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### Section 3-2 : 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