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10.2 Plane Curves and Parametric Equations
628
b. e.
c.
____ 8.
Use the result, "the set of parametric equations for the line passing through
and
is
" to find a set of parametric equations for the
line passing through
and
.
a.
b.
c.
d.
e.
629
Chapter 10: Conics, Parametric Equations, and Polar Coordinates
____
9.
Use the result, “the set of parametric equations for the circle is
”
to find a set of parametric equations for the circle with a center
and radius .
a.
b.
c.
d.
e.
____
10.
Use the result, “the set of parametric equations for the ellipse is
” to find a set of parametric equations for the ellipse with vertices
and
and with foci at
and
.
a.
b.
c.
d.
e.
____
11.
Find a set of parametric equations for the rectangular equation
.
a.
b.
c.
d.
e.
____ 12. Find a set of parametric equations for the rectangular equation that
satisfies the condition at the point .
a.
b.
c.
d.
e.
10.2 Plane Curves and Parametric Equations
630
____ 13.
Find a set of parametric equations for the rectangular equation
that satisfies
the condition at the point .
a.
b.
c.
d.
e.
____ 14. Identify any points at which the cycloid is not smooth.
not smooth when
smooth everywhere
not smooth when
not smooth when
not smooth when
____ 15. Identify any points at which the Folium of Descartes is not
smooth. Round your answer to two decimal places, if necessary.
not smooth when
not smooth when
not smooth when
smooth everywhere
not smooth when
631
Chapter 10: Conics, Parametric Equations, and Polar Coordinates
____
16.
The parametric equations for the path of a projectile launched at a height h feet above
the ground, at an angle
with the horizontal and having an initial velocity of
feet per second is
given by
and
. The center field fence in a ballpark is
feet high and
feet from home plate. The ball is hit 2 feet above the ground. It leaves the bat at an
angle of
degrees with the horizontal at a speed of 100 miles per hour as shown in the figure. Write
a set of parametric equations for the path of the ball.
a.
b.
c.
d.
e.
10.2 Plane Curves and Parametric Equations
632
____
17.
The parametric equations for the path of a projectile launched at a height h feet above
the ground, at an angle
with the horizontal and having an initial velocity of
feet per second is
given by
and
. The center field fence in a ballpark is
feet high and
feet from home plate. The ball is hit 3 feet above the ground. It leaves the bat at an
angle of
degrees with the horizontal at a speed of 100 miles per hour as shown in the figure. Find
the minimum angle at which the ball must leave the bat in order for the hit to be a home run using the
parametric equations and . Round your answer to one
decimal place.
a.
b.
c.
d.
e.
633 Chapter 10: Conics, Parametric Equations, and Polar Coordinates
10.2 Plane Curves and Parametric Equations
Answer Section
10.3 Parametric Equations and Calculus
634
10.3 Parametric Equations and Calculus
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.
635 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
10.3 Parametric Equations and Calculus
636
____ 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.
637 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
.
horizontal tangents: , vertical tangent: none
horizontal tangent: none, vertical tangents:
horizontal tangents: , vertical tangent: none
horizontal tangent: none, vertical tangents:
horizontal tangent: none, vertical tangent: none
____ 10. Find all points (if any) of horizontal and vertical tangency to the curve
.
horizontal tangents: , vertical tangents:
horizontal tangents: , vertical tangents:
horizontal tangent: , vertical tangent:
horizontal tangents: , vertical tangents:
horizontal tangent: , vertical tangent:
10.3 Parametric Equations and Calculus
638
____ 11.
Determine the t intervals on which the curve
is concave
downward or concave upward.
concave downward: ; concave upward:
concave downward: ; concave upward:
concave downward: ; concave upward:
concave downward: ; concave upward:
concave downward: ; concave upward:
____ 12. Determine the t intervals on which the curve is concave downward
or concave upward.
concave downward: ; concave upward:
concave downward: ; concave upward:
concave downward:
concave upward:
concave downward:
____ 13. Find the arc length of the curve on the given interval.
a.
b.
c.
d.
e.
639 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.
287.453
191.635
193.606
66.480
99.721
____ 15. Find the arc length of the curve on the given interval.
a.
b.
c.
d.
e.
10.3 Parametric Equations and Calculus
640
____ 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.
312.85 ft
521.42 ft
131.09 ft
391.06 ft
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.
387.2 ft
595.8 ft
335.0 ft
269.9 ft
465.4 ft
641
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.
1427.12
1178.96
1049.37
589.48
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.
10.3 Parametric Equations and Calculus
642
10.3 Parametric Equations and Calculus
Answer Section
643
Chapter 10: Conics, Parametric Equations, and Polar Coordinates
10.4 Polar Coordinates and Polar Graphs
644
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.
645 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
646
____ 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.
647 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
.
horizontal tangents: , vertical tangent: none
horizontal tangent: none, vertical tangents:
horizontal tangents: , vertical tangent: none
horizontal tangent: none, vertical tangents:
horizontal tangent: none, vertical tangent: none
____ 10. Find all points (if any) of horizontal and vertical tangency to the curve
.
horizontal tangents: , vertical tangents:
horizontal tangents: , vertical tangents:
horizontal tangent: , vertical tangent:
horizontal tangents: , vertical tangents:
horizontal tangent: , vertical tangent:
