BC - 3
Vector Calculus
Name:
Calculator Allowed.
1.
r
r
r
If v  2 i  3 j , find each of the follow.
r
r
v
a.
b.
3v
d.
2.
c
r
4v
r
a unit vector in the direction of v .
Find the position function R(t) for each acceleration vector and initial conditions.
a.
a(t) = -3t i, v(0) = 2j, r(0) = 4i
b.
1
1
a(t) = 1  t  2 i - et j, v(0) = – i + j, r(0) = i – j
3
c.
r
r
r
a (t )  1  cos  2t  , sin(3t ) , v  0   1, 2 and r  0   0, 0
BC - 3
Vector Calculus
Name:
3. (AP 2010 #3) Set up of all derivatives and integrals should be and shown, you may calculate the
answers using calculator.
A particle is moving along a curve so that its position at time t is given by ( x(t ), y (t )), where
x(t )  t 2  4t  8 and y(t) is not explicitly given. Bot x and y are measured in meters and t is measured in
dy
 tet 3  1 .
seconds. It is known that
dt
a. Find the speed of the particle at time t = 3 seconds.
b. Find the total distance traveled by the particle for 0  t  4 seconds.
c. Find the time t , 0  t  4 , when the line tangent to the path of the particle is horizontal. Is
the direction of the particle to the left or right at this time? Give a reason for your answer.
d. There is a point with x – coordinate 5 through which the particle passes twice. Find each
of the following:
i.
The two values of t when that occurs.
ii.
The slopes of the two lines tangent to the particles path at tese points.
iii.
The y – coordinate of that point given y (2)  3 
1
e
BC - 3
Vector Calculus
Name:
4.
If (x) = x3 – 2x + 3, find a unit tangent vector and a unit normal vector to the graph of (x) at the
point where x = -2.
5.
Determine each of the following integrals
a.

 t i  t j  dt
1
cos 3t ,  sin 3t dt
b.
2
3
0
6. At time t, a particle moving in the xy-plane is at position (x(t), y(t)), where x(t) and y(t) are not explicitly given.
For t ≥ 0,
( )
dx
dy
= 4t + 1 and
= sin t 2 . At time t = 0, x(0) = 0 and y(0) = –4.
dt
dt
(a) Find the speed of the particle at time t = 3, and find the acceleration vector of the particle at time t = 3.
(b) Find the slope of the line tangent to the path of the particle at time t = 3.
(c) Find the position of the particle at time t = 3.
(d) Find the total distance traveled by the particle over the time interval 0 ≤ t ≤ 3.