BC Calculus Chapter 10 Review
Section _______
Name ____________________
Date _____________________
Parametric Equations
1. Plot points to sketch the curve described by the parametric equations. Mark the orientation on the
curve.
y

x  t2  5
t

t
y
x
2
x










3  t  2
y


2. Change the following to rectangular form by eliminating the parameter. Then graph.
1
t
x
and y 
, t  1
t 1
t 1
y
x


3. Eliminate the parameter to sketch the curve.
x  cos  and y  3sin  , 0    2

y


x







4. If x  cos t and y  3sin t find
dy
d2y
and
.
dx
dx 2
dy
dx

dy
dt
dx
dt
d2y
and

dx 2
d  dy 
 
dt  dx 
dx
dt
5. Find the slope and concavity of x  t and y  4 t 2  1, t  0 at the point  2,3 .
1
1
6. Write an equation of a tangent line to the curve defined by x  t  1 and y  t  1 at the
point when t  1.
Arc Length: If a curve is smooth and does not intersect itself the length of an arc is given by
arc length  
b
a
2
2
 dx   dy 
     dt
 dt   dt 
7. Using the parametric equations from example 6, find the arc length on the interval 1  t  3 .
Polar Graphs
Plotting points in polar form
8. Use the polar grid to plot these polar points.
A
C
 3, 
3,  

6
5
6
B
 4,  
D
 2, 
Point conversions
9. Write the rectangular point
a. r  0,   0
3
2
 1, 3 




in polar form such that:
b. r  0,   0
c. r  0,   0
d. r  0,   0
10. Change the point  2,  
to rectangular form.
11. Change the point  3,3
to polar form.
Equation conversions (rectangular to polar)
12. y  4
13. 3x  y  2  0
14. x 2  y 2  2 x  0
Equation conversions (polar to rectangular)
15. r  2
16. r  3cos
17. r  2csc
Sketching polar graphs:
Circles: r  d cos (x-axis symmetry)
r  d sin  (y-axis symmetry)
Rose petal curves:
d is the diameter
r  a cos  n  (x-axis symmetry)
r  a sin  n  (y-axis symmetry)
Limaçons:
r  a  b cos
r  a  b sin 
(x-axis symmetry)
(y-axis symmetry)
a is the maximum r ,
n petals if n is odd,
2n petals if n is even
may have inner loop
may or may not include the pole
Sketching polar graphs (use a calculator only on those in bold)
18. Circles
a. r  2cos
20. Limaçons
a. r  2  3cos 
b. r  5sin 
19. Rose petal curves
a. r  3cos  2 
b. r  3  3sin 
b. r  4sin  3 
21. Lemniscate
r 2  9sin  2 
Tangent lines
22. Find an equation of the tangent line to the graph of r  2 1  sin   at the point  2,0  .
23. Find the points at which the graph of r  2  2cos has horizontal tangents.





Tangent lines at the pole
24. Find the equations of the lines tangent to r  4sin  3  at the pole.








Polar Area
25. Find the area of one petal of the curve r  3cos  3  .
Polar area =
1  2
r d
2 




Intersections of Polar Graphs
26. Find the points of intersection of the graphs of r  1  2cos and r  1 .



Area between two curves
27. Find the area of the region common to the two regions bounded by r  6cos and
r  2  2cos .





28.. Find the area between the loops of r  2 1  2sin   .





29. Find the length of the arc from
  0 to   2 for the curve r  2  2cos .
Arc Length  


r2 
 
dr
d
2
30. Given a position vector 3t 2 , t 3  3t 2  4 for a particle moving in the xy-plane find the
following.
a. graph the path of the particle
on the interval 0  t  2
b. the velocity vector
at time t  1
c. the speed of the particle
at time t  1
y



x







d. the distance traveled
between t  0 and t  3
e. the time(s) when the
particle is at rest
g. the direction of the particle at time t  1 and when t  2
f. the acceleration vector
at time t  2
d