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Chapter 9

PARAMETRIC EQUATIONS AND POLAR COORDINATES

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Problem 1

$1-4=$ Sketch the curve by using the parametric equations to
plot points. Indicate with an arrow the direction in which the
curve is traced as $t$ increases.
$$x=t^{2}+t, \quad y=t^{2}-t, \quad-2 \leqslant t \leqslant 2$$

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Problem 2

$1-4=$ Sketch the curve by using the parametric equations to
plot points. Indicate with an arrow the direction in which the
curve is traced as $t$ increases.
$$x=t^{2}, \quad y=t^{3}-4 t, \quad-3 \leq t \leqslant 3$$

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Problem 3

$1-4=$ Sketch the curve by using the parametric equations to
plot points. Indicate with an arrow the direction in which the
curve is traced as $t$ increases.
$$x=\cos ^{2} t, \quad y=1-\sin t, \quad 0 \leqslant t \leqslant \pi / 2$$

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Problem 4

$1-4=$ Sketch the curve by using the parametric equations to
plot points. Indicate with an arrow the direction in which the
curve is traced as $t$ increases.
$$x=e^{-t}+t, \quad y=e^{t}-t, \quad-2 \leqslant t \leqslant 2$$

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Problem 5

$5-8$ "
(a) Sketch the curve by using the parametric equations to plot
points. Indicate with an arrow the direction in which the
curve is traced as $t$ increases.
(b) Eliminate the parameter to find a Cartesian equation of the
curve.
$$x=3-4 t, \quad y=2-3 t$$

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Problem 6

$5-8$ "
(a) Sketch the curve by using the parametric equations to plot
points. Indicate with an arrow the direction in which the
curve is traced as $t$ increases.
(b) Eliminate the parameter to find a Cartesian equation of the
curve.
$$x=t-1, \quad y=t^{3}+1, \quad-2 \leqslant t \leqslant 2$$

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Problem 7

$5-8$ "
(a) Sketch the curve by using the parametric equations to plot
points. Indicate with an arrow the direction in which the
curve is traced as $t$ increases.
(b) Eliminate the parameter to find a Cartesian equation of the
curve.
$$x=\sqrt{t}, \quad y=1-t$$

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Problem 8

$5-8=$
(a) Sketch the curve by using the parametric equations to plot
points. Indicate with an arrow the direction in which the
curve is traced as $t$ increases.
(b) Eliminate the parameter to find a Cartesian equation of the
curve.
$$x=t^{2}, \quad y=t^{3}$$

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Problem 9

$9-14=$
(a) Eliminate the parameter to find a Cartesian equation of the
curve.
(b) Sketch the curve and indicate with an arrow the direction in
which the curve is traced as the parameter increases.
$$x=\sin \frac{1}{2} \theta, \quad y=\cos \frac{1}{2} \theta, \quad-\pi \leqslant \theta \leqslant \pi$$

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Problem 10

$9-14=$
(a) Eliminate the parameter to find a Cartesian equation of the
curve.
(b) Sketch the curve and indicate with an arrow the direction in
which the curve is traced as the parameter increases.
$$x=\frac{1}{2} \cos \theta, \quad y=2 \sin \theta, \quad 0 \leqslant \theta \leqslant \pi$$

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Problem 11

$9-14=$
(a) Eliminate the parameter to find a Cartesian equation of the
curve.
(b) Sketch the curve and indicate with an arrow the direction in
which the curve is traced as the parameter increases.
$x=\sin t, \quad y=\csc t, \quad 0 < t < \pi / 2$

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Problem 12

$9-14=$
(a) Eliminate the parameter to find a Cartesian equation of the
curve.
(b) Sketch the curve and indicate with an arrow the direction in
which the curve is traced as the parameter increases.
$$x=e^{t}-1, \quad y=e^{2 t}$$

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Problem 13

$9-14=$
(a) Eliminate the parameter to find a Cartesian equation of the
curve.
(b) Sketch the curve and indicate with an arrow the direction in
which the curve is traced as the parameter increases.
$$x=e^{2 t}, \quad y=t+1$$

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Problem 14

$9-14=$
(a) Eliminate the parameter to find a Cartesian equation of the
curve.
(b) Sketch the curve and indicate with an arrow the direction in
which the curve is traced as the parameter increases.
$$y=\sqrt{t+1}, \quad y=\sqrt{t-1}$$

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Problem 15

$15-18=$ Describe the motion of a particle with position $(x, y)$
as $t$ varies in the given interval.
$$x=3+2 \cos t, \quad y=1+2 \sin t, \quad \pi / 2 \leqslant t \leqslant 3 \pi / 2$$

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Problem 16

$15-18=$ Describe the motion of a particle with position $(x, y)$
as $t$ varies in the given interval.
$$x=2 \sin t, \quad y=4+\cos t, \quad 0 \leqslant t \leqslant 3 \pi / 2$$

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Problem 17

$15-18=$ Describe the motion of a particle with position $(x, y)$
as $t$ varies in the given interval.
$$x=5 \sin t, \quad y=2 \cos t, \quad-\pi \leqslant t \leqslant 5 \pi$$

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Problem 18

$15-18=$ Describe the motion of a particle with position $(x, y)$
as $t$ varies in the given interval.
$x=\sin t, \quad y=\cos ^{2} t, \quad-2 \pi \leqslant t \leqslant 2 \pi$

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Problem 19

$19-21=$ Use the graphs of $x=f(t)$ and $y=g(t)$ to sketch the
parametric curve $x=f(t), y=g(t) .$ Indicate with arrows the
direction in which the curve is traced as $t$ increases.

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Problem 20

$19-21=$ Use the graphs of $x=f(t)$ and $y=g(t)$ to sketch the
parametric curve $x=f(t), y=g(t) .$ Indicate with arrows the
direction in which the curve is traced as $t$ increases.

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Problem 21

$19-21=$ Use the graphs of $x=f(t)$ and $y=g(t)$ to sketch the
parametric curve $x=f(t), y=g(t) .$ Indicate with arrows the
direction in which the curve is traced as $t$ increases.

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Problem 22

Match the parametric equations with the graphs labeled
I-VI. Give reasons for your choices. (Do not use a graphing device.)
(a) $x=t^{4}-t+1, \quad y=t^{2}$
(b) $x=t^{2}-2 t, \quad y=\sqrt{t}$
(b) $x=t^{2}-2 t, \quad y=\sin (t+\sin 2 t)$
(d) $x=\cos 5 t, \quad y=\sin 2 t$
(d) $x=\cos 5 t, \quad y=t^{2}+\cos 3 t$
(e) $x=\frac{\sin 2 t}{4+t^{2}}, \quad y=\frac{\cos 2 t}{4+t^{2}}$

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Problem 23

Graph the curve $x=y-2 \sin \pi y$

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Problem 24

Graph the curves $y=x^{3}-4 x$ and $x=y^{3}-4 y$ and find
their points of intersection correct to one decimal place.

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Problem 25

(a) Show that the parametric equations
$$x=x_{1}+\left(x_{2}-x_{1}\right) t \quad y=y_{1}+\left(y_{2}-y_{1}\right) t$$
where $0 \leqslant t \leqslant 1,$ describe the line segment that joins
the points $P_{1}\left(x_{1}, y_{1}\right)$ and $P_{2}\left(x_{2}, y_{2}\right) .$
(b) Find parametric equations to represent the line segment
from $(-2,7)$ to $(3,-1)$ .

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Problem 26

Use a graphing device and the result of Exercise 25$(\mathrm{a})$ to
draw the triangle with vertices $A(1,1), B(4,2),$ and
$C(1,5)$ .

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Problem 27

Find parametric equations for the path of a particle that
moves along the circle $x^{2}+(y-1)^{2}=4$ in the manner
described.
(a) Once around clockwise, starting at $(2,1)$
(b) Three times around counterclockwise, starting at ( $(2,1)$
(c) Halfway around counterclockwise, starting at ( $0,3 )$

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Problem 28

(a) Find parametric equations for the ellipse
$x^{2} / a^{2}+y^{2} / b^{2}=1 .$ [Hint: Modify the equations of
the circle in Example $2 . ]$
(b) Use these parametric equations to graph the ellipse
when $a=3$ and $b=1,2,4,$ and $8 .$
(c) How does the shape of the ellipse change as $b$ varies?

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Problem 29

$29-30=$ Use a graphing calculator or computer to reproduce
the picture.

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Problem 30

$29-30=$ Use a graphing calculator or computer to reproduce
the picture.

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Problem 31

$31-32=$ Compare the curves represented by the parametric
equations. How do they differ?
$$\text (a) x=t^{3}, \quad y=t^{2} \quad (b) x=t^{6}, \quad y=t^{4}\text (c) x=e^{-3 t}, \quad y=e^{-2 t}$$

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Problem 32

$31-32=$ Compare the curves represented by the parametric
equations. How do they differ?
$$
\begin{array}{ll}{\text { (a) } x=t,} & {y=t^{-2} \quad \text { (b) } x=\cos t, \quad y=\sec ^{2} t} \\ {\text { (c) } x=e^{t},} & {y=e^{-2 t}}\end{array}
$$

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Problem 33

Derive Equations 1 for the case $\pi / 2<\theta<\pi$

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Problem 34

Let $P$ be a point at a distance $d$ from the center of a circle
of radius $r .$ The curve traced out by $P$ as the circle rolls
along a straight line is called a trochoid. (Think of the
motion of a point on a spoke of a bicycle wheel.) The
cycloid is the special case of a trochoid with $d=r .$ Using
the same parameter $\theta$ as for the cycloid and, assuming the
line is the $x$ -axis and $\theta=0$ when $P$ is at one of its lowest
points, show that parametric equations of the trochoid are
$$x=r \theta-d \sin \theta \quad y=r-d \cos \theta$$
Sketch the trochoid for the cases $d<r$ and $d>r$

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Problem 35

If $a$ and $b$ are fixed numbers, find parametric equations for
the curve that consists of all possible positions of the point
$P$ in the figure, using the angle $\theta$ as the parameter. Then
eliminate the parameter and identify the curve.

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Problem 36

A curve, called a witch of Maria Agnesi, consists of all
possible positions of the point $P$ in the figure. Show that
parametric equations for this curve can be written as
$$x=2 a \cot \theta \quad y=2 a \sin ^{2} \theta$$
Sketch the curve.

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Problem 37

Suppose that the position of one particle at time $t$ is given
by
$$\quad x_{1}=3 \sin t \quad y_{1}=2 \cos t \quad 0 \leqslant t \leqslant 2 \pi$$
and the position of a second particle is given by
$$x_{2}=-3+\cos t \quad y_{2}=1+\sin t \quad 0 \leqslant t \leqslant 2 \pi$$
(a) Graph the paths of both particles. How many points
of intersection are there?
(b) Are any of these points of intersection collision points?
In other words, are the particles ever at the same place
at the same time? If so, find the collision points.
(c) Describe what happens if the second particle is given by
$$x_{2}=3+\cos t \quad y_{2}=1+\sin t \quad 0 \leqslant t \leqslant 2 \pi$$

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Problem 38

If a projectile is fired with an initial velocity of $v_{0}$ meters
per second at an angle $\alpha$ above the horizontal and air
resistance is assumed to be negligible, then its position
after $t$ seconds is given by the parametric equations
$$x=\left(v_{0} \cos \alpha\right) t \quad y=\left(v_{0} \sin \alpha\right) t-\frac{1}{2} g t^{2}$$
where $g$ is the acceleration due to gravity $\left(9.8 \mathrm{m} / \mathrm{s}^{2}\right)$
(a) If a gun is fired with $\alpha=30^{\circ}$ and $v_{0}=500 \mathrm{m} / \mathrm{s},$ when
will the bullet hit the ground? How far from the gun
will it hit the ground? What is the maximum height
reached by the bullet?
(b) Use a graphing device to check your answers to part (a). Then graph the path of the projectile for several
other values of the angle $\alpha$ to see where it hits the
ground. Summarize your findings.
(c) Show that the path is parabolic by eliminating the
parameter.

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Problem 39

Investigate the family of curves defined by the parametric
equations $x=t^{2}, y=t^{3}-$ ct. How does the shape change
as $c$ increases? Illustrate by graphing several members of
the family.

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Problem 40

The swallowtail catastrophe curves are defined by the
parametric equations $x=2 c t-4 t^{3}, y=-c t^{2}+3 t^{4}$ .
Graph several of these curves. What features do the curves
have in common? How do they change when $c$ increases?

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Problem 41

Graph several members of the family of curves with parametric equations $x=t+a \cos t, y=t+a \sin t,$ where
$a > 0 .$ How does the shape change as $a$ increases? For
what values of $a$ does the curve have a loop?

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Problem 42

Graph several members of the family of curves
$x=\sin t+\sin n t, y=\cos t+\cos n t,$ where $n$ is a positive integer. What features do the curves have in common?
What happens as $n$ increases?

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Problem 43

The curves with equations $x=a \sin n t, y=b \cos t$ are
called Lissajous figures. Investigate how these curves
vary when $a, b,$ and $n$ vary. (Take $n$ to be a positive
integer.)

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Problem 44

Investigate the family of curves defined by the parametric
equations $x=\cos t, y=\sin t-\sin c t,$ where $c>0 .$ Start
by letting $c$ be a positive integer and see what happens to
the shape as $c$ increases. Then explore some of the possibilities that occur when $c$ is a fraction.

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