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Assignment Problems Notice
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If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
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.
- \(x = 7{t^2} - 9t\hspace{0.5in}y = {t^6} + 2{t^2}\)
- \(x = \tan \left( {2t} \right) - 12\hspace{0.5in}y = 3\sin \left( {2t} \right) + \sec \left( {2t} \right) + 4t\)
- \(x = \ln \left( {3{t^2}} \right) + 8t\hspace{0.5in}y = \ln \left( {{t^4}} \right) - 6\ln \left( {{t^2}} \right)\)
For problems 4 – 7 find the equation of the tangent line(s) to the given set of parametric equations at the given point.
- \(x = {t^3} + \cos \left( {\pi \,t} \right)\hspace{0.25in}y = 4t + \sin \left( {2t + 6} \right)\) at \(t = - 3\)
- \(x = {t^2} + 2t - 1\hspace{0.25in}y = {t^3} + 7{t^2} + 8t\) at \(t = 1\)
- \(x = 6 - {{\bf{e}}^{{t^{\,3}} - 9t}}\hspace{0.25in}y = {t^3} + 3{t^2} - 18t + 2\) at \(\left( {5,2} \right)\)
- \(x = {t^2} + 5t - 6\hspace{0.25in}y = {t^2} + 2t - 8\) 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.
- \(x = {t^3} - 5{t^2} + t + 1\hspace{0.25in}y = {t^4} + 8{t^3} + 3{t^2}\)
- \(x = 7{t^2} + {{\bf{e}}^{2 - {t^{\,2}}}}\hspace{0.25in}y = 10\sin \left( {\displaystyle \frac{1}{2}t} \right) - 1\)