Paul's Online Notes
Home / Calculus II / Parametric Equations and Polar Coordinates / Tangents with Parametric Equations
Show Mobile Notice Show All Notes Hide All Notes
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 views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

### Section 9.2 : Tangents with Parametric Equations

For problems 1 and 2 compute $$\displaystyle \frac{{dy}}{{dx}}$$ and $$\displaystyle \frac{{{d^2}y}}{{d{x^2}}}$$ for the given set of parametric equations.

1. $$x = 4{t^3} - {t^2} + 7t\hspace{0.5in}\,\,y = {t^4} - 6$$ Solution
2. $$x = {{\bf{e}}^{ - 7t}} + 2\hspace{0.5in}\,\,y = 6{{\bf{e}}^{2t}} + {{\bf{e}}^{ - 3t}} - 4t$$ Solution

For problems 3 and 4 find the equation of the tangent line(s) to the given set of parametric equations at the given point.

1. $$x = 2\cos \left( {3t} \right) - 4\sin \left( {3t} \right)\hspace{0.25in}y = 3\tan \left( {6t} \right)$$ at $$\displaystyle t = \frac{\pi }{2}$$ Solution
2. $$x = {t^2} - 2t - 11\hspace{0.25in}y = t{\left( {t - 4} \right)^3} - 3{t^2}{\left( {t - 4} \right)^2} + 7$$ at $$\left( { - 3,7} \right)$$ Solution
3. Find the values of t that will have horizontal or vertical tangent lines for the following set of parametric equations. $$x = {t^5} - 7{t^4} - 3{t^3}\hspace{0.25in}y = 2\cos \left( {3t} \right) + 4t$$ Solution