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Lesson 10. Parametric Curves
(A) Parametric Curves
If a curve fails the Vertical Line Test, it can’t be expressed by a function. In this case you will encounter a problem if you try to find the slope of a tangent to the curve, or the area enclosed by the curve. In calculus this problem can be solved if the curve is expressed by a pair of parametric equations: x = f (t) , y = g(t) , where t is the parameter with the range of a ≤ t ≤ b in general.
Each value of the parameter t determines a point (x, y) = ( f (t), g(t)). The initial point is
()f (a), g(a) and the terminal point is ( f (b), g(b)). As t varies, the point ()x, y will trace out a curve that is called parametric curve. In many applications of parametric curves, t stands for time, but does not always necessarily represent time. In this section we will learn how to sketch the parametric curves using Mathematica.
Example1 Sketch and identify the curve given by the parametric equations x = t 2 − 2t, y = t +1 for − 2 ≤ t ≤ 4 .
Each value of the parametert gives a point on the curve, which can be evaluated by: