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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a) find the parametric equations to describe the ellipse.
b) find an interval of the parameter to describe the path of a particle which moves counter-clockwise halfway along the ellipse starting at (0,6). Justification is not required.
![Certainly! Below is the transcription of the text you provided, formatted to appear on an educational website:
---
### Equation of an Ellipse
The given mathematical expression represents the equation of an ellipse:
\[
\frac{x^2}{9} + \frac{(y-1)^2}{25} = 1
\]
#### Explanation:
This is a standard form of the ellipse equation.
- **Center of the Ellipse:**
The ellipse is centered at the point \( (h, k) \). From the equation, \( h = 0 \) and \( k = 1 \). Thus, the center is at \( (0, 1) \).
- **Semi-major Axis:**
The value under the \( y \)-term is the square of the length of the semi-major axis. Here, \( 25 = b^2 \), so \( b = \sqrt{25} = 5 \). This indicates that the semi-major axis length is 5 units.
- **Semi-minor Axis:**
The value under the \( x \)-term is the square of the length of the semi-minor axis. Here, \( 9 = a^2 \), so \( a = \sqrt{9} = 3 \). This indicates that the semi-minor axis length is 3 units.
#### Graphical Representation:
The graph of this ellipse would show an elongated shape centered at the point \( (0, 1) \), stretching further along the vertical direction (since the semi-major axis is longer) compared to the horizontal direction. The vertices on the major axis are at \( (0, 1 \pm 5) \) and the vertices on the minor axis are at \( ( \pm 3, 1) \).
Understanding the components and characteristics of the ellipse equation is foundational in the study of conic sections in algebra and geometry.
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe41e8fc6-899f-4d69-814e-58571aaa20e4%2F5a043271-da1d-4c2e-936c-90408968fb3a%2Fu71xyg.jpeg&w=3840&q=75)
Transcribed Image Text:Certainly! Below is the transcription of the text you provided, formatted to appear on an educational website:
---
### Equation of an Ellipse
The given mathematical expression represents the equation of an ellipse:
\[
\frac{x^2}{9} + \frac{(y-1)^2}{25} = 1
\]
#### Explanation:
This is a standard form of the ellipse equation.
- **Center of the Ellipse:**
The ellipse is centered at the point \( (h, k) \). From the equation, \( h = 0 \) and \( k = 1 \). Thus, the center is at \( (0, 1) \).
- **Semi-major Axis:**
The value under the \( y \)-term is the square of the length of the semi-major axis. Here, \( 25 = b^2 \), so \( b = \sqrt{25} = 5 \). This indicates that the semi-major axis length is 5 units.
- **Semi-minor Axis:**
The value under the \( x \)-term is the square of the length of the semi-minor axis. Here, \( 9 = a^2 \), so \( a = \sqrt{9} = 3 \). This indicates that the semi-minor axis length is 3 units.
#### Graphical Representation:
The graph of this ellipse would show an elongated shape centered at the point \( (0, 1) \), stretching further along the vertical direction (since the semi-major axis is longer) compared to the horizontal direction. The vertices on the major axis are at \( (0, 1 \pm 5) \) and the vertices on the minor axis are at \( ( \pm 3, 1) \).
Understanding the components and characteristics of the ellipse equation is foundational in the study of conic sections in algebra and geometry.
---
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