Ch 4.8 Wkst AP Calc BC
Name:
Parametric Equations Day 2
Parametric Equations as Particles in Motion and Vector-valued Functions
1. A particle moves along a curve on a plane such that at any time t > 0 its x-coordinate is defined as x(t )  2t 3  1
and y-coordinate is defined as y(t )  t 2  t .
a) Make a table of values and sketch the graph for 0  t  5 .
b) Convert the parametric equations into rectangular form.
c) Find the velocity vector of the particle at t = 2.
d) Find the speed of the particle at t = 2.
e) Find the acceleration vector of the particle at t = 2.
2. A particle moves along a curve on a plane such that at any time t > 0 its x-coordinate is defined as x(t )  2 cos t
and y-coordinate is defined as y (t )  sin t . The speed of the particle at t =
(A)
3
50
(B)
1
16
(C)
13
2
(D)
b) Use the equations in (a) to sketch the graph for 0  t  2 .
c) Find the speed of the particle at t =
2
.
3
1
8
2
is
3
(E)
3
32
3. A particle moves in the xy-plane with x  2t 3  9t 2  12t and y  3t 4  16t 3  18t 2 , where t is time.
a) At what time is the particle stopped?
b) At what time is the particle moving parallel to the x- or y-axis?
c) Find the speed of the particle at time t.
4. A particle moves along a curve on a plane such that at any time t > 0 its position is defined by the parametric
t2
equations x(t )  3t  t and y (t ) 
. The acceleration vector of the particle at t = 2 is
4t  1
3
(A)
36 ,
4
729
(B)
36 ,
2
729
(D)
18 ,
4
729
(E)
36 ,
6
729
5. If f is a vector-valued function defined by f (t ) 
(A)
2
t
(D)
2
1
t
3
, cosh t
(B)
1
t
, sinh t
(E)
3
2
t3
(C)
18 ,
2
729
1
, cosh t , then f " (t ) 
t
, cosh t
, cosh t
(C)
2
t2
, sinh t
6. If f is a vector-valued function defined by f (t )  sin 2 t , e 0.5t
(A)
 2 sin 2 t  2 cos 2 t , 0.5e 0.5t
(B)
 2 sin t cos t , e 0.5t
(C)
 2 sin 2 t  2 cos 2 t , 0.25e 0.5t
(D)
 2 sin 2 t cos t , 0.25e 0.5t
(E)
 2 , 0.5e 0.5t
, then f " (t ) 
ANSWERS:
1a) 30
b)
2
c) 24,5
d)
251
 x 1 
x 1
 3
y   3

2
2


601
e) 24,2
2
2a) x
 y2  1
4
1
b) -2
2
4 a) t = 1
b) parallel to x-axis at t = 0 and t = 3
parallel to y-axis at t = 2
c)
(6t 2  18t  12)2  (12t 3  48t 2  36t )2
5) B
6) E
-1
c)
13
2
7) C