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Chapter 11
1.
Eliminate the parameter t to find a Cartesian equation of the curve.
y  3t  1, x  t  7
2.
Find an equation for the conic that satisfies the given conditions.


Ellipse, foci  1, 5 , length of major axis 8
Select the correct answer.
a.
x  32  y 2
16
4x 2  y
3.
15
1
b.
x2 y 2

1
16 15
c.
x 2  y  52

1
16
15
d.
e. x 2  y
True or False?
If the parametric curve x  f (t ), y  g (t ) satisfies g '(1)  0, then it has a horizontal tangent when
t  1.
4.
Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve.
x  cos  , y  sec , 0     / 2
5.
Find d 2 y / dx 2 .
x  1 t2 , y  t  t3
6.
Find the points on the curve where the tangent is horizontal.

x  2 cos  cos2 
,

y  2 sin   sin  cos

7.
Try to estimate the coordinates of the highest point on the curve x  15tet , y  3tet . Round the
answer to the nearest hundredth.
8.
Find the polar equation for the curve represented by the given Cartesian equation.
x y  2
9.
Find equations of the tangents to the curve x  3t 2  1, y  2t 3  1 that pass through the point ( 4 , 3
).
10. Find the points of intersection of the curves r  2 and r  4 sin  .
11. True or False?
The exact length of the curve x  et  e t , y  5  12t , 0  t  6 is
1 12
(e  24  e 12 ) .
2
12. Find the length of the curve.
x  3t 2 , y  2t 3 , 0  t  3
13. Find the area enclosed by the curve r 2  9cos5 .
14. The graph of the following curve is given. Find the area that it encloses.
r  2  cos 6
15. Find an equation of the ellipse with foci ( 5,0) and vertices ( 3,0) .
16. Find an equation for the conic that satisfies the given conditions.
Hyperbola,
vertices (3, 0), (-3, 0),
asymptotes y = ±2x
17. Write a polar equation in r and  of a hyperbola with the focus at the origin, with the eccentricity
6 and the directrix x = - 2.
18. Find the equation of the directrix of the conic.
r
6
3  sin 
Select the correct answer.
a. y  6
e. x  2
b.
y  3
19. Find the eccentricity of the conic.
r
9
2  11 cos
c.
x  6
d.
x3
20.
Find the area of the shaded region.
r  1  sin 
1.
y( x)  3x  20
2.
c
3.
True
4.
y
5.

6.
 3

  ,  3 3 ,
 2
2 

7.
(40.77, 1.10)
8.
r
9.
y  2( x  13)  15, y  ( x  4)  3
10.
(2,  / 6), (2, 5 / 6)
11.
F
12.
20 10  2
13.
18
14.
15.
16.
17.
1
x
3t 2  1
4t 3
 3 3 3
 ,

 2
2 

2
cos   sin 
9
2
x2 y 2

1
9
4
x2 y 2

1
9 36
12
r
(1  6 cos )
18.
a
19.
11/2
20.
3
1
8
1.
Eliminate the parameter to find a Cartesian equation of the curve.
x(t )  cos 2 5t , y  sin 2 5t
2.
Find parametric equations to represent the line segment from (-9, 2) to (7, -8).
3.
If a projectile is fired with an initial velocity of v 0 meters per second at an angle  above the
horizontal and air resistance is assumed to be negligible, then its position after t seconds is given by
1
the parametric equations x  v 0 cos   t , y  v 0 sin   t  gt 2 , where g is the acceleration of
2
9.8 m/s  .
2
gravity
ground?
4.
Round the result to the nearest tenth.
Find an equation of the tangent line to the curve at the point corresponding to the value of the
parameter.
xe
5.
If a gun is fired with   20  and v0  470 m/s when will the bullet hit the
t
, y  t  ln t 3 ; t  1
Find the points on the curve where the tangent line is horizontal.
x  5(cos   cos2  ), y  5(sin   sin  cos )
6.
Find equations of the tangent lines to the curve x  3t 2  1, y  2t 3  1 that pass through the point (13, 17) .
7.
Find the area of the region enclosed by each loop of the curve.
x  sin t  2 cos t , y  2  2 sin t cos t
Select the correct answer.
a.
e.
8.
5
5
b.
4

5
c.
2
5
5
d.
4
5
5
2
5
5
Set up, but do not evaluate, an integral that represents the length of the parametric curve.
9
x  t  t 9 , y  t 8 / 7 , 3  t  12
8
9.
True or False?
The exact length of the curve x  et  e t , y  10  2t , 0  t  6 is (e12  24  e12 ) / 2 .
10.
Find d 2 y / dx 2 .
x  t  sin t , y  t  cos t
11. Sketch the polar curve.
r  cos3
12. Find the polar equation for the curve represented by the given Cartesian equation.
x y 3
13. Find the points of intersection of the curves r  2 and r  4 sin  .
14. Using the arc length formula, set up, but do not evaluate, an integral equal to the total arc length of the
ellipse.
x  4 sin  , y  2 cos
2
15. Find the surface area generated by rotating the lemniscate r  5 cos 2 about the line  
Select the correct answer.
a.
L   2 /10
L  5 2
b.
e.
L  10 2
c.
L   /10
L  2 2
16. Find the area that encloses the curve.
r  11 sin 
17. Find an equation of the hyperbola with foci
 0, 6  and asymptotes y   x / 3 .
18. Find an equation of the parabola with focus
9 / 2, 0 and directrix x  7 / 2 .
19.
Find an equation of the ellipse with foci ( 5,0) and vertices ( 3,0) .
d.

2
.
20. Write a polar equation in r and  of an ellipse with the focus at the origin, with the eccentricity
6
and directrix x  8 .
7
Select the correct answer.
a.
e.
48
7  6 cos
48
r
1  7 sin 
r
b.
r
8
3  2 sin 
c.
r
8
7  6 sin 
d.
r
48
6  7 sin 
1.
x  1 y
2.
x  9  16t , y  2  10t , 0  t  1
3.
32.8
4.
y
5.
 15

  ,  15 3 ,
 4
4 

6.
y  127  4( x  49), y  17  2( x  13)
7.
d
12
8.

3
4
( x  e)  1
e
 15 15 3 
 ,

 4

4


2
9
(1  9t 8 ) 2    t 2 / 7 dt
7
9.
T
10.
sin t  cos t  1
(1  cos t )3
11.
3
cos   sin 
12.
r
13.
(2,  / 6), (2, 5 / 6)
2
14.

16 cos2   4 sin 2  d
0
15. b
16.
121
4
17.
y 2 x2

1
36 324
18.
y 2  16 x  8
19.
x2 y 2

1
9
4
20. a