Parametrizations of Plane Curves PDF

Answer  10.1 Parametrizations of Plane Curves Parametric Equation Cartesian equation: An equation in terms of x and y only Example 1: Sketch the curve defined by the parametric equations t x sin , y t ,...

Answer  10.1 Parametrizations of Plane Curves Parametric Equation Cartesian equation: An equation in terms of x and y only Example 1: Sketch the curve defined by the parametric equations t x sin , y t , 0 t 6 2 Cartesian equation of the parametric equations is: y x sin 2 Example 2: Sketch the curve defined by the parametric equations x t 2 , y t  1,    t   Cartesian equation of the parametric equations is: Sinc y t  1, thust  y  1. Then sub it into the first e equation 2 x  y  1 or x  y 2  2 y 1 parabol a Example 3: x cos t , y sin t , 0 t 2 t x y 0 1 0  0 1 2  -1 0 3 0 -1 2 2 1 0 Cartesian equation of the parametric equations is: Note that from the identities, Thus, 2 2 x  y 1 Circle of radius 1, center at (0,0) Example 4: Example 5: x t , y t 2 ,  t   Exercises Exercise 10.1 The exercises give parametric equations and parameter intervals for the motion of a particle in in the x-y plane. a) Identify the particles' path by finding a Cartesian equation for it. b) Graph the Cartesian Equation. c) Indicate the portion of the graph traced by the particle and the direction of motion 2 1. x 3t , y 9t ,  t  2. x t / 2, y t  1,  t  3. x sin 2t , y cos 2t , 0  t  2 4. x cos   t , y sin   t  0 t  5.x 4 cos t , y 4 sin t 0  t  2   3 6.x 3  2 cos t , y 1  2 sin t t  2 2  0 t  7.x 1  sin 5t , y cos 5t  2