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Eliminate the parameter to (a simple!) Cartesian equation for the given parametric equations. Sketch the parametric curve and indicate the direction with arrows. x(1) = cos(1) y(t) = sec(1) 0 <1 < x/2

Eliminate the parameter to (a simple!) Cartesian equation for the given parametric equations. Sketch the parametric curve and indicate the direction with arrows. x(1) = cos(1) y(t) = sec(1) 0 <1 < x/2

Calculus: Early Transcendentals
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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**Title: Converting Parametric Equations to Cartesian Form** **Objective:** To eliminate the parameter and derive a Cartesian equation from given parametric equations. Additionally, sketch the parametric curve and indicate the direction with arrows. **Given Parametric Equations:** - \( x(t) = \cos(t) \) - \( y(t) = \sec(t) \) **Interval:** - \( 0 < t < \pi/2 \) **Instructions:** 1. **Eliminate the Parameter \( t \):** - Start with the identity \(\sec(t) = \frac{1}{\cos(t)}\). - Replace \( y(t) \) using this identity: \( y = \sec(t) = \frac{1}{\cos(t)} \). - Substitute \( x(t) = \cos(t) \) into \( y = \frac{1}{\cos(t)} \) to express \( y \) in terms of \( x \). - The resulting equation is \( y = \frac{1}{x} \). 2. **Expression in Cartesian Coordinates:** - **Cartesian Equation:** \( y = \frac{1}{x} \). 3. **Graphical Representation:** - Sketch the curve based on the Cartesian equation \( y = \frac{1}{x} \). - Indicate the direction of increasing \( t \) with arrows. As \( t \) increases from \( 0 \) to \(\pi/2\), \( x(t) = \cos(t) \) decreases from \( 1 \) to \( 0 \), while \( y(t) = \sec(t) \) increases from \( 1 \) to \( \infty \). **Summary:** This exercise demonstrates how to convert parametric equations to a Cartesian form, which can simplify graphing and analyzing the relationship between variables. The given curve represents a portion of the hyperbola \( y = \frac{1}{x} \) in the specified interval, highlighting the changing direction and behavior of the parametric equations as \( t \) varies.
Transcribed Image Text:**Title: Converting Parametric Equations to Cartesian Form** **Objective:** To eliminate the parameter and derive a Cartesian equation from given parametric equations. Additionally, sketch the parametric curve and indicate the direction with arrows. **Given Parametric Equations:** - \( x(t) = \cos(t) \) - \( y(t) = \sec(t) \) **Interval:** - \( 0 < t < \pi/2 \) **Instructions:** 1. **Eliminate the Parameter \( t \):** - Start with the identity \(\sec(t) = \frac{1}{\cos(t)}\). - Replace \( y(t) \) using this identity: \( y = \sec(t) = \frac{1}{\cos(t)} \). - Substitute \( x(t) = \cos(t) \) into \( y = \frac{1}{\cos(t)} \) to express \( y \) in terms of \( x \). - The resulting equation is \( y = \frac{1}{x} \). 2. **Expression in Cartesian Coordinates:** - **Cartesian Equation:** \( y = \frac{1}{x} \). 3. **Graphical Representation:** - Sketch the curve based on the Cartesian equation \( y = \frac{1}{x} \). - Indicate the direction of increasing \( t \) with arrows. As \( t \) increases from \( 0 \) to \(\pi/2\), \( x(t) = \cos(t) \) decreases from \( 1 \) to \( 0 \), while \( y(t) = \sec(t) \) increases from \( 1 \) to \( \infty \). **Summary:** This exercise demonstrates how to convert parametric equations to a Cartesian form, which can simplify graphing and analyzing the relationship between variables. The given curve represents a portion of the hyperbola \( y = \frac{1}{x} \) in the specified interval, highlighting the changing direction and behavior of the parametric equations as \( t \) varies.
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