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Parametric Equations Quiz

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This quiz tests the work covered in Lecture 20 and corresponds to Section 4.8 of the textbook Calculus: Single and Multivariable (Hughes-Hallett, Gleason, McCallum et al.).
There are more web quizzes at Wiley, select Section 8.

There is an interesting applet at http://www.rfbarrow.btinternet.co.uk/htmasa2/Param1.htm which allows you to play with some equations and see how it changes the parametric curve. There is a slightly more general one at http://www.math.ucla.edu/ ronmiech/crew/eugene/java/ParamPlotter/ParamPlotter.html but you must read the instructions to see how to make it work.

There is a good discussion at http://tutorial.math.lamar.edu/AllBrowsers/2414/ParametricEqn.asp with fully worked examples of plotting parametric curves.

Question 1

Which of the following is the most suitable set of parametric equations to describe the curve below? (Make sure you note the direction of motion.)
PIC

a)
x = 2sint, y = 1 +2 cost
  b)
x = 1+ 2sint, y = 2cost
c)
x = 1 +2 cost, y = 2 sint
  d)
x = 2cost, y = 1+ 2sint

 

Not correct. Choice (a) is false.
Try again, at t = 0 the curve is at (0,3) and at π t = 2 it is at (2,1) so these equations give the following curve
PIC
Not correct. Choice (b) is false.
Try again, at t = 0 the curve is at (1,2) and at t = π2 it is at (3,0) so these equations give the following curve
PIC
which travels in the wrong direction.
Your answer is correct.
At t = 0 the curve is at (3,0) and at t = π2 it is at (1,2) so it is the correct circle.
Not correct. Choice (d) is false.
Try again, at t = 0 the curve is at (2,1) and at t = π2 it is at (0,3) so these equations give the following curve
PIC

Question 2

Which of the following are the parametric equations to describe an object travelling along the line passing through (1,0) and (2,3)?
There may be more than one correct answer.

a)
x = 2 + t, y = 3 + 3t
  b)
x = 2+ 3t, y = 3+ 3t
c)
x = 1 + t, y = 3t
  d)
1+ 3t, y = t

 

There is at least one mistake.
For example, choice (a) should be true.
If we consider an object moving from (1,0) to (2,3) in one unit of time then
dx-= 2--1-= 1 dt 1 and dy = 3--0-= 3 dt 1 and the point (2,3) is on the line,
so x = 2 +t, y = 3 +3t.
There is at least one mistake.
For example, choice (b) should be false.
This is not the correct speed for the object in the y direction.
There is at least one mistake.
For example, choice (c) should be true.
If we consider an object moving from (1,0) to (2,3) in one unit of time then
dx 2- 1 ---= -----= 1 dt 1 and dy 3- 0 -- = -----= 3 dt 1 and the point (1,0) is on the line,
so x = 1 +t, y = 3t.
There is at least one mistake.
For example, choice (d) should be false.
These are not the correct speeds for the objects in either direction.
Your answers are correct
  1. True. If we consider an object moving from (1,0) to (2,3) in one unit of time then
    dx-= 2--1-= 1 dt 1 and dy = 3--0-= 3 dt 1 and the point (2,3) is on the line,
    so x = 2 +t, y = 3 +3t.
  2. False. This is not the correct speed for the object in the y direction.
  3. True. If we consider an object moving from (1,0) to (2,3) in one unit of time then
    dx 2- 1 ---= -----= 1 dt 1 and dy 3- 0 -- = -----= 3 dt 1 and the point (1,0) is on the line,
    so x = 1 +t, y = 3t.
  4. False. These are not the correct speeds for the objects in either direction.

Question 3

A curve is traced out by the parametric equations x = 2t3 - 4t and y = t2 + 4t.
Which of the following is the slope of the curve?

a)
dy-= 1- dx 3t
  b)
dy-= -t+-2- dx 3t2 - 2
c)
-dy= 6t2 - 4 dx
  d)
dy-= 2t3 --4t dx t2 + 4t

 

Not correct. Choice (a) is false.
Try again, you may not have cancelled correctly.
Your answer is correct.
dy-= dy∕dt = 2t+-4-= -t+-2-. dx dx∕dt 6t2 - 4 3t2 - 2
Not correct. Choice (c) is false.
Try again, this is dx. dt
Not correct. Choice (d) is false.
Try again, this is x. y

Question 4

A particle moves in the xy -plane with x = t3 - 6t2 + 9t+ 1 and y = 2t3 + 9t2 - 24t+ 6.
At what time(s) is it stopped?

a)
t = 1, t = 3 and t = 4.
  b)
t = - 4, t = 1 and t = 3.
c)
t = - 1.
  d)
t = 1.

 

Not correct. Choice (a) is false.
Try again, you need dx dy dt-= dt = 0 for the particle to be stopped.
Not correct. Choice (b) is false.
Try again, you need dx- dy dt = dt = 0 for the particle to be stopped.
Not correct. Choice (c) is false.
Try again, neither dx, dt nor dy = 0 dt at t = - 1.
Your answer is correct.
dx-= 3t2 - 12t+ 9 = 3(t2 - 4t+ 3) = 3(t- 3)(t - 1) dt and
dy -- = 6t2 + 18t- 24 = 6(t2 + 3t- 4) = 6(t- 1)(t+ 4) dt
so the particle is stopped at t = 1.
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