Parametric Equations 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 =...

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: Since y = t + 1, thus t = y − 1. Then sub it into the first equation x = ( y − 1) x = y2 − 2 y +1 2 or parabola 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, x2 + y 2 = 1 Circle of radius 1, center at (0,0) Example 4: Example 5: x = t, y = t2, −  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 1. x = 3t , y = 9t 2 , −  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