Chapter 10 - 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES...

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620So far we have described plane curves by givingas a function oforas a function ofor by giving a relation betweenandthat definesimplicitly as a function of. In this chapter we discuss two new methodsfor describing curves.Some curves, such as the cycloid, are best handled when bothandare given interms of a third variablecalled a parameter. Other curves, such asthe cardioid, have their most convenient description when we use a new coordinatesystem, called the polar coordinate system.xf t,ytttyxf x,y0xyyxxtyyxyf xxyParametric equations and polar coordinates enable us todescribe a great variety of new curves—some practical,some beautiful, some fanciful, some strange.PARAMETRIC EQUATIONSAND POLAR COORDINATES10
CURVES DEFINED BY PARAMETRIC EQUATIONSImagine that a particle moves along the curveCshown in Figure 1. It is impossible todescribeCby an equation of the formbecauseCfails the Vertical Line Test. Butthex- andy-coordinates of the particle are functions of time and so we can writeand. Such a pair of equations is often a convenient way of describing a curve andgives rise to the following definition.Suppose thatandare both given as functions of a third variable(called aparam-eter) by the equations(calledparametric equations). Each value ofdetermines a point, which we canplot in a coordinate plane. Asvaries, the pointvaries and traces outa curve, which we call aparametric curve. The parametertdoes not necessarily repre-sent time and, in fact, we could use a letter other thantfor the parameter. But in manyapplications of parametric curves,tdoes denote time and therefore we can interpretas the position of a particle at timet.EXAMPLE 1Sketch and identify the curve defined by the parametric equationsSOLUTIONEach value ofgives 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 thepointsdetermined by several values of the parameter and we join them to producea curve.A particle whose position is given by the parametric equations moves along the curvein the direction of the arrows asincreases. Notice that the consecutive points marked onthe curve appear at equal time intervals but not at equal distances. That is because theparticle slows down and then speeds up asincreases.It appears from Figure 2 that the curve traced out by the particle may be a parabola.This can be confirmed by eliminating the parameteras follows. We obtainfrom the second equation and substitute into the first equation. This givesand so the curve represented by the given parametric equations is the parabola.Mxy24y3xt22ty122y1y24y3ty1tttFIGURE 20t=0t=1t=2t=3t=4t=_1t=_2(0,1)yx8x,y0, 1y1x0t0tyt1xt22tx,yf t,ttCx,yf t,tttx,ytyttxf ttyxyttxf tyf x10.1txy281130001112203334485NThis equation inanddescribeswheretheparticle has been, but it doesn’t tell uswhenthe

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Term
Spring
Professor
LYLES
Tags
Calculus, Equations, Cartesian Coordinate System, Parametric Equations, Polar Coordinates, Polar coordinate system, Parametric equation, Conic section

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