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B. Find the Cartesian equation for the following parametric equations (a) x = e", y = t+1 (b) x = 1-t2, y = t- 2 %3D

B. Find the Cartesian equation for the following parametric equations (a) x = e", y = t+1 (b) x = 1-t2, y = t- 2 %3D

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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**Exercises for Chapter Review** 8. **Find the Cartesian equation for the following parametric equations:** (a) \( x = e^{2t}, \, y = t + 1 \) (b) \( x = 1 - t^2, \, y = t - 2 \) 9. **Find the area of the region bounded by the polar curve** \( r = \tan \theta \) **in the sector** \( \frac{\pi}{6} \leq \theta \leq \frac{\pi}{3} \). 10. **On the attached direction field, sketch the solution curve for the** \( y' = \sin(x) \sin(y) \) **with the initial condition:** (a) \( y(0) = 1 \) (b) \( y(0) = -2 \) (c) \( y(0) = 0 \) **Direction Field Diagram Explanation:** The diagram displays a direction field, which consists of small arrows or line segments indicating the slope of the solution curves at various points in the coordinate plane. The grid spans from \(-3\) to \(4\) on both the x and y axes. The red arrows suggest the behavior of solutions to the differential equation \( y' = \sin(x) \sin(y) \) at given points, showing how the slope varies across different regions of the plane.
Transcribed Image Text:**Exercises for Chapter Review** 8. **Find the Cartesian equation for the following parametric equations:** (a) \( x = e^{2t}, \, y = t + 1 \) (b) \( x = 1 - t^2, \, y = t - 2 \) 9. **Find the area of the region bounded by the polar curve** \( r = \tan \theta \) **in the sector** \( \frac{\pi}{6} \leq \theta \leq \frac{\pi}{3} \). 10. **On the attached direction field, sketch the solution curve for the** \( y' = \sin(x) \sin(y) \) **with the initial condition:** (a) \( y(0) = 1 \) (b) \( y(0) = -2 \) (c) \( y(0) = 0 \) **Direction Field Diagram Explanation:** The diagram displays a direction field, which consists of small arrows or line segments indicating the slope of the solution curves at various points in the coordinate plane. The grid spans from \(-3\) to \(4\) on both the x and y axes. The red arrows suggest the behavior of solutions to the differential equation \( y' = \sin(x) \sin(y) \) at given points, showing how the slope varies across different regions of the plane.
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