Essential Calculus Early Transcendentals

$1-4=$ 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.
$$x=t^{2}+t, \quad y=t^{2}-t, \quad-2 \leqslant t \leqslant 2$$

$1-4=$ 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.
$$x=t^{2}, \quad y=t^{3}-4 t, \quad-3 \leq t \leqslant 3$$

$1-4=$ 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.
$$x=\cos ^{2} t, \quad y=1-\sin t, \quad 0 \leqslant t \leqslant \pi / 2$$

$1-4=$ 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.
$$x=e^{-t}+t, \quad y=e^{t}-t, \quad-2 \leqslant t \leqslant 2$$

$5-8$ "
(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.
$$x=3-4 t, \quad y=2-3 t$$

$5-8$ "
(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.
$$x=t-1, \quad y=t^{3}+1, \quad-2 \leqslant t \leqslant 2$$

$5-8$ "
(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.
$$x=\sqrt{t}, \quad y=1-t$$

$5-8=$
(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.
$$x=t^{2}, \quad y=t^{3}$$

$9-14=$
(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.
$$x=\sin \frac{1}{2} \theta, \quad y=\cos \frac{1}{2} \theta, \quad-\pi \leqslant \theta \leqslant \pi$$

$9-14=$
(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.
$$x=\frac{1}{2} \cos \theta, \quad y=2 \sin \theta, \quad 0 \leqslant \theta \leqslant \pi$$

$9-14=$
(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.
$x=\sin t, \quad y=\csc t, \quad 0 < t < \pi / 2$

$9-14=$
(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.
$$x=e^{t}-1, \quad y=e^{2 t}$$

$9-14=$
(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.
$$x=e^{2 t}, \quad y=t+1$$

$9-14=$
(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.
$$y=\sqrt{t+1}, \quad y=\sqrt{t-1}$$

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

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

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

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

$19-21=$ 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.

$19-21=$ 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.

$19-21=$ 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.

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) $x=\frac{\sin 2 t}{4+t^{2}}, \quad y=\frac{\cos 2 t}{4+t^{2}}$

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

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.

(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 $(3,-1)$ .

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
$C(1,5)$ .

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 ( $0,3 )$

(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?

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

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

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

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

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

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 $d>r$

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.

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.

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
$$x_{2}=3+\cos t \quad y_{2}=1+\sin t \quad 0 \leqslant t \leqslant 2 \pi$$

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.

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.

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?

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?

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?

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.)

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.