sec10_1 - 10.1 Curves Dened by Parametric Equations 1...

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10.1 Curves Defined by Parametric Equations 1. Parametric Equations Definition 1.1. A set of equations that are defined using a single independent variable are called parametric equations . Typically, we use t , called the parameter , as the independent variable to define the functions x ( t ) and y ( t ) (and perhaps z ( t ) ). These equations define a parametric curve , C , in the plane (or in 3D-space if z ( t ) is given) such that the points on C are the set of points given by ( x ( t ) , y ( t )) . Remark 1.1. All functions can be represented using parametric equations. However, not all parametric equations define curves that can be represented using functions. 2. Examples Example 2.1. Define the curve given by y = x 2 - 4 using parametric equations with t as the parameter. Sketch the curve for t 0 . Remark 2.1. The same curve may be represented using many different parametric equations. 1
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Section 10.1 Curves Defined by Parametric Equations 2 Example 2.2. Eliminate the parameter to find a Cartesian equation of the curve x = cos t , y = sin t and sketch the parametric curve for 0 t 2 π. Include an arrow to indicate the direction of the curve as
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