9.7 Plane Curves & Parametric Equations Objectives

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1 . Graph Parametric Equations 9.7 Plane Curves & Parametric Equations Objectives. Find a Rectangular Equation for a Curve Defined Parametrically. Use Time as a Parameter in Parametric Equations 4. Find Parametric Equations for Curves Defined by Rectangular Equations 4 December, 07 Kidoguchi Kenneth

2 9.7 Plane Curves & Parametric Equations Review The location of a point in space can be specified using: Rectangular Coordinates, often written (x, y) where x represents the signed horizontal distance of the point from the vertical axis and y represents the signed vertical distance of the point from the horizontal axis. Polar Coordinates, (r, q) where r represents the signed distance of the point from the pole (i.e., a position vector) and q represents the angle position of r. Parametric Equations, (x(t), y(t)) where x(t) represents the horizontal location as a function of t and y(t) represents the vertical location as a function of t. 4 December, 07 Kidoguchi Kenneth

3 y = r(q) sin(q) 9.7 Plane Curves & Parametric Equations 7.6 Polar Coordinates: Locating a Point in Polar Coordinates on the Cartesian Plane Given r(q), sin(q) = y/r y = r(q) sin(q) 4p/6 p/6 p/6 (r,q ) cos(q) = x/r x = r(q) cos(q) 5p/6 p/6 x + y = r If: (r, q) = (4, p/) ( x, y) 4 p 4cos,,, 4 p 4sin p 7p/6 q x = r(q) cos(q) 4 p/6 8p/6 0p/6 9p/6 4 December, 07 Kidoguchi Kenneth

4 9.7 Plane Curves & Parametric Equations 7.6 Polar Coordinates: Locating a Point in Polar Coordinates q = angular displacement of the point. r = linear distance of a point from pole. r < 0 means the point is located by a reflection through the pole. r = 0 means the point is at the pole. (r,q ) = (, p/6) (r,q ) = (4, p/) (r,q ) = (-4, p/) p 5p/6 7p/6 (r,q ) 4p/6 p/6 p/6 p/6 4 8p/6 0p/6 9p/6 (r,q ) (r,q ) p/6 4 December, 07 4 Kidoguchi Kenneth

5 9.7 Plane Curves & Parametric Equations Locating a Point in Polar Coordinates on the Cartesian Plane Let the angle q be a function of time t so that: 4p/6 y p/6 q(t) = wt - f t + 0 Consider a point located in -space by a polar equation 5p/6 p/6 r(t) = + cos(t), where r(t) represents the distance of the point from the pole at time t. p 0p/ 9p/ 6p/ 5p/ p/ 4p/ 8p/ 7p/ p/6 4 x Then: x(t) = r(t)cos(t) y(t) = r(t)sin(t) 7p/6 p/6 x(0p/) x(4p/) x(p/) x(p/) x(p/) x(5p/) x(6p/) x(7p/) x(8p/) x(9p/) x(0p/) = r(4p/)cos(4p/) r(0p/)cos(0p/) r(p/)cos(p/) r(p/)cos(p/) r(p/)cos(p/) r(5p/)cos(5p/) r(6p/)cos(6p/) r(7p/)cos(7p/) r(8p/)cos(8p/) r(9p/)cos(9p/) r(0p/)cos(0p/) y(0p/) y(4p/) y(p/) y(p/) y(p/) y(5p/) y(6p/) y(7p/) y(8p/) y(9p/) y(0p/) = r(4p/)sin(4p/) r(0p/)sin(0p/) r(p/)sin(p/) r(p/)sin(p/) r(p/)sin(p/) r(5p/)sin(5p/) r(6p/)sin(6p/) r(7p/)sin(7p/) r(8p/)sin(8p/) r(9p/)sin(9p/) r(0p/)sin(0p/) 8p/6 0p/6 4 December, 07 6 Kidoguchi Kenneth

6 9.7 Plane Curves & Parametric Equations Comparison of Graphs in Cartesian and Polar Coordinates Polar form: r(t) = t Parametric form: x(t) = r(t) cos(t) = t cos(t) y(t) = r(t) sin(t) = t sin(t) x(p/) x(p/) x(5p/6) = p/ p/ 5p/6 cos(p/) cos(p/) cos(5p/6) y(p/) y(p/) y(5p/6) = p/ p/ 5p/6 sin(p/) sin(p/) sin(5p/6) p 5p/6 7p/6 Archimedes Spiral 4p/6 p/6 p/6 4 p/6 0 p/6 8p/6 0p/6 9p/6 4 December, 07 7 Kidoguchi Kenneth

7 9.7 Plane Curves & Parametric Equations A System of Linear Equations Consider the system of parametric equations: x(t) = t, horizontal coordinate as a function of the parameter t. y(t) = t, vertical coordinate as a function of the parameter t. In the xy-plane (aka phase plane) we can create a "parametric plot" with the ordered pairs (x(t), y(t)). t x y 5 4 y x 4 December, 07 9 Kidoguchi Kenneth

8 x(t) or y(t) Plane Curves & Parametric Equations A System of Linear Equations Consider the system of parametric equations: x(t) = t, horizontal coordinate as a function of the parameter t. y(t) = t, vertical coordinate as a function of the parameter t. In the xy-plane (aka phase plane) we can create a "parametric plot" with the ordered pairs (x(t), y(t)). Note that in this case, the system can be decoupled so that: y = ½ x. y t December, 07 0 Kidoguchi Kenneth 5 4 t = 0 (0,0) t = (,) t = (4,) x

9 x(t) or y(t) Plane Curves & Parametric Equations An Ellipse Consider the system of parametric equations: x(t) = cos(pt) - y(t) = sin(pt) + In the xy-plane (aka phase plane) we can create a "parametric plot" with the ordered pairs (x(t), y(t)). x + y - cos( pt) sin( pt) x y cos ( pt) + sin t (-,) (-,4) (-,0) 4 December, 07 Kidoguchi Kenneth y ( pt) (0,) x

10 9.7 Plane Curves & Parametric Equations Graphs of Parametric Equations Graph the motion of a particle whose position at time t is given x(t) and y(t) in the xy-plane. x(t) 4 y(t) t t 4 December, 07 Kidoguchi Kenneth

11 x(t) Plane Curves & Parametric Equations Graphs of Parametric Equations t y(t) t December, 07 Kidoguchi Kenneth x

12 x(t) 4 y Plane Curves & Parametric Equations Graphs of Parametric Equations y(t) t December, 07 4 Kidoguchi Kenneth x 0 < t < < t < < t < < t < 4 4 < t < 5 5 < t < 6 t

13 9.7 Plane Curves & Parametric Equations A Coupled System The Parametric Equations of Motion Consider Galileo s kinematic equations, x(t) = v 0 cos(a) t + x 0 y(t) = -½ gt + v 0 sin(a) t + y 0 where x(t) and y(t) are the parametric equations for the coordinates of a particle at time t and x(0) = x 0, y(0)=y 0. The particle s initial speed and initial trajectory angle are v 0 and q 0 respectively. The acceleration due to gravity g is assumed constant. y v 0 a The trajectory of a projectile is described by Galileo s kinematic equations. Present the analysis to find an expression for the vertical position of the projectile, y, as a function of its horizontal position, x. x 4 December, 07 5 Kidoguchi Kenneth

14 9.7 Plane Curves & Parametric Equations Decoupling A System of Parametric Equations Equations {} & {} are the kinematic equations with (x 0, y 0 ) = (0, 0), {} x(t) = v 0 cos(a) t {} y(t) = -½ gt + v 0 sin(a) t Show position coordinates as functions of time. The system can be decoupled as y(x), vertical position as a function of horizontal position. 4 December, 07 6 Kidoguchi Kenneth

15 The hour hand and the minute hand of a clock are cm and 4 cm long, respectively. The origin of the xy-plane is at the centre of the clock face and the positive x-axis and zero radians, goes through the three o'clock position. Let t = 0 be a time when both hands are co-terminal (e.g., both hands are coterminal at midnight). Present the analysis to find an exact value of t for the next occurrence of angular coincidence. 9.7 Plane Curves & Parametric Equations Parametric Equations of Sorts December, 07 8 Kidoguchi Kenneth

16 4 December, 07 9 Kidoguchi Kenneth

17 5 m 9.7 Plane Curves & Parametric Equations Graphs of Parametric Equations A Double Ferris Wheel has a 0 metre rotating arm attached at its centre to a 5 metre main support. At each end of the rotating arm is attached a Ferris Wheel measuring 0 metres in diameter. Each wheel rotates in the direction indicated. It takes the rotating arm 6 minutes to complete one revolution and it takes 4 minutes for each Ferris Wheel to complete a revolution about its hub. Main Support At t = 0, a reference point is at the position indicated in the figure. Present the analysis to: Reference t = 0 y (0,0) 0 m 0m x a) find x(t) and y(t), the horizontal and vertical coordinates of the reference point as functions of t, time in minutes, and b) sketch a graph of x vs y over one complete cycle. 4 December, 07 0 Kidoguchi Kenneth

18 5 m 9.7 Plane Curves & Parametric Equations Graphs of Parametric Equations Step : Analyse the motion of the reference point with respect to the center of the wheel rotation, i.e., ignoring the arm rotation. Let x W (t) and y W (t) be the horizontal and vertical coordinates reference position as functions of t. x W (t) = y W (t) = Main Support Reference t = 0 y (0,0) 0 m 0m x 4 December, 07 Kidoguchi Kenneth

19 5 m 9.7 Plane Curves & Parametric Equations Graphs of Parametric Equations Step : Analyse the motion of the reference point with respect to the center of the wheel rotation, i.e., ignoring the arm rotation. Let x W (t) and y W (t) be the horizontal and vertical coordinates reference position as functions of t. x W (t) = 0 cos(p/ t) y W (t) = 0 sin(p/ t) Sanity Check: x W (0) = 0 cos(p/ 0) = 0 y W (0) = 0 sin(p/ 0) = 0 x W () = 0 cos(p/ ) = 0 y W () = 0 sin(p/ ) = 0 Main Support Reference t = 0 4 December, 07 Kidoguchi Kenneth y (0,0) 0 m 0m x

20 5 m 9.7 Plane Curves & Parametric Equations Graphs of Parametric Equations Step : Analyse the motion of the hub, i.e., the appropriate end point of the rotating arm. Let M x (t) and M y (t) be the horizontal and vertical coordinates of the hub as a function of t. M x (t) = Main Support Hub t = 0 y 0m M y (t) = (0,0) 0 m x 4 December, 07 Kidoguchi Kenneth

21 5 m 9.7 Plane Curves & Parametric Equations Graphs of Parametric Equations Step : Analyse the motion of the hub, i.e., the appropriate end point of the rotating arm. Let M x (t) and M y (t) be the horizontal and vertical coordinates of the hub as a function of t. M x (t) = 5 cos(p/ t) M y (t) = 5 sin(p/ t) + 5 Sanity Check: M x (0) = 5 cos(p/ 0) = 5 M y (0) = 5 sin(p/ 0) + 5 = 5 M x () = 5 cos(p/ ) = -5 M y () = 5 sin(p/ ) + 5 = 5 Main Support Hub t = 0 y (0,0) 0 m 0m x 4 December, 07 4 Kidoguchi Kenneth

9.7 Plane Curves & Parametric Equations Graphs of Parametric Equations Step : Replace the midline of the wheels equations with the hub equation to get the coordinates of the reference position as a

22 9.7 Plane Curves & Parametric Equations Graphs of Parametric Equations Step : Replace the midline of the wheels equations with the hub equation to get the coordinates of the reference position as a function of time: x(t) = x W (t) + M x (t) = 0 cos(p/ t) + 5 cos(p/ t) y(t) = y W (t) + M y (t) = 0 sin(p/ t) + 5 sin(p/ t) + 5 Double Ferris Wheel Parametric Plot 4 December, 07 5 Kidoguchi Kenneth

23 9.7 Plane Curves & Parametric Equations Graphs of Parametric Equations Given: x(t) = cos( p t), y(t) = sin( p t), 0 < t < a) Graph the given system of parametric equations in the xy-plane and show its orientation. b) Find the rectangular equation for the curve. 4 December, 07 6 Kidoguchi Kenneth

24 9.7 Plane Curves & Parametric Equations Graphs of Parametric Equations Given: x(t) = sin( p t), y(t) = cos( p t), 0 < t < a) Graph the given system of parametric equations in the xy-plane and show its orientation. b) Find the rectangular equation for the curve. 4 December, 07 7 Kidoguchi Kenneth

25 9.7 Plane Curves & Parametric Equations Graphs of Parametric Equations Given: x(t) = pt - sin( p t), y(t) = - cos( p t), 0 < t < a) Graph the given system of parametric equations in the xy-plane and show its orientation. b) Find the rectangular equation for the curve. 4 December, 07 8 Kidoguchi Kenneth

Goals: Course Unit: Describing Moving Objects Different Ways of Representing Functions Vector-valued Functions, or Parametric Curves

Block #1: Vector-Valued Functions Goals: Course Unit: Describing Moving Objects Different Ways of Representing Functions Vector-valued Functions, or Parametric Curves 1 The Calculus of Moving Objects Problem.