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Thomas Calculus

George B. Thomas, Jr.

Chapter 11

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

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

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

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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

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 in the back of the book.)

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Problem 20

Find parametric equations and a parameter interval for the motion of a particle that starts at $(a, 0)$ and traces the ellipse $\left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)=1$
$\begin{array}{ll}{\text { a. once clockwise. }} & {\text { b. once counterclockwise. }} \\ {\text { c. twice clockwise. }} & {\text { d. twice counterclockwise. }}\end{array}$
(As in Exercise $19,$ there are many correct answers.)

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Problem 21

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

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Problem 22

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

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Problem 23

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

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Problem 24

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

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Problem 25

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 26

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 27

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 28

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 travels back and forth from $(3,9)$ to $(-3,9)$ infinitely many times.

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Problem 29

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 to the curve at $(x, y) .$

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Problem 30

Find parametric equations for the circle
$$x^{2}+y^{2}=a^{2},$$
using as parameter the arc length $s$ measured counterclockwise from the point $(a, 0)$ to the point $(x, y) .$

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Problem 31

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.

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Problem 32

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

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Problem 33

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.

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Problem 34

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.

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Problem 35

The witch of Maria Agnesi 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 $O A$ makes with the positive $x$ -axis. The following equalities (which you may assume) will help.
$$\begin{array}{ll}{\text { a. } x=A Q} & {\text { b. } y=2-A B \sin t} \\ {\text { c. } A B \cdot O A=(A Q)^{2}}\end{array}$$

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Problem 36

Hypocycloid 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, if $b=a / 4$ as in the accompanying figure, show that the hypocycloid is the astroid
$$x=a \cos ^{3} \theta, \quad y=a \sin ^{3} \theta$$

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Problem 37

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.

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Problem 38

Trochoids $A$ 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 39

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 40

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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Problem 41

If you have a parametric equation grapher, graph the equations over the given intervals.
Ellipse $\quad x=4 \cos t, \quad y=2 \sin t, \quad$ over
$$\begin{array}{l}{\text { a. } 0 \leq t \leq 2 \pi} \\ {\text { b. } 0 \leq t \leq \pi} \\ {\text { c. }-\pi / 2 \leq t \leq \pi / 2}\end{array}$$

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Problem 42

If you have a parametric equation grapher, graph the equations over the given intervals.
Hyperbola branch $x=\sec t$ (enter as $1 / \cos (t) ), y=\tan t$
(enter as $\sin (t) / \cos (t) ),$ over
$$\begin{array}{l}{\text { a. }-1.5 \leq t \leq 1.5} \\ {\text { b. }-0.5 \leq t \leq 0.5} \\ {\text { c. }-0.1 \leq t \leq 0.1}\end{array}$$

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Problem 43

If you have a parametric equation grapher, graph the equations over the given intervals.
Parabola $x=2 t+3, \quad y=t^{2}-1,-2 \leq t \leq 2$

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Problem 44

If you have a parametric equation grapher, graph the equations over the given intervals.
Cycloid $\quad x=t-\sin t, \quad y=1-\cos t, \quad$ over
$$\begin{array}{l}{\text { a. } 0 \leq t \leq 2 \pi} \\ {\text { b. } 0 \leq t \leq 4 \pi} \\ {\text { c. } \pi \leq t \leq 3 \pi}\end{array}$$

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Problem 45

Deltoid
$$x=2 \cos t+\cos 2 t, \quad y=2 \sin t-\sin 2 t ; \quad 0 \leq t \leq 2 \pi$$
What happens if you replace 2 with $-2$ in the equations for $x$ and $y ?$ Graph the new equations and find out.

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Problem 46

A nice curve
$$x=3 \cos t+\cos 3 t, \quad y=3 \sin t-\sin 3 t ; \quad 0 \leq t \leq 2 \pi$$
What happens if you replace 3 with $-3$ in the equations for $x$ and $y ?$ Graph the new equations and find out.

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Problem 47

a. Epicycloid
$$x=9 \cos t-\cos 9 t, \quad y=9 \sin t-\sin 9 t ; \quad 0 \leq t \leq 2 \pi$$
b. Hypocycloid
$$x=8 \cos t+2 \cos 4 t, \quad y=8 \sin t-2 \sin 4 t ; \quad 0 \leq t \leq 2 \pi$$
c. Hypotrochoid
$$x=\cos t+5 \cos 3 t, \quad y=6 \cos t-5 \sin 3 t ; \quad 0 \leq t \leq 2 \pi$$

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Problem 48

$$\begin{aligned} \text { a. } & x=6 \cos t+5 \cos 3 t, \quad y=6 \sin t-5 \sin 3 t \\ & 0 \leq t \leq 2 \pi \\ \text { b. } & x=6 \cos 2 t+5 \cos 6 t, \quad y=6 \sin 2 t-5 \sin 6 t \\ & 0 \leq t \leq \pi \end{aligned}$$
$$\begin{aligned} \text { c. } & x=6 \cos t+5 \cos 3 t, \quad y=6 \sin 2 t-5 \sin 3 t \\ & 0 \leq t \leq 2 \pi \\ \text { d. } & x=6 \cos 2 t+5 \cos 6 t, \quad y=6 \sin 4 t-5 \sin 6 t \\ & 0 \leq t \leq \pi \end{aligned}$$

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