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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![### Graph the Parametric Equations
Consider the following parametric equations:
\[
\begin{cases}
x(t) = \cot t \\
y(t) = \sin^2 t
\end{cases}
\]
To graph these equations, we need to understand the relationships between \( x \) and \( t \), and \( y \) and \( t \).
- **\( x(t) = \cot t \)**: This represents the cotangent of \( t \). The cotangent function is the reciprocal of the tangent function and can be expressed as \( \frac{\cos t}{\sin t} \).
- **\( y(t) = \sin^2 t \)**: This represents the square of the sine function.
When graphing these equations:
1. **Range and Domain**:
- For \( \cot t \), \( t \) must not be an integer multiple of \( \pi \) (where \( t \neq k\pi \) for any integer \( k \)) because the sine function will be zero at those points, causing \( \cot t \) to be undefined. The cotangent function oscillates between positive and negative infinity with vertical asymptotes at these points.
- For \( \sin^2 t \), the function always yields values between 0 and 1 since \( \sin t \) ranges from -1 to 1.
2. **Key Points and Behavior**:
- As \( t \to 0 \) (where \( t \) approaches zero from the right), \( \cot t \to \infty \) and \( y \) is small but positive since \( \sin t \) is approaching zero but squared.
- As \( t \to \pi/2 \) or any odd multiple of \( \pi/2 \), \( \cot t \to 0 \) and \( y \to 1 \).
- As \( t \to \pi \), \( \cot t \to -\infty \).
3. **Interpreting the Graph**:
- Plot \( x(t) \) on the horizontal axis and \( y(t) \) on the vertical axis. The graph will display the trajectory of the point \((x(t), y(t))\) as \( t \) varies.
- The graph will present a periodic behavior due to the periodic nature of](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcdbdf4c8-5459-45de-99f8-cd8d71031beb%2F27d6de36-2c81-47da-bd58-b4582865e11a%2Ftl5fdu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Graph the Parametric Equations
Consider the following parametric equations:
\[
\begin{cases}
x(t) = \cot t \\
y(t) = \sin^2 t
\end{cases}
\]
To graph these equations, we need to understand the relationships between \( x \) and \( t \), and \( y \) and \( t \).
- **\( x(t) = \cot t \)**: This represents the cotangent of \( t \). The cotangent function is the reciprocal of the tangent function and can be expressed as \( \frac{\cos t}{\sin t} \).
- **\( y(t) = \sin^2 t \)**: This represents the square of the sine function.
When graphing these equations:
1. **Range and Domain**:
- For \( \cot t \), \( t \) must not be an integer multiple of \( \pi \) (where \( t \neq k\pi \) for any integer \( k \)) because the sine function will be zero at those points, causing \( \cot t \) to be undefined. The cotangent function oscillates between positive and negative infinity with vertical asymptotes at these points.
- For \( \sin^2 t \), the function always yields values between 0 and 1 since \( \sin t \) ranges from -1 to 1.
2. **Key Points and Behavior**:
- As \( t \to 0 \) (where \( t \) approaches zero from the right), \( \cot t \to \infty \) and \( y \) is small but positive since \( \sin t \) is approaching zero but squared.
- As \( t \to \pi/2 \) or any odd multiple of \( \pi/2 \), \( \cot t \to 0 \) and \( y \to 1 \).
- As \( t \to \pi \), \( \cot t \to -\infty \).
3. **Interpreting the Graph**:
- Plot \( x(t) \) on the horizontal axis and \( y(t) \) on the vertical axis. The graph will display the trajectory of the point \((x(t), y(t))\) as \( t \) varies.
- The graph will present a periodic behavior due to the periodic nature of
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