微積分作業解答 Chapter 9 Parametric Equations and Polar Coordinates 9.1-9.4
9.1
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
5 sin t , y
t 2 ,S d t d S
t, y 1 t
7. x
(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.
t , y 1 t Ÿ y 1 x 2 , t t 0, x t 0
(b) x
13. x
e 2t , y
t 1
(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.
(a) x
e 2t Ÿ t
1
ln x Ÿ y
2
t 1
1
ln x 1 ， x ! 0
2
17. x 5 sin t , y 2 cos t ,S d t d 5S . Describe the motion of a particle with position ( x, y ) as t
varies in the given interval.
1
廖彥法 整理
微積分作業解答 Chapter 9 Parametric Equations and Polar Coordinates 9.1-9.4
由此參數式可知 sin t
x
, cos t
5
y
Ÿ sin 2 t cos 2 t
2
為中心在 (0,0) ，長軸長10、短軸長4的橢圓，當 t
圈。
x2 y2
25 4
1
S 時，起點為 (0,2) ，逆時針旋繞3
19. Use the graphs of x f (t ) and y g (t ) to sketch the parametric curve x
Indicate with arrows the direction in which the curve is traced as t increases.
f (t ) , y
g (t ) .
27. Find parametric equations for the path of a particle that moves along the circle x 2 ( y 1) 2
in the manner described.
4
(a) Once around clockwise, starting at (2,1) .
(b) Three times around counterclockwise, starting at (2,1) .
(c) Half way around counterclockwise, starting at (0,3) .
(a) x 2 ( y 1) 2
順時針 T
4Ÿ x
t Ÿ x
2 cosT , y 1 2 sin T ，為逆時針旋繞，起點為 (2,1)
2 cos t , y 1 2 sin t ,0 d t d 2S
(b) 繞轉三次，即周期減為三分之一，所以 T
(c) 轉半圈，且起點為 (0,3) ，故 T
Ÿx
2 sin
t
S
Ÿx
2
3t Ÿ x
2 cos 3t , y 1 2 sin 3t ,0 d t d 2S
2 cos(t S
S
), y 1 2 sin(t ),0 d t d S
2
2
S
, y 1 2 cos t ,0 d t d S
2
31. Compare the curves represented by the parametric equations. How do they differ?
(a) x
t3, y
t2
(b) x
t6, y
t4
(a) Ÿ y
x 2 / 3 , x  R, y  R ‰ {0}
(c) Ÿ y
x2 / 3 , x  R , y  R
(c) x
e 3t , y
(b) Ÿ y
e 2 t
x 2 / 3 , x  R ‰ {0}, y  R ‰ {0}
2
廖彥法 整理
微積分作業解答 Chapter 9 Parametric Equations and Polar Coordinates 9.1-9.4
9.2
3. x t 4 1, y t 3 t; t 1 . Find an equation of the tangent to the curve at the point
corresponding to the given value of the parameter.
dx
dt
4t 3 ,
dy
dt
3t 2 1 Ÿ
dy
dx
3t 2 1
dy
Ÿ
3
4t
dx t
切點為 (2,2) ，所以切線為 y 2
3 1
4
1
( x 2) Ÿ y
1
x
7. Find an equation of the tangent to the curve x et , y (t 1) 2 at the point (1,1) by two
methods: (a) without eliminating the parameter and (b) by first eliminating the parameter.
dx
dt
(a)
dy
dt
et ,
2(t 1) Ÿ
dy
dx
2t 2
dy
Ÿ
t
e
dx t
0
切點為 (1,1) ，所以切線為 y 1 2( x 1) Ÿ y
et Ÿ t
(b) x
Ÿ
9. x
dy
dx x
dy / dt
dx / dt
d2y
dx 2
當
2
1
1
4 t2, y
dy
dx
ln x Ÿ y
2
2 x 3
2(ln x 1) ˜
1
x
2 ，切點為 (1,1) ，所以切線為 y 1 2( x 1) Ÿ y
t 2 t 3 . Find
2t 3t 2
2t
d § dy ·
¨ ¸
dx © dx ¹
dy
dx
(ln x 1) 2 Ÿ
2
1
2 x 3
dy
d2y
and
. For which values of t is the curve concave upward?
dx
dx 2
3
1 t
2
d ( dy / dx ) / dt
dx / dt
3/ 2
2t
3
4t
d2y
! 0 時，曲線凹向上，則 t ! 0 。
dx 2
15. x 2 cosT , y sin 2T . Find the points on the curve where the tangent is horizontal or vertical.
If you have a graphing device, graph the curve to check your work.
dy
dx
當
當
dy / dT
dx / dT
dy
dx
2 cos 2T
2 sin T
cos 2T
sin T
0 時有水平切線，則 cos 2T
0ŸT
S 3S 5S 7S
, , ,
4 4 4 4
微積分作業解答 Chapter 9 Parametric Equations and Polar Coordinates 9.1-9.4
t 3 4t , y
25. At what points on the curve x
equations x 7t , y 12t 5 ?
x
7t , y 12t 5 Ÿ
x
t 3 4t , y
12t
3t 2 4
6t 2 Ÿ
12
Ÿt
7
dy
dx
dy / dt
dx / dt
dy
dx
1,
dy / dt
dx / dt
6t 2 is the tangent parallel to the line with
12
7
12t
3t 2 4
4
Ÿ ( x, y )
3
( 5,6), (
208 32
, )
27 3
27. Use the parametric equations of an ellipse, x
area that it encloses.
4 ³ ydx
可用第一象限的區域面積計算，則 A
29. Find the area bounded by the curve x
x
a cos T , y
³
S /2
0
cos t , y
b sin T , 0 d T d 2S , to find the
y (T )dx(T )
et , 0 d t d
4ab ³
S /2
0
sin 2 TdT
Sab
S
, and the lines y 1 and
2
0.
S /2
A
³
1
0
³
( y 1)dx
0
S /2
(e t 1)( sin t ) dt
ª1 t
º
«¬ 2 e (sin t cos t ) cos t »¼
0
1 S /2
(e 1)
2
33-36. Set up, but do not evaluate, an integral that represents the length of the curve.
33. x
Ÿ
L
35. x
Ÿ
L
39. x
Ÿ
L
t t2 , y
4 3/ 2
t , 1d t d 2
3
dx
dt
dy
dt
³
1 2t ,
2
1
(
2t 1 / 2 Ÿ (
dx 2 dy 2
) ( ) dt
dt
dt
³
2
dx 2 dy 2
) ( )
dt
dt
1 4t 2 dt
1
t cos t , y
t sin t , 0 d t d 2S
dx
dt
dy
dt
³
1 sin t ,
2S
(
0
t
, y
1 t
dx
dt
³
0
(
³
2S
0
dx 2 dy 2
) ( )
dt
dt
3 2 sin t 2 cos t
3 2 sin t 2 cos t dt
ln(1 t ) , 0 d t d 2 . Find the length of the curve.
1
dy
,
2
(1 t ) dt
2
1 cos t Ÿ (
dx 2 dy 2
) ( ) dt
dt
dt
1 4t 2
t 2 2t 2
(1 t ) 4
1
dx
dy
Ÿ ( )2 ( )2
1 t
dt
dt
dx 2 dy 2
) ( ) dt
dt
dt
³
2
0
t 2 2t 2
dt ，
(1 t ) 2
4
廖彥法 整理
微積分作業解答 Chapter 9 Parametric Equations and Polar Coordinates 9.1-9.4
dt ，則 ³
令 u 1 t Ÿ du
0
tan T Ÿ du
令u
t 2 2t 2
dt
(1 t ) 2
2
sec 2 TdT ，則 ³
3
1
³
u2 1
du
u2
3
1
u2 1
du
u2
³
1
tan 1
tan 1 3
1 º
ª
«¬ln | secT tan T | sin T »¼ 1
tan 1
tan 1 3
sec 3 T
dT
tan 2 T
³
tan 1 3
1
tan 1
(secT cosT
) dT
sin 2 T
10
ln( 2 1) 2
3
ln( 10 3) 10 3
10
2 ln(
)
3
2 1
47. Find the distance traveled by a particle with position ( x, y ) as t varies in the given time
interval. Compare with the length of the curve.
x
sin 2 t , y
L
³
cos 2 t , 0 d t d 3S
3S
(
0
dx 2 dy 2
) ( ) dt
dt
dt
3S
4³
S /2
0
(
6 2³
0
dx 2
dy
) ( ) 2 dT
dT
dT
0
a cos3 T , y
50. Find the total length of the astroid x
L
S /2
2 ³ | sin 2t | dt
12a ³
S /2
0
sin T cosTdT
sin 2tdt
3 2 cos 2t
S /2
0
6 2
a sin 3 T , where a ! 0 .
6a ³
S /2
0
sin 2TdT
S /2
3a cos 2t 0
6a
9.3
5. (a) (1,1) (b) (2 3 ,2)
The Cartesian coordinates of a point are given.
(i) Find polar coordinates (r ,T ) of the point, where r ! 0 and 0 d T 2S .
(ii) Find polar coordinates (r ,T ) of the point, where r 0 and 0 d T 2S .
(a) (i) r cosT
(ii) r cosT
(b) (i) r cosT
(ii) r cosT
r sin T
r sin T
1Ÿ r
1Ÿ r
2 3 , r sin T
2 3 , r sin T
S
4
2 ,T
5S
4
2 ,T
2 Ÿ r
2 Ÿ r
11S
6
4,T
4,T
5S
6
7-12. Sketch the region in the plane consisting of points whose polar coordinates satisfy the given
conditions.
5
廖彥法 整理
13. r
r
3 sin T . Identify the curve by finding a Cartesian equation for the curve.
3 sin T Ÿ x
3
Ÿ x 2 ( y )2
2
3 sin T cosT , y
3 sin 2 T Ÿ x
3
sin 2T , y
2
3 3
cos 2T
2 2
3
3
3
( ) 2 為一圓心 (0, ) 半徑 的圓。
2
2
2
T
6
廖彥法 整理
微積分作業解答 Chapter 9 Parametric Equations and Polar Coordinates 9.1-9.4
47. r 2 sin T ,T S / 6 . Find the slope of the tangent line to the given polar curve at the point
specified by the value of T .
dy
dx
dy / dT
dx / dT
r
2 sin T Ÿ x
Ÿ
dy / dT
dx / dT
2 sin T cosT
2 sin 2T
2 cos 2T
sin 2T , y
2 sin 2 T
dy
dx T
3
tan 2T Ÿ
1 cos 2T
S
6
53. r 1 cos T . Find the points on the given curve where the tangent line is horizontal or vertical.
dy
dx
dy / dT
dx / dT
r 1 cosT Ÿ x
dy / dT
Ÿ
dx / dT
(1 cosT ) cosT , y
(1 cosT ) sin T
cosT 2 cos2 T 1
sin T 2 sin T cosT
水平切線，斜率為零，即 cosT 2 cos 2 T 1 0 Ÿ cosT
1
,1 Ÿ T
2
S
5S
,S ,
3
3
3 S
3 5S
此切點為 ( , ), (0, S ), ( , )
2 3
2 3
垂直切線，斜率不存在，即 sin T 2 sin T cos T
Ÿ sin T
0, cosT
1
ŸT
2
0,
2S
4S
1 2S 1 4S
,S ,
，此切點為 (2,0), ( , ), ( , )
3
3
2 3
2 3
55. Show that the polar equation r
its center and radius.
r
a sin T b cosT Ÿ r 2
a sin T b cos T , where ab z 0 , represents a circle, and find
ar sin T br cosT Ÿ x 2 y 2
b2
a2
2
Ÿ ( x bx ) ( y ay )
4
4
2
0
ay bx
a2 b2
b
a
Ÿ ( x )2 ( y )2
4
2
2
a 2 b2 2
(
)
2
a 2 b2
b a
的圓。
即圓心 ( , ) ，半徑
2 2
2
7
廖彥法 整理
微積分作業解答 Chapter 9 Parametric Equations and Polar Coordinates 9.1-9.4
9.4
3. r sin T , S / 3 d T d 2S / 3 . Find the area of the region that is bounded by the given curve and
lies in the specified sector.
A
7. r
1 2
r dT
2³
1 2S / 3 2
sin T dT
2 ³S / 3
1
1
(T sin 2T )S2S/ 3/ 3
4
2
S
3
12 8
4 3 sin T . Find the area of the shaded region.
A
1 2
r dT
2³
1 S /2
(4 3 sin T ) 2 dT
³
S
/
2
2
(
41
9
T sin 2T )SS/ 2/ 2
4
8
9-12. Sketch the curve and find the area that it encloses.
9. r 2
4 cos 2T
11.
A
15. r
sin 2T
17. r
A
1 11S / 6
(1 2 sin T ) 2 dT
³
7
S
/
6
2
1
(3T 4 cos
2
41
S
4
微積分作業解答 Chapter 9 Parametric Equations and Polar Coordinates 9.1-9.4
19.
21.
r
A
23. r
29-32. Find all points of intersection of the given curves.
29. r
r
cosT , r 1 cosT
cosT , r 1 cosT Ÿ cosT
1
1 cosT Ÿ cosT
1 S
1 5S
所以交點為 ( , ) 、 ( , )
2 3
2 3
31. r
sin T , r
sin 2T
r
sin T , r
sin 2T Ÿ sin T
sin 2T Ÿ sin T (1 2 cos
所以交點為 (0,0) 、 (0, S ) 、 (
3 S
3 5S
, ) 、 (
, )
2 3
2 3
33-36. Find the exact length of the polar curve.
S
33. r 3 sin T , 0 d T d
3
³
L
r 2 ( dr / dT ) 2 dT
³
S /3
³
2S
0
3dT
S /3
3T 0
S
35. r T 2 , 0 d T d 2S
³
L
r 2 ( dr / dT ) 2 dT
令 u T 2 4 Ÿ du
2S
則 ³ T T 4 dT
0
2
0
T T 2 4 dT
2TdT
1 4S 2 4 1 / 2
u du
2 ³4
4S 2 4
1 3/ 2
u
3
4
9
8 2
[(S 1) 3 / 2 1]
3
廖彥法 整理