Calculus 3 question cost 1 et -1 sint (5) Compute lim You may need to review L\'

Question

Calculus 3 question

cost 1 et -1 sint (5) Compute lim You may need to review L'Hopital's Rule to do this problem.) (6) (Bonus question: worth 10 points. Total points for assignment not to exceed 100.) If a particle moving along a curved track (like a roller coaster) flies off the track, it continues in the direction of the tangent vector at the point where it left the track, at least if there are no external forces such as gravity or friction acting on it. Suppose a particle rolling down (towards the ry-plane) the helix shaped track given by T (t) sint, flies off the track when t 4. Determine the point where the particle w ill hit the ry-plane.

Explanation / Answer

(5)

limt->0 <(cost -1)/t2 ,(et-1)/t ,sint/t>

<limt->0 (cost -1)/t2 ,limt->0 (et-1)/t ,limt->0 sint/t>

apply l hospital rule. we get 0/0 in in every term. differentiate numerator and denominator with respect to t

<limt->0 (-sint -0)/2t ,limt->0 (et-0)/1 ,limt->0 cost/1>

<limt->0 (-1/2)sint/t , (e0-0)/1 ,cos0/1>

<limt->0 (-1/2)sint/t , 1 ,1>

vapply l hospital rule. we get 0/0 in first term. differentiate numerator and denominator with respect to t

<limt->0 (-1/2)cost/1 , 1 ,1>

<(-1/2)cos0/1 , 1 ,1>

<(-1/2) , 1 ,1>

limt->0 <(cost -1)/t2 ,(et-1)/t ,sint/t> =<-1/2 , 1 ,1>

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