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11PARAMETRIC EQUATIONS,POLAR COORDINATES, ANDCONIC SECTIONS11.1Parametric Equations(LT Section 12.1)Preliminary Questions1.Describe the shape of the curvex=3 cost,y=3 sint.9.Also, each point on the circlex2+y2=9 can be representedin the form(3 cost,3 sint)for some value oft. We conclude that the curvex=3 cost,y=3 sintis thecircle of radius 3 centered at the origin.2.How doesx=4+3 cost,y=5+3 sintdiffer from the curve in the previous question?3.What is the maximum height of a particle whose path has parametric equationsx=t9,y=4−t2?4.4.Can the parametric curve(t,sint)be represented as a graphy=f(x)? What about(sint,t)?5. (a)Describe the path of an ant that is crawling along the plane according toc1(t)=(f(t),f(t)), wheref(t)is an increasing function.(b)Compare that path to the path of a second ant crawling according toc2(t)=f(2t),f(2t)).solution(a)At any timet, the ant’sx-coordinate is equal to itsy-coordinate, since they are both equal tof(t). Thusthe ant is always on the liney=x; sincef(t)is increasing, the ant is always moving up and to the right.(b)At any timet, the ant’sx-coordinate is equal to itsy-coordinate, since they are both equal tof(t). Thusthe ant is always on the liney=x; sincef(t)is increasing, the ant is always moving up and to the right.This is the same path as in part (a), but the ant is moving at a different speed. For example, fromt=0 tot=1, the first ant moves from(f(0),f(0))to(f(1),f(1)), while the second moves from(f(0),f(0))to(f(2),f(2)), passing through(f(1),f(1))whent=12.6.Find three different parametrizations of the graph ofy=x3.solutionFor example, if we letx=t, we get the parametrization(t,t3). Lettingx=2tgives(2t,8t3).Finally, lety=t, then the parametrization is(t1/3,t).1501
1502C H A P T E R11PARAMETRIC EQUATIONS, POLAR COORDINATES, AND CONIC SECTIONS(LT CHAPTER 12)7.Match the derivatives with a verbal description:(a)dxdt(b)dydt(c)dydx(i)Slope of the tangent line to the curve(ii)Vertical rate of change with respect to time(iii)Horizontal rate of change with respect to timesolution(a)The derivativedxdtis the horizontal rate of change with respect to time.