Single Variable Calculus: Early Transcendentals

James Stewart, Daniel K. Clegg, Saleem Watson, Lothar Redlin

Chapter 10 Parametric Equations and Polar Coordinates - all with Video Answers

Educators

Section 1

Curves Defined by Parametric Equations

03:15

Problem 1

For the given parametric equations, find the points $(x, y)$ corresponding to the parameter values $t=-2,-1,0,1,2$.
$x=t^2+t, \quad y=3^{t+1}$

Willis James

Numerade Educator

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Problem 2

For the given parametric equations, find the points $(x, y)$ corresponding to the parameter values $t=-2,-1,0,1,2$.
$x=\ln \left(t^2+1\right), \quad y=t /(t+4)$

Donna Densmore

Numerade Educator

01:38

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.
$x=1-t^2, \quad y=2 t-t^2, \quad-1 \approx t \leqslant 2$

Wen Zheng

Numerade Educator

02:59

Problem 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.
$x=t^3+t, \quad y=t^2+2, \quad-2 \leqslant t \leqslant 2$

Wen Zheng

Numerade Educator

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Problem 5

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

Carson Merrill

Numerade Educator

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Problem 6

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.
$x=\cos ^2 t, \quad y=1+\cos t, \quad 0 \leqslant t \leqslant \pi$

Carson Merrill

Numerade Educator

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Problem 7

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

Carson Merrill

Numerade Educator

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Problem 8

Carson Merrill

Numerade Educator

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Problem 9

Carson Merrill

Numerade Educator

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Problem 10

Carson Merrill

Numerade Educator

01:22

Problem 11

Carson Merrill

Numerade Educator

05:04

Problem 12

Alexandra Ali

Numerade Educator

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Problem 13

(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.
$x=3 \cos t, \quad y=3 \sin t, \quad 0 \leqslant t \leqslant \pi$

Carson Merrill

Numerade Educator

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Problem 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.
$x=\sin 4 \theta, \quad y=\cos 4 \theta, \quad 0 \leqslant \theta \leqslant \pi / 2$

Carson Merrill

Numerade Educator

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Problem 15

(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.
$x=\cos \theta, \quad y=\sec ^2 \theta, \quad 0 \leqslant \theta<\pi / 2$

Carson Merrill

Numerade Educator

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Problem 16

Carson Merrill

Numerade Educator

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Problem 17

Carson Merrill

Numerade Educator

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Problem 18

Carson Merrill

Numerade Educator

01:46

Problem 19

Carson Merrill

Numerade Educator

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Problem 20

Carson Merrill

Numerade Educator

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Problem 21

Carson Merrill

Numerade Educator

01:09

Problem 22

Wen Zheng

Numerade Educator

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Problem 23

The position of an object in circular motion is modeled by the given parametric equations, where $t$ is measured in seconds. How long does it take to complete one revolution? Is the motion clockwise or counterclockwise?
$x=5 \cos t, \quad y=-5 \sin t$

Carson Merrill

Numerade Educator

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Problem 24

Carson Merrill

Numerade Educator

Problem 25

Describe the motion of a particle with position ( $x, y$ ) as $t$ varies in the given interval.
$x=5+2 \cos \pi t, \quad y=3+2 \sin \pi t, \quad 1 \leqslant t \leqslant 2$

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Problem 26

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

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02:56

Problem 27

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

Wen Zheng

Numerade Educator

Problem 28

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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01:08

Problem 29

Suppose a curve is given by the parametric equations $x=f(t), y=g(t)$, where the range of $f$ is $[1,4]$ and the range of $g$ is $[2,3]$. What can you say about the curve?

Chris Trentman

Numerade Educator

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Problem 30

Match each pair of graphs of equations $x=f(t), y=g(t)$ in (a)-(d) with one of the parametric curves $x=f(t), y=g(t)$ labeled I-IV. Give reasons for your choices.

Carson Merrill

Numerade Educator

Problem 31

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.
Graph can't copy

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Problem 32

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.
Graph can't copy

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Problem 33

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.
Graph can't copy

Check back soon!

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Problem 34

Match the parametric equations with the graphs labeled I-VI. Give reasons for your choices.
(a) $x=t^4-t+1, \quad y=t^2$
(b) $x=t^2-2 t, \quad y=\sqrt{t}$
(c) $x=t^3-2 t, \quad y=t^2-t$
(d) $x=\cos 5 t, \quad y=\sin 2 t$
(e) $x=t+\sin 4 t, \quad y=t^2+\cos 3 t$

Carson Merrill

Numerade Educator

00:22

Problem 35

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

Wen Zheng

Numerade Educator

01:39

Problem 36

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.

Wen Zheng

Numerade Educator

04:25

Problem 37

(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 $0 \approx t \leqslant 1$, 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 $(3,-1)$.

Jeffrey Russell

Numerade Educator

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Problem 38

Use a graphing calculator or computer and the result of Exercise 37 (a) to draw the triangle with vertices $A(1,1)$, $B(4,2)$, and $C(1,5)$.

Carson Merrill

Numerade Educator

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Problem 39

Find parametric equations for the position of a particle moving along a circle as described.
The particle travels clockwise around a circle centered at the origin with radius 5 and completes a revolution in $4 \pi$ seconds.

Carson Merrill

Numerade Educator

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Problem 40

Find parametric equations for the position of a particle moving along a circle as described.
The particle travels counterclockwise around a circle with center $(1,3)$ and radius 1 and completes a revolution in three seconds.

Carson Merrill

Numerade Educator

06:32

Problem 41

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 $(0,3)$

James Kiley

Numerade Educator

03:18

Problem 42

(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 2.1
(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?

Jeffrey Russell

Numerade Educator

01:15

Problem 43

Use a graphing calculator or computer to reproduce the picture.
Graph can't copy

Carson Merrill

Numerade Educator

01:15

Problem 44

Use a graphing calculator or computer to reproduce the picture.
Graph can't copy

Carson Merrill

Numerade Educator

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Problem 45

(a) Show that the points on all four of the given parametric curves satisfy the same Cartesian equation.
(i) $x=t^2, \quad y=t$
(ii) $x=t, \quad y=\sqrt{t}$
(iii) $x=\cos ^2 t, \quad y=\cos t$
(iv) $x=3^{2 t}, \quad y=3^t$
(b) Sketch the graph of each curve in part (a) and explain how the curves differ from one another.

Carson Merrill

Numerade Educator

01:09

Problem 46

Compare the curves represented by the parametric equations. How do they differ?

(a) $x=t, \quad y=t^{-2}$
(b) $x=\cos t, \quad y=\sec ^2 t$
(c) $x=e^t, \quad y=e^{-2 t}$

Carson Merrill

Numerade Educator

01:09

Problem 47

Compare the curves represented by the parametric equations. How do they differ?

(a) $x=t^3, \quad y=t^2$
(b) $x=t^6, \quad y=t^4$
(c) $x=e^{-3 t}, \quad y=e^{-2 t}$

Carson Merrill

Numerade Educator

01:58

Problem 48

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

Wen Zheng

Numerade Educator

02:46

Problem 49

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

$$
x=r \theta-d \sin \theta
$$

$$
y=r-d \cos \theta
$$

Sketch the trochoid for the cases $d<r$ and $d>r$.

Wen Zheng

Numerade Educator

03:04

Problem 50

In the figure, the circle of radius $a$ is stationary, and for every $\theta$, the point $P$ is the midpoint of the segment $Q R$. The curve traced out by $P$ for $0<\theta<\pi$ is called the longbow curve. Find parametric equations for this curve.

Harmender Singh Yadav

Numerade Educator

03:45

Problem 51

If $a$ and $h$ 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.

Linda Hand

Numerade Educator

03:25

Problem 52

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. The line segment $A B$ is tangent to the larger circle.

Chris Trentman

Numerade Educator

03:55

Problem 53

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.

Jeffrey Russell

Numerade Educator

01:25

Problem 54

(a) Find parametric equations for the set of all points $P$ as shown in the figure such that $|O P|=|A B|$. (This curve is called the cissoid of Diocles after the Greek scholar Diocles, who introduced the cissoid as a graphical
method for constructing the edge of a cube whose volume is twice that of a given cube.)
(b) Use the geometric description of the curve to draw a rough sketch of the curve by hand. Check your work by using the parametric equations to graph the curve.

Carson Merrill

Numerade Educator

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Problem 55

The position of a red particle at time $t$ is given by

$$
x=t+5 \quad y=t^2+4 t+6
$$

and the position of a blue particle is given by

$$
x=2 t+1 \quad y=2 t+6
$$

Their paths are shown in the graph.
(a) Verify that the paths of the particles intersect at the points $(1,6)$ and $(6,11)$. Is either of these points a collision point? If so, at what time do the particles collide?
(b) Suppose that the position of a green particle is given by

$$
x=2 t+4 \quad y=2 t+9
$$

Show that this particle moves along the same path as the blue particle. Do the red and green particles collide? If so, at what point and at what time?

Carson Merrill

Numerade Educator

01:06

Problem 56

The position of one particle at time $t$ is given by

$$
x=3 \sin t \quad y=2 \cos t \quad 0 \leqslant t \leqslant 2 \pi
$$

and the position of a second particle is given by

$$
x=-3+\cos t \quad y=1+\sin t \quad 0 \leqslant t \leqslant 2 \pi
$$

(a) Graph the paths of both particles. At how many points do the graphs intersect?
(b) Do the particles collide? If so, find the collision points.
(c) Describe what happens if the path of the second particle is given by

$$
x=3+\cos t \quad y=1+\sin t \quad 0 \leqslant t \leqslant 2 \pi
$$

Carson Merrill

Numerade Educator

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Problem 57

Find the point at which the parametric curve intersects itself and the corresponding values of $t$.
(a) $x=1-t^2, \quad y=t-t^3$
(b) $x=2 t-t^3, \quad y=t-t^2$

Carson Merrill

Numerade Educator

01:05

Problem 58

If a projectile is fired from the origin 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 graph 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.

Carson Merrill

Numerade Educator