Give parametric equations and parameter intervals for the motion of a particle in the $x y$ -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

$$x=3 t, \quad y=9 t^{2}, \quad-\infty<t<\infty$$

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Give parametric equations and parameter intervals for the motion of a particle in the $x y$ -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

$$x=-\sqrt{t}, \quad y=t, \quad t \geq 0$$

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Give parametric equations and parameter intervals for the motion of a particle in the $x y$ -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

$$x=2 t-5, \quad y=4 t-7, \quad-\infty<t<\infty$$

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Give parametric equations and parameter intervals for the motion of a particle in the $x y$ -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

$$x=3-3 t, \quad y=2 t, \quad 0 \leq t \leq 1$$

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Give parametric equations and parameter intervals for the motion of a particle in the $x y$ -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

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

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Give parametric equations and parameter intervals for the motion of a particle in the $x y$ -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

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

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Give parametric equations and parameter intervals for the motion of a particle in the $x y$ -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

$$x=4 \cos t, \quad y=2 \sin t, \quad 0 \leq t \leq 2 \pi$$

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Give parametric equations and parameter intervals for the motion of a particle in the $x y$ -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

$$x=4 \sin t, \quad y=5 \cos t, \quad 0 \leq t \leq 2 \pi$$

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Give parametric equations and parameter intervals for the motion of a particle in the $x y$ -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

$$x=\sin t, \quad y=\cos 2 t, \quad-\frac{\pi}{2} \leq t \leq \frac{\pi}{2}$$

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Give parametric equations and parameter intervals for the motion of a particle in the $x y$ -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

$$x=1+\sin t, \quad y=\cos t-2, \quad 0 \leq t \leq \pi$$

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Give parametric equations and parameter intervals for the motion of a particle in the $x y$ -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

$$x=t^{2}, \quad y=t^{6}-2 t^{4}, \quad-\infty<t<\infty$$

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Give parametric equations and parameter intervals for the motion of a particle in the $x y$ -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

$$x=\frac{t}{t-1}, \quad y=\frac{t-2}{t+1}, \quad-1<t<1$$

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Give parametric equations and parameter intervals for the motion of a particle in the $x y$ -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

$$x=t, \quad y=\sqrt{1-t^{2}}, \quad-1 \leq t \leq 0$$

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Give parametric equations and parameter intervals for the motion of a particle in the $x y$ -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

$$x=\sqrt{t+1}, \quad y=\sqrt{t}, \quad t \geq 0$$

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Give parametric equations and parameter intervals for the motion of a particle in the $x y$ -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

$$x=\sec ^{2} t-1, \quad y=\tan t, \quad-\pi / 2<t<\pi / 2$$

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Give parametric equations and parameter intervals for the motion of a particle in the $x y$ -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

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

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Give parametric equations and parameter intervals for the motion of a particle in the $x y$ -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

$$x=-\cosh t, \quad y=\sinh t, \quad-\infty<t<\infty$$

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Give parametric equations and parameter intervals for the motion of a particle in the $x y$ -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

$$x=2 \sinh t, \quad y=2 \cosh t, \quad-\infty<t<\infty$$

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Match the parametric equations with the parametric curves labeled A through F.

(GRAPH CAN'T COPY)

$$x=1-\sin t, \quad y=1+\cos t$$

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Match the parametric equations with the parametric curves labeled A through F.

(GRAPH CAN'T COPY)

$$x=\cos t, \quad y=2 \sin t$$

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Match the parametric equations with the parametric curves labeled A through F.

(GRAPH CAN'T COPY)

$$x=\frac{1}{4} t \cos t, \quad y=\frac{1}{4} t \sin t$$

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Match the parametric equations with the parametric curves labeled A through F.

(GRAPH CAN'T COPY)

$$x=\sqrt{t}, \quad y=\sqrt{t} \cos t$$

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Match the parametric equations with the parametric curves labeled A through F.

(GRAPH CAN'T COPY)

$$x=\ln t, \quad y=3 e^{-t / 2}$$

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Match the parametric equations with the parametric curves labeled A through F.

(GRAPH CAN'T COPY)

$$x=\cos t, \quad y=\sin 3 t$$

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Use the given graphs of $x=f(t)$ and $y=g(t)$ to sketch the corresponding parametric curve in the $x y$ -plane.

(GRAPH CAN'T COPY)

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(GRAPH CAN'T COPY)

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(GRAPH CAN'T COPY)

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(GRAPH CAN'T COPY)

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Find parametric equations and a parameter interval for the motion of a particle that starts at $(a, 0)$ and traces the circle $x^{2}+y^{2}=a^{2}$

a. once clockwise.

b. once counterclockwise.

c. twice clockwise.

d. twice counterclockwise.

(There are many ways to do these, so your answers may not be the same as the ones at the back of the text.)

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Find parametric equations and a parameter interval for the motion of a particle that starts at $(a, 0)$ and traces the ellipse $\left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)=1$

a. once clockwise.

b. once counterclockwise.

c. twice clockwise.

d. twice counterclockwise.

(As in Exercise 29 , there are many correct answers.)

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Find a parametrization for the curve.

the line segment with endpoints (-1,-3) and (4,1)

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Find a parametrization for the curve.

the line segment with endpoints (-1,3) and (3,-2)

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Find a parametrization for the curve.

the lower half of the parabola $x-1=y^{2}$

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Find a parametrization for the curve.

the left half of the parabola $y=x^{2}+2 x$

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Find a parametrization for the curve.

the ray (half line) with initial point ( 2,3 ) that passes through the point (-1,-1)

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Find a parametrization for the curve.

the ray (half line) with initial point (-1,2) that passes through the point (0,0)

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Find parametric equations and a parameter interval for the motion of a particle starting at the point (2,0) and tracing the top half of the circle $x^{2}+y^{2}=4$ four times.

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Find parametric equations and a parameter interval for the motion of a particle that moves along the graph of $y=x^{2}$ in the following way: Beginning at ( 0.0 ) it moves to $(3,9),$ and then it travels back and forth from (3,9) to (-3,9) infinitely many times.

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Find parametric equations for the semicircle

$$

x^{2}+y^{2}=a^{2}, \quad y>0

$$

using as parameter the slope $t=d y / d x$ of the tangent line to the curve at $(x, y)$

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Find parametric equations for the circle

$$

x^{2}+y^{2}=a^{2}

$$

using as parameter the are length $s$ measured counterclockwise from the point $(a, 0)$ to the point $(x, y)$

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Find a parametrization for the line segment joining points (0,2) and (4,0) using the angle $\theta$ in the accompanying figure as the parameter.

(GRAPH CAN'T COPY)

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Find a parametrization for the curve $y=\sqrt{x}$ with terminal point (0,0) using the angle $\theta$ in the accompanying figure as the parameter.

(GRAPH CAN'T COPY)

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Find a parametrization for the circle $(x-2)^{2}+y^{2}=1$ starting at (1,0) and moving clockwise once around the circle, using the central angle $\theta$ in the accompanying figure as the parameter.

(GRAPH CAN'T COPY)

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Find a parametrization for the circle $x^{2}+y^{2}=1$ starting at (1,0) and moving counterclockwise to the terminal point $(0,1),$ using the angle $\theta$ in the accompanying figure as the parameter.

(GRAPH CAN'T COPY)

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The bell-shaped witch of Maria Agnesi can be constructed in the following way. Start with a circle of radius $1,$ centered at the point $(0,1),$ as shown in the accompanying figure. Choose a point $A$ on the line $y=2$ and connect it to the origin with a line segment. Call the point where the segment crosses the circle $B$. Let $P$ be the point where the vertical line through $A$ crosses the horizontal line through $B$. The witch is the curve traced by $P$ as $A$ moves along the line $y=2 .$ Find parametric equations and a parameter interval for the witch by expressing the coordinates of $P$ in terms of $t,$ the radian measure of the angle that segment OA makes with the positive $x$ -axis. The following equalities (which you may assume) will help.

(GRAPH CAN'T COPY)

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When a circle rolls on the inside of a fixed circle, any point $P$ on the circumference of the rolling circle describes a hypocycloid. Let the fixed circle be $x^{2}+y^{2}=a^{2}$, let the radius of the rolling circle be $b$, and let the initial position of the tracing point $P$ be $A(a, 0) .$ Find parametric equations for the hypocycloid, using as the parameter the angle $\theta$ from the positive $x$ -axis to the line joining the circles centers. In particular accompanying figure, show that the hypocycloid is the astroid

(FIGURE CAN'T COPY)

$$

x=a \cos ^{3} \theta, \quad y=a \sin ^{3} \theta

$$

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As the point $N$ moves along the line $y=a$ in the accompanying figure, $P$ moves in such a way that $O P=M N .$ Find parametric equations for the coordinates of $P$ as functions of the angle $t$ that the line $O N$ makes with the positive $y$ -axis.

(FIGURE CAN'T COPY)

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A wheel of radius $a$ rolls along a horizontal straight line without slipping. Find parametric equations for the curve traced out by a point $P$ on a spoke of the wheel $b$ units from its center. As parameter, use the angle $\theta$ through which the wheel turns. The curve is called a trochoid, which is a cycloid when $$b=a$$

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Find the point on the parabola $x=t, y=t^{2},-\infty<t<\infty$ closest to the point $(2,1 / 2) .$ (Hint: Minimize the square of the distance as a function of $t$ )

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Find the point on the ellipse $x=2 \cos t, y=\sin t, 0 \leq t \leq 2 \pi$

closest to the point $(3 / 4,0) .$ (Hint: Minimize the square of the distance as a function of $t$ )

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