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Chapter 10

Parametric Equations and Polar Coordinates

Educators

WZ

Problem 1

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 = 2t - t^2 $, $ \quad -1 \leqslant t \leqslant 2 $

WZ
Wen Z.
Numerade Educator

Problem 2

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 $

WZ
Wen Z.
Numerade Educator

Problem 3

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 + \sin t $, $ \quad y = \cos t $, $ \quad -\pi \leqslant t \leqslant \pi $

WZ
Wen Z.
Numerade Educator

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 = e^{-a} + t $, $ \quad y = e^a - t $, $ \quad -2 \leqslant t \leqslant 2 $

WZ
Wen Z.
Numerade Educator

Problem 5

(a) Sketch the curve by using the parametric equations to plot points. Indicat 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 = 2t - 1 $, $ \quad y = \dfrac{1}{2}t + 1 $

WZ
Wen Z.
Numerade Educator

Problem 6

(a) Sketch the curve by using the parametric equations to plot points. Indicat 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 = 3t + 2 $, $ \quad y = 2t + 3 $

WZ
Wen Z.
Numerade Educator

Problem 7

(a) Sketch the curve by using the parametric equations to plot points. Indicat 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 - 3 $, $ \quad y = t + 2 $, $ \quad -3 \leqslant t \leqslant 3 $

WZ
Wen Z.
Numerade Educator

Problem 8

(a) Sketch the curve by using the parametric equations to plot points. Indicat 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 = \sin t $, $ \quad y = 1 - \cos t $, $ \quad 0 \leqslant t \leqslant 2\pi $

WZ
Wen Z.
Numerade Educator

Problem 9

(a) Sketch the curve by using the parametric equations to plot points. Indicat 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 $

WZ
Wen Z.
Numerade Educator

Problem 10

(a) Sketch the curve by using the parametric equations to plot points. Indicat 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 $

WZ
Wen Z.
Numerade Educator

Problem 11

(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 parameter increases.

$ x = \sin\dfrac{1}{2}\theta $, $ \quad y = \cos\dfrac{1}{2}\theta $, $ \quad -\pi \leqslant \theta \leqslant \pi $

WZ
Wen Z.
Numerade Educator

Problem 12

(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 parameter increases.

$ x = \dfrac{1}{2}\cos\theta $, $ \quad y = 2\sin\theta $, $ \quad 0 \leqslant \theta \leqslant \pi $

WZ
Wen Z.
Numerade Educator

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 parameter increases.

$ x = \sin t $, $ \quad y = \csc t $, $ \quad 0 \leqslant t \leqslant \pi/2 $

WZ
Wen Z.
Numerade Educator

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 parameter increases.

$ x = e^t $, $ \quad y = e^{-2t} $

WZ
Wen Z.
Numerade Educator

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 parameter increases.

$ x = t^2 $, $ \quad y = \ln t $

WZ
Wen Z.
Numerade Educator

Problem 16

(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 parameter increases.

$ x = \sqrt{t + 1} $, $ \quad y = \sqrt{t - 1} $

WZ
Wen Z.
Numerade Educator

Problem 17

(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 parameter increases.

$ x = \sinh t $, $ \quad y = \cosh t $

WZ
Wen Z.
Numerade Educator

Problem 18

(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 parameter increases.

$ x = \tan^2 \theta $, $ \quad y = \sec \theta $, $ \quad -\pi/2 < \theta < \pi/2 $

WZ
Wen Z.
Numerade Educator

Problem 19

Describe the motion of a particle with position $ (x, y) $ as $ t $ varies in a given interval.

$ x = 5 + 2\cos \pi t $, $ \; y = 3 + 2 \sin \pi t $, $ \; 1 \leqslant t \leqslant 2 $

WZ
Wen Z.
Numerade Educator

Problem 20

Describe the motion of a particle with position $ (x, y) $ as $ t $ varies in a given interval.

$ x = 2 + \sin t $, $ \; y = 1 + 3\cos t $, $ \; \pi/2 \leqslant t \leqslant 2\pi $

WZ
Wen Z.
Numerade Educator

Problem 21

Describe the motion of a particle with position $ (x, y) $ as $ t $ varies in a given interval.

$ x = 5\sin t $, $ \; y = 2\cos t $, $ \; -\pi \leqslant t \leqslant 5\pi $

WZ
Wen Z.
Numerade Educator

Problem 22

Describe the motion of a particle with position $ (x, y) $ as $ t $ varies in a given interval.

$ x = \sin t $, $ \; y = \cos^2 t $, $ \; -2\pi \leqslant t \leqslant 2\pi $

WZ
Wen Z.
Numerade Educator

Problem 23

Suppose a curve is a 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?

WZ
Wen Z.
Numerade Educator

Problem 24

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

WZ
Wen Z.
Numerade Educator

Problem 25

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 cuve is traced as $ t $ increases.

WZ
Wen Z.
Numerade Educator

Problem 26

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 cuve is traced as $ t $ increases.

WZ
Wen Z.
Numerade Educator

Problem 27

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 cuve is traced as $ t $ increases.

WZ
Wen Z.
Numerade Educator

Problem 28

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 $, $ \; y = t^2 $
(b) $ x = t^2 - 2t $, $ \; y = \sqrt{t} $
(c) $ x = \sin 2t $, $ \; y = \sin(t + \sin 2t) $
(d) $ x = \cos 5t $, $ \; y = \sin 2t $
(e) $ x = t + \sin 4t $, $ y = t^2 + \cos 3t $
(f) $ x = \dfrac{\sin 2t}{4 + t^2} $, $ \; y = \dfrac{\cos 2t}{4 + t^2} $

WZ
Wen Z.
Numerade Educator

Problem 29

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

WZ
Wen Z.
Numerade Educator

Problem 30

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

WZ
Wen Z.
Numerade Educator

Problem 31

(a) Show that the parametric equations
$ x = x_1 + (x_2 - x_1)t $ $ \quad y = y_1 + (y_2 - y_1)t $
where $ 0 \leqslant t \leqslant 1 $, describe the line segment that joins the points $ P_1(x_1, y_1) $ and $ P_2(x_2, y_2) $.
(b) Find parametric equations to represent the line segment from $ (-2, 7) $ to $ (3, -1) $.

WZ
Wen Z.
Numerade Educator

Problem 32

Use a graphing device and the result of Exercise 31(a) to draw the triangle with vertices $ A(1, 1) $, $ B(4, 2) $, and $ C(1, 5) $.

WZ
Wen Z.
Numerade Educator

Problem 33

Find parametric equations for the parth 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) $.

WZ
Wen Z.
Numerade Educator

Problem 34

(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?

WZ
Wen Z.
Numerade Educator

Problem 35

Use a graphic calculator or computer to reproduce the picture.

WZ
Wen Z.
Numerade Educator

Problem 36

Use a graphic calculator or computer to reproduce the picture.

WZ
Wen Z.
Numerade Educator

Problem 37

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

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

WZ
Wen Z.
Numerade Educator

Problem 38

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

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

WZ
Wen Z.
Numerade Educator

Problem 39

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

WZ
Wen Z.
Numerade Educator

Problem 40

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 $.

WZ
Wen Z.
Numerade Educator

Problem 41

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.

WZ
Wen Z.
Numerade Educator

Problem 42

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 $ AB $ is tangent to the larger circle.

WZ
Wen Z.
Numerade Educator

Problem 43

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

WZ
Wen Z.
Numerade Educator

Problem 44

(a) Find parametric equations for the set of all points $ P $ as shown in the figure such taht $ | OP | = | AB | $. (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.

WZ
Wen Z.
Numerade Educator

Problem 45

Suppose that the position of one particle at time $ t $ is given by
$$ x_1 = 3 \sin t \quad y_1 = 1 + \sin t \quad 0 \leqslant t \leqslant 2 \pi $$
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 path of the second particle given by
$$ x_2 = 3 + \cos t \quad y_2 = 1 + \sin t \quad 0 \leqslant t \leqslant 2\pi $$

WZ
Wen Z.
Numerade Educator

Problem 46

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, the its position after $ t $ seconds is given by the parametric equations
$$ x = (v_0 \cos \alpha) t \quad y = (v_0 \sin \alpha)t - \dfrac{1}{2}gt^2 $$
where $ g $ is the acceleration due to gravity $ (9.8 m/s^2) $.
(a) If a gun is fired with $ \alpha - 30^{\circ} $ and $ v_0 = 500\;m/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.

WZ
Wen Z.
Numerade Educator

Problem 47

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.

WZ
Wen Z.
Numerade Educator

Problem 48

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

WZ
Wen Z.
Numerade Educator

Problem 49

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

WZ
Wen Z.
Numerade Educator

Problem 50

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

WZ
Wen Z.
Numerade Educator

Problem 51

The curves with equations $ x = \alpha \sin nt $, $ 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.)

WZ
Wen Z.
Numerade Educator

Problem 52

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

WZ
Wen Z.
Numerade Educator