Fundamentals of Parametric Equations

For the following exercises (1-2), sketch the curves below by eliminating the parameter [latex]t[/latex]. Give the orientation of the curve.

1. [latex]x={t}^{2}+2t[/latex], [latex]y=t+1[/latex]
2. [latex]x=2t+4,y=t - 1[/latex]

For the following exercise, eliminate the parameter and sketch the graphs.

3. [latex]x=2{t}^{2},y={t}^{4}+1[/latex]

For the following exercises (4-5), use technology (CAS or calculator) to sketch the parametric equations.

4. [latex]\begin{array}{cc}x={e}^{\text{-}t},\hfill & y={e}^{2t}-1\hfill \end{array}[/latex]
5. [latex]\begin{array}{cc}x=\sec{t},\hfill & y=\cos{t}\hfill \end{array}[/latex]

For the following exercises (6-10), sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph.

6. [latex]x=6\sin\left(2\theta \right),y=4\cos\left(2\theta \right)[/latex]
7. [latex]\begin{array}{cc}x=3 - 2\cos\theta ,\hfill & y=-5+3\sin\theta \hfill \end{array}[/latex]
8. [latex]\begin{array}{cc}x=\sec{t},\hfill & y=\tan{t}\hfill \end{array}[/latex]
9. [latex]\begin{array}{cc}x={e}^{t},\hfill & y={e}^{2t}\hfill \end{array}[/latex]
10. [latex]\begin{array}{cc}x={t}^{3},\hfill & y=3\text{ln}t\hfill \end{array}[/latex]

For the following exercises (11-19), convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form.

11. [latex]\begin{array}{cc}x={t}^{2}-1,\hfill & y=\dfrac{t}{2}\hfill \end{array}[/latex]
12. [latex]x=4\cos\theta ,y=3\sin\theta ,t\in \left(0,2\pi \right][/latex]
13. [latex]\begin{array}{cc}x=2t - 3,\hfill & y=6t - 7\hfill \end{array}[/latex]
14. [latex]\begin{array}{cc}x=1+\cos{t},\hfill & y=3-\sin{t}\hfill \end{array}[/latex]
15. [latex]\begin{array}{cc}x=\sec{t},\hfill & y=\tan{t},\pi \le t<\dfrac{3\pi }{2}\hfill \end{array}[/latex]
16. [latex]\begin{array}{cc}x=\cos\left(2t\right),\hfill & y=\sin{t}\hfill \end{array}[/latex]
17. [latex]\begin{array}{cc}x={t}^{2},\hfill & y=2\text{ln}t,t\ge 1\hfill \end{array}[/latex]
18. [latex]\begin{array}{cc}x={t}^{n},\hfill & y=n\text{ln}t,t\ge 1,\hfill \end{array}[/latex] where n is a natural number
19. [latex]\begin{array}{c}x=2\sin\left(8t\right)\hfill \ y=2\cos\left(8t\right)\hfill \end{array}[/latex]

For the following exercises (20-24), the pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents.

20. [latex]\begin{array}{c}x=3t+4\hfill \ y=5t - 2\hfill \end{array}[/latex]
21. [latex]\begin{array}{c}x=2t+1\hfill \ y={t}^{2}-3\hfill \end{array}[/latex]
22. [latex]\begin{array}{c}x=2\cos\left(3t\right)\hfill \ y=2\sin\left(3t\right)\hfill \end{array}[/latex]
23. [latex]\begin{array}{c}x=3\cos{t}\hfill \ y=4\sin{t}\hfill \end{array}[/latex]
24. [latex]\begin{array}{c}x=3\text{cosh}\left(4t\right)\hfill \ y=4\text{sinh}\left(4t\right)\hfill \end{array}[/latex]

For the following exercises (25-26), use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation.

25. [latex]\begin{array}{c}x=\theta +\sin\theta \hfill \ y=1-\cos\theta \hfill \end{array}[/latex]
26. [latex]\begin{array}{c}x=t - 0.5\sin{t}\hfill \ y=1 - 1.5\cos{t}\hfill \end{array}[/latex]

Calculus with Parametric Curves

For the following exercises (1-2), each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line.

1. [latex]\begin{array}{cc}x=8+2t,\hfill & y=1\hfill \end{array}[/latex]
2. [latex]\begin{array}{cc}x=-5t+7,\hfill & y=3t - 1\hfill \end{array}[/latex]

For the following exercises (3-4), determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter.

3. [latex]\begin{array}{cc}x=\cos{t},\hfill & y=8\sin{t},\hfill \end{array}t=\dfrac{\pi }{2}[/latex]
4. [latex]\begin{array}{cc}x=t+\dfrac{1}{t},\hfill & y=t-\dfrac{1}{t},t=1\hfill \end{array}[/latex]

For the following exercises (5-6), find all points on the curve that have the given slope.

5. [latex]\begin{array}{cc}x=4\cos{t},\hfill & y=4\sin{t},\hfill \end{array}[/latex] slope = [latex]0.5[/latex]
6. [latex]\begin{array}{cc}x=t+\dfrac{1}{t},\hfill & y=t-\dfrac{1}{t},\text{slope}=1\hfill \end{array}[/latex]

For the following exercises (7-8), write the equation of the tangent line in Cartesian coordinates for the given parameter t.

7. [latex]\begin{array}{cc}x={e}^{\sqrt{t}},\hfill & y=1-\text{ln}{t}^{2},t=1\hfill \end{array}[/latex]
8. [latex]\begin{array}{cc}x={e}^{t},\hfill & y={\left(t - 1\right)}^{2},\text{at}\left(1,1\right)\hfill \end{array}[/latex]

For the following exercises (9-13), solve each problem.

9. For [latex]x=\sin\left(2t\right),y=2\sin{t}[/latex] where [latex]0\le t<2\pi[/latex]. Find all values of [latex]t[/latex] at which a vertical tangent line exists.
10. Find [latex]\dfrac{dy}{dx}[/latex] for [latex]x=\sin\left(t\right),y=\cos\left(t\right)[/latex].
11. For the curve [latex]x=4t,y=3t - 2[/latex], find the slope and concavity of the curve at [latex]t=3[/latex].
12. Find the slope and concavity for the curve whose equation is [latex]x=2+\sec\theta ,y=1+2\tan\theta[/latex] at [latex]\theta =\dfrac{\pi }{6}[/latex].
13. Find all points on the curve [latex]x=\sec\theta ,y=\tan\theta[/latex] at which horizontal and vertical tangents exist.

For the following exercise, find [latex]\dfrac{{d}^{2}y}{d{x}^{2}}[/latex].

14. [latex]\begin{array}{cc}x=\sin\left(\pi t\right),\hfill & y=\cos\left(\pi t\right)\hfill \end{array}[/latex]

For the following exercise, find points on the curve at which tangent line is horizontal or vertical.

15. [latex]\begin{array}{cc}x=t\left({t}^{2}-3\right),\hfill & y=3\left({t}^{2}-3\right)\hfill \end{array}[/latex]

For the following exercises (16-17), find [latex]\dfrac{dy}{dx}[/latex] at the value of the parameter.

16. [latex]\begin{array}{cc}x=\cos{t},\hfill & y=\sin{t},t=\dfrac{3\pi }{4}\hfill \end{array}[/latex]
17. [latex]\begin{array}{cc}x=4\cos{(2\pi s)},\hfill & y=3\sin{(2\pi s)}\hfill \end{array} ,s=-\dfrac{1}{4}[/latex]

For the following exercise, find [latex]\dfrac{{d}^{2}y}{d{x}^{2}}[/latex] at the given point without eliminating the parameter.

18. [latex]x=\sqrt{t},y=2t+4,t=1[/latex]

For the following exercises (19-21), solve each problem.

19. Determine the concavity of the curve [latex]x=2t+\text{ln}t,y=2t-\text{ln}t[/latex].
20. Find the area bounded by the curve [latex]x=\cos{t},y={e}^{t},0\le t\le \dfrac{\pi }{2}[/latex] and the lines [latex]y=1[/latex] and [latex]x=0[/latex].
21. Find the area of the region bounded by [latex]x=2{\sin}^{2}\theta ,y=2{\sin}^{2}\theta \tan\theta[/latex], for [latex]0\le \theta \le \dfrac{\pi }{2}[/latex].

For the following exercises (22-23), find the area of the regions bounded by the parametric curves and the indicated values of the parameter.

22.  [latex]x=2a\cos{t}-a\cos\left(2t\right),y=2a\sin{t}-a\sin\left(2t\right),0\le t<2\pi[/latex]
23. [latex]x=2a\cos{t}-a\sin\left(2t\right),y=b\sin{t},0\le t<2\pi[/latex] (the “teardrop”)

For the following exercises (24-26), find the arc length of the curve on the indicated interval of the parameter.

24. [latex]\begin{array}{ccc}x=\dfrac{1}{3}{t}^{3},\hfill & y=\dfrac{1}{2}{t}^{2},\hfill & 0\le t\le 1\hfill \end{array}[/latex]
25. [latex]\begin{array}{ccc}x=1+{t}^{2},\hfill & y={\left(1+t\right)}^{3},\hfill & 0\le t\le 1\hfill \end{array}[/latex]
26. [latex]x=a{\cos}^{3}\theta ,y=a{\sin}^{3}\theta[/latex] on the interval [latex]\left[0,2\pi \right)[/latex] (the hypocycloid)

For the following exercises (27-29), find the area of the surface obtained by rotating the given curve about the [latex]x[/latex]-axis.

27. [latex]\begin{array}{ccc}x={t}^{3},\hfill & y={t}^{2},\hfill & 0\le t\le 1\hfill \end{array}[/latex]
28. Use a CAS to find the area of the surface generated by rotating [latex]x=t+{t}^{3},y=t-\dfrac{1}{{t}^{2}},1\le t\le 2[/latex] about the [latex]x[/latex]-axis. (Answer to three decimal places.)
29. Find the area of the surface generated by revolving [latex]x={t}^{2},y=2t,0\le t\le 4[/latex] about the [latex]x[/latex]-axis.