Thomas Calculus

George B. Thomas Jr.

Chapter 11

Parametric Equations and Polar Coordinates - all with Video Answers

Educators

Section 1

Parametrizations of Plane Curves

Problem 1

Exercises $1-18$ 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.
$$
x=3 t, \quad y=9 t^{2}, \quad-\infty < t < \infty
$$

Luuk Verhoeven

Luuk Verhoeven

Numerade Educator

Problem 2

Exercises $1-18$ 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.
$$
x=-\sqrt{t}, \quad y=t, \quad t \geq 0
$$

Matt Just

Numerade Educator

Problem 3

Exercises $1-18$ 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.
$$
x=2 t-5, \quad y=4 t-7, \quad-\infty < t < \infty
$$

Luuk Verhoeven

Numerade Educator

Problem 4

Exercises $1-18$ 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.
$$
x=3-3 t, \quad y=2 t, \quad 0 \leq t \leq 1
$$

Matt Just

Numerade Educator

Problem 5

Exercises $1-18$ 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.
$$
x=\cos 2 t, \quad y=\sin 2 t, \quad 0 \leq t \leq \pi
$$

Luuk Verhoeven

Numerade Educator

Problem 6

Exercises $1-18$ 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.
$$
x=\cos (\pi-t), \quad y=\sin (\pi-t), \quad 0 \leq t \leq \pi
$$

Matt Just

Numerade Educator

Problem 7

Exercises $1-18$ 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.
$$
x=4 \cos t, \quad y=2 \sin t, \quad 0 \leq t \leq 2 \pi
$$

Luuk Verhoeven

Numerade Educator

Problem 8

Exercises $1-18$ 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.
$$
x=4 \sin t, \quad y=5 \cos t, \quad 0 \leq t \leq 2 \pi
$$

Matt Just

Numerade Educator

Problem 9

Exercises $1-18$ 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.
$$
x=\sin t, \quad y=\cos 2 t, \quad-\frac{\pi}{2} \leq t \leq \frac{\pi}{2}
$$

Luuk Verhoeven

Numerade Educator

Problem 10

Exercises $1-18$ 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.
$$
x=1+\sin t, \quad y=\cos t-2, \quad 0 \leq t \leq \pi
$$

Matt Just

Numerade Educator

Problem 11

Exercises $1-18$ 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.
$$
x=t^{2}, \quad y=t^{6}-2 t^{4}, \quad-\infty < t < \infty
$$

Luuk Verhoeven

Numerade Educator

Problem 12

Exercises $1-18$ 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.
$$
x=\frac{t}{t-1}, \quad y=\frac{t-2}{t+1}, \quad-1 < t < 1
$$

Matt Just

Numerade Educator

Problem 13

Exercises $1-18$ 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.
$$
x=t, \quad y=\sqrt{1-t^{2}}, \quad-1 \leq t \leq 0
$$

Luuk Verhoeven

Numerade Educator

Problem 14

Exercises $1-18$ 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.
$$
x=\sqrt{t+1}, \quad y=\sqrt{t}, \quad t \geq 0
$$

Matt Just

Numerade Educator

Problem 15

Exercises $1-18$ 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.
$$
x=\sec ^{2} t-1, \quad y=\tan t, \quad-\pi / 2 < t < \pi / 2
$$

Luuk Verhoeven

Numerade Educator

Problem 16

Exercises $1-18$ 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.
$$
x=-\sec t, \quad y=\tan t, \quad-\pi / 2 < t < \pi / 2
$$

Matt Just

Numerade Educator

Problem 17

Exercises $1-18$ 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.
$$
x=-\cosh t, \quad y=\sinh t, \quad-\infty < t < \infty
$$

Luuk Verhoeven

Numerade Educator

Problem 18

Exercises $1-18$ 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.
$$
x=2 \sinh t, \quad y=2 \cosh t, \quad-\infty < t < \infty
$$

Matt Just

Numerade Educator

Problem 19

In Exercises $19-24,$ match the parametric equations with the parametric curves labeled A through F.
$$
x=1-\sin t, \quad y=1+\cos t
$$

Luuk Verhoeven

Numerade Educator

Problem 20

In Exercises $19-24,$ match the parametric equations with the parametric curves labeled A through F.
$$
x=\cos t, \quad y=2 \sin t
$$

Matt Just

Numerade Educator

Problem 21

In Exercises $19-24,$ match the parametric equations with the parametric curves labeled A through F.
$$
x=\frac{1}{4} t \cos t, \quad y=\frac{1}{4} t \sin t
$$

Luuk Verhoeven

Numerade Educator

Problem 22

In Exercises $19-24,$ match the parametric equations with the parametric curves labeled A through F.
$$
x=\sqrt{t}, \quad y=\sqrt{t} \cos t
$$

Matt Just

Numerade Educator

Problem 23

In Exercises $19-24,$ match the parametric equations with the parametric curves labeled A through F.
$$
x=\ln t, \quad y=3 e^{-t / 2}
$$

Luuk Verhoeven

Numerade Educator

Problem 24

In Exercises $19-24,$ match the parametric equations with the parametric curves labeled A through F.
$$
x=\cos t, \quad y=\sin 3 t
$$

Matt Just

Numerade Educator

Problem 25

In Exercises $25-28,$ use the given graphs of $x=f(t)$ and $y=g(t)$ to
sketch the corresponding parametric curve in the $x y$ -plane.

Luuk Verhoeven

Numerade Educator

Problem 26

In Exercises $25-28,$ use the given graphs of $x=f(t)$ and $y=g(t)$ to
sketch the corresponding parametric curve in the $x y$ -plane.

Matt Just

Numerade Educator

Problem 27

In Exercises $25-28,$ use the given graphs of $x=f(t)$ and $y=g(t)$ to
sketch the corresponding parametric curve in the $x y$ -plane.

Luuk Verhoeven

Numerade Educator

Problem 28

In Exercises $25-28,$ use the given graphs of $x=f(t)$ and $y=g(t)$ to
sketch the corresponding parametric curve in the $x y$ -plane.

Matt Just

Numerade Educator

Problem 29

$\begin{array}{l}{\text { Find parametric equations and a parameter interval for the motion }} \\ {\text { of a particle that starts at }(a, 0) \text { and traces the circle } x^{2}+y^{2}=a^{2}}\end{array}$
$\begin{array}{l}{\text { a. once clockwise. }} \\ {\text { b. once counterclockwise. }} \\ {\text { c. twice clockwise. }} \\ {\text { d. twice counterclockwise. }}\end{array}$
$\begin{array}{l}{\text { (There are many ways to do these, so your answers may not be the }} \\ {\text { same as the ones in the back of the book.) }}\end{array}$

Luuk Verhoeven

Numerade Educator

Problem 30

$\begin{array}{l}{\text { Find parametric equations and a parameter interval for the mo- }} \\ {\text { tion of a particle that starts at }(a, 0) \text { and traces the ellipse }} \\ {\left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)=1}\end{array}$
$\begin{array}{l}{\text { a. } \text { once clockwise. }} \\ {\text { b. once counterclockwise. }} \\ {\text { c. twice clockwise. }} \\ {\text { d. twice counterclockwise. }} \\ {\text { (As in Exercise } 29, \text { there are many correct answers.) }}\end{array}$

Matt Just

Numerade Educator

Problem 31

$\begin{array}{l}{\text { In Exercises } 31-36, \text { find a parametrization for the curve. }} \\ {\text { the line segment with endpoints }(-1,-3) \text { and }(4,1)}\end{array}$

Luuk Verhoeven

Numerade Educator

Problem 32

In Exercises $31-36,$ find a parametrization for the curve.
the line segment with endpoints $(-1,3)$ and $(3,-2)$

Matt Just

Numerade Educator

Problem 33

In Exercises $31-36,$ find a parametrization for the curve.
the lower half of the parabola $x-1=y^{2}$

Luuk Verhoeven

Numerade Educator

Problem 34

In Exercises $31-36,$ find a parametrization for the curve.
the left half of the parabola $y=x^{2}+2 x$

Matt Just

Numerade Educator

Problem 35

In Exercises $31-36,$ find a parametrization for the curve.
the ray (half line) with initial point $(2,3)$ that passes through the
point $(-1,-1)$

Luuk Verhoeven

Numerade Educator

Problem 36

In Exercises $31-36,$ find a parametrization for the curve.
the ray (half line) with initial point $(-1,2)$ that passes through the
point $(0,0)$

Matt Just

Numerade Educator

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.

Luuk Verhoeven

Numerade Educator

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

Matt Just

Numerade Educator

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

Luuk Verhoeven

Numerade Educator

Problem 40

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 $(x, y)$ .

Matt Just

Numerade Educator

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.

Luuk Verhoeven

Numerade Educator

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.

Matt Just

Numerade Educator

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.

Luuk Verhoeven

Numerade Educator

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.

Matt Just

Numerade Educator

Problem 45

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.
$\begin{array}{ll}{\text { a. }} & {x=A Q} & {\text { b. } y=2-A B \sin t} \\ {\text { c. }} & {A B \cdot O A=(A Q)^{2}}\end{array}$

Luuk Verhoeven

Numerade Educator

Problem 46

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

Matt Just

Numerade Educator

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.

Yingtai Xiao

Numerade Educator

Problem 48

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
$b=a .$

Matt Just

Numerade Educator

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 $t .$ )

Luuk Verhoeven

Numerade Educator

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 $t . )$

Luuk Verhoeven

Numerade Educator

Problem 51

\begin{equation}
\begin{array}{l}{\text { If you have a parametric equation grapher, graph the equations over }} \\ {\text { the given intervals in Exercises } 51-58 .}\end{array}
\end{equation}\begin{equation}
\begin{array}{l}{\text { Ellipse } x=4 \cos t, \quad y=2 \sin t, \quad \text { over }} \\ {\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}
\end{equation}

Luuk Verhoeven

Numerade Educator

Problem 52

If you have a parametric equation grapher, graph the equations over
the given intervals in Exercises $51-58 .$
Hyperbola branch $x=\sec t$ (enter as $1 / \cos (t) ), y=\tan t$
$($ enter as $\sin (t) / \cos (t)),$ over
$$
\begin{array}{l}{\text { a. }-1.5 \leq t \leq 1.5} \\ {\text { b. }-0.5 \leq t \leq 0.5} \\ {\text { c. }-0.1 \leq t \leq 0.1}\end{array}
$$

Luuk Verhoeven

Numerade Educator

Problem 53

If you have a parametric equation grapher, graph the equations over
the given intervals in Exercises $51-58 .$
$$
\quad x=2 t+3, \quad y=t^{2}-1, \quad-2 \leq t \leq 2
$$

Luuk Verhoeven

Numerade Educator

Problem 54

If you have a parametric equation grapher, graph the equations over
the given intervals in Exercises $51-58 .$
$$
\begin{array}{l}{\text { Cycloid } x=t-\sin t, \quad y=1-\cos t, \quad \text { over }} \\ {\text { a. } 0 \leq t \leq 2 \pi} \\ {\text { b. } 0 \leq t \leq 4 \pi} \\ {\text { c. } \pi \leq t \leq 3 \pi}\end{array}
$$

Luuk Verhoeven

Numerade Educator

Problem 55

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

Luuk Verhoeven

Numerade Educator

Problem 56

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.

Luuk Verhoeven

Numerade Educator

Problem 57

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
$$x=\cos t+5 \cos 3 t, \quad y=6 \cos t-5 \sin 3 t ; \quad 0 \leq t \leq 2 \pi$$

Luuk Verhoeven

Numerade Educator

Problem 58

\begin{equation}
\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 \\ \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}
\end{equation}

Luuk Verhoeven

Numerade Educator