equations. Use arrows to indicate the direction of the curve as t increases. y y X -2 4. -2
A pair of parametric equations is given. x = 4 cos(t), y = 5 sin(t), Osts 2n (a) Sketch the curve represented by the parametric equations. Use arrows to indicate the direction of the curve as t increases. y y X -2 4. -2
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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Transcribed Image Text:The image contains four graphs of ellipses on a coordinate plane, each with red arrows indicating the direction of traversal. Here's a detailed description of each graph:
1. **Top Left Ellipse**:
- The ellipse is vertically oriented.
- The major axis is along the y-axis, stretching from -4 to 4.
- The minor axis is along the x-axis, stretching from -2 to 2.
- Arrows indicate a counter-clockwise traversal.
2. **Top Right Ellipse**:
- The ellipse is vertically oriented.
- The major axis is along the y-axis, stretching from -2 to 2.
- The minor axis is along the x-axis, stretching from -1 to 1.
- Arrows indicate a clockwise traversal.
3. **Bottom Left Ellipse**:
- The ellipse is horizontally oriented.
- The major axis is along the x-axis, stretching from -2 to 2.
- The minor axis is along the y-axis, stretching from -1 to 1.
- Arrows indicate a counter-clockwise traversal.
4. **Bottom Right Ellipse**:
- The ellipse is horizontally oriented.
- The major axis is along the x-axis, stretching from -4 to 4.
- The minor axis is along the y-axis, stretching from -2 to 2.
- Arrows indicate a clockwise traversal.
Below the graphs, there is a prompt related to converting a curve parameter:
(b) Find a rectangular-coordinate equation for the curve by eliminating the parameter.
This prompt suggests an exercise where students eliminate the parameter from parametric equations to derive a standard equation for the ellipses.
![**Parametric Equations and their Graphs**
A pair of parametric equations is given:
\[ x = 4 \cos(t), \quad y = 5 \sin(t), \quad 0 \leq t \leq 2\pi \]
**Task:**
(a) Sketch the curve represented by the parametric equations. Use arrows to indicate the direction of the curve as \( t \) increases.
**Graph Descriptions:**
The image contains four plots of ellipses on Cartesian coordinates, each illustrating different orientations of the curve defined by the parametric equations.
1. **Top Left Graph:**
- Displays an ellipse centered at the origin (0,0).
- The major axis is along the vertical \( y \)-axis, extending from -5 to 5.
- The minor axis is along the horizontal \( x \)-axis, extending from -4 to 4.
- Red arrows indicate a counterclockwise direction of the curve as \( t \) increases.
2. **Top Right Graph:**
- Similar ellipse oriented downward.
- The major axis remains along the vertical \( y \)-axis.
- The elliptical path is traced in a counterclockwise direction from the top of the ellipse.
3. **Bottom Left Graph:**
- The ellipse is oriented horizontally.
- The major axis is along the \( x \)-axis and extends from -5 to 5.
- The minor axis along the \( y \)-axis extends from -4 to 4.
- The path is traced in a counterclockwise direction, starting from the far right.
4. **Bottom Right Graph:**
- Displays a rightward rotated ellipse.
- Arrows indicate the path is traced in a counterclockwise direction.
**Illustrative Notes:**
- The arrows in each graph illustrate how the parameter \( t \) affects the direction of the trace on each curve.
- Understanding these directions and shapes helps in visualizing how parametric equations determine the path of an object in space.
This visualization is crucial for the study of parametric equations and their application in modeling periodic and oscillatory motions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F02406fc0-0b75-44cf-ad17-4f97ce3b4869%2F99342899-25a5-4b69-9606-fb6c44cc7ac8%2F6654tyn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Parametric Equations and their Graphs**
A pair of parametric equations is given:
\[ x = 4 \cos(t), \quad y = 5 \sin(t), \quad 0 \leq t \leq 2\pi \]
**Task:**
(a) Sketch the curve represented by the parametric equations. Use arrows to indicate the direction of the curve as \( t \) increases.
**Graph Descriptions:**
The image contains four plots of ellipses on Cartesian coordinates, each illustrating different orientations of the curve defined by the parametric equations.
1. **Top Left Graph:**
- Displays an ellipse centered at the origin (0,0).
- The major axis is along the vertical \( y \)-axis, extending from -5 to 5.
- The minor axis is along the horizontal \( x \)-axis, extending from -4 to 4.
- Red arrows indicate a counterclockwise direction of the curve as \( t \) increases.
2. **Top Right Graph:**
- Similar ellipse oriented downward.
- The major axis remains along the vertical \( y \)-axis.
- The elliptical path is traced in a counterclockwise direction from the top of the ellipse.
3. **Bottom Left Graph:**
- The ellipse is oriented horizontally.
- The major axis is along the \( x \)-axis and extends from -5 to 5.
- The minor axis along the \( y \)-axis extends from -4 to 4.
- The path is traced in a counterclockwise direction, starting from the far right.
4. **Bottom Right Graph:**
- Displays a rightward rotated ellipse.
- Arrows indicate the path is traced in a counterclockwise direction.
**Illustrative Notes:**
- The arrows in each graph illustrate how the parameter \( t \) affects the direction of the trace on each curve.
- Understanding these directions and shapes helps in visualizing how parametric equations determine the path of an object in space.
This visualization is crucial for the study of parametric equations and their application in modeling periodic and oscillatory motions.
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