Chapter10_ Parametric Equations and Polar Coordinates (1) -...

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Chapter 1 / Exercise 1.3
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Parametric Equations and Polar Coordinates 10 So far we have described plane curves by giving as a function of or as a function of or by giving a relation between and that defines implicitly as a function of . In this chapter we discuss two new methods for describing curves. Some curves, such as the cycloid, are best handled when both and are given in terms of a third variable called a parameter . Other curves, such as the cardioid, have their most convenient description when we use a new coordinate system, called the polar coordinate system. y x y f x x y x t y x y y x f x , y 0 x y t x f t , y t t 659 © Dean Ketelsen The Hale-Bopp comet, with its blue ion tail and white dust tail, appeared in the sky in March 1997. In Section 10.6 you will see how polar coordinates provide a convenient equation for the path of this comet.
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Chapter 1 / Exercise 1.3
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660 CHAPTER 10 Imagine that a particle moves along the curve C shown in Figure 1. It is impossible to describe C by an equation of the form because C fails the Vertical Line Test. But the x - and y -coordinates of the particle are functions of time and so we can write and . Such a pair of equations is often a convenient way of describing a curve and gives rise to the following definition. Suppose that and are both given as functions of a third variable (called a param- eter ) by the equations (called parametric equations ). Each value of determines a point , which we can plot in a coordinate plane. As varies, the point varies and traces out a curve , which we call a parametric curve . The parameter t does not necessarily represent time and, in fact, we could use a letter other than t for the parameter. But in many applications of parametric curves, t does denote time and therefore we can interpret as the position of a particle at time t . Sketch and identify the curve defined by the parametric equations SOLUTION Each value of gives a point on the curve, as shown in the table. For instance, if , then , and so the corresponding point is . In Figure 2 we plot the points determined by several values of the parameter and we join them to pro- duce a curve. A particle whose position is given by the parametric equations moves along the curve in the direction of the arrows as increases. Notice that the consecutive points marked on the curve appear at equal time intervals but not at equal distances. That is because the particle slows down and then speeds up as increases. It appears from Figure 2 that the curve traced out by the particle may be a parabola. This can be confirmed by eliminating the parameter as follows. We obtain from the second equation and substitute into the first equation. This gives and so the curve represented by the given parametric equations is the parabola . y f x x f t y t t x y t x f t y t t t x , y t x , y f t , t t C x , y f t , t t x t 2 2 t y t 1 t t 0 x 0 y 1 0, 1 x , y FIGURE 2 0 t=0 t=1 t=2 t=3 t=4 t=_1 t=_2 (0, 1) y x 8 t t t t y 1 x t 2 2 t y 1 2 2 y 1 y 2 4 y 3 EXAMPLE 1 x y 2 4 y 3 10.1 Curves Defined by Parametric Equations C 0 (x, y)={f(t), g(t)} FIGURE 1 y x t x y 2 8 1 1 3 0 0 0 1 1 1 2 2 0 3 3 3 4 4 8 5

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