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Unformatted text preview: Monday, March 26 − Lecture 33 : Parametric Equations, curves and tangents (Refers to Section 9.1, 9.2) After having practiced the problems associated to the concepts of this lecture the student should be able to : Plot simple parametric equations in the plane, find the equations of all tangent lines to a curve in the plane which is expressed as parametric equations, compute curve lengths. 33.1 Definition − A curve C in the Cartesian plane is called a parametric curve if the points ( x , y ) on the curve C can be represented as ( x ( t ), y ( t )) where t is a variable in an interval [ a , b ] on the real line R . The variable t i s called a parameter . The point ( x ( a ), y ( a )) is called the initial point of the curve, while the point ( x ( b ), y ( b )) is called the terminal point of the curve. The " direction of the curve " is from the initial point to the terminal point. The two equations x = x ( t ) and y = y ( t ) are called its parametric equations of the curve C . A simple example of this is a curve C which represents the trajectory of an object in two-space during a time interval [0, 10], where ( x ( t ), y ( t )) gives the position of the object at time t, t between 0 seconds and 10 seconds. 33.1.1 Examples − Graphing parametric equations in a rudimentary way. Plot the curve whose parametric equations are x = t − 3 sin t and y = 4 − 3 cos t , 0 ≤ t ≤ 10. By drawing out a table and plotting the points we get: (The values of t are in radians, values of x and y are computed with a calculator.) t = 0 x = 0.0 y = 1.0. t = 1 x = − 1.5 y = 2.4. t = 2 x = − 0.7 y = 5.2. t = 3 x = 2.6 y = 7.0. t = 4 x = 6.3 y = 6.0. t = 5 x = 7.9 y = 3.1. t = 6 x = 6.8 y = 1.1. t = 7 x = 5.0 y = 1.7. 33.1.2 Note on how to plot parametric curves − Sometimes parametric equations can easily be transformed into and xy-equation (but this may be difficult or even impossible to do). When trying to plot a parametric curve it is a good idea, if it is possible, to eliminate the parameter t to obtain an expression of a familiar curve in x and y. Example – Plot the parametric curve ( x ( t ) , y ( t )) where and t is in [0,1], by eliminating the parameter and indicate the direction of the curve....
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