9.1 Parametric Equations

1-8. (a) Find the Cartesian equation of the curve.

      (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.


1. , , .

 From the second equation, we have . Now putting the first equation, we set . Notice that if , we have . Thus the Cartesian equation of the given curve is ,

 (a) .

  (b) http://matrix.skku.ac.kr/cal-lab/cal-9-1-1.html           

t=var('t');

parametric_plot((t^2-2*t,3-t),(t,0,1)).show(aspect_ratio=1, xmin=-3, xmax=3, ymin=-3, ymax=3)

parametric_plot((t^2-2*t,3-t),(t,0,2)).show(aspect_ratio=1, xmin=-3, xmax=3, ymin=-3, ymax=3)

parametric_plot((t^2-2*t,3-t),(t,0,3)).show(aspect_ratio=1, xmin=-3, xmax=3, ymin=-3, ymax=3)


  


2. , , .

 (a) .

(b) http://matrix.skku.ac.kr/cal-lab/cal-9-1-2.html

t=var('t');

parametric_plot((sin(t)^2,cos(3*t)),(t,0,1/4*pi)).show(aspect_ratio=1, xmin=-1.5, xmax=1.5, ymin=-0.5, ymax=1.5)

parametric_plot((sin(t)^2,cos(3*t)),(t,0,2/4*pi)).show(aspect_ratio=1, xmin=-1.5, xmax=1.5, ymin=-0.5, ymax=1.5)

parametric_plot((sin(t)^2,cos(3*t)),(t,0,3/4*pi)).show(aspect_ratio=1, xmin=-1.5, xmax=1.5, ymin=-0.5, ymax=1.5)

parametric_plot((sin(t)^2,cos(3*t)),(t,0,4/4*pi)).show(aspect_ratio=1, xmin=-1.5, xmax=1.5, ymin=-1, ymax=1.5)


                

                




3. , .

 (a) .

 (b) http://matrix.skku.ac.kr/cal-lab/cal-9-1-3.html

t=var('t');

parametric_plot((sin(t)^2,cos(t)^2),(t,0,1/2)).show(aspect_ratio=1, xmin=-2, xmax=2, ymin=-1, ymax=2)

parametric_plot((sin(t)^2,cos(t)^2),(t,0,1)).show(aspect_ratio=1, xmin=-2, xmax=2, ymin=-1, ymax=2)

parametric_plot((sin(t)^2,cos(t)^2),(t,0,2)).show(aspect_ratio=1, xmin=-2, xmax=2, ymin=-1, ymax=2)


4. , , .7

 (a) .

 (b) http://matrix.skku.ac.kr/cal-lab/cal-9-1-4.html

t=var('t');

parametric_plot((ln(t),t-1),(t,1,3/2)).show(aspect_ratio=1, xmin=-1, xmax=1, ymin=-1, ymax=1.5)

parametric_plot((ln(t),t-1),(t,1,2)).show(aspect_ratio=1, xmin=-1, xmax=1, ymin=-1, ymax=1.5)

parametric_plot((ln(t),t-1),(t,1,5/2)).show(aspect_ratio=1, xmin=-1, xmax=1, ymin=-1, ymax=1.5)



5. , .


6. , .


7. , , .

 (a) Since .

 (b) http://matrix.skku.ac.kr/cal-lab/cal-9-1-7.html

tau = var('tau');

parametric_plot((sin(tau),tan(tau)),tau,0,4*pi).show(aspect_ratio=1, xmin=0, xmax=0.7, ymin=0, ymax=1)



8. , .


9. Find a parametric equation for the path of a particle that moves along in the manner described below.

   (a) Once around clockwise, starting at (3, 1).

   (b) Twice around counterclockwise, starting at (3, 1).

   (c) Halfway around counterclockwise, starting at (1, 3).

   (d) Graph the semicircle traced by the particle.

 The circle has center (1, 1) and radius 2, so it can be represented by . This representation gives us the circle with a counterclockwise orientation starting (3, 1).

      (a) To get a clockwise orientation, we can replace by in the equations to get,

.

      (b) To get twice around in the counterclockwise direction, we use the original equations

 with the domain expanded to .

      (c) To start at (1, 3) using the original equations, we must have ; that is, and . We use the original equations with the domain to be a counterclockwise orientation starting (1, 3).

      (d) Graph it by Sage.


10.(a)Find parametric equation for the ellipse .

    (b) Sketch the ellipse when , and

    (c) How does the shape of the ellipse change as varies?

 (a) Let , and to obtain with as possible parametric equations for the ellipse .

 (b) http://matrix.skku.ac.kr/cal-lab/cal-9-1-10.html

t=var('t');

parametric_plot((3*cos(t),sin(t)),(t,0,2*pi)).show(aspect_ratio=1, xmin=-3, xmax=3, ymin=-4, ymax=4)

parametric_plot((3*cos(t),2*sin(t)),(t,0,2*pi)).show(aspect_ratio=1, xmin=-3, xmax=3, ymin=-4, ymax=4)

parametric_plot((3*cos(t),4*sin(t)),(t,0,2*pi)).show(aspect_ratio=1, xmin=-3, xmax=3, ymin=-4, ymax=4)



(c) As b increases, the ellipse stretches vertically.


11. Find three different sets of parametric equations to represent the curve , .

  ⅰ) ,  ⅱ) ,  ⅲ) .


12. Suppose that the position of the particle at time is given by , ,   and the position of the second particle is given by , , .

    (a) Graph the paths of both particles. How many points of intersection are there?

    (b) Are any of these points of intersection collision points? In other words, are the particles ever at the same place at same time? If so, find the collision points.

    (c) Describe what happens if the path of the second particle is given by , .

 

(a) There are 4-points of intersection :

http://matrix.skku.ac.kr/cal-lab/cal-9-1-12.html 

t=var('t');

p1=parametric_plot((sin(t),3*cos(t)),(t,0,2*pi)).show(aspect_ratio=1, xmin=-5, xmax=3, ymin=-3, ymax=3)

p2=parametric_plot((-1+2*cos(t),1+sin(t)),(t,0,2*pi)).show(aspect_ratio=1, xmin=-5, xmax=3, ymin=-3, ymax=3)


(b) A collision point occurs when for the same . Consider the following equations :

   

   

   From ②, . Substituting into [1],

   we get

                                .

   We check that satisfies [1], [2], but does not.

   So the only collision point occurs when , and this gives the point (-2, 0).

   [We could check our work by graphing together as functions of and on another plot, as functions of . If we do so, we see that the only value of for which both pairs of graphs intersect is .]

(c) The equation , is the circle centered at (2, 1) instead of (-2, 1). There are still 4 intersection points.

        However there are no collision points, since (*) in part (b) becomes .


13. Investigate the family of curves defined by the parametric equations

    .

         How dose the shape change as changes? In particular, you should identify values of for which the basic shape of the curve changes.

 http://matrix.skku.ac.kr/cal-lab/cal-9-1-13.html

t=var('t');

parametric_plot((sin(t)*(3/2-sin(t)),cos(t)*(3/2-sin(t))),(t,0,2*pi)).show(aspect_ratio=1, xmin=-4, xmax=2, ymin=-2, ymax=2)

parametric_plot((sin(t)*(1-sin(t)),cos(t)*(1-sin(t))),(t,0,2*pi)).show(aspect_ratio=1, xmin=-4, xmax=2, ymin=-2, ymax=2)

parametric_plot((sin(t)*(1/2-sin(t)),cos(t)*(1/2-sin(t))),(t,0,2*pi)).show(aspect_ratio=1, xmin=-4, xmax=2, ymin=-2, ymax=2)

parametric_plot((sin(t)*(1/4-sin(t)),cos(t)*(1/4-sin(t))),(t,0,2*pi)).show(aspect_ratio=1, xmin=-4, xmax=2, ymin=-2, ymax=2)