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.