University Calculus: Early Transcendentals 4th

Joel Hass, Christopher Heil, Przemyslaw Bogacki

Chapter 10

Parametric Equations and Polar Coordinates

Educators


Problem 1

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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Problem 2

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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Problem 3

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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Problem 4

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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Problem 5

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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Problem 6

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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Problem 7

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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Problem 8

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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Problem 9

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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Problem 10

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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Problem 11

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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Problem 12

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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Problem 13

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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Problem 14

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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Problem 15

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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Problem 16

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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Problem 17

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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Problem 18

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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Problem 19

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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Problem 20

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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Problem 21

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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Problem 22

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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Problem 23

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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Problem 24

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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Problem 25

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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Problem 26

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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Problem 27

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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Problem 28

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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Problem 29

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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Problem 30

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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Problem 31

Find a parametrization for the curve.
the line segment with endpoints (-1,-3) and (4,1)

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Problem 32

Find a parametrization for the curve.
the line segment with endpoints (-1,3) and (3,-2)

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Problem 33

Find a parametrization for the curve.
the lower half of the parabola $x-1=y^{2}$

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Problem 34

Find a parametrization for the curve.
the left half of the parabola $y=x^{2}+2 x$

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Problem 35

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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Problem 36

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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Problem 37

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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Problem 38

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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Problem 39

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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Problem 40

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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Problem 41

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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Problem 42

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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Problem 43

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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Problem 44

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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Problem 45

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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Problem 46

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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Problem 47

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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Problem 48

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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Problem 49

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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Problem 50

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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