RogawskiET3E_InstructorsSolutionsManual_ch11 - 11 PARAMETRIC EQUATIONS POLAR COORDINATES AND CONIC SECTIONS 11.1 Parametric Equations(LT Section 12.1

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11 PARAMETRIC EQUATIONS, POLAR COORDINATES, AND CONIC SECTIONS 11.1 Parametric Equations (LT Section 12.1) Preliminary Questions 1.Describe the shape of the curvex=3 cost,y=3 sint. 9.Also, each point on the circle x 2 + y 2 = 9 can be represented in the form ( 3 cos t, 3 sin t) for some value of t . We conclude that the curve x = 3 cos t , y = 3 sin t is the circle 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=4t2? 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 to c 1 (t) = (f(t),f(t)) , where f(t) is an increasing function. (b) Compare that path to the path of a second ant crawling according to c 2 (t) = f( 2 t),f( 2 t)) . solution (a) At any time t , the ant’s x -coordinate is equal to its y -coordinate, since they are both equal to f(t) . Thus the ant is always on the line y = x ; since f(t) is increasing, the ant is always moving up and to the right. (b) At any time t , the ant’s x -coordinate is equal to its y -coordinate, since they are both equal to f(t) . Thus the ant is always on the line y = x ; since f(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, from t = 0 to t = 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 )) when t = 1 2 . 6. Find three different parametrizations of the graph of y = x 3 . solution For example, if we let x = t , we get the parametrization (t,t 3 ) . Letting x = 2 t gives ( 2 t, 8 t 3 ) . Finally, let y = t , then the parametrization is (t 1 / 3 ,t) . 1501
1502 C H A P T E R 11 PARAMETRIC EQUATIONS, POLAR COORDINATES, AND CONIC SECTIONS (LT CHAPTER 12) 7. Match the derivatives with a verbal description: (a) dx dt (b) dy dt (c) dy dx (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 time solution (a) The derivative dx dt is the horizontal rate of change with respect to time.