## Problem Set: Calculus of Parametric Curves

For the following exercises, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line.

1. $\begin{array}{cc}x=3+t,\hfill & y=1-t\hfill \end{array}$

2. $\begin{array}{cc}x=8+2t,\hfill & y=1\hfill \end{array}$

3. $\begin{array}{cc}x=4 - 3t,\hfill & y=-2+6t\hfill \end{array}$

4. $\begin{array}{cc}x=-5t+7,\hfill & y=3t - 1\hfill \end{array}$

For the following exercises, determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter.

5. $\begin{array}{cc}x=3\sin{t},\hfill & y=3\cos{t},t=\frac{\pi }{4}\hfill \end{array}$

6. $\begin{array}{cc}x=\cos{t},\hfill & y=8\sin{t},\hfill \end{array}t=\frac{\pi }{2}$

7. $\begin{array}{cc}x=2t,\hfill & y={t}^{3},t=-1\hfill \end{array}$

8. $\begin{array}{cc}x=t+\frac{1}{t},\hfill & y=t-\frac{1}{t},t=1\hfill \end{array}$

9. $\begin{array}{cc}x=\sqrt{t},\hfill & y=2t,t=4\hfill \end{array}$

For the following exercises, find all points on the curve that have the given slope.

10. $\begin{array}{cc}x=4\cos{t},\hfill & y=4\sin{t},\hfill \end{array}$ slope = 0.5

11. $\begin{array}{cc}x=2\cos{t},\hfill & y=8\sin{t},\text{slope}=-1\hfill \end{array}$

12. $\begin{array}{cc}x=t+\frac{1}{t},\hfill & y=t-\frac{1}{t},\text{slope}=1\hfill \end{array}$

13. $\begin{array}{cc}x=2+\sqrt{t},\hfill & y=2 - 4t,\text{slope}=0\hfill \end{array}$

For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t.

14. $\begin{array}{cc}x={e}^{\sqrt{t}},\hfill & y=1-\text{ln}{t}^{2},t=1\hfill \end{array}$

15. $\begin{array}{cc}x=t\text{ln}t,\hfill & y={\sin}^{2}t,\hfill \end{array}t=\frac{\pi }{4}$

16. $\begin{array}{cc}x={e}^{t},\hfill & y={\left(t - 1\right)}^{2},\text{at}\left(1,1\right)\hfill \end{array}$

17. For $x=\sin\left(2t\right),y=2\sin{t}$ where $0\le t<2\pi$. Find all values of t at which a horizontal tangent line exists.

18. For $x=\sin\left(2t\right),y=2\sin{t}$ where $0\le t<2\pi$. Find all values of t at which a vertical tangent line exists.

19. Find all points on the curve $x=4\sin\left(t\right),y=4\cos\left(t\right)$ that have the slope of $0.5$.

20. Find $\frac{dy}{dx}$ for $x=\sin\left(t\right),y=\cos\left(t\right)$.

21. Find the equation of the tangent line to $x=\sin\left(t\right),y=\cos\left(t\right)$ at $t=\frac{\pi }{4}$.

22. For the curve $x=4t,y=3t - 2$, find the slope and concavity of the curve at $t=3$.

23. For the parametric curve whose equation is $x=4\cos\theta ,y=4\sin\theta$, find the slope and concavity of the curve at $\theta =\frac{\pi }{4}$.

24. Find the slope and concavity for the curve whose equation is $x=2+\sec\theta ,y=1+2\tan\theta$ at $\theta =\frac{\pi }{6}$.

25. Find all points on the curve $x=t+4,y={t}^{3}-3t$ at which there are vertical and horizontal tangents.

26. Find all points on the curve $x=\sec\theta ,y=\tan\theta$ at which horizontal and vertical tangents exist.

For the following exercises, find $\frac{{d}^{2}y}{d{x}^{2}}$.

27. $\begin{array}{cc}x={t}^{4}-1,\hfill & y=t-{t}^{2}\hfill \end{array}$

28. $\begin{array}{cc}x=\sin\left(\pi t\right),\hfill & y=\cos\left(\pi t\right)\hfill \end{array}$

29. $\begin{array}{cc}x={e}^{\text{-}t},\hfill & y=t\hfill \end{array}{e}^{2t}$

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

30. $\begin{array}{cc}x=t\left({t}^{2}-3\right),\hfill & y=3\left({t}^{2}-3\right)\hfill \end{array}$

31. $\begin{array}{cc}x=\frac{3t}{1+{t}^{3}},\hfill & y=\frac{3{t}^{2}}{1+{t}^{3}}\hfill \end{array}$

For the following exercises, find $\frac{dy}{dx}$ at the value of the parameter.

32. $\begin{array}{cc}x=\cos{t},\hfill & y=\sin{t},t=\frac{3\pi }{4}\hfill \end{array}$

33. $\begin{array}{cc}x=\sqrt{t},\hfill & y=2t+4,t=9\hfill \end{array}$

34. $\begin{array}{cc}x=4\cos{(2\pi s)},\hfill & y=3\sin{(2\pi s)}\hfill \end{array} ,s=-\frac{1}{4}$

For the following exercises, find $\frac{{d}^{2}y}{d{x}^{2}}$ at the given point without eliminating the parameter.

35. $\begin{array}{ccc}x=\frac{1}{2}{t}^{2},\hfill & y=\frac{1}{3}{t}^{3},\hfill & t=2\hfill \end{array}$

36. $x=\sqrt{t},y=2t+4,t=1$

37. Find t intervals on which the curve $x=3{t}^{2},y={t}^{3}-t$ is concave up as well as concave down.

38. Determine the concavity of the curve $x=2t+\text{ln}t,y=2t-\text{ln}t$.

39. Sketch and find the area under one arch of the cycloid $x=r\left(\theta -\sin\theta \right),y=r\left(1-\cos\theta \right)$.

40. Find the area bounded by the curve $x=\cos{t},y={e}^{t},0\le t\le \frac{\pi }{2}$ and the lines $y=1$ and $x=0$.

41. Find the area enclosed by the ellipse $x=a\cos\theta ,y=b\sin\theta ,0\le \theta <2\pi$.

42. Find the area of the region bounded by $x=2{\sin}^{2}\theta ,y=2{\sin}^{2}\theta \tan\theta$, for $0\le \theta \le \frac{\pi }{2}$.

For the following exercises, find the area of the regions bounded by the parametric curves and the indicated values of the parameter.

43. $x=2\cot\theta ,y=2{\sin}^{2}\theta ,0\le \theta \le \pi$

44. [T] $x=2a\cos{t}-a\cos\left(2t\right),y=2a\sin{t}-a\sin\left(2t\right),0\le t<2\pi$

45. [T] $x=a\sin\left(2t\right),y=b\sin\left(t\right),0\le t<2\pi$ (the “hourglass”)

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

For the following exercises, find the arc length of the curve on the indicated interval of the parameter.

47. $x=4t+3,y=3t - 2,0\le t\le 2$

48. $\begin{array}{ccc}x=\frac{1}{3}{t}^{3},\hfill & y=\frac{1}{2}{t}^{2},\hfill & 0\le t\le 1\hfill \end{array}$

49. $\begin{array}{ccc}x=\cos\left(2t\right),\hfill & y=\sin\left(2t\right),\hfill & 0\le t\le \frac{\pi }{2}\hfill \end{array}$

50. $\begin{array}{ccc}x=1+{t}^{2},\hfill & y={\left(1+t\right)}^{3},\hfill & 0\le t\le 1\hfill \end{array}$

51. $\begin{array}{ccc}x={e}^{t}\cos{t},\hfill & y={e}^{t}\sin{t},\hfill & 0\le t\le \frac{\pi }{2}\hfill \end{array}$ (express answer as a decimal rounded to three places)

52. $x=a{\cos}^{3}\theta ,y=a{\sin}^{3}\theta$ on the interval $\left[0,2\pi \right)$ (the hypocycloid)

53. Find the length of one arch of the cycloid $x=4\left(t-\sin{t}\right),y=4\left(1-\cos{t}\right)$.

54. Find the distance traveled by a particle with position $\left(x,y\right)$ as t varies in the given time interval: $\begin{array}{ccc}x={\sin}^{2}t,\hfill & y={\cos}^{2}t,\hfill & 0\le t\le 3\pi \hfill \end{array}$.

55. Find the length of one arch of the cycloid $x=\theta -\sin\theta ,y=1-\cos\theta$.
56. Show that the total length of the ellipse $x=4\sin\theta ,y=3\cos\theta$ is $L=16{\displaystyle\int }_{0}^{\frac{\pi}{2}}\sqrt{1-{e}^{2}{\sin}^{2}\theta }d\theta$, where $e=\frac{c}{a}$ and $c=\sqrt{{a}^{2}-{b}^{2}}$.
57. Find the length of the curve $x={e}^{t}-t,y=4{e}^{\frac{t}{2}},-8\le t\le 3$.

For the following exercises, find the area of the surface obtained by rotating the given curve about the x-axis.

58. $\begin{array}{ccc}x={t}^{3},\hfill & y={t}^{2},\hfill & 0\le t\le 1\hfill \end{array}$

59. $\begin{array}{ccc}x=a{\cos}^{3}\theta ,\hfill & y=a{\sin}^{3}\theta ,\hfill & 0\le \theta \le \hfill \end{array}\frac{\pi }{2}$

60. [T] Use a CAS to find the area of the surface generated by rotating $x=t+{t}^{3},y=t-\frac{1}{{t}^{2}},1\le t\le 2$ about the x-axis. (Answer to three decimal places.)

61. Find the surface area obtained by rotating $x=3{t}^{2},y=2{t}^{3},0\le t\le 5$ about the y-axis.

62. Find the area of the surface generated by revolving $x={t}^{2},y=2t,0\le t\le 4$ about the x-axis.

63. Find the surface area generated by revolving $x={t}^{2},y=2{t}^{2},0\le t\le 1$ about the y-axis.