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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