10.4 Polar Coordinates and Polar Graphs 617
10.4 Polar Coordinates and Polar Graphs
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find
.
a.
b.
c.
d.
e.
____ 2. Find .
a.
b.
c.
d.
e.
618 Chapter 10: Conics, Parametric Equations, and Polar Coordinates
____ 3. Find .
a.
b.
c.
d.
e.
____ 4. Find the second derivative of the parametric equations . Round
your answer to two decimal places, if necessary.
a. 0
b. 2.00
c.
d. 0.50
e.
____ 5. Find and if possible, and find the slope and concavity (if possible) at the
point corresponding to t = 5.
a.
: slope and concave up
b.
slope –6 and concave down
c.
slope 14 and concave up
d.
: slope 6 and concave down
e.
: slope and concave up
10.4 Polar Coordinates and Polar Graphs 619
____ 6. Find and if possible, and find the slope and concavity (if possible) at the
point corresponding to .
a.
at : slope 1 and concave down
b.
at : slope and concave down
c.
at : slope and concave up
d.
at : slope 1 and concave up
e.
at : slope of and concave down
____ 7. Find the second derivative of the parametric equations .
a.
b.
c.
d.
e.
620 Chapter 10: Conics, Parametric Equations, and Polar Coordinates
____ 8. Find an equation of the tangent line at a point on the curve
.
a.
b.
c.
d.
e.
____ 9. Find all points (if any) of horizontal and vertical tangency to the curve
.
a. horizontal tangents: , vertical tangent: none
b. horizontal tangent: none, vertical tangents:
c. horizontal tangents: , vertical tangent: none
d. horizontal tangent: none, vertical tangents:
e. horizontal tangent: none, vertical tangent: none
____ 10. Find all points (if any) of horizontal and vertical tangency to the curve
.
a. horizontal tangents: , vertical tangents:
b. horizontal tangents: , vertical tangents:
c. horizontal tangent: , vertical tangent:
d. horizontal tangents: , vertical tangents:
e. horizontal tangent: , vertical tangent:
10.4 Polar Coordinates and Polar Graphs 621
____ 11. Determine the t intervals on which the curve is concave
downward or concave upward.
a. concave downward: ; concave upward:
b. concave downward: ; concave upward:
c. concave downward: ; concave upward:
d. concave downward: ; concave upward:
e. concave downward: ; concave upward:
____ 12. Determine the t intervals on which the curve is concave downward or
concave upward.
a. concave downward: ; concave upward:
b. concave downward: ; concave upward:
c. concave downward:
d. concave upward:
e. concave downward:
____ 13. Find the arc length of the curve on the given interval.
a.
b.
c.
d.
e.
622 Chapter 10: Conics, Parametric Equations, and Polar Coordinates
____ 14. Find the arc length of the curve on the interval . Round your
answer to three decimal places.
a. 287.453
b. 191.635
c. 193.606
d. 66.480
e. 99.721
____ 15. Find the arc length of the curve on the given interval.
a.
b.
c.
d.
e.
10.4 Polar Coordinates and Polar Graphs 623
____ 16. Find the arc length of the curve on the given interval.
a.
b.
c.
d.
e.
____ 17. The path of a projectile is modeled by the parametric equations and
where x and y are measured in feet. Use a graphing utility to approximate the
range of the projectile. Round your answer to two decimal places.
a. 312.85 ft
b. 521.42 ft
c. 131.09 ft
d. 391.06 ft
e. 195.53 ft
____ 18. The path of a projectile is modeled by the parametric equations and
where x and y are measured in feet. Use the integration capabilities of a
graphing utility to approximate the arc length of the path. Round your answer to one decimal place.
a. 387.2 ft
b. 595.8 ft
c. 335.0 ft
d. 269.9 ft
e. 465.4 ft
624 Chapter 10: Conics, Parametric Equations, and Polar Coordinates
____ 19. Find the area of the surface generated by revolving the curve about the
x-axis on the interval .
a.
b.
c.
d.
e.
____ 20. Find the area of the surface generated by revolving the curve about the given axis.
(i) x-axis; (ii) y-axis
a.
b.
c.
d.
e.
____ 21. Find the area of the surface generated by revolving the curve about the
y-axis on the interval . Round your answer to two decimal places.
a. 1427.12
b. 1178.96
c. 1049.37
d. 589.48
e. 1898.04
____ 22. Find the area of the surface generated by revolving the curve about the given axis.
(i) x-axis; (ii) y-axis
a.
b.
c.
d.
e.
10.4 Polar Coordinates and Polar Graphs 625
10.4 Polar Coordinates and Polar Graphs
Answer Section
626 Chapter 10: Conics, Parametric Equations, and Polar Coordinates