SOLUTIONWeuse a graphing device to produce the graphs for the cases a = -2, -1, -0.5, -0.2,0,0.5, 1, and 2 shown inFigure 17.Notice that allof these curves (except the case a =0) have two branches, and both branches approach the vertical asymptote x =a as x approaches a from the left or right.
When a
<
-1,both branches are smooth; but when areaches - I,the right branch acquires asharp point, called acusp. For abetween -1 and 0the cusp turns into aloop, which becomes larger asa approaches O.When a =0, both branches come together and form a circle (see Example 2). For abetween 0and I, the left branch has a loop, which shrinks tobecome acusp when a=
1.For a>
1,the branches become smooth again, and asaincreases further, they become less curved. Notice that the curves with aposi- tive are reflections about the y-axis of the corresponding curves with anegative.These curves are called conchoids of Nicomedes after the ancient Greek scholar Nicomedes. He called them conchoids because the shape of their outer branches
resembles that of aconch shell ormussel shell. 0
FIGURE 17 Members ofthe family x
=
a+
cosf,Y=
atant+
sinf,allgraphed in the viewing rectangle [-4,4] by[-4,4]
-8
EXERCISES1-4 Sketch the curve byusing the parametric equations toplot points. Indicate with anarrow the direction inwhich the curve is traced as tincreases.
5-10
(a) Sketch the curve by using the parametric equations to plot points. Indicate with anarrow the direction inwhich the curve is traced ast increases.
(b) Eliminate the parameter to find aCartesian equation of the curve.
5. x = 3t - 5, Y= 2t
+
16. x = 1
+
t, Y= 5- 2t, -2 ~ t~ 3 7.x = t2 - 2, y =5 - 2t, -3 ~ t ~ 48. x
=
I+
3t, Y=
2- t200
X =jt,
y = I - t 10. x=
t2, Y=
tJ11-18
(a) Eliminate the parameter to find a Cartesian equation ofthe curve.
(b) Sketch the curve and indicate with anarrow the direction in which the curve istraced asthe parameter increases.
II. x = sin e, y = cose, 0~ e ~ 7T
12. x=4cose, y=5sine, -7T/2~e~7T/2
[QJ
x = sin t, y = csc t, 0<
t<
7T/2 14.x = sec e, y = tan e, -7T/2<
e<
7T/2 15.x=e', y=e-'16. x =In t, y =
jt,
t~ I 17.x = sinh f, y = cosh f19-22 Describe the motion of aparticle with position (x, y) as tvaries inthe given interval.
19. x =cos 7Tt, Y=sin 7Tt, I~ t ~ 2
23. 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 ofgis [2,3]. What can you say about the curve?
24. Match the graphs ofthe parametric equations x =f(t) and y =g(t) in (a)-(d) with theparametric curves labeled I-IV.
Give reasons for your choices.
28. Match the parametric equations with the graphs labeled I-VI.
x Give reasons for your choices. (Do not usea graphing device.)
(a) x= t4 - t
+
I, y =t2 (b)x = t2 - 2t, y =Jt
II (c) x = sin2t, y = sin(t
+
sin2t)(d) x= cos 5t, y = sin2t
(e) x= t
+
sin 4t, y = t2+
COS 3t sin2t cos2t (f) x = --"4+
t- y = 4+
t2II III
y y y
III
y x
I
x
IV V VI
25-27 Use the graphs ofx =f(t) and y =g(t) to sketch the parametric curve x =f(t), y = g(t). Indicate with arrows the direction inwhich the curve is traced astincreases.
ffi
29. Graph the curve x =y - 3y3+
y5.ffi
30. Graph the curves y=
x5 and x=
y(y - 1)2and find their points of intersection correct to one decimal place.where
°~
t~ 1, describe the line segment that joins the points P1(XI, Yl) and P2(X2, Y2).(b) Find parametric equations to represent the line segment from (-2,7) to (3, -1).
ffi
32. Use a graphing device and the result of Exercise 31(a) to draw the triangle with vertices A(l, 1), B(4, 2), and C(l, 5).ITIJ
Find parametric equations for the path of a particle that moves along the circle x2+
(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)
ffi [B:]
(a) Find parametric equations for the ellipsex2/a2
+
y2/b2 = 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?
ffi
35-36 Use a graphing calculator or computer to reproduce the picture.37-38 Compare the curves represented by the parametric equa- tions. How do they differ?
37. (a) x =t3, Y=t2 (b) x =t6, Y=t4
(c) X =e-3', y =e-2'
38. (a) x =t, Y =t-2 (c) x=e', y =e-2'
Let Pbe a point at a distance dfrom the center of a circle of radius r.The curve traced out by P as the circle rolls along a straight line iscalled a trochoid. (Think of the motion of a point on a spoke of abicycle wheel.) The cycloid is the spe- cial case of a trochoid with d=r. Using the same parameter (J as for the cycloid and, assuming the line is the x-axis and (J=
°
when Pis atone of its lowest points, show that parametric equations of the trochoid are11[]
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 (J as the parameter. Then elimi- nate the parameter and identify the curve.42. If a and b are fixed numbers, find parametric equations for the curve that consists ofall possible positions of the point P in the figure, using the angle (J as the parameter. The line seg- ment AS is tangent to the larger circle.
A curve, called a witch of Maria Agnesi, consists ofall pos- sible positions of the point P in the figure. Show that para- metric equations for this curve can be written as
x = 2a cot (J Sketch the curve.
(a) Find parametric equations for the set of all points P as shown in the figure such that lOP 1= 1ABI. (This curve is called the cissoid of Diodes 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 ofthe curve to draw a rough sketch of the curve by hand. Check your work by using the parametric equations to graph the curve.
(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 theparticles ever at the same place at the same time? Ifso,find the collision points.
(c) Describe what happens if the path of the second particle is given by
If a projectile isfiredwith an initial velocity ofVo meters per second at an angle 0'above the horizontal and air resistance isassumed to be negligible, then itsposition after t seconds is
l/d~ORATORY
PROJECT(a,0)
A x
ilI3
LookatModule 11.1 Bto see how hypocycloids and epicycloids are formed by the motion of rolling circles.where 9isthe acceleration due to gravity (9.8 m/s2).
(a) If a gun is fired with 0'= 30°and Vo = 500 mis, when will the bullet hit the ground? How far from the gun will it hit the ground? What isthe maximum height reached by the bullet?
~ (b) Use agraphing device tocheck your answers to part (a).
Then graph the path ofthe projectile for several other values of the angle 0'to see where it hits the ground.
Summarize your findings.
(c) Show that the path isparabolic by eliminating the parameter.
~ l1LJ
Investigate the family of curves defined by theparametric equations x =t2, Y=t3 - ct. Howdoes the shape change as cincreases? Illustrate by graphing several members of the family.~ 48. The swallowtail catastrophe curves are defined bythepara- metric equations x
=
2ct - 4t3, Y=
-et2+
3t4• Graph several of these curves. What features do the curves have in common? How do they change when c increases?~ ~ The curves with equations x =a sinnt, y =b cas tare called Lissajous figures. Investigate how these curves vary when a, b,and n vary. (Take nto be apositive integer.)
Investigate the family of curves defined bythe parametric equations x
=
cas t,y=
sin t - sinct,where e>
O. Start by letting cbe a positive integer and see what happens tothe shape asc increases. Then explore some ofthe possibilities that occur when eis afraction.~ RUNNING CIRCLES AROUND CIRCLES
In this project we investigate families of curves, called hypocycloids and epicycloids, that are generated by the motion of a point on acircle that rolls inside oroutside another circle.
I. A hypocycloid is acurve traced out by a fixed point Pona circle Cof radius bas C rolls on the inside ofacircle with center 0and radius a.Show that if theinitial position ofPis(a,0) and the parameter
e
is chosen as in the figure, then parametric equations of the hypocycloid are(a - b )
x =(a - b)cas
e +
bcas -b-e y
=(a- b)sine -
bsin(a-b-- be
)2. Use a graphing device (or the interactive graphic in TEC Module 11.IB) todraw the graphs ofhypocycloids with aa positive integer and b = 1.How does the value ofaaffect the graph?
Show that if we take a=4,then the parametric equations ofthe hypocycloid reduce to
about the x-axis. Therefore, from Formula 7, we get S=So" 27fTsin t
J( -r
sint)2+ (r
cos t)2dt=27T
fo'T r
sin tJr
2(sin
2t+ cos
2t) dt =27TSo" r
sin t •r
dto
EXERCISES1-2 Find dy/dx.
I. x =t - t3, Y =2 - 5r
3-6 Find anequation ofthe tangent to the curve at the point cor- responding tothe given value of the parameter.
3. x = r2
+
t, Y= t2 - t; t= 0 4. x = t -r-
1, y = I+
t2; t=W
x = eJi, y =t - Int2; t = I 6. x =tsin t, y = t cos t; t= 7T7-8 Find an equation ofthe tangent to the curve at the given point by two methods: (a) without eliminating theparameter and (b) by first eliminating the parameter.
7. x = I
+
Int, y = t2+
2; (1,3) 8. x = tane, y = sec e;(I,-li)
~ 9-10 Find anequation of the tangent(s) tothe curve at the given point. Then graph the curve and the tangent(s).
9. x =6 sin t, y =t2
+
t; (0,0)10.x=cost+cos2r, y=sint+sin2t; (-1,1)
11-16 Find dy/dx and d2y/dx2• For which values of t isthe curve concave upward?
12. x=t3-12t, y=t2-
14. x = t
+
Int, y =t - Int 15. x =2 sint, y =3 cost, 0<
t<
27T16.x =cos2r, y =cosr, 0<r<7T
17-20 Find the points on the curve where the tangent is horizon- tal orvertical. If you have a graphing device, graph the curve to check your work.
17. x = 10 - t2, Y=r3 - 12t
18. x =2r3
+
3t2 - 12t, Y=2t3+
3t2 +19. x =2 cose, y =sin2e 20. x =cos 3e, y =2sine
~ 21. Use a graph toestimate the coordinates of the rightmost point on the curve x=t - t6, Y=e'.Then use calculus tofind the exact coordinates.
~ 22. Use a graph to estimate the coordinates of the lowest point and the leftmost point on the curve x =t4 - 2t, y =r
+
t4•Then findthe exact coordinates.
~ 23-24 Graph the curve in a viewing rectangle that displays all the important aspects of the curve.
InJ
x = r4 - 2t3 - 2t2, Y=t3 - t 24. x =t4+
4r3 - 8t2, y =2t2 - t~ Show that the curve x =cos t,Y=sin t cost has two tangents at (0,0) and find their equations. Sketch the curve.
~ 26. Graph the curve x =cos t
+
2 cos 2t, y =sin t+
2sin2r to discover where itcrosses itself. Then find equations of both tangents at that point.27. (a) Find the slope of the tangent line tothe trochoid x =
re -
dsine, y
=r -
dcose
in terms ofe.
(SeeExercise 40 in Section 11.1.)
(b) Show that ifd
<
r,then thetrochoid does not have a vertical tangent.28. (a) Find the slope of the tangent tothe astroid x =acos3
e,
y =asin3
e
interms of e.(Astroids are explored in the Laboratory Project on page 665.)(b) Atwhat points isthetangent horizontal orvertical?
(c) At what points does the tangent have slope I or - I?
29. At what points on the curve x =2r3, y =I
+
4r - t2 does the tangent line have slope I?30. Find equations of thetangents to the curve x = 3r2
+
I, Y=2t3+
1 that pass through the point (4,3).1m
Use the parametric equations ofanellipse, x = a cose,
y
=bsine,
0~e ~
27T, tofind the area that itencloses.32. Find the area enclosed by the curve x
=
t2 - 2t, Y= Jt
andthey-axis.
33. Find the area enclosed by thex-axis and the curve x = 1
+
e', y = t - t2•34. Find the area of the region enclosed by the astroid
x = a cos3(), y =a sin3(). (Astroids areexplored in the Labo- ratory Project on page 665.)
y
35. Find the area under one arch of the trochoid of Exercise 40 in Section 11.1 for the case d
<
r.36. Let m be the region enclosed by the loop of the curve in Example 1.
(a) Find the area ofm.
(b) Ifm isrotated about the x-axis, find the volume of the resulting solid.
(c) Find the centroid ofm.
37-40 Set up an integral that represents the length ofthe curve.
Then use your calculator to find the length correct to four decimal places.
37. x = t - t\ y = ~t3/\ I",; t",;2 38. x = 1
+
e', y =t2, -3"'; t ",;3 39. x = t+
cos t, Y= t- sint, 0",; t ",;27T 40. x = In t, y =Jt+l,
1",; t",;541-44 Find the exact length of the curve.
IiQ
x = 1+
3t2, Y =4+
2t3, 0",; t ",;42. x = e'
+
e-', y = 5 - 2t, 0",; t",;3 43. x = --t y =In(1+
t), 0",; t",;2I
+
t'~ 45-47 Graph the curve and find itslength.
~ x =e'cost, y =e' sin t, 0",; t ",;7T
46. x =cost
+
In(tan ~t), y = sint, 7T/4"'; t",; 37T/4 47. x =e' - t, Y=4e,/2, -8"'; t",;348. Find the length of the loop of the curve x =3t - t3, Y = 3t2•
49. Use Simpson's Rule with n =6toestimate the length ofthe curve x =t - e', y =t
+
e', -6"'; t ",;6.50. InExercise 43 in Section I 1.1you were asked toderive the parametric equations x =2a cot (), y =2a sin2() for the curve called the witch of Maria Agnesi. Use Simpson's Rule with n =4 to estimate the length of the arc ofthis curve given by 7T/4 ",; ()",;7T/2.
51-52 Find the distance traveled bya particle with position (x, y) astvaries in the given time interval. Compare with the length of the curve.
51. x
=
sin2t, y=
cos2t, 0",; t",; 37T 52. x=
cos2t, Y=
cos t, 0",; t ",;47T53. Show that the total length ofthe ellipse x =a sin (), y =b cos(),a
>
b>
0,isL=4aSo"/2 JI - e2 sin2() d()
where eisthe eccentricity ofthe ellipse (e =c /a,where c=
Ja
2 - b2 ).54. Find the total length of the astroid x =acos3(), y =a sin3(),
where a> O.
[ill]
55. (a) Graph the epitrochoid with equationsx = II cos t - 4 cos(llt/2) Y= II sin t - 4sin(llt/2)
What parameter interval gives the complete curve?
(b) Use your CAS tofind the approximate length ofthis curve.
[ill]
56. Acurve called Cornu's spiral isdefined by the parametric equationsx = C(t) =
f~
COS(7Tu2/2) duy = S(t) =
t
sin(7Tu2/2) duwhere CandSare the Fresnel functions that were introduced inChapter 5.
(a) Graph this curve. What happens ast~ 00and as t ~ -oo?
(b) Find the length ofCornu's spiral from the origin tothe point with parameter value t.
57-58 Set upan integral that represents the area ofthe surface obtained by rotating the given curve about the x-axis. Then use your calculator tofind the surface area correct tofour decimal places.
57. x = 1
+
tel, y = (t2+
I)e', 0",; t",;I 58. x =sin2t, y =sin 3t, 0",; t ",; 7T/359-61 Find the exact area of the surface obtained by rotating the given curve about the x-axis.
59. x = t3, Y= t2, 0 ~ t ~ I
60. x = 3t - t', y = 3t2, 0 ~ t ~ I
~ x =a cos3
e, y
=a sin3e,
0 ~e ~
7[/2ffi
62. Graph the curvex =2 cos
e -
cos2e
If this curve isrotated about the x-axis, find the area ofthe resulting surface. (Use your graph to help find the correct parameter interva1.)
63. Ifthe curve
y=t-- I t2
is rotated about the x-axis, use your calculator to estimate the area of the resulting surface to three decimal places.
64. If the arc of the curve in Exercise 50 is rotated about the x-axis, estimate the area of the resulting surface using Simp- son's Rule with n=4.
65-66 Find the surface area generated by rotating the given curve about the y-axis.
~ x = 3t2, Y=2t3, 0 ~ t ~ 5 66. x = e' - t, Y =4e'/2, 0 ~ t ~ I
67. If
l'
is continuous and1'(t) ~ 0 for a ~ t ~ b,show that the parametric curve x =f(t), y = g(t), a ~ t~ b, can be put in the form y =F(x). [Hint: Show that f-I exists.]68. Use Formula 2 to derive Formula 7 from Formula 9.2.5 for the case inwhich the curve can be represented in the form y =F(x), a ~ x ~ b.
69. The curvature at a point P of a curve is defined as
I
d¢1
K- -
ds
where ¢is the angle of inclination of the tangent line atP, as shown in the figure. Thus the curvature is the absolute value of the rate of change of ¢with respect to arc length. It can be regarded as a measure of the rate of change of direc- tion of the curve atPand will be studied in greater detail in Chapter 14.
(a) For aparametric curve x = x(t), Y = y(t), derive the formula
where the dots indicate derivatives with respect to t, so
x
=dx/dt. [Hint: Use ¢=tan-I(dy/dx) and Formula 2 to find d¢/dt. Then use the Chain Rule to find d¢/ds.](b) By regarding a curve y =f(x) as the parametric curve x =x, y =f(x), with parameter x, show that the formula in part (a) becomes
70. (a) Use the formula in Exercise 69(b) to find the curvature of the parabola y =x2 at the point (I, I).
(b) At what point does this parabola have maximum curvature?
71. Use the formula in Exercise 69(a) to find the curvature ofthe cycloid x =
e -
sine, y
= I - cose
at the top of one ofits arches.72. (a) Show that the curvature at each point of a straight line is K =O.
(b) Show that the curvature at each point of a circle of radius r is K = l/r.
73. A string is wound around a circle and then unwound while being held taut. The curve traced by the point Pat the end of the string is called the involute of the circle. If the circle has radius r and center 0 and the initial position of Pis (r, 0), and if the parameter
e
is chosen as in the figure, show that parametric equations of the involute are74. A cow is tied to a silo with radius r by a rope just long enough to reach the opposite side of the silo. Find the area available for grazing bythe cow.
o
EXERCISES1-2 Plot the point whose polar coordinates are given. Then find
two other pairs ofpolar coordinates ofthis point, one with r
>
0andone with r
<
O.FIGURE 18
r=sin(88/5)
• In Exercise 55 you areasked to proveanalyti- cally what we have discovered from the graphs in Figure19.
FIGURE 19
Members ofthe family of lima90ns r=1
+
csin 8I. (a) (2,7f/3) 2. (a) (1,77f/4)
(b) (I, -37f/4)
(b) (-3,7f/6)
and so we require that
16n7T/5
be an even multiple of 7T.This will first occur when n =5.Therefore we will graph the entire curve if we specify that 0 <:;:e
<:;: 107T.Switching from
e
to t,we have the equationsi.!J
EXAMPLE II Investigate the family of polar curves given byr
=1+
c sine.
How does the shape change as c changes? (These curves are called limac;ons, after a French word forsnail, because of the shape of thecurves for certain values ofc.)SOLUTION Figure 19 shows computer-drawn graphs for various values ofc. For c
>
I there is aloop that decreases insize ascdecreases. When c= 1 the loop disappears and the curve becomes thecardioid that we sketched in Example 7. For cbetween 1and ~the cardioid's cusp is smoothed out and becomes a"dimple." When cdecreases from ~ to 0, the lima~on is shaped like an oval. This oval becomes more circular asc ~ 0, and when c=
0 the curve isjust the circler =
1.The remaining parts of Figure 19 show that as cbecomes negative, the shapes change in reverse order. Infact, these curves arereflections about thehorizontal axis of the corre-
sponding curves with positive c. 0
(c) (-1, 7f/2) (c) (I,-I)
5-6 The Cartesian coordinates of apoint are given.
(i) Find polar coordinates (r, 8)ofthepoint, where r>0 and 0,,:; 8
<
27f.(ii) Find polar coordinates (r,8)ofthe point, where r
<
0and 0,,:; 8<27f.3-4 Plot the point whose polar coordinates are given. Then find
the Cartesian coordinates of the point. 5. (a) (2,-2)
6. (a)
(3.)3, 3)
(b)
(-1,.)3)
(b) (I, -2)
7-12 Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions.
7. I "S r"S 2
8. r ~ 0, 7r/3 "S
e
"S27r/39. O"S r
<
4, -7r/2"Se <
7r/6 10. 2<
r"S 5, 37r/4< e <
57r/4[II;]
2<
r<
3, 57r/3 "Se
"S 77r/313. Find the distance between the points with polar coordinates (2, 7r/3) and (4, 27r/3).
14. Find a formula for the distance between the points with polar coordinates (rl, (1) and (r2, (2).
15-20 Identify the curve byfinding a Cartesian equation for the curve.
15. r=2
[ill
r =3 sine
16. r cos
e
= I18. r=2 sin
e +
2 cose
20. r=tan
e
sece
21-26 Find a polar equation for the curve represented by the given Cartesian equation.
21. Y =5 23.x=_y2
~ x2
+
y2 = 2ex24. Y=2x - 1 26. xy =4
27-28 For each of the described curves, decide if the curve would be more easily given by a polar equation or a Cartesian equation.
Then write an equation for the curve.
27. (a) A line through the origin that makes an angle of 7r/6 with the positive x-axis
(b) A vertical line through the point (3,3) 28. (a) A circle with radius 5and center (2,3)
(b) A circle centered atthe origin with radius 4
29-48 Sketch the curve with the given polar equation.
29.
e
= -7r/6 30. r2 - 3r+
2 =031. r =sin
e
33. r =2(1 - sin
e), e ~
0~r=
e, e~o
37. r = sin2e
m
r=2cos4e41. r = 1- 2 sin
e
32. r= -3 cos e 34. r= 1 - 3 cos e 36. r= In e, e~1 38. r= 2 cos3e 40. r= 3 cos 6e 42. r= 2
+
sin e43. r2 = 9 sin2e 45. r=2cos(3e/2) 47. r =I
+
2 cos 2e44. r2 =cos 4e 46. r2
e
= I48. r= 1
+
2 cos(e/2)49-50 The figure shows the graph ofras a function of
e
in Carte-sian coordinates. Use it to sketch the corresponding polar curve.
51. Show that the polar curve r =4
+
2 sece
(called a conchoid) has the line x =2 asavertical asymptote by showing that lim,~"xx =2. Use this fact to help sketch the conchoid.52. Show that the curve r= 2 - csc
e
(also a conchoid) has the line y = -1 as a horizontal asymptote by showing that lim,~"x y =-I. Use this fact to help sketch the conchoid.53. Show that the curve r =sin
e
tane
(called a cissoid of Diodes) has the line x = I as a vertical asymptote. Show also that the curve liesentirely within the vertical strip 0 "Sx<
I.Use these facts to help sketch the cissoid.
54. Sketch the curve (x2
+
y2)J = 4X2y2.~ (a) In Example 11the graphs suggest that the lima<;on r = I
+
e sine
has an inner loop whenI
eI >
I. Prove that thisistrue, and find the values ofe
that correspond to the inner loop.(b) From Figure 19 it appears that the lima<;on loses its dimple when e =~.Prove this.
56. Match the polar equations with the graphs labeled I-VI. Give reasons for your choices. (Don't use a graphing device.) (a) r
=)8,
0 "S e "S 167r (b) r=
e2, 0 "S e"S167r(c) r =cos(e/3) (d) r= I
+
2cos e(e) r=2
+
sin 3e (f) r = 1+
2 sin 3eI II
57-62 Find the slope ofthetangent line to the given polar curve atthe point specified by the value of O.
57. r =2 sinO, 0=17/6
rru
r = I/O, 0= 1761. r=cos 20, 0=17/4
58. r =2 - sin 0, 0= 17/3 60. r=cos(0/3), 0= 17
ffi
80. Afamily of curves is given bythe equations r = I+
c sin nO, where c is a real number and n is a positive integer. How does the graph change asnincreases? How does it change asc changes? Illustrate by graphing enough members of the fam- ilyto support your conclusions.63-68 Find the points on the given curve where the tangent line ishorizontal or vertical.
~ Show that the polar equation r=a sin 0
+
b cos0,where ab # 0,represents a circle, and find its center and radius.70. Show that the curves r=asin0 and r=a cos0 intersect at right angles.
ffi
71-76 Useagraphing device to graph the polar curve. Choose the parameter interval tomake sure that you produce the entire curve.71. r= I
+
2sin(0/2) (nephroid of Freeth) 72. r= JI - 0.8 sin20 (hippopede) 73. r= es;no - 2 cos(40) (butterfly curve) 74. r=sin2(40)+
cos(40)75. r = 2 - 5sin(0/6) 76. r=cos(0/2)
+
cos(0/3)ffi
77. How are the graphs ofr = I+
sin(O - 17/6) andr= I
+
sin(O - 17/3)related tothegraph of r= I+
sin O?Ingeneral, how isthe graph of r =f(O - a)related tothe graph of r=f(O)?
ffi
78. Use agraph toestimate they-coordinate of the highest points on the curve r =sin 20.Then use calculus tofind the exact value.ffi
79. (a) Investigate the family of curves defined bythe polar equa-tions r =sin nO,where n is apositive integer. How isthe
number of loops related to n?
(b) What happens ifthe equation in part (a)is replaced by r= IsinnOI?
- a cos 0 r=
+
a cos0Investigate how thegraph changes asthe number achanges.
In particular, you should identify the transitional values ofa for which the basic shape ofthe curve changes.
ffi
82. The astronomer Giovanni Cassini (1625-1712) studied the family ofcurves with polar equationswhere a and c are positive real numbers. These curves are called the ovals of Cassini even though they are oval shaped only for certain values ofaandc.(Cassini thought that these curves might represent planetary orbits better than Kepler's elli pses.) Investigate the variety ofshapes that these curves may have. In particular, how are aand crelated to each other when the curve splits into two parts?
83. Let P be any point (except the origin) onthe curve r =f(O).
If
l/J
isthe angle between the tangent line at Pand the radial line OF,show thattan l/J= -/-r dr dO
84. (a) UseExercise 83to show that the angle between the tan- gent line and the radial line is
l/J
=17/4 at every point on the curve r =eO.ffi
(b) Illustrate part (a) bygraphing the curve and the tangent lines at the points where 0=0and 17/2.(c) Prove that any polar curve r =f(O) with the property that the angle l/J between the radial line and the tangent line is a constant must be ofthe form r=CekO, where Cand k are constants.
FIGURE 8 r=I
+
sin 0Formula 5gives
L
= J:7T ) r 2 + (* r de = fo 27T
)(1+
sin 8)2+
cos2e de
=
f027T
)2+
2sine de
We could evaluate this integral by multiplying and dividing the integrand by
)2 - 2 sin
e ,
or we could use acomputer algebra system. In anyevent, we find that thelength of the cardioid isL =8. 0
-EJ
EXERCISES1-4 Find the area of the region that isbounded bythe given curve and liesin the specified sector.
5-8 Find the area ofthe shaded region.
5. 6.
17-21 Find the area of the region enclosed by one loop of the curve.
19. r=3cos5e 20. r=2sin68
1m
r = I+
2sine
(inner loop)22. Find the area enclosed by the loop ofthe strophoid r =2cos
e -
sece.
23-28 Find the area of the region that liesinside the firstcurve and outside the second curve.
23. r =2 cos0, r = I 25. r =4sin 0, r =2
~ r
=
3 cos0, r=
1+
cos0 28. r=3sine,
r =2 - sin024. r= I - sin
e,
r =126. r =3 cos
e,
r=2 - cos029-34 Find the area ofthe region that lies inside both curves.
29. r=
J3
cose,
r=sine
30. r= 1
+
cose,
r= I - cose lID
r=
sin20, r=
cos2032. r
=
3+
2 cos0, r=
3+
2sin 0 33. r2 =sin 2e, r2 =cos 2034. r=asin 0, r=b cos0, a
>
0, b>
035. Find the area inside the larger loop and outside the smaller loop of the limac;on r =~
+
cose.
36. Find the area between alarge loop and the enclosed small loop of the curve r = I
+
2 cos3e.37-42 Find allpoints of intersection ofthe given curves.
37. r = I
+
sine,
r =3sine
38. r= 1 - cos 0, r = I
+
sine
39. r
=
2 sin2e, r=
1[ill
r= sin e, r= sin2e40. r=cos30, r=sin30 42. r2
=
sin 2e, r2=
cos2ern
43. The points of intersection of the cardioid r = I+
sin0 and the spiral loop r = 20, -17/2 "" 0""17/2, can't befound exactly. Use a graphing device to find the approximate values of 0 at which they intersect. Then use these values to estimate the area that lies inside both curves.44. When recording live performances, sound engineers often use amicrophone with a cardioid pickup pattern because itsup- presses noise from the audience. Suppose the microphone is placed 4 m from the front ofthe stage (as in the figure) and the boundary ofthe optimal pickup region is given by the car- dioid r =8
+
8 sin0,where r ismeasured in meters and the microphone isat the pole. The musicians want to know the area they will have onstage within the optimal pickup range ofthe microphone. Answer their question.45-48 Find the exact length of the polar curve.
45. r=3 sin0, 0 ""0"" 17/3 46. r=e20, 0 "" 0 ""217
49-52 Use acalculator to find the length of the curve correct to four decimal places.
rn
53-54 Graph the curve and find itslength.53. r
=
cos4(O/4) 54. r=
cos2(O/2)55. (a) Use Formula 11.2.7 to show that the area of the surface generated by rotating the polar curve
(where
f'
is continuous and 0 "" a<
b ""17) about the polar axis iss
=r
217rsin 0 ~ r2+ (:~)
2dO(b) Use theformula in part (a) to find the surface area gener- ated byrotating the lemniscate r2 =cos 20 about the polar axis.
56. (a) Find a formula for the area of the surface generated by rotating the polar curve r =f(O), a"" 0 "" b (where
f'
is continuous and 0 ""a<
b "" 17),about the line 0= 17/2.(b) Find the surface area generated byrotating the lemniscate r2 =cos 20 about the line 0= 17/2.
=================~
CO N IC SECTIO NSIn this section we give geometric definitions of parabolas, ellipses, and hyperbolas and derive their standard equations. They are called conic sections, or conics, because they result from intersecting a cone with aplane as shown in Figure I.
FIGURE I Conics
y-l=-2:(x-4)3 / / / / / (4,4) //
\ /
\ / X(4,1)
FIGURE 15
9x2 - 4l- 72x
+
8y+
176=0B
EXERCISESi.'J EXAMPLE7 Sketch the conic
9x2 - 4y2 - 72x
+
8y+
176 =04(y2 - 2y) - 9(x2 - 8x) = 176
4(y2 - 2y
+
1) - 9(x2 - 8x+
16) = 176+
4 - 144 4(y - 1)2 - 9(x - 4)Z = 36(y - 1)2 (x - 4)2
---= 1
9 4
This is in the form (8) except that x and y are replaced by x - 4 and y - 1.Thus a2 =9, b2 =4, and c2 = 13. The hyperbola is shifted four units to the right and one unit upward. The foci are
(4,
1+ /13)
and(4,
1-/13)
and the vertices are (4, 4) and (4, -2). The asymptotes are y - 1= ±:~(x - 4). The hyperbola is sketched inFigure 15. 0
1-8 Find the vertex, focus, and directrix of the parabola and sketch its graph.
I. x =2y2
3. 4x2=-y
[I]
(x+
2)2 =8(y - 3) 7. l + 2y + 12x +25 = 02. 4y
+
x2 = 0 4. y2 = 12x 6. x-I =(y+
5)2 8. Y+
12x - 2x2= 169-10 Find an equation ofthe parabola. Then find the focus and directrix.
11-16 Find the vertices and foci of the ellipse and sketch its graph.
x2 y2
11.-+-=1
16 4
x2 y2
12.
-+-=
164 100
13. 25x2
+
9y2 = 225 14. 4x2+
25y2 =25~ 9x2 - 18x
+
4y2 =2716. x2 + 3l + 2x - 12y + 10= 0
Y
/V -...
~
-1'I'--
2 V /y.
/ "-
L \
~
10 1
1\ /
'\ V
19-24 Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph.
x2 y2
[ill---=1
144 25
y2 x2
20. - - - = 1
16 36
21. /-x2=4 22.9x2-4y2=36
23. 4x2 - l - 24x - 4y
+
28 =0 24. l - 4x2 - 2y+
16x =3125-30 Identify the type of conic section whose equation isgiven and find the vertices and foci.
25. x2 =Y
+
1ITIJ
x2 = 4y - 2y229. y2
+
2y =4x2+
326. x2 =y2
+
1 28. y2 - 8y =6x - 16 30. 4x2+
4x+
y2 =031-48 Find anequation for theconic that satisfies thegiven conditions.
31. Parabola, vertex (0,0), focus (0, -2) 32. Parabola, vertex (1,0), directrix x =-5
~ Parabola, focus (-4,0), directrix x =2 34. Parabola, focus (3,6), vertex (3,2) 35. Parabola, vertex (0,0), axisthe x-axis,
passing through (I,-4)
36. Parabola, vertical axis, passing through (-2, 3),(0,3), and (I,9)
IEJ
Ellipse, foci (::'::2,0), vertices (::'::5,0) 38. Ellipse, foci (0, ::'::5), vertices (0, ::'::13) 39. Ellipse, foci (0, 2), (0,6), vertices (0,0), (0,8) 40. Ellipse, foci (0, - I), (8, -1), vertex (9, -1)41. Ellipse, center (- I,4), vertex (- I,0), focus (- I,6) 42. Ellipse, foci (::'::4,0), passing through (-4, 1.8) 43. Hyperbola, vertices (::'::3,0), foci (::'::5,0) 44. Hyperbola, vertices (0, ::'::2), foci (0, ::'::5) 45. Hyperbola, vertices (- 3, -4), (- 3,6),
foci (-3, -7), (-3,9)
46. Hyperbola, vertices (-1,2), (7,2), foci (- 2,2), (8,2)
~ Hyperbola, vertices (::'::3,0), asymptotes y =::'::2x
48. Hyperbola, foci (2,0), (2,8),
asymptotes y = 3
+
~xandy = 5 - ~x49. The point inalunar orbit nearest the surface of the moon is called perilune and the point farthest from the surface is called apolune. The Apollo II spacecraft was placed in anelliptical lunar orbit with perilune altitude 110 km and apolune altitude 314 km (above the moon). Find anequation ofthis ellipse if the radius of the moon is 1728km and the center of the moon is atone focus.
50. A cross-section of aparabolic reflector is shown in the figure.
The bulb islocated atthefocus and theopening at the focus is 10 cm.
(a) Find an equation of the parabola.
(b) Find the diameter of the opening
I
CDI,
11 cm from the vertex.5cm 11cm- F 5cm
51. In the LORAN (LOng RAnge Navigation) radio navigation system, two radio stations located atA and Btransmit simul- taneous signals to aship or anaircraft located at P. The onboard computer converts the time difference in receiving these signals into adistance difference
I
PAI - 1
PBI,
and this, according to the definition ofahyperbola, locates the ship or aircraft onone branch ofahyperbola (see the figure).Suppose that station B islocated 600 km due east of station A on acoastline. A ship received the signal from B 1200 micro- seconds (J..ls)before it received the signal from A.
(a) Assuming that radio signals travel ata speed of 300 m/J..ls, find anequation ofthe hyperbola on which the ship lies.
(b) Ifthe ship isdue north of B,howfar offthe coastline is the ship?
600km transmitting stations
52. Use the definition of ahyperbola to derive Equation 6 for a hyperbola with foci (::,::c,0)and vertices (::'::a, 0).
53. Show that the function defined by the upper branch of the hyperbola y2/a2 - x2
/b
2 = I isconcave upward.54. Find an equation for theellipse withfoci (I,1) and (- 1, -1) and major axis oflength 4.
55. Determine the type of curve represented by the equation
x2 y2
-+---=
k k - 16
ineach of the following cases: (a)k > 16,(b)0
<
k<
16, and (c) k <O.(d) Show that all the curves in parts (a) and (b) have the same foci, no matter what the value ofk is.
56. (a) Show that the equation of the tangent line to the parabola y2 =4px atthe point (xo, Yo)can be written as
YoY =2p(x
+
xo)(b) What is thex-intercept of thistangent line? Use this fact to draw the tangent line.
57. Show thatthe tangent lines to the parabola x2 =4py drawn from anypoint on the directrix areperpendicular.
58. Show that if anellipse and ahyperbola have the same foci, then their tangent lines at each point of intersection are perpendicular.
59. Use Simpson's Rule with n = 10toestimate the length ofthe ellipse x2
+
4y2 =4.60. The planet Pluto travels inanelliptical orbit around the sun (at one focus). The length of the major axis is I.18 X 1010km
and the length of the minor axis is 1.14 X 1010km. Use Simp- son's Rule with n = 10to estimate the distance traveled by the planet during one complete orbit around the sun.
61. Find the area of the region enclosed bythe hyperbola x2/a2 -
i/
b2= I and the vertical line through afocus.62. (a) Ifanellipse is rotated about its major axis, find the volume of the resulting solid.
(b) If it isrotated about itsminor axis, find the resulting volume.
64. Let P(XI, Yl) be apoint onthe hyperbola
x
2/a
2 - y2/b
2 = I with foci Fl and F2 and let aand {3be the angles between the lines P Fl, PF2 and the hyperbola as shown in the figure. Prove that a={3.(This isthe reflection property ofthehyperbola. It shows that light aimed at a focus F2 of ahyperbolic mirror is reflected toward the other focus Fl')63. Let PI(XI, yd be apoint onthe ellipse
x
2/a
2+
y2/b2 = I with foci Fl and F2 and let a and {3bethe angles between the lines PFI, PF2 and the ellipse as shown in the figure. Prove that a= {3.This explains how whispering galleries and lithotripsy work. Sound coming from one focus is reflected and passes through the other focus. [Hint: Use the formula in Problem 15 on page 202 to show that tan a =tan {3.]y
In the preceding section we defined the parabola in terms of a focus and directrix, but we defined the ellipse and hyperbola interms of two foci. In this section we give a more uni- fied treatment ofallthree types ofconic sections interms of afocus and directrix. Further- more, ifwe place the focus at the origin, then a conic section has a simple polar equation, which provides a convenient description of the motion of planets, satellites, and comets.
OJ
THEOREM Let F be a fixed point (called the focus) and I bea fixed line (called the directrix) in a plane. Let ebe a fixed positive number (called the eccentricity). The set of all points P in the plane such that---=e
I?FI IPII
(that is, the ratio of thedistance from F to the distance from I isthe constant e) is a conic section. The conic is
(a) anellipse if
e <
I (b) aparabola if e = I (c) ahyperbola ife >
1§
EXERCISES1-8 Write a polar equation of a conic with the focus atthe origin and the given data.
I. Hyperbola, eccentricity;1, directrix y =6 2. Parabola, directrix x = 4
rn
Ellipse, eccentricityt
directrix x = -5 4. Hyperbola, eccentricity 2, directrix y = -25. Parabola, vertex (4,37T/2)
6. Ellipse, eccentricity 0.8, vertex
0,
7T/2) 7. Ellipse, eccentricityt
directrix r =4 sec 89-16 (a) Find theeccentricity, (b) identify the conic, (c) give an equation of the directrix, and (d) sketch the conic.
9. r = 10. r= 12
+
sin 8 3 - 10 cos 812 3
II. r = 12. r=
4 - sin 8 2
+
2 cos 89 8
Iill
r= 14. r=6
+
2 cos8 4+
5 sin 83 10
15. r = 16. r=
4 - 8 cos8 5 - 6 sin 8
ffi
17. (a) Find the eccentricity and directrix of the conicr=
1/0 -
2 sin8) and graph the conic and its directrix.(b) If this conic is rotated counterclockwise about the origin through an angle 37T/4, write the resulting equation and graph its curve.
ffi
18. Graph the conic r= 4/(5+
6 cos 8) and its directrix. Also graph the conic obtained by rotating this curve about the ori- gin through anangle 7T/3.ffi
19. Graph the conics r=e/O - e cos 8) with e= 0.4, 0.6, 0.8, and 1.0 on acommon screen. How does the value of e affect the shape of the curve?ffi
20. (a) Graph the conics r =ed/(l+
e sin 8) for e= 1 and var- ious values ofd.How does the value of d affect the shape of the conic?(b) Graph these conics for d= 1 and various values of e.
How does the value of e affect the shape of the conic?
In;]
Show that a conic with focus at the origin, eccentricity e,and directrix x = -d has polar equationed
22. Show that a conic with focus at the origin, eccentricity e,and directrix y =d has polar equation
ed
+
e sin 823. Show that a conic with focus at the origin, eccentricity e,and directrix y = -d has polar equation
ed
24. Show that the parabolas r =
c/O +
cos 8) and r = d/(l - cos 8) intersect at right angles.25. The orbit of Mars around the sun is an ellipse with eccen- tricity 0.093 and semimajor axis 2.28 X 108km. Find a polar equation for the orbit.
26. Jupiter's orbit has eccentricity 0.048 and the length of the major axis is 1.56 X 109km. Find a polar equation for the orbit.
ITIJ
The orbit of Halley's comet, last seen in 1986 and due to return in 2062, is an ellipse with eccentricity 0.97 and one focus at the sun. The length of its major axis is 36.18 AU.[An astronomical unit (AU) is the mean distance between the earth and the sun, about 1.5 X 108km.] Find a polar equation for the orbit of Halley's comet. What is the maximum distance from the comet to the sun?
28. The Hale-Bopp comet, discovered in 1995, has an elliptical orbit with eccentricity 0.9951 and the length of the major axis is 356.5 AU. Find a polar equation for the orbit of this comet.
How close to thesun does it come?
29. The planet Mercury travels in an elliptical orbit with eccen- tricity 0.206. Its minimum distance from the sun is 4.6 X 107km. Find its maximum distance from the sun.
30. The distance from the planet Pluto to the sun is
4.43 X 109km at perihelion and 7.37 X 109km at aphelion.
Find the eccentricity of Pluto's orbit.
31. Using the data from Exercise 29, find the distance traveled by the planet Mercury during one complete orbit around thesun.
(If your calculator or computer algebra system evaluates defi- nite integrals, use it.Otherwise, use Simpson's Rule.)
=================0
REVI EWI. (a) What is aparametric curve?
(b) How do you sketch a parametric curve?
2. (a) How doyou find the slope of atangent to aparametric curve?
(b) How do you find the area under a parametric curve?
3. Write an expression for each of the following:
(a) The length of a parametric curve
(b) The area ofthe surface obtained byrotating a parametric curve about the x-axis
4. (a) Use adiagram toexplain the meaning ofthepolar coordi- nates (r, 0) of a point.
(b) Write equations that express the Cartesian coordinates (x, y) of apoint in terms of the polar coordinates.
(c) What equations would you use to find thepolar coordinates ofa point ifyou knew the Cartesian coordinates?
5. (a) How do you find the slope of atangent line toapolar curve?
(b) How do you find the area of a region bounded by apolar curve?
(c) How do you find the length of apolar curve?
6. (a) Give a geometric definition ofa parabola.
(b) Write anequation of aparabola with focus (0,p) and direc- trix y = -po What ifthe focus is(p, 0) and the directrix isx= -p?
7. (a) Give a definition ofanellipse in terms of foci.
(b) Write anequation forthe ellipse with foci (±c, 0)and vertices (±a, 0).
8. (a) Give adefinition of ahyperbola interms offoci.
(b) Write anequation for the hyperbola with foci (±c, 0) and vertices (±a, 0).
(c) Write equations for the asymptotes ofthe hyperbola in part (b).
9. (a) What isthe eccentricity of a conic section?
(b) What can you say about theeccentricity ifthe conic section isanellipse? A hyperbola? Aparabola?
(c) Write apolar equation for aconic section with eccentricity eand directrix x = d. What ifthe directrix isx = -d?
y = d? y = -d?
Determine whether the statement is true or false.If itis true. explain why.
Ifit is false,explain why or give an example that disproves the statement.
I. If theparametric curve x = f(t), y = g(t) satisfies g'(I) = 0, then it has ahorizontal tangent when t = I.
2. If x = f(t) and y = get) are twice differentiable, then
d2y d2y/dt2
dx2 d2x/dt2
3. The length ofthecurve x =f(t), y = g(t), a ~ t ~ b,is
.I:J[f'(t)]2
+
[g'(t)]2 dt.4. If apoint isrepresented by (x,y) in Cartesian coordinates (where x ¥- 0) and (r, 0) in polar coordinates, then
0= tan-I(y/X).
5. The polar curves r= 1 - sin28 and r = sin28 - ] havethe same graph.
6. The equations r = 2, x2
+
y2 = 4,and x = 2 sin3t, y = 2 cos 3t (0 ~ t~ 27T) all have the same graph.7. The parametric equations x = t2, Y = t4 have the same graph as x =t3, Y =t6•
