Calculus
3 Hour Exam 1
(continued) - Math 241
Fall, 2010
3a. |
If a(t) = < 0,0,1 > and v(0) = < 1,0,0 >, r(0) = < 0,1,0 >, find r(t). Answer: r(t) = < t, 1, t2/2 > |
3b. |
Use r(t) to find T(t) , the unit tangential vector at any time t. Answer: T(t) = < 1, 0, t > / √ (1 + t2 ) |
3c. |
Find aT and aN at t = 1. (the tangential and normal components of acceleration) Hint: draw a picture. Answers: aT = 1/√2, aN = √(1/2) |
4. |
Joe the bug follows a trail of strawberry jam tracked across the kitchen floor by a four year old along the path x(t) = √(1 + t), y(t) = 4 + t2 . The density of the jam at a point is given by the function JD(x,y). Determine the change in jam density dJ/dt at t = 8 given the following data. ∂JD/∂x(1,4) = 5 ∂JD/∂y(1,4) = 9 ∂JD/∂x (3,68) = 6 ∂JD/∂y(3,68) = 10 ∂JD/∂x(2,3) = 7 ∂JD/∂y(2,3) = 11 JD (2,3) = 8 JD (1,4) = 12 Answer: dJ/dt = 161 |
5a. |
Find the distance from (1,1,1) to the plane x + 2y + 3z = 7. Answer: 1/√14 |
5b. |
If r(t) = < 1, t, t2 > , find ĸ (t). the curvature. Answer: ĸ (t) = 2 / (1 + 4t2 )3/2 |