매개변수 방정식과 곡선

For the following exercises, use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation.

[T] $\begin{array}{c}x=t-0.5\,\sin \,t \\ y=1-1.5\,\cos \,t\end{array}$

[T] Use technology to sketch the curve represented by $x=\sin (4t),y=\sin (3t),0\le t\le 2\pi .$

For the following exercises, use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation.

[T] Use technology to sketch $x=2\,\tan (t),y=3\,\sec (t),-\pi <t<\pi .$

Sketch the curve known as an epitrochoid, which gives the path of a point on a circle of radius b as it rolls on the outside of a circle of radius a. The equations are

$\begin{array}{c}x=(a+b)\cos \,t-c\cdot \cos \left[\frac{(a+b)t}{b}\right] \\ y=(a+b)\sin \,t-c\cdot \sin \left[\frac{(a+b)t}{b}\right].\end{array}$

Let $a=1,b=2,c=1.$

[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 $-2\le t\le 2.$

In Exercises 5–8, sketch the graph of the given parametric equations by hand, making a table of points to plot. Be sure to indicate the orientation of the graph.

$x=t^2+t$, $y=1-t^2$, $-3\leq t\leq 3$

In Exercises 5–8, sketch the graph of the given parametric equations by hand, making a table of points to plot. Be sure to indicate the orientation of the graph.

$x=1$, $y=5\sin t$, $-\pi/2\leq t\leq \pi/2$

In Exercises 5–8, sketch the graph of the given parametric equations by hand, making a table of points to plot. Be sure to indicate the orientation of the graph.

$x=t^2$, $y=2$, $-2\leq t\leq 2$

In Exercises 5–8, sketch the graph of the given parametric equations by hand, making a table of points to plot. Be sure to indicate the orientation of the graph.

$x=t^3-t+3$, $y=t^2+1$, $-2\leq t\leq 2$

In Exercises 9–18, sketch the graph of the given parametric equations; using a graphing utility is advisable. Be sure to indicate the orientation of the graph.

$x=t^3-2t^2$, $y=t^2$, $-2\leq t \leq 3$

In Exercises 9–18, sketch the graph of the given parametric equations; using a graphing utility is advisable. Be sure to indicate the orientation of the graph.

$x=1/t$, $y=\sin t$, $0< t \leq 10$

In Exercises 9–18, sketch the graph of the given parametric equations; using a graphing utility is advisable. Be sure to indicate the orientation of the graph.

$x=3\cos t$, $y=5\sin t$, $0\leq t \leq 2\pi$

In Exercises 9–18, sketch the graph of the given parametric equations; using a graphing utility is advisable. Be sure to indicate the orientation of the graph.

$x=3\cos t+2$, $y=5\sin t+3$, $0\leq t \leq 2\pi$

In Exercises 9–18, sketch the graph of the given parametric equations; using a graphing utility is advisable. Be sure to indicate the orientation of the graph.

$x=\cos t$, $y=\cos(2t)$, $0\leq t \leq \pi$

In Exercises 9–18, sketch the graph of the given parametric equations; using a graphing utility is advisable. Be sure to indicate the orientation of the graph.

$x=\cos t$, $y=\sin(2t)$, $0\leq t \leq 2\pi$

In Exercises 9–18, sketch the graph of the given parametric equations; using a graphing utility is advisable. Be sure to indicate the orientation of the graph.

$x=2\sec t$, $y=3\tan t$, $-\pi/2< t < \pi/2$

In Exercises 9–18, sketch the graph of the given parametric equations; using a graphing utility is advisable. Be sure to indicate the orientation of the graph.

$x=\cosh t$, $y=\sinh t$, $-2\leq t \leq 2$

In Exercises 9–18, sketch the graph of the given parametric equations; using a graphing utility is advisable. Be sure to indicate the orientation of the graph.

$x=\cos t+\frac14\cos(8t)$, $y=\sin t+\frac14\sin(8t)$, $0\leq t \leq 2\pi$