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Section 9.2 : Tangents with Parametric Equations

For problems 1 – 3 compute \(\displaystyle \frac{{dy}}{{dx}}\) and \(\displaystyle \frac{{{d^2}y}}{{d{x^2}}}\) for the given set of parametric equations.

1. \(x = 7{t^2} - 9t\hspace{0.5in}y = {t^6} + 2{t^2}\)
2. \(x = \tan \left( {2t} \right) - 12\hspace{0.5in}y = 3\sin \left( {2t} \right) + \sec \left( {2t} \right) + 4t\)
3. \(x = \ln \left( {3{t^2}} \right) + 8t\hspace{0.5in}y = \ln \left( {{t^4}} \right) - 6\ln \left( {{t^2}} \right)\)

For problems 4 – 7 find the equation of the tangent line(s) to the given set of parametric equations at the given point.

1. \(x = {t^3} + \cos \left( {\pi \,t} \right)\hspace{0.25in}y = 4t + \sin \left( {2t + 6} \right)\) at \(t = - 3\)
2. \(x = {t^2} + 2t - 1\hspace{0.25in}y = {t^3} + 7{t^2} + 8t\) at \(t = 1\)
3. \(x = 6 - {{\bf{e}}^{{t^{\,3}} - 9t}}\hspace{0.25in}y = {t^3} + 3{t^2} - 18t + 2\) at \(\left( {5,2} \right)\)
4. \(x = {t^2} + 5t - 6\hspace{0.25in}y = {t^2} + 2t - 8\) at \(\left( { - 6,7} \right)\)

For problems 8 and 9 find the values of t that will have horizontal or vertical tangent lines for the given set of parametric equations.

1. \(x = {t^3} - 5{t^2} + t + 1\hspace{0.25in}y = {t^4} + 8{t^3} + 3{t^2}\)
2. \(x = 7{t^2} + {{\bf{e}}^{2 - {t^{\,2}}}}\hspace{0.25in}y = 10\sin \left( {\displaystyle \frac{1}{2}t} \right) - 1\)