1. Find parametric equations to represent the line segment from .
a.
b.
c.
d.
e.
ANSWER: d
2. If a projectile is fired with an initial velocity of v0 meters per second at an angle α above the horizontal and air resistance is assumed to be negligible, then its position after t seconds is given by the parametric equations
where g is the acceleration of gravity . If a gun is fired with α = 55° and v0 = 440 m/s when will the bullet hit the ground?
a. t = 244 s
b. t = 74 s
c. t = 344 s
d. t = 124 s
e. t = 224 s
ANSWER: b
3. Describe the motion of a particle with position (x, y) as t varies in the given interval 0 ≤ t ≤ 2π.
a. Moves once counterclockwise along the circle x2 + y2 = 1 starting and ending at (0, –6).
b. Moves once counterclockwise along the ellipse starting and ending at (0, 6).
c. Moves once counterclockwise along the ellipse starting and ending at (–6, 0).
d. Moves once clockwise along the ellipse starting and ending at (0, 6).
e. Moves once clockwise along the circle starting and ending at (0, 6).
ANSWER: d
4. If a and b are fixed numbers, find parametric equations for the set of all points P determined as shown in the figure, using the angle ang as the parameter. Write the equations for a = 15 and b = 6.
ANSWER:
5. Find parametric equations for the path of a particle that moves once clockwise along the circle , starting at (4, 9).
ANSWER:
6. Eliminate the parameter to find a Cartesian equation of the curve.
ANSWER:
7. Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve.
ANSWER:
8. Eliminate the parameter to find a Cartesian equation of the curve.
ANSWER: x = 6 – y
9. Find the point(s) of intersection of the following two parametric curves, by first eliminating the parameter, then solving the system of equations.
and
a.
b.
c.
d. (0, 0) and (1, 1)
e. (0, 0), (1, -1), and (1, 1)
ANSWER: d
10. Find the point(s) of intersection of the following two parametric curves, by first eliminating the parameter, then solving the system of equations.
and
a.
b.
c.
d. (45, 900) and (5, 100)
e. (45, 900) and (900, 100)
ANSWER: d
11. Eliminate the parameter to find a Cartesian equation of the curve.
a.
b.
c.
d.
e.
ANSWER: a
12. Find the parametric equations for the path of a particle that moves two and a half times clockwise around the circle , starting at .
a.
b.
c.
d.
e.
ANSWER: a
13. Find parametric equations for the ellipse .
a.
b.
c.
d.
e.
ANSWER: a
14. Let P be a point at a distance 6 units from the center of a circle of radius 2. 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.) Assuming the line is the x-axis and when P is at one of its lowest points, find the parametric equations of the trochoid. (Hint: use the same parameter as for the cycloid.)
a.
b.
c.
a.
b.
c.
d.
e.
ANSWER: d
2. If a projectile is fired with an initial velocity of v0 meters per second at an angle α above the horizontal and air resistance is assumed to be negligible, then its position after t seconds is given by the parametric equations
where g is the acceleration of gravity . If a gun is fired with α = 55° and v0 = 440 m/s when will the bullet hit the ground?
a. t = 244 s
b. t = 74 s
c. t = 344 s
d. t = 124 s
e. t = 224 s
ANSWER: b
3. Describe the motion of a particle with position (x, y) as t varies in the given interval 0 ≤ t ≤ 2π.
a. Moves once counterclockwise along the circle x2 + y2 = 1 starting and ending at (0, –6).
b. Moves once counterclockwise along the ellipse starting and ending at (0, 6).
c. Moves once counterclockwise along the ellipse starting and ending at (–6, 0).
d. Moves once clockwise along the ellipse starting and ending at (0, 6).
e. Moves once clockwise along the circle starting and ending at (0, 6).
ANSWER: d
4. If a and b are fixed numbers, find parametric equations for the set of all points P determined as shown in the figure, using the angle ang as the parameter. Write the equations for a = 15 and b = 6.
ANSWER:
5. Find parametric equations for the path of a particle that moves once clockwise along the circle , starting at (4, 9).
ANSWER:
6. Eliminate the parameter to find a Cartesian equation of the curve.
ANSWER:
7. Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve.
ANSWER:
8. Eliminate the parameter to find a Cartesian equation of the curve.
ANSWER: x = 6 – y
9. Find the point(s) of intersection of the following two parametric curves, by first eliminating the parameter, then solving the system of equations.
and
a.
b.
c.
d. (0, 0) and (1, 1)
e. (0, 0), (1, -1), and (1, 1)
ANSWER: d
10. Find the point(s) of intersection of the following two parametric curves, by first eliminating the parameter, then solving the system of equations.
and
a.
b.
c.
d. (45, 900) and (5, 100)
e. (45, 900) and (900, 100)
ANSWER: d
11. Eliminate the parameter to find a Cartesian equation of the curve.
a.
b.
c.
d.
e.
ANSWER: a
12. Find the parametric equations for the path of a particle that moves two and a half times clockwise around the circle , starting at .
a.
b.
c.
d.
e.
ANSWER: a
13. Find parametric equations for the ellipse .
a.
b.
c.
d.
e.
ANSWER: a
14. Let P be a point at a distance 6 units from the center of a circle of radius 2. 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.) Assuming the line is the x-axis and when P is at one of its lowest points, find the parametric equations of the trochoid. (Hint: use the same parameter as for the cycloid.)
a.
b.
c.