7.2 Calculus of parametric curves  (Page 6/6)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Find all points on the curve $x=4\text{cos}(t),y=4\text{sin}(t)$ that have the slope of $\frac{1}{2}.$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[T] $x=a\text{sin}(2t),y=b\text{sin}(t),0\le t<2\pi$ (the “hourglass”)

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

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

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

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

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

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

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

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

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

Find the length of one arch of the cycloid $x=\theta -\text{sin}\theta ,y=1-\text{cos}\theta .$

Show that the total length of the ellipse $x=4\text{sin}\theta ,y=3\text{cos}\theta$ is $L=16{\displaystyle {\int }_0^{\pi \text{/}2}\sqrt{1-e^2{\text{sin}}^2\theta }d\theta ,}$ where $e=\frac{c}{a}$ and $c=\sqrt{a^2-b^2}.$

Find the length of the curve $x=e^t-t,y=4e^{t\text{/}2},\mathrm{-8}\le t\le 3.$

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

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

[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.)

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

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

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