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

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

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 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 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 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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