For the following exercises, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line.
1. [latex]\begin{array}{cc}x=3+t,\hfill & y=1-t\hfill \end{array}[/latex]
2. [latex]\begin{array}{cc}x=8+2t,\hfill & y=1\hfill \end{array}[/latex]
3. [latex]\begin{array}{cc}x=4 - 3t,\hfill & y=-2+6t\hfill \end{array}[/latex]
4. [latex]\begin{array}{cc}x=-5t+7,\hfill & y=3t - 1\hfill \end{array}[/latex]
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[latex]\frac{-3}{5}[/latex]
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. [latex]\begin{array}{cc}x=3\sin{t},\hfill & y=3\cos{t},t=\frac{\pi }{4}\hfill \end{array}[/latex]
6. [latex]\begin{array}{cc}x=\cos{t},\hfill & y=8\sin{t},\hfill \end{array}t=\frac{\pi }{2}[/latex]
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[latex]\text{Slope}=0[/latex]; [latex]y=8[/latex].
7. [latex]\begin{array}{cc}x=2t,\hfill & y={t}^{3},t=-1\hfill \end{array}[/latex]
8. [latex]\begin{array}{cc}x=t+\frac{1}{t},\hfill & y=t-\frac{1}{t},t=1\hfill \end{array}[/latex]
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Slope is undefined; [latex]x=2[/latex].
9. [latex]\begin{array}{cc}x=\sqrt{t},\hfill & y=2t,t=4\hfill \end{array}[/latex]
For the following exercises, find all points on the curve that have the given slope.
10. [latex]\begin{array}{cc}x=4\cos{t},\hfill & y=4\sin{t},\hfill \end{array}[/latex] slope = 0.5
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[latex]\tan{t}=\left(-2\right)[/latex] [latex]\left(\frac{4}{\sqrt{5}},\frac{-8}{\sqrt{5}}\right),\left(\frac{4}{\sqrt{5}},\frac{-8}{\sqrt{5}}\right)[/latex].
11. [latex]\begin{array}{cc}x=2\cos{t},\hfill & y=8\sin{t},\text{slope}=-1\hfill \end{array}[/latex]
12. [latex]\begin{array}{cc}x=t+\frac{1}{t},\hfill & y=t-\frac{1}{t},\text{slope}=1\hfill \end{array}[/latex]
Show Solution
No points possible; undefined expression.
13. [latex]\begin{array}{cc}x=2+\sqrt{t},\hfill & y=2 - 4t,\text{slope}=0\hfill \end{array}[/latex]
For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t.
14. [latex]\begin{array}{cc}x={e}^{\sqrt{t}},\hfill & y=1-\text{ln}{t}^{2},t=1\hfill \end{array}[/latex]
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[latex]y=\text{-}\left(\frac{4}{e}\right)x+5[/latex]
15. [latex]\begin{array}{cc}x=t\text{ln}t,\hfill & y={\sin}^{2}t,\hfill \end{array}t=\frac{\pi }{4}[/latex]
16. [latex]\begin{array}{cc}x={e}^{t},\hfill & y={\left(t - 1\right)}^{2},\text{at}\left(1,1\right)\hfill \end{array}[/latex]
Show Solution
[latex]y=-2x + 3[/latex]
17. For [latex]x=\sin\left(2t\right),y=2\sin{t}[/latex] where [latex]0\le t<2\pi [/latex]. Find all values of t at which a horizontal tangent line exists.
18. For [latex]x=\sin\left(2t\right),y=2\sin{t}[/latex] where [latex]0\le t<2\pi [/latex]. Find all values of t at which a vertical tangent line exists.
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[latex]\frac{\pi }{4},\frac{5\pi }{4},\frac{3\pi }{4},\frac{7\pi }{4}[/latex]
19. Find all points on the curve [latex]x=4\sin\left(t\right),y=4\cos\left(t\right)[/latex] that have the slope of [latex]0.5[/latex].
20. Find [latex]\frac{dy}{dx}[/latex] for [latex]x=\sin\left(t\right),y=\cos\left(t\right)[/latex].
Show Solution
[latex]\frac{dy}{dx}=\text{-}\tan\left(t\right)[/latex]
21. Find the equation of the tangent line to [latex]x=\sin\left(t\right),y=\cos\left(t\right)[/latex] at [latex]t=\frac{\pi }{4}[/latex].
22. For the curve [latex]x=4t,y=3t - 2[/latex], find the slope and concavity of the curve at [latex]t=3[/latex].
Show Solution
[latex]\frac{dy}{dx}=\frac{3}{4}[/latex] and [latex]\frac{{d}^{2}y}{d{x}^{2}}=0[/latex], so the curve is neither concave up nor concave down at [latex]t=3[/latex]. Therefore the graph is linear and has a constant slope but no concavity.
23. For the parametric curve whose equation is [latex]x=4\cos\theta ,y=4\sin\theta [/latex], find the slope and concavity of the curve at [latex]\theta =\frac{\pi }{4}[/latex].
24. 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 =\frac{\pi }{6}[/latex].
Show Solution
[latex]\frac{dy}{dx}=4,\frac{{d}^{2}y}{d{x}^{2}}=-6\sqrt{3}[/latex]; the curve is concave down at [latex]\theta =\frac{\pi }{6}[/latex].
25. Find all points on the curve [latex]x=t+4,y={t}^{3}-3t[/latex] at which there are vertical and horizontal tangents.
26. Find all points on the curve [latex]x=\sec\theta ,y=\tan\theta [/latex] at which horizontal and vertical tangents exist.
Show Solution
No horizontal tangents. Vertical tangents at [latex]\left(1,0\right),\left(-1,0\right)[/latex].
For the following exercises, find [latex]\frac{{d}^{2}y}{d{x}^{2}}[/latex].
27. [latex]\begin{array}{cc}x={t}^{4}-1,\hfill & y=t-{t}^{2}\hfill \end{array}[/latex]
28. [latex]\begin{array}{cc}x=\sin\left(\pi t\right),\hfill & y=\cos\left(\pi t\right)\hfill \end{array}[/latex]
Show Solution
[latex]\text{-}{\sec}^{3}\left(\pi t\right)[/latex]
29. [latex]\begin{array}{cc}x={e}^{\text{-}t},\hfill & y=t\hfill \end{array}{e}^{2t}[/latex]
For the following exercises, find points on the curve at which tangent line is horizontal or vertical.
30. [latex]\begin{array}{cc}x=t\left({t}^{2}-3\right),\hfill & y=3\left({t}^{2}-3\right)\hfill \end{array}[/latex]
Show Solution
Horizontal [latex]\left(0,-9\right)[/latex]; vertical [latex]\left(\pm2,-6\right)[/latex].
31. [latex]\begin{array}{cc}x=\frac{3t}{1+{t}^{3}},\hfill & y=\frac{3{t}^{2}}{1+{t}^{3}}\hfill \end{array}[/latex]
For the following exercises, find [latex]\frac{dy}{dx}[/latex] at the value of the parameter.
32. [latex]\begin{array}{cc}x=\cos{t},\hfill & y=\sin{t},t=\frac{3\pi }{4}\hfill \end{array}[/latex]
33. [latex]\begin{array}{cc}x=\sqrt{t},\hfill & y=2t+4,t=9\hfill \end{array}[/latex]
34. [latex]\begin{array}{cc}x=4\cos{(2\pi s)},\hfill & y=3\sin{(2\pi s)}\hfill \end{array} ,s=-\frac{1}{4}[/latex]
For the following exercises, find [latex]\frac{{d}^{2}y}{d{x}^{2}}[/latex] at the given point without eliminating the parameter.
35. [latex]\begin{array}{ccc}x=\frac{1}{2}{t}^{2},\hfill & y=\frac{1}{3}{t}^{3},\hfill & t=2\hfill \end{array}[/latex]
36. [latex]x=\sqrt{t},y=2t+4,t=1[/latex]
37. Find t intervals on which the curve [latex]x=3{t}^{2},y={t}^{3}-t[/latex] is concave up as well as concave down.
38. Determine the concavity of the curve [latex]x=2t+\text{ln}t,y=2t-\text{ln}t[/latex].
Show Solution
Concave up on [latex]t>0[/latex].
39. Sketch and find the area under one arch of the cycloid [latex]x=r\left(\theta -\sin\theta \right),y=r\left(1-\cos\theta \right)[/latex].
40. Find the area bounded by the curve [latex]x=\cos{t},y={e}^{t},0\le t\le \frac{\pi }{2}[/latex] and the lines [latex]y=1[/latex] and [latex]x=0[/latex].
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[latex]\frac{e\frac{1}{2}-1}{2}[/latex]
41. Find the area enclosed by the ellipse [latex]x=a\cos\theta ,y=b\sin\theta ,0\le \theta <2\pi [/latex].
42. 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 \frac{\pi }{2}[/latex].
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[latex]\frac{3\pi }{2}[/latex]
For the following exercises, find the area of the regions bounded by the parametric curves and the indicated values of the parameter.
43. [latex]x=2\cot\theta ,y=2{\sin}^{2}\theta ,0\le \theta \le \pi [/latex]
44. [T] [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]
Show Solution
[latex]6\pi {a}^{2}[/latex]
45. [T] [latex]x=a\sin\left(2t\right),y=b\sin\left(t\right),0\le t<2\pi [/latex] (the “hourglass”)
46. [T] [latex]x=2a\cos{t}-a\sin\left(2t\right),y=b\sin{t},0\le t<2\pi [/latex] (the “teardrop”)
Show Solution
[latex]2\pi ab[/latex]
For the following exercises, find the arc length of the curve on the indicated interval of the parameter.
47. [latex]x=4t+3,y=3t - 2,0\le t\le 2[/latex]
48. [latex]\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}[/latex]
Show Solution
[latex]\frac{1}{3}\left(2\sqrt{2}-1\right)[/latex]
49. [latex]\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}[/latex]
50. [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]
Show Solution
[latex]7.075[/latex]
51. [latex]\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}[/latex] (express answer as a decimal rounded to three places)
52. [latex]x=a{\cos}^{3}\theta ,y=a{\sin}^{3}\theta [/latex] on the interval [latex]\left[0,2\pi \right)[/latex] (the hypocycloid)
Show Solution
[latex]6a[/latex]
53. Find the length of one arch of the cycloid [latex]x=4\left(t-\sin{t}\right),y=4\left(1-\cos{t}\right)[/latex].
54. Find the distance traveled by a particle with position [latex]\left(x,y\right)[/latex] as t varies in the given time interval: [latex]\begin{array}{ccc}x={\sin}^{2}t,\hfill & y={\cos}^{2}t,\hfill & 0\le t\le 3\pi \hfill \end{array}[/latex].
Show Solution
[latex]6\sqrt{2}[/latex]
55. Find the length of one arch of the cycloid [latex]x=\theta -\sin\theta ,y=1-\cos\theta [/latex].
56. Show that the total length of the ellipse [latex]x=4\sin\theta ,y=3\cos\theta [/latex] is [latex]L=16{\displaystyle\int }_{0}^{\frac{\pi}{2}}\sqrt{1-{e}^{2}{\sin}^{2}\theta }d\theta [/latex], where [latex]e=\frac{c}{a}[/latex] and [latex]c=\sqrt{{a}^{2}-{b}^{2}}[/latex].
57. Find the length of the curve [latex]x={e}^{t}-t,y=4{e}^{\frac{t}{2}},-8\le t\le 3[/latex].
For the following exercises, find the area of the surface obtained by rotating the given curve about the x-axis.
58. [latex]\begin{array}{ccc}x={t}^{3},\hfill & y={t}^{2},\hfill & 0\le t\le 1\hfill \end{array}[/latex]
Show Solution
[latex]\frac{2\pi \left(247\sqrt{13}+64\right)}{1215}[/latex]
59. [latex]\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}[/latex]
60. [T] Use a CAS to find the area of the surface generated by rotating [latex]x=t+{t}^{3},y=t-\frac{1}{{t}^{2}},1\le t\le 2[/latex] about the x-axis. (Answer to three decimal places.)
61. Find the surface area obtained by rotating [latex]x=3{t}^{2},y=2{t}^{3},0\le t\le 5[/latex] about the y-axis.
62. Find the area of the surface generated by revolving [latex]x={t}^{2},y=2t,0\le t\le 4[/latex] about the x-axis.
Show Solution
[latex]\frac{8\pi }{3}\left(17\sqrt{17}-1\right)[/latex]
63. Find the surface area generated by revolving [latex]x={t}^{2},y=2{t}^{2},0\le t\le 1[/latex] about the y-axis.
