Calculus-Sec-10-1-Solution

10.1   Parametric Equations                        by SGLee - HSKim - JHLee

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.(a)

From the second equation, we have .

Now from the first equation, we get .

Notice that if , we have .

Thus the Cartesian equation of the given curve is

2.

(a) .

3.

(a) .

4.

(a) .

5.

6.

.

7.

(a) Since .

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 using 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 a and b varies?

(a) Let , and  to obtain

with  as possible parametric equations for the ellipse .

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

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

ⅰ) ,  ⅱ) .

12. Suppose that the position of a 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 :

(b) A collision point occurs when  for the same . Thus we get the following equations :

From ②, . Substituting into [1],

we get

.

Note 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.