Math 241 - Quiz #4

1. r(t) = sqrt(t+3) i + (t-1)/(t^2-1) j + tan(t)/t k
compute lim (t -> 1) r(t)

2. r(t) = t i + 2 sin(t) j + 3 cos(t) k
compute T(pi/6) (unit tangent vector)

3. Evaluate
integral from 0 to 1 (t i + t^2 j + t^3 k) dt

Solutions:

1. limit = 2 i + 1/2 j + tan(1) k
note: (t-1)/(t^2-1) = 1/(t+1)

2. r'(t) = i + 2 cos(t) j - 3 sin(t) k
r'(pi/6) = i + sqrt(3) j - 3/2 k
|r'(pi/6)| = sqrt(1 + 3 + 9/4) = sqrt(25/4) = 5/2
T(pi/6) = r'(pi/6)/|r'(pi/6)| = 2/5 i + 2 sqrt(3)/5 j - 3/5 k

3. (t^2/2 i + t^3/3 j + t^4/4 k) eval at t=1, t=0
1/2 i + 1/3 j + 1/4 k

ccollins@math.utk.edu
Last Modified: Feb. 5, 1999