Math 241 - Quiz #5

1. Find the length of the curve
r(t) = < 2t , 3 sin t, 3 cos t>, a <= t <= b.

2. Find the limit (if it exists)
lim_((x,y)->(0,0)) 2x^2y/(x^4+y^2)

Solutions

1. r'(t) = < 2, 3 cos t, -3 sin t>, so
|r'(t)| = sqrt( 4 + 9 cos^2 t + 9 sin^2 t) = sqrt(13)
Length = L = int_(a to b) |r'| dt = sqrt(13)(b-a)

2. Choosing the path with x=0, we get a limit of 0, if instead we choose a path with y=x^2, we get
2x^2(x^2)/(x^4+x^4) = 1,
so the limit taking this path is 1. Since these two limits are different, we know that the limit does not exist.

ccollins@math.utk.edu
Last Modified: Feb. 16, 1999