BC: Ch 9 Test FREE RESPONSE
The ( r = 1− sin θ )ly________________________________
NO CALCULATOR
Period__________
1. (2 pts) Convert the rectangular equation, y 2 = −x 2 − 3y , into a polar equation. Solve for r.
2. (2 pts) An object moves along a curve in the xy-plane with position x (t ), y (t ) and with
dx
dy
= cost 2 and
= sin t 3 . At t=5, the object is at (4, 7). Set up, but DO NOT EVALUATE, an
dt
dt
expression that represents the y-value at t=2.
3. Consider the polar curves r = 4 and r = 4sin 3θ .
a) (1 pt) SKETCH the two polar curves on the same coordinate plane and SHADE the area inside
the polar curve r = 4 and outside the polar curve r = 4sin 3θ .
b) (4 pts) Set up, but DO NOT EVALUATE, an expression that calculates the area described in
part a).
4. Consider the polar curve r = 2 + 4 cosθ .
a) (4 pts) Find the slope of the polar curve at θ =
3π
.
2
b) (2 pts) Write the equation of the tangent line at θ =
3π
.
2
c) (4 pts) Set up, but DO NOT EVALUATE, an integral expression that can be used to calculate
the area enclosed by the inner loop.
5. A particle is moving along the polar curve r = 2 + 4sin θ .
b) (1 pt) What does this say about the curve at
dy
a) (3 pts) Find
at the instant that θ = π .
θ =π?
dθ
6. A particle moving in the xy-plane is modeled by the position vector r (t ) = t 2 − 3,
a) (2 pts) Sketch the graph of r (t ) = t 2 − 3,
t .
t . Label the orientation.
b) (2 pts) Find v (1) .
c) (2 pts) Sketch and LABEL v (1) on the graph you drew in part a).
d) (1 pt) How fast is the particle moving at t=1?
$" x = et
7. Consider the plane curve defined by the parametric equations: #
$% y = 3e 2t −1
a) (3 pts) Without eliminating the parameter,
b) (3 pts) Without eliminating the parameter,
dy
d2y
find
.
find
.
dx
dx 2
c) (2 pts) State the domain and range in any
notation.
D:_______________________
R:_______________________
d) (2 pts) Find the corresponding rectangular
equation by eliminating the parameter.