In Problems 1–6:
(a) Find the rectangular equation of each curve.
(b) Graph each plane curve whose parametric equations are given and show its orientation.
(c) Determine the restrictions on \(x\) and \(y\) that make the rectangular equation identical to the plane curve.
\(x(t) =4t-2, \, y(t) =1-t;\; -\infty <t<\infty \)
\(x(t) =2t^{2}+6, \, y(t) =5-t;\;-\infty <t<\infty \)
\(x(t) =e^{t}, \, y(t) =e^{-t}\;-\infty < t< \infty \)
\(x(t) =\ln t, \, y(t) =t^{3};\;t>0\)
\(x(t) =\sec ^{2}t, \, y(t) =\tan ^{2}t;\;0\leq t\leq \dfrac{\pi }{4}\)
\(x(t) =t^{3/2}, \, y(t) =2t+4;\;t\geq 0\)
In Problems 7–10, for the parametric equations below:
(a) Find an equation of the tangent line to the curve at \(t\).
(b) Graph the curve and the tangent line.
\(x(t) =t^{2}-4,\; y(t) =t \,\, at \,\, t=1\)
\(x(t) =3\sin t,\;y(t) =4\cos t+2 \,\, at \,\, t=\dfrac{\pi }{4}\)
\(x(t) =\dfrac{1}{t^{2}},\;y(t) =\sqrt{t^{2}+1} \,\, at \,\, t=3\)
\(x(t) =\dfrac{t^{2}}{1+t},\;y(t) =\dfrac{t}{1+t} \,\, at \,\, t=0\)
In Problems 11 and 12, find two different pairs of parametric equations for each rectangular equation.
\(y=-2x+4\)
\(y=2x\)
Describe the motion of an object that moves so that at time \(t\) (in seconds) it has coordinates \(x(t) =2\cos t,\;y(t) =\sin t, \;0\leq t\leq 2\pi\).
In Problems 14–17, the polar coordinates of a point are given. Plot each point in a polar coordinate system, and find its rectangular coordinates.
\(\left( 3,\dfrac{\pi }{6}\right)\)
\(\left( -2,\dfrac{4\pi }{3}\right) \)
\(\left( 3,-\dfrac{\pi }{2}\right) \)
\(\left( -4,-\dfrac{\pi }{4}\right) \)
In Problems 18–21, the rectangular coordinates of a point are given. Find two pairs of polar coordinates \(( r,\theta ) \) for each point, one with \(r>0\) and the other with \(r<0, 0 \le \theta < 2 \pi.\)
\(( 2,0) \)
\(( 3,4) \)
\( ( -5,12) \)
\(( -3,3) \)
In Problems 22–27, the letters \(r, \theta \) represent polar coordinates. Write each equation in terms of the rectangular coordinates \(x, y\).
\(r=4\sin ( 2\theta ) \)
\(r=e^{\theta /2}\)
\(r=\dfrac{1}{1+2\cos \theta }\)
\(r=a-\sin \theta \)
\(r^{2}=4\cos ( 2\theta ) \)
\(r=\theta \)
In Problems 28–31, the letters \(x, y\) represent rectangular coordinates. Write each equation in terms of the polar coordinates \(r, \theta\).
\(x^{2}+y^{2}=x\)
\((x^{2}+y^{2})^{2}=x^{2}-y^{2}\)
\(y^{2}=(x^{2}+y^{2})\cos ^{2}[(x^{2}+y^{2})^{1/2}]\)
\(\dfrac{x^{2}}{2^{2}}+\dfrac{y^{2}}{3^{2}}=1\)
In Problems 32–35, identify and graph each polar equation. Convert it to a rectangular equation if necessary.
\(r\sin \theta =1\)
\(r\sec \theta =2\)
\(r=\sin \theta \)
\(r=-5\cos \theta \)
In Problems 36–45, for each equation:
(a) Graph the equation.
(b) Find parametric equations that represent the equation.
\(r=1-\sin \theta \)
\(r=4\cos ( 2\theta ) \)
\(r=\dfrac{1}{2}-\sin \theta \)
\(r=\dfrac{4}{1-2\cos \theta }\)
\(r=4\sin ( 3\theta ) \)
\(r=2-2\cos \theta \)
\(r=2-\sin \theta \)
\(r=e^{0.5\theta }\)
\(r^{2}=1-\sin ^{2}\theta \)
\(r^{2}=1+\sin ^{2}\theta \)
In Problems 46 and 47, for each polar equation:
(a) Identify and graph the equation.
(b) Convert the polar equation to a rectangular equation.
(c) Find parametric equations for the polar equation.
\(r=\dfrac{2}{1-\cos \theta }\)
\(r=\dfrac{1}{1-\dfrac{1}{6}\cos \theta }\)
In Problems 48 and 49, find parametric equations for an object that moves along the ellipse \(\dfrac{x^{2}}{16}+\dfrac{y^{2}}{9}=1\) with the motion described.
The motion begins at \(( 4,0) \), is counterclockwise, and requires 4 seconds for a complete revolution.
The motion begins at \(( 0, 3) \), is clockwise, and requires 5 seconds for a complete revolution.
In Problems 50 and 51, find the points (if any) on the curve at which the tangent line is vertical or horizontal.
\(x(t) =t^{3}-1,\;y(t) =2t^{2}+1\)
\(x(t) =1-\sin t,\;y(t) =2+3\cos t,\;0\leq t\leq 2\pi \)
In Problems 52–55, find the arc length of each plane curve.
\(x(t) =\sinh ^{-1}t, \, y(t) =\sqrt{t^{2}+1} \, {\rm from} \, t=0 \, {\rm to} \, t=1\)
\(x(t) =\tan t, \, y(t) =\dfrac{1}{3}( \sec ^{2}t+1) \, {\rm from} \, t=0 \, {\rm to} \, t=\dfrac{\pi }{4}\)
\(x(t) =e^{t}, \, y(t) =\dfrac{1}{2} e^{2t}-\dfrac{1}{4}\,t \, {\rm from} \, t=0 \, {\rm to} \, t=2\)
\(x=\dfrac{1}{2}\,y^{2}-\dfrac{1}{4}\ln y \, {\rm from} \, y=1\, {\rm to} \,y=2\)
In Problems 56–59, find the arc length of each curve represented by a polar equation.
\(r=2\sin \theta \, {\rm from} \, \theta =0\, {\rm to} \,\theta =\pi \)
\(r=e^{-\theta } \, {\rm from} \, \theta =0\, {\rm to} \,\theta =2\pi \)
\(r=3\theta \, {\rm from} \, \theta =0\, {\rm to} \,\theta =2\pi \)
\(r=2\sin ^{2}\dfrac{\theta }{2} \, {\rm from} \, \theta =-\dfrac{\pi }{2}\, {\rm to} \,\theta =\dfrac{\pi }{2}\)
Area Find the area of the region inside the circle \(r=4\sin \theta \) and above the line \(r=3\csc \theta\).
Area Find the area of the region that lies inside the rose \(r=4\cos ( 2\theta ) \) and outside the circle \(r=\sqrt{2}\).
Area Find the area of the region common to the graphs of \(r=\cos \theta \) and \(r=1-\cos \theta \).
Surface Area Find the surface area of the solid generated by revolving the smooth curve represented by the parametric equations \(x(t) =\sinh ^{-1}t,\;y(t) =\sqrt{t^{2}+1}\) from \(0\leq \;t\leq 1\) about the \(x\)-axis.
Surface Area Find the surface area of the solid generated by revolving the smooth curve represented by \(x(t) =e^{t}-t,\;y(t) =4e^{t/2}\) from \(t=0\) to \(t=1\) about the \(x\)-axis.
Surface Area Find the surface area of the solid of revolution generated by revolving the curve represented by \(x=y^{2}+1,\;0\leq y\leq 2\), about the \(y\)-axis.
Surface Area Find the surface area of the solid generated by revolving the curve represented by \(y=\cos \, \left( \dfrac{x}{2}\right)\, ,\, 0\leq \, x\leq \pi \) about the \(x\)-axis.
Surface Area Find the surface area of the solid of revolution generated by revolving the arc of the circle \(r=4,\;0\leq \theta \leq \dfrac{\pi }{3}\), about the \(x\)-axis.