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Section 3-1 : Parametric Equations and Curves

For problems 1 – 9 eliminate the parameter for the given set of parametric equations, sketch the graph of the parametric curve and give any limits that might exist on \(x\) and \(y\).

  1. \(x = 2 + t\hspace{0.5in}y = {t^2} - 4t + 7\)
  2. \(x = 2 + t\hspace{0.5in}y = {t^2} - 4t + 7\hspace{0.25in} - 3 \le t \le 1\)
  3. \(x = 1 - {t^2}\hspace{0.5in}y = 3 + 2t\)
  4. \(x = 1 - {t^2}\hspace{0.5in}y = 3 + 2t\hspace{0.25in}\hspace{0.25in} - 2 \le t \le 3\)
  5. \(\displaystyle x = \frac{1}{2}\sqrt t \hspace{0.5in}y = t - \sqrt t - 6\hspace{0.5in}t \ge 0\)
  6. \(\displaystyle x = \frac{1}{2}\sqrt t \hspace{0.5in}y = t - \sqrt t - 6\hspace{0.5in}8 \le t \le 20\)
  7. \(\displaystyle x = - 6\cos \left( {4t} \right)\hspace{0.5in}\,y = 2\sin \left( {4t} \right)\hspace{0.5in} - \frac{\pi }{4} \le t \le \frac{\pi }{8}\)
  8. \(\displaystyle x = 1 - 3\sin \left( {\frac{1}{2}t} \right)\hspace{0.5in}\,y = 4\cos \left( {\frac{1}{2}t} \right)\,\)
  9. \(x = 6 - 7{{\bf{e}}^{ - 2t}}\hspace{0.5in}y = 4 + 3{{\bf{e}}^{ - 2t}}\)
  10. Answer each of the questions about the following set of parametric equations \[x = 3\cos \left( {at} \right)\hspace{0.5in}y = 3\sin \left( {at} \right)\hspace{0.5in}0 \le t \le 2\pi \]
    1. Sketch the graph of the parametric curve for \(a = 1\).
    2. Sketch the graph of the parametric curve for \(a = 6\).
    3. Sketch the graph of the parametric curve for \(a = \frac{1}{5}\).
    4. In general, for \(a > 0\), how does the value of a affect the graph of the parametric curve?

For problems 11 – 21 the path of a particle is given by the set of parametric equations. Completely describe the path of the particle. To completely describe the path of the particle you will need to provide the following information.

  1. A sketch of the parametric curve (including direction of motion) based on the equation you get by eliminating the parameter.
  2. Limits on \(x\) and \(y\).
  3. A range of \(t\)’s for a single trace of the parametric curve.
  4. The number of traces of the curve the particle makes if an overall range of \(t\)’s is provided in the problem.

  1. \(\displaystyle x = 6\cos \left( {\frac{1}{3}t} \right)\hspace{0.5in}y = 2 + \sin \left( {\frac{1}{3}t} \right)\hspace{0.5in} 0 \le t \le 75\pi \)
  2. \(x = 7 - 3\sin \left( {2t} \right)\hspace{0.5in}y = 4 + 2\cos \left( {2t} \right)\)
  3. \(\displaystyle x = 6{\cos ^2}\left( {3t} \right)\hspace{0.5in}y = 2 - 3{\sin ^2}\left( {3t} \right)\hspace{0.25in} - \frac{5}{6}\pi \le t \le 3\pi \)
  4. \(\displaystyle x = \sqrt {2 + {{\cos }^2}\left( {\frac{7}{4}t} \right)} \hspace{0.5in}y = \frac{1}{3}\sin\left( {\frac{7}{4}t} \right)\)
  5. \(x = 6 - {\sin ^3}\left( {4t} \right)\hspace{0.5in}y = 2\sin \left( {4t} \right)\hspace{0.5in} - 127\pi \le t \le 201\pi \)
  6. \(\displaystyle x = 3 + \cos \left( {\frac{1}{6}t} \right)\hspace{0.5in}y = 4 + {\cos ^2}\left( {\frac{1}{6}t} \right)\hspace{0.5in} - 90\pi \le t \le 216\pi \)
  7. \(x = {{\bf{e}}^{ - 4t}}\hspace{0.5in}y = 2{{\bf{e}}^{12t}}\)
  8. \(x = 1 + {{\bf{e}}^{3t}}\hspace{0.5in}y = {{\bf{e}}^{6t}}\hspace{0.5in} - 1 \le t \le 6\)
  9. \(x = 1 - \ln \left( t \right)\hspace{0.5in}y = {\left[ {\ln \left( t \right)} \right]^2}\hspace{0.5in} t > 0\)
  10. \(\displaystyle x = \cos \left( {\frac{1}{2}t} \right)\hspace{0.5in}y = \sec \left( {\frac{1}{2}t} \right)\hspace{0.5in} - \pi < t < \pi \)
  11. \(\displaystyle x = \sin \left( {2t} \right)\hspace{0.5in}y = {\sin ^2}\left( {2t} \right) - 4\sin \left( {2t} \right)\hspace{0.5in} - \frac{{91}}{4}\pi \le t \le \frac{{17}}{4}\pi \)

For problems 22 – 27 write down a set of parametric equations for the given equation that meets the given extra conditions (if any).

  1. \(x = \sin \left( {3 - {y^2}} \right) + {\cos ^2}\left( y \right)\)
  2. \(\displaystyle y = \frac{{6\cos \left( x \right) - 8}}{{{x^2} + 9x}}\)
  3. \({x^2} + {y^2} = 100\) and the parametric curve resulting from the parametric equations should be at \(\left( {0,10} \right)\) when \(t = 0\) and the curve should have a clockwise rotation.
  4. \({x^2} + {y^2} = 100\) and the parametric curve resulting from the parametric equations should be at \(\left( {0,10} \right)\) when \(t = 0\) and the curve should have a counter clockwise rotation.
  5. \(\displaystyle \frac{{{x^2}}}{{25}} + {y^2} = 1\) and the parametric curve resulting from the parametric equations should be at \(\left( { - 5,0} \right)\) when \(t = 0\) and the curve should have a counter clockwise rotation.
  6. \(\displaystyle \frac{{{x^2}}}{{25}} + {y^2} = 1\) and the parametric curve resulting from the parametric equations should be at \(\left( { - 5,0} \right)\) when \(t = 0\) and the curve should have a clockwise rotation.
  7. Eliminate the parameter for the following set of parametric equations and identify the resulting equation. \[x = h + a\cos \left( {\omega \,t} \right)\hspace{0.25in}\hspace{0.25in}x = k + b\sin \left( {\omega \,t} \right)\]