## MATH1111 Quizzes

Parametric Equations Quiz
Web resources available Questions

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.

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.)

Exactly one option must be correct)
 a) $x=2sint\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}y=1+2cost$ b) $x=1+2sint\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}y=2cost$ c) $x=1+2cost\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}y=2sint$ d) $x=2cost\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}y=1+2sint$

Choice (a) is incorrect
Try again, at $t=0$ the curve is at (0,3) and at $t=\frac{\pi }{2}$ it is at (2,1) so these equations give the following curve

Choice (b) is incorrect
Try again, at $t=0$ the curve is at (1,2) and at $t=\frac{\pi }{2}$ it is at (3,0) so these equations give the following curve

which travels in the wrong direction.
Choice (c) is correct!
At $t=0$ the curve is at (3,0) and at $t=\frac{\pi }{2}$ it is at (1,2) so it is the correct circle.
Choice (d) is incorrect
Try again, at $t=0$ the curve is at (2,1) and at $t=\frac{\pi }{2}$ it is at (0,3) so these equations give the following curve

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. (Zero or more options can be correct)
 a) $x=2+t\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}y=3+3t$ b) $x=2+3t\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}y=3+3t$ c) $x=1+t\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}y=3t$ d) $1+3t\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}y=t$

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
$\frac{dx}{dt}=\frac{2-1}{1}=1$ and $\frac{dy}{dt}=\frac{3-0}{1}=3$ and the point (2,3) is on the line,
so $x=2+t\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}y=3+3t\phantom{\rule{0.3em}{0ex}}.$
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
$\frac{dx}{dt}=\frac{2-1}{1}=1$ and $\frac{dy}{dt}=\frac{3-0}{1}=3$ and the point (1,0) is on the line,
so $x=1+t\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}y=3t\phantom{\rule{0.3em}{0ex}}.$
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.
Correct!
1. True If we consider an object moving from (1,0) to (2,3) in one unit of time then
$\frac{dx}{dt}=\frac{2-1}{1}=1$ and $\frac{dy}{dt}=\frac{3-0}{1}=3$ and the point (2,3) is on the line,
so $x=2+t\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}y=3+3t\phantom{\rule{0.3em}{0ex}}.$
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
$\frac{dx}{dt}=\frac{2-1}{1}=1$ and $\frac{dy}{dt}=\frac{3-0}{1}=3$ and the point (1,0) is on the line,
so $x=1+t\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}y=3t\phantom{\rule{0.3em}{0ex}}.$
4. False These are not the correct speeds for the objects in either direction.
A curve is traced out by the parametric equations $x=2{t}^{3}-4t$ and $y={t}^{2}+4t\phantom{\rule{0.3em}{0ex}}.$
Which of the following is the slope of the curve? Exactly one option must be correct)
 a) $\frac{dy}{dx}=\frac{1}{3t}$ b) $\frac{dy}{dx}=\frac{t+2}{3{t}^{2}-2}$ c) $\frac{dy}{dx}=6{t}^{2}-4$ d) $\frac{dy}{dx}=\frac{2{t}^{3}-4t}{{t}^{2}+4t}$

Choice (a) is incorrect
Try again, you may not have cancelled correctly.
Choice (b) is correct!
$\frac{dy}{dx}=\frac{dy∕dt}{dx∕dt}=\frac{2t+4}{6{t}^{2}-4}=\frac{t+2}{3{t}^{2}-2}\phantom{\rule{0.3em}{0ex}}.$
Choice (c) is incorrect
Try again, this is $\frac{dx}{dt}\phantom{\rule{0.3em}{0ex}}.$
Choice (d) is incorrect
Try again, this is $\frac{x}{y}\phantom{\rule{0.3em}{0ex}}.$
A particle moves in the $xy$-plane with $x={t}^{3}-6{t}^{2}+9t+1$ and $y=2{t}^{3}+9{t}^{2}-24t+6\phantom{\rule{0.3em}{0ex}}.$
At what time(s) is it stopped? Exactly one option must be correct)
 a) $t=1\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}t=3$ and $t=4\phantom{\rule{0.3em}{0ex}}.$ b) $t=-4\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}t=1$ and $t=3\phantom{\rule{0.3em}{0ex}}.$ c) $t=-1\phantom{\rule{0.3em}{0ex}}.$ d) $t=1\phantom{\rule{0.3em}{0ex}}.$

Choice (a) is incorrect
Try again, you need $\frac{dx}{dt}=\frac{dy}{dt}=0$ for the particle to be stopped.
Choice (b) is incorrect
Try again, you need $\frac{dx}{dt}=\frac{dy}{dt}=0$ for the particle to be stopped.
Choice (c) is incorrect
Try again, neither $\frac{dx}{dt}\phantom{\rule{0.3em}{0ex}},$ nor $\frac{dy}{dt}=0$ at $t=-1\phantom{\rule{0.3em}{0ex}}.$
Choice (d) is correct!
$\frac{dx}{dt}=3{t}^{2}-12t+9=3\left({t}^{2}-4t+3\right)=3\left(t-3\right)\left(t-1\right)$ and
$\frac{dy}{dt}=6{t}^{2}+18t-24=6\left({t}^{2}+3t-4\right)=6\left(t-1\right)\left(t+4\right)$
so the particle is stopped at $t=1\phantom{\rule{0.3em}{0ex}}.$