is a plane curve and the equations are parametric equations for the curve, with parameter t.

Transcription

1 MATH 2412 Sections 6.3, 6.4, and 6.5 Parametric Equations and Polar Coordinates. Plane Curves and Parametric Equations Suppose t is contained in some interval I of the real numbers, and = f( t), = gt ( ), for t I Then the set of points in the plane with coordinate ( f(), t gt ()) is a plane curve and the equations are parametric equations for the curve, with parameter t. Eample: Sketch the curve defined b the parametric equations 2 t t t = 3, = 1 The Orientation of the Curve: Eliminating the Parameter

2 Eample: Graph the parametric equations sin t, cos t, for t ( 0,2π ] = =. Show the orientation of the graph. Eliminate the parameter, writing an equation in terms of and. 1 =, =, for 1, 4. Show the t orientation of the graph. Eliminate the parameter, writing an equation in terms of and. Eample: Graph the parametric equations t 2 t [ ] Application An object is projected upward from the top of a building 480 feet high with an initial speed of 96 feet per second and at an angle of elevation of 30 degrees. Using the equations given below, determine the maimum height of the object, the length of time the object remains in the air, and the horizontal distance the object travels. 2 ( ) ( ) t ( ) = υ cos θ t, t ( ) = 16t + υ sinθ t+ d 0 0

3 Application: The Ccloid As a circle rolls along a straight line, the curve traced out b a fied point P on the circumference of a circle is called a ccloid Let s derive the parametric equations for the ccloid = a( θ sin θ) = a(1 cos θ) The ccloid has a number of interesting phsical properties. It is both the curve of quickest descent and the curve of equal descent

4 After we stud Polar Coordinates, we will return and look at the Parametrization of Polar Coordinates. Parametric Equations-Matching In-Class Problem. Work in groups of 2 or 3. Match the graphs of the parametric equations in (a)-(d) with the parametric curves labeled I-IV. Give reasons for our choices

5 Problem: work in groups of 2 or 3 Compare the curves represented b the parametric equations. How do the differ? How are the alike? Show the orientation of each curve on a separate graph. Eliminate the parameter in part (c). (a) (b) (c) = t, = t t = e, = e 2 2t 2 = cos t, = sec t

6 Polar Coordinates Developed b Isaac Newton Defined b two coordinates, r and θ Plotting points in Polar Coordinates: Eample: Plot the point 2π 2, on the polar graph above. 3

7 An Infinite Number of Representations Eample: Find three other polar representations of this point. (a) π, 3 (b) ( 2, ) (c) ( 2, ) Translation Equations: Find the Cartesian coordinates of 2π 2, 3 Graphs in Polar Coordinates: Eample: Graph the polar equation (a) π θ = b) r = 4 4

8 Eample: Graph the polar equation r = 4 cosθ for θ [ )., on the polar graph. Eample: Find an equivalent equation in rectangular coordinates for the following (a) r = 1 (b) 6 r = 3cosθ 2sinθ Eample: Find an equivalent equation in polar coordinates for =

9 Smmetr in Polar Graphs Look at the following graphs in polar coordinates. The are superimposed on the plane. Notice the smmetr. Are ou able to offer an eplanation considering our knowledge of the sine and cosine functions? Tests for smmetr:

10 Graphing Polar Equations with a calculator: Consider the following polar equations r = cos( 2θ 3), r = sin ( 8θ 5), or r = sinθ + sin 3 ( 5θ 2) In order to graph these accuratel, we need to determine the domain for θ. In other words, how man times must θ go through a complete rotation ( 2π radians) before the graph starts to repeat itself? ( θ + nπ) 2 2 In the first eample, we need to find an integer n so that cos = cos( 2θ 3). 3 For this inequalit to hold 4 nπ must be a multiple of 2π. This happens when n = 3. 3 Therefore, we can obtain the entire graph if we choose values of θ between θ = 0 and θ = 6π Students, work together on the remaining 2 polar equations. Then, have fun with our graphing calculator! Do not memorize the common polar curves and their equations. You should know the polar equations for a circle centered at the origin and a line passing through the origin.