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.

\begin{matrix} x = 3 + t, & y = 1 - t \end{matrix}

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

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

Show Solution

\begin{matrix} x = 4 - 3 t, & y = - 2 + 6 t \end{matrix}

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

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

Show Solution

\frac{- 3}{5}

$\frac{-3}{5}$

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

Show Solution

Slope = 0

$\text{Slope}=0$ ;

y = 8

$y=8$ .

\begin{matrix} x = 2 t, & y = t^{3}, t = - 1 \end{matrix}

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

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

Show Solution

Slope is undefined;

x = 2

$x=2$ .

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

Show Solution

tan t = (- 2)

$\tan{t}=\left(-2\right)$

(\frac{4}{\sqrt{5}}, \frac{- 8}{\sqrt{5}}), (\frac{4}{\sqrt{5}}, \frac{- 8}{\sqrt{5}})

$\left(\frac{4}{\sqrt{5}},\frac{-8}{\sqrt{5}}\right),\left(\frac{4}{\sqrt{5}},\frac{-8}{\sqrt{5}}\right)$ .

11.

\begin{matrix} x = 2 cos t, & y = 8 sin t, slope = - 1 \end{matrix}

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

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

Show Solution

13.

\begin{matrix} x = 2 + \sqrt{t}, & y = 2 - 4 t, slope = 0 \end{matrix}

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

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

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

Show Solution

y = - (\frac{4}{e}) x + 5

$y=\text{-}\left(\frac{4}{e}\right)x+5$

15.

\begin{matrix} x = t ln t, & y = {sin}^{2} t, \end{matrix} t = \frac{π}{4}

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

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

Show Solution

y = - 2 x + 3

$y=-2x + 3$

17. For

x = sin (2 t), y = 2 sin t

$x=\sin\left(2t\right),y=2\sin{t}$ where

0 \leq t < 2 π

$0\le t<2\pi$ . 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.

Show Solution

\frac{π}{4}, \frac{5 π}{4}, \frac{3 π}{4}, \frac{7 π}{4}

$\frac{\pi }{4},\frac{5\pi }{4},\frac{3\pi }{4},\frac{7\pi }{4}$

19. Find all points on the curve

x = 4 sin (t), y = 4 cos (t)

$x=4\sin\left(t\right),y=4\cos\left(t\right)$ that have the slope of

0.5

$0.5$ .

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

Show Solution

\frac{d y}{d x} = - tan (t)

$\frac{dy}{dx}=\text{-}\tan\left(t\right)$

21. Find the equation of the tangent line to

x = sin (t), y = cos (t)

$x=\sin\left(t\right),y=\cos\left(t\right)$ at

t = \frac{π}{4}

$t=\frac{\pi }{4}$ .

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

Show Solution

\frac{d y}{d x} = \frac{3}{4}

$\frac{dy}{dx}=\frac{3}{4}$ and

\frac{d^{2} y}{d x^{2}} = 0

$\frac{{d}^{2}y}{d{x}^{2}}=0$ , so the curve is neither concave up nor concave down at

t = 3

$t=3$ . Therefore the graph is linear and has a constant slope but no concavity.

23. For the parametric curve whose equation is

x = 4 cos θ, y = 4 sin θ

$x=4\cos\theta ,y=4\sin\theta$ , find the slope and concavity of the curve at

θ = \frac{π}{4}

$\theta =\frac{\pi }{4}$ .

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

Show Solution

\frac{d y}{d x} = 4, \frac{d^{2} y}{d x^{2}} = - 6 \sqrt{3}

$\frac{dy}{dx}=4,\frac{{d}^{2}y}{d{x}^{2}}=-6\sqrt{3}$ ; the curve is concave down at

θ = \frac{π}{6}

$\theta =\frac{\pi }{6}$ .

25. Find all points on the curve

x = t + 4, y = t^{3} - 3 t

$x=t+4,y={t}^{3}-3t$ 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.

Show Solution

No horizontal tangents. Vertical tangents at

(1, 0), (- 1, 0)

$\left(1,0\right),\left(-1,0\right)$ .

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

27.

\begin{matrix} x = t^{4} - 1, & y = t - t^{2} \end{matrix}

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

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

Show Solution

- {sec}^{3} (π t)

$\text{-}{\sec}^{3}\left(\pi t\right)$

29.

\begin{matrix} x = e^{- t}, & y = t \end{matrix} e^{2 t}

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

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

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

Show Solution

Horizontal

(0, - 9)

$\left(0,-9\right)$ ; vertical

(\pm 2, - 6)

$\left(\pm2,-6\right)$ .

31.

\begin{matrix} x = \frac{3 t}{1 + t^{3}}, & y = \frac{3 t^{2}}{1 + t^{3}} \end{matrix}

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

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

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

Show Solution

33.

\begin{matrix} x = \sqrt{t}, & y = 2 t + 4, t = 9 \end{matrix}

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

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

Show Solution

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

35.

\begin{matrix} x = \frac{1}{2} t^{2}, & y = \frac{1}{3} t^{3}, & t = 2 \end{matrix}

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

36. $x=\sqrt{t},y=2t+4,t=1$

Show Solution

37. Find t intervals on which the curve

x = 3 t^{2}, y = t^{3} - t

$x=3{t}^{2},y={t}^{3}-t$ is concave up as well as concave down.

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

Show Solution

Concave up on

t > 0

$t>0$ .

39. Sketch and find the area under one arch of the cycloid

x = r (θ - sin θ), y = r (1 - cos θ)

$x=r\left(\theta -\sin\theta \right),y=r\left(1-\cos\theta \right)$ .

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 $x=0$ .

Show Solution

\frac{e \frac{1}{2} - 1}{2}

$\frac{e\frac{1}{2}-1}{2}$

41. Find the area enclosed by the ellipse

x = a cos θ, y = b sin θ, 0 \leq θ < 2 π

$x=a\cos\theta ,y=b\sin\theta ,0\le \theta <2\pi$ .

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

Show Solution

\frac{3 π}{2}

$\frac{3\pi }{2}$

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 θ, y = 2 {sin}^{2} θ, 0 \leq θ \leq π

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

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

Show Solution

6 π a^{2}

$6\pi {a}^{2}$

45. [T]

x = a sin (2 t), y = b sin (t), 0 \leq t < 2 π

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

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

Show Solution

2 π a b

$2\pi ab$

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

47.

x = 4 t + 3, y = 3 t - 2, 0 \leq t \leq 2

$x=4t+3,y=3t - 2,0\le t\le 2$

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

Show Solution

$\frac{1}{3}\left(2\sqrt{2}-1\right)$

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

Show Solution

$7.075$

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)

Show Solution

$6a$

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: $\begin{array}{ccc}x={\sin}^{2}t,\hfill & y={\cos}^{2}t,\hfill & 0\le t\le 3\pi \hfill \end{array}$ .

Show Solution

$6\sqrt{2}$

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

Show Solution

$\frac{2\pi \left(247\sqrt{13}+64\right)}{1215}$

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.)

Show Solution

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.

Show Solution

$\frac{8\pi }{3}\left(17\sqrt{17}-1\right)$

63. Find the surface area generated by revolving

$x={t}^{2},y=2{t}^{2},0\le t\le 1$ about the y-axis.

Parametric Equations and Polar Coordinates: Supplemental Content

Problem Set: Calculus of Parametric Curves

Candela Citations