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Let \(x=x( t) \) and \(y=y( t) \) be two functions whose common domain is some interval \(I\). The collection of points defined by \(( x, y) =( x( t), y( t) ) \) is called a plane __________. The variable \(t\) is called a(n) __________.

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Multiple Choice The parametric equations \(x( t) =2\sin t\), \(y( t) =3\cos t\) define a(n) [(a) line, (b) hyperbola, (c) ellipse, (d) parabola].

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Multiple Choice The parametric equations \(x( t) =a\sin t\), \(y( t) =a\cos t\), \(a > 0\), define a [(a) line, (b) hyperbola, (c) parabola, (d) circle].

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If a circle rolls along a horizontal line without slipping, a point \(P\) on the circle traces out a curve called a(n) ______________.

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True or False The parametric equations defining a curve are unique.

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True or False Plane curves represented using parametric equations have an orientation.

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\(x(t) =2t+1,\quad y(t) =t+2;\quad -\infty \lt t \lt \infty\)

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\(x(t) =t-2,\quad y(t) =3t+1;\quad -\infty \lt t\lt \infty\)

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\(x(t) =2t+1,\quad y(t) =t+2;\quad \ 0\leq t\leq 2\)

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\(x(t) =t-2,\quad y(t) =3t+1;\quad 0\leq t\leq 2\)

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\(x(t) =e^{t},\quad y(t) =t;\quad -\infty \lt t \lt \infty\)

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\(x(t) =t,\quad y(t) =\dfrac{1}{t};\quad -\infty \lt t \lt \infty,t\neq 0\)

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

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\(x(t) =\cos t,\quad y(t) =\sin t;\quad 0\leq t\leq \pi\)

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\(x(t) =2\sin t,\quad y(t) =3\cos t;\quad \ 0\leq t\leq 2\pi\)

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\(x(t) =4\cos t,\quad y(t) =3\sin t;\quad 0\leq t\leq 2\pi\)

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\(x(t) =2\sin t-3,\quad y(t) =2\cos t+1;\quad 0\leq t\leq \pi\)

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\(x(t) =4\cos t+1,\quad y(t) =4\sin t-3;\quad 0\leq t\leq \pi\)

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\(x(t) =3,\quad y(t) =2t;\quad -\infty \lt t \lt \infty\)

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\(x(t) =4t+1,\ \ y(t) =2t;\quad -\infty \lt t \lt \infty\)

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\(x(t) =2,\quad y(t) =t^{2}+4;\;t>0\)

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\(x(t) =t+3,\quad y(t) =t^{3};\quad -4\leq t\leq 4\)

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\(x(t) =t+5,\quad y(t) = \sqrt{t};\quad t\geq 0\)

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\(x(t) =2t^{2},\quad y(t) =2t^{3};\quad 0\leq t\leq 3\)

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\(x(t) =t^{1/2}+1,\quad y(t) =t^{3/2};\quad t\geq 1\)

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\(x(t) =2e^{t},\quad y(t) =1-e^{t};\quad t\geq 0\)

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\(x(t) =\sec t,\quad y(t) =\tan t;\quad -\dfrac{\pi }{2} \lt t \lt \dfrac{\pi }{2} \)

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\(x(t) =3\sinh t,\quad y(t) =2\cosh t;\quad -\infty \lt t\lt \infty \)

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\(x(t) =t^{4},\quad y(t) =t^{2}\)

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\(x(t) =t^{2},\quad y(t) =t^{4}\)

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\(x(t) =t^{2},\quad y(t) =2t-1\)

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\(x(t) =e^{2t},\quad y(t) =2e^{t}-1\)

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\(x(t) =\left( \dfrac{1}{t}\right) ^{2},\quad y(t) =\dfrac{2}{t}-1\)

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\(x(t) =t^{4},\quad y(t) =2t^{2}-1\)

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\(x(t) =3\sin ^{2}t-2,\quad y(t) =2\cos t;\quad 0\leq t\leq \pi \)

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\(x(t) =1+2\sin ^{2}t,\quad y(t) =2-\cos t;\quad 0\leq t\leq 2\pi \)

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\(x(t) =\dfrac{1}{t^{2}},\quad y(t) =\dfrac{2}{t^{2}+1};\quad t> 0\)

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\(x(t) =\dfrac{3t}{ \sqrt{t^{2}+1}},\quad y(t) =\dfrac{3}{ \sqrt{t^{2}+1}}\)

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\(x(t) =\dfrac{4}{ \sqrt{4-t^{2}}},\quad y(t) =\dfrac{4t}{ \sqrt{4-t^{2}}};\quad 0\leq t \lt 2\)

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\(x(t) = \sqrt{t-3},\quad y(t) = \sqrt{t+1};\quad t\geq 3\)

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\(x(t) =\sin t-2,\quad y(t) =4-2\cos t;\quad 0\leq t\leq 2\pi\)

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\(x(t) =2+\tan t,\quad y(t) =3-2\sec t;\quad -\pi/2 \lt t \lt \pi/2\)

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\(y=4x-2\)

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\(y=-8x+3\)

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\(y=-2x^{2}+1\)

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\(y=x^{2}+1\)

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\(y=4x^{3}\)

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\(y=2x^{2}\)

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\(x=\dfrac{1}{3} \sqrt{y}-3\)

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\(x=y^{3/2}\)

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the motion begins at \((3,0)\), is counterclockwise, and requires 3 seconds for 1 revolution.

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the motion begins at \((3,0)\), is clockwise, and requires 3 seconds for 1 revolution.

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The motion begins at \((0,2)\), is clockwise, and requires 2 seconds for 1 revolution.

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the motion begins at \((0,2)\), is counterclockwise, and requires 1 second for 1 revolution.

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1. \(x(t) =t,\quad y(t) =t^{2};\quad -4\leq t\leq 4\)
2. \(x(t) = \sqrt{t},\quad \ y(t) =t;\quad 0\leq t\leq 16\)
3. \(x(t) =e^{t},\quad \ y(t) =e^{2t};\quad 0\leq t\leq \ln 4\)
4. \(x(t) =\cos t,\quad \ y(t) =1-\sin ^{2}t;\quad 0\leq t\leq \pi \)

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1. \(x(t) =t,\quad \ y(t) = \sqrt{1-t^{2}};\quad -1\leq t\leq 1\)
2. \(x(t) =\sin t,\quad \ y(t) =\cos t;\quad 0\leq t\leq 2\pi \)
3. \(x(t) =\cos t,\quad \ y(t) =\sin t;\quad 0\leq t\leq 2\pi \)
4. \(x(t) = \sqrt{1-t^{2}},\quad y(t) =t;\quad -1\leq t\leq 1\)

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1. \(\left[ -\dfrac{\pi }{2},\dfrac{\pi }{2}\right] \)
2. \(\left[ -\pi ,\dfrac{\pi }{2}\right] \)
3. \(\left[ -\pi ,0\right] \)
4. \(\left[ 0,\pi \right] \)

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1. \(\left[ \dfrac{\pi }{2},\pi \right] \)
2. \(\left[ 0 ,\dfrac{\pi}{3} \right] \)
3. \(\left[ 0,\dfrac{\pi }{2}\right] \)
4. \(\left[ \dfrac{\pi }{3},\pi \right] \)

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1. \(\left[ \dfrac{\pi }{2}, \dfrac{3\pi}{2} \right] \)
2. \(\left[ \pi ,2\pi \right] \)
3. \([ 0,\pi ] \)
4. \(\left[ \dfrac{5\pi }{4},2\pi \right] \)

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1. \(\left[ 0,\dfrac{5\pi }{6}\right] \)
2. \( [ \pi ,2\pi ] \)
3. \(\left[ 0,\dfrac{\pi }{2}\right] \)
4. \(\left[ \dfrac{\pi }{6},\dfrac{5\pi }{6}\right] \)

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1. Graph the plane curve represented by the parametric equations \(x(t) =7( t-\sin t)\), \(y(t) =7( 1-\cos t), 0\leq t\leq 2\pi\).
2. Find the coordinates of the point \(( x,y)\) on the curve when \(t=2.1\).

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1. Graph the plane curve represented by the parametric equations \(x(t) =t-e^{t}, y(t) =2e^{t/2}, -8\leq t\leq 2\).
2. Find the coordinates of the point \(( x,y) \) on the curve when \(t=1.5\).

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Find parametric equations for the ellipse \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\).

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Find parametric equations for the hyperbola \(\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1\).

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Trajectory of a Baseball A baseball is hit with an initial speed of \(125\;\rm{ft}/\!\rm{s}\) at an angle of \(40^{\circ}\) to the horizontal. The ball is hit at a height of \(3\;\rm{ft}\) above the ground.

1. Find parametric equations that model the position of the ball as a function of the time \(t\) in seconds.
2. What is the height of the baseball after \(2\) seconds?
3. What horizontal distance \(x\) has the ball traveled after 2 seconds?
4. How long does it take the baseball to travel \(x=300\;\rm{ft}?\)
5. What is the height of the baseball at the time found in (d)?
6. How long is the ball in the air before it hits the ground?
7. How far has the baseball traveled horizontally when it hits the ground?

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Trajectory of a Baseball A pitcher throws a baseball with an initial speed of \(145\;\rm{ft}/\!\rm{s}\) at an angle of \(20^{\circ}\) to the horizontal. The ball leaves his hand at a height of \(5\;\rm{ft}\).

1. Find parametric equations that model the position of the ball as a function of the time \(t\) in seconds.
2. What is the height of the baseball after \(\dfrac{1}{2}\) second?
3. What horizontal distance \(x\) has the ball traveled after \(\dfrac{1}{2}\) second?
4. How long does it take the baseball to travel \(x=60\;\rm{ft}?\)
5. What is the height of the baseball at the time found in (d)?

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Trajectory of a Football A quarterback throws a football with an initial speed of \(80\;\rm{ft}/\!\rm{s}\) at an angle of \(35^{\circ}\) to the horizontal. The ball leaves the quarterback's hand at a height of \(6\;\rm{ft}\).

1. Find parametric equations that model the position of the ball as a function of time \(t\) in seconds.
2. What is the height of the football after \(1\) second?
3. What horizontal distance \(x\) has the ball traveled after 1 second?
4. How long does it take the football to travel \(x=120\;\rm{ft}?\)
5. What is the height of the football at the time found in (d)?

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The plane curve represented by the parametric equations \(x(t) =t,\quad y(t) =t^{2}\), and the plane curve represented by the parametric equations \(x(t) =t^{2},\quad y(t) =t^{4}\) (where, in each case, time \(t\) is the parameter) appear to be identical, but they differ in an important aspect. Identify the difference, and explain its meaning.

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The circle \(x^{2}+y^{2}=4\) can be represented by the parametric equations \(x(\theta) =2\cos \theta ,\quad y(\theta) =2\sin \theta ,\quad 0\leq \theta \leq 2\pi \), or by the parametric equations \(x(\theta) =2\sin \theta ,\quad y(\theta) =2\cos \theta \), \(0\leq \theta \leq 2\pi\). But the plane curves represented by each pair of parametric equations are different. Identify and explain the difference.

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Uniform Motion A train leaves a station at \(7:15\) a.m. and accelerates at the rate of \(3\;\rm{mi}/\!\rm{h}^{2}\). Mary, who can run \(6 \;\rm{mi}/\!\rm{h}\), arrives at the station \(2\) seconds after the train has left. Find parametric equations that model the motion of the train and of Mary as a function of time. (Hint: The position \(s\) at time \(t\) of an object having acceleration \(a\) is \(s=\dfrac{1}{2}a t^{2}\).)

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Challenge Problems

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Find parametric equations for the circle \(x^{2}+y^{2}=R^{2}\), using as the parameter the slope \(m\) of the line through the point \((-R,0)\) and a general point \(P=(x,y)\) on the circle.

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Find parametric equations for the parabola \(y=x^{2}\), using as the parameter the slope \(m\) of the line joining the point \((1,1)\) to a general point \(P=(x,y)\) on the parabola.

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Hypocycloid Let a circle of radius \(b\) roll, without slipping, inside a fixed circle with radius \(a\), where \(a > b\). A fixed point \(P\) on the circle of radius \(b\) traces out a curve, called a hypocycloid, as shown in the figure. If \(A=(a,0)\) is the initial position of the point \(P\) and if \(t\) denotes the angle from the positive \(x\)-axis to the line segment from the origin to the center of the circle, show that the parametric equations of the hypocycloid are \[ \begin{eqnarray*} x(t) &=&(a-b)\cos t+b\cos \left( \frac{a-b}{b}t\right) \\[4pt] y(t) &=&(a-b)\sin t-b\sin \left( \frac{a-b}{b}t\right)\;0\leq t\leq 2\pi \end{eqnarray*} \]

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Hypocycloid Show that the rectangular equation of a hypocycloid with \(a=4b\) is \(x^{2/3}+y^{2/3}=a^{2/3}\).

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Epicycloid Suppose a circle of radius \(b\) rolls on the outside of a second circle, as shown in the figure. Find the parametric equations of the curve, called an epicycloid, traced out by the point \(P\).