Calculus II - Parametric Equations and Curves (Practice Problems)

Show General Notice Show Mobile Notice Show All Notes Hide All Notes

General Notice

I have been informed that on March 7th from 6:00am to 6:00pm Central Time Lamar University will be doing some maintenance to replace a faulty UPS component and to do this they will be completely powering down their data center.

Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.

Paul
February 18, 2026

Mobile Notice

You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll/swipe to see them) and some of the menu items will be cut off due to the narrow screen width.

Section 9.1 : Parametric Equations and Curves

For problems 1 – 6 eliminate the parameter for the given set of parametric equations, sketch the graph of the parametric curve and give any limits that might exist on \(x\) and \(y\).

\(x = 4 - 2t\hspace{0.5in}y = 3 + 6t - 4{t^2}\) Solution
\(x = 4 - 2t\hspace{0.5in}y = 3 + 6t - 4{t^2}\hspace{0.5in}0 \le t \le 3\) Solution
\(\displaystyle x = \sqrt {t + 1} \hspace{0.5in}y = \frac{1}{{t + 1}} \hspace{0.5in} t > - 1\) Solution
\(x = 3\sin \left( t \right)\hspace{0.5in}y = - 4\cos \left( t \right) \hspace{0.5in} 0 \le t \le 2\pi \) Solution
\(x = 3\sin \left( {2t} \right)\hspace{0.5in}y = - 4\cos \left( {2t} \right) \hspace{0.5in}0 \le t \le 2\pi \) Solution
\(\displaystyle x = 3\sin \left( {\frac{1}{3}t} \right)\hspace{0.5in}y = - 4\cos \left( {\frac{1}{3}t} \right) \hspace{0.5in}0 \le t \le 2\pi \) Solution

For problems 7 – 11 the path of a particle is given by the set of parametric equations. Completely describe the path of the particle. To completely describe the path of the particle you will need to provide the following information.

A sketch of the parametric curve (including direction of motion) based on the equation you get by eliminating the parameter.
Limits on \(x\) and \(y\).
A range of \(t\)’s for a single trace of the parametric curve.
The number of traces of the curve the particle makes if an overall range of \(t\)’s is provided in the problem.

\(x = 3 - 2\cos \left( {3t} \right)\hspace{0.5in}y = 1 + 4\sin \left( {3t} \right)\) Solution
\(x = 4\sin \left( {\frac{1}{4}t} \right)\hspace{0.5in}y = 1 - 2{\cos ^2}\left( {\frac{1}{4}t} \right) \hspace{0.5in} - 52\pi \le t \le 34\pi \) Solution
\(\displaystyle x = \sqrt {4 + \cos \left( {\frac{5}{2}t} \right)} \hspace{0.5in}y = 1 + \frac{1}{3}\cos \left( {\frac{5}{2}t} \right) \hspace{0.5in} - 48\pi \le t \le 2\pi \) Solution
\(\displaystyle x = 2{{\bf{e}}^t}\hspace{0.5in}y = \cos \left( {1 + {{\bf{e}}^{3t}}} \right) \hspace{0.5in} 0 \le t \le \frac{3}{4}\) Solution
\(\displaystyle x = \frac{1}{2}{{\bf{e}}^{ - 3t}}\hspace{0.5in}y = {{\bf{e}}^{ - 6t}} + 2{{\bf{e}}^{ - 3t}} - 8\) Solution

For problems 12 – 14 write down a set of parametric equations for the given equation that meets the given extra conditions (if any).

\(y = 3{x^2} - \ln \left( {4x + 2} \right)\) Solution
\({x^2} + {y^2} = 36\) and the parametric curve resulting from the parametric equations should be at \(\left( {6,0} \right)\) when \(t = 0\) and the curve should have a counter clockwise rotation. Solution
\(\displaystyle \frac{{{x^2}}}{4} + \frac{{{y^2}}}{{49}} = 1\) and the parametric curve resulting from the parametric equations should be at \(\left( {0, - 7} \right)\) when \(t = 0\) and the curve should have a clockwise rotation. Solution