매개변수와 극좌표 전체 문제

[T] Use technology to sketch the spiral curve given by $x=t\,\cos (t),y=t\,\sin (t)$ from $-2\pi \le t\le 2\pi .$

[T] Use technology to graph the curve given by the parametric equations $x=2\,\cot (t),y=1-\cos (2t),-\pi /2\le t\le \pi /2.$ This curve is known as the witch of Agnesi.

[T] Sketch the curve given by parametric equations $\begin{array}{c}x=\cosh (t) \\ y=\sinh (t),\end{array}$ where $-2\le t\le 2.$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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